1018 files changed, 177237 insertions, 1 deletions

diff --git a/polly/lib/CMakeLists.txt b/polly/lib/CMakeLists.txt
index 96254580496..6e9a8ba534b 100644
--- a/polly/lib/CMakeLists.txt
+++ b/polly/lib/CMakeLists.txt
@@ -20,6 +20,85 @@ if (GPU_CODEGEN)
        CodeGen/PTXGenerator.cpp)
 endif (GPU_CODEGEN)
 
+set (ISL_FILES
+    External/isl/basis_reduction_tab.c
+    External/isl/isl_aff.c
+    External/isl/isl_affine_hull.c
+    External/isl/isl_arg.c
+    External/isl/isl_ast_build.c
+    External/isl/isl_ast_build_expr.c
+    External/isl/isl_ast.c
+    External/isl/isl_ast_codegen.c
+    External/isl/isl_ast_graft.c
+    External/isl/isl_band.c
+    External/isl/isl_bernstein.c
+    External/isl/isl_blk.c
+    External/isl/isl_bound.c
+    External/isl/isl_coalesce.c
+    External/isl/isl_constraint.c
+    External/isl/isl_convex_hull.c
+    External/isl/isl_ctx.c
+    External/isl/isl_deprecated.c
+    External/isl/isl_dim.c
+    External/isl/isl_dim_map.c
+    External/isl/isl_equalities.c
+    External/isl/isl_factorization.c
+    External/isl/isl_farkas.c
+    External/isl/isl_flow.c
+    External/isl/isl_fold.c
+    External/isl/isl_hash.c
+    External/isl/isl_id.c
+    External/isl/isl_id_to_ast_expr.c
+    External/isl/isl_id_to_pw_aff.c
+    External/isl/isl_ilp.c
+    External/isl/isl_imath.c
+    External/isl/isl_input.c
+    External/isl/isl_local_space.c
+    External/isl/isl_lp.c
+    External/isl/isl_map.c
+    External/isl/isl_map_simplify.c
+    External/isl/isl_map_subtract.c
+    External/isl/isl_map_to_basic_set.c
+    External/isl/isl_mat.c
+    External/isl/isl_morph.c
+    External/isl/isl_obj.c
+    External/isl/isl_options.c
+    External/isl/isl_output.c
+    External/isl/isl_point.c
+    External/isl/isl_polynomial.c
+    External/isl/isl_printer.c
+    External/isl/isl_range.c
+    External/isl/isl_reordering.c
+    External/isl/isl_sample.c
+    External/isl/isl_scan.c
+    External/isl/isl_schedule.c
+    External/isl/isl_scheduler.c
+    External/isl/isl_seq.c
+    External/isl/isl_set_list.c
+    External/isl/isl_sort.c
+    External/isl/isl_space.c
+    External/isl/isl_stream.c
+    External/isl/isl_tab.c
+    External/isl/isl_tab_pip.c
+    External/isl/isl_tarjan.c
+    External/isl/isl_transitive_closure.c
+    External/isl/isl_union_map.c
+    External/isl/isl_val.c
+    External/isl/isl_val_imath.c
+    External/isl/isl_vec.c
+    External/isl/isl_version.c
+    External/isl/isl_vertices.c
+    External/isl/print.c
+    External/isl/imath/gmp_compat.c
+    External/isl/imath/imath.c
+    External/isl/imath/imdrover.c
+    External/isl/imath/imrat.c
+    External/isl/imath/iprime.c
+    External/isl/imath/pi.c
+    External/isl/imath/rsamath.c
+    )
+
+
 add_polly_library(Polly
   Analysis/Dependences.cpp
   Analysis/ScopDetection.cpp
@@ -47,6 +126,7 @@ add_polly_library(Polly
   Transform/IndependentBlocks.cpp
   Transform/ScheduleOptimizer.cpp
   ${POLLY_PLUTO_FILES}
+  ${ISL_FILES}
   )
 
 # Build a monolithic Polly.a and a thin module LLVMPolly.moduleext that links to
diff --git a/polly/lib/External/README.txt b/polly/lib/External/README.txt
new file mode 100644
index 00000000000..81ef08f767b
--- /dev/null
+++ b/polly/lib/External/README.txt
@@ -0,0 +1,17 @@
+The libraries in this directory are mirrored from external projects.
+
+Patches to them should first be contributed upstream and then return to Polly
+as normal (re)imports of these updated libraries.
+
+We currently have the following external libraries.
+
+# isl
+License: MIT-STYLE
+Details: isl/LICENSE
+Version: b3e0fa7a05d
+
+# imath
+License: MIT-STYLE
+Details: isl/imath/LICENSE
+Version: 4d707e5ef2
+
diff --git a/polly/lib/External/isl/.gitignore b/polly/lib/External/isl/.gitignore
new file mode 100644
index 00000000000..b2fd552a482
--- /dev/null
+++ b/polly/lib/External/isl/.gitignore
@@ -0,0 +1,53 @@
+# ignore objects and archives
+*.[oa]
+*.lo
+*.la
+
+.deps/
+.libs/
+Makefile
+Makefile.in
+aclocal.m4
+autom4te.cache/
+config.guess
+isl_config.h.in
+isl_config.h.in~
+compile
+config.log
+config.status
+config.sub
+configure
+depcomp
+doc/Makefile
+doc/Makefile.in
+include/isl/config.h
+include/isl/stdint.h
+include/stamp-h2
+include/isl/stamp-h2
+install-sh
+isl-uninstalled.pc
+isl-uninstalled.sh
+isl.pc
+isl.pc.in
+isl_bound
+isl_cat
+isl_closure
+isl_pip
+isl_polyhedron_detect_equalities
+isl_polyhedron_minimize
+isl_polyhedron_sample
+isl_polytope_scan
+isl_test
+libtool
+ltmain.sh
+m4/libtool.m4
+m4/ltoptions.m4
+m4/ltsugar.m4
+m4/ltversion.m4
+m4/lt~obsolete.m4
+missing
+stamp-h1
+test-driver
+
+bound_test.sh
+pip_test.sh
diff --git a/polly/lib/External/isl/AUTHORS b/polly/lib/External/isl/AUTHORS
new file mode 100644
index 00000000000..a08f5b12fb0
--- /dev/null
+++ b/polly/lib/External/isl/AUTHORS
@@ -0,0 +1,42 @@
+isl was written by
+
+	    Sven Verdoolaege
+2006-2007   Leiden Institute of Advanced Computer Science
+	    Universiteit Leiden
+	    Niels Bohrweg 1
+	    2333 CA Leiden
+	    The Netherlands
+2008-2009   K.U.Leuven
+	    Departement Computerwetenschappen
+	    Celestijnenlaan 200A
+	    B-3001 Leuven
+	    Belgium
+2010-2011   INRIA Saclay - Ile-de-France
+	    Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod
+	    91893 Orsay
+	    France
+2011-2012   consultant for Leiden Institute of Advanced Computer Science
+2012-2014   Ecole Normale Superieure
+	    45 rue d'Ulm, 75230 Paris
+	    France
+2014	    INRIA Rocquencourt
+	    Domaine de Voluceau - Rocquencourt, B.P. 105
+	    78153 Le Chesnay
+	    France
+
+Contributions by
+
+Mythri Alle
+Riyadh Baghdadi
+Serge Belyshev
+Ray Donnelly
+Johannes Doerfert
+Tobias Grosser
+Alexandre Isoard
+Andreas Kloeckner
+Sebastian Pop
+Louis-Noel Pouchet
+Uday Kumar Reddy
+Sven van Haastregt
+
+The merge sort implementation was written by Jeffrey Stedfast.
diff --git a/polly/lib/External/isl/ChangeLog b/polly/lib/External/isl/ChangeLog
new file mode 100644
index 00000000000..a802e45909a
--- /dev/null
+++ b/polly/lib/External/isl/ChangeLog
@@ -0,0 +1,134 @@
+version: 0.14
+date: Sat Oct 25 16:08:47 CEST 2014
+changes:
+	- support IMath as an optional replacement for GMP
+	- minor AST generator improvements
+---
+version: 0.13
+date: Mon Apr 14 11:08:45 CEST 2014
+changes:
+	- deprecate isl_int
+	- improved support for multi piecewise quasi-affine expressions
+	- allow the user to impose a bound on the number of low-level operations
+	- add isl_id_to_ast_expr and isl_id_to_pw_aff
+	- add isl_schedule_constraints
+	- hide internal structure of isl_vec
+	- remove support for piplib
+---
+version: 0.12.2
+date: Sun Jan 12 12:09:46 CET 2014
+changes:
+	- MinGW-w64 build fix
+	- fix simplification bug
+---
+version: 0.12.1
+date: Wed Jul 24 12:54:46 CEST 2013
+changes:
+	- handle malloc returning NULL on zero-size allocation
+	- fix regression in AST generator
+---
+version: 0.12
+date: Sun Jun 23 20:23:05 CEST 2013
+changes:
+	- add isl_val abstraction
+---
+version: 0.11.2
+date: Tue Apr  9 18:45:10 CEST 2013
+changes:
+	- make code generation output the same on Solaris
+	- fix some hard to trigger bugs
+---
+version: 0.11.1
+date: Mon Dec 10 11:55:30 CET 2012
+changes:
+	- add LICENSE file to distribution
+	- make code generation output independent of endianness
+---
+version: 0.11
+date: Mon Dec  3 08:17:18 CET 2012
+changes:
+	- change license from LGPL 2.1 to MIT
+	- add support for multi piecewise quasi-affine expressions
+	- add code generation
+	- various minor bug fixes
+---
+version: 0.10
+date: Sun Jun  3 18:00:16 CEST 2012
+changes:
+	- support for interaction with dependence analysis
+	- add public API for vectors
+	- improved support for (piecewise) multi quasi-affine expressions
+	- various minor bug fixes
+---
+version: 0.09
+date: Sat Dec 17 18:19:26 CET 2011
+changes:
+	- improved argument parsing
+	- hide internal structure of isl_options
+	- improved support for parameter sets
+	- configurable scheduling
+---
+version: 0.08
+date: Fri Oct 21 12:36:20 CEST 2011
+changes:
+	- improved parsing
+	- drop isl_div abstraction
+	- rename isl_dim to isl_space
+	- |-
+	  explicitly differentiate between spaces of maps,
+	  sets and parameter sets
+	- add support for identifiers
+	- add support for (piecewise) multi quasi-affine expressions
+	- preliminary Python bindings
+---
+version: 0.07
+date: Tue Jul 12 19:34:51 CEST 2011
+changes:
+	- hide internal structures of isl_div and isl_constraint
+	- preliminary scheduling
+	- add support for local spaces and (piecewise) quasi-affine expressions
+---
+version: 0.06
+date: Fri Mar 18 15:59:16 CET 2011
+changes:
+	- improved parsing
+	- consistency changes in API
+	- hide internal structure of isl_ctx
+---
+version: 0.05.1
+date: Wed Jan  5 10:21:42 CET 2011
+changes:
+	- fix simple symmetry detection in parametric integer programming
+---
+version: 0.05
+date: Thu Dec 23 17:03:14 CET 2010
+changes:
+	- rename header files from isl_header.h to isl/header.h
+	- add higher level interface for dependence analysis
+	- improved argument parsing
+	- optionally triangulate domains during Bernstein expansion
+	- support extended PolyLib format
+	- hide internal structure of some data types
+	- improved coalescing
+	- add simple symmetry detection in parametric integer programming
+---
+version: 0.04
+date: Fri Sep 10 12:57:50 CEST 2010
+changes:
+	- rename isl_pw_qpolynomial_fold_add
+	- add isl_map_apply_pw_qpolynomial_fold
+	- support named and nested spaces
+	- support union sets and maps
+	- add public API for matrices
+---
+version: 0.03
+date: Tue Jun 29 13:16:46 CEST 2010
+changes:
+	- new printing functions
+	- support for "may" accesses in dependence analysis
+	- improved coalescing
+	- improved transitive closure
+	- fix several hard to trigger bugs
+	- improved argument parsing
+	- support parametric vertex enumeration for barvinok
+	- optionally use Bernstein expansion to compute bounds
diff --git a/polly/lib/External/isl/LICENSE b/polly/lib/External/isl/LICENSE
new file mode 100644
index 00000000000..e93f5973e00
--- /dev/null
+++ b/polly/lib/External/isl/LICENSE
@@ -0,0 +1,19 @@
+MIT License (MIT)
+
+Permission is hereby granted, free of charge, to any person obtaining a copy of
+this software and associated documentation files (the "Software"), to deal in
+the Software without restriction, including without limitation the rights to
+use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies
+of the Software, and to permit persons to whom the Software is furnished to do
+so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.
diff --git a/polly/lib/External/isl/Makefile.am b/polly/lib/External/isl/Makefile.am
new file mode 100644
index 00000000000..f975d6a2499
--- /dev/null
+++ b/polly/lib/External/isl/Makefile.am
@@ -0,0 +1,329 @@
+if HAVE_CLANG
+    MAYBE_INTERFACE = interface
+endif
+SUBDIRS = . $(MAYBE_INTERFACE) doc
+DIST_SUBDIRS = $(MAYBE_INTERFACE) doc
+
+ACLOCAL_AMFLAGS = -I m4
+AUTOMAKE_OPTIONS = nostdinc subdir-objects
+
+lib_LTLIBRARIES = libisl.la
+noinst_PROGRAMS = isl_test isl_polyhedron_sample isl_pip \
+	isl_polyhedron_minimize isl_polytope_scan \
+	isl_polyhedron_detect_equalities isl_cat \
+	isl_closure isl_bound isl_codegen
+TESTS = isl_test codegen_test.sh pip_test.sh bound_test.sh
+
+if IMATH_FOR_MP
+
+MP_SRC = \
+	isl_hide_deprecated.h \
+	isl_imath.c \
+	isl_imath.h \
+	isl_int_imath.h \
+	isl_val_imath.c \
+	imath_wrap/gmp_compat.h \
+	imath_wrap/imath.h \
+	imath_wrap/imrat.h \
+	imath_wrap/wrap.h \
+	imath_wrap/gmp_compat.c \
+	imath_wrap/imath.c \
+	imath_wrap/imrat.c
+
+DEPRECATED_SRC =
+MP_INCLUDE_H =
+endif
+
+if GMP_FOR_MP
+if NEED_GET_MEMORY_FUNCTIONS
+GET_MEMORY_FUNCTIONS=mp_get_memory_functions.c
+endif
+
+MP_SRC = \
+	$(GET_MEMORY_FUNCTIONS) \
+	isl_int_gmp.h \
+	isl_gmp.c \
+	isl_val_gmp.c
+
+DEPRECATED_SRC = isl_ast_int.c
+MP_INCLUDE_H = include/isl/val_gmp.h
+endif
+
+AM_CPPFLAGS = -I. -I$(srcdir) -I$(srcdir)/include -Iinclude/ @MP_CPPFLAGS@
+AM_CFLAGS = @WARNING_FLAGS@
+
+libisl_la_SOURCES = \
+	$(MP_SRC) \
+	$(DEPRECATED_SRC) \
+	isl_aff.c \
+	isl_aff_private.h \
+	isl_affine_hull.c \
+	isl_arg.c \
+	isl_ast.c \
+	isl_ast_private.h \
+	isl_ast_build.c \
+	isl_ast_build_private.h \
+	isl_ast_build_expr.c \
+	isl_ast_build_expr.h \
+	isl_ast_codegen.c \
+	isl_ast_graft.c \
+	isl_ast_graft_private.h \
+	isl_band.c \
+	isl_band_private.h \
+	isl_basis_reduction.h \
+	basis_reduction_tab.c \
+	isl_bernstein.c \
+	isl_bernstein.h \
+	isl_blk.c \
+	isl_blk.h \
+	isl_bound.c \
+	isl_bound.h \
+	isl_coalesce.c \
+	isl_constraint.c \
+	isl_constraint_private.h \
+	isl_convex_hull.c \
+	isl_ctx.c \
+	isl_ctx_private.h \
+	isl_deprecated.c \
+	isl_dim.c \
+	isl_dim_map.h \
+	isl_dim_map.c \
+	isl_equalities.c \
+	isl_equalities.h \
+	isl_factorization.c \
+	isl_factorization.h \
+	isl_farkas.c \
+	isl_flow.c \
+	isl_fold.c \
+	isl_hash.c \
+	isl_id_to_ast_expr.c \
+	isl_id_to_pw_aff.c \
+	isl_ilp.c \
+	isl_ilp_private.h \
+	isl_input.c \
+	isl_int.h \
+	isl_local_space_private.h \
+	isl_local_space.c \
+	isl_lp.c \
+	isl_lp_private.h \
+	isl_map.c \
+	isl_map_simplify.c \
+	isl_map_subtract.c \
+	isl_map_private.h \
+	isl_map_to_basic_set.c \
+	isl_mat.c \
+	isl_mat_private.h \
+	isl_morph.c \
+	isl_morph.h \
+	isl_id.c \
+	isl_id_private.h \
+	isl_obj.c \
+	isl_options.c \
+	isl_options_private.h \
+	isl_output.c \
+	isl_point_private.h \
+	isl_point.c \
+	isl_polynomial_private.h \
+	isl_polynomial.c \
+	isl_printer_private.h \
+	isl_printer.c \
+	print.c \
+	isl_range.c \
+	isl_range.h \
+	isl_reordering.c \
+	isl_reordering.h \
+	isl_sample.h \
+	isl_sample.c \
+	isl_scan.c \
+	isl_scan.h \
+	isl_schedule.c \
+	isl_schedule_private.h \
+	isl_scheduler.c \
+	isl_set_list.c \
+	isl_sort.c \
+	isl_sort.h \
+	isl_space.c \
+	isl_space_private.h \
+	isl_stream.c \
+	isl_stream_private.h \
+	isl_seq.c \
+	isl_seq.h \
+	isl_tab.c \
+	isl_tab.h \
+	isl_tab_pip.c \
+	isl_tarjan.c \
+	isl_tarjan.h \
+	isl_transitive_closure.c \
+	isl_union_map.c \
+	isl_union_map_private.h \
+	isl_val.c \
+	isl_val_private.h \
+	isl_vec_private.h \
+	isl_vec.c \
+	isl_version.c \
+	isl_vertices_private.h \
+	isl_vertices.c
+libisl_la_LIBADD = @MP_LIBS@
+libisl_la_LDFLAGS = -version-info @versioninfo@ \
+	@MP_LDFLAGS@
+
+isl_test_LDFLAGS = @MP_LDFLAGS@
+isl_test_LDADD = libisl.la @MP_LIBS@
+
+isl_polyhedron_sample_LDADD = libisl.la
+isl_polyhedron_sample_SOURCES = \
+	polyhedron_sample.c
+
+isl_pip_LDFLAGS = @MP_LDFLAGS@
+isl_pip_LDADD = libisl.la @MP_LIBS@
+isl_pip_SOURCES = \
+	pip.c
+
+isl_codegen_LDFLAGS = @MP_LDFLAGS@
+isl_codegen_LDADD = libisl.la @MP_LIBS@
+isl_codegen_SOURCES = \
+	codegen.c
+
+isl_bound_LDFLAGS = @MP_LDFLAGS@
+isl_bound_LDADD = libisl.la @MP_LIBS@
+isl_bound_SOURCES = \
+	bound.c
+
+isl_polyhedron_minimize_LDFLAGS = @MP_LDFLAGS@
+isl_polyhedron_minimize_LDADD = libisl.la @MP_LIBS@
+isl_polyhedron_minimize_SOURCES = \
+	polyhedron_minimize.c
+
+isl_polytope_scan_LDADD = libisl.la
+isl_polytope_scan_SOURCES = \
+	polytope_scan.c
+
+isl_polyhedron_detect_equalities_LDADD = libisl.la
+isl_polyhedron_detect_equalities_SOURCES = \
+	polyhedron_detect_equalities.c
+
+isl_cat_LDADD = libisl.la
+isl_cat_SOURCES = \
+	cat.c
+
+isl_closure_LDADD = libisl.la
+isl_closure_SOURCES = \
+	closure.c
+
+nodist_pkginclude_HEADERS = \
+	include/isl/stdint.h
+pkginclude_HEADERS = \
+	$(MP_INCLUDE_H) \
+	include/isl/aff.h \
+	include/isl/aff_type.h \
+	include/isl/arg.h \
+	include/isl/ast.h \
+	include/isl/ast_type.h \
+	include/isl/ast_build.h \
+	include/isl/band.h \
+	include/isl/constraint.h \
+	include/isl/ctx.h \
+	include/isl/dim.h \
+	include/isl/flow.h \
+	include/isl/id.h \
+	include/isl/id_to_ast_expr.h \
+	include/isl/id_to_pw_aff.h \
+	include/isl/ilp.h \
+	include/isl/hash.h \
+	include/isl/hmap.h \
+	include/isl/list.h \
+	include/isl/local_space.h \
+	include/isl/lp.h \
+	include/isl/mat.h \
+	include/isl/map.h \
+	include/isl/map_to_basic_set.h \
+	include/isl/map_type.h \
+	include/isl/multi.h \
+	include/isl/obj.h \
+	include/isl/options.h \
+	include/isl/point.h \
+	include/isl/polynomial.h \
+	include/isl/polynomial_type.h \
+	include/isl/printer.h \
+	include/isl/schedule.h \
+	include/isl/set.h \
+	include/isl/set_type.h \
+	include/isl/space.h \
+	include/isl/stream.h \
+	include/isl/union_map.h \
+	include/isl/union_map_type.h \
+	include/isl/union_set.h \
+	include/isl/union_set_type.h \
+	include/isl/val.h \
+	include/isl/vec.h \
+	include/isl/version.h \
+	include/isl/vertices.h
+deprecateddir = $(pkgincludedir)/deprecated
+deprecated_HEADERS = \
+	include/isl/deprecated/int.h \
+	include/isl/deprecated/aff_int.h \
+	include/isl/deprecated/ast_int.h \
+	include/isl/deprecated/constraint_int.h \
+	include/isl/deprecated/ilp_int.h \
+	include/isl/deprecated/map_int.h \
+	include/isl/deprecated/mat_int.h \
+	include/isl/deprecated/point_int.h \
+	include/isl/deprecated/polynomial_int.h \
+	include/isl/deprecated/set_int.h \
+	include/isl/deprecated/union_map_int.h \
+	include/isl/deprecated/val_int.h \
+	include/isl/deprecated/vec_int.h
+
+EXTRA_DIST = \
+	LICENSE \
+	isl_config_post.h \
+	basis_reduction_templ.c \
+	isl_hmap_templ.c \
+	isl_list_templ.c \
+	isl_list_templ.h \
+	isl_map_lexopt_templ.c \
+	isl_multi_templ.c \
+	isl_multi_templ.h \
+	print_templ.c \
+	isl_power_templ.c \
+	isl_pw_templ.c \
+	isl_union_templ.c \
+	isl.py \
+	doc/SubmittingPatches \
+	doc/chicago.bst \
+	doc/chicago.sty \
+	doc/implementation.tex \
+	doc/isl.bib \
+	doc/mypod2latex \
+	doc/manual.tex \
+	doc/user.pod \
+	imath/gmp_compat.c \
+	imath/gmp_compat.h \
+	imath/imath.c \
+	imath/imath.h \
+	imath/imrat.c \
+	imath/imrat.h \
+	interface/all.h \
+	interface/isl.py.top \
+	test_inputs
+
+dist-hook:
+	echo @GIT_HEAD_VERSION@ > $(distdir)/GIT_HEAD_ID
+	(cd doc; make manual.pdf)
+	cp doc/manual.pdf $(distdir)/doc/
+
+pkgconfigdir=$(pkgconfig_libdir)
+pkgconfig_DATA = $(pkgconfig_libfile)
+
+gitversion.h: @GIT_HEAD@
+	$(AM_V_GEN)echo '#define GIT_HEAD_ID "'@GIT_HEAD_VERSION@'"' > $@
+
+install-data-local: $(srcdir)/isl.py
+	@libisl=`sed -ne "/^library_names=/{s/.*='//;s/'$$//;s/ .*//;p;}" \
+		 $(builddir)/libisl.la`; \
+	case $$libisl in \
+	'') echo Cannot find isl library name. GDB bindings not installed.;; \
+	*) echo $(INSTALL_DATA) $(srcdir)/isl.py \
+		$(DESTDIR)$(libdir)/$$libisl-gdb.py; \
+	test -z "$(libdir)" || $(MKDIR_P) "$(DESTDIR)$(libdir)"; \
+	$(INSTALL_DATA) $(srcdir)/isl.py $(DESTDIR)$(libdir)/$$libisl-gdb.py; esac
diff --git a/polly/lib/External/isl/README b/polly/lib/External/isl/README
new file mode 100644
index 00000000000..9f48a0a76f9
--- /dev/null
+++ b/polly/lib/External/isl/README
@@ -0,0 +1,53 @@
+isl is a thread-safe C library for manipulating sets and relations
+of integer points bounded by affine constraints.  The descriptions of
+the sets and relations may involve both parameters and existentially
+quantified variables.  All computations are performed in exact integer
+arithmetic using GMP.
+
+isl is released under the MIT license, but depends on the LGPL GMP
+library.
+
+Minimal compilation instructions:
+
+	./configure
+	make
+	make install
+
+If you are taking the source from the git repository, then you first
+need to do
+
+	git clone git://repo.or.cz/isl.git
+	./autogen.sh
+
+For more information, see doc/user.pod or the generated documentation.
+
+New releases are announced on http://freecode.com/projects/isl
+
+If you use isl, you can let me know by stacking
+https://www.ohloh.net/p/isl on ohloh.
+
+For bug reports, feature requests and questions,
+contact http://groups.google.com/group/isl-development
+
+Whenever you report a bug, please mention the exact version of isl
+that you are using (output of "./isl_cat --version").  If you are unable
+to compile isl, then report the git version (output of "git describe")
+or the version included in the name of the tarball.
+
+If you use isl for your research, you are invited do cite
+the following paper and/or the paper(s) describing the specific
+operations you use.
+
+@incollection{Verdoolaege2010isl,
+   author = {Verdoolaege, Sven},
+   title = {isl: An Integer Set Library for the Polyhedral Model},
+   booktitle = {Mathematical Software - ICMS 2010},
+   series = {Lecture Notes in Computer Science},
+   editor = {Fukuda, Komei and Hoeven, Joris and Joswig, Michael and
+		Takayama, Nobuki},
+   publisher = {Springer},
+   isbn = {978-3-642-15581-9},
+   pages = {299-302},
+   volume = {6327},
+   year = {2010}
+}
diff --git a/polly/lib/External/isl/autogen.sh b/polly/lib/External/isl/autogen.sh
new file mode 100755
index 00000000000..68f4a174b46
--- /dev/null
+++ b/polly/lib/External/isl/autogen.sh
@@ -0,0 +1,2 @@
+#!/bin/sh
+autoreconf -i
diff --git a/polly/lib/External/isl/basis_reduction_tab.c b/polly/lib/External/isl/basis_reduction_tab.c
new file mode 100644
index 00000000000..112d423dbe0
--- /dev/null
+++ b/polly/lib/External/isl/basis_reduction_tab.c
@@ -0,0 +1,267 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <assert.h>
+#include <isl_map_private.h>
+#include <isl_seq.h>
+#include "isl_tab.h"
+#include <isl_int.h>
+#include <isl_config.h>
+
+struct tab_lp {
+	struct isl_ctx  *ctx;
+	struct isl_vec  *row;
+	struct isl_tab  *tab;
+	struct isl_tab_undo	**stack;
+	isl_int		*obj;
+	isl_int		 opt;
+	isl_int		 opt_denom;
+	isl_int		 tmp;
+	isl_int		 tmp2;
+	int	         neq;
+	unsigned	 dim;
+	/* number of constraints in initial product tableau */
+	int		 con_offset;
+	/* objective function has fixed or no integer value */
+	int		 is_fixed;
+};
+
+#ifdef USE_GMP_FOR_MP
+#define GBR_type		    	    mpq_t
+#define GBR_init(v)		    	    mpq_init(v)
+#define GBR_clear(v)		    	    mpq_clear(v)
+#define GBR_set(a,b)			    mpq_set(a,b)
+#define GBR_set_ui(a,b)			    mpq_set_ui(a,b,1)
+#define GBR_mul(a,b,c)			    mpq_mul(a,b,c)
+#define GBR_lt(a,b)			    (mpq_cmp(a,b) < 0)
+#define GBR_is_zero(a)			    (mpq_sgn(a) == 0)
+#define GBR_numref(a)			    mpq_numref(a)
+#define GBR_denref(a)			    mpq_denref(a)
+#define GBR_floor(a,b)			    mpz_fdiv_q(a,GBR_numref(b),GBR_denref(b))
+#define GBR_ceil(a,b)			    mpz_cdiv_q(a,GBR_numref(b),GBR_denref(b))
+#endif /* USE_GMP_FOR_MP */
+
+#ifdef USE_IMATH_FOR_MP
+#include <imrat.h>
+
+#define GBR_type		    	    mp_rat
+#define GBR_init(v)		    	    v = mp_rat_alloc()
+#define GBR_clear(v)		    	    mp_rat_free(v)
+#define GBR_set(a,b)			    mp_rat_copy(b,a)
+#define GBR_set_ui(a,b)			    mp_rat_set_uvalue(a,b,1)
+#define GBR_mul(a,b,c)			    mp_rat_mul(b,c,a)
+#define GBR_lt(a,b)			    (mp_rat_compare(a,b) < 0)
+#define GBR_is_zero(a)			    (mp_rat_compare_zero(a) == 0)
+#define GBR_numref(a)			    mp_rat_numer_ref(a)
+#define GBR_denref(a)			    mp_rat_denom_ref(a)
+#define GBR_floor(a,b)			    impz_fdiv_q(a,GBR_numref(b),GBR_denref(b))
+#define GBR_ceil(a,b)			    impz_cdiv_q(a,GBR_numref(b),GBR_denref(b))
+#endif /* USE_IMATH_FOR_MP */
+
+static struct tab_lp *init_lp(struct isl_tab *tab);
+static void set_lp_obj(struct tab_lp *lp, isl_int *row, int dim);
+static int solve_lp(struct tab_lp *lp);
+static void get_obj_val(struct tab_lp* lp, GBR_type *F);
+static void delete_lp(struct tab_lp *lp);
+static int add_lp_row(struct tab_lp *lp, isl_int *row, int dim);
+static void get_alpha(struct tab_lp* lp, int row, GBR_type *alpha);
+static int del_lp_row(struct tab_lp *lp) WARN_UNUSED;
+static int cut_lp_to_hyperplane(struct tab_lp *lp, isl_int *row);
+
+#define GBR_LP			    	    struct tab_lp
+#define GBR_lp_init(P)		    	    init_lp(P)
+#define GBR_lp_set_obj(lp, obj, dim)	    set_lp_obj(lp, obj, dim)
+#define GBR_lp_solve(lp)		    solve_lp(lp)
+#define GBR_lp_get_obj_val(lp, F)	    get_obj_val(lp, F)
+#define GBR_lp_delete(lp)		    delete_lp(lp)
+#define GBR_lp_next_row(lp)		    lp->neq
+#define GBR_lp_add_row(lp, row, dim)	    add_lp_row(lp, row, dim)
+#define GBR_lp_get_alpha(lp, row, alpha)    get_alpha(lp, row, alpha)
+#define GBR_lp_del_row(lp)		    del_lp_row(lp)
+#define GBR_lp_is_fixed(lp)		    (lp)->is_fixed
+#define GBR_lp_cut(lp, obj)	    	    cut_lp_to_hyperplane(lp, obj)
+#include "basis_reduction_templ.c"
+
+/* Set up a tableau for the Cartesian product of bset with itself.
+ * This could be optimized by first setting up a tableau for bset
+ * and then performing the Cartesian product on the tableau.
+ */
+static struct isl_tab *gbr_tab(struct isl_tab *tab, struct isl_vec *row)
+{
+	unsigned dim;
+	struct isl_tab *prod;
+
+	if (!tab || !row)
+		return NULL;
+
+	dim = tab->n_var;
+	prod = isl_tab_product(tab, tab);
+	if (isl_tab_extend_cons(prod, 3 * dim + 1) < 0) {
+		isl_tab_free(prod);
+		return NULL;
+	}
+	return prod;
+}
+
+static struct tab_lp *init_lp(struct isl_tab *tab)
+{
+	struct tab_lp *lp = NULL;
+
+	if (!tab)
+		return NULL;
+
+	lp = isl_calloc_type(tab->mat->ctx, struct tab_lp);
+	if (!lp)
+		return NULL;
+
+	isl_int_init(lp->opt);
+	isl_int_init(lp->opt_denom);
+	isl_int_init(lp->tmp);
+	isl_int_init(lp->tmp2);
+
+	lp->dim = tab->n_var;
+
+	lp->ctx = tab->mat->ctx;
+	isl_ctx_ref(lp->ctx);
+
+	lp->stack = isl_alloc_array(lp->ctx, struct isl_tab_undo *, lp->dim);
+
+	lp->row = isl_vec_alloc(lp->ctx, 1 + 2 * lp->dim);
+	if (!lp->row)
+		goto error;
+	lp->tab = gbr_tab(tab, lp->row);
+	if (!lp->tab)
+		goto error;
+	lp->con_offset = lp->tab->n_con;
+	lp->obj = NULL;
+	lp->neq = 0;
+
+	return lp;
+error:
+	delete_lp(lp);
+	return NULL;
+}
+
+static void set_lp_obj(struct tab_lp *lp, isl_int *row, int dim)
+{
+	lp->obj = row;
+}
+
+static int solve_lp(struct tab_lp *lp)
+{
+	enum isl_lp_result res;
+	unsigned flags = 0;
+
+	lp->is_fixed = 0;
+
+	isl_int_set_si(lp->row->el[0], 0);
+	isl_seq_cpy(lp->row->el + 1, lp->obj, lp->dim);
+	isl_seq_neg(lp->row->el + 1 + lp->dim, lp->obj, lp->dim);
+	if (lp->neq)
+		flags = ISL_TAB_SAVE_DUAL;
+	res = isl_tab_min(lp->tab, lp->row->el, lp->ctx->one,
+			  &lp->opt, &lp->opt_denom, flags);
+	isl_int_mul_ui(lp->opt_denom, lp->opt_denom, 2);
+	if (isl_int_abs_lt(lp->opt, lp->opt_denom)) {
+		struct isl_vec *sample = isl_tab_get_sample_value(lp->tab);
+		if (!sample)
+			return -1;
+		isl_seq_inner_product(lp->obj, sample->el + 1, lp->dim, &lp->tmp);
+		isl_seq_inner_product(lp->obj, sample->el + 1 + lp->dim, lp->dim, &lp->tmp2);
+		isl_int_cdiv_q(lp->tmp, lp->tmp, sample->el[0]);
+		isl_int_fdiv_q(lp->tmp2, lp->tmp2, sample->el[0]);
+		if (isl_int_ge(lp->tmp, lp->tmp2))
+			lp->is_fixed = 1;
+		isl_vec_free(sample);
+	}
+	isl_int_divexact_ui(lp->opt_denom, lp->opt_denom, 2);
+	if (res != isl_lp_ok)
+		return -1;
+	return 0;
+}
+
+/* The current objective function has a fixed (or no) integer value.
+ * Cut the tableau to the hyperplane that fixes this value in
+ * both halves of the tableau.
+ * Return 1 if the resulting tableau is empty.
+ */
+static int cut_lp_to_hyperplane(struct tab_lp *lp, isl_int *row)
+{
+	enum isl_lp_result res;
+
+	isl_int_set_si(lp->row->el[0], 0);
+	isl_seq_cpy(lp->row->el + 1, row, lp->dim);
+	isl_seq_clr(lp->row->el + 1 + lp->dim, lp->dim);
+	res = isl_tab_min(lp->tab, lp->row->el, lp->ctx->one,
+			  &lp->tmp, NULL, 0);
+	if (res != isl_lp_ok)
+		return -1;
+
+	isl_int_neg(lp->row->el[0], lp->tmp);
+	if (isl_tab_add_eq(lp->tab, lp->row->el) < 0)
+		return -1;
+
+	isl_seq_cpy(lp->row->el + 1 + lp->dim, row, lp->dim);
+	isl_seq_clr(lp->row->el + 1, lp->dim);
+	if (isl_tab_add_eq(lp->tab, lp->row->el) < 0)
+		return -1;
+
+	lp->con_offset += 2;
+
+	return lp->tab->empty;
+}
+
+static void get_obj_val(struct tab_lp* lp, GBR_type *F)
+{
+	isl_int_neg(GBR_numref(*F), lp->opt);
+	isl_int_set(GBR_denref(*F), lp->opt_denom);
+}
+
+static void delete_lp(struct tab_lp *lp)
+{
+	if (!lp)
+		return;
+
+	isl_int_clear(lp->opt);
+	isl_int_clear(lp->opt_denom);
+	isl_int_clear(lp->tmp);
+	isl_int_clear(lp->tmp2);
+	isl_vec_free(lp->row);
+	free(lp->stack);
+	isl_tab_free(lp->tab);
+	isl_ctx_deref(lp->ctx);
+	free(lp);
+}
+
+static int add_lp_row(struct tab_lp *lp, isl_int *row, int dim)
+{
+	lp->stack[lp->neq] = isl_tab_snap(lp->tab);
+
+	isl_int_set_si(lp->row->el[0], 0);
+	isl_seq_cpy(lp->row->el + 1, row, lp->dim);
+	isl_seq_neg(lp->row->el + 1 + lp->dim, row, lp->dim);
+
+	if (isl_tab_add_valid_eq(lp->tab, lp->row->el) < 0)
+		return -1;
+
+	return lp->neq++;
+}
+
+static void get_alpha(struct tab_lp* lp, int row, GBR_type *alpha)
+{
+	row += lp->con_offset;
+	isl_int_neg(GBR_numref(*alpha), lp->tab->dual->el[1 + row]);
+	isl_int_set(GBR_denref(*alpha), lp->tab->dual->el[0]);
+}
+
+static int del_lp_row(struct tab_lp *lp)
+{
+	lp->neq--;
+	return isl_tab_rollback(lp->tab, lp->stack[lp->neq]);
+}
diff --git a/polly/lib/External/isl/basis_reduction_templ.c b/polly/lib/External/isl/basis_reduction_templ.c
new file mode 100644
index 00000000000..98d8141372b
--- /dev/null
+++ b/polly/lib/External/isl/basis_reduction_templ.c
@@ -0,0 +1,358 @@
+/*
+ * Copyright 2006-2007 Universiteit Leiden
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, Leiden Institute of Advanced Computer Science,
+ * Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
+ * and K.U.Leuven, Departement Computerwetenschappen, Celestijnenlaan 200A,
+ * B-3001 Leuven, Belgium
+ */
+
+#include <stdlib.h>
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl_vec_private.h>
+#include <isl_options_private.h>
+#include "isl_basis_reduction.h"
+
+static void save_alpha(GBR_LP *lp, int first, int n, GBR_type *alpha)
+{
+	int i;
+
+	for (i = 0; i < n; ++i)
+		GBR_lp_get_alpha(lp, first + i, &alpha[i]);
+}
+
+/* Compute a reduced basis for the set represented by the tableau "tab".
+ * tab->basis, which must be initialized by the calling function to an affine
+ * unimodular basis, is updated to reflect the reduced basis.
+ * The first tab->n_zero rows of the basis (ignoring the constant row)
+ * are assumed to correspond to equalities and are left untouched.
+ * tab->n_zero is updated to reflect any additional equalities that
+ * have been detected in the first rows of the new basis.
+ * The final tab->n_unbounded rows of the basis are assumed to correspond
+ * to unbounded directions and are also left untouched.
+ * In particular this means that the remaining rows are assumed to
+ * correspond to bounded directions.
+ *
+ * This function implements the algorithm described in
+ * "An Implementation of the Generalized Basis Reduction Algorithm
+ *  for Integer Programming" of Cook el al. to compute a reduced basis.
+ * We use \epsilon = 1/4.
+ *
+ * If ctx->opt->gbr_only_first is set, the user is only interested
+ * in the first direction.  In this case we stop the basis reduction when
+ * the width in the first direction becomes smaller than 2.
+ */
+struct isl_tab *isl_tab_compute_reduced_basis(struct isl_tab *tab)
+{
+	unsigned dim;
+	struct isl_ctx *ctx;
+	struct isl_mat *B;
+	int unbounded;
+	int i;
+	GBR_LP *lp = NULL;
+	GBR_type F_old, alpha, F_new;
+	int row;
+	isl_int tmp;
+	struct isl_vec *b_tmp;
+	GBR_type *F = NULL;
+	GBR_type *alpha_buffer[2] = { NULL, NULL };
+	GBR_type *alpha_saved;
+	GBR_type F_saved;
+	int use_saved = 0;
+	isl_int mu[2];
+	GBR_type mu_F[2];
+	GBR_type two;
+	GBR_type one;
+	int empty = 0;
+	int fixed = 0;
+	int fixed_saved = 0;
+	int mu_fixed[2];
+	int n_bounded;
+	int gbr_only_first;
+
+	if (!tab)
+		return NULL;
+
+	if (tab->empty)
+		return tab;
+
+	ctx = tab->mat->ctx;
+	gbr_only_first = ctx->opt->gbr_only_first;
+	dim = tab->n_var;
+	B = tab->basis;
+	if (!B)
+		return tab;
+
+	n_bounded = dim - tab->n_unbounded;
+	if (n_bounded <= tab->n_zero + 1)
+		return tab;
+
+	isl_int_init(tmp);
+	isl_int_init(mu[0]);
+	isl_int_init(mu[1]);
+
+	GBR_init(alpha);
+	GBR_init(F_old);
+	GBR_init(F_new);
+	GBR_init(F_saved);
+	GBR_init(mu_F[0]);
+	GBR_init(mu_F[1]);
+	GBR_init(two);
+	GBR_init(one);
+
+	b_tmp = isl_vec_alloc(ctx, dim);
+	if (!b_tmp)
+		goto error;
+
+	F = isl_alloc_array(ctx, GBR_type, n_bounded);
+	alpha_buffer[0] = isl_alloc_array(ctx, GBR_type, n_bounded);
+	alpha_buffer[1] = isl_alloc_array(ctx, GBR_type, n_bounded);
+	alpha_saved = alpha_buffer[0];
+
+	if (!F || !alpha_buffer[0] || !alpha_buffer[1])
+		goto error;
+
+	for (i = 0; i < n_bounded; ++i) {
+		GBR_init(F[i]);
+		GBR_init(alpha_buffer[0][i]);
+		GBR_init(alpha_buffer[1][i]);
+	}
+
+	GBR_set_ui(two, 2);
+	GBR_set_ui(one, 1);
+
+	lp = GBR_lp_init(tab);
+	if (!lp)
+		goto error;
+
+	i = tab->n_zero;
+
+	GBR_lp_set_obj(lp, B->row[1+i]+1, dim);
+	ctx->stats->gbr_solved_lps++;
+	unbounded = GBR_lp_solve(lp);
+	isl_assert(ctx, !unbounded, goto error);
+	GBR_lp_get_obj_val(lp, &F[i]);
+
+	if (GBR_lt(F[i], one)) {
+		if (!GBR_is_zero(F[i])) {
+			empty = GBR_lp_cut(lp, B->row[1+i]+1);
+			if (empty)
+				goto done;
+			GBR_set_ui(F[i], 0);
+		}
+		tab->n_zero++;
+	}
+
+	do {
+		if (i+1 == tab->n_zero) {
+			GBR_lp_set_obj(lp, B->row[1+i+1]+1, dim);
+			ctx->stats->gbr_solved_lps++;
+			unbounded = GBR_lp_solve(lp);
+			isl_assert(ctx, !unbounded, goto error);
+			GBR_lp_get_obj_val(lp, &F_new);
+			fixed = GBR_lp_is_fixed(lp);
+			GBR_set_ui(alpha, 0);
+		} else
+		if (use_saved) {
+			row = GBR_lp_next_row(lp);
+			GBR_set(F_new, F_saved);
+			fixed = fixed_saved;
+			GBR_set(alpha, alpha_saved[i]);
+		} else {
+			row = GBR_lp_add_row(lp, B->row[1+i]+1, dim);
+			GBR_lp_set_obj(lp, B->row[1+i+1]+1, dim);
+			ctx->stats->gbr_solved_lps++;
+			unbounded = GBR_lp_solve(lp);
+			isl_assert(ctx, !unbounded, goto error);
+			GBR_lp_get_obj_val(lp, &F_new);
+			fixed = GBR_lp_is_fixed(lp);
+
+			GBR_lp_get_alpha(lp, row, &alpha);
+
+			if (i > 0)
+				save_alpha(lp, row-i, i, alpha_saved);
+
+			if (GBR_lp_del_row(lp) < 0)
+				goto error;
+		}
+		GBR_set(F[i+1], F_new);
+
+		GBR_floor(mu[0], alpha);
+		GBR_ceil(mu[1], alpha);
+
+		if (isl_int_eq(mu[0], mu[1]))
+			isl_int_set(tmp, mu[0]);
+		else {
+			int j;
+
+			for (j = 0; j <= 1; ++j) {
+				isl_int_set(tmp, mu[j]);
+				isl_seq_combine(b_tmp->el,
+						ctx->one, B->row[1+i+1]+1,
+						tmp, B->row[1+i]+1, dim);
+				GBR_lp_set_obj(lp, b_tmp->el, dim);
+				ctx->stats->gbr_solved_lps++;
+				unbounded = GBR_lp_solve(lp);
+				isl_assert(ctx, !unbounded, goto error);
+				GBR_lp_get_obj_val(lp, &mu_F[j]);
+				mu_fixed[j] = GBR_lp_is_fixed(lp);
+				if (i > 0)
+					save_alpha(lp, row-i, i, alpha_buffer[j]);
+			}
+
+			if (GBR_lt(mu_F[0], mu_F[1]))
+				j = 0;
+			else
+				j = 1;
+
+			isl_int_set(tmp, mu[j]);
+			GBR_set(F_new, mu_F[j]);
+			fixed = mu_fixed[j];
+			alpha_saved = alpha_buffer[j];
+		}
+		isl_seq_combine(B->row[1+i+1]+1, ctx->one, B->row[1+i+1]+1,
+				tmp, B->row[1+i]+1, dim);
+
+		if (i+1 == tab->n_zero && fixed) {
+			if (!GBR_is_zero(F[i+1])) {
+				empty = GBR_lp_cut(lp, B->row[1+i+1]+1);
+				if (empty)
+					goto done;
+				GBR_set_ui(F[i+1], 0);
+			}
+			tab->n_zero++;
+		}
+
+		GBR_set(F_old, F[i]);
+
+		use_saved = 0;
+		/* mu_F[0] = 4 * F_new; mu_F[1] = 3 * F_old */
+		GBR_set_ui(mu_F[0], 4);
+		GBR_mul(mu_F[0], mu_F[0], F_new);
+		GBR_set_ui(mu_F[1], 3);
+		GBR_mul(mu_F[1], mu_F[1], F_old);
+		if (GBR_lt(mu_F[0], mu_F[1])) {
+			B = isl_mat_swap_rows(B, 1 + i, 1 + i + 1);
+			if (i > tab->n_zero) {
+				use_saved = 1;
+				GBR_set(F_saved, F_new);
+				fixed_saved = fixed;
+				if (GBR_lp_del_row(lp) < 0)
+					goto error;
+				--i;
+			} else {
+				GBR_set(F[tab->n_zero], F_new);
+				if (gbr_only_first && GBR_lt(F[tab->n_zero], two))
+					break;
+
+				if (fixed) {
+					if (!GBR_is_zero(F[tab->n_zero])) {
+						empty = GBR_lp_cut(lp, B->row[1+tab->n_zero]+1);
+						if (empty)
+							goto done;
+						GBR_set_ui(F[tab->n_zero], 0);
+					}
+					tab->n_zero++;
+				}
+			}
+		} else {
+			GBR_lp_add_row(lp, B->row[1+i]+1, dim);
+			++i;
+		}
+	} while (i < n_bounded - 1);
+
+	if (0) {
+done:
+		if (empty < 0) {
+error:
+			isl_mat_free(B);
+			B = NULL;
+		}
+	}
+
+	GBR_lp_delete(lp);
+
+	if (alpha_buffer[1])
+		for (i = 0; i < n_bounded; ++i) {
+			GBR_clear(F[i]);
+			GBR_clear(alpha_buffer[0][i]);
+			GBR_clear(alpha_buffer[1][i]);
+		}
+	free(F);
+	free(alpha_buffer[0]);
+	free(alpha_buffer[1]);
+
+	isl_vec_free(b_tmp);
+
+	GBR_clear(alpha);
+	GBR_clear(F_old);
+	GBR_clear(F_new);
+	GBR_clear(F_saved);
+	GBR_clear(mu_F[0]);
+	GBR_clear(mu_F[1]);
+	GBR_clear(two);
+	GBR_clear(one);
+
+	isl_int_clear(tmp);
+	isl_int_clear(mu[0]);
+	isl_int_clear(mu[1]);
+
+	tab->basis = B;
+
+	return tab;
+}
+
+/* Compute an affine form of a reduced basis of the given basic
+ * non-parametric set, which is assumed to be bounded and not
+ * include any integer divisions.
+ * The first column and the first row correspond to the constant term.
+ *
+ * If the input contains any equalities, we first create an initial
+ * basis with the equalities first.  Otherwise, we start off with
+ * the identity matrix.
+ */
+struct isl_mat *isl_basic_set_reduced_basis(struct isl_basic_set *bset)
+{
+	struct isl_mat *basis;
+	struct isl_tab *tab;
+
+	if (!bset)
+		return NULL;
+
+	if (isl_basic_set_dim(bset, isl_dim_div) != 0)
+		isl_die(bset->ctx, isl_error_invalid,
+			"no integer division allowed", return NULL);
+	if (isl_basic_set_dim(bset, isl_dim_param) != 0)
+		isl_die(bset->ctx, isl_error_invalid,
+			"no parameters allowed", return NULL);
+
+	tab = isl_tab_from_basic_set(bset, 0);
+	if (!tab)
+		return NULL;
+
+	if (bset->n_eq == 0)
+		tab->basis = isl_mat_identity(bset->ctx, 1 + tab->n_var);
+	else {
+		isl_mat *eq;
+		unsigned nvar = isl_basic_set_total_dim(bset);
+		eq = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, bset->n_eq,
+					1, nvar);
+		eq = isl_mat_left_hermite(eq, 0, NULL, &tab->basis);
+		tab->basis = isl_mat_lin_to_aff(tab->basis);
+		tab->n_zero = bset->n_eq;
+		isl_mat_free(eq);
+	}
+	tab = isl_tab_compute_reduced_basis(tab);
+	if (!tab)
+		return NULL;
+
+	basis = isl_mat_copy(tab->basis);
+
+	isl_tab_free(tab);
+
+	return basis;
+}
diff --git a/polly/lib/External/isl/bound.c b/polly/lib/External/isl/bound.c
new file mode 100644
index 00000000000..4aaaf13e5ce
--- /dev/null
+++ b/polly/lib/External/isl/bound.c
@@ -0,0 +1,291 @@
+#include <assert.h>
+#include <isl/stream.h>
+#include <isl_map_private.h>
+#include <isl_polynomial_private.h>
+#include <isl_scan.h>
+#include <isl/options.h>
+#include <isl/deprecated/point_int.h>
+#include <isl/deprecated/polynomial_int.h>
+
+struct bound_options {
+	struct isl_options	*isl;
+	unsigned		 verify;
+	int			 print_all;
+	int			 continue_on_error;
+};
+
+ISL_ARGS_START(struct bound_options, bound_options_args)
+ISL_ARG_CHILD(struct bound_options, isl, "isl", &isl_options_args,
+	"isl options")
+ISL_ARG_BOOL(struct bound_options, verify, 'T', "verify", 0, NULL)
+ISL_ARG_BOOL(struct bound_options, print_all, 'A', "print-all", 0, NULL)
+ISL_ARG_BOOL(struct bound_options, continue_on_error, '\0', "continue-on-error", 0, NULL)
+ISL_ARGS_END
+
+ISL_ARG_DEF(bound_options, struct bound_options, bound_options_args)
+
+static __isl_give isl_set *set_bounds(__isl_take isl_set *set)
+{
+	unsigned nparam;
+	int i, r;
+	isl_point *pt, *pt2;
+	isl_set *box;
+
+	nparam = isl_set_dim(set, isl_dim_param);
+	r = nparam >= 8 ? 5 : nparam >= 5 ? 15 : 50;
+
+	pt = isl_set_sample_point(isl_set_copy(set));
+	pt2 = isl_point_copy(pt);
+
+	for (i = 0; i < nparam; ++i) {
+		pt = isl_point_add_ui(pt, isl_dim_param, i, r);
+		pt2 = isl_point_sub_ui(pt2, isl_dim_param, i, r);
+	}
+
+	box = isl_set_box_from_points(pt, pt2);
+
+	return isl_set_intersect(set, box);
+}
+
+struct verify_point_bound {
+	struct bound_options *options;
+	int stride;
+	int n;
+	int exact;
+	int error;
+
+	isl_pw_qpolynomial_fold *pwf;
+	isl_pw_qpolynomial_fold *bound;
+};
+
+static int verify_point(__isl_take isl_point *pnt, void *user)
+{
+	int i;
+	unsigned nvar;
+	unsigned nparam;
+	struct verify_point_bound *vpb = (struct verify_point_bound *) user;
+	isl_int t;
+	isl_ctx *ctx;
+	isl_pw_qpolynomial_fold *pwf;
+	isl_val *bound = NULL;
+	isl_val *opt = NULL;
+	isl_set *dom = NULL;
+	isl_printer *p;
+	const char *minmax;
+	int bounded;
+	int sign;
+	int ok;
+	FILE *out = vpb->options->print_all ? stdout : stderr;
+
+	vpb->n--;
+
+	if (1) {
+		minmax = "ub";
+		sign = 1;
+	} else {
+		minmax = "lb";
+		sign = -1;
+	}
+
+	ctx = isl_point_get_ctx(pnt);
+	p = isl_printer_to_file(ctx, out);
+
+	isl_int_init(t);
+
+	pwf = isl_pw_qpolynomial_fold_copy(vpb->pwf);
+
+	nparam = isl_pw_qpolynomial_fold_dim(pwf, isl_dim_param);
+	for (i = 0; i < nparam; ++i) {
+		isl_point_get_coordinate(pnt, isl_dim_param, i, &t);
+		pwf = isl_pw_qpolynomial_fold_fix_dim(pwf, isl_dim_param, i, t);
+	}
+
+	bound = isl_pw_qpolynomial_fold_eval(
+				    isl_pw_qpolynomial_fold_copy(vpb->bound),
+				    isl_point_copy(pnt));
+
+	dom = isl_pw_qpolynomial_fold_domain(isl_pw_qpolynomial_fold_copy(pwf));
+	bounded = isl_set_is_bounded(dom);
+
+	if (bounded < 0)
+		goto error;
+
+	if (!bounded)
+		opt = isl_pw_qpolynomial_fold_eval(
+				    isl_pw_qpolynomial_fold_copy(pwf),
+				    isl_set_sample_point(isl_set_copy(dom)));
+	else if (sign > 0)
+		opt = isl_pw_qpolynomial_fold_max(isl_pw_qpolynomial_fold_copy(pwf));
+	else
+		opt = isl_pw_qpolynomial_fold_min(isl_pw_qpolynomial_fold_copy(pwf));
+
+	nvar = isl_set_dim(dom, isl_dim_set);
+	if (vpb->exact && bounded)
+		ok = isl_val_eq(opt, bound);
+	else if (sign > 0)
+		ok = isl_val_le(opt, bound);
+	else
+		ok = isl_val_le(bound, opt);
+	if (ok < 0)
+		goto error;
+
+	if (vpb->options->print_all || !ok) {
+		p = isl_printer_print_str(p, minmax);
+		p = isl_printer_print_str(p, "(");
+		for (i = 0; i < nparam; ++i) {
+			if (i)
+				p = isl_printer_print_str(p, ", ");
+			isl_point_get_coordinate(pnt, isl_dim_param, i, &t);
+			p = isl_printer_print_isl_int(p, t);
+		}
+		p = isl_printer_print_str(p, ") = ");
+		p = isl_printer_print_val(p, bound);
+		p = isl_printer_print_str(p, ", ");
+		p = isl_printer_print_str(p, bounded ? "opt" : "sample");
+		p = isl_printer_print_str(p, " = ");
+		p = isl_printer_print_val(p, opt);
+		if (ok)
+			p = isl_printer_print_str(p, ". OK");
+		else
+			p = isl_printer_print_str(p, ". NOT OK");
+		p = isl_printer_end_line(p);
+	} else if ((vpb->n % vpb->stride) == 0) {
+		p = isl_printer_print_str(p, "o");
+		p = isl_printer_flush(p);
+	}
+
+	if (0) {
+error:
+		ok = 0;
+	}
+
+	isl_pw_qpolynomial_fold_free(pwf);
+	isl_val_free(bound);
+	isl_val_free(opt);
+	isl_point_free(pnt);
+	isl_set_free(dom);
+
+	isl_int_clear(t);
+
+	isl_printer_free(p);
+
+	if (!ok)
+		vpb->error = 1;
+
+	if (vpb->options->continue_on_error)
+		ok = 1;
+
+	return (vpb->n >= 1 && ok) ? 0 : -1;
+}
+
+static int check_solution(__isl_take isl_pw_qpolynomial_fold *pwf,
+	__isl_take isl_pw_qpolynomial_fold *bound, int exact,
+	struct bound_options *options)
+{
+	struct verify_point_bound vpb;
+	isl_int count, max;
+	isl_set *dom;
+	isl_set *context;
+	int i, r, n;
+
+	dom = isl_pw_qpolynomial_fold_domain(isl_pw_qpolynomial_fold_copy(pwf));
+	context = isl_set_params(isl_set_copy(dom));
+	context = isl_set_remove_divs(context);
+	context = set_bounds(context);
+
+	isl_int_init(count);
+	isl_int_init(max);
+
+	isl_int_set_si(max, 200);
+	r = isl_set_count_upto(context, max, &count);
+	assert(r >= 0);
+	n = isl_int_get_si(count);
+
+	isl_int_clear(max);
+	isl_int_clear(count);
+
+	vpb.options = options;
+	vpb.pwf = pwf;
+	vpb.bound = bound;
+	vpb.n = n;
+	vpb.stride = n > 70 ? 1 + (n + 1)/70 : 1;
+	vpb.error = 0;
+	vpb.exact = exact;
+
+	if (!options->print_all) {
+		for (i = 0; i < vpb.n; i += vpb.stride)
+			printf(".");
+		printf("\r");
+		fflush(stdout);
+	}
+
+	isl_set_foreach_point(context, verify_point, &vpb);
+
+	isl_set_free(context);
+	isl_set_free(dom);
+	isl_pw_qpolynomial_fold_free(pwf);
+	isl_pw_qpolynomial_fold_free(bound);
+
+	if (!options->print_all)
+		printf("\n");
+
+	if (vpb.error) {
+		fprintf(stderr, "Check failed !\n");
+		return -1;
+	}
+
+	return 0;
+}
+
+int main(int argc, char **argv)
+{
+	isl_ctx *ctx;
+	isl_pw_qpolynomial_fold *copy;
+	isl_pw_qpolynomial_fold *pwf;
+	struct isl_stream *s;
+	struct isl_obj obj;
+	struct bound_options *options;
+	int exact;
+	int r = 0;
+
+	options = bound_options_new_with_defaults();
+	assert(options);
+	argc = bound_options_parse(options, argc, argv, ISL_ARG_ALL);
+
+	ctx = isl_ctx_alloc_with_options(&bound_options_args, options);
+
+	s = isl_stream_new_file(ctx, stdin);
+	obj = isl_stream_read_obj(s);
+	if (obj.type == isl_obj_pw_qpolynomial)
+		pwf = isl_pw_qpolynomial_fold_from_pw_qpolynomial(isl_fold_max,
+								  obj.v);
+	else if (obj.type == isl_obj_pw_qpolynomial_fold)
+		pwf = obj.v;
+	else {
+		obj.type->free(obj.v);
+		isl_die(ctx, isl_error_invalid, "invalid input", goto error);
+	}
+
+	if (options->verify)
+		copy = isl_pw_qpolynomial_fold_copy(pwf);
+
+	pwf = isl_pw_qpolynomial_fold_bound(pwf, &exact);
+	pwf = isl_pw_qpolynomial_fold_coalesce(pwf);
+
+	if (options->verify) {
+		r = check_solution(copy, pwf, exact, options);
+	} else {
+		if (!exact)
+			printf("# NOT exact\n");
+		isl_pw_qpolynomial_fold_print(pwf, stdout, 0);
+		fprintf(stdout, "\n");
+		isl_pw_qpolynomial_fold_free(pwf);
+	}
+
+error:
+	isl_stream_free(s);
+
+	isl_ctx_free(ctx);
+
+	return r;
+}
diff --git a/polly/lib/External/isl/bound_test.sh.in b/polly/lib/External/isl/bound_test.sh.in
new file mode 100755
index 00000000000..e3fc037ecaf
--- /dev/null
+++ b/polly/lib/External/isl/bound_test.sh.in
@@ -0,0 +1,35 @@
+#!/bin/sh
+
+EXEEXT=@EXEEXT@
+
+BOUND_TESTS="\
+	basicLinear2.pwqp \
+	basicLinear.pwqp \
+	basicTestParameterPosNeg.pwqp \
+	basicTest.pwqp \
+	devos.pwqp \
+	equality1.pwqp \
+	equality2.pwqp \
+	equality3.pwqp \
+	equality4.pwqp \
+	equality5.pwqp \
+	faddeev.pwqp \
+	linearExample.pwqp \
+	neg.pwqp \
+	philippe3vars3pars.pwqp \
+	philippe3vars.pwqp \
+	philippeNeg.pwqp \
+	philippePolynomialCoeff1P.pwqp \
+	philippePolynomialCoeff.pwqp \
+	philippe.pwqp \
+	product.pwqp \
+	split.pwqp \
+	test3Deg3Var.pwqp \
+	toplas.pwqp \
+	unexpanded.pwqp"
+
+for i in $BOUND_TESTS; do
+	echo $i;
+	./isl_bound$EXEEXT -T --bound=bernstein < $srcdir/test_inputs/$i || exit
+	./isl_bound$EXEEXT -T --bound=range < $srcdir/test_inputs/$i || exit
+done
diff --git a/polly/lib/External/isl/cat.c b/polly/lib/External/isl/cat.c
new file mode 100644
index 00000000000..63131420592
--- /dev/null
+++ b/polly/lib/External/isl/cat.c
@@ -0,0 +1,58 @@
+#include <assert.h>
+#include <isl/obj.h>
+#include <isl/printer.h>
+#include <isl/stream.h>
+#include <isl/options.h>
+
+struct isl_arg_choice cat_format[] = {
+	{"isl",		ISL_FORMAT_ISL},
+	{"omega",	ISL_FORMAT_OMEGA},
+	{"polylib",	ISL_FORMAT_POLYLIB},
+	{"ext-polylib",	ISL_FORMAT_EXT_POLYLIB},
+	{"latex",	ISL_FORMAT_LATEX},
+	{0}
+};
+
+struct cat_options {
+	struct isl_options	*isl;
+	unsigned		 format;
+};
+
+ISL_ARGS_START(struct cat_options, cat_options_args)
+ISL_ARG_CHILD(struct cat_options, isl, "isl", &isl_options_args, "isl options")
+ISL_ARG_CHOICE(struct cat_options, format, 0, "format", \
+	cat_format,	ISL_FORMAT_ISL, "output format")
+ISL_ARGS_END
+
+ISL_ARG_DEF(cat_options, struct cat_options, cat_options_args)
+
+int main(int argc, char **argv)
+{
+	struct isl_ctx *ctx;
+	struct isl_stream *s;
+	struct isl_obj obj;
+	struct cat_options *options;
+	isl_printer *p;
+
+	options = cat_options_new_with_defaults();
+	assert(options);
+	argc = cat_options_parse(options, argc, argv, ISL_ARG_ALL);
+
+	ctx = isl_ctx_alloc_with_options(&cat_options_args, options);
+
+	s = isl_stream_new_file(ctx, stdin);
+	obj = isl_stream_read_obj(s);
+	isl_stream_free(s);
+
+	p = isl_printer_to_file(ctx, stdout);
+	p = isl_printer_set_output_format(p, options->format);
+	p = obj.type->print(p, obj.v);
+	p = isl_printer_end_line(p);
+	isl_printer_free(p);
+
+	obj.type->free(obj.v);
+
+	isl_ctx_free(ctx);
+
+	return 0;
+}
diff --git a/polly/lib/External/isl/closure.c b/polly/lib/External/isl/closure.c
new file mode 100644
index 00000000000..a0faa15820a
--- /dev/null
+++ b/polly/lib/External/isl/closure.c
@@ -0,0 +1,34 @@
+#include <assert.h>
+#include <isl/map.h>
+#include <isl/options.h>
+
+int main(int argc, char **argv)
+{
+	struct isl_ctx *ctx;
+	struct isl_map *map;
+	struct isl_options *options;
+	int exact;
+
+	options = isl_options_new_with_defaults();
+	assert(options);
+	argc = isl_options_parse(options, argc, argv, ISL_ARG_ALL);
+
+	ctx = isl_ctx_alloc_with_options(&isl_options_args, options);
+
+	map = isl_map_read_from_file(ctx, stdin);
+	map = isl_map_transitive_closure(map, &exact);
+	if (!exact)
+		printf("# NOT exact\n");
+	isl_map_print(map, stdout, 0, ISL_FORMAT_ISL);
+	printf("\n");
+	map = isl_map_compute_divs(map);
+	map = isl_map_coalesce(map);
+	printf("# coalesced\n");
+	isl_map_print(map, stdout, 0, ISL_FORMAT_ISL);
+	printf("\n");
+	isl_map_free(map);
+
+	isl_ctx_free(ctx);
+
+	return 0;
+}
diff --git a/polly/lib/External/isl/codegen.c b/polly/lib/External/isl/codegen.c
new file mode 100644
index 00000000000..e9cd692fe8d
--- /dev/null
+++ b/polly/lib/External/isl/codegen.c
@@ -0,0 +1,138 @@
+/*
+ * Copyright 2012      Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege,
+ * Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+/* This program prints an AST that scans the domain elements of
+ * the domain of a given schedule in the order of their image(s).
+ *
+ * The input consists of three sets/relations.
+ * - a schedule
+ * - a context
+ * - a relation describing AST generation options
+ */
+
+#include <assert.h>
+#include <isl/ast.h>
+#include <isl/ast_build.h>
+#include <isl/options.h>
+#include <isl/set.h>
+
+struct options {
+	struct isl_options	*isl;
+	unsigned		 atomic;
+	unsigned		 separate;
+};
+
+ISL_ARGS_START(struct options, options_args)
+ISL_ARG_CHILD(struct options, isl, "isl", &isl_options_args, "isl options")
+ISL_ARG_BOOL(struct options, atomic, 0, "atomic", 0,
+	"globally set the atomic option")
+ISL_ARG_BOOL(struct options, separate, 0, "separate", 0,
+	"globally set the separate option")
+ISL_ARGS_END
+
+ISL_ARG_DEF(options, struct options, options_args)
+
+/* Return a universal, 1-dimensional set with the given name.
+ */
+static __isl_give isl_union_set *universe(isl_ctx *ctx, const char *name)
+{
+	isl_space *space;
+
+	space = isl_space_set_alloc(ctx, 0, 1);
+	space = isl_space_set_tuple_name(space, isl_dim_set, name);
+	return isl_union_set_from_set(isl_set_universe(space));
+}
+
+/* Set the "name" option for the entire schedule domain.
+ */
+static __isl_give isl_union_map *set_universe(__isl_take isl_union_map *opt,
+	__isl_keep isl_union_map *schedule, const char *name)
+{
+	isl_ctx *ctx;
+	isl_union_set *domain, *target;
+	isl_union_map *option;
+
+	ctx = isl_union_map_get_ctx(opt);
+
+	domain = isl_union_map_range(isl_union_map_copy(schedule));
+	domain = isl_union_set_universe(domain);
+	target = universe(ctx, name);
+	option = isl_union_map_from_domain_and_range(domain, target);
+	opt = isl_union_map_union(opt, option);
+
+	return opt;
+}
+
+/* Update the build options based on the user-specified options.
+ *
+ * If the --separate or --atomic options were specified, then
+ * we clear any separate or atomic options that may already exist in "opt".
+ */
+static __isl_give isl_ast_build *set_options(__isl_take isl_ast_build *build,
+	__isl_take isl_union_map *opt, struct options *options,
+	__isl_keep isl_union_map *schedule)
+{
+	if (options->separate || options->atomic) {
+		isl_ctx *ctx;
+		isl_union_set *target;
+
+		ctx = isl_union_map_get_ctx(schedule);
+
+		target = universe(ctx, "separate");
+		opt = isl_union_map_subtract_range(opt, target);
+		target = universe(ctx, "atomic");
+		opt = isl_union_map_subtract_range(opt, target);
+	}
+
+	if (options->separate)
+		opt = set_universe(opt, schedule, "separate");
+	if (options->atomic)
+		opt = set_universe(opt, schedule, "atomic");
+
+	build = isl_ast_build_set_options(build, opt);
+
+	return build;
+}
+
+int main(int argc, char **argv)
+{
+	isl_ctx *ctx;
+	isl_set *context;
+	isl_union_map *schedule;
+	isl_union_map *options_map;
+	isl_ast_build *build;
+	isl_ast_node *tree;
+	struct options *options;
+	isl_printer *p;
+
+	options = options_new_with_defaults();
+	assert(options);
+	argc = options_parse(options, argc, argv, ISL_ARG_ALL);
+
+	ctx = isl_ctx_alloc_with_options(&options_args, options);
+
+	schedule = isl_union_map_read_from_file(ctx, stdin);
+	context = isl_set_read_from_file(ctx, stdin);
+	options_map = isl_union_map_read_from_file(ctx, stdin);
+
+	build = isl_ast_build_from_context(context);
+	build = set_options(build, options_map, options, schedule);
+	tree = isl_ast_build_ast_from_schedule(build, schedule);
+	isl_ast_build_free(build);
+
+	p = isl_printer_to_file(ctx, stdout);
+	p = isl_printer_set_output_format(p, ISL_FORMAT_C);
+	p = isl_printer_print_ast_node(p, tree);
+	isl_printer_free(p);
+
+	isl_ast_node_free(tree);
+
+	isl_ctx_free(ctx);
+	return 0;
+}
diff --git a/polly/lib/External/isl/codegen_test.sh.in b/polly/lib/External/isl/codegen_test.sh.in
new file mode 100644
index 00000000000..36a1815b762
--- /dev/null
+++ b/polly/lib/External/isl/codegen_test.sh.in
@@ -0,0 +1,21 @@
+#!/bin/sh
+
+EXEEXT=@EXEEXT@
+srcdir=@srcdir@
+
+failed=0
+
+for i in $srcdir/test_inputs/codegen/*.in \
+		$srcdir/test_inputs/codegen/cloog/*.in \
+		$srcdir/test_inputs/codegen/omega/*.in \
+		$srcdir/test_inputs/codegen/pldi2012/*.in; do
+	echo $i;
+	base=`basename $i .in`
+	test=test-$base.c
+	dir=`dirname $i`
+	ref=$dir/$base.c
+	(./isl_codegen$EXEEXT < $i > $test &&
+	 diff -uw $ref $test && rm $test) || failed=1
+done
+
+test $failed -eq 0 || exit
diff --git a/polly/lib/External/isl/configure.ac b/polly/lib/External/isl/configure.ac
new file mode 100644
index 00000000000..f5d684b2d82
--- /dev/null
+++ b/polly/lib/External/isl/configure.ac
@@ -0,0 +1,236 @@
+AC_INIT([isl], [0.14], [isl-development@googlegroups.com])
+AC_CONFIG_AUX_DIR([.])
+AC_CONFIG_MACRO_DIR([m4])
+AM_INIT_AUTOMAKE([foreign])
+m4_ifdef([AM_SILENT_RULES],[AM_SILENT_RULES([yes])])
+AC_SUBST(versioninfo)
+versioninfo=14:0:1
+
+if test "x$prefix" != "xNONE"; then
+	prefix_wd=`cd $prefix && pwd`
+	srcdir_wd=`cd $srcdir && pwd`
+	wd=`pwd`
+	if test "x$prefix_wd" = "x$srcdir_wd"; then
+		AC_MSG_ERROR(Installation in source directory not supported)
+	fi
+	if test "x$prefix_wd" = "x$wd"; then
+		AC_MSG_ERROR(Installation in build directory not supported)
+	fi
+fi
+
+AC_PROG_CC
+AC_PROG_CXX
+
+AX_CC_MAXOPT
+AX_GCC_WARN_UNUSED_RESULT
+AX_C___ATTRIBUTE__
+
+AC_PROG_LIBTOOL
+
+AC_CHECK_PROG(PERL, perl, perl, [])
+AC_CHECK_PROG(PDFLATEX, pdflatex, pdflatex, [])
+AC_CHECK_PROG(POD2HTML, pod2html, pod2html, [])
+
+AM_CONDITIONAL(GENERATE_DOC, test -n "$PERL" -a -n "$PDFLATEX" -a -n "$POD2HTML")
+
+AX_CREATE_STDINT_H(include/isl/stdint.h)
+
+AC_ARG_WITH([int],
+	    [AS_HELP_STRING([--with-int=gmp|imath],
+			    [Which package to use to represent
+				multi-precision integers [default=gmp]])],
+	    [], [with_int=gmp])
+case "$with_int" in
+gmp|imath)
+	;;
+*)
+	AC_MSG_ERROR([bad value ${withval} for --with-int (use gmp or imath)])
+esac
+
+AC_SUBST(MP_CPPFLAGS)
+AC_SUBST(MP_LDFLAGS)
+AC_SUBST(MP_LIBS)
+case "$with_int" in
+gmp)
+	AX_DETECT_GMP
+	;;
+imath)
+	AX_DETECT_IMATH
+	;;
+esac
+
+AM_CONDITIONAL(IMATH_FOR_MP, test x$with_int = ximath)
+AM_CONDITIONAL(GMP_FOR_MP, test x$with_int = xgmp)
+AC_CHECK_DECLS(ffs,[],[],[#include <strings.h>])
+AC_CHECK_DECLS(__builtin_ffs,[],[],[])
+
+AC_SUBST(CLANG_CXXFLAGS)
+AC_SUBST(CLANG_LDFLAGS)
+AC_SUBST(CLANG_LIBS)
+AX_SUBMODULE(clang,system|no,no)
+case "$with_clang" in
+system)
+	AC_PROG_GREP
+	llvm_config="llvm-config"
+	AC_CHECK_PROG([llvm_config_found], ["$llvm_config"], [yes])
+	if test "x$with_clang_prefix" != "x"; then
+		llvm_config="$with_clang_prefix/bin/llvm-config"
+		if test -x "$llvm_config"; then
+			llvm_config_found=yes
+		fi
+	fi
+	if test "$llvm_config_found" != yes; then
+		AC_MSG_ERROR([llvm-config not found])
+	fi
+	CLANG_CXXFLAGS=`$llvm_config --cxxflags`
+	CLANG_LDFLAGS=`$llvm_config --ldflags`
+	targets=`$llvm_config --targets-built`
+	components="$targets asmparser bitreader support mc"
+	$llvm_config --components | $GREP option > /dev/null 2> /dev/null
+	if test $? -eq 0; then
+		components="$components option"
+	fi
+	CLANG_LIBS=`$llvm_config --libs $components`
+	systemlibs=`$llvm_config --system-libs 2> /dev/null | tail -1`
+	if test $? -eq 0; then
+		CLANG_LIBS="$CLANG_LIBS $systemlibs"
+	fi
+	CLANG_PREFIX=`$llvm_config --prefix`
+	AC_DEFINE_UNQUOTED(CLANG_PREFIX, ["$CLANG_PREFIX"],
+				[Clang installation prefix])
+
+	SAVE_CPPFLAGS="$CPPFLAGS"
+	CPPFLAGS="$CLANG_CXXFLAGS $CPPFLAGS"
+	AC_LANG_PUSH(C++)
+	AC_CHECK_HEADER([clang/Basic/SourceLocation.h], [],
+		[AC_ERROR([clang header file not found])])
+	AC_EGREP_HEADER([getDefaultTargetTriple], [llvm/Support/Host.h], [],
+		[AC_DEFINE([getDefaultTargetTriple], [getHostTriple],
+		[Define to getHostTriple for older versions of clang])])
+	AC_EGREP_HEADER([getExpansionLineNumber],
+		[clang/Basic/SourceLocation.h], [],
+		[AC_DEFINE([getExpansionLineNumber],
+			[getInstantiationLineNumber],
+	    [Define to getInstantiationLineNumber for older versions of clang])])
+	AC_EGREP_HEADER([DiagnosticsEngine],
+		[clang/Basic/Diagnostic.h], [],
+		[AC_DEFINE([DiagnosticsEngine],
+			[Diagnostic],
+	    [Define to Diagnostic for older versions of clang])])
+	AC_EGREP_HEADER([ArrayRef], [clang/Driver/Driver.h],
+		[AC_DEFINE([USE_ARRAYREF], [],
+			[Define if Driver::BuildCompilation takes ArrayRef])])
+	AC_EGREP_HEADER([CXXIsProduction], [clang/Driver/Driver.h],
+		[AC_DEFINE([HAVE_CXXISPRODUCTION], [],
+		[Define if Driver constructor takes CXXIsProduction argument])])
+	AC_EGREP_HEADER([ IsProduction], [clang/Driver/Driver.h],
+		[AC_DEFINE([HAVE_ISPRODUCTION], [],
+		[Define if Driver constructor takes IsProduction argument])])
+	AC_TRY_COMPILE([#include <clang/Driver/Driver.h>], [
+		using namespace clang;
+		DiagnosticsEngine *Diags;
+		new driver::Driver("", "", "", *Diags);
+	], [AC_DEFINE([DRIVER_CTOR_TAKES_DEFAULTIMAGENAME], [],
+		      [Define if Driver constructor takes default image name])])
+	AC_EGREP_HEADER([void HandleTopLevelDecl\(], [clang/AST/ASTConsumer.h],
+		[AC_DEFINE([HandleTopLevelDeclReturn], [void],
+			   [Return type of HandleTopLevelDeclReturn])
+		 AC_DEFINE([HandleTopLevelDeclContinue], [],
+			   [Return type of HandleTopLevelDeclReturn])],
+		[AC_DEFINE([HandleTopLevelDeclReturn], [bool],
+			   [Return type of HandleTopLevelDeclReturn])
+		 AC_DEFINE([HandleTopLevelDeclContinue], [true],
+			   [Return type of HandleTopLevelDeclReturn])])
+	AC_CHECK_HEADER([clang/Basic/DiagnosticOptions.h],
+		[AC_DEFINE([HAVE_BASIC_DIAGNOSTICOPTIONS_H], [],
+			   [Define if clang/Basic/DiagnosticOptions.h exists])])
+	AC_TRY_COMPILE([#include <clang/Basic/TargetInfo.h>], [
+		using namespace clang;
+		std::shared_ptr<TargetOptions> TO;
+		DiagnosticsEngine *Diags;
+		TargetInfo::CreateTargetInfo(*Diags, TO);
+	], [AC_DEFINE([CREATETARGETINFO_TAKES_SHARED_PTR], [],
+		    [Define if TargetInfo::CreateTargetInfo takes shared_ptr])])
+	AC_TRY_COMPILE([#include <clang/Basic/TargetInfo.h>], [
+		using namespace clang;
+		TargetOptions *TO;
+		DiagnosticsEngine *Diags;
+		TargetInfo::CreateTargetInfo(*Diags, TO);
+	], [AC_DEFINE([CREATETARGETINFO_TAKES_POINTER], [],
+		      [Define if TargetInfo::CreateTargetInfo takes pointer])])
+	AC_TRY_COMPILE([#include <clang/Frontend/CompilerInstance.h>], [
+		using namespace clang;
+		DiagnosticConsumer *client;
+		CompilerInstance *Clang;
+		Clang->createDiagnostics(client);
+	], [], [AC_DEFINE([CREATEDIAGNOSTICS_TAKES_ARG], [],
+	[Define if CompilerInstance::createDiagnostics takes argc and argv])])
+	AC_TRY_COMPILE([#include <clang/Lex/HeaderSearchOptions.h>], [
+		using namespace clang;
+		HeaderSearchOptions HSO;
+		HSO.AddPath("", frontend::Angled, false, false);
+	], [AC_DEFINE([ADDPATH_TAKES_4_ARGUMENTS], [],
+		[Define if HeaderSearchOptions::AddPath takes 4 arguments])])
+	AC_EGREP_HEADER([getNumParams],
+		[clang/AST/CanonicalType.h],
+		[AC_DEFINE([getNumArgs], [getNumParams],
+		    [Define to getNumParams for newer versions of clang])
+		 AC_DEFINE([getArgType], [getParamType],
+		    [Define to getParamType for newer versions of clang])])
+	AC_EGREP_HEADER([getReturnType],
+		[clang/AST/CanonicalType.h], [],
+		[AC_DEFINE([getReturnType], [getResultType],
+		    [Define to getResultType for older versions of clang])])
+	AC_TRY_COMPILE([#include <clang/Frontend/CompilerInstance.h>], [
+		using namespace clang;
+		CompilerInstance *Clang;
+		Clang->createPreprocessor(TU_Complete);
+	], [AC_DEFINE([CREATEPREPROCESSOR_TAKES_TUKIND], [],
+	[Define if CompilerInstance::createPreprocessor takes
+	 TranslationUnitKind])])
+	AC_EGREP_HEADER([setMainFileID], [clang/Basic/SourceManager.h],
+		[AC_DEFINE([HAVE_SETMAINFILEID], [],
+		[Define if SourceManager has a setMainFileID method])])
+	AC_CHECK_HEADER([llvm/ADT/OwningPtr.h],
+		[AC_DEFINE([HAVE_ADT_OWNINGPTR_H], [],
+			   [Define if llvm/ADT/OwningPtr.h exists])])
+	AC_LANG_POP
+	CPPFLAGS="$SAVE_CPPFLAGS"
+
+	SAVE_LDFLAGS="$LDFLAGS"
+	LDFLAGS="$CLANG_LDFLAGS $LDFLAGS"
+	AC_SUBST(LIB_CLANG_EDIT)
+	AC_CHECK_LIB([clangEdit], [main], [LIB_CLANG_EDIT=-lclangEdit], [])
+	LDFLAGS="$SAVE_LDFLAGS"
+	;;
+esac
+AM_CONDITIONAL(HAVE_CLANG, test $with_clang = system)
+
+AX_SET_WARNING_FLAGS
+
+AC_SUBST(WARNING_FLAGS)
+
+PACKAGE_CFLAGS="$MP_CPPFLAGS"
+PACKAGE_LDFLAGS="$MP_LDFLAGS"
+PACKAGE_LIBS="-lisl $MP_LIBS"
+AX_CREATE_PKGCONFIG_INFO
+
+AX_DETECT_GIT_HEAD
+echo '#define GIT_HEAD_ID "'$GIT_HEAD_ID'"' > gitversion.h
+
+AH_BOTTOM([#include <isl_config_post.h>])
+AC_CONFIG_HEADERS(isl_config.h)
+AC_CONFIG_FILES(Makefile)
+AC_CONFIG_FILES(doc/Makefile)
+if test $with_clang = system; then
+	AC_CONFIG_FILES(interface/Makefile)
+fi
+AC_CONFIG_FILES([bound_test.sh], [chmod +x bound_test.sh])
+AC_CONFIG_FILES([codegen_test.sh], [chmod +x codegen_test.sh])
+AC_CONFIG_FILES([pip_test.sh], [chmod +x pip_test.sh])
+AC_CONFIG_COMMANDS_POST([
+	dnl pass on arguments to subdir configures, but don't
+	dnl add them to config.status
+	ac_configure_args="$ac_configure_args $isl_configure_args"
+])
+AC_OUTPUT
diff --git a/polly/lib/External/isl/doc/Makefile.am b/polly/lib/External/isl/doc/Makefile.am
new file mode 100644
index 00000000000..aa79a6c9516
--- /dev/null
+++ b/polly/lib/External/isl/doc/Makefile.am
@@ -0,0 +1,17 @@
+if GENERATE_DOC
+export TEXINPUTS := $(srcdir):$(TEXINPUTS)
+export BIBINPUTS := $(srcdir):$(BIBINPUTS)
+export BSTINPUTS := $(srcdir):$(BSTINPUTS)
+
+user.tex: user.pod
+	$(PERL) $(srcdir)/mypod2latex $< $@
+manual.pdf: manual.tex user.tex $(srcdir)/implementation.tex
+	(cd ..; echo "@GIT_HEAD_VERSION@") > version.tex
+	$(PDFLATEX) $<
+	bibtex manual
+	$(PDFLATEX) $<
+	$(PDFLATEX) $<
+user.html: user.pod
+	(cd ..; echo "@GIT_HEAD_VERSION@") > version
+	$(POD2HTML) --infile=$< --outfile=$@ --title="Integer Set Library: Manual [version `cat version`]"
+endif
diff --git a/polly/lib/External/isl/doc/SubmittingPatches b/polly/lib/External/isl/doc/SubmittingPatches
new file mode 100644
index 00000000000..2c2b93881bc
--- /dev/null
+++ b/polly/lib/External/isl/doc/SubmittingPatches
@@ -0,0 +1,52 @@
+[Mostly copied from git's SubmittingPatches]
+
+	Commits:
+
+	- make commits of logical units
+	- check for unnecessary whitespace with "git diff --check"
+	  before committing
+	- do not check in commented out code or unneeded files
+	- the first line of the commit message should be a short
+	  description and should skip the full stop
+	- the body should provide a meaningful commit message, which
+	  includes motivation for the change, and contrasts
+	  its implementation with previous behaviour
+	- if you want your work included in isl.git, add a
+	  "Signed-off-by: Your Name <you@example.com>" line to the
+	  commit message (or just use the option "-s" when
+	  committing) to confirm that you agree to the Developer's
+	  Certificate of Origin
+	- make sure that you have tests for the bug you are fixing
+	- make sure that the test suite passes after your commit
+
+	Patch:
+
+	- use "git format-patch -M" to create the patch
+	- do not PGP sign your patch
+	- send a single patch per mail, e.g., using git-send-email(1)
+	- do not attach your patch, but read in the mail
+	  body, unless you cannot teach your mailer to
+	  leave the formatting of the patch alone.
+	- be careful doing cut & paste into your mailer, not to
+	  corrupt whitespaces.
+	- provide additional information (which is unsuitable for
+	  the commit message) between the "---" and the diffstat
+	- if you change, add, or remove a command line option or
+	  make some other user interface change, the associated
+	  documentation should be updated as well.
+	- if your name is not writable in ASCII, make sure that
+	  you send off a message in the correct encoding.
+	- send the patch to the development mailing list
+	  (isl-development@googlegroups.com).  If you use
+	  git-send-email(1), please test it first by sending email
+	  to yourself.
+
+	Revisions:
+
+	- add the revision number inside square brackets to
+	  the subject line (e.g., use --subject-prefix='PATCH v2'
+	  when creating the patch)
+	- recall the major issues discovered during the previous
+	  review and explain how you addressed them or why you
+	  disagree.  Do so either in a cover letter, between the
+	  "---" and the diffstat or in a separate message.
diff --git a/polly/lib/External/isl/doc/chicago.bst b/polly/lib/External/isl/doc/chicago.bst
new file mode 100644
index 00000000000..ba058338b97
--- /dev/null
+++ b/polly/lib/External/isl/doc/chicago.bst
@@ -0,0 +1,1726 @@
+%%% ====================================================================
+%%%  @BibTeX-style-file{
+%%%     author          = "Glenn Paulley",
+%%%     version         = "4",
+%%%     date            = "28 August 1992",
+%%%     time            = "10:23:39 199",
+%%%     filename        = "chicago.bst",
+%%%     address         = "Data Structuring Group
+%%%                        Department of Computer Science
+%%%                        University of Waterloo
+%%%                        Waterloo, Ontario, Canada
+%%%                        N2L 3G1",
+%%%     telephone       = "(519) 885-1211",
+%%%     FAX             = "(519) 885-1208",
+%%%     checksum        = "26323 1654 5143 37417",
+%%%     email           = "gnpaulle@bluebox.uwaterloo.ca",
+%%%     codetable       = "ISO/ASCII",
+%%%     keywords        = "",
+%%%     supported       = "yes",
+%%%     abstract        = "A BibTeX bibliography style that follows the
+%%%                        `B' reference style of the 13th Edition of
+%%%                        the Chicago Manual of Style. A detailed
+%%%                        feature list is given below.",
+%%%     docstring       = "The checksum field above contains a CRC-16
+%%%                        checksum as the first value, followed by the
+%%%                        equivalent of the standard UNIX wc (word
+%%%                        count) utility output of lines, words, and
+%%%                        characters.  This is produced by Robert
+%%%                        Solovay's checksum utility.",
+%%%  }
+%%% ====================================================================
+%
+% "Chicago" BibTeX style, chicago.bst
+% ===================================
+%
+% BibTeX `chicago' style file for BibTeX version 0.99c, LaTeX version 2.09
+% Place it in a file called chicago.bst in the BibTeX search path.
+% You need to include chicago.sty as a \documentstyle option.
+% (Placing it in the same directory as the LaTeX document should also work.)
+% This "chicago" style is based on newapa.bst (American Psych. Assoc.)
+% found at ymir.claremont.edu.
+%
+%   Citation format: (author-last-name year)
+%             (author-last-name and author-last-name year)
+%             (author-last-name, author-last-name, and author-last-name year)
+%             (author-last-name et al. year)
+%             (author-last-name)
+%             author-last-name (year)
+%             (author-last-name and author-last-name)
+%             (author-last-name et al.)
+%             (year) or (year,year)
+%             year or year,year
+%
+%   Reference list ordering: alphabetical by author or whatever passes
+%    for author in the absence of one.
+%
+% This BibTeX style has support for abbreviated author lists and for
+%    year-only citations.  This is done by having the citations
+%    actually look like
+%
+%    \citeauthoryear{full-author-info}{abbrev-author-info}{year}
+%
+% The LaTeX style has to have the following (or similar)
+%
+%     \let\@internalcite\cite
+%     \def\fullcite{\def\citeauthoryear##1##2##3{##1, ##3}\@internalcite}
+%     \def\fullciteA{\def\citeauthoryear##1##2##3{##1}\@internalcite}
+%     \def\shortcite{\def\citeauthoryear##1##2##3{##2, ##3}\@internalcite}
+%     \def\shortciteA{\def\citeauthoryear##1##2##3{##2}\@internalcite}
+%     \def\citeyear{\def\citeauthoryear##1##2##3{##3}\@internalcite}
+%
+% These TeX macro definitions are found in chicago.sty. Additional
+% commands to manipulate different components of a citation can be defined
+% so that, for example, you can list author's names without parentheses
+% if using a citation as a noun or object in a sentence.
+%
+% This file was originally copied from newapa.bst at ymir.claremont.edu.
+%
+% Features of chicago.bst:
+% =======================
+%
+% - full names used in citations, but abbreviated citations are available
+%   (see above)
+% - if an entry has a "month", then the month and year are also printed
+%   as part of that bibitem.
+% - all conjunctions use "and" instead of "\&"
+% - major modification from Chicago Manual of Style (13th ed.) is that
+%   only the first author in a reference appears last name first-
+%   additional authors appear as J. Q. Public.
+% - pages are listed as "pp. xx-xx" in all entry types except
+%   article entries.
+% - book, inbook, and manual use "location: publisher" (or organization)
+%   for address and publisher. All other types list publishers separately.
+% - "pp." are used to identify page numbers for all entry types except
+%   articles.
+% - organization is used as a citation label if neither author nor editor
+%   is present (for manuals).
+% - "et al." is used for long author and editor lists, or when "others"
+%   is used.
+%
+% Modifications and bug fixes from newapa.bst:
+% ===========================================
+%
+%   - added month, year to bib entries if month is present
+%   - fixed bug with In proceedings, added necessary comma after title
+%   - all conjunctions changed to "and" from "\&"
+%   - fixed bug with author labels in my.full.label: "et al." now is
+%        generated when "others" is an author name
+%   - major modification from Chicago Manual of Style (13th ed.) is that
+%     only the first author in a reference appears last name first-
+%     additional authors appear as J. Q. Public.
+%   - pages are listed as "pp. xx-xx" in all entry types except
+%     article entries. Unnecessary (IMHO) "()" around page numbers
+%     were removed, and page numbers now don't end with a period.
+%   - created chicago.sty for use with this bibstyle (required).
+%   - fixed bugs in FUNCTION {format.vol.num.pages} for missing volume,
+%     number, and /or pages. Renamed to format.jour.vol.
+%   - fixed bug in formatting booktitles: additional period an error if
+%     book has a volume.
+%   - fixed bug: editors usually given redundant period before next clause
+%     (format.editors.dot) removed.
+%   - added label support for organizations, if both author and editor
+%     are missing (from alpha.bst). If organization is too long, then
+%     the key field is used for abbreviated citations.
+%   - In proceedings or books of several volumes, no comma was written
+%     between the "Volume x" and the page numbers (this was intentional
+%     in newapa.bst). Fixed.
+%   - Some journals may not have volumes/numbers, only month/year (eg.
+%     IEEE Computer). Fixed bug in article style that assumed volume/number
+%     was always present.
+%
+% Original documentation for newapa.sty:
+% =====================================
+%
+% This version was made by modifying the master file made by
+% Oren Patashnik (PATASHNIK@SCORE.STANFORD.EDU), and the 'named' BibTeX
+% style of Peter F. Patel-Schneider.
+%
+% Copyright (C) 1985, all rights reserved.
+% Copying of this file is authorized only if either
+% (1) you make absolutely no changes to your copy, including name, or
+% (2) if you do make changes, you name it something other than 'newapa.bst'.
+% There are undoubtably bugs in this style.  If you make bug fixes,
+% improvements, etc.  please let me know.  My e-mail address is:
+%    spencer@cgrg.ohio.state.edu or 71160.3141@compuserve.com
+%
+% This style was made from 'plain.bst', 'named.bst', and 'apalike.bst',
+% with lots of tweaking to make it look like APA style, along with tips
+% from Young Ryu and Brian Reiser's modifications of 'apalike.bst'.
+
+ENTRY
+  { address
+    author
+    booktitle
+    chapter
+    edition
+    editor
+    fjournal
+    howpublished
+    institution
+    journal
+    key
+    month
+    note
+    number
+    organization
+    pages
+    publisher
+    school
+    series
+    title
+    type
+    volume
+    year
+  }
+  {}
+  { label.year extra.label sort.year sort.label }
+
+INTEGERS { output.state before.all mid.sentence after.sentence after.block }
+
+FUNCTION {init.state.consts}
+{ #0 'before.all :=
+  #1 'mid.sentence :=
+  #2 'after.sentence :=
+  #3 'after.block :=
+}
+
+STRINGS { s t u }
+
+FUNCTION {output.nonnull}
+{ 's :=
+  output.state mid.sentence =
+    { ", " * write$ }
+    { output.state after.block =
+    { add.period$ write$
+      newline$
+      "\newblock " write$
+    }
+    { output.state before.all =
+        'write$
+        { add.period$ " " * write$ }
+      if$
+    }
+      if$
+      mid.sentence 'output.state :=
+    }
+  if$
+  s
+}
+
+% Use a colon to separate output. Used only for address/publisher
+% combination in book/inbook types, address/institution for manuals,
+% and organization:publisher for proceedings (inproceedings).
+%
+FUNCTION {output.nonnull.colon}
+{ 's :=
+  output.state mid.sentence =
+    { ": " * write$ }
+    { output.state after.block =
+    { add.period$ write$
+      newline$
+      "\newblock " write$
+    }
+    { output.state before.all =
+        'write$
+        { add.period$ " " * write$ }
+      if$
+    }
+      if$
+      mid.sentence 'output.state :=
+    }
+  if$
+  s
+}
+
+FUNCTION {output}
+{ duplicate$ empty$
+    'pop$
+    'output.nonnull
+  if$
+}
+
+FUNCTION {output.colon}
+{ duplicate$ empty$
+    'pop$
+    'output.nonnull.colon
+  if$
+}
+
+FUNCTION {output.check}
+{ 't :=
+  duplicate$ empty$
+    { pop$ "empty " t * " in " * cite$ * warning$ }
+    'output.nonnull
+  if$
+}
+
+FUNCTION {output.check.colon}
+{ 't :=
+  duplicate$ empty$
+    { pop$ "empty " t * " in " * cite$ * warning$ }
+    'output.nonnull.colon
+  if$
+}
+
+FUNCTION {output.year.check}
+{ year empty$
+     { "empty year in " cite$ * warning$ }
+     { write$
+        " (" year * extra.label *
+       month empty$
+          { ")" * }
+          { ", " * month * ")" * }
+       if$
+       mid.sentence 'output.state :=
+     }
+  if$
+}
+
+
+FUNCTION {fin.entry}
+{ add.period$
+  write$
+  newline$
+}
+
+FUNCTION {new.block}
+{ output.state before.all =
+    'skip$
+    { after.block 'output.state := }
+  if$
+}
+
+FUNCTION {new.sentence}
+{ output.state after.block =
+    'skip$
+    { output.state before.all =
+    'skip$
+    { after.sentence 'output.state := }
+      if$
+    }
+  if$
+}
+
+FUNCTION {not}
+{   { #0 }
+    { #1 }
+  if$
+}
+
+FUNCTION {and}
+{   'skip$
+    { pop$ #0 }
+  if$
+}
+
+FUNCTION {or}
+{   { pop$ #1 }
+    'skip$
+  if$
+}
+
+FUNCTION {new.block.checka}
+{ empty$
+    'skip$
+    'new.block
+  if$
+}
+
+FUNCTION {new.block.checkb}
+{ empty$
+  swap$ empty$
+  and
+    'skip$
+    'new.block
+  if$
+}
+
+FUNCTION {new.sentence.checka}
+{ empty$
+    'skip$
+    'new.sentence
+  if$
+}
+
+FUNCTION {new.sentence.checkb}
+{ empty$
+  swap$ empty$
+  and
+    'skip$
+    'new.sentence
+  if$
+}
+
+FUNCTION {field.or.null}
+{ duplicate$ empty$
+    { pop$ "" }
+    'skip$
+  if$
+}
+
+%
+% Emphasize the top string on the stack.
+%
+FUNCTION {emphasize}
+{ duplicate$ empty$
+    { pop$ "" }
+    { "{\em " swap$ * "}" * }
+  if$
+}
+
+%
+% Emphasize the top string on the stack, but add a trailing space.
+%
+FUNCTION {emphasize.space}
+{ duplicate$ empty$
+    { pop$ "" }
+    { "{\em " swap$ * "\/}" * }
+  if$
+}
+
+INTEGERS { nameptr namesleft numnames }
+%
+% Format bibliographical entries with the first author last name first,
+% and subsequent authors with initials followed by last name.
+% All names are formatted in this routine.
+%
+FUNCTION {format.names}
+{ 's :=
+  #1 'nameptr :=               % nameptr = 1;
+  s num.names$ 'numnames :=    % numnames = num.name$(s);
+  numnames 'namesleft :=
+    { namesleft #0 > }
+
+    { nameptr #1 =
+        {s nameptr "{vv~}{ll}{, jj}{, f.}" format.name$ 't := }
+        {s nameptr "{f.~}{vv~}{ll}{, jj}" format.name$ 't := }
+      if$
+      nameptr #1 >
+        { namesleft #1 >
+              { ", " * t * }
+              { numnames #2 >
+                  { "," * }
+                  'skip$
+                if$
+                t "others" =
+                    { " et~al." * }
+                    { " and " * t * } % from Chicago Manual of Style
+                  if$
+               }
+               if$
+             }
+            't
+        if$
+        s nameptr "{vv~}{ll}{, jj}{, f.}" format.name$ 't := 
+	"\protect \index {" * t * "|hyperemph}" *
+        nameptr #1 + 'nameptr :=          % nameptr += 1;
+        namesleft #1 - 'namesleft :=      % namesleft =- 1;
+    }
+  while$
+}
+
+FUNCTION {my.full.label}
+{ 's :=
+  #1 'nameptr :=               % nameptr = 1;
+  s num.names$ 'numnames :=    % numnames = num.name$(s);
+  numnames 'namesleft :=
+    { namesleft #0 > }
+
+    { s nameptr "{vv~}{ll}" format.name$ 't :=  % get the next name
+      nameptr #1 >
+        { namesleft #1 >
+              { ", " * t * }
+              { numnames #2 >
+                  { "," * }
+                  'skip$
+                if$
+                t "others" =
+                    { " et~al." * }
+                    { " and " * t * } % from Chicago Manual of Style
+                  if$
+               }
+               if$
+             }
+            't
+        if$
+        s nameptr "{vv~}{ll}{, jj}{, f.}" format.name$ 't := 
+	"\protect \index {" * t * "|bold}" *
+        nameptr #1 + 'nameptr :=          % nameptr += 1;
+        namesleft #1 - 'namesleft :=      % namesleft =- 1;
+    }
+  while$
+
+}
+
+FUNCTION {format.names.fml}
+%
+% Format names in "familiar" format, with first initial followed by
+% last name. Like format.names, ALL names are formatted.
+%
+{ 's :=
+  #1 'nameptr :=               % nameptr = 1;
+  s num.names$ 'numnames :=    % numnames = num.name$(s);
+  numnames 'namesleft :=
+    { namesleft #0 > }
+
+    { s nameptr "{f.~}{vv~}{ll}{, jj}" format.name$ 't :=
+
+      nameptr #1 >
+        { namesleft #1 >
+              { ", " * t * }
+               { numnames #2 >
+                    { "," * }
+                    'skip$
+                  if$
+                  t "others" =
+                        { " et~al." * }
+                        { " and " * t * }
+%                       { " \& " * t * }
+                      if$
+                }
+               if$
+             }
+            't
+        if$
+        nameptr #1 + 'nameptr :=          % nameptr += 1;
+        namesleft #1 - 'namesleft :=      % namesleft =- 1;
+    }
+  while$
+}
+
+FUNCTION {format.authors}
+{ author empty$
+    { "" }
+    { author format.names }
+  if$
+}
+
+FUNCTION {format.key}
+{ empty$
+    { key field.or.null }
+    { "" }
+  if$
+}
+
+%
+% Format editor names for use in the "in" types: inbook, incollection,
+% inproceedings: first initial, then last names. When editors are the
+% LABEL for an entry, then format.editor is used which lists editors
+% by last name first.
+%
+FUNCTION {format.editors.fml}
+{ editor empty$
+    { "" }
+    { editor format.names.fml
+      editor num.names$ #1 >
+    { " (Eds.)" * }
+    { " (Ed.)" * }
+      if$
+    }
+  if$
+}
+
+%
+% Format editor names for use in labels, last names first.
+%
+FUNCTION {format.editors}
+{ editor empty$
+    { "" }
+    { editor format.names
+      editor num.names$ #1 >
+    { " (Eds.)" * }
+    { " (Ed.)" * }
+      if$
+    }
+  if$
+}
+
+FUNCTION {format.title}
+{ title empty$
+    { "" }
+    { title "t" change.case$ }
+  if$
+}
+
+% Note that the APA style requres case changes
+% in article titles. The following does not
+% change cases. If you perfer it, uncomment the
+% following and comment out the above.
+
+%FUNCTION {format.title}
+%{ title empty$
+%    { "" }
+%    { title }
+%  if$
+%}
+
+FUNCTION {n.dashify}
+{ 't :=
+  ""
+    { t empty$ not }
+    { t #1 #1 substring$ "-" =
+    { t #1 #2 substring$ "--" = not
+        { "--" *
+          t #2 global.max$ substring$ 't :=
+        }
+        {   { t #1 #1 substring$ "-" = }
+        { "-" *
+          t #2 global.max$ substring$ 't :=
+        }
+          while$
+        }
+      if$
+    }
+    { t #1 #1 substring$ *
+      t #2 global.max$ substring$ 't :=
+    }
+      if$
+    }
+  while$
+}
+
+FUNCTION {format.btitle}
+{ edition empty$
+  { title emphasize }
+  { title empty$
+    { title emphasize }
+    { volume empty$     % gnp - check for volume, then don't need period
+       { "{\em " title * "\/} (" * edition * " ed.)" * "." * }
+       { "{\em " title * "\/} (" * edition * " ed.)" * }
+      if$
+    }
+    if$
+  }
+  if$
+}
+
+FUNCTION {format.emphasize.booktitle}
+{ edition empty$
+  { booktitle emphasize }
+  { booktitle empty$
+    { booktitle emphasize }
+    { volume empty$    % gnp - extra period an error if book has a volume
+        { "{\em " booktitle * "\/} (" * edition * " ed.)" * "." *}
+        { "{\em " booktitle * "\/} (" * edition * " ed.)" * }
+      if$
+      }
+    if$
+    }
+  if$
+  }
+
+
+FUNCTION {tie.or.space.connect}
+{ duplicate$ text.length$ #3 <
+    { "~" }
+    { " " }
+  if$
+  swap$ * *
+}
+
+FUNCTION {either.or.check}
+{ empty$
+    'pop$
+    { "can't use both " swap$ * " fields in " * cite$ * warning$ }
+  if$
+}
+
+FUNCTION {format.bvolume}
+{ volume empty$
+    { "" }
+    { "Volume" volume tie.or.space.connect % gnp - changed to mixed case
+      series empty$
+        'skip$
+        { " of " * series emphasize * }
+      if$
+      "volume and number" number either.or.check
+    }
+  if$
+}
+
+FUNCTION {format.number.series}
+{ volume empty$
+    { number empty$
+    { series field.or.null }
+    { output.state mid.sentence =
+        { "Number" } % gnp - changed to mixed case always
+        { "Number" }
+      if$
+      number tie.or.space.connect
+      series empty$
+        { "there's a number but no series in " cite$ * warning$ }
+        { " in " * series * }
+      if$
+    }
+      if$
+    }
+    { "" }
+  if$
+}
+
+INTEGERS { multiresult }
+
+FUNCTION {multi.page.check}
+{ 't :=
+  #0 'multiresult :=
+    { multiresult not
+      t empty$ not
+      and
+    }
+    { t #1 #1 substring$
+      duplicate$ "-" =
+      swap$ duplicate$ "," =
+      swap$ "+" =
+      or or
+    { #1 'multiresult := }
+    { t #2 global.max$ substring$ 't := }
+      if$
+    }
+  while$
+  multiresult
+}
+
+FUNCTION {format.pages}
+{ pages empty$
+  { "" }
+  { pages multi.page.check
+	{ "pp.\ " pages n.dashify tie.or.space.connect } % gnp - removed ()
+	{ "pp.\ " pages tie.or.space.connect }
+    if$
+  }
+  if$
+}
+
+% By Young (and Spencer)
+% GNP - fixed bugs with missing volume, number, and/or pages
+%
+% Format journal, volume, number, pages for article types.
+%
+FUNCTION {format.jour.vol}
+{ fjournal empty$
+    { journal empty$
+	{ "no journal in " cite$ * warning$
+	  "" }
+	{ journal emphasize.space }
+	if$
+    } 
+    { fjournal emphasize.space }
+    if$
+  number empty$
+    { volume empty$
+       { "no number and no volume in " cite$ * warning$
+         "" * }
+       { "~{\em " * Volume * "}" * }
+      if$
+    }
+    { volume empty$
+      {"no volume for " cite$ * warning$
+       "~(" * number * ")" * }
+      { "~" *
+        volume emphasize.space
+        "(" * number * ")" * * }
+      if$
+    }
+  if$
+  pages empty$
+    {"page numbers missing in " cite$ * warning$
+     "" * } % gnp - place a null string on the stack for output
+    { duplicate$ empty$
+      { pop$ format.pages }
+      { ", " *  pages n.dashify * } % gnp - removed pp. for articles
+      if$
+    }
+  if$
+}
+
+FUNCTION {format.chapter.pages}
+{ chapter empty$
+    'format.pages
+    { type empty$
+        { "Chapter" } % gnp - changed to mixed case
+        { type "t" change.case$ }
+      if$
+      chapter tie.or.space.connect
+      pages empty$
+        {"page numbers missing in " cite$ * warning$} % gnp - added check
+        { ", " * format.pages * }
+      if$
+    }
+  if$
+}
+
+FUNCTION {format.in.ed.booktitle}
+{ booktitle empty$
+  { "" }
+  { editor empty$
+    { "In " format.emphasize.booktitle * }
+    { "In " format.editors.fml * ", " * format.emphasize.booktitle * }
+    if$
+  }
+  if$
+}
+
+FUNCTION {format.thesis.type}
+{ type empty$
+    'skip$
+    { pop$
+      type "t" change.case$
+    }
+  if$
+}
+
+FUNCTION {format.tr.number}
+{ type empty$
+    { "Technical Report" }
+    'type
+  if$
+  number empty$
+    { "t" change.case$ }
+    { number tie.or.space.connect }
+  if$
+}
+
+FUNCTION {format.article.crossref}
+{ "See"
+  "\citeN{" * crossref * "}" *
+}
+
+FUNCTION {format.crossref.editor}
+{ editor #1 "{vv~}{ll}" format.name$
+  editor num.names$ duplicate$
+  #2 >
+    { pop$ " et~al." * }
+    { #2 <
+    'skip$
+    { editor #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" =
+        { " et~al." * }
+        { " and " * editor #2 "{vv~}{ll}" format.name$ * }
+      if$
+    }
+      if$
+    }
+  if$
+}
+
+FUNCTION {format.book.crossref}
+{ volume empty$
+    { "empty volume in " cite$ * "'s crossref of " * crossref * warning$
+      "In "
+    }
+    { "Volume" volume tie.or.space.connect % gnp - changed to mixed case
+      " of " *
+    }
+  if$
+  editor empty$
+  editor field.or.null author field.or.null =
+  or
+    { key empty$
+    { series empty$
+        { "need editor, key, or series for " cite$ * " to crossref " *
+          crossref * warning$
+          "" *
+        }
+        { "{\em " * series * "\/}" * }
+      if$
+    }
+    { key * }
+      if$
+    }
+    { format.crossref.editor * }
+  if$
+  " \citeN{" * crossref * "}" *
+}
+
+FUNCTION {format.incoll.inproc.crossref}
+{ "See"
+  " \citeN{" * crossref * "}" *
+}
+
+% format.lab.names:
+%
+% determines "short" names for the abbreviated author information.
+% "Long" labels are created in calc.label, using the routine my.full.label
+% to format author and editor fields.
+%
+% There are 4 cases for labels.   (n=3 in the example)
+% a) one author             Foo
+% b) one to n               Foo, Bar and Baz
+% c) use of "and others"    Foo, Bar et al.
+% d) more than n            Foo et al.
+%
+FUNCTION {format.lab.names}
+{ 's :=
+  s num.names$ 'numnames :=
+  numnames #2 >    % change number to number of others allowed before
+		   % forcing "et al".
+    { s #1 "{vv~}{ll}" format.name$ 
+      "\protect \index {" * 
+      s #1 "{vv~}{ll}{, jj}{, f.}" format.name$ *
+      "}" *
+	"\protect\chicagoetal/" * }
+    {
+      numnames #1 - 'namesleft :=
+      #2 'nameptr :=
+      s #1 "{vv~}{ll}" format.name$
+      "\protect \index {" * 
+      s #1 "{vv~}{ll}{, jj}{, f.}" format.name$ *
+      "}" *
+	{ namesleft #0 > }
+	{ nameptr numnames =
+	    { s nameptr "{ff }{vv }{ll}{ jj}" format.name$ "others" =
+		{ "\protect\chicagoetal/" * }
+		{ "\protect\chicagoand/" * s nameptr "{vv~}{ll}" format.name$ * 
+		  "\protect \index {" * 
+		  s nameptr "{vv~}{ll}{, jj}{, f.}" format.name$ *
+		  "}" *
+                }
+	      if$
+	    }
+	    { ", " * s nameptr "{vv~}{ll}" format.name$ * }
+	  if$
+	  nameptr #1 + 'nameptr :=
+	  namesleft #1 - 'namesleft :=
+	}
+      while$
+    }
+  if$
+}
+
+FUNCTION {author.key.label}
+{ author empty$
+    { key empty$
+          { "no key, author in " cite$ * warning$
+            cite$ #1 #3 substring$ }
+         'key
+      if$
+    }
+    { author format.lab.names }
+  if$
+}
+
+FUNCTION {editor.key.label}
+{ editor empty$
+    { key empty$
+          { "no key, editor in " cite$ * warning$
+            cite$ #1 #3 substring$ }
+          'key
+        if$
+     }
+     { editor format.lab.names }
+  if$
+}
+
+FUNCTION {author.key.organization.label}
+%
+% added - gnp. Provide label formatting by organization if author is null.
+%
+{ author empty$
+    { organization empty$
+	{ key empty$
+	    { "no key, author or organization in " cite$ * warning$
+              cite$ #1 #3 substring$ }
+	    'key
+	  if$
+	}
+        { organization }
+      if$
+    }
+    { author format.lab.names }
+  if$
+}
+
+FUNCTION {editor.key.organization.label}
+%
+% added - gnp. Provide label formatting by organization if editor is null.
+%
+{ editor empty$
+    { organization empty$
+	{ key empty$
+	    { "no key, editor or organization in " cite$ * warning$
+              cite$ #1 #3 substring$ }
+	    'key
+	  if$
+	}
+        { organization }
+      if$
+    }
+    { editor format.lab.names }
+  if$
+}
+
+FUNCTION {author.editor.key.label}
+{ author empty$
+    { editor empty$
+          { key empty$
+               { "no key, author, or editor in " cite$ * warning$
+                 cite$ #1 #3 substring$ }
+             'key
+           if$
+         }
+          { editor format.lab.names }
+      if$
+    }
+    { author format.lab.names }
+  if$
+}
+
+FUNCTION {calc.label.orig}
+%
+% Changed - GNP. See also author.organization.sort, editor.organization.sort
+% Form label for BibTeX entry. The classification of which fields are used
+% for which type of entry (book, inbook, etc.) are taken from alpha.bst.
+% The change here from newapa is to also include organization as a
+% citation label if author or editor is missing.
+%
+{ type$ "book" =
+  type$ "inbook" =
+  or
+    'author.editor.key.label
+    { type$ "proceedings" =
+	'editor.key.organization.label
+	{ type$ "manual" =
+	    'author.key.organization.label
+	    'author.key.label
+	  if$
+	}
+      if$
+    }
+  if$
+
+  author empty$  % generate the full label citation information.
+    { editor empty$
+        { organization empty$
+           { "no author, editor, or organization in " cite$ * warning$
+             "??" }
+           { organization }
+           if$
+        }
+        { editor my.full.label }
+        if$
+    }
+    { author.key.label }
+  if$
+
+% leave label on the stack, to be popped when required.
+
+  "}{" * swap$ *
+%  year field.or.null purify$ #-1 #4 substring$ *
+%
+% save the year for sort processing afterwards (adding a, b, c, etc.)
+%
+  year field.or.null purify$ #-1 #4 substring$
+  'label.year :=
+}
+
+FUNCTION {calc.label}
+%
+% Changed - GNP. See also author.organization.sort, editor.organization.sort
+% Form label for BibTeX entry. The classification of which fields are used
+% for which type of entry (book, inbook, etc.) are taken from alpha.bst.
+% The change here from newapa is to also include organization as a
+% citation label if author or editor is missing.
+%
+{ type$ "book" =
+  type$ "inbook" =
+  or
+    'author.editor.key.label
+    { type$ "proceedings" =
+	'editor.key.organization.label
+	{ type$ "manual" =
+	    'author.key.organization.label
+	    'author.key.label
+	  if$
+	}
+      if$
+    }
+  if$
+
+  author empty$  % generate the full label citation information.
+    { editor empty$
+        { organization empty$
+           { "no author, editor, or organization in " cite$ * warning$
+             "??" }
+           { organization }
+           if$
+        }
+        { editor my.full.label }
+        if$
+    }
+    { author my.full.label }
+  if$
+
+% leave label on the stack, to be popped when required.
+
+  "}{" * swap$ * "}{" * title * "}{" *
+%  year field.or.null purify$ #-1 #4 substring$ *
+%
+% save the year for sort processing afterwards (adding a, b, c, etc.)
+%
+  year field.or.null purify$ #-1 #4 substring$
+  'label.year :=
+}
+
+FUNCTION {output.bibitem}
+{ newline$
+
+  "\bibitem[\protect\citeauthortitleyear{" write$
+  calc.label write$
+  sort.year write$
+  "}]{" write$
+
+  cite$ write$
+  "}" write$
+  newline$
+  ""
+  before.all 'output.state :=
+}
+
+FUNCTION {article}
+{ output.bibitem
+  format.authors
+  "author" output.check
+  author format.key output          % added
+  output.year.check                 % added
+  new.block
+  format.title
+  "title" output.check
+  new.block
+  crossref missing$
+    { format.jour.vol output
+    }
+    { format.article.crossref output.nonnull
+      format.pages output
+    }
+  if$
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {book}
+{ output.bibitem
+  author empty$
+    { format.editors
+	  "author and editor" output.check }
+    { format.authors
+	  output.nonnull
+      crossref missing$
+    	{ "author and editor" editor either.or.check }
+    	'skip$
+      if$
+    }
+  if$
+  output.year.check       % added
+  new.block
+  format.btitle
+  "title" output.check
+  crossref missing$
+    { format.bvolume output
+      new.block
+      format.number.series output
+      new.sentence
+      address output
+      publisher "publisher" output.check.colon
+    }
+    { new.block
+      format.book.crossref output.nonnull
+    }
+  if$
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {booklet}
+{ output.bibitem
+  format.authors output
+  author format.key output          % added
+  output.year.check                 % added
+  new.block
+  format.title
+  "title" output.check
+  new.block
+  howpublished output
+  address output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {inbook}
+{ output.bibitem
+  author empty$
+    { format.editors
+      "author and editor" output.check
+    }
+    { format.authors output.nonnull
+      crossref missing$
+    { "author and editor" editor either.or.check }
+    'skip$
+      if$
+    }
+  if$
+  output.year.check                 % added
+  new.block
+  format.btitle
+  "title" output.check
+  crossref missing$
+    { format.bvolume output
+      format.chapter.pages
+      "chapter and pages" output.check
+      new.block
+      format.number.series output
+      new.sentence
+      address output
+      publisher
+      "publisher" output.check.colon
+    }
+    { format.chapter.pages "chapter and pages" output.check
+      new.block
+      format.book.crossref output.nonnull
+    }
+  if$
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {incollection}
+{ output.bibitem
+  format.authors
+  "author" output.check
+  author format.key output       % added
+  output.year.check              % added
+  new.block
+  format.title
+  "title" output.check
+  new.block
+  crossref missing$
+  { format.in.ed.booktitle
+    "booktitle" output.check
+    format.bvolume output
+    format.number.series output
+    format.chapter.pages output % gnp - was special.output.nonnull
+%                                 left out comma before page numbers
+    new.sentence
+    address output
+    publisher "publisher" output.check.colon
+  }
+  { format.incoll.inproc.crossref
+	output.nonnull
+    format.chapter.pages output
+  }
+  if$
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {inproceedings}
+{ output.bibitem
+  format.authors
+  "author" output.check
+  author format.key output            % added
+  output.year.check                   % added
+  new.block
+  format.title
+  "title" output.check
+  new.block
+  crossref missing$
+    { format.in.ed.booktitle
+	  "booktitle" output.check
+      format.bvolume output
+      format.number.series output
+      address output
+      format.pages output
+      new.sentence
+      organization output
+      publisher output.colon
+      }
+    { format.incoll.inproc.crossref output.nonnull
+      format.pages output
+    }
+  if$
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {conference} { inproceedings }
+
+FUNCTION {manual}
+{ output.bibitem
+  author empty$
+    { editor empty$
+      { organization "organization" output.check
+        organization format.key output }  % if all else fails, use key
+      { format.editors "author and editor" output.check }
+      if$
+    }
+    { format.authors output.nonnull }
+    if$
+  output.year.check                 % added
+  new.block
+  format.btitle
+  "title" output.check
+  organization address new.block.checkb
+% Reversed the order of "address" and "organization", added the ":".
+  address output
+  organization "organization" output.check.colon
+%  address output
+%  ":" output
+%  organization output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {mastersthesis}
+{ output.bibitem
+  format.authors
+  "author" output.check
+  author format.key output          % added
+  output.year.check                 % added
+  new.block
+  format.title
+  "title" output.check
+  new.block
+  "Master's thesis" format.thesis.type output.nonnull
+  school "school" output.check
+  address output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {misc}
+{ output.bibitem
+  format.authors output
+  author format.key output            % added
+  output.year.check                   % added
+  title howpublished new.block.checkb
+  format.title output
+  new.block
+  howpublished output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {phdthesis}
+{ output.bibitem
+  format.authors
+  "author" output.check
+  author format.key output            % added
+  output.year.check                   % added
+  new.block
+  format.btitle
+  "title" output.check
+  new.block
+  "Ph.\ D. thesis" format.thesis.type output.nonnull
+  school "school" output.check
+  address output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {proceedings}
+{ output.bibitem
+  editor empty$
+    { organization output
+      organization format.key output }  % gnp - changed from author format.key
+    { format.editors output.nonnull }
+  if$
+% author format.key output             % gnp - removed (should be either
+%                                        editor or organization
+  output.year.check                    % added (newapa)
+  new.block
+  format.btitle
+  "title" output.check
+  format.bvolume output
+  format.number.series output
+  address output
+  new.sentence
+  organization output
+  publisher output.colon
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {techreport}
+{ output.bibitem
+  format.authors
+  "author" output.check
+  author format.key output             % added
+  output.year.check                    % added
+  new.block
+  format.title
+  "title" output.check
+  new.block
+  format.tr.number output.nonnull
+  institution
+  "institution" output.check
+  address output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {unpublished}
+{ output.bibitem
+  format.authors
+  "author" output.check
+  author format.key output              % added
+  output.year.check                      % added
+  new.block
+  format.title
+  "title" output.check
+  new.block
+  note "note" output.check
+  fin.entry
+}
+
+FUNCTION {default.type} { misc }
+
+MACRO {jan} {"January"}
+
+MACRO {feb} {"February"}
+
+MACRO {mar} {"March"}
+
+MACRO {apr} {"April"}
+
+MACRO {may} {"May"}
+
+MACRO {jun} {"June"}
+
+MACRO {jul} {"July"}
+
+MACRO {aug} {"August"}
+
+MACRO {sep} {"September"}
+
+MACRO {oct} {"October"}
+
+MACRO {nov} {"November"}
+
+MACRO {dec} {"December"}
+
+MACRO {acmcs} {"ACM Computing Surveys"}
+
+MACRO {acta} {"Acta Informatica"}
+
+MACRO {ai} {"Artificial Intelligence"}
+
+MACRO {cacm} {"Communications of the ACM"}
+
+MACRO {ibmjrd} {"IBM Journal of Research and Development"}
+
+MACRO {ibmsj} {"IBM Systems Journal"}
+
+MACRO {ieeese} {"IEEE Transactions on Software Engineering"}
+
+MACRO {ieeetc} {"IEEE Transactions on Computers"}
+
+MACRO {ieeetcad}
+ {"IEEE Transactions on Computer-Aided Design of Integrated Circuits"}
+
+MACRO {ipl} {"Information Processing Letters"}
+
+MACRO {jacm} {"Journal of the ACM"}
+
+MACRO {jcss} {"Journal of Computer and System Sciences"}
+
+MACRO {scp} {"Science of Computer Programming"}
+
+MACRO {sicomp} {"SIAM Journal on Computing"}
+
+MACRO {tocs} {"ACM Transactions on Computer Systems"}
+
+MACRO {tods} {"ACM Transactions on Database Systems"}
+
+MACRO {tog} {"ACM Transactions on Graphics"}
+
+MACRO {toms} {"ACM Transactions on Mathematical Software"}
+
+MACRO {toois} {"ACM Transactions on Office Information Systems"}
+
+MACRO {toplas} {"ACM Transactions on Programming Languages and Systems"}
+
+MACRO {tcs} {"Theoretical Computer Science"}
+
+READ
+
+FUNCTION {sortify}
+{ purify$
+  "l" change.case$
+}
+
+INTEGERS { len }
+
+FUNCTION {chop.word}
+{ 's :=
+  'len :=
+  s #1 len substring$ =
+    { s len #1 + global.max$ substring$ }
+    's
+  if$
+}
+
+
+
+FUNCTION {sort.format.names}
+{ 's :=
+  #1 'nameptr :=
+  ""
+  s num.names$ 'numnames :=
+  numnames 'namesleft :=
+    { namesleft #0 > }
+    { nameptr #2 =
+	{ year field.or.null purify$ #-1 #4 substring$ * }
+	'skip$
+      if$
+      nameptr #1 >
+          { "   " * }
+         'skip$
+      if$
+      s nameptr "{vv{ } }{ll{ }}{  f{ }}{  jj{ }}" format.name$ 't :=
+      nameptr numnames = t "others" = and
+          { " et~al" * }
+          { t sortify * }
+      if$
+      nameptr #1 + 'nameptr :=
+      namesleft #1 - 'namesleft :=
+    }
+  while$
+}
+
+FUNCTION {sort.format.title}
+{ 't :=
+  "A " #2
+    "An " #3
+      "The " #4 t chop.word
+    chop.word
+  chop.word
+  sortify
+  #1 global.max$ substring$
+}
+
+FUNCTION {author.sort}
+{ author empty$
+    { key empty$
+         { "to sort, need author or key in " cite$ * warning$
+           "" }
+         { key sortify }
+      if$
+    }
+    { author sort.format.names }
+  if$
+}
+
+FUNCTION {editor.sort}
+{ editor empty$
+    { key empty$
+         { "to sort, need editor or key in " cite$ * warning$
+           ""
+         }
+         { key sortify }
+      if$
+    }
+    { editor sort.format.names }
+  if$
+}
+
+FUNCTION {author.editor.sort}
+{ author empty$
+    { "missing author in " cite$ * warning$
+      editor empty$
+         { key empty$
+             { "to sort, need author, editor, or key in " cite$ * warning$
+               ""
+             }
+             { key sortify }
+           if$
+         }
+         { editor sort.format.names }
+      if$
+    }
+    { author sort.format.names }
+  if$
+}
+
+FUNCTION {author.organization.sort}
+%
+% added - GNP. Stack author or organization for sorting (from alpha.bst).
+% Unlike alpha.bst, we need entire names, not abbreviations
+%
+{ author empty$
+    { organization empty$
+	{ key empty$
+	    { "to sort, need author, organization, or key in " cite$ * warning$
+	      ""
+	    }
+	    { key sortify }
+	  if$
+	}
+	{ organization sortify }
+      if$
+    }
+    { author sort.format.names }
+  if$
+}
+
+FUNCTION {editor.organization.sort}
+%
+% added - GNP. Stack editor or organization for sorting (from alpha.bst).
+% Unlike alpha.bst, we need entire names, not abbreviations
+%
+{ editor empty$
+    { organization empty$
+	{ key empty$
+	    { "to sort, need editor, organization, or key in " cite$ * warning$
+	      ""
+	    }
+	    { key sortify }
+	  if$
+	}
+	{ organization sortify }
+      if$
+    }
+    { editor sort.format.names }
+  if$
+}
+
+FUNCTION {presort}
+%
+% Presort creates the bibentry's label via a call to calc.label, and then
+% sorts the entries based on entry type. Chicago.bst adds support for
+% including organizations as the sort key; the following is stolen from
+% alpha.bst.
+%
+{ %calc.label sortify % recalculate bibitem label
+  %year field.or.null purify$ #-1 #4 substring$ * % add year
+  %duplicate$ warning$
+  %"    "
+  %*
+  type$ "book" =
+  type$ "inbook" =
+  or
+    'author.editor.sort
+    { type$ "proceedings" =
+	'editor.organization.sort
+	{ type$ "manual" =
+	    'author.organization.sort
+	    'author.sort
+	  if$
+	}
+      if$
+    }
+  if$
+  #1 entry.max$ substring$        % added for newapa
+  'sort.label :=                  % added for newapa
+  sort.label                      % added for newapa
+  %*
+  "    "
+  *
+  title field.or.null
+  sort.format.title
+  *
+  #1 entry.max$ substring$
+  'sort.key$ :=
+}
+
+ITERATE {presort}
+
+SORT             % by label, year, author/editor, title
+
+STRINGS { last.label next.extra }
+
+INTEGERS { last.extra.num }
+
+FUNCTION {initialize.extra.label.stuff}
+{ #0 int.to.chr$ 'last.label :=
+  "" 'next.extra :=
+  #0 'last.extra.num :=
+}
+
+FUNCTION {forward.pass}
+%
+% Pass through all entries, comparing current entry to last one.
+% Need to concatenate year to the stack (done by calc.label) to determine
+% if two entries are the same (see presort)
+%
+{ last.label
+  calc.label.orig year field.or.null purify$ #-1 #4 substring$ * % add year
+  #1 entry.max$ substring$ =     % are they equal?
+     { last.extra.num #1 + 'last.extra.num :=
+       last.extra.num int.to.chr$ 'extra.label :=
+     }
+     { "a" chr.to.int$ 'last.extra.num :=
+       "" 'extra.label :=
+       calc.label.orig year field.or.null purify$ #-1 #4 substring$ * % add year
+       #1 entry.max$ substring$ 'last.label := % assign to last.label
+     }
+  if$
+}
+
+FUNCTION {reverse.pass}
+{ next.extra "b" =
+    { "a" 'extra.label := }
+     'skip$
+  if$
+  label.year extra.label * 'sort.year :=
+  extra.label 'next.extra :=
+}
+
+EXECUTE {initialize.extra.label.stuff}
+
+ITERATE {forward.pass}
+
+REVERSE {reverse.pass}
+
+FUNCTION {bib.sort.order}
+{ sort.label
+  "    "
+  *
+  year field.or.null sortify
+  *
+  "    "
+  *
+  title field.or.null
+  sort.format.title
+  *
+  #1 entry.max$ substring$
+  'sort.key$ :=
+}
+
+ITERATE {bib.sort.order}
+
+SORT             % by sort.label, year, title --- giving final bib. order.
+
+FUNCTION {begin.bib}
+
+{ preamble$ empty$
+    'skip$
+    { preamble$ write$ newline$ }
+  if$
+  "\begin{thebibliography}{}" write$ newline$
+}
+
+
+EXECUTE {begin.bib}
+
+EXECUTE {init.state.consts}
+
+ITERATE {call.type$}
+
+FUNCTION {end.bib}
+{ newline$
+  "\end{thebibliography}" write$ newline$
+}
+
+EXECUTE {end.bib}
+
diff --git a/polly/lib/External/isl/doc/chicago.sty b/polly/lib/External/isl/doc/chicago.sty
new file mode 100644
index 00000000000..33588dba923
--- /dev/null
+++ b/polly/lib/External/isl/doc/chicago.sty
@@ -0,0 +1,320 @@
+% -*- LaTeX -*-
+%%% ====================================================================
+%%%  @LaTeX-style-file{
+%%%     author          = "Glenn Paulley",
+%%%     version         = "4",
+%%%     date            = "31 August 1992",
+%%%     time            = "09:42:44 199",
+%%%     filename        = "chicago.sty",
+%%%     address         = "Data Structuring Group
+%%%                        Department of Computer Science
+%%%                        University of Waterloo
+%%%                        Waterloo, Ontario, Canada
+%%%                        N2L 3G1",
+%%%     telephone       = "(519) 885-1211",
+%%%     FAX             = "(519) 885-1208",
+%%%     checksum        = "44674 264 1050 10394",
+%%%     email           = "gnpaulle@bluebox.uwaterloo.ca",
+%%%     codetable       = "ISO/ASCII",
+%%%     keywords        = "",
+%%%     supported       = "yes",
+%%%     abstract        = "Contains the LaTeX style command definitions
+%%%                        for the Chicago BibTeX styles chicago.bst and
+%%%                        chicagoa.bst. For details, see below.",
+%%%     docstring       = "The checksum field above contains a CRC-16
+%%%                        checksum as the first value, followed by the
+%%%                        equivalent of the standard UNIX wc (word
+%%%                        count) utility output of lines, words, and
+%%%                        characters.  This is produced by Robert
+%%%                        Solovay's checksum utility.",
+%%%  }
+%%% ====================================================================
+%
+% chicago.sty: Style file for use with bibtex style chicago.bst, for
+% bibliographies formatted according to the 13th Edition of the Chicago
+% Manual of Style.
+%
+% 'newapa.bst' was made from 'plain.bst', 'named.bst', and 'apalike.bst',
+% with lots of tweaking to make it look like APA style, along with tips
+% from Young Ryu and Brian Reiser's modifications of 'apalike.bst'.
+% newapa.sty formed the basis of this style, chicago.sty. Author-date
+% references in newapa.bst formed the basis for chicago.bst. Chicagoa.bst
+% supports annotations.
+%
+% Version 4 (August, 1992):
+% - fixed chicago.bst and chicagoa.bst to handle long author lists in
+%   sorting
+% - fixed chicago.bst and chicagoa.bst so that missing page numbers in
+%   ``article'' entries are handled correctly
+% - modified chicago.sty to format entries with 2nd and subsequent lines
+%   indented.
+%
+%   Citation format: (author-last-name year)
+%             (author-last-name and author-last-name year)
+%             (author-last-name et al. year)
+%             (author-last-name)
+%             author-last-name
+%             author-last-name (year)
+%             (author-last-name and author-last-name)
+%             (author-last-name et al.)
+%             (year) or (year,year)
+%             year or year,year
+%
+%   Reference list ordering: alphabetical by author or whatever passes
+%    for author in the absence of one.
+%
+% This BibTeX style has support for abbreviated author lists and for
+%    year-only citations.  This is done by having the citations
+%    actually look like
+%
+%    \citeauthoryear{full-author-info}{abbrev-author-info}{year}
+%
+% The LaTeX style has to have the following (or similar)
+%
+%     \let\@internalcite\cite
+%     \def\fullcite{\def\citeauthoryear##1##2##3{##1, ##3}\@internalcite}
+%     \def\fullciteA{\def\citeauthoryear##1##2##3{##1}\@internalcite}
+%     \def\shortcite{\def\citeauthoryear##1##2##3{##2, ##3}\@internalcite}
+%     \def\shortciteA{\def\citeauthoryear##1##2##3{##2}\@internalcite}
+%     \def\citeyear{\def\citeauthoryear##1##2##3{##3}\@internalcite}
+%
+% -------------------------------------------------------------------------
+% This file implements citations for the ``chicago'' bibliography style.
+%  Place it in a file called chicago.sty in the TeX search path.
+%(Placing it in the same directory as the LaTeX document should also work.)
+%
+%    This file is a modification of the ``newapa'' LaTeX style,
+%    originally adapted by Steven Spencer from the ``apalike'' LaTeX style.
+%    It was originally modified by Stephen N. Spencer, with further
+%    modifications by Young U. Ryu.
+%
+% The ``chicago'' BibTeX bibliography style creates citations with labels:
+%       \citeauthoryear{author-info}{abbrev. author-info}{year}
+%
+% These labels are processed by the following LaTeX commands:
+%
+%  \cite{key}
+%    which produces citations with full author list and year.
+%    eg. (Brown 1978; Jarke, Turner, Stohl, et al. 1985)
+%  \citeNP{key}
+%    which produces citations with full author list and year, but without
+%    enclosing parentheses:
+%    eg. Brown 1978; Jarke, Turner and Stohl 1985
+%  \citeA{key}
+%    which produces citations with only the full author list.
+%    eg. (Brown; Jarke, Turner and Stohl)
+%  \citeANP{key}
+%    which produces citations with only the full author list, without
+%    parentheses eg. Brown; Jarke, Turner and Stohl
+%  \citeN{key}
+%    which produces citations with the full author list and year, but
+%    can be used as nouns in a sentence; no parentheses appear around
+%    the author names, but only around the year.
+%      eg. Shneiderman (1978) states that......
+%    \citeN should only be used for a single citation.
+%  \shortcite{key}
+%    which produces citations with abbreviated author list and year.
+%  \shortciteNP{key}
+%    which produces citations with abbreviated author list and year.
+%  \shortciteA{key}
+%    which produces only the abbreviated author list.
+%  \shortciteANP{key}
+%    which produces only the abbreviated author list.
+%  \shortciteN{key}
+%    which produces the abbreviated author list and year, with only the
+%    year in parentheses. Use with only one citation.
+%  \citeyear{key}
+%    which produces the year information only, within parentheses.
+%  \citeyearNP{key}
+%    which produces the year information only.
+%
+% Abbreviated author lists use the ``et al.'' construct.
+%
+% `NP' means `no parentheses'.
+%
+% This LaTeX style file must be used with the ``chicago'' or ``chicagoa''
+% (annotated chicago style) BibTeX styles.
+%
+\typeout{Using Chicago Manual of Style bibliography: 31 August 1992}
+%
+% -------------------------------------------------------------------------
+%
+% Citation macros.
+%
+\def\chicagoand/{ and }
+\def\chicagoetal/{ et~al.}
+%
+\let\@internalcite\cite
+%
+\def\cite{\def\@citeseppen{-1000}%
+    \def\@cite##1##2{(##1\if@tempswa , ##2\fi)}%
+    \def\citeauthortitleyear##1##2##3##4{##1\ ##4}\@internalcite}
+\def\citeNP{\def\@citeseppen{-1000}%
+    \def\@cite##1##2{##1\if@tempswa , ##2\fi}%
+    \def\citeauthortitleyear##1##2##3##4{##1\ ##4}\@internalcite}
+\def\citetitleN{\def\@citeseppen{-1000}%
+    \def\@cite##1##2{##1\if@tempswa , ##2)\else{)}\fi}%
+    \def\citeauthortitleyear##1##2##3##4{##3\ (##1; ##4}\@citedata}
+\def\citeN{\def\@citeseppen{-1000}%
+    \def\@cite##1##2{##1\if@tempswa , ##2)\else{)}\fi}%
+    \def\citeauthortitleyear##1##2##3##4{##1\ (##4}\@citedata}
+\def\citeA{\def\@citeseppen{-1000}%
+    \def\@cite##1##2{(##1\if@tempswa , ##2\fi)}%
+    \def\citeauthortitleyear##1##2##3##4{##1}\@internalcite}
+\def\citeANP{\def\@citeseppen{-1000}%
+    \def\@cite##1##2{##1\if@tempswa , ##2\fi}%
+    \def\citeauthortitleyear##1##2##3##4{##1}\@internalcite}
+%
+\def\shortcite{\def\@citeseppen{-1000}%
+    \def\@cite##1##2{(##1\if@tempswa , ##2\fi)}%
+    \def\citeauthortitleyear##1##2##3##4{##2\ ##4}\@internalcite}
+\def\shortciteNP{\def\@citeseppen{-1000}%
+    \def\@cite##1##2{##1\if@tempswa , ##2\fi}%
+    \def\citeauthortitleyear##1##2##3##4{##2\ ##4}\@internalcite}
+\def\shortciteN{\def\@citeseppen{-1000}%
+    \def\@cite##1##2{##1\if@tempswa , ##2)\else{)}\fi}%
+    \def\citeauthortitleyear##1##2##3##4{##2\ (##4}\@citedata}
+\def\shortciteA{\def\@citeseppen{-1000}%
+    \def\@cite##1##2{(##1\if@tempswa , ##2\fi)}%
+    \def\citeauthortitleyear##1##2##3##4{##2}\@internalcite}
+\def\shortciteANP{\def\@citeseppen{-1000}%
+    \def\@cite##1##2{##1\if@tempswa , ##2\fi}%
+    \def\citeauthortitleyear##1##2##3##4{##2}\@internalcite}
+%
+\def\citeyear{\def\@citeseppen{-1000}%
+    \def\@cite##1##2{(##1\if@tempswa , ##2\fi)}%
+    \def\citeauthortitleyear##1##2##3##4{##4}\@citedata}
+\def\citeyearNP{\def\@citeseppen{-1000}%
+    \def\@cite##1##2{##1\if@tempswa , ##2\fi}%
+    \def\citeauthortitleyear##1##2##3##4{##4}\@citedata}
+
+%
+% \@citedata and \@citedatax:
+%
+% Place commas in-between citations in the same \citeyear, \citeyearNP,
+% \citeN, or \shortciteN command.
+% Use something like \citeN{ref1,ref2,ref3} and \citeN{ref4} for a list.
+%
+\def\@citedata{%
+	\@ifnextchar [{\@tempswatrue\@citedatax}%
+				  {\@tempswafalse\@citedatax[]}%
+}
+
+\def\@citedatax[#1]#2{%
+\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi%
+  \def\@citea{}\@cite{\@for\@citeb:=#2\do%
+    {\@citea\def\@citea{), }\@ifundefined% by Young
+       {b@\@citeb}{{\bf ?}%
+       \@warning{Citation `\@citeb' on page \thepage \space undefined}}%
+{\csname b@\@citeb\endcsname}}}{#1}}%
+
+\@ifpackageloaded{hyperref}{%
+  \let\BRorg@citedatax\@citedatax 
+  \def\@citedatax[#1]#2{%
+    \BRorg@citedatax[#1]{#2}%
+    \Hy@backout{#2}%
+  }%
+}{}
+\@ifpackageloaded{hyperref}{%
+\def\hyperemph#1{{\em\hyperpage{#1}}}%
+\def\bold#1{{\bf\hyperpage{#1}}}%
+}{%
+\def\hyperemph#1{{\em #1}}%
+\def\bold#1{{\bf #1}}%
+}
+
+\def\BR@@lbibitem[#1]#2#3\par{%
+  \BRorg@bibitem[#1]{#2}#3\hfill\penalty100\hbox{}
+  \newblock
+  \backref\hfill[{\csname br@#2\endcsname}%
+  ]\parskip=-10pt\penalty-10000\hbox{}\nobreak\par
+}%
+\def\BR@@bibitem#1#2\par{%
+  \BRorg@bibitem{#1}#2
+  \newblock
+  \backref\penalty-100\hbox{}\nobreak\hfill[\hbox{\csname br@#2\endcsname}%
+  ]\par
+}
+\def\thepageorcolor{\thepage}
+\def\Hy@backout#1{%
+  \@bsphack
+  \ifx\@empty\@currentlabel
+    \protected@write\@auxout{}{%
+      \string\@writefile{brf}{%
+        \string\backcite{#1}{{\thepageorcolor}{(document)}{Doc-Start}}%
+      }%
+    }%
+  \else
+   \protected@write\@auxout{}{%
+     \string\@writefile{brf}{%
+       \string\backcite{#1}{{\thepageorcolor}{\@currentlabel}{\@currentHref}}%
+     }%
+   }%
+  \fi
+  \@esphack
+}
+
+% don't box citations, separate with ; and a space
+% also, make the penalty between citations negative: a good place to break.
+%
+\def\@citex[#1]#2{%
+\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi%
+  \def\@citea{}\@cite{\@for\@citeb:=#2\do%
+    {\@citea\def\@citea{; }\@ifundefined% by Young
+       {b@\@citeb}{{\bf ?}%
+       \@warning{Citation `\@citeb' on page \thepage \space undefined}}%
+{\csname b@\@citeb\endcsname}}}{#1}}%
+
+% (from apalike.sty)
+% No labels in the bibliography.
+%
+\def\@biblabel#1{}
+
+% (from apalike.sty)
+% Set length of hanging indentation for bibliography entries.
+%
+\newlength{\bibhang}
+\setlength{\bibhang}{2em}
+
+% Indent second and subsequent lines of bibliographic entries. Stolen
+% from openbib.sty: \newblock is set to {}.
+
+\newdimen\bibindent
+\bibindent=1.5em
+\@ifundefined{refname}%
+   {\@ifundefined{chapter}%
+     {\newcommand{\refname}{References}}%
+     {\newcommand{\refname}{Bibliography}}%
+   }%
+   {}%
+\@ifundefined{chapter}%
+ {\def\thebibliography#1{\section*{\refname\@mkboth
+   {\uppercase{\refname}}{\uppercase{\refname}}}
+   \addcontentsline{toc}{section}{References}
+   \list
+   {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}
+   \leftmargin\labelwidth
+   \advance\leftmargin\labelsep
+   \advance\leftmargin\bibindent
+   \itemindent -\bibindent
+   \listparindent \itemindent
+   \parsep \z@
+   \usecounter{enumi}}
+   \def\newblock{}
+   \sloppy
+   \sfcode`\.=1000\relax}}
+ {\def\thebibliography#1{\chapter*{\refname\@mkboth
+   {\refname}{\refname}}
+   \addcontentsline{toc}{chapter}{References}
+   \list
+   {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}
+   \leftmargin\labelwidth
+   \advance\leftmargin\labelsep
+   \advance\leftmargin\bibindent
+   \itemindent -\bibindent
+   \listparindent \itemindent
+   \parsep \z@
+   \usecounter{enumi}}
+   \def\newblock{}
+   \sloppy
+   \sfcode`\.=1000\relax}}
diff --git a/polly/lib/External/isl/doc/implementation.tex b/polly/lib/External/isl/doc/implementation.tex
new file mode 100644
index 00000000000..76d1acd5896
--- /dev/null
+++ b/polly/lib/External/isl/doc/implementation.tex
@@ -0,0 +1,2036 @@
+\section{Sets and Relations}
+
+\begin{definition}[Polyhedral Set]
+A {\em polyhedral set}\index{polyhedral set} $S$ is a finite union of basic sets
+$S = \bigcup_i S_i$, each of which can be represented using affine
+constraints
+$$
+S_i : \Z^n \to 2^{\Z^d} : \vec s \mapsto
+S_i(\vec s) =
+\{\, \vec x \in \Z^d \mid \exists \vec z \in \Z^e :
+A \vec x + B \vec s + D \vec z + \vec c \geq \vec 0 \,\}
+,
+$$
+with $A \in \Z^{m \times d}$,
+$B \in \Z^{m \times n}$,
+$D \in \Z^{m \times e}$
+and $\vec c \in \Z^m$.
+\end{definition}
+
+\begin{definition}[Parameter Domain of a Set]
+Let $S \in \Z^n \to 2^{\Z^d}$ be a set.
+The {\em parameter domain} of $S$ is the set
+$$\pdom S \coloneqq \{\, \vec s \in \Z^n \mid S(\vec s) \ne \emptyset \,\}.$$
+\end{definition}
+
+\begin{definition}[Polyhedral Relation]
+A {\em polyhedral relation}\index{polyhedral relation}
+$R$ is a finite union of basic relations
+$R = \bigcup_i R_i$ of type
+$\Z^n \to 2^{\Z^{d_1+d_2}}$,
+each of which can be represented using affine
+constraints
+$$
+R_i = \vec s \mapsto
+R_i(\vec s) =
+\{\, \vec x_1 \to \vec x_2 \in \Z^{d_1} \times \Z^{d_2}
+\mid \exists \vec z \in \Z^e :
+A_1 \vec x_1 + A_2 \vec x_2 + B \vec s + D \vec z + \vec c \geq \vec 0 \,\}
+,
+$$
+with $A_i \in \Z^{m \times d_i}$,
+$B \in \Z^{m \times n}$,
+$D \in \Z^{m \times e}$
+and $\vec c \in \Z^m$.
+\end{definition}
+
+\begin{definition}[Parameter Domain of a Relation]
+Let $R \in \Z^n \to 2^{\Z^{d+d}}$ be a relation.
+The {\em parameter domain} of $R$ is the set
+$$\pdom R \coloneqq \{\, \vec s \in \Z^n \mid R(\vec s) \ne \emptyset \,\}.$$
+\end{definition}
+
+\begin{definition}[Domain of a Relation]
+Let $R \in \Z^n \to 2^{\Z^{d+d}}$ be a relation.
+The {\em domain} of $R$ is the polyhedral set
+$$\domain R \coloneqq \vec s \mapsto
+\{\, \vec x_1 \in \Z^{d_1} \mid \exists \vec x_2 \in \Z^{d_2} :
+(\vec x_1, \vec x_2) \in R(\vec s) \,\}
+.
+$$
+\end{definition}
+
+\begin{definition}[Range of a Relation]
+Let $R \in \Z^n \to 2^{\Z^{d+d}}$ be a relation.
+The {\em range} of $R$ is the polyhedral set
+$$
+\range R \coloneqq \vec s \mapsto
+\{\, \vec x_2 \in \Z^{d_2} \mid \exists \vec x_1 \in \Z^{d_1} :
+(\vec x_1, \vec x_2) \in R(\vec s) \,\}
+.
+$$
+\end{definition}
+
+\begin{definition}[Composition of Relations]
+Let $R \in \Z^n \to 2^{\Z^{d_1+d_2}}$ and
+$S \in \Z^n \to 2^{\Z^{d_2+d_3}}$ be two relations,
+then the composition of
+$R$ and $S$ is defined as
+$$
+S \circ R \coloneqq
+\vec s \mapsto
+\{\, \vec x_1 \to \vec x_3 \in \Z^{d_1} \times \Z^{d_3}
+\mid \exists \vec x_2 \in \Z^{d_2} :
+\vec x_1 \to \vec x_2 \in R(\vec s) \wedge
+\vec x_2 \to \vec x_3 \in S(\vec s)
+\,\}
+.
+$$
+\end{definition}
+
+\begin{definition}[Difference Set of a Relation]
+Let $R \in \Z^n \to 2^{\Z^{d+d}}$ be a relation.
+The difference set ($\Delta \, R$) of $R$ is the set
+of differences between image elements and the corresponding
+domain elements,
+$$
+\diff R \coloneqq
+\vec s \mapsto
+\{\, \vec \delta \in \Z^{d} \mid \exists \vec x \to \vec y \in R :
+\vec \delta = \vec y - \vec x
+\,\}
+$$
+\end{definition}
+
+\section{Simple Hull}\label{s:simple hull}
+
+It is sometimes useful to have a single
+basic set or basic relation that contains a given set or relation.
+For rational sets, the obvious choice would be to compute the
+(rational) convex hull.  For integer sets, the obvious choice
+would be the integer hull.
+However, {\tt isl} currently does not support an integer hull operation
+and even if it did, it would be fairly expensive to compute.
+The convex hull operation is supported, but it is also fairly
+expensive to compute given only an implicit representation.
+
+Usually, it is not required to compute the exact integer hull,
+and an overapproximation of this hull is sufficient.
+The ``simple hull'' of a set is such an overapproximation
+and it is defined as the (inclusion-wise) smallest basic set
+that is described by constraints that are translates of
+the constraints in the input set.
+This means that the simple hull is relatively cheap to compute
+and that the number of constraints in the simple hull is no
+larger than the number of constraints in the input.
+\begin{definition}[Simple Hull of a Set]
+The {\em simple hull} of a set
+$S = \bigcup_{1 \le i \le v} S_i$, with
+$$
+S : \Z^n \to 2^{\Z^d} : \vec s \mapsto
+S(\vec s) =
+\left\{\, \vec x \in \Z^d \mid \exists \vec z \in \Z^e :
+\bigvee_{1 \le i \le v}
+A_i \vec x + B_i \vec s + D_i \vec z + \vec c_i \geq \vec 0 \,\right\}
+$$
+is the set
+$$
+H : \Z^n \to 2^{\Z^d} : \vec s \mapsto
+S(\vec s) =
+\left\{\, \vec x \in \Z^d \mid \exists \vec z \in \Z^e :
+\bigwedge_{1 \le i \le v}
+A_i \vec x + B_i \vec s + D_i \vec z + \vec c_i + \vec K_i \geq \vec 0
+\,\right\}
+,
+$$
+with $\vec K_i$ the (component-wise) smallest non-negative integer vectors
+such that $S \subseteq H$.
+\end{definition}
+The $\vec K_i$ can be obtained by solving a number of
+LP problems, one for each element of each $\vec K_i$.
+If any LP problem is unbounded, then the corresponding constraint
+is dropped.
+
+\section{Parametric Integer Programming}
+
+\subsection{Introduction}\label{s:intro}
+
+Parametric integer programming \shortcite{Feautrier88parametric}
+is used to solve many problems within the context of the polyhedral model.
+Here, we are mainly interested in dependence analysis \shortcite{Fea91}
+and in computing a unique representation for existentially quantified
+variables.  The latter operation has been used for counting elements
+in sets involving such variables
+\shortcite{BouletRe98,Verdoolaege2005experiences} and lies at the core
+of the internal representation of {\tt isl}.
+
+Parametric integer programming was first implemented in \texttt{PipLib}.
+An alternative method for parametric integer programming
+was later implemented in {\tt barvinok} \cite{barvinok-0.22}.
+This method is not based on Feautrier's algorithm, but on rational
+generating functions \cite{Woods2003short} and was inspired by the
+``digging'' technique of \shortciteN{DeLoera2004Three} for solving
+non-parametric integer programming problems.
+
+In the following sections, we briefly recall the dual simplex
+method combined with Gomory cuts and describe some extensions
+and optimizations.  The main algorithm is applied to a matrix
+data structure known as a tableau.  In case of parametric problems,
+there are two tableaus, one for the main problem and one for
+the constraints on the parameters, known as the context tableau.
+The handling of the context tableau is described in \autoref{s:context}.
+
+\subsection{The Dual Simplex Method}
+
+Tableaus can be represented in several slightly different ways.
+In {\tt isl}, the dual simplex method uses the same representation
+as that used by its incremental LP solver based on the \emph{primal}
+simplex method.  The implementation of this LP solver is based
+on that of {\tt Simplify} \shortcite{Detlefs2005simplify}, which, in turn,
+was derived from the work of \shortciteN{Nelson1980phd}.
+In the original \shortcite{Nelson1980phd}, the tableau was implemented
+as a sparse matrix, but neither {\tt Simplify} nor the current
+implementation of {\tt isl} does so.
+
+Given some affine constraints on the variables,
+$A \vec x + \vec b \ge \vec 0$, the tableau represents the relationship
+between the variables $\vec x$ and non-negative variables
+$\vec y = A \vec x + \vec b$ corresponding to the constraints.
+The initial tableau contains $\begin{pmatrix}
+\vec b & A
+\end{pmatrix}$ and expresses the constraints $\vec y$ in the rows in terms
+of the variables $\vec x$ in the columns.  The main operation defined
+on a tableau exchanges a column and a row variable and is called a pivot.
+During this process, some coefficients may become rational.
+As in the \texttt{PipLib} implementation,
+{\tt isl} maintains a shared denominator per row.
+The sample value of a tableau is one where each column variable is assigned
+zero and each row variable is assigned the constant term of the row.
+This sample value represents a valid solution if each constraint variable
+is assigned a non-negative value, i.e., if the constant terms of
+rows corresponding to constraints are all non-negative.
+
+The dual simplex method starts from an initial sample value that
+may be invalid, but that is known to be (lexicographically) no
+greater than any solution, and gradually increments this sample value
+through pivoting until a valid solution is obtained.
+In particular, each pivot exchanges a row variable
+$r = -n + \sum_i a_i \, c_i$ with negative
+sample value $-n$ with a column variable $c_j$
+such that $a_j > 0$.  Since $c_j = (n + r - \sum_{i\ne j} a_i \, c_i)/a_j$,
+the new row variable will have a positive sample value $n$.
+If no such column can be found, then the problem is infeasible.
+By always choosing the column that leads to the (lexicographically)
+smallest increment in the variables $\vec x$,
+the first solution found is guaranteed to be the (lexicographically)
+minimal solution \cite{Feautrier88parametric}.
+In order to be able to determine the smallest increment, the tableau
+is (implicitly) extended with extra rows defining the original
+variables in terms of the column variables.
+If we assume that all variables are non-negative, then we know
+that the zero vector is no greater than the minimal solution and
+then the initial extended tableau looks as follows.
+$$
+\begin{tikzpicture}
+\matrix (m) [matrix of math nodes]
+{
+& {} & 1 & \vec c \\
+\vec x && |(top)| \vec 0 & I \\
+\vec r && \vec b & |(bottom)|A \\
+};
+\begin{pgfonlayer}{background}
+\node (core) [inner sep=0pt,fill=black!20,right delimiter=),left delimiter=(,fit=(top)(bottom)] {};
+\end{pgfonlayer}
+\end{tikzpicture}
+$$
+Each column in this extended tableau is lexicographically positive
+and will remain so because of the column choice explained above.
+It is then clear that the value of $\vec x$ will increase in each step.
+Note that there is no need to store the extra rows explicitly.
+If a given $x_i$ is a column variable, then the corresponding row
+is the unit vector $e_i$.  If, on the other hand, it is a row variable,
+then the row already appears somewhere else in the tableau.
+
+In case of parametric problems, the sign of the constant term
+may depend on the parameters.  Each time the constant term of a constraint row
+changes, we therefore need to check whether the new term can attain
+negative and/or positive values over the current set of possible
+parameter values, i.e., the context.
+If all these terms can only attain non-negative values, the current
+state of the tableau represents a solution.  If one of the terms
+can only attain non-positive values and is not identically zero,
+the corresponding row can be pivoted.
+Otherwise, we pick one of the terms that can attain both positive
+and negative values and split the context into a part where
+it only attains non-negative values and a part where it only attains
+negative values.
+
+\subsection{Gomory Cuts}
+
+The solution found by the dual simplex method may have
+non-integral coordinates.  If so, some rational solutions
+(including the current sample value), can be cut off by
+applying a (parametric) Gomory cut.
+Let $r = b(\vec p) + \sp {\vec a} {\vec c}$ be the row
+corresponding to the first non-integral coordinate of $\vec x$,
+with $b(\vec p)$ the constant term, an affine expression in the
+parameters $\vec p$, i.e., $b(\vec p) = \sp {\vec f} {\vec p} + g$.
+Note that only row variables can attain
+non-integral values as the sample value of the column variables is zero.
+Consider the expression
+$b(\vec p) - \ceil{b(\vec p)} + \sp {\fract{\vec a}} {\vec c}$,
+with $\ceil\cdot$ the ceiling function and $\fract\cdot$ the
+fractional part.  This expression is negative at the sample value
+since $\vec c = \vec 0$ and $r = b(\vec p)$ is fractional, i.e.,
+$\ceil{b(\vec p)} > b(\vec p)$.  On the other hand, for each integral
+value of $r$ and $\vec c \ge 0$, the expression is non-negative
+because $b(\vec p) - \ceil{b(\vec p)} > -1$.
+Imposing this expression to be non-negative therefore does not
+invalidate any integral solutions, while it does cut away the current
+fractional sample value.  To be able to formulate this constraint,
+a new variable $q = \floor{-b(\vec p)} = - \ceil{b(\vec p)}$ is added
+to the context.  This integral variable is uniquely defined by the constraints
+$0 \le -d \, b(\vec p) - d \, q \le d - 1$, with $d$ the common
+denominator of $\vec f$ and $g$.  In practice, the variable
+$q' = \floor{\sp {\fract{-f}} {\vec p} + \fract{-g}}$ is used instead
+and the coefficients of the new constraint are adjusted accordingly.
+The sign of the constant term of this new constraint need not be determined
+as it is non-positive by construction.
+When several of these extra context variables are added, it is important
+to avoid adding duplicates.
+Recent versions of {\tt PipLib} also check for such duplicates.
+
+\subsection{Negative Unknowns and Maximization}
+
+There are two places in the above algorithm where the unknowns $\vec x$
+are assumed to be non-negative: the initial tableau starts from
+sample value $\vec x = \vec 0$ and $\vec c$ is assumed to be non-negative
+during the construction of Gomory cuts.
+To deal with negative unknowns, \shortciteN[Appendix A.2]{Fea91}
+proposed to use a ``big parameter'', say $M$, that is taken to be
+an arbitrarily large positive number.  Instead of looking for the
+lexicographically minimal value of $\vec x$, we search instead
+for the lexicographically minimal value of $\vec x' = \vec M + \vec x$.
+The sample value $\vec x' = \vec 0$ of the initial tableau then
+corresponds to $\vec x = -\vec M$, which is clearly not greater than
+any potential solution.  The sign of the constant term of a row
+is determined lexicographically, with the coefficient of $M$ considered
+first.  That is, if the coefficient of $M$ is not zero, then its sign
+is the sign of the entire term.  Otherwise, the sign is determined
+by the remaining affine expression in the parameters.
+If the original problem has a bounded optimum, then the final sample
+value will be of the form $\vec M + \vec v$ and the optimal value
+of the original problem is then $\vec v$.
+Maximization problems can be handled in a similar way by computing
+the minimum of $\vec M - \vec x$.
+
+When the optimum is unbounded, the optimal value computed for
+the original problem will involve the big parameter.
+In the original implementation of {\tt PipLib}, the big parameter could
+even appear in some of the extra variables $\vec q$ created during
+the application of a Gomory cut.  The final result could then contain
+implicit conditions on the big parameter through conditions on such
+$\vec q$ variables.  This problem was resolved in later versions
+of {\tt PipLib} by taking $M$ to be divisible by any positive number.
+The big parameter can then never appear in any $\vec q$ because
+$\fract {\alpha M } = 0$.  It should be noted, though, that an unbounded
+problem usually (but not always)
+indicates an incorrect formulation of the problem.
+
+The original version of {\tt PipLib} required the user to ``manually''
+add a big parameter, perform the reformulation and interpret the result
+\shortcite{Feautrier02}.  Recent versions allow the user to simply
+specify that the unknowns may be negative or that the maximum should
+be computed and then these transformations are performed internally.
+Although there are some application, e.g.,
+that of \shortciteN{Feautrier92multi},
+where it is useful to have explicit control over the big parameter,
+negative unknowns and maximization are by far the most common applications
+of the big parameter and we believe that the user should not be bothered
+with such implementation issues.
+The current version of {\tt isl} therefore does not
+provide any interface for specifying big parameters.  Instead, the user
+can specify whether a maximum needs to be computed and no assumptions
+are made on the sign of the unknowns.  Instead, the sign of the unknowns
+is checked internally and a big parameter is automatically introduced when
+needed.  For compatibility with {\tt PipLib}, the {\tt isl\_pip} tool
+does explicitly add non-negativity constraints on the unknowns unless
+the \verb+Urs_unknowns+ option is specified.
+Currently, there is also no way in {\tt isl} of expressing a big
+parameter in the output.  Even though
+{\tt isl} makes the same divisibility assumption on the big parameter
+as recent versions of {\tt PipLib}, it will therefore eventually
+produce an error if the problem turns out to be unbounded.
+
+\subsection{Preprocessing}
+
+In this section, we describe some transformations that are
+or can be applied in advance to reduce the running time
+of the actual dual simplex method with Gomory cuts.
+
+\subsubsection{Feasibility Check and Detection of Equalities}
+
+Experience with the original {\tt PipLib} has shown that Gomory cuts
+do not perform very well on problems that are (non-obviously) empty,
+i.e., problems with rational solutions, but no integer solutions.
+In {\tt isl}, we therefore first perform a feasibility check on
+the original problem considered as a non-parametric problem
+over the combined space of unknowns and parameters.
+In fact, we do not simply check the feasibility, but we also
+check for implicit equalities among the integer points by computing
+the integer affine hull.  The algorithm used is the same as that
+described in \autoref{s:GBR} below.
+Computing the affine hull is fairly expensive, but it can
+bring huge benefits if any equalities can be found or if the problem
+turns out to be empty.
+
+\subsubsection{Constraint Simplification}
+
+If the coefficients of the unknown and parameters in a constraint
+have a common factor, then this factor should be removed, possibly
+rounding down the constant term.  For example, the constraint
+$2 x - 5 \ge 0$ should be simplified to $x - 3 \ge 0$.
+{\tt isl} performs such simplifications on all sets and relations.
+Recent versions of {\tt PipLib} also perform this simplification
+on the input.
+
+\subsubsection{Exploiting Equalities}\label{s:equalities}
+
+If there are any (explicit) equalities in the input description,
+{\tt PipLib} converts each into a pair of inequalities.
+It is also possible to write $r$ equalities as $r+1$ inequalities
+\shortcite{Feautrier02}, but it is even better to \emph{exploit} the
+equalities to reduce the dimensionality of the problem.
+Given an equality involving at least one unknown, we pivot
+the row corresponding to the equality with the column corresponding
+to the last unknown with non-zero coefficient.  The new column variable
+can then be removed completely because it is identically zero,
+thereby reducing the dimensionality of the problem by one.
+The last unknown is chosen to ensure that the columns of the initial
+tableau remain lexicographically positive.  In particular, if
+the equality is of the form $b + \sum_{i \le j} a_i \, x_i = 0$ with
+$a_j \ne 0$, then the (implicit) top rows of the initial tableau
+are changed as follows
+$$
+\begin{tikzpicture}
+\matrix [matrix of math nodes]
+{
+ & {} & |(top)| 0 & I_1 & |(j)| &  \\
+j && 0 & & 1 & \\
+  && 0 & & & |(bottom)|I_2 \\
+};
+\node[overlay,above=2mm of j,anchor=south]{j};
+\begin{pgfonlayer}{background}
+\node (m) [inner sep=0pt,fill=black!20,right delimiter=),left delimiter=(,fit=(top)(bottom)] {};
+\end{pgfonlayer}
+\begin{scope}[xshift=4cm]
+\matrix [matrix of math nodes]
+{
+ & {} & |(top)| 0 & I_1 &  \\
+j && |(left)| -b/a_j & -a_i/a_j & \\
+  && 0 & & |(bottom)|I_2 \\
+};
+\begin{pgfonlayer}{background}
+\node (m2) [inner sep=0pt,fill=black!20,right delimiter=),left delimiter=(,fit=(top)(bottom)(left)] {};
+\end{pgfonlayer}
+\end{scope}
+ \draw [shorten >=7mm,-to,thick,decorate,
+        decoration={snake,amplitude=.4mm,segment length=2mm,
+                    pre=moveto,pre length=5mm,post length=8mm}]
+   (m) -- (m2);
+\end{tikzpicture}
+$$
+Currently, {\tt isl} also eliminates equalities involving only parameters
+in a similar way, provided at least one of the coefficients is equal to one.
+The application of parameter compression (see below)
+would obviate the need for removing parametric equalities.
+
+\subsubsection{Offline Symmetry Detection}\label{s:offline}
+
+Some problems, notably those of \shortciteN{Bygde2010licentiate},
+have a collection of constraints, say
+$b_i(\vec p) + \sp {\vec a} {\vec x} \ge 0$,
+that only differ in their (parametric) constant terms.
+These constant terms will be non-negative on different parts
+of the context and this context may have to be split for each
+of the constraints.  In the worst case, the basic algorithm may
+have to consider all possible orderings of the constant terms.
+Instead, {\tt isl} introduces a new parameter, say $u$, and
+replaces the collection of constraints by the single
+constraint $u + \sp {\vec a} {\vec x} \ge 0$ along with
+context constraints $u \le b_i(\vec p)$.
+Any solution to the new system is also a solution
+to the original system since
+$\sp {\vec a} {\vec x} \ge -u \ge -b_i(\vec p)$.
+Conversely, $m = \min_i b_i(\vec p)$ satisfies the constraints
+on $u$ and therefore extends a solution to the new system.
+It can also be plugged into a new solution.
+See \autoref{s:post} for how this substitution is currently performed
+in {\tt isl}.
+The method described in this section can only detect symmetries
+that are explicitly available in the input.
+See \autoref{s:online} for the detection
+and exploitation of symmetries that appear during the course of
+the dual simplex method.
+
+\subsubsection{Parameter Compression}\label{s:compression}
+
+It may in some cases be apparent from the equalities in the problem
+description that there can only be a solution for a sublattice
+of the parameters.  In such cases ``parameter compression''
+\shortcite{Meister2004PhD,Meister2008} can be used to replace
+the parameters by alternative ``dense'' parameters.
+For example, if there is a constraint $2x = n$, then the system
+will only have solutions for even values of $n$ and $n$ can be replaced
+by $2n'$.  Similarly, the parameters $n$ and $m$ in a system with
+the constraint $2n = 3m$ can be replaced by a single parameter $n'$
+with $n=3n'$ and $m=2n'$.
+It is also possible to perform a similar compression on the unknowns,
+but it would be more complicated as the compression would have to
+preserve the lexicographical order.  Moreover, due to our handling
+of equalities described above there should be
+no need for such variable compression.
+Although parameter compression has been implemented in {\tt isl},
+it is currently not yet used during parametric integer programming.
+
+\subsection{Postprocessing}\label{s:post}
+
+The output of {\tt PipLib} is a quast (quasi-affine selection tree).
+Each internal node in this tree corresponds to a split of the context
+based on a parametric constant term in the main tableau with indeterminate
+sign.  Each of these nodes may introduce extra variables in the context
+corresponding to integer divisions.  Each leaf of the tree prescribes
+the solution in that part of the context that satisfies all the conditions
+on the path leading to the leaf.
+Such a quast is a very economical way of representing the solution, but
+it would not be suitable as the (only) internal representation of
+sets and relations in {\tt isl}.  Instead, {\tt isl} represents
+the constraints of a set or relation in disjunctive normal form.
+The result of a parametric integer programming problem is then also
+converted to this internal representation.  Unfortunately, the conversion
+to disjunctive normal form can lead to an explosion of the size
+of the representation.
+In some cases, this overhead would have to be paid anyway in subsequent
+operations, but in other cases, especially for outside users that just
+want to solve parametric integer programming problems, we would like
+to avoid this overhead in future.  That is, we are planning on introducing
+quasts or a related representation as one of several possible internal
+representations and on allowing the output of {\tt isl\_pip} to optionally
+be printed as a quast.
+
+Currently, {\tt isl} also does not have an internal representation
+for expressions such as $\min_i b_i(\vec p)$ from the offline
+symmetry detection of \autoref{s:offline}.
+Assume that one of these expressions has $n$ bounds $b_i(\vec p)$.
+If the expression
+does not appear in the affine expression describing the solution,
+but only in the constraints, and if moreover, the expression
+only appears with a positive coefficient, i.e.,
+$\min_i b_i(\vec p) \ge f_j(\vec p)$, then each of these constraints
+can simply be reduplicated $n$ times, once for each of the bounds.
+Otherwise, a conversion to disjunctive normal form
+leads to $n$ cases, each described as $u = b_i(\vec p)$ with constraints
+$b_i(\vec p) \le b_j(\vec p)$ for $j > i$
+and
+$b_i(\vec p)  < b_j(\vec p)$ for $j < i$.
+Note that even though this conversion leads to a size increase
+by a factor of $n$, not detecting the symmetry could lead to
+an increase by a factor of $n!$ if all possible orderings end up being
+considered.
+
+\subsection{Context Tableau}\label{s:context}
+
+The main operation that a context tableau needs to provide is a test
+on the sign of an affine expression over the elements of the context.
+This sign can be determined by solving two integer linear feasibility
+problems, one with a constraint added to the context that enforces
+the expression to be non-negative and one where the expression is
+negative.  As already mentioned by \shortciteN{Feautrier88parametric},
+any integer linear feasibility solver could be used, but the {\tt PipLib}
+implementation uses a recursive call to the dual simplex with Gomory
+cuts algorithm to determine the feasibility of a context.
+In {\tt isl}, two ways of handling the context have been implemented,
+one that performs the recursive call and one, used by default, that
+uses generalized basis reduction.
+We start with some optimizations that are shared between the two
+implementations and then discuss additional details of each of them.
+
+\subsubsection{Maintaining Witnesses}\label{s:witness}
+
+A common feature of both integer linear feasibility solvers is that
+they will not only say whether a set is empty or not, but if the set
+is non-empty, they will also provide a \emph{witness} for this result,
+i.e., a point that belongs to the set.  By maintaining a list of such
+witnesses, we can avoid many feasibility tests during the determination
+of the signs of affine expressions.  In particular, if the expression
+evaluates to a positive number on some of these points and to a negative
+number on some others, then no feasibility test needs to be performed.
+If all the evaluations are non-negative, we only need to check for the
+possibility of a negative value and similarly in case of all
+non-positive evaluations.  Finally, in the rare case that all points
+evaluate to zero or at the start, when no points have been collected yet,
+one or two feasibility tests need to be performed depending on the result
+of the first test.
+
+When a new constraint is added to the context, the points that
+violate the constraint are temporarily removed.  They are reconsidered
+when we backtrack over the addition of the constraint, as they will
+satisfy the negation of the constraint.  It is only when we backtrack
+over the addition of the points that they are finally removed completely.
+When an extra integer division is added to the context,
+the new coordinates of the
+witnesses can easily be computed by evaluating the integer division.
+The idea of keeping track of witnesses was first used in {\tt barvinok}.
+
+\subsubsection{Choice of Constant Term on which to Split}
+
+Recall that if there are no rows with a non-positive constant term,
+but there are rows with an indeterminate sign, then the context
+needs to be split along the constant term of one of these rows.
+If there is more than one such row, then we need to choose which row
+to split on first.  {\tt PipLib} uses a heuristic based on the (absolute)
+sizes of the coefficients.  In particular, it takes the largest coefficient
+of each row and then selects the row where this largest coefficient is smaller
+than those of the other rows.
+
+In {\tt isl}, we take that row for which non-negativity of its constant
+term implies non-negativity of as many of the constant terms of the other
+rows as possible.  The intuition behind this heuristic is that on the
+positive side, we will have fewer negative and indeterminate signs,
+while on the negative side, we need to perform a pivot, which may
+affect any number of rows meaning that the effect on the signs
+is difficult to predict.  This heuristic is of course much more
+expensive to evaluate than the heuristic used by {\tt PipLib}.
+More extensive tests are needed to evaluate whether the heuristic is worthwhile.
+
+\subsubsection{Dual Simplex + Gomory Cuts}
+
+When a new constraint is added to the context, the first steps
+of the dual simplex method applied to this new context will be the same
+or at least very similar to those taken on the original context, i.e.,
+before the constraint was added.  In {\tt isl}, we therefore apply
+the dual simplex method incrementally on the context and backtrack
+to a previous state when a constraint is removed again.
+An initial implementation that was never made public would also
+keep the Gomory cuts, but the current implementation backtracks
+to before the point where Gomory cuts are added before adding
+an extra constraint to the context.
+Keeping the Gomory cuts has the advantage that the sample value
+is always an integer point and that this point may also satisfy
+the new constraint.  However, due to the technique of maintaining
+witnesses explained above,
+we would not perform a feasibility test in such cases and then
+the previously added cuts may be redundant, possibly resulting
+in an accumulation of a large number of cuts.
+
+If the parameters may be negative, then the same big parameter trick
+used in the main tableau is applied to the context.  This big parameter
+is of course unrelated to the big parameter from the main tableau.
+Note that it is not a requirement for this parameter to be ``big'',
+but it does allow for some code reuse in {\tt isl}.
+In {\tt PipLib}, the extra parameter is not ``big'', but this may be because
+the big parameter of the main tableau also appears
+in the context tableau.
+
+Finally, it was reported by \shortciteN{Galea2009personal}, who
+worked on a parametric integer programming implementation
+in {\tt PPL} \shortcite{PPL},
+that it is beneficial to add cuts for \emph{all} rational coordinates
+in the context tableau.  Based on this report,
+the initial {\tt isl} implementation was adapted accordingly.
+
+\subsubsection{Generalized Basis Reduction}\label{s:GBR}
+
+The default algorithm used in {\tt isl} for feasibility checking
+is generalized basis reduction \shortcite{Cook1991implementation}.
+This algorithm is also used in the {\tt barvinok} implementation.
+The algorithm is fairly robust, but it has some overhead.
+We therefore try to avoid calling the algorithm in easy cases.
+In particular, we incrementally keep track of points for which
+the entire unit hypercube positioned at that point lies in the context.
+This set is described by translates of the constraints of the context
+and if (rationally) non-empty, any rational point
+in the set can be rounded up to yield an integer point in the context.
+
+A restriction of the algorithm is that it only works on bounded sets.
+The affine hull of the recession cone therefore needs to be projected
+out first.  As soon as the algorithm is invoked, we then also
+incrementally keep track of this recession cone.  The reduced basis
+found by one call of the algorithm is also reused as initial basis
+for the next call.
+
+Some problems lead to the
+introduction of many integer divisions.  Within a given context,
+some of these integer divisions may be equal to each other, even
+if the expressions are not identical, or they may be equal to some
+affine combination of other variables.
+To detect such cases, we compute the affine hull of the context
+each time a new integer division is added.  The algorithm used
+for computing this affine hull is that of \shortciteN{Karr1976affine},
+while the points used in this algorithm are obtained by performing
+integer feasibility checks on that part of the context outside
+the current approximation of the affine hull.
+The list of witnesses is used to construct an initial approximation
+of the hull, while any extra points found during the construction
+of the hull is added to this list.
+Any equality found in this way that expresses an integer division
+as an \emph{integer} affine combination of other variables is
+propagated to the main tableau, where it is used to eliminate that
+integer division.
+
+\subsection{Experiments}
+
+\autoref{t:comparison} compares the execution times of {\tt isl}
+(with both types of context tableau)
+on some more difficult instances to those of other tools,
+run on an Intel Xeon W3520 @ 2.66GHz.
+Easier problems such as the
+test cases distributed with {\tt Pip\-Lib} can be solved so quickly
+that we would only be measuring overhead such as input/output and conversions
+and not the running time of the actual algorithm.
+We compare the following versions:
+{\tt piplib-1.4.0-5-g0132fd9},
+{\tt barvinok-0.32.1-73-gc5d7751},
+{\tt isl-0.05.1-82-g3a37260}
+and {\tt PPL} version 0.11.2.
+
+The first test case is the following dependence analysis problem
+originating from the Phideo project \shortcite{Verhaegh1995PhD}
+that was communicated to us by Bart Kienhuis:
+\begin{lstlisting}[flexiblecolumns=true,breaklines=true]{}
+lexmax { [j1,j2] -> [i1,i2,i3,i4,i5,i6,i7,i8,i9,i10] : 1 <= i1,j1 <= 8 and 1 <= i2,i3,i4,i5,i6,i7,i8,i9,i10 <= 2 and 1 <= j2 <= 128 and i1-1 = j1-1 and i2-1+2*i3-2+4*i4-4+8*i5-8+16*i6-16+32*i7-32+64*i8-64+128*i9-128+256*i10-256=3*j2-3+66 };
+\end{lstlisting}
+This problem was the main inspiration
+for some of the optimizations in \autoref{s:GBR}.
+The second group of test cases are projections used during counting.
+The first nine of these come from \shortciteN{Seghir2006minimizing}.
+The remaining two come from \shortciteN{Verdoolaege2005experiences} and
+were used to drive the first, Gomory cuts based, implementation
+in {\tt isl}.
+The third and final group of test cases are borrowed from
+\shortciteN{Bygde2010licentiate} and inspired the offline symmetry detection
+of \autoref{s:offline}.  Without symmetry detection, the running times
+are 11s and 5.9s.
+All running times of {\tt barvinok} and {\tt isl} include a conversion
+to disjunctive normal form.  Without this conversion, the final two
+cases can be solved in 0.07s and 0.21s.
+The {\tt PipLib} implementation has some fixed limits and will
+sometimes report the problem to be too complex (TC), while on some other
+problems it will run out of memory (OOM).
+The {\tt barvinok} implementation does not support problems
+with a non-trivial lineality space (line) nor maximization problems (max).
+The Gomory cuts based {\tt isl} implementation was terminated after 1000
+minutes on the first problem.  The gbr version introduces some
+overhead on some of the easier problems, but is overall the clear winner.
+
+\begin{table}
+\begin{center}
+\begin{tabular}{lrrrrr}
+    & {\tt PipLib} & {\tt barvinok} & {\tt isl} cut & {\tt isl} gbr & {\tt PPL} \\
+\hline
+\hline
+% bart.pip
+Phideo & TC    & 793m   & $>$999m &   2.7s  & 372m \\
+\hline
+e1 & 0.33s & 3.5s & 0.08s & 0.11s & 0.18s \\
+e3 & 0.14s & 0.13s & 0.10s & 0.10s & 0.17s \\
+e4 & 0.24s & 9.1s & 0.09s & 0.11s & 0.70s \\
+e5 & 0.12s & 6.0s & 0.06s & 0.14s & 0.17s \\
+e6 & 0.10s & 6.8s & 0.17s & 0.08s & 0.21s \\
+e7 & 0.03s & 0.27s & 0.04s & 0.04s & 0.03s \\
+e8 & 0.03s & 0.18s & 0.03s & 0.04s & 0.01s \\
+e9 & OOM & 70m & 2.6s & 0.94s & 22s \\
+vd & 0.04s & 0.10s & 0.03s & 0.03s & 0.03s \\
+bouleti & 0.25s & line & 0.06s & 0.06s & 0.15s \\
+difficult & OOM & 1.3s & 1.7s & 0.33s & 1.4s \\
+\hline
+cnt/sum & TC & max & 2.2s & 2.2s & OOM \\
+jcomplex & TC & max & 3.7s & 3.9s & OOM \\
+\end{tabular}
+\caption{Comparison of Execution Times}
+\label{t:comparison}
+\end{center}
+\end{table}
+
+\subsection{Online Symmetry Detection}\label{s:online}
+
+Manual experiments on small instances of the problems of
+\shortciteN{Bygde2010licentiate} and an analysis of the results
+by the approximate MPA method developed by \shortciteN{Bygde2010licentiate}
+have revealed that these problems contain many more symmetries
+than can be detected using the offline method of \autoref{s:offline}.
+In this section, we present an online detection mechanism that has
+not been implemented yet, but that has shown promising results
+in manual applications.
+
+Let us first consider what happens when we do not perform offline
+symmetry detection.  At some point, one of the
+$b_i(\vec p) + \sp {\vec a} {\vec x} \ge 0$ constraints,
+say the $j$th constraint, appears as a column
+variable, say $c_1$, while the other constraints are represented
+as rows of the form $b_i(\vec p) - b_j(\vec p) + c$.
+The context is then split according to the relative order of
+$b_j(\vec p)$ and one of the remaining $b_i(\vec p)$.
+The offline method avoids this split by replacing all $b_i(\vec p)$
+by a single newly introduced parameter that represents the minimum
+of these $b_i(\vec p)$.
+In the online method the split is similarly avoided by the introduction
+of a new parameter.  In particular, a new parameter is introduced
+that represents
+$\left| b_j(\vec p) - b_i(\vec p) \right|_+ =
+\max(b_j(\vec p) - b_i(\vec p), 0)$.
+
+In general, let $r = b(\vec p) + \sp {\vec a} {\vec c}$ be a row
+of the tableau such that the sign of $b(\vec p)$ is indeterminate
+and such that exactly one of the elements of $\vec a$ is a $1$,
+while all remaining elements are non-positive.
+That is, $r = b(\vec p) + c_j - f$ with $f = -\sum_{i\ne j} a_i c_i \ge 0$.
+We introduce a new parameter $t$ with
+context constraints $t \ge -b(\vec p)$ and $t \ge 0$ and replace
+the column variable $c_j$ by $c' + t$.  The row $r$ is now equal
+to $b(\vec p) + t + c' - f$.  The constant term of this row is always
+non-negative because any negative value of $b(\vec p)$ is compensated
+by $t \ge -b(\vec p)$ while and non-negative value remains non-negative
+because $t \ge 0$.
+
+We need to show that this transformation does not eliminate any valid
+solutions and that it does not introduce any spurious solutions.
+Given a valid solution for the original problem, we need to find
+a non-negative value of $c'$ satisfying the constraints.
+If $b(\vec p) \ge 0$, we can take $t = 0$ so that
+$c' = c_j - t = c_j \ge 0$.
+If $b(\vec p) < 0$, we can take $t = -b(\vec p)$.
+Since $r = b(\vec p) + c_j - f \ge 0$ and $f \ge 0$, we have 
+$c' = c_j + b(\vec p) \ge 0$.
+Note that these choices amount to plugging in
+$t = \left|-b(\vec p)\right|_+ = \max(-b(\vec p), 0)$.
+Conversely, given a solution to the new problem, we need to find
+a non-negative value of $c_j$, but this is easy since $c_j = c' + t$
+and both of these are non-negative.
+
+Plugging in $t = \max(-b(\vec p), 0)$ can be performed as in
+\autoref{s:post}, but, as in the case of offline symmetry detection,
+it may be better to provide a direct representation for such
+expressions in the internal representation of sets and relations
+or at least in a quast-like output format.
+
+\section{Coalescing}\label{s:coalescing}
+
+See \shortciteN{Verdoolaege2009isl}, for now.
+More details will be added later.
+
+\section{Transitive Closure}
+
+\subsection{Introduction}
+
+\begin{definition}[Power of a Relation]
+Let $R \in \Z^n \to 2^{\Z^{d+d}}$ be a relation and
+$k \in \Z_{\ge 1}$
+a positive number, then power $k$ of relation $R$ is defined as
+\begin{equation}
+\label{eq:transitive:power}
+R^k \coloneqq
+\begin{cases}
+R & \text{if $k = 1$}
+\\
+R \circ R^{k-1} & \text{if $k \ge 2$}
+.
+\end{cases}
+\end{equation}
+\end{definition}
+
+\begin{definition}[Transitive Closure of a Relation]
+Let $R \in \Z^n \to 2^{\Z^{d+d}}$ be a relation,
+then the transitive closure $R^+$ of $R$ is the union
+of all positive powers of $R$,
+$$
+R^+ \coloneqq \bigcup_{k \ge 1} R^k
+.
+$$
+\end{definition}
+Alternatively, the transitive closure may be defined
+inductively as
+\begin{equation}
+\label{eq:transitive:inductive}
+R^+ \coloneqq R \cup \left(R \circ R^+\right)
+.
+\end{equation}
+
+Since the transitive closure of a polyhedral relation
+may no longer be a polyhedral relation \shortcite{Kelly1996closure},
+we can, in the general case, only compute an approximation
+of the transitive closure.
+Whereas \shortciteN{Kelly1996closure} compute underapproximations,
+we, like \shortciteN{Beletska2009}, compute overapproximations.
+That is, given a relation $R$, we will compute a relation $T$
+such that $R^+ \subseteq T$.  Of course, we want this approximation
+to be as close as possible to the actual transitive closure
+$R^+$ and we want to detect the cases where the approximation is
+exact, i.e., where $T = R^+$.
+
+For computing an approximation of the transitive closure of $R$,
+we follow the same general strategy as \shortciteN{Beletska2009}
+and first compute an approximation of $R^k$ for $k \ge 1$ and then project
+out the parameter $k$ from the resulting relation.
+
+\begin{example}
+As a trivial example, consider the relation
+$R = \{\, x \to x + 1 \,\}$.  The $k$th power of this map
+for arbitrary $k$ is
+$$
+R^k = k \mapsto \{\, x \to x + k \mid k \ge 1 \,\}
+.
+$$
+The transitive closure is then
+$$
+\begin{aligned}
+R^+ & = \{\, x \to y \mid \exists k \in \Z_{\ge 1} : y = x + k \,\}
+\\
+& = \{\, x \to y \mid y \ge x + 1 \,\}
+.
+\end{aligned}
+$$
+\end{example}
+
+\subsection{Computing an Approximation of $R^k$}
+\label{s:power}
+
+There are some special cases where the computation of $R^k$ is very easy.
+One such case is that where $R$ does not compose with itself,
+i.e., $R \circ R = \emptyset$ or $\domain R \cap \range R = \emptyset$.
+In this case, $R^k$ is only non-empty for $k=1$ where it is equal
+to $R$ itself.
+
+In general, it is impossible to construct a closed form
+of $R^k$ as a polyhedral relation.
+We will therefore need to make some approximations.
+As a first approximations, we will consider each of the basic
+relations in $R$ as simply adding one or more offsets to a domain element
+to arrive at an image element and ignore the fact that some of these
+offsets may only be applied to some of the domain elements.
+That is, we will only consider the difference set $\Delta\,R$ of the relation.
+In particular, we will first construct a collection $P$ of paths
+that move through
+a total of $k$ offsets and then intersect domain and range of this
+collection with those of $R$.
+That is, 
+\begin{equation}
+\label{eq:transitive:approx}
+K = P \cap \left(\domain R \to \range R\right)
+,
+\end{equation}
+with
+\begin{equation}
+\label{eq:transitive:path}
+P = \vec s \mapsto \{\, \vec x \to \vec y \mid
+\exists k_i \in \Z_{\ge 0}, \vec\delta_i \in k_i \, \Delta_i(\vec s) :
+\vec y = \vec x + \sum_i \vec\delta_i
+\wedge
+\sum_i k_i = k > 0
+\,\}
+\end{equation}
+and with $\Delta_i$ the basic sets that compose
+the difference set $\Delta\,R$.
+Note that the number of basic sets $\Delta_i$ need not be
+the same as the number of basic relations in $R$.
+Also note that since addition is commutative, it does not
+matter in which order we add the offsets and so we are allowed
+to group them as we did in \eqref{eq:transitive:path}.
+
+If all the $\Delta_i$s are singleton sets
+$\Delta_i = \{\, \vec \delta_i \,\}$ with $\vec \delta_i \in \Z^d$,
+then \eqref{eq:transitive:path} simplifies to
+\begin{equation}
+\label{eq:transitive:singleton}
+P = \{\, \vec x \to \vec y \mid
+\exists k_i \in \Z_{\ge 0} :
+\vec y = \vec x + \sum_i k_i \, \vec \delta_i
+\wedge
+\sum_i k_i = k > 0
+\,\}
+\end{equation}
+and then the approximation computed in \eqref{eq:transitive:approx}
+is essentially the same as that of \shortciteN{Beletska2009}.
+If some of the $\Delta_i$s are not singleton sets or if
+some of $\vec \delta_i$s are parametric, then we need
+to resort to further approximations.
+
+To ease both the exposition and the implementation, we will for
+the remainder of this section work with extended offsets
+$\Delta_i' = \Delta_i \times \{\, 1 \,\}$.
+That is, each offset is extended with an extra coordinate that is
+set equal to one.  The paths constructed by summing such extended
+offsets have the length encoded as the difference of their
+final coordinates.  The path $P'$ can then be decomposed into
+paths $P_i'$, one for each $\Delta_i$,
+\begin{equation}
+\label{eq:transitive:decompose}
+P' = \left(
+(P_m' \cup \identity) \circ \cdots \circ
+(P_2' \cup \identity) \circ
+(P_1' \cup \identity)
+\right) \cap
+\{\,
+\vec x' \to \vec y' \mid y_{d+1} - x_{d+1} = k > 0
+\,\}
+,
+\end{equation}
+with
+$$
+P_i' = \vec s \mapsto \{\, \vec x' \to \vec y' \mid
+\exists k \in \Z_{\ge 1}, \vec \delta \in k \, \Delta_i'(\vec s) :
+\vec y' = \vec x' + \vec \delta
+\,\}
+.
+$$
+Note that each $P_i'$ contains paths of length at least one.
+We therefore need to take the union with the identity relation
+when composing the $P_i'$s to allow for paths that do not contain
+any offsets from one or more $\Delta_i'$.
+The path that consists of only identity relations is removed
+by imposing the constraint $y_{d+1} - x_{d+1} > 0$.
+Taking the union with the identity relation means that
+that the relations we compose in \eqref{eq:transitive:decompose}
+each consist of two basic relations.  If there are $m$
+disjuncts in the input relation, then a direct application
+of the composition operation may therefore result in a relation
+with $2^m$ disjuncts, which is prohibitively expensive.
+It is therefore crucial to apply coalescing (\autoref{s:coalescing})
+after each composition.
+
+Let us now consider how to compute an overapproximation of $P_i'$.
+Those that correspond to singleton $\Delta_i$s are grouped together
+and handled as in \eqref{eq:transitive:singleton}.
+Note that this is just an optimization.  The procedure described
+below would produce results that are at least as accurate.
+For simplicity, we first assume that no constraint in $\Delta_i'$
+involves any existentially quantified variables.
+We will return to existentially quantified variables at the end
+of this section.
+Without existentially quantified variables, we can classify
+the constraints of $\Delta_i'$ as follows
+\begin{enumerate}
+\item non-parametric constraints
+\begin{equation}
+\label{eq:transitive:non-parametric}
+A_1 \vec x + \vec c_1 \geq \vec 0
+\end{equation}
+\item purely parametric constraints
+\begin{equation}
+\label{eq:transitive:parametric}
+B_2 \vec s + \vec c_2 \geq \vec 0
+\end{equation}
+\item negative mixed constraints
+\begin{equation}
+\label{eq:transitive:mixed}
+A_3 \vec x + B_3 \vec s + \vec c_3 \geq \vec 0
+\end{equation}
+such that for each row $j$ and for all $\vec s$,
+$$
+\Delta_i'(\vec s) \cap
+\{\, \vec \delta' \mid B_{3,j} \vec s + c_{3,j} > 0 \,\}
+= \emptyset
+$$
+\item positive mixed constraints
+$$
+A_4 \vec x + B_4 \vec s + \vec c_4 \geq \vec 0
+$$
+such that for each row $j$, there is at least one $\vec s$ such that
+$$
+\Delta_i'(\vec s) \cap
+\{\, \vec \delta' \mid B_{4,j} \vec s + c_{4,j} > 0 \,\}
+\ne \emptyset
+$$
+\end{enumerate}
+We will use the following approximation $Q_i$ for $P_i'$:
+\begin{equation}
+\label{eq:transitive:Q}
+\begin{aligned}
+Q_i = \vec s \mapsto
+\{\,
+\vec x' \to \vec y'
+\mid {} & \exists k \in \Z_{\ge 1}, \vec f \in \Z^d :
+\vec y' = \vec x' + (\vec f, k)
+\wedge {}
+\\
+&
+A_1 \vec f + k \vec c_1 \geq \vec 0
+\wedge
+B_2 \vec s + \vec c_2 \geq \vec 0
+\wedge
+A_3 \vec f + B_3 \vec s + \vec c_3 \geq \vec 0
+\,\}
+.
+\end{aligned}
+\end{equation}
+To prove that $Q_i$ is indeed an overapproximation of $P_i'$,
+we need to show that for every $\vec s \in \Z^n$, for every
+$k \in \Z_{\ge 1}$ and for every $\vec f \in k \, \Delta_i(\vec s)$
+we have that
+$(\vec f, k)$ satisfies the constraints in \eqref{eq:transitive:Q}.
+If $\Delta_i(\vec s)$ is non-empty, then $\vec s$ must satisfy
+the constraints in \eqref{eq:transitive:parametric}.
+Each element $(\vec f, k) \in k \, \Delta_i'(\vec s)$ is a sum
+of $k$ elements $(\vec f_j, 1)$ in $\Delta_i'(\vec s)$.
+Each of these elements satisfies the constraints in
+\eqref{eq:transitive:non-parametric}, i.e.,
+$$
+\left[
+\begin{matrix}
+A_1 & \vec c_1
+\end{matrix}
+\right]
+\left[
+\begin{matrix}
+\vec f_j \\ 1
+\end{matrix}
+\right]
+\ge \vec 0
+.
+$$
+The sum of these elements therefore satisfies the same set of inequalities,
+i.e., $A_1 \vec f + k \vec c_1 \geq \vec 0$.
+Finally, the constraints in \eqref{eq:transitive:mixed} are such
+that for any $\vec s$ in the parameter domain of $\Delta$,
+we have $-\vec r(\vec s) \coloneqq B_3 \vec s + \vec c_3 \le \vec 0$,
+i.e., $A_3 \vec f_j \ge \vec r(\vec s) \ge \vec 0$
+and therefore also $A_3 \vec f \ge \vec r(\vec s)$.
+Note that if there are no mixed constraints and if the
+rational relaxation of $\Delta_i(\vec s)$, i.e.,
+$\{\, \vec x \in \Q^d \mid A_1 \vec x + \vec c_1 \ge \vec 0\,\}$,
+has integer vertices, then the approximation is exact, i.e.,
+$Q_i = P_i'$.  In this case, the vertices of $\Delta'_i(\vec s)$
+generate the rational cone
+$\{\, \vec x' \in \Q^{d+1} \mid \left[
+\begin{matrix}
+A_1 & \vec c_1
+\end{matrix}
+\right] \vec x' \,\}$ and therefore $\Delta'_i(\vec s)$ is
+a Hilbert basis of this cone \shortcite[Theorem~16.4]{Schrijver1986}.
+
+Note however that, as pointed out by \shortciteN{DeSmet2010personal},
+if there \emph{are} any mixed constraints, then the above procedure may
+not compute the most accurate affine approximation of
+$k \, \Delta_i(\vec s)$ with $k \ge 1$.
+In particular, we only consider the negative mixed constraints that
+happen to appear in the description of $\Delta_i(\vec s)$, while we
+should instead consider \emph{all} valid such constraints.
+It is also sufficient to consider those constraints because any
+constraint that is valid for $k \, \Delta_i(\vec s)$ is also
+valid for $1 \, \Delta_i(\vec s) = \Delta_i(\vec s)$.
+Take therefore any constraint
+$\spv a x + \spv b s + c \ge 0$ valid for $\Delta_i(\vec s)$.
+This constraint is also valid for $k \, \Delta_i(\vec s)$ iff
+$k \, \spv a x + \spv b s + c \ge 0$.
+If $\spv b s + c$ can attain any positive value, then $\spv a x$
+may be negative for some elements of $\Delta_i(\vec s)$.
+We then have $k \, \spv a x < \spv a x$ for $k > 1$ and so the constraint
+is not valid for $k \, \Delta_i(\vec s)$.
+We therefore need to impose $\spv b s + c \le 0$ for all values
+of $\vec s$ such that $\Delta_i(\vec s)$ is non-empty, i.e.,
+$\vec b$ and $c$ need to be such that $- \spv b s - c \ge 0$ is a valid
+constraint of $\Delta_i(\vec s)$.  That is, $(\vec b, c)$ are the opposites
+of the coefficients of a valid constraint of $\Delta_i(\vec s)$.
+The approximation of $k \, \Delta_i(\vec s)$ can therefore be obtained
+using three applications of Farkas' lemma.  The first obtains the coefficients
+of constraints valid for $\Delta_i(\vec s)$.  The second obtains
+the coefficients of constraints valid for the projection of $\Delta_i(\vec s)$
+onto the parameters.  The opposite of the second set is then computed
+and intersected with the first set.  The result is the set of coefficients
+of constraints valid for $k \, \Delta_i(\vec s)$.  A final application
+of Farkas' lemma is needed to obtain the approximation of
+$k \, \Delta_i(\vec s)$ itself.
+
+\begin{example}
+Consider the relation
+$$
+n \to \{\, (x, y) \to (1 + x, 1 - n + y) \mid n \ge 2 \,\}
+.
+$$
+The difference set of this relation is
+$$
+\Delta = n \to \{\, (1, 1 - n) \mid n \ge 2 \,\}
+.
+$$
+Using our approach, we would only consider the mixed constraint
+$y - 1 + n \ge 0$, leading to the following approximation of the
+transitive closure:
+$$
+n \to \{\, (x, y) \to (o_0, o_1) \mid n \ge 2 \wedge o_1 \le 1 - n + y \wedge o_0 \ge 1 + x \,\}
+.
+$$
+If, instead, we apply Farkas's lemma to $\Delta$, i.e.,
+\begin{verbatim}
+D := [n] -> { [1, 1 - n] : n >= 2 };
+CD := coefficients D;
+CD;
+\end{verbatim}
+we obtain
+\begin{verbatim}
+{ rat: coefficients[[c_cst, c_n] -> [i2, i3]] : i3 <= c_n and
+  i3 <= c_cst + 2c_n + i2 }
+\end{verbatim}
+The pure-parametric constraints valid for $\Delta$,
+\begin{verbatim}
+P := { [a,b] -> [] }(D);
+CP := coefficients P;
+CP;
+\end{verbatim}
+are
+\begin{verbatim}
+{ rat: coefficients[[c_cst, c_n] -> []] : c_n >= 0 and 2c_n >= -c_cst }
+\end{verbatim}
+Negating these coefficients and intersecting with \verb+CD+,
+\begin{verbatim}
+NCP := { rat: coefficients[[a,b] -> []]
+              -> coefficients[[-a,-b] -> []] }(CP);
+CK := wrap((unwrap CD) * (dom (unwrap NCP)));
+CK;
+\end{verbatim}
+we obtain
+\begin{verbatim}
+{ rat: [[c_cst, c_n] -> [i2, i3]] : i3 <= c_n and
+  i3 <= c_cst + 2c_n + i2 and c_n <= 0 and 2c_n <= -c_cst }
+\end{verbatim}
+The approximation for $k\,\Delta$,
+\begin{verbatim}
+K := solutions CK;
+K;
+\end{verbatim}
+is then
+\begin{verbatim}
+[n] -> { rat: [i0, i1] : i1 <= -i0 and i0 >= 1 and i1 <= 2 - n - i0 }
+\end{verbatim}
+Finally, the computed approximation for $R^+$,
+\begin{verbatim}
+T := unwrap({ [dx,dy] -> [[x,y] -> [x+dx,y+dy]] }(K));
+R := [n] -> { [x,y] -> [x+1,y+1-n] : n >= 2 };
+T := T * ((dom R) -> (ran R));
+T;
+\end{verbatim}
+is
+\begin{verbatim}
+[n] -> { [x, y] -> [o0, o1] : o1 <= x + y - o0 and
+         o0 >= 1 + x and o1 <= 2 - n + x + y - o0 and n >= 2 }
+\end{verbatim}
+\end{example}
+
+Existentially quantified variables can be handled by
+classifying them into variables that are uniquely
+determined by the parameters, variables that are independent
+of the parameters and others.  The first set can be treated
+as parameters and the second as variables.  Constraints involving
+the other existentially quantified variables are removed.
+
+\begin{example}
+Consider the relation
+$$
+R =
+n \to \{\, x \to y \mid \exists \, \alpha_0, \alpha_1: 7\alpha_0 = -2 + n \wedge 5\alpha_1 = -1 - x + y \wedge y \ge 6 + x \,\}
+.
+$$
+The difference set of this relation is
+$$
+\Delta = \Delta \, R =
+n \to \{\, x \mid \exists \, \alpha_0, \alpha_1: 7\alpha_0 = -2 + n \wedge 5\alpha_1 = -1 + x \wedge x \ge 6 \,\}
+.
+$$
+The existentially quantified variables can be defined in terms
+of the parameters and variables as
+$$
+\alpha_0 = \floor{\frac{-2 + n}7}
+\qquad
+\text{and}
+\qquad
+\alpha_1 = \floor{\frac{-1 + x}5}
+.
+$$
+$\alpha_0$ can therefore be treated as a parameter,
+while $\alpha_1$ can be treated as a variable.
+This in turn means that $7\alpha_0 = -2 + n$ can be treated as
+a purely parametric constraint, while the other two constraints are
+non-parametric.
+The corresponding $Q$~\eqref{eq:transitive:Q} is therefore
+$$
+\begin{aligned}
+n \to \{\, (x,z) \to (y,w) \mid
+\exists\, \alpha_0, \alpha_1, k, f : {} &
+k \ge 1 \wedge
+y = x + f \wedge
+w = z + k \wedge {} \\
+&
+7\alpha_0 = -2 + n \wedge
+5\alpha_1 = -k + x \wedge
+x \ge 6 k
+\,\}
+.
+\end{aligned}
+$$
+Projecting out the final coordinates encoding the length of the paths,
+results in the exact transitive closure
+$$
+R^+ =
+n \to \{\, x \to y \mid \exists \, \alpha_0, \alpha_1: 7\alpha_1 = -2 + n \wedge 6\alpha_0 \ge -x + y \wedge 5\alpha_0 \le -1 - x + y \,\}
+.
+$$
+\end{example}
+
+The fact that we ignore some impure constraints clearly leads
+to a loss of accuracy.  In some cases, some of this loss can be recovered
+by not considering the parameters in a special way.
+That is, instead of considering the set
+$$
+\Delta = \diff R =
+\vec s \mapsto
+\{\, \vec \delta \in \Z^{d} \mid \exists \vec x \to \vec y \in R :
+\vec \delta = \vec y - \vec x
+\,\}
+$$
+we consider the set
+$$
+\Delta' = \diff R' =
+\{\, \vec \delta \in \Z^{n+d} \mid \exists
+(\vec s, \vec x) \to (\vec s, \vec y) \in R' :
+\vec \delta = (\vec s - \vec s, \vec y - \vec x)
+\,\}
+.
+$$
+The first $n$ coordinates of every element in $\Delta'$ are zero.
+Projecting out these zero coordinates from $\Delta'$ is equivalent
+to projecting out the parameters in $\Delta$.
+The result is obviously a superset of $\Delta$, but all its constraints
+are of type \eqref{eq:transitive:non-parametric} and they can therefore
+all be used in the construction of $Q_i$.
+
+\begin{example}
+Consider the relation
+$$
+% [n] -> { [x, y] -> [1 + x, 1 - n + y] | n >= 2 }
+R = n \to \{\, (x, y) \to (1 + x, 1 - n + y) \mid n \ge 2 \,\}
+.
+$$
+We have
+$$
+\diff R = n \to \{\, (1, 1 - n) \mid n \ge 2 \,\}
+$$
+and so, by treating the parameters in a special way, we obtain
+the following approximation for $R^+$:
+$$
+n \to \{\, (x, y) \to (x', y') \mid n \ge 2 \wedge y' \le 1 - n + y \wedge x' \ge 1 + x \,\}
+.
+$$
+If we consider instead
+$$
+R' = \{\, (n, x, y) \to (n, 1 + x, 1 - n + y) \mid n \ge 2 \,\}
+$$
+then
+$$
+\diff R' = \{\, (0, 1, y) \mid y \le -1 \,\}
+$$
+and we obtain the approximation
+$$
+n \to \{\, (x, y) \to (x', y') \mid n \ge 2 \wedge x' \ge 1 + x \wedge y' \le x + y - x' \,\}
+.
+$$
+If we consider both $\diff R$ and $\diff R'$, then we obtain
+$$
+n \to \{\, (x, y) \to (x', y') \mid n \ge 2 \wedge y' \le 1 - n + y \wedge x' \ge 1 + x \wedge y' \le x + y - x' \,\}
+.
+$$
+Note, however, that this is not the most accurate affine approximation that
+can be obtained.  That would be
+$$
+n \to \{\, (x, y) \to (x', y') \mid y' \le 2 - n + x + y - x' \wedge n \ge 2 \wedge x' \ge 1 + x \,\}
+.
+$$
+\end{example}
+
+\subsection{Checking Exactness}
+
+The approximation $T$ for the transitive closure $R^+$ can be obtained
+by projecting out the parameter $k$ from the approximation $K$
+\eqref{eq:transitive:approx} of the power $R^k$.
+Since $K$ is an overapproximation of $R^k$, $T$ will also be an
+overapproximation of $R^+$.
+To check whether the results are exact, we need to consider two
+cases depending on whether $R$ is {\em cyclic}, where $R$ is defined
+to be cyclic if $R^+$ maps any element to itself, i.e.,
+$R^+ \cap \identity \ne \emptyset$.
+If $R$ is acyclic, then the inductive definition of
+\eqref{eq:transitive:inductive} is equivalent to its completion,
+i.e.,
+$$
+R^+ = R \cup \left(R \circ R^+\right)
+$$
+is a defining property.
+Since $T$ is known to be an overapproximation, we only need to check
+whether
+$$
+T \subseteq R \cup \left(R \circ T\right)
+.
+$$
+This is essentially Theorem~5 of \shortciteN{Kelly1996closure}.
+The only difference is that they only consider lexicographically
+forward relations, a special case of acyclic relations.
+
+If, on the other hand, $R$ is cyclic, then we have to resort
+to checking whether the approximation $K$ of the power is exact.
+Note that $T$ may be exact even if $K$ is not exact, so the check
+is sound, but incomplete.
+To check exactness of the power, we simply need to check
+\eqref{eq:transitive:power}.  Since again $K$ is known
+to be an overapproximation, we only need to check whether
+$$
+\begin{aligned}
+K'|_{y_{d+1} - x_{d+1} = 1} & \subseteq R'
+\\
+K'|_{y_{d+1} - x_{d+1} \ge 2} & \subseteq R' \circ K'|_{y_{d+1} - x_{d+1} \ge 1}
+,
+\end{aligned}
+$$
+where $R' = \{\, \vec x' \to \vec y' \mid \vec x \to \vec y \in R
+\wedge y_{d+1} - x_{d+1} = 1\,\}$, i.e., $R$ extended with path
+lengths equal to 1.
+
+All that remains is to explain how to check the cyclicity of $R$.
+Note that the exactness on the power is always sound, even
+in the acyclic case, so we only need to be careful that we find
+all cyclic cases.  Now, if $R$ is cyclic, i.e.,
+$R^+ \cap \identity \ne \emptyset$, then, since $T$ is
+an overapproximation of $R^+$, also
+$T \cap \identity \ne \emptyset$.  This in turn means
+that $\Delta \, K'$ contains a point whose first $d$ coordinates
+are zero and whose final coordinate is positive.
+In the implementation we currently perform this test on $P'$ instead of $K'$.
+Note that if $R^+$ is acyclic and $T$ is not, then the approximation
+is clearly not exact and the approximation of the power $K$
+will not be exact either.
+
+\subsection{Decomposing $R$ into strongly connected components}
+
+If the input relation $R$ is a union of several basic relations
+that can be partially ordered
+then the accuracy of the approximation may be improved by computing
+an approximation of each strongly connected components separately.
+For example, if $R = R_1 \cup R_2$ and $R_1 \circ R_2 = \emptyset$,
+then we know that any path that passes through $R_2$ cannot later
+pass through $R_1$, i.e.,
+\begin{equation}
+\label{eq:transitive:components}
+R^+ = R_1^+ \cup R_2^+ \cup \left(R_2^+ \circ R_1^+\right)
+.
+\end{equation}
+We can therefore compute (approximations of) transitive closures
+of $R_1$ and $R_2$ separately.
+Note, however, that the condition $R_1 \circ R_2 = \emptyset$
+is actually too strong.
+If $R_1 \circ R_2$ is a subset of $R_2 \circ R_1$
+then we can reorder the segments
+in any path that moves through both $R_1$ and $R_2$ to
+first move through $R_1$ and then through $R_2$.
+
+This idea can be generalized to relations that are unions
+of more than two basic relations by constructing the
+strongly connected components in the graph with as vertices
+the basic relations and an edge between two basic relations
+$R_i$ and $R_j$ if $R_i$ needs to follow $R_j$ in some paths.
+That is, there is an edge from $R_i$ to $R_j$ iff
+\begin{equation}
+\label{eq:transitive:edge}
+R_i \circ R_j
+\not\subseteq
+R_j \circ R_i
+.
+\end{equation}
+The components can be obtained from the graph by applying
+Tarjan's algorithm \shortcite{Tarjan1972}.
+
+In practice, we compute the (extended) powers $K_i'$ of each component
+separately and then compose them as in \eqref{eq:transitive:decompose}.
+Note, however, that in this case the order in which we apply them is
+important and should correspond to a topological ordering of the
+strongly connected components.  Simply applying Tarjan's
+algorithm will produce topologically sorted strongly connected components.
+The graph on which Tarjan's algorithm is applied is constructed on-the-fly.
+That is, whenever the algorithm checks if there is an edge between
+two vertices, we evaluate \eqref{eq:transitive:edge}.
+The exactness check is performed on each component separately.
+If the approximation turns out to be inexact for any of the components,
+then the entire result is marked inexact and the exactness check
+is skipped on the components that still need to be handled.
+
+It should be noted that \eqref{eq:transitive:components}
+is only valid for exact transitive closures.
+If overapproximations are computed in the right hand side, then the result will
+still be an overapproximation of the left hand side, but this result
+may not be transitively closed.  If we only separate components based
+on the condition $R_i \circ R_j = \emptyset$, then there is no problem,
+as this condition will still hold on the computed approximations
+of the transitive closures.  If, however, we have exploited
+\eqref{eq:transitive:edge} during the decomposition and if the
+result turns out not to be exact, then we check whether
+the result is transitively closed.  If not, we recompute
+the transitive closure, skipping the decomposition.
+Note that testing for transitive closedness on the result may
+be fairly expensive, so we may want to make this check
+configurable.
+
+\begin{figure}
+\begin{center}
+\begin{tikzpicture}[x=0.5cm,y=0.5cm,>=stealth,shorten >=1pt]
+\foreach \x in {1,...,10}{
+    \foreach \y in {1,...,10}{
+	\draw[->] (\x,\y) -- (\x,\y+1);
+    }
+}
+\foreach \x in {1,...,20}{
+    \foreach \y in {5,...,15}{
+	\draw[->] (\x,\y) -- (\x+1,\y);
+    }
+}
+\end{tikzpicture}
+\end{center}
+\caption{The relation from \autoref{ex:closure4}}
+\label{f:closure4}
+\end{figure}
+\begin{example}
+\label{ex:closure4}
+Consider the relation in example {\tt closure4} that comes with
+the Omega calculator~\shortcite{Omega_calc}, $R = R_1 \cup R_2$,
+with
+$$
+\begin{aligned}
+R_1 & = \{\, (x,y) \to (x,y+1) \mid 1 \le x,y \le 10 \,\}
+\\
+R_2 & = \{\, (x,y) \to (x+1,y) \mid 1 \le x \le 20 \wedge 5 \le y \le 15 \,\}
+.
+\end{aligned}
+$$
+This relation is shown graphically in \autoref{f:closure4}.
+We have
+$$
+\begin{aligned}
+R_1 \circ R_2 &=
+\{\, (x,y) \to (x+1,y+1) \mid 1 \le x \le 9 \wedge 5 \le y \le 10 \,\}
+\\
+R_2 \circ R_1 &=
+\{\, (x,y) \to (x+1,y+1) \mid 1 \le x \le 10 \wedge 4 \le y \le 10 \,\}
+.
+\end{aligned}
+$$
+Clearly, $R_1 \circ R_2 \subseteq R_2 \circ R_1$ and so
+$$
+\left(
+R_1 \cup R_2
+\right)^+
+=
+\left(R_2^+ \circ R_1^+\right)
+\cup R_1^+
+\cup R_2^+
+.
+$$
+\end{example}
+
+\begin{figure}
+\newcounter{n}
+\newcounter{t1}
+\newcounter{t2}
+\newcounter{t3}
+\newcounter{t4}
+\begin{center}
+\begin{tikzpicture}[>=stealth,shorten >=1pt]
+\setcounter{n}{7}
+\foreach \i in {1,...,\value{n}}{
+    \foreach \j in {1,...,\value{n}}{
+	\setcounter{t1}{2 * \j - 4 - \i + 1}
+	\setcounter{t2}{\value{n} - 3 - \i + 1}
+	\setcounter{t3}{2 * \i - 1 - \j + 1}
+	\setcounter{t4}{\value{n} - \j + 1}
+	\ifnum\value{t1}>0\ifnum\value{t2}>0
+	\ifnum\value{t3}>0\ifnum\value{t4}>0
+	    \draw[thick,->] (\i,\j) to[out=20] (\i+3,\j);
+	\fi\fi\fi\fi
+	\setcounter{t1}{2 * \j - 1 - \i + 1}
+	\setcounter{t2}{\value{n} - \i + 1}
+	\setcounter{t3}{2 * \i - 4 - \j + 1}
+	\setcounter{t4}{\value{n} - 3 - \j + 1}
+	\ifnum\value{t1}>0\ifnum\value{t2}>0
+	\ifnum\value{t3}>0\ifnum\value{t4}>0
+	    \draw[thick,->] (\i,\j) to[in=-20,out=20] (\i,\j+3);
+	\fi\fi\fi\fi
+	\setcounter{t1}{2 * \j - 1 - \i + 1}
+	\setcounter{t2}{\value{n} - 1 - \i + 1}
+	\setcounter{t3}{2 * \i - 1 - \j + 1}
+	\setcounter{t4}{\value{n} - 1 - \j + 1}
+	\ifnum\value{t1}>0\ifnum\value{t2}>0
+	\ifnum\value{t3}>0\ifnum\value{t4}>0
+	    \draw[thick,->] (\i,\j) to (\i+1,\j+1);
+	\fi\fi\fi\fi
+    }
+}
+\end{tikzpicture}
+\end{center}
+\caption{The relation from \autoref{ex:decomposition}}
+\label{f:decomposition}
+\end{figure}
+\begin{example}
+\label{ex:decomposition}
+Consider the relation on the right of \shortciteN[Figure~2]{Beletska2009},
+reproduced in \autoref{f:decomposition}.
+The relation can be described as $R = R_1 \cup R_2 \cup R_3$,
+with
+$$
+\begin{aligned}
+R_1 &= n \mapsto \{\, (i,j) \to (i+3,j) \mid
+i \le 2 j - 4 \wedge
+i \le n - 3 \wedge
+j \le 2 i - 1 \wedge
+j \le n \,\}
+\\
+R_2 &= n \mapsto \{\, (i,j) \to (i,j+3) \mid
+i \le 2 j - 1 \wedge
+i \le n \wedge
+j \le 2 i - 4 \wedge
+j \le n - 3 \,\}
+\\
+R_3 &= n \mapsto \{\, (i,j) \to (i+1,j+1) \mid
+i \le 2 j - 1 \wedge
+i \le n - 1 \wedge
+j \le 2 i - 1 \wedge
+j \le n - 1\,\}
+.
+\end{aligned}
+$$
+The figure shows this relation for $n = 7$.
+Both
+$R_3 \circ R_1 \subseteq R_1 \circ R_3$
+and
+$R_3 \circ R_2 \subseteq R_2 \circ R_3$,
+which the reader can verify using the {\tt iscc} calculator:
+\begin{verbatim}
+R1 := [n] -> { [i,j] -> [i+3,j] : i <= 2 j - 4 and i <= n - 3 and
+                                  j <= 2 i - 1 and j <= n };
+R2 := [n] -> { [i,j] -> [i,j+3] : i <= 2 j - 1 and i <= n and
+                                  j <= 2 i - 4 and j <= n - 3 };
+R3 := [n] -> { [i,j] -> [i+1,j+1] : i <= 2 j - 1 and i <= n - 1 and
+                                    j <= 2 i - 1 and j <= n - 1 };
+(R1 . R3) - (R3 . R1);
+(R2 . R3) - (R3 . R2);
+\end{verbatim}
+$R_3$ can therefore be moved forward in any path.
+For the other two basic relations, we have both
+$R_2 \circ R_1 \not\subseteq R_1 \circ R_2$
+and
+$R_1 \circ R_2 \not\subseteq R_2 \circ R_1$
+and so $R_1$ and $R_2$ form a strongly connected component.
+By computing the power of $R_3$ and $R_1 \cup R_2$ separately
+and composing the results, the power of $R$ can be computed exactly
+using \eqref{eq:transitive:singleton}.
+As explained by \shortciteN{Beletska2009}, applying the same formula
+to $R$ directly, without a decomposition, would result in
+an overapproximation of the power.
+\end{example}
+
+\subsection{Partitioning the domains and ranges of $R$}
+
+The algorithm of \autoref{s:power} assumes that the input relation $R$
+can be treated as a union of translations.
+This is a reasonable assumption if $R$ maps elements of a given
+abstract domain to the same domain.
+However, if $R$ is a union of relations that map between different
+domains, then this assumption no longer holds.
+In particular, when an entire dependence graph is encoded
+in a single relation, as is done by, e.g.,
+\shortciteN[Section~6.1]{Barthou2000MSE}, then it does not make
+sense to look at differences between iterations of different domains.
+Now, arguably, a modified Floyd-Warshall algorithm should
+be applied to the dependence graph, as advocated by
+\shortciteN{Kelly1996closure}, with the transitive closure operation
+only being applied to relations from a given domain to itself.
+However, it is also possible to detect disjoint domains and ranges
+and to apply Floyd-Warshall internally.
+
+\LinesNumbered
+\begin{algorithm}
+\caption{The modified Floyd-Warshall algorithm of
+\protect\shortciteN{Kelly1996closure}}
+\label{a:Floyd}
+\SetKwInput{Input}{Input}
+\SetKwInput{Output}{Output}
+\Input{Relations $R_{pq}$, $0 \le p, q < n$}
+\Output{Updated relations $R_{pq}$ such that each relation
+$R_{pq}$ contains all indirect paths from $p$ to $q$ in the input graph}
+%
+\BlankLine
+\SetAlgoVlined
+\DontPrintSemicolon
+%
+\For{$r \in [0, n-1]$}{
+    $R_{rr} \coloneqq R_{rr}^+$ \nllabel{l:Floyd:closure}\;
+    \For{$p \in [0, n-1]$}{
+	\For{$q \in [0, n-1]$}{
+	    \If{$p \ne r$ or $q \ne r$}{
+		$R_{pq} \coloneqq R_{pq} \cup \left(R_{rq} \circ R_{pr}\right)
+			     \cup \left(R_{rq} \circ R_{rr} \circ R_{pr}\right)$
+	     \nllabel{l:Floyd:update}
+	     }
+	}
+    }
+}
+\end{algorithm}
+
+Let the input relation $R$ be a union of $m$ basic relations $R_i$.
+Let $D_{2i}$ be the domains of $R_i$ and $D_{2i+1}$ the ranges of $R_i$.
+The first step is to group overlapping $D_j$ until a partition is
+obtained.  If the resulting partition consists of a single part,
+then we continue with the algorithm of \autoref{s:power}.
+Otherwise, we apply Floyd-Warshall on the graph with as vertices
+the parts of the partition and as edges the $R_i$ attached to
+the appropriate pairs of vertices.
+In particular, let there be $n$ parts $P_k$ in the partition.
+We construct $n^2$ relations
+$$
+R_{pq} \coloneqq \bigcup_{i \text{ s.t. } \domain R_i \subseteq P_p \wedge
+				 \range R_i \subseteq P_q} R_i
+,
+$$
+apply \autoref{a:Floyd} and return the union of all resulting
+$R_{pq}$ as the transitive closure of $R$.
+Each iteration of the $r$-loop in \autoref{a:Floyd} updates
+all relations $R_{pq}$ to include paths that go from $p$ to $r$,
+possibly stay there for a while, and then go from $r$ to $q$.
+Note that paths that ``stay in $r$'' include all paths that
+pass through earlier vertices since $R_{rr}$ itself has been updated
+accordingly in previous iterations of the outer loop.
+In principle, it would be sufficient to use the $R_{pr}$
+and $R_{rq}$ computed in the previous iteration of the
+$r$-loop in Line~\ref{l:Floyd:update}.
+However, from an implementation perspective, it is easier
+to allow either or both of these to have been updated
+in the same iteration of the $r$-loop.
+This may result in duplicate paths, but these can usually
+be removed by coalescing (\autoref{s:coalescing}) the result of the union
+in Line~\ref{l:Floyd:update}, which should be done in any case.
+The transitive closure in Line~\ref{l:Floyd:closure}
+is performed using a recursive call.  This recursive call
+includes the partitioning step, but the resulting partition will
+usually be a singleton.
+The result of the recursive call will either be exact or an
+overapproximation.  The final result of Floyd-Warshall is therefore
+also exact or an overapproximation.
+
+\begin{figure}
+\begin{center}
+\begin{tikzpicture}[x=1cm,y=1cm,>=stealth,shorten >=3pt]
+\foreach \x/\y in {0/0,1/1,3/2} {
+    \fill (\x,\y) circle (2pt);
+}
+\foreach \x/\y in {0/1,2/2,3/3} {
+    \draw (\x,\y) circle (2pt);
+}
+\draw[->] (0,0) -- (0,1);
+\draw[->] (0,1) -- (1,1);
+\draw[->] (2,2) -- (3,2);
+\draw[->] (3,2) -- (3,3);
+\draw[->,dashed] (2,2) -- (3,3);
+\draw[->,dotted] (0,0) -- (1,1);
+\end{tikzpicture}
+\end{center}
+\caption{The relation (solid arrows) on the right of Figure~1 of
+\protect\shortciteN{Beletska2009} and its transitive closure}
+\label{f:COCOA:1}
+\end{figure}
+\begin{example}
+Consider the relation on the right of Figure~1 of
+\shortciteN{Beletska2009},
+reproduced in \autoref{f:COCOA:1}.
+This relation can be described as
+$$
+\begin{aligned}
+\{\, (x, y) \to (x_2, y_2) \mid {} & (3y = 2x \wedge x_2 = x \wedge 3y_2 = 3 + 2x \wedge x \ge 0 \wedge x \le 3) \vee {} \\
+& (x_2 = 1 + x \wedge y_2 = y \wedge x \ge 0 \wedge 3y \ge 2 + 2x \wedge x \le 2 \wedge 3y \le 3 + 2x) \,\}
+.
+\end{aligned}
+$$
+Note that the domain of the upward relation overlaps with the range
+of the rightward relation and vice versa, but that the domain
+of neither relation overlaps with its own range or the domain of
+the other relation.
+The domains and ranges can therefore be partitioned into two parts,
+$P_0$ and $P_1$, shown as the white and black dots in \autoref{f:COCOA:1},
+respectively.
+Initially, we have
+$$
+\begin{aligned}
+R_{00} & = \emptyset
+\\
+R_{01} & = 
+\{\, (x, y) \to (x+1, y) \mid 
+(x \ge 0 \wedge 3y \ge 2 + 2x \wedge x \le 2 \wedge 3y \le 3 + 2x) \,\}
+\\
+R_{10} & =
+\{\, (x, y) \to (x_2, y_2) \mid (3y = 2x \wedge x_2 = x \wedge 3y_2 = 3 + 2x \wedge x \ge 0 \wedge x \le 3) \,\}
+\\
+R_{11} & = \emptyset
+.
+\end{aligned}
+$$
+In the first iteration, $R_{00}$ remains the same ($\emptyset^+ = \emptyset$).
+$R_{01}$ and $R_{10}$ are therefore also unaffected, but
+$R_{11}$ is updated to include $R_{01} \circ R_{10}$, i.e.,
+the dashed arrow in the figure.
+This new $R_{11}$ is obviously transitively closed, so it is not
+changed in the second iteration and it does not have an effect
+on $R_{01}$ and $R_{10}$.  However, $R_{00}$ is updated to
+include $R_{10} \circ R_{01}$, i.e., the dotted arrow in the figure.
+The transitive closure of the original relation is then equal to
+$R_{00} \cup R_{01} \cup R_{10} \cup R_{11}$.
+\end{example}
+
+\subsection{Incremental Computation}
+\label{s:incremental}
+
+In some cases it is possible and useful to compute the transitive closure
+of union of basic relations incrementally.  In particular,
+if $R$ is a union of $m$ basic maps,
+$$
+R = \bigcup_j R_j
+,
+$$
+then we can pick some $R_i$ and compute the transitive closure of $R$ as
+\begin{equation}
+\label{eq:transitive:incremental}
+R^+ = R_i^+ \cup
+\left(
+\bigcup_{j \ne i}
+R_i^* \circ R_j \circ R_i^*
+\right)^+
+.
+\end{equation}
+For this approach to be successful, it is crucial that each
+of the disjuncts in the argument of the second transitive
+closure in \eqref{eq:transitive:incremental} be representable
+as a single basic relation, i.e., without a union.
+If this condition holds, then by using \eqref{eq:transitive:incremental},
+the number of disjuncts in the argument of the transitive closure
+can be reduced by one.
+Now, $R_i^* = R_i^+ \cup \identity$, but in some cases it is possible
+to relax the constraints of $R_i^+$ to include part of the identity relation,
+say on domain $D$.  We will use the notation
+${\cal C}(R_i,D) = R_i^+ \cup \identity_D$ to represent
+this relaxed version of $R^+$.
+\shortciteN{Kelly1996closure} use the notation $R_i^?$.
+${\cal C}(R_i,D)$ can be computed by allowing $k$ to attain
+the value $0$ in \eqref{eq:transitive:Q} and by using
+$$
+P \cap \left(D \to D\right)
+$$
+instead of \eqref{eq:transitive:approx}.
+Typically, $D$ will be a strict superset of both $\domain R_i$
+and $\range R_i$.  We therefore need to check that domain
+and range of the transitive closure are part of ${\cal C}(R_i,D)$,
+i.e., the part that results from the paths of positive length ($k \ge 1$),
+are equal to the domain and range of $R_i$.
+If not, then the incremental approach cannot be applied for
+the given choice of $R_i$ and $D$.
+
+In order to be able to replace $R^*$ by ${\cal C}(R_i,D)$
+in \eqref{eq:transitive:incremental}, $D$ should be chosen
+to include both $\domain R$ and $\range R$, i.e., such
+that $\identity_D \circ R_j \circ \identity_D = R_j$ for all $j\ne i$.
+\shortciteN{Kelly1996closure} say that they use
+$D = \domain R_i \cup \range R_i$, but presumably they mean that
+they use $D = \domain R \cup \range R$.
+Now, this expression of $D$ contains a union, so it not directly usable.
+\shortciteN{Kelly1996closure} do not explain how they avoid this union.
+Apparently, in their implementation,
+they are using the convex hull of $\domain R \cup \range R$
+or at least an approximation of this convex hull.
+We use the simple hull (\autoref{s:simple hull}) of $\domain R \cup \range R$.
+
+It is also possible to use a domain $D$ that does {\em not\/}
+include $\domain R \cup \range R$, but then we have to
+compose with ${\cal C}(R_i,D)$ more selectively.
+In particular, if we have
+\begin{equation}
+\label{eq:transitive:right}
+\text{for each $j \ne i$ either }
+\domain R_j \subseteq D \text{ or } \domain R_j \cap \range R_i = \emptyset
+\end{equation}
+and, similarly,
+\begin{equation}
+\label{eq:transitive:left}
+\text{for each $j \ne i$ either }
+\range R_j \subseteq D \text{ or } \range R_j \cap \domain R_i = \emptyset
+\end{equation}
+then we can refine \eqref{eq:transitive:incremental} to
+$$
+R_i^+ \cup
+\left(
+\left(
+\bigcup_{\shortstack{$\scriptstyle\domain R_j \subseteq D $\\
+		     $\scriptstyle\range R_j \subseteq D$}}
+{\cal C} \circ R_j \circ {\cal C}
+\right)
+\cup
+\left(
+\bigcup_{\shortstack{$\scriptstyle\domain R_j \cap \range R_i = \emptyset$\\
+		     $\scriptstyle\range R_j \subseteq D$}}
+\!\!\!\!\!
+{\cal C} \circ R_j
+\right)
+\cup
+\left(
+\bigcup_{\shortstack{$\scriptstyle\domain R_j \subseteq D $\\
+		     $\scriptstyle\range R_j \cap \domain R_i = \emptyset$}}
+\!\!\!\!\!
+R_j \circ {\cal C}
+\right)
+\cup
+\left(
+\bigcup_{\shortstack{$\scriptstyle\domain R_j \cap \range R_i = \emptyset$\\
+		     $\scriptstyle\range R_j \cap \domain R_i = \emptyset$}}
+\!\!\!\!\!
+R_j
+\right)
+\right)^+
+.
+$$
+If only property~\eqref{eq:transitive:right} holds,
+we can use
+$$
+R_i^+ \cup
+\left(
+\left(
+R_i^+ \cup \identity
+\right)
+\circ
+\left(
+\left(
+\bigcup_{\shortstack{$\scriptstyle\domain R_j \subseteq D $}}
+R_j \circ {\cal C}
+\right)
+\cup
+\left(
+\bigcup_{\shortstack{$\scriptstyle\domain R_j \cap \range R_i = \emptyset$}}
+\!\!\!\!\!
+R_j
+\right)
+\right)^+
+\right)
+,
+$$
+while if only property~\eqref{eq:transitive:left} holds,
+we can use
+$$
+R_i^+ \cup
+\left(
+\left(
+\left(
+\bigcup_{\shortstack{$\scriptstyle\range R_j \subseteq D $}}
+{\cal C} \circ R_j
+\right)
+\cup
+\left(
+\bigcup_{\shortstack{$\scriptstyle\range R_j \cap \domain R_i = \emptyset$}}
+\!\!\!\!\!
+R_j
+\right)
+\right)^+
+\circ
+\left(
+R_i^+ \cup \identity
+\right)
+\right)
+.
+$$
+
+It should be noted that if we want the result of the incremental
+approach to be transitively closed, then we can only apply it
+if all of the transitive closure operations involved are exact.
+If, say, the second transitive closure in \eqref{eq:transitive:incremental}
+contains extra elements, then the result does not necessarily contain
+the composition of these extra elements with powers of $R_i$.
+
+\subsection{An {\tt Omega}-like implementation}
+
+While the main algorithm of \shortciteN{Kelly1996closure} is
+designed to compute and underapproximation of the transitive closure,
+the authors mention that they could also compute overapproximations.
+In this section, we describe our implementation of an algorithm
+that is based on their ideas.
+Note that the {\tt Omega} library computes underapproximations
+\shortcite[Section 6.4]{Omega_lib}.
+
+The main tool is Equation~(2) of \shortciteN{Kelly1996closure}.
+The input relation $R$ is first overapproximated by a ``d-form'' relation
+$$
+\{\, \vec i \to \vec j \mid \exists \vec \alpha :
+\vec L \le \vec j - \vec i \le \vec U
+\wedge
+(\forall p : j_p - i_p = M_p \alpha_p)
+\,\}
+,
+$$
+where $p$ ranges over the dimensions and $\vec L$, $\vec U$ and
+$\vec M$ are constant integer vectors.  The elements of $\vec U$
+may be $\infty$, meaning that there is no upper bound corresponding
+to that element, and similarly for $\vec L$.
+Such an overapproximation can be obtained by computing strides,
+lower and upper bounds on the difference set $\Delta \, R$.
+The transitive closure of such a ``d-form'' relation is
+\begin{equation}
+\label{eq:omega}
+\{\, \vec i \to \vec j \mid \exists \vec \alpha, k :
+k \ge 1 \wedge
+k \, \vec L \le \vec j - \vec i \le k \, \vec U
+\wedge
+(\forall p : j_p - i_p = M_p \alpha_p)
+\,\}
+.
+\end{equation}
+The domain and range of this transitive closure are then
+intersected with those of the input relation.
+This is a special case of the algorithm in \autoref{s:power}.
+
+In their algorithm for computing lower bounds, the authors
+use the above algorithm as a substep on the disjuncts in the relation.
+At the end, they say
+\begin{quote}
+If an upper bound is required, it can be calculated in a manner
+similar to that of a single conjunct [sic] relation.
+\end{quote}
+Presumably, the authors mean that a ``d-form'' approximation
+of the whole input relation should be used.
+However, the accuracy can be improved by also trying to
+apply the incremental technique from the same paper,
+which is explained in more detail in \autoref{s:incremental}.
+In this case, ${\cal C}(R_i,D)$ can be obtained by
+allowing the value zero for $k$ in \eqref{eq:omega},
+i.e., by computing
+$$
+\{\, \vec i \to \vec j \mid \exists \vec \alpha, k :
+k \ge 0 \wedge
+k \, \vec L \le \vec j - \vec i \le k \, \vec U
+\wedge
+(\forall p : j_p - i_p = M_p \alpha_p)
+\,\}
+.
+$$
+In our implementation we take as $D$ the simple hull
+(\autoref{s:simple hull}) of $\domain R \cup \range R$.
+To determine whether it is safe to use ${\cal C}(R_i,D)$,
+we check the following conditions, as proposed by
+\shortciteN{Kelly1996closure}:
+${\cal C}(R_i,D) - R_i^+$ is not a union and for each $j \ne i$
+the condition
+$$
+\left({\cal C}(R_i,D) - R_i^+\right)
+\circ
+R_j
+\circ
+\left({\cal C}(R_i,D) - R_i^+\right)
+=
+R_j
+$$
+holds.
diff --git a/polly/lib/External/isl/doc/isl.bib b/polly/lib/External/isl/doc/isl.bib
new file mode 100644
index 00000000000..cfe8081e56b
--- /dev/null
+++ b/polly/lib/External/isl/doc/isl.bib
@@ -0,0 +1,313 @@
+@inproceedings{Kelly1996closure,
+  author    = {Wayne Kelly and
+               William Pugh and
+               Evan Rosser and
+               Tatiana Shpeisman},
+  title     = {Transitive Closure of Infinite Graphs and Its Applications},
+  pages     = {126-140},
+  editor    = {Chua-Huang Huang and
+               P. Sadayappan and
+               Utpal Banerjee and
+               David Gelernter and
+               Alexandru Nicolau and
+               David A. Padua},
+  booktitle = {Languages and Compilers for Parallel Computing, 8th International
+               Workshop, LCPC'95, Columbus, Ohio, USA, August 10-12, 1995,
+               Proceedings},
+  publisher = {Springer},
+  series    = {Lecture Notes in Computer Science},
+  volume    = {1033},
+  year      = {1996},
+  isbn      = {3-540-60765-X},
+}
+
+@inproceedings{Beletska2009,
+  author = {Beletska, Anna and Barthou, Denis and Bielecki, Wlodzimierz and Cohen, Albert},
+  title = {Computing the Transitive Closure of a Union of Affine Integer Tuple Relations},
+  booktitle = {COCOA '09: Proceedings of the 3rd International Conference on Combinatorial Optimization and Applications},
+  year = {2009},
+  isbn = {978-3-642-02025-4},
+  pages = {98--109},
+  location = {Huangshan, China},
+  doi = {10.1007/978-3-642-02026-1_9},
+  publisher = {Springer-Verlag},
+  address = {Berlin, Heidelberg},
+}
+
+@book{Schrijver1986,
+    author  =	"Schrijver, Alexander",
+    title   =	"Theory of Linear and Integer Programming",
+    publisher	=   "John Wiley \& Sons",
+    year    =	1986
+}
+
+@article{Tarjan1972,
+    author = {Tarjan, Robert},
+    journal = {SIAM Journal on Computing},
+    number = {2},
+    pages = {146--160},
+    publisher = {SIAM},
+    title = {Depth-First Search and Linear Graph Algorithms},
+    volume = {1},
+    year = {1972}
+}
+
+@TechReport{ Omega_calc,
+    author = "Wayne Kelly and Vadim Maslov and William Pugh and Evan Rosser and Tatiana Shpeisman and Dave Wonnacott",
+    title = "The {Omega} Calculator and Library",
+    month = nov,
+    institution = "University of Maryland",
+    year = 1996
+}
+
+@TechReport{ Omega_lib,
+    author = "Wayne Kelly and Vadim Maslov and William Pugh and Evan Rosser and Tatiana Shpeisman and Dave Wonnacott",
+    title = "The {Omega} Library",
+    month = nov,
+    institution = "University of Maryland",
+    year = 1996
+}
+
+@unpublished{Verdoolaege2009isl,
+  author = "Verdoolaege, Sven",
+  title = "An integer set library for program analysis",
+  note = "Advances in the Theory of Integer Linear Optimization and its Extensions,AMS 2009 Spring Western Section Meeting, San Francisco, California, 25-26 April 2009",
+  month = Apr,
+  year = "2009",
+  url = "https://lirias.kuleuven.be/handle/123456789/228373",
+}
+
+@article{Barthou2000MSE,
+  author = {Barthou, Denis and Cohen, Albert and Collard, Jean-Fran\c{c}ois},
+  title = {Maximal Static Expansion},
+  journal = {Int. J. Parallel Program.},
+  volume = {28},
+  number = {3},
+  year = {2000},
+  issn = {0885-7458},
+  pages = {213--243},
+  doi = {10.1023/A:1007500431910},
+  publisher = {Kluwer Academic Publishers},
+  address = {Norwell, MA, USA},
+}
+
+@article{ Feautrier88parametric,
+    author = "P. Feautrier",
+    title = "Parametric Integer Programming",
+    journal = "RAIRO Recherche Op\'erationnelle",
+    volume = "22",
+    number = "3",
+    pages = "243--268",
+    year = "1988",
+}
+
+@Article{ Fea91,
+  author =       {Feautrier, P.},
+  title =        {Dataflow analysis of array and scalar references},
+  journal =      {International Journal of Parallel Programming},
+  year =         {1991},
+  OPTkey =         {},
+  volume =       {20},
+  number =       {1},
+  OPTmonth =     {},
+  pages =        {23--53},
+  OPTnote =      {},
+  OPTannote =    {},
+}
+
+@INPROCEEDINGS{BouletRe98,
+  AUTHOR = {Pierre Boulet and Xavier Redon},
+  TITLE = {Communication Pre-evaluation in {HPF}},
+  BOOKTITLE = {EUROPAR'98},
+  PAGES = {263--272},
+  YEAR = 1998,
+  VOLUME = 1470,
+  series =	 {Lecture Notes in Computer Science},
+  PUBLISHER = {Springer-Verlag, Berlin},
+  ABSTRACT = {  Parallel computers are difficult to program efficiently.  We believe
+  that a good way to help programmers write efficient programs is to
+  provide them with tools that show them how their programs behave on
+  a parallel computer.  Data distribution is the major performance
+  factor of data-parallel programs and so automatic data layout for
+  HPF programs has been studied by many researchers recently.  The
+  communication volume induced by a data distribution is a good
+  estimator of the efficiency of this data distribution.
+
+  We present here a symbolic method to compute the communication
+  volume generated by a given data distribution during the program
+  writing phase (before compilation). We stay machine-independent to
+  assure portability.  Our goal is to help the programmer understand
+  the data movements its program generates and thus find a good data
+  distribution. Our method is based on parametric polyhedral
+  computations. It can be applied to a large class of regular codes.},
+}
+
+@INPROCEEDINGS {Verdoolaege2005experiences,
+  AUTHOR = "Verdoolaege, Sven and Beyls, Kristof and Bruynooghe, Maurice and Catthoor, Francky",
+  TITLE = {{E}xperiences with enumeration of integer projections of parametric polytopes},
+  BOOKTITLE = {{P}roceedings of 14th {I}nternational {C}onference on {C}ompiler {C}onstruction, {E}dinburgh, {S}cotland},
+  YEAR = {2005},
+  EDITOR = {Bodik, R.},
+  VOLUME = 3443,
+    pages = "91-105",
+    series = "Lecture Notes in Computer Science",
+    publisher = "Springer-Verlag",
+    address = "Berlin",
+    doi = "10.1007/b107108",
+}
+
+@article{Detlefs2005simplify,
+ author = {David Detlefs and Greg Nelson and James B. Saxe},
+ title = {Simplify: a theorem prover for program checking},
+ journal = {J. ACM},
+ volume = {52},
+ number = {3},
+ year = {2005},
+ issn = {0004-5411},
+ pages = {365--473},
+ doi = {10.1145/1066100.1066102},
+ publisher = {ACM},
+ address = {New York, NY, USA},
+ }
+
+@phdthesis{Nelson1980phd,
+ author = {Charles Gregory Nelson},
+ title = {Techniques for program verification},
+ year = {1980},
+ order_no = {AAI8011683},
+ school = {Stanford University},
+ address = {Stanford, CA, USA},
+ }
+
+@article{Woods2003short,
+    year = 2003,
+    Journal = "J. Amer. Math. Soc.",
+    volume =  16,
+    pages = "957--979",
+    month = apr,
+    title = {{Short rational generating functions for lattice point
+        problems}},
+    author = {Alexander Barvinok and Kevin Woods},
+}
+
+@misc{barvinok-0.22,
+  author = {Sven Verdoolaege},
+  title = {{\texttt{barvinok}}, version 0.22},
+  howpublished = {Available from \url{http://freshmeat.net/projects/barvinok/}},
+  year = 2006
+}
+
+@inproceedings{DeLoera2004Three,
+    title = "Three Kinds of Integer Programming Algorithms based on Barvinok's Rational Functions",
+    author = "De Loera, J. A. and D. Haws and R. Hemmecke and P. Huggins and R. Yoshida",
+    booktitle = "Integer Programming and Combinatorial Optimization: 10th International IPCO Conference",
+    year = "2004",
+    month = jan,
+    series = "Lecture Notes in Computer Science",
+    Volume = 3064,
+    Pages = "244-255",
+}
+
+@TechReport{Feautrier02,
+  author = 	 {P. Feautrier and J. Collard and C. Bastoul},
+  title = 	 {Solving systems of affine (in)equalities},
+  institution =  {PRiSM, Versailles University},
+  year = 	 2002
+}
+
+@article{ Feautrier92multi,
+    author = "Paul Feautrier",
+    title = "Some Efficient Solutions to the Affine Scheduling Problem. {P}art {II}. Multidimensional Time",
+    journal = "International Journal of Parallel Programming",
+    volume = "21",
+    number = "6",
+    pages = "389--420",
+    year = "1992",
+    month = dec,
+    url = "citeseer.nj.nec.com/article/feautrier92some.html",
+}
+
+@misc{Bygde2010licentiate,
+   author = {Stefan Bygde},
+   title = {Static {WCET} Analysis based on Abstract Interpretation and Counting of Elements},
+   month = {March},
+   year = {2010},
+   howpublished = {Licentiate thesis},
+   publisher = {M{\"{a}}lardalen University Press},
+   url = {http://www.mrtc.mdh.se/index.php?choice=publications&id=2144},
+}
+
+@phdthesis{Meister2004PhD,
+	title = {Stating and Manipulating Periodicity in the Polytope Model. Applications to Program Analysis and Optimization},
+	author= {Beno\^it Meister},
+ 	school = {Universit\'e Louis Pasteur},
+	month = Dec,
+	year  = {2004},
+}
+
+@inproceedings{Meister2008,
+  author = {Beno\^it Meister and Sven Verdoolaege},
+  title = {Polynomial Approximations in the Polytope Model: Bringing the Power
+  of Quasi-Polynomials to the Masses},
+  year = {2008},
+  booktitle = {Digest of the 6th Workshop on Optimization for DSP and Embedded Systems, ODES-6},
+  editor = "Jagadeesh Sankaran and Vander Aa, Tom",
+  month = apr,
+}
+
+@misc{Galea2009personal,
+    author = "Fran\c{c}ois Galea",
+    title = "personal communication",
+    year = 2009,
+    month = nov,
+}
+
+@misc{PPL,
+  author = "R. Bagnara and P. M. Hill and E. Zaffanella",
+  title = "The {Parma Polyhedra Library}",
+  howpublished = {\url{http://www.cs.unipr.it/ppl/}},
+}
+
+@TECHREPORT{Cook1991implementation,
+AUTHOR={William Cook and Thomas Rutherford and Herbert E. Scarf and David F. Shallcross},
+TITLE={An Implementation of the Generalized Basis Reduction Algorithm for Integer Programming},
+YEAR=1991,
+MONTH=Aug,
+INSTITUTION={Cowles Foundation, Yale University},
+TYPE={Cowles Foundation Discussion Papers},
+NOTE={available at \url{http://ideas.repec.org/p/cwl/cwldpp/990.html}},
+NUMBER={990},
+}
+
+ @article{Karr1976affine,
+author={ Michael Karr},
+title={ Affine Relationships Among Variables of a Program },
+journal={Acta Informatica},
+Volume={6},
+pages={133-151},
+year={1976},
+publisher={Springer-Verlag},
+ignore={ },
+}
+
+@PhdThesis{Verhaegh1995PhD,
+	title = "Multidimensional Periodic Scheduling",
+	author = "Wim F. J. Verhaegh",
+	school = "Technische Universiteit Eindhoven",
+	year = 1995,
+}
+
+@INPROCEEDINGS{Seghir2006minimizing,
+  AUTHOR = "Rachid Seghir and Vincent Loechner",
+  TITLE = {Memory Optimization by Counting Points in Integer Transformations of Parametric Polytopes},
+  BOOKTITLE = {{P}roceedings of the {I}nternational {C}onference on {C}ompilers, {A}rchitectures, and {S}ynthesis for {E}mbedded Systems, CASES 2006, {S}eoul, {K}orea},
+  month = oct,
+  YEAR = {2006}
+}
+
+@misc{DeSmet2010personal,
+    author = "De Smet, Sven",
+    title = "personal communication",
+    year = 2010,
+    month = apr,
+}
diff --git a/polly/lib/External/isl/doc/manual.tex b/polly/lib/External/isl/doc/manual.tex
new file mode 100644
index 00000000000..91b3db0e640
--- /dev/null
+++ b/polly/lib/External/isl/doc/manual.tex
@@ -0,0 +1,75 @@
+\documentclass{report}
+\usepackage[plainpages=false,pdfpagelabels,breaklinks,pagebackref]{hyperref}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{txfonts}
+\usepackage{chicago}
+\usepackage{aliascnt}
+\usepackage{tikz}
+\usepackage{calc}
+\usepackage[ruled]{algorithm2e}
+\usetikzlibrary{matrix,fit,backgrounds,decorations.pathmorphing,positioning}
+\usepackage{listings}
+
+\lstset{basicstyle=\tt,flexiblecolumns=false}
+
+\def\vec#1{\mathchoice{\mbox{\boldmath$\displaystyle\bf#1$}}
+{\mbox{\boldmath$\textstyle\bf#1$}}
+{\mbox{\boldmath$\scriptstyle\bf#1$}}
+{\mbox{\boldmath$\scriptscriptstyle\bf#1$}}}
+
+\providecommand{\fract}[1]{\left\{#1\right\}}
+\providecommand{\floor}[1]{\left\lfloor#1\right\rfloor}
+\providecommand{\ceil}[1]{\left\lceil#1\right\rceil}
+\def\sp#1#2{\langle #1, #2 \rangle}
+\def\spv#1#2{\langle\vec #1,\vec #2\rangle}
+
+\newtheorem{theorem}{Theorem}
+\newaliascnt{example}{theorem}
+\newtheorem{example}[example]{Example}
+\newaliascnt{def}{theorem}
+\newtheorem{definition}[def]{Definition}
+\aliascntresetthe{example}
+\aliascntresetthe{def}
+\numberwithin{theorem}{section}
+\numberwithin{def}{section}
+\numberwithin{example}{section}
+
+\newcommand{\algocflineautorefname}{Algorithm}
+\newcommand{\exampleautorefname}{Example}
+\newcommand{\lstnumberautorefname}{Line}
+\renewcommand{\sectionautorefname}{Section}
+\renewcommand{\subsectionautorefname}{Section}
+
+\def\Z{\mathbb{Z}}
+\def\Q{\mathbb{Q}}
+
+\def\pdom{\mathop{\rm pdom}\nolimits}
+\def\domain{\mathop{\rm dom}\nolimits}
+\def\range{\mathop{\rm ran}\nolimits}
+\def\identity{\mathop{\rm Id}\nolimits}
+\def\diff{\mathop{\Delta}\nolimits}
+
+\providecommand{\floor}[1]{\left\lfloor#1\right\rfloor}
+
+\begin{document}
+
+\title{Integer Set Library: Manual\\
+\small Version: \input{version} }
+\author{Sven Verdoolaege}
+
+\maketitle
+\tableofcontents
+
+\chapter{User Manual}
+
+\input{user}
+
+\chapter{Implementation Details}
+
+\input{implementation}
+
+\bibliography{isl}
+\bibliographystyle{chicago}
+
+\end{document}
diff --git a/polly/lib/External/isl/doc/mypod2latex b/polly/lib/External/isl/doc/mypod2latex
new file mode 100755
index 00000000000..b11c0593d01
--- /dev/null
+++ b/polly/lib/External/isl/doc/mypod2latex
@@ -0,0 +1,14 @@
+#!/usr/bin/perl
+
+use strict;
+use Pod::LaTeX;
+
+my ($in, $out) = @ARGV;
+
+my $parser = new Pod::LaTeX(
+		AddPreamble => 0,
+		AddPostamble => 0,
+		LevelNoNum => 5,
+	     );
+
+$parser->parse_from_file($in, $out);
diff --git a/polly/lib/External/isl/doc/user.pod b/polly/lib/External/isl/doc/user.pod
new file mode 100644
index 00000000000..f5c0c4dfe42
--- /dev/null
+++ b/polly/lib/External/isl/doc/user.pod
@@ -0,0 +1,8039 @@
+=head1 Introduction
+
+C<isl> is a thread-safe C library for manipulating
+sets and relations of integer points bounded by affine constraints.
+The descriptions of the sets and relations may involve
+both parameters and existentially quantified variables.
+All computations are performed in exact integer arithmetic
+using C<GMP> or C<imath>.
+The C<isl> library offers functionality that is similar
+to that offered by the C<Omega> and C<Omega+> libraries,
+but the underlying algorithms are in most cases completely different.
+
+The library is by no means complete and some fairly basic
+functionality is still missing.
+Still, even in its current form, the library has been successfully
+used as a backend polyhedral library for the polyhedral
+scanner C<CLooG> and as part of an equivalence checker of
+static affine programs.
+For bug reports, feature requests and questions,
+visit the discussion group at
+L<http://groups.google.com/group/isl-development>.
+
+=head2 Backward Incompatible Changes
+
+=head3 Changes since isl-0.02
+
+=over
+
+=item * The old printing functions have been deprecated
+and replaced by C<isl_printer> functions, see L<Input and Output>.
+
+=item * Most functions related to dependence analysis have acquired
+an extra C<must> argument.  To obtain the old behavior, this argument
+should be given the value 1.  See L<Dependence Analysis>.
+
+=back
+
+=head3 Changes since isl-0.03
+
+=over
+
+=item * The function C<isl_pw_qpolynomial_fold_add> has been
+renamed to C<isl_pw_qpolynomial_fold_fold>.
+Similarly, C<isl_union_pw_qpolynomial_fold_add> has been
+renamed to C<isl_union_pw_qpolynomial_fold_fold>.
+
+=back
+
+=head3 Changes since isl-0.04
+
+=over
+
+=item * All header files have been renamed from C<isl_header.h>
+to C<isl/header.h>.
+
+=back
+
+=head3 Changes since isl-0.05
+
+=over
+
+=item * The functions C<isl_printer_print_basic_set> and
+C<isl_printer_print_basic_map> no longer print a newline.
+
+=item * The functions C<isl_flow_get_no_source>
+and C<isl_union_map_compute_flow> now return
+the accesses for which no source could be found instead of
+the iterations where those accesses occur.
+
+=item * The functions C<isl_basic_map_identity> and
+C<isl_map_identity> now take a B<map> space as input.  An old call
+C<isl_map_identity(space)> can be rewritten to
+C<isl_map_identity(isl_space_map_from_set(space))>.
+
+=item * The function C<isl_map_power> no longer takes
+a parameter position as input.  Instead, the exponent
+is now expressed as the domain of the resulting relation.
+
+=back
+
+=head3 Changes since isl-0.06
+
+=over
+
+=item * The format of C<isl_printer_print_qpolynomial>'s
+C<ISL_FORMAT_ISL> output has changed.
+Use C<ISL_FORMAT_C> to obtain the old output.
+
+=item * The C<*_fast_*> functions have been renamed to C<*_plain_*>.
+Some of the old names have been kept for backward compatibility,
+but they will be removed in the future.
+
+=back
+
+=head3 Changes since isl-0.07
+
+=over
+
+=item * The function C<isl_pw_aff_max> has been renamed to
+C<isl_pw_aff_union_max>.
+Similarly, the function C<isl_pw_aff_add> has been renamed to
+C<isl_pw_aff_union_add>.
+
+=item * The C<isl_dim> type has been renamed to C<isl_space>
+along with the associated functions.
+Some of the old names have been kept for backward compatibility,
+but they will be removed in the future.
+
+=item * Spaces of maps, sets and parameter domains are now
+treated differently.  The distinction between map spaces and set spaces
+has always been made on a conceptual level, but proper use of such spaces
+was never checked.  Furthermore, up until isl-0.07 there was no way
+of explicitly creating a parameter space.  These can now be created
+directly using C<isl_space_params_alloc> or from other spaces using
+C<isl_space_params>.
+
+=item * The space in which C<isl_aff>, C<isl_pw_aff>, C<isl_qpolynomial>,
+C<isl_pw_qpolynomial>, C<isl_qpolynomial_fold> and C<isl_pw_qpolynomial_fold>
+objects live is now a map space
+instead of a set space.  This means, for example, that the dimensions
+of the domain of an C<isl_aff> are now considered to be of type
+C<isl_dim_in> instead of C<isl_dim_set>.  Extra functions have been
+added to obtain the domain space.  Some of the constructors still
+take a domain space and have therefore been renamed.
+
+=item * The functions C<isl_equality_alloc> and C<isl_inequality_alloc>
+now take an C<isl_local_space> instead of an C<isl_space>.
+An C<isl_local_space> can be created from an C<isl_space>
+using C<isl_local_space_from_space>.
+
+=item * The C<isl_div> type has been removed.  Functions that used
+to return an C<isl_div> now return an C<isl_aff>.
+Note that the space of an C<isl_aff> is that of relation.
+When replacing a call to C<isl_div_get_coefficient> by a call to
+C<isl_aff_get_coefficient> any C<isl_dim_set> argument needs
+to be replaced by C<isl_dim_in>.
+A call to C<isl_aff_from_div> can be replaced by a call
+to C<isl_aff_floor>.
+A call to C<isl_qpolynomial_div(div)> call be replaced by
+the nested call
+
+	isl_qpolynomial_from_aff(isl_aff_floor(div))
+
+The function C<isl_constraint_div> has also been renamed
+to C<isl_constraint_get_div>.
+
+=item * The C<nparam> argument has been removed from
+C<isl_map_read_from_str> and similar functions.
+When reading input in the original PolyLib format,
+the result will have no parameters.
+If parameters are expected, the caller may want to perform
+dimension manipulation on the result.
+
+=back
+
+=head3 Changes since isl-0.09
+
+=over
+
+=item * The C<schedule_split_parallel> option has been replaced
+by the C<schedule_split_scaled> option.
+
+=item * The first argument of C<isl_pw_aff_cond> is now
+an C<isl_pw_aff> instead of an C<isl_set>.
+A call C<isl_pw_aff_cond(a, b, c)> can be replaced by
+
+	isl_pw_aff_cond(isl_set_indicator_function(a), b, c)
+
+=back
+
+=head3 Changes since isl-0.10
+
+=over
+
+=item * The functions C<isl_set_dim_has_lower_bound> and
+C<isl_set_dim_has_upper_bound> have been renamed to
+C<isl_set_dim_has_any_lower_bound> and
+C<isl_set_dim_has_any_upper_bound>.
+The new C<isl_set_dim_has_lower_bound> and
+C<isl_set_dim_has_upper_bound> have slightly different meanings.
+
+=back
+
+=head3 Changes since isl-0.12
+
+=over
+
+=item * C<isl_int> has been replaced by C<isl_val>.
+Some of the old functions are still available in C<isl/deprecated/*.h>
+but they will be removed in the future.
+
+=item * The functions C<isl_pw_qpolynomial_eval>,
+C<isl_union_pw_qpolynomial_eval>, C<isl_pw_qpolynomial_fold_eval>
+and C<isl_union_pw_qpolynomial_fold_eval> have been changed to return
+an C<isl_val> instead of an C<isl_qpolynomial>.
+
+=item * The function C<isl_band_member_is_zero_distance>
+has been removed.  Essentially the same functionality is available
+through C<isl_band_member_is_coincident>, except that is requires
+setting up coincidence constraints.
+The option C<schedule_outer_zero_distance> has accordingly been
+replaced by the option C<schedule_outer_coincidence>.
+
+=item * The function C<isl_vertex_get_expr> has been changed
+to return an C<isl_multi_aff> instead of a rational C<isl_basic_set>.
+The function C<isl_vertex_get_domain> has been changed to return
+a regular basic set, rather than a rational basic set.
+
+=back
+
+=head3 Changes since isl-0.14
+
+=over
+
+=item * Objects of type C<isl_union_pw_multi_aff> can no longer contain
+two or more C<isl_pw_multi_aff> objects with the same domain space.
+
+=item * The function C<isl_union_pw_multi_aff_add> now consistently
+computes the sum on the shared definition domain.
+The function C<isl_union_pw_multi_aff_union_add> has been added
+to compute the sum on the union of definition domains.
+The original behavior of C<isl_union_pw_multi_aff_add> was
+confused and is no longer available.
+
+=back
+
+=head1 License
+
+C<isl> is released under the MIT license.
+
+=over
+
+Permission is hereby granted, free of charge, to any person obtaining a copy of
+this software and associated documentation files (the "Software"), to deal in
+the Software without restriction, including without limitation the rights to
+use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies
+of the Software, and to permit persons to whom the Software is furnished to do
+so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.
+
+=back
+
+Note that by default C<isl> requires C<GMP>, which is released
+under the GNU Lesser General Public License (LGPL).  This means
+that code linked against C<isl> is also linked against LGPL code.
+
+When configuring with C<--with-int=imath>, C<isl> will link against C<imath>, a
+library for exact integer arithmetic released under the MIT license.
+
+=head1 Installation
+
+The source of C<isl> can be obtained either as a tarball
+or from the git repository.  Both are available from
+L<http://freshmeat.net/projects/isl/>.
+The installation process depends on how you obtained
+the source.
+
+=head2 Installation from the git repository
+
+=over
+
+=item 1 Clone or update the repository
+
+The first time the source is obtained, you need to clone
+the repository.
+
+	git clone git://repo.or.cz/isl.git
+
+To obtain updates, you need to pull in the latest changes
+
+	git pull
+
+=item 2 Optionally get C<imath> submodule
+
+To build C<isl> with C<imath>, you need to obtain the C<imath>
+submodule by running in the git source tree of C<isl>
+
+       git submodule init
+       git submodule update
+
+This will fetch the required version of C<imath> in a subdirectory of C<isl>.
+
+=item 2 Generate C<configure>
+
+	./autogen.sh
+
+=back
+
+After performing the above steps, continue
+with the L<Common installation instructions>.
+
+=head2 Common installation instructions
+
+=over
+
+=item 1 Obtain C<GMP>
+
+By default, building C<isl> requires C<GMP>, including its headers files.
+Your distribution may not provide these header files by default
+and you may need to install a package called C<gmp-devel> or something
+similar.  Alternatively, C<GMP> can be built from
+source, available from L<http://gmplib.org/>.
+C<GMP> is not needed if you build C<isl> with C<imath>.
+
+=item 2 Configure
+
+C<isl> uses the standard C<autoconf> C<configure> script.
+To run it, just type
+
+	./configure
+
+optionally followed by some configure options.
+A complete list of options can be obtained by running
+
+	./configure --help
+
+Below we discuss some of the more common options.
+
+=over
+
+=item C<--prefix>
+
+Installation prefix for C<isl>
+
+=item C<--with-int=[gmp|imath]>
+
+Select the integer library to be used by C<isl>, the default is C<gmp>.
+Note that C<isl> may run significantly slower if you use C<imath>.
+
+=item C<--with-gmp-prefix>
+
+Installation prefix for C<GMP> (architecture-independent files).
+
+=item C<--with-gmp-exec-prefix>
+
+Installation prefix for C<GMP> (architecture-dependent files).
+
+=back
+
+=item 3 Compile
+
+	make
+
+=item 4 Install (optional)
+
+	make install
+
+=back
+
+=head1 Integer Set Library
+
+=head2 Memory Management
+
+Since a high-level operation on isl objects usually involves
+several substeps and since the user is usually not interested in
+the intermediate results, most functions that return a new object
+will also release all the objects passed as arguments.
+If the user still wants to use one or more of these arguments
+after the function call, she should pass along a copy of the
+object rather than the object itself.
+The user is then responsible for making sure that the original
+object gets used somewhere else or is explicitly freed.
+
+The arguments and return values of all documented functions are
+annotated to make clear which arguments are released and which
+arguments are preserved.  In particular, the following annotations
+are used
+
+=over
+
+=item C<__isl_give>
+
+C<__isl_give> means that a new object is returned.
+The user should make sure that the returned pointer is
+used exactly once as a value for an C<__isl_take> argument.
+In between, it can be used as a value for as many
+C<__isl_keep> arguments as the user likes.
+There is one exception, and that is the case where the
+pointer returned is C<NULL>.  Is this case, the user
+is free to use it as an C<__isl_take> argument or not.
+When applied to a C<char *>, the returned pointer needs to be
+freed using C<free>.
+
+=item C<__isl_null>
+
+C<__isl_null> means that a C<NULL> value is returned.
+
+=item C<__isl_take>
+
+C<__isl_take> means that the object the argument points to
+is taken over by the function and may no longer be used
+by the user as an argument to any other function.
+The pointer value must be one returned by a function
+returning an C<__isl_give> pointer.
+If the user passes in a C<NULL> value, then this will
+be treated as an error in the sense that the function will
+not perform its usual operation.  However, it will still
+make sure that all the other C<__isl_take> arguments
+are released.
+
+=item C<__isl_keep>
+
+C<__isl_keep> means that the function will only use the object
+temporarily.  After the function has finished, the user
+can still use it as an argument to other functions.
+A C<NULL> value will be treated in the same way as
+a C<NULL> value for an C<__isl_take> argument.
+This annotation may also be used on return values of
+type C<const char *>, in which case the returned pointer should
+not be freed by the user and is only valid until the object
+from which it was derived is updated or freed.
+
+=back
+
+=head2 Initialization
+
+All manipulations of integer sets and relations occur within
+the context of an C<isl_ctx>.
+A given C<isl_ctx> can only be used within a single thread.
+All arguments of a function are required to have been allocated
+within the same context.
+There are currently no functions available for moving an object
+from one C<isl_ctx> to another C<isl_ctx>.  This means that
+there is currently no way of safely moving an object from one
+thread to another, unless the whole C<isl_ctx> is moved.
+
+An C<isl_ctx> can be allocated using C<isl_ctx_alloc> and
+freed using C<isl_ctx_free>.
+All objects allocated within an C<isl_ctx> should be freed
+before the C<isl_ctx> itself is freed.
+
+	isl_ctx *isl_ctx_alloc();
+	void isl_ctx_free(isl_ctx *ctx);
+
+The user can impose a bound on the number of low-level I<operations>
+that can be performed by an C<isl_ctx>.  This bound can be set and
+retrieved using the following functions.  A bound of zero means that
+no bound is imposed.  The number of operations performed can be
+reset using C<isl_ctx_reset_operations>.  Note that the number
+of low-level operations needed to perform a high-level computation
+may differ significantly across different versions
+of C<isl>, but it should be the same across different platforms
+for the same version of C<isl>.
+
+Warning: This feature is experimental.  C<isl> has good support to abort and
+bail out during the computation, but this feature may exercise error code paths
+that are normally not used that much. Consequently, it is not unlikely that
+hidden bugs will be exposed.
+
+	void isl_ctx_set_max_operations(isl_ctx *ctx,
+		unsigned long max_operations);
+	unsigned long isl_ctx_get_max_operations(isl_ctx *ctx);
+	void isl_ctx_reset_operations(isl_ctx *ctx);
+
+In order to be able to create an object in the same context
+as another object, most object types (described later in
+this document) provide a function to obtain the context
+in which the object was created.
+
+	#include <isl/val.h>
+	isl_ctx *isl_val_get_ctx(__isl_keep isl_val *val);
+	isl_ctx *isl_multi_val_get_ctx(
+		__isl_keep isl_multi_val *mv);
+
+	#include <isl/id.h>
+	isl_ctx *isl_id_get_ctx(__isl_keep isl_id *id);
+
+	#include <isl/local_space.h>
+	isl_ctx *isl_local_space_get_ctx(
+		__isl_keep isl_local_space *ls);
+
+	#include <isl/set.h>
+	isl_ctx *isl_set_list_get_ctx(
+		__isl_keep isl_set_list *list);
+
+	#include <isl/aff.h>
+	isl_ctx *isl_aff_get_ctx(__isl_keep isl_aff *aff);
+	isl_ctx *isl_multi_aff_get_ctx(
+		__isl_keep isl_multi_aff *maff);
+	isl_ctx *isl_pw_aff_get_ctx(__isl_keep isl_pw_aff *pa);
+	isl_ctx *isl_pw_multi_aff_get_ctx(
+		__isl_keep isl_pw_multi_aff *pma);
+	isl_ctx *isl_multi_pw_aff_get_ctx(
+		__isl_keep isl_multi_pw_aff *mpa);
+	isl_ctx *isl_union_pw_multi_aff_get_ctx(
+		__isl_keep isl_union_pw_multi_aff *upma);
+
+	#include <isl/id_to_ast_expr.h>
+	isl_ctx *isl_id_to_ast_expr_get_ctx(
+		__isl_keep id_to_ast_expr *id2expr);
+
+	#include <isl/point.h>
+	isl_ctx *isl_point_get_ctx(__isl_keep isl_point *pnt);
+
+	#include <isl/vec.h>
+	isl_ctx *isl_vec_get_ctx(__isl_keep isl_vec *vec);
+
+	#include <isl/mat.h>
+	isl_ctx *isl_mat_get_ctx(__isl_keep isl_mat *mat);
+
+	#include <isl/vertices.h>
+	isl_ctx *isl_vertices_get_ctx(
+		__isl_keep isl_vertices *vertices);
+	isl_ctx *isl_vertex_get_ctx(__isl_keep isl_vertex *vertex);
+	isl_ctx *isl_cell_get_ctx(__isl_keep isl_cell *cell);
+
+	#include <isl/flow.h>
+	isl_ctx *isl_restriction_get_ctx(
+		__isl_keep isl_restriction *restr);
+
+	#include <isl/schedule.h>
+	isl_ctx *isl_schedule_constraints_get_ctx(
+		__isl_keep isl_schedule_constraints *sc);
+
+	#include <isl/band.h>
+	isl_ctx *isl_band_get_ctx(__isl_keep isl_band *band);
+
+	#include <isl/ast_build.h>
+	isl_ctx *isl_ast_build_get_ctx(
+		__isl_keep isl_ast_build *build);
+
+	#include <isl/ast.h>
+	isl_ctx *isl_ast_expr_get_ctx(
+		__isl_keep isl_ast_expr *expr);
+	isl_ctx *isl_ast_node_get_ctx(
+		__isl_keep isl_ast_node *node);
+
+=head2 Values
+
+An C<isl_val> represents an integer value, a rational value
+or one of three special values, infinity, negative infinity and NaN.
+Some predefined values can be created using the following functions.
+
+	#include <isl/val.h>
+	__isl_give isl_val *isl_val_zero(isl_ctx *ctx);
+	__isl_give isl_val *isl_val_one(isl_ctx *ctx);
+	__isl_give isl_val *isl_val_negone(isl_ctx *ctx);
+	__isl_give isl_val *isl_val_nan(isl_ctx *ctx);
+	__isl_give isl_val *isl_val_infty(isl_ctx *ctx);
+	__isl_give isl_val *isl_val_neginfty(isl_ctx *ctx);
+
+Specific integer values can be created using the following functions.
+
+	#include <isl/val.h>
+	__isl_give isl_val *isl_val_int_from_si(isl_ctx *ctx,
+		long i);
+	__isl_give isl_val *isl_val_int_from_ui(isl_ctx *ctx,
+		unsigned long u);
+	__isl_give isl_val *isl_val_int_from_chunks(isl_ctx *ctx,
+		size_t n, size_t size, const void *chunks);
+
+The function C<isl_val_int_from_chunks> constructs an C<isl_val>
+from the C<n> I<digits>, each consisting of C<size> bytes, stored at C<chunks>.
+The least significant digit is assumed to be stored first.
+
+Value objects can be copied and freed using the following functions.
+
+	#include <isl/val.h>
+	__isl_give isl_val *isl_val_copy(__isl_keep isl_val *v);
+	__isl_null isl_val *isl_val_free(__isl_take isl_val *v);
+
+They can be inspected using the following functions.
+
+	#include <isl/val.h>
+	long isl_val_get_num_si(__isl_keep isl_val *v);
+	long isl_val_get_den_si(__isl_keep isl_val *v);
+	double isl_val_get_d(__isl_keep isl_val *v);
+	size_t isl_val_n_abs_num_chunks(__isl_keep isl_val *v,
+		size_t size);
+	int isl_val_get_abs_num_chunks(__isl_keep isl_val *v,
+		size_t size, void *chunks);
+
+C<isl_val_n_abs_num_chunks> returns the number of I<digits>
+of C<size> bytes needed to store the absolute value of the
+numerator of C<v>.
+C<isl_val_get_abs_num_chunks> stores these digits at C<chunks>,
+which is assumed to have been preallocated by the caller.
+The least significant digit is stored first.
+Note that C<isl_val_get_num_si>, C<isl_val_get_den_si>,
+C<isl_val_get_d>, C<isl_val_n_abs_num_chunks>
+and C<isl_val_get_abs_num_chunks> can only be applied to rational values.
+
+An C<isl_val> can be modified using the following function.
+
+	#include <isl/val.h>
+	__isl_give isl_val *isl_val_set_si(__isl_take isl_val *v,
+		long i);
+
+The following unary properties are defined on C<isl_val>s.
+
+	#include <isl/val.h>
+	int isl_val_sgn(__isl_keep isl_val *v);
+	int isl_val_is_zero(__isl_keep isl_val *v);
+	int isl_val_is_one(__isl_keep isl_val *v);
+	int isl_val_is_negone(__isl_keep isl_val *v);
+	int isl_val_is_nonneg(__isl_keep isl_val *v);
+	int isl_val_is_nonpos(__isl_keep isl_val *v);
+	int isl_val_is_pos(__isl_keep isl_val *v);
+	int isl_val_is_neg(__isl_keep isl_val *v);
+	int isl_val_is_int(__isl_keep isl_val *v);
+	int isl_val_is_rat(__isl_keep isl_val *v);
+	int isl_val_is_nan(__isl_keep isl_val *v);
+	int isl_val_is_infty(__isl_keep isl_val *v);
+	int isl_val_is_neginfty(__isl_keep isl_val *v);
+
+Note that the sign of NaN is undefined.
+
+The following binary properties are defined on pairs of C<isl_val>s.
+
+	#include <isl/val.h>
+	int isl_val_lt(__isl_keep isl_val *v1,
+		__isl_keep isl_val *v2);
+	int isl_val_le(__isl_keep isl_val *v1,
+		__isl_keep isl_val *v2);
+	int isl_val_gt(__isl_keep isl_val *v1,
+		__isl_keep isl_val *v2);
+	int isl_val_ge(__isl_keep isl_val *v1,
+		__isl_keep isl_val *v2);
+	int isl_val_eq(__isl_keep isl_val *v1,
+		__isl_keep isl_val *v2);
+	int isl_val_ne(__isl_keep isl_val *v1,
+		__isl_keep isl_val *v2);
+	int isl_val_abs_eq(__isl_keep isl_val *v1,
+		__isl_keep isl_val *v2);
+
+The function C<isl_val_abs_eq> checks whether its two arguments
+are equal in absolute value.
+
+For integer C<isl_val>s we additionally have the following binary property.
+
+	#include <isl/val.h>
+	int isl_val_is_divisible_by(__isl_keep isl_val *v1,
+		__isl_keep isl_val *v2);
+
+An C<isl_val> can also be compared to an integer using the following
+function.  The result is undefined for NaN.
+
+	#include <isl/val.h>
+	int isl_val_cmp_si(__isl_keep isl_val *v, long i);
+
+The following unary operations are available on C<isl_val>s.
+
+	#include <isl/val.h>
+	__isl_give isl_val *isl_val_abs(__isl_take isl_val *v);
+	__isl_give isl_val *isl_val_neg(__isl_take isl_val *v);
+	__isl_give isl_val *isl_val_floor(__isl_take isl_val *v);
+	__isl_give isl_val *isl_val_ceil(__isl_take isl_val *v);
+	__isl_give isl_val *isl_val_trunc(__isl_take isl_val *v);
+	__isl_give isl_val *isl_val_inv(__isl_take isl_val *v);
+	__isl_give isl_val *isl_val_2exp(__isl_take isl_val *v);
+
+The following binary operations are available on C<isl_val>s.
+
+	#include <isl/val.h>
+	__isl_give isl_val *isl_val_min(__isl_take isl_val *v1,
+		__isl_take isl_val *v2);
+	__isl_give isl_val *isl_val_max(__isl_take isl_val *v1,
+		__isl_take isl_val *v2);
+	__isl_give isl_val *isl_val_add(__isl_take isl_val *v1,
+		__isl_take isl_val *v2);
+	__isl_give isl_val *isl_val_add_ui(__isl_take isl_val *v1,
+		unsigned long v2);
+	__isl_give isl_val *isl_val_sub(__isl_take isl_val *v1,
+		__isl_take isl_val *v2);
+	__isl_give isl_val *isl_val_sub_ui(__isl_take isl_val *v1,
+		unsigned long v2);
+	__isl_give isl_val *isl_val_mul(__isl_take isl_val *v1,
+		__isl_take isl_val *v2);
+	__isl_give isl_val *isl_val_mul_ui(__isl_take isl_val *v1,
+		unsigned long v2);
+	__isl_give isl_val *isl_val_div(__isl_take isl_val *v1,
+		__isl_take isl_val *v2);
+
+On integer values, we additionally have the following operations.
+
+	#include <isl/val.h>
+	__isl_give isl_val *isl_val_2exp(__isl_take isl_val *v);
+	__isl_give isl_val *isl_val_mod(__isl_take isl_val *v1,
+		__isl_take isl_val *v2);
+	__isl_give isl_val *isl_val_gcd(__isl_take isl_val *v1,
+		__isl_take isl_val *v2);
+	__isl_give isl_val *isl_val_gcdext(__isl_take isl_val *v1,
+		__isl_take isl_val *v2, __isl_give isl_val **x,
+		__isl_give isl_val **y);
+
+The function C<isl_val_gcdext> returns the greatest common divisor g
+of C<v1> and C<v2> as well as two integers C<*x> and C<*y> such
+that C<*x> * C<v1> + C<*y> * C<v2> = g.
+
+=head3 GMP specific functions
+
+These functions are only available if C<isl> has been compiled with C<GMP>
+support.
+
+Specific integer and rational values can be created from C<GMP> values using
+the following functions.
+
+	#include <isl/val_gmp.h>
+	__isl_give isl_val *isl_val_int_from_gmp(isl_ctx *ctx,
+		mpz_t z);
+	__isl_give isl_val *isl_val_from_gmp(isl_ctx *ctx,
+		const mpz_t n, const mpz_t d);
+
+The numerator and denominator of a rational value can be extracted as
+C<GMP> values using the following functions.
+
+	#include <isl/val_gmp.h>
+	int isl_val_get_num_gmp(__isl_keep isl_val *v, mpz_t z);
+	int isl_val_get_den_gmp(__isl_keep isl_val *v, mpz_t z);
+
+=head2 Sets and Relations
+
+C<isl> uses six types of objects for representing sets and relations,
+C<isl_basic_set>, C<isl_basic_map>, C<isl_set>, C<isl_map>,
+C<isl_union_set> and C<isl_union_map>.
+C<isl_basic_set> and C<isl_basic_map> represent sets and relations that
+can be described as a conjunction of affine constraints, while
+C<isl_set> and C<isl_map> represent unions of
+C<isl_basic_set>s and C<isl_basic_map>s, respectively.
+However, all C<isl_basic_set>s or C<isl_basic_map>s in the union need
+to live in the same space.  C<isl_union_set>s and C<isl_union_map>s
+represent unions of C<isl_set>s or C<isl_map>s in I<different> spaces,
+where spaces are considered different if they have a different number
+of dimensions and/or different names (see L<"Spaces">).
+The difference between sets and relations (maps) is that sets have
+one set of variables, while relations have two sets of variables,
+input variables and output variables.
+
+=head2 Error Handling
+
+C<isl> supports different ways to react in case a runtime error is triggered.
+Runtime errors arise, e.g., if a function such as C<isl_map_intersect> is called
+with two maps that have incompatible spaces. There are three possible ways
+to react on error: to warn, to continue or to abort.
+
+The default behavior is to warn. In this mode, C<isl> prints a warning, stores
+the last error in the corresponding C<isl_ctx> and the function in which the
+error was triggered returns C<NULL>. An error does not corrupt internal state,
+such that isl can continue to be used. C<isl> also provides functions to
+read the last error and to reset the memory that stores the last error. The
+last error is only stored for information purposes. Its presence does not
+change the behavior of C<isl>. Hence, resetting an error is not required to
+continue to use isl, but only to observe new errors.
+
+	#include <isl/ctx.h>
+	enum isl_error isl_ctx_last_error(isl_ctx *ctx);
+	void isl_ctx_reset_error(isl_ctx *ctx);
+
+Another option is to continue on error. This is similar to warn on error mode,
+except that C<isl> does not print any warning. This allows a program to
+implement its own error reporting.
+
+The last option is to directly abort the execution of the program from within
+the isl library. This makes it obviously impossible to recover from an error,
+but it allows to directly spot the error location. By aborting on error,
+debuggers break at the location the error occurred and can provide a stack
+trace. Other tools that automatically provide stack traces on abort or that do
+not want to continue execution after an error was triggered may also prefer to
+abort on error.
+
+The on error behavior of isl can be specified by calling
+C<isl_options_set_on_error> or by setting the command line option
+C<--isl-on-error>. Valid arguments for the function call are
+C<ISL_ON_ERROR_WARN>, C<ISL_ON_ERROR_CONTINUE> and C<ISL_ON_ERROR_ABORT>. The
+choices for the command line option are C<warn>, C<continue> and C<abort>.
+It is also possible to query the current error mode.
+
+	#include <isl/options.h>
+	int isl_options_set_on_error(isl_ctx *ctx, int val);
+	int isl_options_get_on_error(isl_ctx *ctx);
+
+=head2 Identifiers
+
+Identifiers are used to identify both individual dimensions
+and tuples of dimensions.  They consist of an optional name and an optional
+user pointer.  The name and the user pointer cannot both be C<NULL>, however.
+Identifiers with the same name but different pointer values
+are considered to be distinct.
+Similarly, identifiers with different names but the same pointer value
+are also considered to be distinct.
+Equal identifiers are represented using the same object.
+Pairs of identifiers can therefore be tested for equality using the
+C<==> operator.
+Identifiers can be constructed, copied, freed, inspected and printed
+using the following functions.
+
+	#include <isl/id.h>
+	__isl_give isl_id *isl_id_alloc(isl_ctx *ctx,
+		__isl_keep const char *name, void *user);
+	__isl_give isl_id *isl_id_set_free_user(
+		__isl_take isl_id *id,
+		__isl_give void (*free_user)(void *user));
+	__isl_give isl_id *isl_id_copy(isl_id *id);
+	__isl_null isl_id *isl_id_free(__isl_take isl_id *id);
+
+	void *isl_id_get_user(__isl_keep isl_id *id);
+	__isl_keep const char *isl_id_get_name(__isl_keep isl_id *id);
+
+	__isl_give isl_printer *isl_printer_print_id(
+		__isl_take isl_printer *p, __isl_keep isl_id *id);
+
+The callback set by C<isl_id_set_free_user> is called on the user
+pointer when the last reference to the C<isl_id> is freed.
+Note that C<isl_id_get_name> returns a pointer to some internal
+data structure, so the result can only be used while the
+corresponding C<isl_id> is alive.
+
+=head2 Spaces
+
+Whenever a new set, relation or similar object is created from scratch,
+the space in which it lives needs to be specified using an C<isl_space>.
+Each space involves zero or more parameters and zero, one or two
+tuples of set or input/output dimensions.  The parameters and dimensions
+are identified by an C<isl_dim_type> and a position.
+The type C<isl_dim_param> refers to parameters,
+the type C<isl_dim_set> refers to set dimensions (for spaces
+with a single tuple of dimensions) and the types C<isl_dim_in>
+and C<isl_dim_out> refer to input and output dimensions
+(for spaces with two tuples of dimensions).
+Local spaces (see L</"Local Spaces">) also contain dimensions
+of type C<isl_dim_div>.
+Note that parameters are only identified by their position within
+a given object.  Across different objects, parameters are (usually)
+identified by their names or identifiers.  Only unnamed parameters
+are identified by their positions across objects.  The use of unnamed
+parameters is discouraged.
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_alloc(isl_ctx *ctx,
+		unsigned nparam, unsigned n_in, unsigned n_out);
+	__isl_give isl_space *isl_space_params_alloc(isl_ctx *ctx,
+		unsigned nparam);
+	__isl_give isl_space *isl_space_set_alloc(isl_ctx *ctx,
+		unsigned nparam, unsigned dim);
+	__isl_give isl_space *isl_space_copy(__isl_keep isl_space *space);
+	__isl_null isl_space *isl_space_free(__isl_take isl_space *space);
+
+The space used for creating a parameter domain
+needs to be created using C<isl_space_params_alloc>.
+For other sets, the space
+needs to be created using C<isl_space_set_alloc>, while
+for a relation, the space
+needs to be created using C<isl_space_alloc>.
+
+To check whether a given space is that of a set or a map
+or whether it is a parameter space, use these functions:
+
+	#include <isl/space.h>
+	int isl_space_is_params(__isl_keep isl_space *space);
+	int isl_space_is_set(__isl_keep isl_space *space);
+	int isl_space_is_map(__isl_keep isl_space *space);
+
+Spaces can be compared using the following functions:
+
+	#include <isl/space.h>
+	int isl_space_is_equal(__isl_keep isl_space *space1,
+		__isl_keep isl_space *space2);
+	int isl_space_is_domain(__isl_keep isl_space *space1,
+		__isl_keep isl_space *space2);
+	int isl_space_is_range(__isl_keep isl_space *space1,
+		__isl_keep isl_space *space2);
+	int isl_space_tuple_is_equal(
+		__isl_keep isl_space *space1,
+		enum isl_dim_type type1,
+		__isl_keep isl_space *space2,
+		enum isl_dim_type type2);
+
+C<isl_space_is_domain> checks whether the first argument is equal
+to the domain of the second argument.  This requires in particular that
+the first argument is a set space and that the second argument
+is a map space.  C<isl_space_tuple_is_equal> checks whether the given
+tuples (C<isl_dim_in>, C<isl_dim_out> or C<isl_dim_set>) of the given
+spaces are the same.  That is, it checks if they have the same
+identifier (if any), the same dimension and the same internal structure
+(if any).
+
+It is often useful to create objects that live in the
+same space as some other object.  This can be accomplished
+by creating the new objects
+(see L</"Creating New Sets and Relations"> or
+L</"Functions">) based on the space
+of the original object.
+
+	#include <isl/set.h>
+	__isl_give isl_space *isl_basic_set_get_space(
+		__isl_keep isl_basic_set *bset);
+	__isl_give isl_space *isl_set_get_space(__isl_keep isl_set *set);
+
+	#include <isl/union_set.h>
+	__isl_give isl_space *isl_union_set_get_space(
+		__isl_keep isl_union_set *uset);
+
+	#include <isl/map.h>
+	__isl_give isl_space *isl_basic_map_get_space(
+		__isl_keep isl_basic_map *bmap);
+	__isl_give isl_space *isl_map_get_space(__isl_keep isl_map *map);
+
+	#include <isl/union_map.h>
+	__isl_give isl_space *isl_union_map_get_space(
+		__isl_keep isl_union_map *umap);
+
+	#include <isl/constraint.h>
+	__isl_give isl_space *isl_constraint_get_space(
+		__isl_keep isl_constraint *constraint);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_space *isl_qpolynomial_get_domain_space(
+		__isl_keep isl_qpolynomial *qp);
+	__isl_give isl_space *isl_qpolynomial_get_space(
+		__isl_keep isl_qpolynomial *qp);
+	__isl_give isl_space *isl_qpolynomial_fold_get_space(
+		__isl_keep isl_qpolynomial_fold *fold);
+	__isl_give isl_space *isl_pw_qpolynomial_get_domain_space(
+		__isl_keep isl_pw_qpolynomial *pwqp);
+	__isl_give isl_space *isl_pw_qpolynomial_get_space(
+		__isl_keep isl_pw_qpolynomial *pwqp);
+	__isl_give isl_space *isl_pw_qpolynomial_fold_get_domain_space(
+		__isl_keep isl_pw_qpolynomial_fold *pwf);
+	__isl_give isl_space *isl_pw_qpolynomial_fold_get_space(
+		__isl_keep isl_pw_qpolynomial_fold *pwf);
+	__isl_give isl_space *isl_union_pw_qpolynomial_get_space(
+		__isl_keep isl_union_pw_qpolynomial *upwqp);
+	__isl_give isl_space *isl_union_pw_qpolynomial_fold_get_space(
+		__isl_keep isl_union_pw_qpolynomial_fold *upwf);
+
+	#include <isl/val.h>
+	__isl_give isl_space *isl_multi_val_get_space(
+		__isl_keep isl_multi_val *mv);
+
+	#include <isl/aff.h>
+	__isl_give isl_space *isl_aff_get_domain_space(
+		__isl_keep isl_aff *aff);
+	__isl_give isl_space *isl_aff_get_space(
+		__isl_keep isl_aff *aff);
+	__isl_give isl_space *isl_pw_aff_get_domain_space(
+		__isl_keep isl_pw_aff *pwaff);
+	__isl_give isl_space *isl_pw_aff_get_space(
+		__isl_keep isl_pw_aff *pwaff);
+	__isl_give isl_space *isl_multi_aff_get_domain_space(
+		__isl_keep isl_multi_aff *maff);
+	__isl_give isl_space *isl_multi_aff_get_space(
+		__isl_keep isl_multi_aff *maff);
+	__isl_give isl_space *isl_pw_multi_aff_get_domain_space(
+		__isl_keep isl_pw_multi_aff *pma);
+	__isl_give isl_space *isl_pw_multi_aff_get_space(
+		__isl_keep isl_pw_multi_aff *pma);
+	__isl_give isl_space *isl_union_pw_multi_aff_get_space(
+		__isl_keep isl_union_pw_multi_aff *upma);
+	__isl_give isl_space *isl_multi_pw_aff_get_domain_space(
+		__isl_keep isl_multi_pw_aff *mpa);
+	__isl_give isl_space *isl_multi_pw_aff_get_space(
+		__isl_keep isl_multi_pw_aff *mpa);
+
+	#include <isl/point.h>
+	__isl_give isl_space *isl_point_get_space(
+		__isl_keep isl_point *pnt);
+
+The number of dimensions of a given type of space
+may be read off from a space or an object that lives
+in a space using the following functions.
+In case of C<isl_space_dim>, type may be
+C<isl_dim_param>, C<isl_dim_in> (only for relations),
+C<isl_dim_out> (only for relations), C<isl_dim_set>
+(only for sets) or C<isl_dim_all>.
+
+	#include <isl/space.h>
+	unsigned isl_space_dim(__isl_keep isl_space *space,
+		enum isl_dim_type type);
+
+	#include <isl/local_space.h>
+	int isl_local_space_dim(__isl_keep isl_local_space *ls,
+		enum isl_dim_type type);
+
+	#include <isl/set.h>
+	unsigned isl_basic_set_dim(__isl_keep isl_basic_set *bset,
+		enum isl_dim_type type);
+	unsigned isl_set_dim(__isl_keep isl_set *set,
+		enum isl_dim_type type);
+
+	#include <isl/union_set.h>
+	unsigned isl_union_set_dim(__isl_keep isl_union_set *uset,
+		enum isl_dim_type type);
+
+	#include <isl/map.h>
+	unsigned isl_basic_map_dim(__isl_keep isl_basic_map *bmap,
+		enum isl_dim_type type);
+	unsigned isl_map_dim(__isl_keep isl_map *map,
+		enum isl_dim_type type);
+
+	#include <isl/union_map.h>
+	unsigned isl_union_map_dim(__isl_keep isl_union_map *umap,
+		enum isl_dim_type type);
+
+	#include <isl/val.h>
+	unsigned isl_multi_val_dim(__isl_keep isl_multi_val *mv,
+		enum isl_dim_type type);
+
+	#include <isl/aff.h>
+	int isl_aff_dim(__isl_keep isl_aff *aff,
+		enum isl_dim_type type);
+	unsigned isl_multi_aff_dim(__isl_keep isl_multi_aff *maff,
+		enum isl_dim_type type);
+	unsigned isl_pw_aff_dim(__isl_keep isl_pw_aff *pwaff,
+		enum isl_dim_type type);
+	unsigned isl_pw_multi_aff_dim(
+		__isl_keep isl_pw_multi_aff *pma,
+		enum isl_dim_type type);
+	unsigned isl_multi_pw_aff_dim(
+		__isl_keep isl_multi_pw_aff *mpa,
+		enum isl_dim_type type);
+	unsigned isl_union_pw_multi_aff_dim(
+		__isl_keep isl_union_pw_multi_aff *upma,
+		enum isl_dim_type type);
+
+	#include <isl/polynomial.h>
+	unsigned isl_union_pw_qpolynomial_dim(
+		__isl_keep isl_union_pw_qpolynomial *upwqp,
+		enum isl_dim_type type);
+	unsigned isl_union_pw_qpolynomial_fold_dim(
+		__isl_keep isl_union_pw_qpolynomial_fold *upwf,
+		enum isl_dim_type type);
+
+Note that an C<isl_union_set>, an C<isl_union_map>,
+an C<isl_union_pw_multi_aff>,
+an C<isl_union_pw_qpolynomial> and
+an C<isl_union_pw_qpolynomial_fold>
+only have parameters.
+
+The identifiers or names of the individual dimensions of spaces
+may be set or read off using the following functions on spaces
+or objects that live in spaces.
+These functions are mostly useful to obtain the identifiers, positions
+or names of the parameters.  Identifiers of individual dimensions are
+essentially only useful for printing.  They are ignored by all other
+operations and may not be preserved across those operations.
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_set_dim_id(
+		__isl_take isl_space *space,
+		enum isl_dim_type type, unsigned pos,
+		__isl_take isl_id *id);
+	int isl_space_has_dim_id(__isl_keep isl_space *space,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_id *isl_space_get_dim_id(
+		__isl_keep isl_space *space,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_space *isl_space_set_dim_name(
+		__isl_take isl_space *space,
+		 enum isl_dim_type type, unsigned pos,
+		 __isl_keep const char *name);
+	int isl_space_has_dim_name(__isl_keep isl_space *space,
+		enum isl_dim_type type, unsigned pos);
+	__isl_keep const char *isl_space_get_dim_name(
+		__isl_keep isl_space *space,
+		enum isl_dim_type type, unsigned pos);
+
+	#include <isl/local_space.h>
+	__isl_give isl_local_space *isl_local_space_set_dim_id(
+		__isl_take isl_local_space *ls,
+		enum isl_dim_type type, unsigned pos,
+		__isl_take isl_id *id);
+	int isl_local_space_has_dim_id(
+		__isl_keep isl_local_space *ls,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_id *isl_local_space_get_dim_id(
+		__isl_keep isl_local_space *ls,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_local_space *isl_local_space_set_dim_name(
+		__isl_take isl_local_space *ls,
+		enum isl_dim_type type, unsigned pos, const char *s);
+	int isl_local_space_has_dim_name(
+		__isl_keep isl_local_space *ls,
+		enum isl_dim_type type, unsigned pos)
+	const char *isl_local_space_get_dim_name(
+		__isl_keep isl_local_space *ls,
+		enum isl_dim_type type, unsigned pos);
+
+	#include <isl/constraint.h>
+	const char *isl_constraint_get_dim_name(
+		__isl_keep isl_constraint *constraint,
+		enum isl_dim_type type, unsigned pos);
+
+	#include <isl/set.h>
+	__isl_give isl_id *isl_basic_set_get_dim_id(
+		__isl_keep isl_basic_set *bset,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_set *isl_set_set_dim_id(
+		__isl_take isl_set *set, enum isl_dim_type type,
+		unsigned pos, __isl_take isl_id *id);
+	int isl_set_has_dim_id(__isl_keep isl_set *set,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_id *isl_set_get_dim_id(
+		__isl_keep isl_set *set, enum isl_dim_type type,
+		unsigned pos);
+	const char *isl_basic_set_get_dim_name(
+		__isl_keep isl_basic_set *bset,
+		enum isl_dim_type type, unsigned pos);
+	int isl_set_has_dim_name(__isl_keep isl_set *set,
+		enum isl_dim_type type, unsigned pos);
+	const char *isl_set_get_dim_name(
+		__isl_keep isl_set *set,
+		enum isl_dim_type type, unsigned pos);
+
+	#include <isl/map.h>
+	__isl_give isl_map *isl_map_set_dim_id(
+		__isl_take isl_map *map, enum isl_dim_type type,
+		unsigned pos, __isl_take isl_id *id);
+	int isl_basic_map_has_dim_id(
+		__isl_keep isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned pos);
+	int isl_map_has_dim_id(__isl_keep isl_map *map,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_id *isl_map_get_dim_id(
+		__isl_keep isl_map *map, enum isl_dim_type type,
+		unsigned pos);
+	__isl_give isl_id *isl_union_map_get_dim_id(
+		__isl_keep isl_union_map *umap,
+		enum isl_dim_type type, unsigned pos);
+	const char *isl_basic_map_get_dim_name(
+		__isl_keep isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned pos);
+	int isl_map_has_dim_name(__isl_keep isl_map *map,
+		enum isl_dim_type type, unsigned pos);
+	const char *isl_map_get_dim_name(
+		__isl_keep isl_map *map,
+		enum isl_dim_type type, unsigned pos);
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_set_dim_id(
+		__isl_take isl_multi_val *mv,
+		enum isl_dim_type type, unsigned pos,
+		__isl_take isl_id *id);
+	__isl_give isl_id *isl_multi_val_get_dim_id(
+		__isl_keep isl_multi_val *mv,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_multi_val *isl_multi_val_set_dim_name(
+		__isl_take isl_multi_val *mv,
+		enum isl_dim_type type, unsigned pos, const char *s);
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_aff_set_dim_id(
+		__isl_take isl_aff *aff, enum isl_dim_type type,
+		unsigned pos, __isl_take isl_id *id);
+	__isl_give isl_multi_aff *isl_multi_aff_set_dim_id(
+		__isl_take isl_multi_aff *maff,
+		enum isl_dim_type type, unsigned pos,
+		__isl_take isl_id *id);
+	__isl_give isl_pw_aff *isl_pw_aff_set_dim_id(
+		__isl_take isl_pw_aff *pma,
+		enum isl_dim_type type, unsigned pos,
+		__isl_take isl_id *id);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_set_dim_id(
+		__isl_take isl_multi_pw_aff *mpa,
+		enum isl_dim_type type, unsigned pos,
+		__isl_take isl_id *id);
+	__isl_give isl_id *isl_multi_aff_get_dim_id(
+		__isl_keep isl_multi_aff *ma,
+		enum isl_dim_type type, unsigned pos);
+	int isl_pw_aff_has_dim_id(__isl_keep isl_pw_aff *pa,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_id *isl_pw_aff_get_dim_id(
+		__isl_keep isl_pw_aff *pa,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_id *isl_pw_multi_aff_get_dim_id(
+		__isl_keep isl_pw_multi_aff *pma,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_id *isl_multi_pw_aff_get_dim_id(
+		__isl_keep isl_multi_pw_aff *mpa,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_aff *isl_aff_set_dim_name(
+		__isl_take isl_aff *aff, enum isl_dim_type type,
+		unsigned pos, const char *s);
+	__isl_give isl_multi_aff *isl_multi_aff_set_dim_name(
+		__isl_take isl_multi_aff *maff,
+		enum isl_dim_type type, unsigned pos, const char *s);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_set_dim_name(
+		__isl_take isl_multi_pw_aff *mpa,
+		enum isl_dim_type type, unsigned pos, const char *s);
+	const char *isl_aff_get_dim_name(__isl_keep isl_aff *aff,
+		enum isl_dim_type type, unsigned pos);
+	const char *isl_pw_aff_get_dim_name(
+		__isl_keep isl_pw_aff *pa,
+		enum isl_dim_type type, unsigned pos);
+	const char *isl_pw_multi_aff_get_dim_name(
+		__isl_keep isl_pw_multi_aff *pma,
+		enum isl_dim_type type, unsigned pos);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_qpolynomial *isl_qpolynomial_set_dim_name(
+		__isl_take isl_qpolynomial *qp,
+		enum isl_dim_type type, unsigned pos,
+		const char *s);
+	__isl_give isl_pw_qpolynomial *
+	isl_pw_qpolynomial_set_dim_name(
+		__isl_take isl_pw_qpolynomial *pwqp,
+		enum isl_dim_type type, unsigned pos,
+		const char *s);
+	__isl_give isl_pw_qpolynomial_fold *
+	isl_pw_qpolynomial_fold_set_dim_name(
+		__isl_take isl_pw_qpolynomial_fold *pwf,
+		enum isl_dim_type type, unsigned pos,
+		const char *s);
+
+Note that C<isl_space_get_name> returns a pointer to some internal
+data structure, so the result can only be used while the
+corresponding C<isl_space> is alive.
+Also note that every function that operates on two sets or relations
+requires that both arguments have the same parameters.  This also
+means that if one of the arguments has named parameters, then the
+other needs to have named parameters too and the names need to match.
+Pairs of C<isl_set>, C<isl_map>, C<isl_union_set> and/or C<isl_union_map>
+arguments may have different parameters (as long as they are named),
+in which case the result will have as parameters the union of the parameters of
+the arguments.
+
+Given the identifier or name of a dimension (typically a parameter),
+its position can be obtained from the following functions.
+
+	#include <isl/space.h>
+	int isl_space_find_dim_by_id(__isl_keep isl_space *space,
+		enum isl_dim_type type, __isl_keep isl_id *id);
+	int isl_space_find_dim_by_name(__isl_keep isl_space *space,
+		enum isl_dim_type type, const char *name);
+
+	#include <isl/local_space.h>
+	int isl_local_space_find_dim_by_name(
+		__isl_keep isl_local_space *ls,
+		enum isl_dim_type type, const char *name);
+
+	#include <isl/val.h>
+	int isl_multi_val_find_dim_by_id(
+		__isl_keep isl_multi_val *mv,
+		enum isl_dim_type type, __isl_keep isl_id *id);
+	int isl_multi_val_find_dim_by_name(
+		__isl_keep isl_multi_val *mv,
+		enum isl_dim_type type, const char *name);
+
+	#include <isl/set.h>
+	int isl_set_find_dim_by_id(__isl_keep isl_set *set,
+		enum isl_dim_type type, __isl_keep isl_id *id);
+	int isl_set_find_dim_by_name(__isl_keep isl_set *set,
+		enum isl_dim_type type, const char *name);
+
+	#include <isl/map.h>
+	int isl_map_find_dim_by_id(__isl_keep isl_map *map,
+		enum isl_dim_type type, __isl_keep isl_id *id);
+	int isl_basic_map_find_dim_by_name(
+		__isl_keep isl_basic_map *bmap,
+		enum isl_dim_type type, const char *name);
+	int isl_map_find_dim_by_name(__isl_keep isl_map *map,
+		enum isl_dim_type type, const char *name);
+	int isl_union_map_find_dim_by_name(
+		__isl_keep isl_union_map *umap,
+		enum isl_dim_type type, const char *name);
+
+	#include <isl/aff.h>
+	int isl_multi_aff_find_dim_by_id(
+		__isl_keep isl_multi_aff *ma,
+		enum isl_dim_type type, __isl_keep isl_id *id);
+	int isl_multi_pw_aff_find_dim_by_id(
+		__isl_keep isl_multi_pw_aff *mpa,
+		enum isl_dim_type type, __isl_keep isl_id *id);
+	int isl_aff_find_dim_by_name(__isl_keep isl_aff *aff,
+		enum isl_dim_type type, const char *name);
+	int isl_multi_aff_find_dim_by_name(
+		__isl_keep isl_multi_aff *ma,
+		enum isl_dim_type type, const char *name);
+	int isl_pw_aff_find_dim_by_name(__isl_keep isl_pw_aff *pa,
+		enum isl_dim_type type, const char *name);
+	int isl_multi_pw_aff_find_dim_by_name(
+		__isl_keep isl_multi_pw_aff *mpa,
+		enum isl_dim_type type, const char *name);
+	int isl_pw_multi_aff_find_dim_by_name(
+		__isl_keep isl_pw_multi_aff *pma,
+		enum isl_dim_type type, const char *name);
+	int isl_union_pw_multi_aff_find_dim_by_name(
+		__isl_keep isl_union_pw_multi_aff *upma,
+		enum isl_dim_type type, const char *name);
+
+	#include <isl/polynomial.h>
+	int isl_pw_qpolynomial_find_dim_by_name(
+		__isl_keep isl_pw_qpolynomial *pwqp,
+		enum isl_dim_type type, const char *name);
+	int isl_pw_qpolynomial_fold_find_dim_by_name(
+		__isl_keep isl_pw_qpolynomial_fold *pwf,
+		enum isl_dim_type type, const char *name);
+	int isl_union_pw_qpolynomial_find_dim_by_name(
+		__isl_keep isl_union_pw_qpolynomial *upwqp,
+		enum isl_dim_type type, const char *name);
+	int isl_union_pw_qpolynomial_fold_find_dim_by_name(
+		__isl_keep isl_union_pw_qpolynomial_fold *upwf,
+		enum isl_dim_type type, const char *name);
+
+The identifiers or names of entire spaces may be set or read off
+using the following functions.
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_set_tuple_id(
+		__isl_take isl_space *space,
+		enum isl_dim_type type, __isl_take isl_id *id);
+	__isl_give isl_space *isl_space_reset_tuple_id(
+		__isl_take isl_space *space, enum isl_dim_type type);
+	int isl_space_has_tuple_id(__isl_keep isl_space *space,
+		enum isl_dim_type type);
+	__isl_give isl_id *isl_space_get_tuple_id(
+		__isl_keep isl_space *space, enum isl_dim_type type);
+	__isl_give isl_space *isl_space_set_tuple_name(
+		__isl_take isl_space *space,
+		enum isl_dim_type type, const char *s);
+	int isl_space_has_tuple_name(__isl_keep isl_space *space,
+		enum isl_dim_type type);
+	const char *isl_space_get_tuple_name(__isl_keep isl_space *space,
+		enum isl_dim_type type);
+
+	#include <isl/local_space.h>
+	__isl_give isl_local_space *isl_local_space_set_tuple_id(
+		__isl_take isl_local_space *ls,
+		enum isl_dim_type type, __isl_take isl_id *id);
+
+	#include <isl/set.h>
+	__isl_give isl_basic_set *isl_basic_set_set_tuple_id(
+		__isl_take isl_basic_set *bset,
+		__isl_take isl_id *id);
+	__isl_give isl_set *isl_set_set_tuple_id(
+		__isl_take isl_set *set, __isl_take isl_id *id);
+	__isl_give isl_set *isl_set_reset_tuple_id(
+		__isl_take isl_set *set);
+	int isl_set_has_tuple_id(__isl_keep isl_set *set);
+	__isl_give isl_id *isl_set_get_tuple_id(
+		__isl_keep isl_set *set);
+	__isl_give isl_basic_set *isl_basic_set_set_tuple_name(
+		__isl_take isl_basic_set *set, const char *s);
+	__isl_give isl_set *isl_set_set_tuple_name(
+		__isl_take isl_set *set, const char *s);
+	const char *isl_basic_set_get_tuple_name(
+		__isl_keep isl_basic_set *bset);
+	int isl_set_has_tuple_name(__isl_keep isl_set *set);
+	const char *isl_set_get_tuple_name(
+		__isl_keep isl_set *set);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *isl_basic_map_set_tuple_id(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type, __isl_take isl_id *id);
+	__isl_give isl_map *isl_map_set_tuple_id(
+		__isl_take isl_map *map, enum isl_dim_type type,
+		__isl_take isl_id *id);
+	__isl_give isl_map *isl_map_reset_tuple_id(
+		__isl_take isl_map *map, enum isl_dim_type type);
+	int isl_map_has_tuple_id(__isl_keep isl_map *map,
+		enum isl_dim_type type);
+	__isl_give isl_id *isl_map_get_tuple_id(
+		__isl_keep isl_map *map, enum isl_dim_type type);
+	__isl_give isl_map *isl_map_set_tuple_name(
+		__isl_take isl_map *map,
+		enum isl_dim_type type, const char *s);
+	const char *isl_basic_map_get_tuple_name(
+		__isl_keep isl_basic_map *bmap,
+		enum isl_dim_type type);
+	__isl_give isl_basic_map *isl_basic_map_set_tuple_name(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type, const char *s);
+	int isl_map_has_tuple_name(__isl_keep isl_map *map,
+		enum isl_dim_type type);
+	const char *isl_map_get_tuple_name(
+		__isl_keep isl_map *map,
+		enum isl_dim_type type);
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_set_tuple_id(
+		__isl_take isl_multi_val *mv,
+		enum isl_dim_type type, __isl_take isl_id *id);
+	__isl_give isl_multi_val *isl_multi_val_reset_tuple_id(
+		__isl_take isl_multi_val *mv,
+		enum isl_dim_type type);
+	int isl_multi_val_has_tuple_id(__isl_keep isl_multi_val *mv,
+		enum isl_dim_type type);
+	__isl_give isl_id *isl_multi_val_get_tuple_id(
+		__isl_keep isl_multi_val *mv,
+		enum isl_dim_type type);
+	__isl_give isl_multi_val *isl_multi_val_set_tuple_name(
+		__isl_take isl_multi_val *mv,
+		enum isl_dim_type type, const char *s);
+	const char *isl_multi_val_get_tuple_name(
+		__isl_keep isl_multi_val *mv,
+		enum isl_dim_type type);
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_aff_set_tuple_id(
+		__isl_take isl_aff *aff,
+		enum isl_dim_type type, __isl_take isl_id *id);
+	__isl_give isl_multi_aff *isl_multi_aff_set_tuple_id(
+		__isl_take isl_multi_aff *maff,
+		enum isl_dim_type type, __isl_take isl_id *id);
+	__isl_give isl_pw_aff *isl_pw_aff_set_tuple_id(
+		__isl_take isl_pw_aff *pwaff,
+		enum isl_dim_type type, __isl_take isl_id *id);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_set_tuple_id(
+		__isl_take isl_pw_multi_aff *pma,
+		enum isl_dim_type type, __isl_take isl_id *id);
+	__isl_give isl_multi_aff *isl_multi_aff_reset_tuple_id(
+		__isl_take isl_multi_aff *ma,
+		enum isl_dim_type type);
+	__isl_give isl_pw_aff *isl_pw_aff_reset_tuple_id(
+		__isl_take isl_pw_aff *pa,
+		enum isl_dim_type type);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_reset_tuple_id(
+		__isl_take isl_multi_pw_aff *mpa,
+		enum isl_dim_type type);
+	__isl_give isl_pw_multi_aff *
+	isl_pw_multi_aff_reset_tuple_id(
+		__isl_take isl_pw_multi_aff *pma,
+		enum isl_dim_type type);
+	int isl_multi_aff_has_tuple_id(__isl_keep isl_multi_aff *ma,
+		enum isl_dim_type type);
+	__isl_give isl_id *isl_multi_aff_get_tuple_id(
+		__isl_keep isl_multi_aff *ma,
+		enum isl_dim_type type);
+	int isl_pw_aff_has_tuple_id(__isl_keep isl_pw_aff *pa,
+		enum isl_dim_type type);
+	__isl_give isl_id *isl_pw_aff_get_tuple_id(
+		__isl_keep isl_pw_aff *pa,
+		enum isl_dim_type type);
+	int isl_pw_multi_aff_has_tuple_id(
+		__isl_keep isl_pw_multi_aff *pma,
+		enum isl_dim_type type);
+	__isl_give isl_id *isl_pw_multi_aff_get_tuple_id(
+		__isl_keep isl_pw_multi_aff *pma,
+		enum isl_dim_type type);
+	int isl_multi_pw_aff_has_tuple_id(
+		__isl_keep isl_multi_pw_aff *mpa,
+		enum isl_dim_type type);
+	__isl_give isl_id *isl_multi_pw_aff_get_tuple_id(
+		__isl_keep isl_multi_pw_aff *mpa,
+		enum isl_dim_type type);
+	__isl_give isl_multi_aff *isl_multi_aff_set_tuple_name(
+		__isl_take isl_multi_aff *maff,
+		enum isl_dim_type type, const char *s);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_set_tuple_name(
+		__isl_take isl_multi_pw_aff *mpa,
+		enum isl_dim_type type, const char *s);
+	const char *isl_multi_aff_get_tuple_name(
+		__isl_keep isl_multi_aff *multi,
+		enum isl_dim_type type);
+	int isl_pw_multi_aff_has_tuple_name(
+		__isl_keep isl_pw_multi_aff *pma,
+		enum isl_dim_type type);
+	const char *isl_pw_multi_aff_get_tuple_name(
+		__isl_keep isl_pw_multi_aff *pma,
+		enum isl_dim_type type);
+
+The C<type> argument needs to be one of C<isl_dim_in>, C<isl_dim_out>
+or C<isl_dim_set>.  As with C<isl_space_get_name>,
+the C<isl_space_get_tuple_name> function returns a pointer to some internal
+data structure.
+Binary operations require the corresponding spaces of their arguments
+to have the same name.
+
+To keep the names of all parameters and tuples, but reset the user pointers
+of all the corresponding identifiers, use the following function.
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_reset_user(
+		__isl_take isl_space *space);
+
+	#include <isl/set.h>
+	__isl_give isl_set *isl_set_reset_user(
+		__isl_take isl_set *set);
+
+	#include <isl/map.h>
+	__isl_give isl_map *isl_map_reset_user(
+		__isl_take isl_map *map);
+
+	#include <isl/union_set.h>
+	__isl_give isl_union_set *isl_union_set_reset_user(
+		__isl_take isl_union_set *uset);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *isl_union_map_reset_user(
+		__isl_take isl_union_map *umap);
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_reset_user(
+		__isl_take isl_multi_val *mv);
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *isl_multi_aff_reset_user(
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_reset_user(
+		__isl_take isl_multi_pw_aff *mpa);
+
+Spaces can be nested.  In particular, the domain of a set or
+the domain or range of a relation can be a nested relation.
+This process is also called I<wrapping>.
+The functions for detecting, constructing and deconstructing
+such nested spaces can be found in the wrapping properties
+of L</"Unary Properties">, the wrapping operations
+of L</"Unary Operations"> and the Cartesian product operations
+of L</"Basic Operations">.
+
+Spaces can be created from other spaces
+using the functions described in L</"Unary Operations">
+and L</"Binary Operations">.
+
+=head2 Local Spaces
+
+A local space is essentially a space with
+zero or more existentially quantified variables.
+The local space of various objects can be obtained
+using the following functions.
+
+	#include <isl/constraint.h>
+	__isl_give isl_local_space *isl_constraint_get_local_space(
+		__isl_keep isl_constraint *constraint);
+
+	#include <isl/set.h>
+	__isl_give isl_local_space *isl_basic_set_get_local_space(
+		__isl_keep isl_basic_set *bset);
+
+	#include <isl/map.h>
+	__isl_give isl_local_space *isl_basic_map_get_local_space(
+		__isl_keep isl_basic_map *bmap);
+
+	#include <isl/aff.h>
+	__isl_give isl_local_space *isl_aff_get_domain_local_space(
+		__isl_keep isl_aff *aff);
+	__isl_give isl_local_space *isl_aff_get_local_space(
+		__isl_keep isl_aff *aff);
+
+A new local space can be created from a space using
+
+	#include <isl/local_space.h>
+	__isl_give isl_local_space *isl_local_space_from_space(
+		__isl_take isl_space *space);
+
+They can be inspected, modified, copied and freed using the following functions.
+
+	#include <isl/local_space.h>
+	int isl_local_space_is_params(
+		__isl_keep isl_local_space *ls);
+	int isl_local_space_is_set(__isl_keep isl_local_space *ls);
+	__isl_give isl_space *isl_local_space_get_space(
+		__isl_keep isl_local_space *ls);
+	__isl_give isl_aff *isl_local_space_get_div(
+		__isl_keep isl_local_space *ls, int pos);
+	__isl_give isl_local_space *isl_local_space_copy(
+		__isl_keep isl_local_space *ls);
+	__isl_null isl_local_space *isl_local_space_free(
+		__isl_take isl_local_space *ls);
+
+Note that C<isl_local_space_get_div> can only be used on local spaces
+of sets.
+
+Two local spaces can be compared using
+
+	int isl_local_space_is_equal(__isl_keep isl_local_space *ls1,
+		__isl_keep isl_local_space *ls2);
+
+Local spaces can be created from other local spaces
+using the functions described in L</"Unary Operations">
+and L</"Binary Operations">.
+
+=head2 Creating New Sets and Relations
+
+C<isl> has functions for creating some standard sets and relations.
+
+=over
+
+=item * Empty sets and relations
+
+	__isl_give isl_basic_set *isl_basic_set_empty(
+		__isl_take isl_space *space);
+	__isl_give isl_basic_map *isl_basic_map_empty(
+		__isl_take isl_space *space);
+	__isl_give isl_set *isl_set_empty(
+		__isl_take isl_space *space);
+	__isl_give isl_map *isl_map_empty(
+		__isl_take isl_space *space);
+	__isl_give isl_union_set *isl_union_set_empty(
+		__isl_take isl_space *space);
+	__isl_give isl_union_map *isl_union_map_empty(
+		__isl_take isl_space *space);
+
+For C<isl_union_set>s and C<isl_union_map>s, the space
+is only used to specify the parameters.
+
+=item * Universe sets and relations
+
+	__isl_give isl_basic_set *isl_basic_set_universe(
+		__isl_take isl_space *space);
+	__isl_give isl_basic_map *isl_basic_map_universe(
+		__isl_take isl_space *space);
+	__isl_give isl_set *isl_set_universe(
+		__isl_take isl_space *space);
+	__isl_give isl_map *isl_map_universe(
+		__isl_take isl_space *space);
+	__isl_give isl_union_set *isl_union_set_universe(
+		__isl_take isl_union_set *uset);
+	__isl_give isl_union_map *isl_union_map_universe(
+		__isl_take isl_union_map *umap);
+
+The sets and relations constructed by the functions above
+contain all integer values, while those constructed by the
+functions below only contain non-negative values.
+
+	__isl_give isl_basic_set *isl_basic_set_nat_universe(
+		__isl_take isl_space *space);
+	__isl_give isl_basic_map *isl_basic_map_nat_universe(
+		__isl_take isl_space *space);
+	__isl_give isl_set *isl_set_nat_universe(
+		__isl_take isl_space *space);
+	__isl_give isl_map *isl_map_nat_universe(
+		__isl_take isl_space *space);
+
+=item * Identity relations
+
+	__isl_give isl_basic_map *isl_basic_map_identity(
+		__isl_take isl_space *space);
+	__isl_give isl_map *isl_map_identity(
+		__isl_take isl_space *space);
+
+The number of input and output dimensions in C<space> needs
+to be the same.
+
+=item * Lexicographic order
+
+	__isl_give isl_map *isl_map_lex_lt(
+		__isl_take isl_space *set_space);
+	__isl_give isl_map *isl_map_lex_le(
+		__isl_take isl_space *set_space);
+	__isl_give isl_map *isl_map_lex_gt(
+		__isl_take isl_space *set_space);
+	__isl_give isl_map *isl_map_lex_ge(
+		__isl_take isl_space *set_space);
+	__isl_give isl_map *isl_map_lex_lt_first(
+		__isl_take isl_space *space, unsigned n);
+	__isl_give isl_map *isl_map_lex_le_first(
+		__isl_take isl_space *space, unsigned n);
+	__isl_give isl_map *isl_map_lex_gt_first(
+		__isl_take isl_space *space, unsigned n);
+	__isl_give isl_map *isl_map_lex_ge_first(
+		__isl_take isl_space *space, unsigned n);
+
+The first four functions take a space for a B<set>
+and return relations that express that the elements in the domain
+are lexicographically less
+(C<isl_map_lex_lt>), less or equal (C<isl_map_lex_le>),
+greater (C<isl_map_lex_gt>) or greater or equal (C<isl_map_lex_ge>)
+than the elements in the range.
+The last four functions take a space for a map
+and return relations that express that the first C<n> dimensions
+in the domain are lexicographically less
+(C<isl_map_lex_lt_first>), less or equal (C<isl_map_lex_le_first>),
+greater (C<isl_map_lex_gt_first>) or greater or equal (C<isl_map_lex_ge_first>)
+than the first C<n> dimensions in the range.
+
+=back
+
+A basic set or relation can be converted to a set or relation
+using the following functions.
+
+	__isl_give isl_set *isl_set_from_basic_set(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_map *isl_map_from_basic_map(
+		__isl_take isl_basic_map *bmap);
+
+Sets and relations can be converted to union sets and relations
+using the following functions.
+
+	__isl_give isl_union_set *isl_union_set_from_basic_set(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_union_map *isl_union_map_from_basic_map(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_union_set *isl_union_set_from_set(
+		__isl_take isl_set *set);
+	__isl_give isl_union_map *isl_union_map_from_map(
+		__isl_take isl_map *map);
+
+The inverse conversions below can only be used if the input
+union set or relation is known to contain elements in exactly one
+space.
+
+	__isl_give isl_set *isl_set_from_union_set(
+		__isl_take isl_union_set *uset);
+	__isl_give isl_map *isl_map_from_union_map(
+		__isl_take isl_union_map *umap);
+
+Sets and relations can be copied and freed again using the following
+functions.
+
+	__isl_give isl_basic_set *isl_basic_set_copy(
+		__isl_keep isl_basic_set *bset);
+	__isl_give isl_set *isl_set_copy(__isl_keep isl_set *set);
+	__isl_give isl_union_set *isl_union_set_copy(
+		__isl_keep isl_union_set *uset);
+	__isl_give isl_basic_map *isl_basic_map_copy(
+		__isl_keep isl_basic_map *bmap);
+	__isl_give isl_map *isl_map_copy(__isl_keep isl_map *map);
+	__isl_give isl_union_map *isl_union_map_copy(
+		__isl_keep isl_union_map *umap);
+	__isl_null isl_basic_set *isl_basic_set_free(
+		__isl_take isl_basic_set *bset);
+	__isl_null isl_set *isl_set_free(__isl_take isl_set *set);
+	__isl_null isl_union_set *isl_union_set_free(
+		__isl_take isl_union_set *uset);
+	__isl_null isl_basic_map *isl_basic_map_free(
+		__isl_take isl_basic_map *bmap);
+	__isl_null isl_map *isl_map_free(__isl_take isl_map *map);
+	__isl_null isl_union_map *isl_union_map_free(
+		__isl_take isl_union_map *umap);
+
+Other sets and relations can be constructed by starting
+from a universe set or relation, adding equality and/or
+inequality constraints and then projecting out the
+existentially quantified variables, if any.
+Constraints can be constructed, manipulated and
+added to (or removed from) (basic) sets and relations
+using the following functions.
+
+	#include <isl/constraint.h>
+	__isl_give isl_constraint *isl_equality_alloc(
+		__isl_take isl_local_space *ls);
+	__isl_give isl_constraint *isl_inequality_alloc(
+		__isl_take isl_local_space *ls);
+	__isl_give isl_constraint *isl_constraint_set_constant_si(
+		__isl_take isl_constraint *constraint, int v);
+	__isl_give isl_constraint *isl_constraint_set_constant_val(
+		__isl_take isl_constraint *constraint,
+		__isl_take isl_val *v);
+	__isl_give isl_constraint *isl_constraint_set_coefficient_si(
+		__isl_take isl_constraint *constraint,
+		enum isl_dim_type type, int pos, int v);
+	__isl_give isl_constraint *
+	isl_constraint_set_coefficient_val(
+		__isl_take isl_constraint *constraint,
+		enum isl_dim_type type, int pos,
+		__isl_take isl_val *v);
+	__isl_give isl_basic_map *isl_basic_map_add_constraint(
+		__isl_take isl_basic_map *bmap,
+		__isl_take isl_constraint *constraint);
+	__isl_give isl_basic_set *isl_basic_set_add_constraint(
+		__isl_take isl_basic_set *bset,
+		__isl_take isl_constraint *constraint);
+	__isl_give isl_map *isl_map_add_constraint(
+		__isl_take isl_map *map,
+		__isl_take isl_constraint *constraint);
+	__isl_give isl_set *isl_set_add_constraint(
+		__isl_take isl_set *set,
+		__isl_take isl_constraint *constraint);
+	__isl_give isl_basic_set *isl_basic_set_drop_constraint(
+		__isl_take isl_basic_set *bset,
+		__isl_take isl_constraint *constraint);
+
+For example, to create a set containing the even integers
+between 10 and 42, you would use the following code.
+
+	isl_space *space;
+	isl_local_space *ls;
+	isl_constraint *c;
+	isl_basic_set *bset;
+
+	space = isl_space_set_alloc(ctx, 0, 2);
+	bset = isl_basic_set_universe(isl_space_copy(space));
+	ls = isl_local_space_from_space(space);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	c = isl_constraint_set_coefficient_si(c, isl_dim_set, 0, -1);
+	c = isl_constraint_set_coefficient_si(c, isl_dim_set, 1, 2);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	c = isl_inequality_alloc(isl_local_space_copy(ls));
+	c = isl_constraint_set_constant_si(c, -10);
+	c = isl_constraint_set_coefficient_si(c, isl_dim_set, 0, 1);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	c = isl_inequality_alloc(ls);
+	c = isl_constraint_set_constant_si(c, 42);
+	c = isl_constraint_set_coefficient_si(c, isl_dim_set, 0, -1);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	bset = isl_basic_set_project_out(bset, isl_dim_set, 1, 1);
+
+Or, alternatively,
+
+	isl_basic_set *bset;
+	bset = isl_basic_set_read_from_str(ctx,
+		"{[i] : exists (a : i = 2a and i >= 10 and i <= 42)}");
+
+A basic set or relation can also be constructed from two matrices
+describing the equalities and the inequalities.
+
+	__isl_give isl_basic_set *isl_basic_set_from_constraint_matrices(
+		__isl_take isl_space *space,
+		__isl_take isl_mat *eq, __isl_take isl_mat *ineq,
+		enum isl_dim_type c1,
+		enum isl_dim_type c2, enum isl_dim_type c3,
+		enum isl_dim_type c4);
+	__isl_give isl_basic_map *isl_basic_map_from_constraint_matrices(
+		__isl_take isl_space *space,
+		__isl_take isl_mat *eq, __isl_take isl_mat *ineq,
+		enum isl_dim_type c1,
+		enum isl_dim_type c2, enum isl_dim_type c3,
+		enum isl_dim_type c4, enum isl_dim_type c5);
+
+The C<isl_dim_type> arguments indicate the order in which
+different kinds of variables appear in the input matrices
+and should be a permutation of C<isl_dim_cst>, C<isl_dim_param>,
+C<isl_dim_set> and C<isl_dim_div> for sets and
+of C<isl_dim_cst>, C<isl_dim_param>,
+C<isl_dim_in>, C<isl_dim_out> and C<isl_dim_div> for relations.
+
+A (basic or union) set or relation can also be constructed from a
+(union) (piecewise) (multiple) affine expression
+or a list of affine expressions
+(See L</"Functions">).
+
+	__isl_give isl_basic_map *isl_basic_map_from_aff(
+		__isl_take isl_aff *aff);
+	__isl_give isl_map *isl_map_from_aff(
+		__isl_take isl_aff *aff);
+	__isl_give isl_set *isl_set_from_pw_aff(
+		__isl_take isl_pw_aff *pwaff);
+	__isl_give isl_map *isl_map_from_pw_aff(
+		__isl_take isl_pw_aff *pwaff);
+	__isl_give isl_basic_map *isl_basic_map_from_aff_list(
+		__isl_take isl_space *domain_space,
+		__isl_take isl_aff_list *list);
+	__isl_give isl_basic_map *isl_basic_map_from_multi_aff(
+		__isl_take isl_multi_aff *maff)
+	__isl_give isl_map *isl_map_from_multi_aff(
+		__isl_take isl_multi_aff *maff)
+	__isl_give isl_set *isl_set_from_pw_multi_aff(
+		__isl_take isl_pw_multi_aff *pma);
+	__isl_give isl_map *isl_map_from_pw_multi_aff(
+		__isl_take isl_pw_multi_aff *pma);
+	__isl_give isl_set *isl_set_from_multi_pw_aff(
+		__isl_take isl_multi_pw_aff *mpa);
+	__isl_give isl_map *isl_map_from_multi_pw_aff(
+		__isl_take isl_multi_pw_aff *mpa);
+	__isl_give isl_union_map *
+	isl_union_map_from_union_pw_multi_aff(
+		__isl_take isl_union_pw_multi_aff *upma);
+
+The C<domain_space> argument describes the domain of the resulting
+basic relation.  It is required because the C<list> may consist
+of zero affine expressions.
+
+=head2 Inspecting Sets and Relations
+
+Usually, the user should not have to care about the actual constraints
+of the sets and maps, but should instead apply the abstract operations
+explained in the following sections.
+Occasionally, however, it may be required to inspect the individual
+coefficients of the constraints.  This section explains how to do so.
+In these cases, it may also be useful to have C<isl> compute
+an explicit representation of the existentially quantified variables.
+
+	__isl_give isl_set *isl_set_compute_divs(
+		__isl_take isl_set *set);
+	__isl_give isl_map *isl_map_compute_divs(
+		__isl_take isl_map *map);
+	__isl_give isl_union_set *isl_union_set_compute_divs(
+		__isl_take isl_union_set *uset);
+	__isl_give isl_union_map *isl_union_map_compute_divs(
+		__isl_take isl_union_map *umap);
+
+This explicit representation defines the existentially quantified
+variables as integer divisions of the other variables, possibly
+including earlier existentially quantified variables.
+An explicitly represented existentially quantified variable therefore
+has a unique value when the values of the other variables are known.
+If, furthermore, the same existentials, i.e., existentials
+with the same explicit representations, should appear in the
+same order in each of the disjuncts of a set or map, then the user should call
+either of the following functions.
+
+	__isl_give isl_set *isl_set_align_divs(
+		__isl_take isl_set *set);
+	__isl_give isl_map *isl_map_align_divs(
+		__isl_take isl_map *map);
+
+Alternatively, the existentially quantified variables can be removed
+using the following functions, which compute an overapproximation.
+
+	__isl_give isl_basic_set *isl_basic_set_remove_divs(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_basic_map *isl_basic_map_remove_divs(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_set *isl_set_remove_divs(
+		__isl_take isl_set *set);
+	__isl_give isl_map *isl_map_remove_divs(
+		__isl_take isl_map *map);
+
+It is also possible to only remove those divs that are defined
+in terms of a given range of dimensions or only those for which
+no explicit representation is known.
+
+	__isl_give isl_basic_set *
+	isl_basic_set_remove_divs_involving_dims(
+		__isl_take isl_basic_set *bset,
+		enum isl_dim_type type,
+		unsigned first, unsigned n);
+	__isl_give isl_basic_map *
+	isl_basic_map_remove_divs_involving_dims(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type,
+		unsigned first, unsigned n);
+	__isl_give isl_set *isl_set_remove_divs_involving_dims(
+		__isl_take isl_set *set, enum isl_dim_type type,
+		unsigned first, unsigned n);
+	__isl_give isl_map *isl_map_remove_divs_involving_dims(
+		__isl_take isl_map *map, enum isl_dim_type type,
+		unsigned first, unsigned n);
+
+	__isl_give isl_basic_set *
+	isl_basic_set_remove_unknown_divs(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_set *isl_set_remove_unknown_divs(
+		__isl_take isl_set *set);
+	__isl_give isl_map *isl_map_remove_unknown_divs(
+		__isl_take isl_map *map);
+
+To iterate over all the sets or maps in a union set or map, use
+
+	int isl_union_set_foreach_set(__isl_keep isl_union_set *uset,
+		int (*fn)(__isl_take isl_set *set, void *user),
+		void *user);
+	int isl_union_map_foreach_map(__isl_keep isl_union_map *umap,
+		int (*fn)(__isl_take isl_map *map, void *user),
+		void *user);
+
+The number of sets or maps in a union set or map can be obtained
+from
+
+	int isl_union_set_n_set(__isl_keep isl_union_set *uset);
+	int isl_union_map_n_map(__isl_keep isl_union_map *umap);
+
+To extract the set or map in a given space from a union, use
+
+	__isl_give isl_set *isl_union_set_extract_set(
+		__isl_keep isl_union_set *uset,
+		__isl_take isl_space *space);
+	__isl_give isl_map *isl_union_map_extract_map(
+		__isl_keep isl_union_map *umap,
+		__isl_take isl_space *space);
+
+To iterate over all the basic sets or maps in a set or map, use
+
+	int isl_set_foreach_basic_set(__isl_keep isl_set *set,
+		int (*fn)(__isl_take isl_basic_set *bset, void *user),
+		void *user);
+	int isl_map_foreach_basic_map(__isl_keep isl_map *map,
+		int (*fn)(__isl_take isl_basic_map *bmap, void *user),
+		void *user);
+
+The callback function C<fn> should return 0 if successful and
+-1 if an error occurs.  In the latter case, or if any other error
+occurs, the above functions will return -1.
+
+It should be noted that C<isl> does not guarantee that
+the basic sets or maps passed to C<fn> are disjoint.
+If this is required, then the user should call one of
+the following functions first.
+
+	__isl_give isl_set *isl_set_make_disjoint(
+		__isl_take isl_set *set);
+	__isl_give isl_map *isl_map_make_disjoint(
+		__isl_take isl_map *map);
+
+The number of basic sets in a set can be obtained
+or the number of basic maps in a map can be obtained
+from
+
+	#include <isl/set.h>
+	int isl_set_n_basic_set(__isl_keep isl_set *set);
+
+	#include <isl/map.h>
+	int isl_map_n_basic_map(__isl_keep isl_map *map);
+
+To iterate over the constraints of a basic set or map, use
+
+	#include <isl/constraint.h>
+
+	int isl_basic_set_n_constraint(
+		__isl_keep isl_basic_set *bset);
+	int isl_basic_set_foreach_constraint(
+		__isl_keep isl_basic_set *bset,
+		int (*fn)(__isl_take isl_constraint *c, void *user),
+		void *user);
+	int isl_basic_map_n_constraint(
+		__isl_keep isl_basic_map *bmap);
+	int isl_basic_map_foreach_constraint(
+		__isl_keep isl_basic_map *bmap,
+		int (*fn)(__isl_take isl_constraint *c, void *user),
+		void *user);
+	__isl_null isl_constraint *isl_constraint_free(
+		__isl_take isl_constraint *c);
+
+Again, the callback function C<fn> should return 0 if successful and
+-1 if an error occurs.  In the latter case, or if any other error
+occurs, the above functions will return -1.
+The constraint C<c> represents either an equality or an inequality.
+Use the following function to find out whether a constraint
+represents an equality.  If not, it represents an inequality.
+
+	int isl_constraint_is_equality(
+		__isl_keep isl_constraint *constraint);
+
+It is also possible to obtain a list of constraints from a basic
+map or set
+
+	#include <isl/constraint.h>
+	__isl_give isl_constraint_list *
+	isl_basic_map_get_constraint_list(
+		__isl_keep isl_basic_map *bmap);
+	__isl_give isl_constraint_list *
+	isl_basic_set_get_constraint_list(
+		__isl_keep isl_basic_set *bset);
+
+These functions require that all existentially quantified variables
+have an explicit representation.
+The returned list can be manipulated using the functions in L<"Lists">.
+
+The coefficients of the constraints can be inspected using
+the following functions.
+
+	int isl_constraint_is_lower_bound(
+		__isl_keep isl_constraint *constraint,
+		enum isl_dim_type type, unsigned pos);
+	int isl_constraint_is_upper_bound(
+		__isl_keep isl_constraint *constraint,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_val *isl_constraint_get_constant_val(
+		__isl_keep isl_constraint *constraint);
+	__isl_give isl_val *isl_constraint_get_coefficient_val(
+		__isl_keep isl_constraint *constraint,
+		enum isl_dim_type type, int pos);
+
+The explicit representations of the existentially quantified
+variables can be inspected using the following function.
+Note that the user is only allowed to use this function
+if the inspected set or map is the result of a call
+to C<isl_set_compute_divs> or C<isl_map_compute_divs>.
+The existentially quantified variable is equal to the floor
+of the returned affine expression.  The affine expression
+itself can be inspected using the functions in
+L</"Functions">.
+
+	__isl_give isl_aff *isl_constraint_get_div(
+		__isl_keep isl_constraint *constraint, int pos);
+
+To obtain the constraints of a basic set or map in matrix
+form, use the following functions.
+
+	__isl_give isl_mat *isl_basic_set_equalities_matrix(
+		__isl_keep isl_basic_set *bset,
+		enum isl_dim_type c1, enum isl_dim_type c2,
+		enum isl_dim_type c3, enum isl_dim_type c4);
+	__isl_give isl_mat *isl_basic_set_inequalities_matrix(
+		__isl_keep isl_basic_set *bset,
+		enum isl_dim_type c1, enum isl_dim_type c2,
+		enum isl_dim_type c3, enum isl_dim_type c4);
+	__isl_give isl_mat *isl_basic_map_equalities_matrix(
+		__isl_keep isl_basic_map *bmap,
+		enum isl_dim_type c1,
+		enum isl_dim_type c2, enum isl_dim_type c3,
+		enum isl_dim_type c4, enum isl_dim_type c5);
+	__isl_give isl_mat *isl_basic_map_inequalities_matrix(
+		__isl_keep isl_basic_map *bmap,
+		enum isl_dim_type c1,
+		enum isl_dim_type c2, enum isl_dim_type c3,
+		enum isl_dim_type c4, enum isl_dim_type c5);
+
+The C<isl_dim_type> arguments dictate the order in which
+different kinds of variables appear in the resulting matrix.
+For set inputs, they should be a permutation of
+C<isl_dim_cst>, C<isl_dim_param>, C<isl_dim_set> and C<isl_dim_div>.
+For map inputs, they should be a permutation of
+C<isl_dim_cst>, C<isl_dim_param>,
+C<isl_dim_in>, C<isl_dim_out> and C<isl_dim_div>.
+
+=head2 Points
+
+Points are elements of a set.  They can be used to construct
+simple sets (boxes) or they can be used to represent the
+individual elements of a set.
+The zero point (the origin) can be created using
+
+	__isl_give isl_point *isl_point_zero(__isl_take isl_space *space);
+
+The coordinates of a point can be inspected, set and changed
+using
+
+	__isl_give isl_val *isl_point_get_coordinate_val(
+		__isl_keep isl_point *pnt,
+		enum isl_dim_type type, int pos);
+	__isl_give isl_point *isl_point_set_coordinate_val(
+		__isl_take isl_point *pnt,
+		enum isl_dim_type type, int pos,
+		__isl_take isl_val *v);
+
+	__isl_give isl_point *isl_point_add_ui(
+		__isl_take isl_point *pnt,
+		enum isl_dim_type type, int pos, unsigned val);
+	__isl_give isl_point *isl_point_sub_ui(
+		__isl_take isl_point *pnt,
+		enum isl_dim_type type, int pos, unsigned val);
+
+Points can be copied or freed using
+
+	__isl_give isl_point *isl_point_copy(
+		__isl_keep isl_point *pnt);
+	void isl_point_free(__isl_take isl_point *pnt);
+
+A singleton set can be created from a point using
+
+	__isl_give isl_basic_set *isl_basic_set_from_point(
+		__isl_take isl_point *pnt);
+	__isl_give isl_set *isl_set_from_point(
+		__isl_take isl_point *pnt);
+
+and a box can be created from two opposite extremal points using
+
+	__isl_give isl_basic_set *isl_basic_set_box_from_points(
+		__isl_take isl_point *pnt1,
+		__isl_take isl_point *pnt2);
+	__isl_give isl_set *isl_set_box_from_points(
+		__isl_take isl_point *pnt1,
+		__isl_take isl_point *pnt2);
+
+All elements of a B<bounded> (union) set can be enumerated using
+the following functions.
+
+	int isl_set_foreach_point(__isl_keep isl_set *set,
+		int (*fn)(__isl_take isl_point *pnt, void *user),
+		void *user);
+	int isl_union_set_foreach_point(__isl_keep isl_union_set *uset,
+		int (*fn)(__isl_take isl_point *pnt, void *user),
+		void *user);
+
+The function C<fn> is called for each integer point in
+C<set> with as second argument the last argument of
+the C<isl_set_foreach_point> call.  The function C<fn>
+should return C<0> on success and C<-1> on failure.
+In the latter case, C<isl_set_foreach_point> will stop
+enumerating and return C<-1> as well.
+If the enumeration is performed successfully and to completion,
+then C<isl_set_foreach_point> returns C<0>.
+
+To obtain a single point of a (basic) set, use
+
+	__isl_give isl_point *isl_basic_set_sample_point(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_point *isl_set_sample_point(
+		__isl_take isl_set *set);
+
+If C<set> does not contain any (integer) points, then the
+resulting point will be ``void'', a property that can be
+tested using
+
+	int isl_point_is_void(__isl_keep isl_point *pnt);
+
+=head2 Functions
+
+Besides sets and relation, C<isl> also supports various types of functions.
+Each of these types is derived from the value type (see L</"Values">)
+or from one of two primitive function types
+through the application of zero or more type constructors.
+We first describe the primitive type and then we describe
+the types derived from these primitive types.
+
+=head3 Primitive Functions
+
+C<isl> support two primitive function types, quasi-affine
+expressions and quasipolynomials.
+A quasi-affine expression is defined either over a parameter
+space or over a set and is composed of integer constants,
+parameters and set variables, addition, subtraction and
+integer division by an integer constant.
+For example, the quasi-affine expression
+
+	[n] -> { [x] -> [2*floor((4 n + x)/9] }
+
+maps C<x> to C<2*floor((4 n + x)/9>.
+A quasipolynomial is a polynomial expression in quasi-affine
+expression.  That is, it additionally allows for multiplication.
+Note, though, that it is not allowed to construct an integer
+division of an expression involving multiplications.
+Here is an example of a quasipolynomial that is not
+quasi-affine expression
+
+	[n] -> { [x] -> (n*floor((4 n + x)/9) }
+
+Note that the external representations of quasi-affine expressions
+and quasipolynomials are different.  Quasi-affine expressions
+use a notation with square brackets just like binary relations,
+while quasipolynomials do not.  This might change at some point.
+
+If a primitive function is defined over a parameter space,
+then the space of the function itself is that of a set.
+If it is defined over a set, then the space of the function
+is that of a relation.  In both cases, the set space (or
+the output space) is single-dimensional, anonymous and unstructured.
+To create functions with multiple dimensions or with other kinds
+of set or output spaces, use multiple expressions
+(see L</"Multiple Expressions">).
+
+=over
+
+=item * Quasi-affine Expressions
+
+Besides the expressions described above, a quasi-affine
+expression can also be set to NaN.  Such expressions
+typically represent a failure to represent a result
+as a quasi-affine expression.
+
+The zero quasi affine expression or the quasi affine expression
+that is equal to a given value or
+a specified dimension on a given domain can be created using
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_aff_zero_on_domain(
+		__isl_take isl_local_space *ls);
+	__isl_give isl_aff *isl_aff_val_on_domain(
+		__isl_take isl_local_space *ls,
+		__isl_take isl_val *val);
+	__isl_give isl_aff *isl_aff_var_on_domain(
+		__isl_take isl_local_space *ls,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_aff *isl_aff_nan_on_domain(
+		__isl_take isl_local_space *ls);
+
+Quasi affine expressions can be copied and freed using
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_aff_copy(
+		__isl_keep isl_aff *aff);
+	__isl_null isl_aff *isl_aff_free(
+		__isl_take isl_aff *aff);
+
+A (rational) bound on a dimension can be extracted from an C<isl_constraint>
+using the following function.  The constraint is required to have
+a non-zero coefficient for the specified dimension.
+
+	#include <isl/constraint.h>
+	__isl_give isl_aff *isl_constraint_get_bound(
+		__isl_keep isl_constraint *constraint,
+		enum isl_dim_type type, int pos);
+
+The entire affine expression of the constraint can also be extracted
+using the following function.
+
+	#include <isl/constraint.h>
+	__isl_give isl_aff *isl_constraint_get_aff(
+		__isl_keep isl_constraint *constraint);
+
+Conversely, an equality constraint equating
+the affine expression to zero or an inequality constraint enforcing
+the affine expression to be non-negative, can be constructed using
+
+	__isl_give isl_constraint *isl_equality_from_aff(
+		__isl_take isl_aff *aff);
+	__isl_give isl_constraint *isl_inequality_from_aff(
+		__isl_take isl_aff *aff);
+
+The coefficients and the integer divisions of an affine expression
+can be inspected using the following functions.
+
+	#include <isl/aff.h>
+	__isl_give isl_val *isl_aff_get_constant_val(
+		__isl_keep isl_aff *aff);
+	__isl_give isl_val *isl_aff_get_coefficient_val(
+		__isl_keep isl_aff *aff,
+		enum isl_dim_type type, int pos);
+	int isl_aff_coefficient_sgn(__isl_keep isl_aff *aff,
+		enum isl_dim_type type, int pos);
+	__isl_give isl_val *isl_aff_get_denominator_val(
+		__isl_keep isl_aff *aff);
+	__isl_give isl_aff *isl_aff_get_div(
+		__isl_keep isl_aff *aff, int pos);
+
+They can be modified using the following functions.
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_aff_set_constant_si(
+		__isl_take isl_aff *aff, int v);
+	__isl_give isl_aff *isl_aff_set_constant_val(
+		__isl_take isl_aff *aff, __isl_take isl_val *v);
+	__isl_give isl_aff *isl_aff_set_coefficient_si(
+		__isl_take isl_aff *aff,
+		enum isl_dim_type type, int pos, int v);
+	__isl_give isl_aff *isl_aff_set_coefficient_val(
+		__isl_take isl_aff *aff,
+		enum isl_dim_type type, int pos,
+		__isl_take isl_val *v);
+
+	__isl_give isl_aff *isl_aff_add_constant_si(
+		__isl_take isl_aff *aff, int v);
+	__isl_give isl_aff *isl_aff_add_constant_val(
+		__isl_take isl_aff *aff, __isl_take isl_val *v);
+	__isl_give isl_aff *isl_aff_add_constant_num_si(
+		__isl_take isl_aff *aff, int v);
+	__isl_give isl_aff *isl_aff_add_coefficient_si(
+		__isl_take isl_aff *aff,
+		enum isl_dim_type type, int pos, int v);
+	__isl_give isl_aff *isl_aff_add_coefficient_val(
+		__isl_take isl_aff *aff,
+		enum isl_dim_type type, int pos,
+		__isl_take isl_val *v);
+
+Note that C<isl_aff_set_constant_si> and C<isl_aff_set_coefficient_si>
+set the I<numerator> of the constant or coefficient, while
+C<isl_aff_set_constant_val> and C<isl_aff_set_coefficient_val> set
+the constant or coefficient as a whole.
+The C<add_constant> and C<add_coefficient> functions add an integer
+or rational value to
+the possibly rational constant or coefficient.
+The C<add_constant_num> functions add an integer value to
+the numerator.
+
+=item * Quasipolynomials
+
+Some simple quasipolynomials can be created using the following functions.
+
+	#include <isl/polynomial.h>
+	__isl_give isl_qpolynomial *isl_qpolynomial_zero_on_domain(
+		__isl_take isl_space *domain);
+	__isl_give isl_qpolynomial *isl_qpolynomial_one_on_domain(
+		__isl_take isl_space *domain);
+	__isl_give isl_qpolynomial *isl_qpolynomial_infty_on_domain(
+		__isl_take isl_space *domain);
+	__isl_give isl_qpolynomial *isl_qpolynomial_neginfty_on_domain(
+		__isl_take isl_space *domain);
+	__isl_give isl_qpolynomial *isl_qpolynomial_nan_on_domain(
+		__isl_take isl_space *domain);
+	__isl_give isl_qpolynomial *isl_qpolynomial_val_on_domain(
+		__isl_take isl_space *domain,
+		__isl_take isl_val *val);
+	__isl_give isl_qpolynomial *isl_qpolynomial_var_on_domain(
+		__isl_take isl_space *domain,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_qpolynomial *isl_qpolynomial_from_aff(
+		__isl_take isl_aff *aff);
+
+Recall that the space in which a quasipolynomial lives is a map space
+with a one-dimensional range.  The C<domain> argument in some of
+the functions above corresponds to the domain of this map space.
+
+Quasipolynomials can be copied and freed again using the following
+functions.
+
+	#include <isl/polynomial.h>
+	__isl_give isl_qpolynomial *isl_qpolynomial_copy(
+		__isl_keep isl_qpolynomial *qp);
+	__isl_null isl_qpolynomial *isl_qpolynomial_free(
+		__isl_take isl_qpolynomial *qp);
+
+The constant term of a quasipolynomial can be extracted using
+
+	__isl_give isl_val *isl_qpolynomial_get_constant_val(
+		__isl_keep isl_qpolynomial *qp);
+
+To iterate over all terms in a quasipolynomial,
+use
+
+	int isl_qpolynomial_foreach_term(
+		__isl_keep isl_qpolynomial *qp,
+		int (*fn)(__isl_take isl_term *term,
+			  void *user), void *user);
+
+The terms themselves can be inspected and freed using
+these functions
+
+	unsigned isl_term_dim(__isl_keep isl_term *term,
+		enum isl_dim_type type);
+	__isl_give isl_val *isl_term_get_coefficient_val(
+		__isl_keep isl_term *term);
+	int isl_term_get_exp(__isl_keep isl_term *term,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_aff *isl_term_get_div(
+		__isl_keep isl_term *term, unsigned pos);
+	void isl_term_free(__isl_take isl_term *term);
+
+Each term is a product of parameters, set variables and
+integer divisions.  The function C<isl_term_get_exp>
+returns the exponent of a given dimensions in the given term.
+
+=back
+
+=head3 Reductions
+
+A reduction represents a maximum or a minimum of its
+base expressions.
+The only reduction type defined by C<isl> is
+C<isl_qpolynomial_fold>.
+
+There are currently no functions to directly create such
+objects, but they do appear in the piecewise quasipolynomial
+reductions returned by the C<isl_pw_qpolynomial_bound> function.
+See
+L</"Bounds on Piecewise Quasipolynomials and Piecewise Quasipolynomial Reductions">.
+
+Reductions can be copied and freed using
+the following functions.
+
+	#include <isl/polynomial.h>
+	__isl_give isl_qpolynomial_fold *
+	isl_qpolynomial_fold_copy(
+		__isl_keep isl_qpolynomial_fold *fold);
+	void isl_qpolynomial_fold_free(
+		__isl_take isl_qpolynomial_fold *fold);
+
+To iterate over all quasipolynomials in a reduction, use
+
+	int isl_qpolynomial_fold_foreach_qpolynomial(
+		__isl_keep isl_qpolynomial_fold *fold,
+		int (*fn)(__isl_take isl_qpolynomial *qp,
+			  void *user), void *user);
+
+=head3 Multiple Expressions
+
+A multiple expression represents a sequence of zero or
+more base expressions, all defined on the same domain space.
+The domain space of the multiple expression is the same
+as that of the base expressions, but the range space
+can be any space.  In case the base expressions have
+a set space, the corresponding multiple expression
+also has a set space.
+Objects of the value type do not have an associated space.
+The space of a multiple value is therefore always a set space.
+
+The multiple expression types defined by C<isl>
+are C<isl_multi_val>, C<isl_multi_aff> and C<isl_multi_pw_aff>.
+
+A multiple expression with the value zero for
+each output (or set) dimension can be created
+using the following functions.
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_zero(
+		__isl_take isl_space *space);
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *isl_multi_aff_zero(
+		__isl_take isl_space *space);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_zero(
+		__isl_take isl_space *space);
+
+An identity function can be created using the following
+functions.  The space needs to be that of a relation
+with the same number of input and output dimensions.
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *isl_multi_aff_identity(
+		__isl_take isl_space *space);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_identity(
+		__isl_take isl_space *space);
+
+A function that performs a projection on a universe
+relation or set can be created using the following functions.
+See also the corresponding
+projection operations in L</"Unary Operations">.
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *isl_multi_aff_domain_map(
+		__isl_take isl_space *space);
+	__isl_give isl_multi_aff *isl_multi_aff_range_map(
+		__isl_take isl_space *space);
+	__isl_give isl_multi_aff *isl_multi_aff_project_out_map(
+		__isl_take isl_space *space,
+		enum isl_dim_type type,
+		unsigned first, unsigned n);
+
+A multiple expression can be created from a single
+base expression using the following functions.
+The space of the created multiple expression is the same
+as that of the base expression.
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *isl_multi_aff_from_aff(
+		__isl_take isl_aff *aff);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_from_pw_aff(
+		__isl_take isl_pw_aff *pa);
+
+A multiple expression can be created from a list
+of base expression in a specified space.
+The domain of this space needs to be the same
+as the domains of the base expressions in the list.
+If the base expressions have a set space (or no associated space),
+then this space also needs to be a set space.
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_from_val_list(
+		__isl_take isl_space *space,
+		__isl_take isl_val_list *list);
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *isl_multi_aff_from_aff_list(
+		__isl_take isl_space *space,
+		__isl_take isl_aff_list *list);
+
+As a convenience, a multiple piecewise expression can
+also be created from a multiple expression.
+Each piecewise expression in the result has a single
+universe cell.
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_from_multi_aff(
+		__isl_take isl_multi_aff *ma);
+
+A multiple quasi-affine expression can be created from
+a multiple value with a given domain space using the following
+function.
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *
+	isl_multi_aff_multi_val_on_space(
+		__isl_take isl_space *space,
+		__isl_take isl_multi_val *mv);
+
+Multiple expressions can be copied and freed using
+the following functions.
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_copy(
+		__isl_keep isl_multi_val *mv);
+	__isl_null isl_multi_val *isl_multi_val_free(
+		__isl_take isl_multi_val *mv);
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *isl_multi_aff_copy(
+		__isl_keep isl_multi_aff *maff);
+	__isl_null isl_multi_aff *isl_multi_aff_free(
+		__isl_take isl_multi_aff *maff);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_copy(
+		__isl_keep isl_multi_pw_aff *mpa);
+	__isl_null isl_multi_pw_aff *isl_multi_pw_aff_free(
+		__isl_take isl_multi_pw_aff *mpa);
+
+The base expression at a given position of a multiple
+expression can be extracted using the following functions.
+
+	#include <isl/val.h>
+	__isl_give isl_val *isl_multi_val_get_val(
+		__isl_keep isl_multi_val *mv, int pos);
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_multi_aff_get_aff(
+		__isl_keep isl_multi_aff *multi, int pos);
+	__isl_give isl_pw_aff *isl_multi_pw_aff_get_pw_aff(
+		__isl_keep isl_multi_pw_aff *mpa, int pos);
+
+It can be replaced using the following functions.
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_set_val(
+		__isl_take isl_multi_val *mv, int pos,
+		__isl_take isl_val *val);
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *isl_multi_aff_set_aff(
+		__isl_take isl_multi_aff *multi, int pos,
+		__isl_take isl_aff *aff);
+
+=head3 Piecewise Expressions
+
+A piecewise expression is an expression that is described
+using zero or more base expression defined over the same
+number of cells in the domain space of the base expressions.
+All base expressions are defined over the same
+domain space and the cells are disjoint.
+The space of a piecewise expression is the same as
+that of the base expressions.
+If the union of the cells is a strict subset of the domain
+space, then the value of the piecewise expression outside
+this union is different for types derived from quasi-affine
+expressions and those derived from quasipolynomials.
+Piecewise expressions derived from quasi-affine expressions
+are considered to be undefined outside the union of their cells.
+Piecewise expressions derived from quasipolynomials
+are considered to be zero outside the union of their cells.
+
+Piecewise quasipolynomials are mainly used by the C<barvinok>
+library for representing the number of elements in a parametric set or map.
+For example, the piecewise quasipolynomial
+
+	[n] -> { [x] -> ((1 + n) - x) : x <= n and x >= 0 }
+
+represents the number of points in the map
+
+	[n] -> { [x] -> [y] : x,y >= 0 and 0 <= x + y <= n }
+
+The piecewise expression types defined by C<isl>
+are C<isl_pw_aff>, C<isl_pw_multi_aff>,
+C<isl_pw_qpolynomial> and C<isl_pw_qpolynomial_fold>.
+
+A piecewise expression with no cells can be created using
+the following functions.
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_aff *isl_pw_aff_empty(
+		__isl_take isl_space *space);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_empty(
+		__isl_take isl_space *space);
+
+A piecewise expression with a single universe cell can be
+created using the following functions.
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_aff *isl_pw_aff_from_aff(
+		__isl_take isl_aff *aff);
+	__isl_give isl_pw_multi_aff *
+	isl_pw_multi_aff_from_multi_aff(
+		__isl_take isl_multi_aff *ma);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_pw_qpolynomial *
+	isl_pw_qpolynomial_from_qpolynomial(
+		__isl_take isl_qpolynomial *qp);
+
+A piecewise expression with a single specified cell can be
+created using the following functions.
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_aff *isl_pw_aff_alloc(
+		__isl_take isl_set *set, __isl_take isl_aff *aff);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_alloc(
+		__isl_take isl_set *set,
+		__isl_take isl_multi_aff *maff);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_alloc(
+		__isl_take isl_set *set,
+		__isl_take isl_qpolynomial *qp);
+
+The following convenience functions first create a base expression and
+then create a piecewise expression over a universe domain.
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_aff *isl_pw_aff_zero_on_domain(
+		__isl_take isl_local_space *ls);
+	__isl_give isl_pw_aff *isl_pw_aff_var_on_domain(
+		__isl_take isl_local_space *ls,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_pw_aff *isl_pw_aff_nan_on_domain(
+		__isl_take isl_local_space *ls);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_zero(
+		__isl_take isl_space *space);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_identity(
+		__isl_take isl_space *space);
+	__isl_give isl_pw_multi_aff *
+	isl_pw_multi_aff_project_out_map(
+		__isl_take isl_space *space,
+		enum isl_dim_type type,
+		unsigned first, unsigned n);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_zero(
+		__isl_take isl_space *space);
+
+The following convenience functions first create a base expression and
+then create a piecewise expression over a given domain.
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_aff *isl_pw_aff_val_on_domain(
+		__isl_take isl_set *domain,
+		__isl_take isl_val *v);
+	__isl_give isl_pw_multi_aff *
+	isl_pw_multi_aff_multi_val_on_domain(
+		__isl_take isl_set *domain,
+		__isl_take isl_multi_val *mv);
+
+As a convenience, a piecewise multiple expression can
+also be created from a piecewise expression.
+Each multiple expression in the result is derived
+from the corresponding base expression.
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_pw_aff(
+		__isl_take isl_pw_aff *pa);
+
+Similarly, a piecewise quasipolynomial can be
+created from a piecewise quasi-affine expression using
+the following function.
+
+	#include <isl/polynomial.h>
+	__isl_give isl_pw_qpolynomial *
+	isl_pw_qpolynomial_from_pw_aff(
+		__isl_take isl_pw_aff *pwaff);
+
+Piecewise expressions can be copied and freed using the following functions.
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_aff *isl_pw_aff_copy(
+		__isl_keep isl_pw_aff *pwaff);
+	__isl_null isl_pw_aff *isl_pw_aff_free(
+		__isl_take isl_pw_aff *pwaff);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_copy(
+		__isl_keep isl_pw_multi_aff *pma);
+	__isl_null isl_pw_multi_aff *isl_pw_multi_aff_free(
+		__isl_take isl_pw_multi_aff *pma);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_copy(
+		__isl_keep isl_pw_qpolynomial *pwqp);
+	__isl_null isl_pw_qpolynomial *isl_pw_qpolynomial_free(
+		__isl_take isl_pw_qpolynomial *pwqp);
+	__isl_give isl_pw_qpolynomial_fold *
+	isl_pw_qpolynomial_fold_copy(
+		__isl_keep isl_pw_qpolynomial_fold *pwf);
+	__isl_null isl_pw_qpolynomial_fold *
+	isl_pw_qpolynomial_fold_free(
+		__isl_take isl_pw_qpolynomial_fold *pwf);
+
+To iterate over the different cells of a piecewise expression,
+use the following functions.
+
+	#include <isl/aff.h>
+	int isl_pw_aff_is_empty(__isl_keep isl_pw_aff *pwaff);
+	int isl_pw_aff_n_piece(__isl_keep isl_pw_aff *pwaff);
+	int isl_pw_aff_foreach_piece(__isl_keep isl_pw_aff *pwaff,
+		int (*fn)(__isl_take isl_set *set,
+			  __isl_take isl_aff *aff,
+			  void *user), void *user);
+	int isl_pw_multi_aff_foreach_piece(
+		__isl_keep isl_pw_multi_aff *pma,
+		int (*fn)(__isl_take isl_set *set,
+			    __isl_take isl_multi_aff *maff,
+			    void *user), void *user);
+
+	#include <isl/polynomial.h>
+	int isl_pw_qpolynomial_foreach_piece(
+		__isl_keep isl_pw_qpolynomial *pwqp,
+		int (*fn)(__isl_take isl_set *set,
+			  __isl_take isl_qpolynomial *qp,
+			  void *user), void *user);
+	int isl_pw_qpolynomial_foreach_lifted_piece(
+		__isl_keep isl_pw_qpolynomial *pwqp,
+		int (*fn)(__isl_take isl_set *set,
+			  __isl_take isl_qpolynomial *qp,
+			  void *user), void *user);
+	int isl_pw_qpolynomial_fold_foreach_piece(
+		__isl_keep isl_pw_qpolynomial_fold *pwf,
+		int (*fn)(__isl_take isl_set *set,
+			  __isl_take isl_qpolynomial_fold *fold,
+			  void *user), void *user);
+	int isl_pw_qpolynomial_fold_foreach_lifted_piece(
+		__isl_keep isl_pw_qpolynomial_fold *pwf,
+		int (*fn)(__isl_take isl_set *set,
+			  __isl_take isl_qpolynomial_fold *fold,
+			  void *user), void *user);
+
+As usual, the function C<fn> should return C<0> on success
+and C<-1> on failure.  The difference between
+C<isl_pw_qpolynomial_foreach_piece> and
+C<isl_pw_qpolynomial_foreach_lifted_piece> is that
+C<isl_pw_qpolynomial_foreach_lifted_piece> will first
+compute unique representations for all existentially quantified
+variables and then turn these existentially quantified variables
+into extra set variables, adapting the associated quasipolynomial
+accordingly.  This means that the C<set> passed to C<fn>
+will not have any existentially quantified variables, but that
+the dimensions of the sets may be different for different
+invocations of C<fn>.
+Similarly for C<isl_pw_qpolynomial_fold_foreach_piece>
+and C<isl_pw_qpolynomial_fold_foreach_lifted_piece>.
+
+A piecewise expression consisting of the expressions at a given
+position of a piecewise multiple expression can be extracted
+using the following function.
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_aff *isl_pw_multi_aff_get_pw_aff(
+		__isl_keep isl_pw_multi_aff *pma, int pos);
+
+These expressions can be replaced using the following function.
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_set_pw_aff(
+		__isl_take isl_pw_multi_aff *pma, unsigned pos,
+		__isl_take isl_pw_aff *pa);
+
+Note that there is a difference between C<isl_multi_pw_aff> and
+C<isl_pw_multi_aff> objects.  The first is a sequence of piecewise
+affine expressions, while the second is a piecewise sequence
+of affine expressions.  In particular, each of the piecewise
+affine expressions in an C<isl_multi_pw_aff> may have a different
+domain, while all multiple expressions associated to a cell
+in an C<isl_pw_multi_aff> have the same domain.
+It is possible to convert between the two, but when converting
+an C<isl_multi_pw_aff> to an C<isl_pw_multi_aff>, the domain
+of the result is the intersection of the domains of the input.
+The reverse conversion is exact.
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_multi_aff *
+	isl_pw_multi_aff_from_multi_pw_aff(
+		__isl_take isl_multi_pw_aff *mpa);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_from_pw_multi_aff(
+		__isl_take isl_pw_multi_aff *pma);
+
+=head3 Union Expressions
+
+A union expression collects base expressions defined
+over different domains.  The space of a union expression
+is that of the shared parameter space.
+
+The union expression types defined by C<isl>
+are C<isl_union_pw_multi_aff>, C<isl_union_pw_qpolynomial> and
+C<isl_union_pw_qpolynomial_fold>.
+
+An empty union expression can be created using the following functions.
+
+	#include <isl/aff.h>
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_empty(
+		__isl_take isl_space *space);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_union_pw_qpolynomial *
+	isl_union_pw_qpolynomial_zero(
+		__isl_take isl_space *space);
+
+A union expression containing a single base expression
+can be created using the following functions.
+
+	#include <isl/aff.h>
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_from_pw_multi_aff(
+		__isl_take isl_pw_multi_aff *pma);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_union_pw_qpolynomial *
+	isl_union_pw_qpolynomial_from_pw_qpolynomial(
+		__isl_take isl_pw_qpolynomial *pwqp);
+
+The following function creates a base expression on each
+of the sets in the union set and collects the results.
+
+	#include <isl/aff.h>
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_multi_val_on_domain(
+		__isl_take isl_union_set *domain,
+		__isl_take isl_multi_val *mv);
+
+A base expression can be added to a union expression using
+the following functions.
+
+	#include <isl/aff.h>
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_add_pw_multi_aff(
+		__isl_take isl_union_pw_multi_aff *upma,
+		__isl_take isl_pw_multi_aff *pma);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_union_pw_qpolynomial *
+	isl_union_pw_qpolynomial_add_pw_qpolynomial(
+		__isl_take isl_union_pw_qpolynomial *upwqp,
+		__isl_take isl_pw_qpolynomial *pwqp);
+
+Union expressions can be copied and freed using
+the following functions.
+
+	#include <isl/aff.h>
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_copy(
+		__isl_keep isl_union_pw_multi_aff *upma);
+	__isl_null isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_free(
+		__isl_take isl_union_pw_multi_aff *upma);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_union_pw_qpolynomial *
+	isl_union_pw_qpolynomial_copy(
+		__isl_keep isl_union_pw_qpolynomial *upwqp);
+	__isl_null isl_union_pw_qpolynomial *
+	isl_union_pw_qpolynomial_free(
+		__isl_take isl_union_pw_qpolynomial *upwqp);
+	__isl_give isl_union_pw_qpolynomial_fold *
+	isl_union_pw_qpolynomial_fold_copy(
+		__isl_keep isl_union_pw_qpolynomial_fold *upwf);
+	__isl_null isl_union_pw_qpolynomial_fold *
+	isl_union_pw_qpolynomial_fold_free(
+		__isl_take isl_union_pw_qpolynomial_fold *upwf);
+
+To iterate over the base expressions in a union expression,
+use the following functions.
+
+	#include <isl/aff.h>
+	int isl_union_pw_multi_aff_foreach_pw_multi_aff(
+		__isl_keep isl_union_pw_multi_aff *upma,
+		int (*fn)(__isl_take isl_pw_multi_aff *pma,
+			    void *user), void *user);
+
+	#include <isl/polynomial.h>
+	int isl_union_pw_qpolynomial_foreach_pw_qpolynomial(
+		__isl_keep isl_union_pw_qpolynomial *upwqp,
+		int (*fn)(__isl_take isl_pw_qpolynomial *pwqp,
+			    void *user), void *user);
+	int isl_union_pw_qpolynomial_fold_foreach_pw_qpolynomial_fold(
+		__isl_keep isl_union_pw_qpolynomial_fold *upwf,
+		int (*fn)(__isl_take isl_pw_qpolynomial_fold *pwf,
+			    void *user), void *user);
+
+To extract the base expression in a given space from a union, use
+the following functions.
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_multi_aff *
+	isl_union_pw_multi_aff_extract_pw_multi_aff(
+		__isl_keep isl_union_pw_multi_aff *upma,
+		__isl_take isl_space *space);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_pw_qpolynomial *
+	isl_union_pw_qpolynomial_extract_pw_qpolynomial(
+		__isl_keep isl_union_pw_qpolynomial *upwqp,
+		__isl_take isl_space *space);
+
+=head2 Input and Output
+
+For set and relation,
+C<isl> supports its own input/output format, which is similar
+to the C<Omega> format, but also supports the C<PolyLib> format
+in some cases.
+For other object types, typically only an C<isl> format is supported.
+
+=head3 C<isl> format
+
+The C<isl> format is similar to that of C<Omega>, but has a different
+syntax for describing the parameters and allows for the definition
+of an existentially quantified variable as the integer division
+of an affine expression.
+For example, the set of integers C<i> between C<0> and C<n>
+such that C<i % 10 <= 6> can be described as
+
+	[n] -> { [i] : exists (a = [i/10] : 0 <= i and i <= n and
+				i - 10 a <= 6) }
+
+A set or relation can have several disjuncts, separated
+by the keyword C<or>.  Each disjunct is either a conjunction
+of constraints or a projection (C<exists>) of a conjunction
+of constraints.  The constraints are separated by the keyword
+C<and>.
+
+=head3 C<PolyLib> format
+
+If the represented set is a union, then the first line
+contains a single number representing the number of disjuncts.
+Otherwise, a line containing the number C<1> is optional.
+
+Each disjunct is represented by a matrix of constraints.
+The first line contains two numbers representing
+the number of rows and columns,
+where the number of rows is equal to the number of constraints
+and the number of columns is equal to two plus the number of variables.
+The following lines contain the actual rows of the constraint matrix.
+In each row, the first column indicates whether the constraint
+is an equality (C<0>) or inequality (C<1>).  The final column
+corresponds to the constant term.
+
+If the set is parametric, then the coefficients of the parameters
+appear in the last columns before the constant column.
+The coefficients of any existentially quantified variables appear
+between those of the set variables and those of the parameters.
+
+=head3 Extended C<PolyLib> format
+
+The extended C<PolyLib> format is nearly identical to the
+C<PolyLib> format.  The only difference is that the line
+containing the number of rows and columns of a constraint matrix
+also contains four additional numbers:
+the number of output dimensions, the number of input dimensions,
+the number of local dimensions (i.e., the number of existentially
+quantified variables) and the number of parameters.
+For sets, the number of ``output'' dimensions is equal
+to the number of set dimensions, while the number of ``input''
+dimensions is zero.
+
+=head3 Input
+
+Objects can be read from input using the following functions.
+
+	#include <isl/val.h>
+	__isl_give isl_val *isl_val_read_from_str(isl_ctx *ctx,
+		const char *str);
+
+	#include <isl/set.h>
+	__isl_give isl_basic_set *isl_basic_set_read_from_file(
+		isl_ctx *ctx, FILE *input);
+	__isl_give isl_basic_set *isl_basic_set_read_from_str(
+		isl_ctx *ctx, const char *str);
+	__isl_give isl_set *isl_set_read_from_file(isl_ctx *ctx,
+		FILE *input);
+	__isl_give isl_set *isl_set_read_from_str(isl_ctx *ctx,
+		const char *str);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *isl_basic_map_read_from_file(
+		isl_ctx *ctx, FILE *input);
+	__isl_give isl_basic_map *isl_basic_map_read_from_str(
+		isl_ctx *ctx, const char *str);
+	__isl_give isl_map *isl_map_read_from_file(
+		isl_ctx *ctx, FILE *input);
+	__isl_give isl_map *isl_map_read_from_str(isl_ctx *ctx,
+		const char *str);
+
+	#include <isl/union_set.h>
+	__isl_give isl_union_set *isl_union_set_read_from_file(
+		isl_ctx *ctx, FILE *input);
+	__isl_give isl_union_set *isl_union_set_read_from_str(
+		isl_ctx *ctx, const char *str);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *isl_union_map_read_from_file(
+		isl_ctx *ctx, FILE *input);
+	__isl_give isl_union_map *isl_union_map_read_from_str(
+		isl_ctx *ctx, const char *str);
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_aff_read_from_str(
+		isl_ctx *ctx, const char *str);
+	__isl_give isl_multi_aff *isl_multi_aff_read_from_str(
+		isl_ctx *ctx, const char *str);
+	__isl_give isl_pw_aff *isl_pw_aff_read_from_str(
+		isl_ctx *ctx, const char *str);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_read_from_str(
+		isl_ctx *ctx, const char *str);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_read_from_str(
+		isl_ctx *ctx, const char *str);
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_read_from_str(
+		isl_ctx *ctx, const char *str);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_union_pw_qpolynomial *
+	isl_union_pw_qpolynomial_read_from_str(
+		isl_ctx *ctx, const char *str);
+
+For sets and relations,
+the input format is autodetected and may be either the C<PolyLib> format
+or the C<isl> format.
+
+=head3 Output
+
+Before anything can be printed, an C<isl_printer> needs to
+be created.
+
+	__isl_give isl_printer *isl_printer_to_file(isl_ctx *ctx,
+		FILE *file);
+	__isl_give isl_printer *isl_printer_to_str(isl_ctx *ctx);
+	__isl_null isl_printer *isl_printer_free(
+		__isl_take isl_printer *printer);
+	__isl_give char *isl_printer_get_str(
+		__isl_keep isl_printer *printer);
+
+The printer can be inspected using the following functions.
+
+	FILE *isl_printer_get_file(
+		__isl_keep isl_printer *printer);
+	int isl_printer_get_output_format(
+		__isl_keep isl_printer *p);
+
+The behavior of the printer can be modified in various ways
+
+	__isl_give isl_printer *isl_printer_set_output_format(
+		__isl_take isl_printer *p, int output_format);
+	__isl_give isl_printer *isl_printer_set_indent(
+		__isl_take isl_printer *p, int indent);
+	__isl_give isl_printer *isl_printer_set_indent_prefix(
+		__isl_take isl_printer *p, const char *prefix);
+	__isl_give isl_printer *isl_printer_indent(
+		__isl_take isl_printer *p, int indent);
+	__isl_give isl_printer *isl_printer_set_prefix(
+		__isl_take isl_printer *p, const char *prefix);
+	__isl_give isl_printer *isl_printer_set_suffix(
+		__isl_take isl_printer *p, const char *suffix);
+
+The C<output_format> may be either C<ISL_FORMAT_ISL>, C<ISL_FORMAT_OMEGA>,
+C<ISL_FORMAT_POLYLIB>, C<ISL_FORMAT_EXT_POLYLIB> or C<ISL_FORMAT_LATEX>
+and defaults to C<ISL_FORMAT_ISL>.
+Each line in the output is prefixed by C<indent_prefix>,
+indented by C<indent> (set by C<isl_printer_set_indent>) spaces
+(default: 0), prefixed by C<prefix> and suffixed by C<suffix>.
+In the C<PolyLib> format output,
+the coefficients of the existentially quantified variables
+appear between those of the set variables and those
+of the parameters.
+The function C<isl_printer_indent> increases the indentation
+by the specified amount (which may be negative).
+
+To actually print something, use
+
+	#include <isl/printer.h>
+	__isl_give isl_printer *isl_printer_print_double(
+		__isl_take isl_printer *p, double d);
+
+	#include <isl/val.h>
+	__isl_give isl_printer *isl_printer_print_val(
+		__isl_take isl_printer *p, __isl_keep isl_val *v);
+
+	#include <isl/set.h>
+	__isl_give isl_printer *isl_printer_print_basic_set(
+		__isl_take isl_printer *printer,
+		__isl_keep isl_basic_set *bset);
+	__isl_give isl_printer *isl_printer_print_set(
+		__isl_take isl_printer *printer,
+		__isl_keep isl_set *set);
+
+	#include <isl/map.h>
+	__isl_give isl_printer *isl_printer_print_basic_map(
+		__isl_take isl_printer *printer,
+		__isl_keep isl_basic_map *bmap);
+	__isl_give isl_printer *isl_printer_print_map(
+		__isl_take isl_printer *printer,
+		__isl_keep isl_map *map);
+
+	#include <isl/union_set.h>
+	__isl_give isl_printer *isl_printer_print_union_set(
+		__isl_take isl_printer *p,
+		__isl_keep isl_union_set *uset);
+
+	#include <isl/union_map.h>
+	__isl_give isl_printer *isl_printer_print_union_map(
+		__isl_take isl_printer *p,
+		__isl_keep isl_union_map *umap);
+
+	#include <isl/val.h>
+	__isl_give isl_printer *isl_printer_print_multi_val(
+		__isl_take isl_printer *p,
+		__isl_keep isl_multi_val *mv);
+
+	#include <isl/aff.h>
+	__isl_give isl_printer *isl_printer_print_aff(
+		__isl_take isl_printer *p, __isl_keep isl_aff *aff);
+	__isl_give isl_printer *isl_printer_print_multi_aff(
+		__isl_take isl_printer *p,
+		__isl_keep isl_multi_aff *maff);
+	__isl_give isl_printer *isl_printer_print_pw_aff(
+		__isl_take isl_printer *p,
+		__isl_keep isl_pw_aff *pwaff);
+	__isl_give isl_printer *isl_printer_print_pw_multi_aff(
+		__isl_take isl_printer *p,
+		__isl_keep isl_pw_multi_aff *pma);
+	__isl_give isl_printer *isl_printer_print_multi_pw_aff(
+		__isl_take isl_printer *p,
+		__isl_keep isl_multi_pw_aff *mpa);
+	__isl_give isl_printer *isl_printer_print_union_pw_multi_aff(
+		__isl_take isl_printer *p,
+		__isl_keep isl_union_pw_multi_aff *upma);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_printer *isl_printer_print_qpolynomial(
+		__isl_take isl_printer *p,
+		__isl_keep isl_qpolynomial *qp);
+	__isl_give isl_printer *isl_printer_print_pw_qpolynomial(
+		__isl_take isl_printer *p,
+		__isl_keep isl_pw_qpolynomial *pwqp);
+	__isl_give isl_printer *isl_printer_print_union_pw_qpolynomial(
+		__isl_take isl_printer *p,
+		__isl_keep isl_union_pw_qpolynomial *upwqp);
+
+	__isl_give isl_printer *
+	isl_printer_print_pw_qpolynomial_fold(
+		__isl_take isl_printer *p,
+		__isl_keep isl_pw_qpolynomial_fold *pwf);
+	__isl_give isl_printer *
+	isl_printer_print_union_pw_qpolynomial_fold(
+		__isl_take isl_printer *p,
+		__isl_keep isl_union_pw_qpolynomial_fold *upwf);
+
+For C<isl_printer_print_qpolynomial>,
+C<isl_printer_print_pw_qpolynomial> and
+C<isl_printer_print_pw_qpolynomial_fold>,
+the output format of the printer
+needs to be set to either C<ISL_FORMAT_ISL> or C<ISL_FORMAT_C>.
+For C<isl_printer_print_union_pw_qpolynomial> and
+C<isl_printer_print_union_pw_qpolynomial_fold>, only C<ISL_FORMAT_ISL>
+is supported.
+In case of printing in C<ISL_FORMAT_C>, the user may want
+to set the names of all dimensions first.
+
+When called on a file printer, the following function flushes
+the file.  When called on a string printer, the buffer is cleared.
+
+	__isl_give isl_printer *isl_printer_flush(
+		__isl_take isl_printer *p);
+
+Alternatively, a string representation can be obtained
+directly using the following functions, which always print
+in isl format.
+
+	#include <isl/space.h>
+	__isl_give char *isl_space_to_str(
+		__isl_keep isl_space *space);
+
+	#include <isl/val.h>
+	__isl_give char *isl_val_to_str(__isl_keep isl_val *v);
+	__isl_give char *isl_multi_val_to_str(
+		__isl_keep isl_multi_val *mv);
+
+	#include <isl/set.h>
+	__isl_give char *isl_set_to_str(
+		__isl_keep isl_set *set);
+
+	#include <isl/union_set.h>
+	__isl_give char *isl_union_set_to_str(
+		__isl_keep isl_union_set *uset);
+
+	#include <isl/map.h>
+	__isl_give char *isl_map_to_str(
+		__isl_keep isl_map *map);
+
+	#include <isl/union_map.h>
+	__isl_give char *isl_union_map_to_str(
+		__isl_keep isl_union_map *umap);
+
+	#include <isl/aff.h>
+	__isl_give char *isl_multi_aff_to_str(
+		__isl_keep isl_multi_aff *aff);
+	__isl_give char *isl_union_pw_multi_aff_to_str(
+		__isl_keep isl_union_pw_multi_aff *upma);
+
+=head2 Properties
+
+=head3 Unary Properties
+
+=over
+
+=item * Emptiness
+
+The following functions test whether the given set or relation
+contains any integer points.  The ``plain'' variants do not perform
+any computations, but simply check if the given set or relation
+is already known to be empty.
+
+	int isl_basic_set_plain_is_empty(__isl_keep isl_basic_set *bset);
+	int isl_basic_set_is_empty(__isl_keep isl_basic_set *bset);
+	int isl_set_plain_is_empty(__isl_keep isl_set *set);
+	int isl_set_is_empty(__isl_keep isl_set *set);
+	int isl_union_set_is_empty(__isl_keep isl_union_set *uset);
+	int isl_basic_map_plain_is_empty(__isl_keep isl_basic_map *bmap);
+	int isl_basic_map_is_empty(__isl_keep isl_basic_map *bmap);
+	int isl_map_plain_is_empty(__isl_keep isl_map *map);
+	int isl_map_is_empty(__isl_keep isl_map *map);
+	int isl_union_map_is_empty(__isl_keep isl_union_map *umap);
+
+=item * Universality
+
+	int isl_basic_set_is_universe(__isl_keep isl_basic_set *bset);
+	int isl_basic_map_is_universe(__isl_keep isl_basic_map *bmap);
+	int isl_set_plain_is_universe(__isl_keep isl_set *set);
+
+=item * Single-valuedness
+
+	#include <isl/set.h>
+	int isl_set_is_singleton(__isl_keep isl_set *set);
+
+	#include <isl/map.h>
+	int isl_basic_map_is_single_valued(
+		__isl_keep isl_basic_map *bmap);
+	int isl_map_plain_is_single_valued(
+		__isl_keep isl_map *map);
+	int isl_map_is_single_valued(__isl_keep isl_map *map);
+
+	#include <isl/union_map.h>
+	int isl_union_map_is_single_valued(__isl_keep isl_union_map *umap);
+
+=item * Injectivity
+
+	int isl_map_plain_is_injective(__isl_keep isl_map *map);
+	int isl_map_is_injective(__isl_keep isl_map *map);
+	int isl_union_map_plain_is_injective(
+		__isl_keep isl_union_map *umap);
+	int isl_union_map_is_injective(
+		__isl_keep isl_union_map *umap);
+
+=item * Bijectivity
+
+	int isl_map_is_bijective(__isl_keep isl_map *map);
+	int isl_union_map_is_bijective(__isl_keep isl_union_map *umap);
+
+=item * Position
+
+	__isl_give isl_val *
+	isl_basic_map_plain_get_val_if_fixed(
+		__isl_keep isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_val *isl_set_plain_get_val_if_fixed(
+		__isl_keep isl_set *set,
+		enum isl_dim_type type, unsigned pos);
+	__isl_give isl_val *isl_map_plain_get_val_if_fixed(
+		__isl_keep isl_map *map,
+		enum isl_dim_type type, unsigned pos);
+
+If the set or relation obviously lies on a hyperplane where the given dimension
+has a fixed value, then return that value.
+Otherwise return NaN.
+
+=item * Stride
+
+	int isl_set_dim_residue_class_val(
+		__isl_keep isl_set *set,
+		int pos, __isl_give isl_val **modulo,
+		__isl_give isl_val **residue);
+
+Check if the values of the given set dimension are equal to a fixed
+value modulo some integer value.  If so, assign the modulo to C<*modulo>
+and the fixed value to C<*residue>.  If the given dimension attains only
+a single value, then assign C<0> to C<*modulo> and the fixed value to
+C<*residue>.
+If the dimension does not attain only a single value and if no modulo
+can be found then assign C<1> to C<*modulo> and C<1> to C<*residue>.
+
+=item * Dependence
+
+To check whether the description of a set, relation or function depends
+on one or more given dimensions,
+the following functions can be used.
+
+	#include <isl/constraint.h>
+	int isl_constraint_involves_dims(
+		__isl_keep isl_constraint *constraint,
+		enum isl_dim_type type, unsigned first, unsigned n);
+
+	#include <isl/set.h>
+	int isl_basic_set_involves_dims(
+		__isl_keep isl_basic_set *bset,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	int isl_set_involves_dims(__isl_keep isl_set *set,
+		enum isl_dim_type type, unsigned first, unsigned n);
+
+	#include <isl/map.h>
+	int isl_basic_map_involves_dims(
+		__isl_keep isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	int isl_map_involves_dims(__isl_keep isl_map *map,
+		enum isl_dim_type type, unsigned first, unsigned n);
+
+	#include <isl/union_map.h>
+	int isl_union_map_involves_dims(
+		__isl_keep isl_union_map *umap,
+		enum isl_dim_type type, unsigned first, unsigned n);
+
+	#include <isl/aff.h>
+	int isl_aff_involves_dims(__isl_keep isl_aff *aff,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	int isl_pw_aff_involves_dims(__isl_keep isl_pw_aff *pwaff,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	int isl_multi_aff_involves_dims(
+		__isl_keep isl_multi_aff *ma,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	int isl_multi_pw_aff_involves_dims(
+		__isl_keep isl_multi_pw_aff *mpa,
+		enum isl_dim_type type, unsigned first, unsigned n);
+
+Similarly, the following functions can be used to check whether
+a given dimension is involved in any lower or upper bound.
+
+	#include <isl/set.h>
+	int isl_set_dim_has_any_lower_bound(__isl_keep isl_set *set,
+		enum isl_dim_type type, unsigned pos);
+	int isl_set_dim_has_any_upper_bound(__isl_keep isl_set *set,
+		enum isl_dim_type type, unsigned pos);
+
+Note that these functions return true even if there is a bound on
+the dimension on only some of the basic sets of C<set>.
+To check if they have a bound for all of the basic sets in C<set>,
+use the following functions instead.
+
+	#include <isl/set.h>
+	int isl_set_dim_has_lower_bound(__isl_keep isl_set *set,
+		enum isl_dim_type type, unsigned pos);
+	int isl_set_dim_has_upper_bound(__isl_keep isl_set *set,
+		enum isl_dim_type type, unsigned pos);
+
+=item * Space
+
+To check whether a set is a parameter domain, use this function:
+
+	int isl_set_is_params(__isl_keep isl_set *set);
+	int isl_union_set_is_params(
+		__isl_keep isl_union_set *uset);
+
+=item * Wrapping
+
+The following functions check whether the space of the given
+(basic) set or relation range is a wrapped relation.
+
+	#include <isl/space.h>
+	int isl_space_is_wrapping(
+		__isl_keep isl_space *space);
+	int isl_space_domain_is_wrapping(
+		__isl_keep isl_space *space);
+	int isl_space_range_is_wrapping(
+		__isl_keep isl_space *space);
+
+	#include <isl/set.h>
+	int isl_basic_set_is_wrapping(
+		__isl_keep isl_basic_set *bset);
+	int isl_set_is_wrapping(__isl_keep isl_set *set);
+
+	#include <isl/map.h>
+	int isl_map_domain_is_wrapping(
+		__isl_keep isl_map *map);
+	int isl_map_range_is_wrapping(
+		__isl_keep isl_map *map);
+
+	#include <isl/val.h>
+	int isl_multi_val_range_is_wrapping(
+		__isl_keep isl_multi_val *mv);
+
+	#include <isl/aff.h>
+	int isl_multi_aff_range_is_wrapping(
+		__isl_keep isl_multi_aff *ma);
+	int isl_multi_pw_aff_range_is_wrapping(
+		__isl_keep isl_multi_pw_aff *mpa);
+
+The input to C<isl_space_is_wrapping> should
+be the space of a set, while that of
+C<isl_space_domain_is_wrapping> and
+C<isl_space_range_is_wrapping> should be the space of a relation.
+
+=item * Internal Product
+
+	int isl_basic_map_can_zip(
+		__isl_keep isl_basic_map *bmap);
+	int isl_map_can_zip(__isl_keep isl_map *map);
+
+Check whether the product of domain and range of the given relation
+can be computed,
+i.e., whether both domain and range are nested relations.
+
+=item * Currying
+
+	int isl_basic_map_can_curry(
+		__isl_keep isl_basic_map *bmap);
+	int isl_map_can_curry(__isl_keep isl_map *map);
+
+Check whether the domain of the (basic) relation is a wrapped relation.
+
+	int isl_basic_map_can_uncurry(
+		__isl_keep isl_basic_map *bmap);
+	int isl_map_can_uncurry(__isl_keep isl_map *map);
+
+Check whether the range of the (basic) relation is a wrapped relation.
+
+=item * Special Values
+
+	#include <isl/aff.h>
+	int isl_aff_is_cst(__isl_keep isl_aff *aff);
+	int isl_pw_aff_is_cst(__isl_keep isl_pw_aff *pwaff);
+
+Check whether the given expression is a constant.
+
+	#include <isl/aff.h>
+	int isl_aff_is_nan(__isl_keep isl_aff *aff);
+	int isl_pw_aff_involves_nan(__isl_keep isl_pw_aff *pa);
+
+Check whether the given expression is equal to or involves NaN.
+
+	#include <isl/aff.h>
+	int isl_aff_plain_is_zero(__isl_keep isl_aff *aff);
+
+Check whether the affine expression is obviously zero.
+
+=back
+
+=head3 Binary Properties
+
+=over
+
+=item * Equality
+
+The following functions check whether two objects
+represent the same set, relation or function.
+The C<plain> variants only return true if the objects
+are obviously the same.  That is, they may return false
+even if the objects are the same, but they will never
+return true if the objects are not the same.
+
+	#include <isl/set.h>
+	int isl_basic_set_plain_is_equal(
+		__isl_keep isl_basic_set *bset1,
+		__isl_keep isl_basic_set *bset2);
+	int isl_set_plain_is_equal(__isl_keep isl_set *set1,
+		__isl_keep isl_set *set2);
+	int isl_set_is_equal(__isl_keep isl_set *set1,
+		__isl_keep isl_set *set2);
+
+	#include <isl/map.h>
+	int isl_basic_map_is_equal(
+		__isl_keep isl_basic_map *bmap1,
+		__isl_keep isl_basic_map *bmap2);
+	int isl_map_is_equal(__isl_keep isl_map *map1,
+		__isl_keep isl_map *map2);
+	int isl_map_plain_is_equal(__isl_keep isl_map *map1,
+		__isl_keep isl_map *map2);
+
+	#include <isl/union_set.h>
+	int isl_union_set_is_equal(
+		__isl_keep isl_union_set *uset1,
+		__isl_keep isl_union_set *uset2);
+
+	#include <isl/union_map.h>
+	int isl_union_map_is_equal(
+		__isl_keep isl_union_map *umap1,
+		__isl_keep isl_union_map *umap2);
+
+	#include <isl/aff.h>
+	int isl_aff_plain_is_equal(__isl_keep isl_aff *aff1,
+		__isl_keep isl_aff *aff2);
+	int isl_multi_aff_plain_is_equal(
+		__isl_keep isl_multi_aff *maff1,
+		__isl_keep isl_multi_aff *maff2);
+	int isl_pw_aff_plain_is_equal(
+		__isl_keep isl_pw_aff *pwaff1,
+		__isl_keep isl_pw_aff *pwaff2);
+	int isl_pw_multi_aff_plain_is_equal(
+		__isl_keep isl_pw_multi_aff *pma1,
+		__isl_keep isl_pw_multi_aff *pma2);
+	int isl_multi_pw_aff_plain_is_equal(
+		__isl_keep isl_multi_pw_aff *mpa1,
+		__isl_keep isl_multi_pw_aff *mpa2);
+	int isl_multi_pw_aff_is_equal(
+		__isl_keep isl_multi_pw_aff *mpa1,
+		__isl_keep isl_multi_pw_aff *mpa2);
+	int isl_union_pw_multi_aff_plain_is_equal(
+		__isl_keep isl_union_pw_multi_aff *upma1,
+		__isl_keep isl_union_pw_multi_aff *upma2);
+
+	#include <isl/polynomial.h>
+	int isl_union_pw_qpolynomial_plain_is_equal(
+		__isl_keep isl_union_pw_qpolynomial *upwqp1,
+		__isl_keep isl_union_pw_qpolynomial *upwqp2);
+	int isl_union_pw_qpolynomial_fold_plain_is_equal(
+		__isl_keep isl_union_pw_qpolynomial_fold *upwf1,
+		__isl_keep isl_union_pw_qpolynomial_fold *upwf2);
+
+=item * Disjointness
+
+	int isl_basic_set_is_disjoint(
+		__isl_keep isl_basic_set *bset1,
+		__isl_keep isl_basic_set *bset2);
+	int isl_set_plain_is_disjoint(__isl_keep isl_set *set1,
+		__isl_keep isl_set *set2);
+	int isl_set_is_disjoint(__isl_keep isl_set *set1,
+		__isl_keep isl_set *set2);
+	int isl_basic_map_is_disjoint(
+		__isl_keep isl_basic_map *bmap1,
+		__isl_keep isl_basic_map *bmap2);
+	int isl_map_is_disjoint(__isl_keep isl_map *map1,
+		__isl_keep isl_map *map2);
+
+=item * Subset
+
+	int isl_basic_set_is_subset(
+		__isl_keep isl_basic_set *bset1,
+		__isl_keep isl_basic_set *bset2);
+	int isl_set_is_subset(__isl_keep isl_set *set1,
+		__isl_keep isl_set *set2);
+	int isl_set_is_strict_subset(
+		__isl_keep isl_set *set1,
+		__isl_keep isl_set *set2);
+	int isl_union_set_is_subset(
+		__isl_keep isl_union_set *uset1,
+		__isl_keep isl_union_set *uset2);
+	int isl_union_set_is_strict_subset(
+		__isl_keep isl_union_set *uset1,
+		__isl_keep isl_union_set *uset2);
+	int isl_basic_map_is_subset(
+		__isl_keep isl_basic_map *bmap1,
+		__isl_keep isl_basic_map *bmap2);
+	int isl_basic_map_is_strict_subset(
+		__isl_keep isl_basic_map *bmap1,
+		__isl_keep isl_basic_map *bmap2);
+	int isl_map_is_subset(
+		__isl_keep isl_map *map1,
+		__isl_keep isl_map *map2);
+	int isl_map_is_strict_subset(
+		__isl_keep isl_map *map1,
+		__isl_keep isl_map *map2);
+	int isl_union_map_is_subset(
+		__isl_keep isl_union_map *umap1,
+		__isl_keep isl_union_map *umap2);
+	int isl_union_map_is_strict_subset(
+		__isl_keep isl_union_map *umap1,
+		__isl_keep isl_union_map *umap2);
+
+Check whether the first argument is a (strict) subset of the
+second argument.
+
+=item * Order
+
+Every comparison function returns a negative value if the first
+argument is considered smaller than the second, a positive value
+if the first argument is considered greater and zero if the two
+constraints are considered the same by the comparison criterion.
+
+	#include <isl/constraint.h>
+	int isl_constraint_plain_cmp(
+		__isl_keep isl_constraint *c1,
+		__isl_keep isl_constraint *c2);
+
+This function is useful for sorting C<isl_constraint>s.
+The order depends on the internal representation of the inputs.
+The order is fixed over different calls to the function (assuming
+the internal representation of the inputs has not changed), but may
+change over different versions of C<isl>.
+
+	#include <isl/constraint.h>
+	int isl_constraint_cmp_last_non_zero(
+		__isl_keep isl_constraint *c1,
+		__isl_keep isl_constraint *c2);
+
+This function can be used to sort constraints that live in the same
+local space.  Constraints that involve ``earlier'' dimensions or
+that have a smaller coefficient for the shared latest dimension
+are considered smaller than other constraints.
+This function only defines a B<partial> order.
+
+	#include <isl/set.h>
+	int isl_set_plain_cmp(__isl_keep isl_set *set1,
+		__isl_keep isl_set *set2);
+
+This function is useful for sorting C<isl_set>s.
+The order depends on the internal representation of the inputs.
+The order is fixed over different calls to the function (assuming
+the internal representation of the inputs has not changed), but may
+change over different versions of C<isl>.
+
+	#include <isl/aff.h>
+	int isl_pw_aff_plain_cmp(__isl_keep isl_pw_aff *pa1,
+		__isl_keep isl_pw_aff *pa2);
+
+The function C<isl_pw_aff_plain_cmp> can be used to sort
+C<isl_pw_aff>s.  The order is not strictly defined.
+The current order sorts expressions that only involve
+earlier dimensions before those that involve later dimensions.
+
+=back
+
+=head2 Unary Operations
+
+=over
+
+=item * Complement
+
+	__isl_give isl_set *isl_set_complement(
+		__isl_take isl_set *set);
+	__isl_give isl_map *isl_map_complement(
+		__isl_take isl_map *map);
+
+=item * Inverse map
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_reverse(
+		__isl_take isl_space *space);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *isl_basic_map_reverse(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_map *isl_map_reverse(
+		__isl_take isl_map *map);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *isl_union_map_reverse(
+		__isl_take isl_union_map *umap);
+
+=item * Projection
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_domain(
+		__isl_take isl_space *space);
+	__isl_give isl_space *isl_space_range(
+		__isl_take isl_space *space);
+	__isl_give isl_space *isl_space_params(
+		__isl_take isl_space *space);
+
+	#include <isl/local_space.h>
+	__isl_give isl_local_space *isl_local_space_domain(
+		__isl_take isl_local_space *ls);
+	__isl_give isl_local_space *isl_local_space_range(
+		__isl_take isl_local_space *ls);
+
+	#include <isl/set.h>
+	__isl_give isl_basic_set *isl_basic_set_project_out(
+		__isl_take isl_basic_set *bset,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_set *isl_set_project_out(__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_basic_set *isl_basic_set_params(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_set *isl_set_params(__isl_take isl_set *set);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *isl_basic_map_project_out(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_map *isl_map_project_out(__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_basic_set *isl_basic_map_domain(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_basic_set *isl_basic_map_range(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_set *isl_map_params(__isl_take isl_map *map);
+	__isl_give isl_set *isl_map_domain(
+		__isl_take isl_map *bmap);
+	__isl_give isl_set *isl_map_range(
+		__isl_take isl_map *map);
+
+	#include <isl/union_set.h>
+	__isl_give isl_set *isl_union_set_params(
+		__isl_take isl_union_set *uset);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *isl_union_map_project_out(
+		__isl_take isl_union_map *umap,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_set *isl_union_map_params(
+		__isl_take isl_union_map *umap);
+	__isl_give isl_union_set *isl_union_map_domain(
+		__isl_take isl_union_map *umap);
+	__isl_give isl_union_set *isl_union_map_range(
+		__isl_take isl_union_map *umap);
+
+The function C<isl_union_map_project_out> can only project out
+parameters.
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_aff_project_domain_on_params(
+		__isl_take isl_aff *aff);
+	__isl_give isl_pw_multi_aff *
+	isl_pw_multi_aff_project_domain_on_params(
+		__isl_take isl_pw_multi_aff *pma);
+	__isl_give isl_set *isl_pw_aff_domain(
+		__isl_take isl_pw_aff *pwaff);
+	__isl_give isl_set *isl_pw_multi_aff_domain(
+		__isl_take isl_pw_multi_aff *pma);
+	__isl_give isl_set *isl_multi_pw_aff_domain(
+		__isl_take isl_multi_pw_aff *mpa);
+	__isl_give isl_union_set *isl_union_pw_multi_aff_domain(
+		__isl_take isl_union_pw_multi_aff *upma);
+	__isl_give isl_set *isl_pw_aff_params(
+		__isl_take isl_pw_aff *pwa);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_qpolynomial *
+	isl_qpolynomial_project_domain_on_params(
+		__isl_take isl_qpolynomial *qp);
+	__isl_give isl_pw_qpolynomial *
+	isl_pw_qpolynomial_project_domain_on_params(
+		__isl_take isl_pw_qpolynomial *pwqp);
+	__isl_give isl_pw_qpolynomial_fold *
+	isl_pw_qpolynomial_fold_project_domain_on_params(
+		__isl_take isl_pw_qpolynomial_fold *pwf);
+	__isl_give isl_set *isl_pw_qpolynomial_domain(
+		__isl_take isl_pw_qpolynomial *pwqp);
+	__isl_give isl_union_set *isl_union_pw_qpolynomial_fold_domain(
+		__isl_take isl_union_pw_qpolynomial_fold *upwf);
+	__isl_give isl_union_set *isl_union_pw_qpolynomial_domain(
+		__isl_take isl_union_pw_qpolynomial *upwqp);
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_domain_map(
+		__isl_take isl_space *space);
+	__isl_give isl_space *isl_space_range_map(
+		__isl_take isl_space *space);
+
+	#include <isl/map.h>
+	__isl_give isl_map *isl_set_wrapped_domain_map(
+		__isl_take isl_set *set);
+	__isl_give isl_basic_map *isl_basic_map_domain_map(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_basic_map *isl_basic_map_range_map(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_map *isl_map_domain_map(__isl_take isl_map *map);
+	__isl_give isl_map *isl_map_range_map(__isl_take isl_map *map);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *isl_union_map_domain_map(
+		__isl_take isl_union_map *umap);
+	__isl_give isl_union_map *isl_union_map_range_map(
+		__isl_take isl_union_map *umap);
+	__isl_give isl_union_map *
+	isl_union_set_wrapped_domain_map(
+		__isl_take isl_union_set *uset);
+
+The functions above construct a (basic, regular or union) relation
+that maps (a wrapped version of) the input relation to its domain or range.
+C<isl_set_wrapped_domain_map> maps the input set to the domain
+of its wrapped relation.
+
+=item * Elimination
+
+	__isl_give isl_basic_set *isl_basic_set_eliminate(
+		__isl_take isl_basic_set *bset,
+		enum isl_dim_type type,
+		unsigned first, unsigned n);
+	__isl_give isl_set *isl_set_eliminate(
+		__isl_take isl_set *set, enum isl_dim_type type,
+		unsigned first, unsigned n);
+	__isl_give isl_basic_map *isl_basic_map_eliminate(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type,
+		unsigned first, unsigned n);
+	__isl_give isl_map *isl_map_eliminate(
+		__isl_take isl_map *map, enum isl_dim_type type,
+		unsigned first, unsigned n);
+
+Eliminate the coefficients for the given dimensions from the constraints,
+without removing the dimensions.
+
+=item * Constructing a set from a parameter domain
+
+A zero-dimensional space or (basic) set can be constructed
+on a given parameter domain using the following functions.
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_set_from_params(
+		__isl_take isl_space *space);
+
+	#include <isl/set.h>
+	__isl_give isl_basic_set *isl_basic_set_from_params(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_set *isl_set_from_params(
+		__isl_take isl_set *set);
+
+=item * Constructing a relation from a set
+
+Create a relation with the given set as domain or range.
+The range or domain of the created relation is a zero-dimensional
+flat anonymous space.
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_from_domain(
+		__isl_take isl_space *space);
+	__isl_give isl_space *isl_space_from_range(
+		__isl_take isl_space *space);
+	__isl_give isl_space *isl_space_map_from_set(
+		__isl_take isl_space *space);
+	__isl_give isl_space *isl_space_map_from_domain_and_range(
+		__isl_take isl_space *domain,
+		__isl_take isl_space *range);
+
+	#include <isl/local_space.h>
+	__isl_give isl_local_space *isl_local_space_from_domain(
+		__isl_take isl_local_space *ls);
+
+	#include <isl/map.h>
+	__isl_give isl_map *isl_map_from_domain(
+		__isl_take isl_set *set);
+	__isl_give isl_map *isl_map_from_range(
+		__isl_take isl_set *set);
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_from_range(
+		__isl_take isl_multi_val *mv);
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *isl_multi_aff_from_range(
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_pw_aff *isl_pw_aff_from_range(
+		__isl_take isl_pw_aff *pwa);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_from_range(
+		__isl_take isl_multi_pw_aff *mpa);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_domain(
+		__isl_take isl_set *set);
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_from_domain(
+		__isl_take isl_union_set *uset);
+
+=item * Slicing
+
+	#include <isl/set.h>
+	__isl_give isl_basic_set *isl_basic_set_fix_si(
+		__isl_take isl_basic_set *bset,
+		enum isl_dim_type type, unsigned pos, int value);
+	__isl_give isl_basic_set *isl_basic_set_fix_val(
+		__isl_take isl_basic_set *bset,
+		enum isl_dim_type type, unsigned pos,
+		__isl_take isl_val *v);
+	__isl_give isl_set *isl_set_fix_si(__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned pos, int value);
+	__isl_give isl_set *isl_set_fix_val(
+		__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned pos,
+		__isl_take isl_val *v);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *isl_basic_map_fix_si(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned pos, int value);
+	__isl_give isl_basic_map *isl_basic_map_fix_val(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned pos,
+		__isl_take isl_val *v);
+	__isl_give isl_map *isl_map_fix_si(__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned pos, int value);
+	__isl_give isl_map *isl_map_fix_val(
+		__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned pos,
+		__isl_take isl_val *v);
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_fix_si(
+		__isl_take isl_pw_multi_aff *pma,
+		enum isl_dim_type type, unsigned pos, int value);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_fix_val(
+		__isl_take isl_pw_qpolynomial *pwqp,
+		enum isl_dim_type type, unsigned n,
+		__isl_take isl_val *v);
+
+Intersect the set, relation or function domain
+with the hyperplane where the given
+dimension has the fixed given value.
+
+	__isl_give isl_basic_map *isl_basic_map_lower_bound_si(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned pos, int value);
+	__isl_give isl_basic_map *isl_basic_map_upper_bound_si(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned pos, int value);
+	__isl_give isl_set *isl_set_lower_bound_si(
+		__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned pos, int value);
+	__isl_give isl_set *isl_set_lower_bound_val(
+		__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned pos,
+		__isl_take isl_val *value);
+	__isl_give isl_map *isl_map_lower_bound_si(
+		__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned pos, int value);
+	__isl_give isl_set *isl_set_upper_bound_si(
+		__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned pos, int value);
+	__isl_give isl_set *isl_set_upper_bound_val(
+		__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned pos,
+		__isl_take isl_val *value);
+	__isl_give isl_map *isl_map_upper_bound_si(
+		__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned pos, int value);
+
+Intersect the set or relation with the half-space where the given
+dimension has a value bounded by the fixed given integer value.
+
+	__isl_give isl_set *isl_set_equate(__isl_take isl_set *set,
+		enum isl_dim_type type1, int pos1,
+		enum isl_dim_type type2, int pos2);
+	__isl_give isl_basic_map *isl_basic_map_equate(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type1, int pos1,
+		enum isl_dim_type type2, int pos2);
+	__isl_give isl_map *isl_map_equate(__isl_take isl_map *map,
+		enum isl_dim_type type1, int pos1,
+		enum isl_dim_type type2, int pos2);
+
+Intersect the set or relation with the hyperplane where the given
+dimensions are equal to each other.
+
+	__isl_give isl_map *isl_map_oppose(__isl_take isl_map *map,
+		enum isl_dim_type type1, int pos1,
+		enum isl_dim_type type2, int pos2);
+
+Intersect the relation with the hyperplane where the given
+dimensions have opposite values.
+
+	__isl_give isl_map *isl_map_order_le(
+		__isl_take isl_map *map,
+		enum isl_dim_type type1, int pos1,
+		enum isl_dim_type type2, int pos2);
+	__isl_give isl_basic_map *isl_basic_map_order_ge(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type1, int pos1,
+		enum isl_dim_type type2, int pos2);
+	__isl_give isl_map *isl_map_order_ge(
+		__isl_take isl_map *map,
+		enum isl_dim_type type1, int pos1,
+		enum isl_dim_type type2, int pos2);
+	__isl_give isl_map *isl_map_order_lt(__isl_take isl_map *map,
+		enum isl_dim_type type1, int pos1,
+		enum isl_dim_type type2, int pos2);
+	__isl_give isl_basic_map *isl_basic_map_order_gt(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type1, int pos1,
+		enum isl_dim_type type2, int pos2);
+	__isl_give isl_map *isl_map_order_gt(__isl_take isl_map *map,
+		enum isl_dim_type type1, int pos1,
+		enum isl_dim_type type2, int pos2);
+
+Intersect the relation with the half-space where the given
+dimensions satisfy the given ordering.
+
+=item * Locus
+
+	#include <isl/aff.h>
+	__isl_give isl_basic_set *isl_aff_zero_basic_set(
+		__isl_take isl_aff *aff);
+	__isl_give isl_basic_set *isl_aff_neg_basic_set(
+		__isl_take isl_aff *aff);
+	__isl_give isl_set *isl_pw_aff_nonneg_set(
+		__isl_take isl_pw_aff *pwaff);
+	__isl_give isl_set *isl_pw_aff_zero_set(
+		__isl_take isl_pw_aff *pwaff);
+	__isl_give isl_set *isl_pw_aff_non_zero_set(
+		__isl_take isl_pw_aff *pwaff);
+
+The function C<isl_aff_neg_basic_set> returns a basic set
+containing those elements in the domain space
+of C<aff> where C<aff> is negative.
+The function C<isl_pw_aff_nonneg_set> returns a set
+containing those elements in the domain
+of C<pwaff> where C<pwaff> is non-negative.
+
+=item * Identity
+
+	__isl_give isl_map *isl_set_identity(
+		__isl_take isl_set *set);
+	__isl_give isl_union_map *isl_union_set_identity(
+		__isl_take isl_union_set *uset);
+
+Construct an identity relation on the given (union) set.
+
+=item * Function Extraction
+
+A piecewise quasi affine expression that is equal to 1 on a set
+and 0 outside the set can be created using the following function.
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_aff *isl_set_indicator_function(
+		__isl_take isl_set *set);
+
+A piecewise multiple quasi affine expression can be extracted
+from an C<isl_set> or C<isl_map>, provided the C<isl_set> is a singleton
+and the C<isl_map> is single-valued.
+In case of a conversion from an C<isl_union_map>
+to an C<isl_union_pw_multi_aff>, these properties need to hold
+in each domain space.
+
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_set(
+		__isl_take isl_set *set);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_map(
+		__isl_take isl_map *map);
+
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_from_union_set(
+		__isl_take isl_union_set *uset);
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_from_union_map(
+		__isl_take isl_union_map *umap);
+
+=item * Deltas
+
+	__isl_give isl_basic_set *isl_basic_map_deltas(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_set *isl_map_deltas(__isl_take isl_map *map);
+	__isl_give isl_union_set *isl_union_map_deltas(
+		__isl_take isl_union_map *umap);
+
+These functions return a (basic) set containing the differences
+between image elements and corresponding domain elements in the input.
+
+	__isl_give isl_basic_map *isl_basic_map_deltas_map(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_map *isl_map_deltas_map(
+		__isl_take isl_map *map);
+	__isl_give isl_union_map *isl_union_map_deltas_map(
+		__isl_take isl_union_map *umap);
+
+The functions above construct a (basic, regular or union) relation
+that maps (a wrapped version of) the input relation to its delta set.
+
+=item * Coalescing
+
+Simplify the representation of a set, relation or functions by trying
+to combine pairs of basic sets or relations into a single
+basic set or relation.
+
+	#include <isl/set.h>
+	__isl_give isl_set *isl_set_coalesce(__isl_take isl_set *set);
+
+	#include <isl/map.h>
+	__isl_give isl_map *isl_map_coalesce(__isl_take isl_map *map);
+
+	#include <isl/union_set.h>
+	__isl_give isl_union_set *isl_union_set_coalesce(
+		__isl_take isl_union_set *uset);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *isl_union_map_coalesce(
+		__isl_take isl_union_map *umap);
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_aff *isl_pw_aff_coalesce(
+		__isl_take isl_pw_aff *pwqp);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_coalesce(
+		__isl_take isl_pw_multi_aff *pma);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_coalesce(
+		__isl_take isl_multi_pw_aff *mpa);
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_coalesce(
+		__isl_take isl_union_pw_multi_aff *upma);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_pw_qpolynomial_fold *
+	isl_pw_qpolynomial_fold_coalesce(
+		__isl_take isl_pw_qpolynomial_fold *pwf);
+	__isl_give isl_union_pw_qpolynomial *
+	isl_union_pw_qpolynomial_coalesce(
+		__isl_take isl_union_pw_qpolynomial *upwqp);
+	__isl_give isl_union_pw_qpolynomial_fold *
+	isl_union_pw_qpolynomial_fold_coalesce(
+		__isl_take isl_union_pw_qpolynomial_fold *upwf);
+
+One of the methods for combining pairs of basic sets or relations
+can result in coefficients that are much larger than those that appear
+in the constraints of the input.  By default, the coefficients are
+not allowed to grow larger, but this can be changed by unsetting
+the following option.
+
+	int isl_options_set_coalesce_bounded_wrapping(
+		isl_ctx *ctx, int val);
+	int isl_options_get_coalesce_bounded_wrapping(
+		isl_ctx *ctx);
+
+=item * Detecting equalities
+
+	__isl_give isl_basic_set *isl_basic_set_detect_equalities(
+                __isl_take isl_basic_set *bset);
+	__isl_give isl_basic_map *isl_basic_map_detect_equalities(
+                __isl_take isl_basic_map *bmap);
+	__isl_give isl_set *isl_set_detect_equalities(
+		__isl_take isl_set *set);
+	__isl_give isl_map *isl_map_detect_equalities(
+		__isl_take isl_map *map);
+	__isl_give isl_union_set *isl_union_set_detect_equalities(
+		__isl_take isl_union_set *uset);
+	__isl_give isl_union_map *isl_union_map_detect_equalities(
+		__isl_take isl_union_map *umap);
+
+Simplify the representation of a set or relation by detecting implicit
+equalities.
+
+=item * Removing redundant constraints
+
+	__isl_give isl_basic_set *isl_basic_set_remove_redundancies(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_set *isl_set_remove_redundancies(
+		__isl_take isl_set *set);
+	__isl_give isl_basic_map *isl_basic_map_remove_redundancies(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_map *isl_map_remove_redundancies(
+		__isl_take isl_map *map);
+
+=item * Convex hull
+
+	__isl_give isl_basic_set *isl_set_convex_hull(
+		__isl_take isl_set *set);
+	__isl_give isl_basic_map *isl_map_convex_hull(
+		__isl_take isl_map *map);
+
+If the input set or relation has any existentially quantified
+variables, then the result of these operations is currently undefined.
+
+=item * Simple hull
+
+	#include <isl/set.h>
+	__isl_give isl_basic_set *
+	isl_set_unshifted_simple_hull(
+		__isl_take isl_set *set);
+	__isl_give isl_basic_set *isl_set_simple_hull(
+		__isl_take isl_set *set);
+	__isl_give isl_basic_set *
+	isl_set_unshifted_simple_hull_from_set_list(
+		__isl_take isl_set *set,
+		__isl_take isl_set_list *list);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *
+	isl_map_unshifted_simple_hull(
+		__isl_take isl_map *map);
+	__isl_give isl_basic_map *isl_map_simple_hull(
+		__isl_take isl_map *map);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *isl_union_map_simple_hull(
+		__isl_take isl_union_map *umap);
+
+These functions compute a single basic set or relation
+that contains the whole input set or relation.
+In particular, the output is described by translates
+of the constraints describing the basic sets or relations in the input.
+In case of C<isl_set_unshifted_simple_hull>, only the original
+constraints are used, without any translation.
+In case of C<isl_set_unshifted_simple_hull_from_set_list>, the
+constraints are taken from the elements of the second argument.
+
+=begin latex
+
+(See \autoref{s:simple hull}.)
+
+=end latex
+
+=item * Affine hull
+
+	__isl_give isl_basic_set *isl_basic_set_affine_hull(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_basic_set *isl_set_affine_hull(
+		__isl_take isl_set *set);
+	__isl_give isl_union_set *isl_union_set_affine_hull(
+		__isl_take isl_union_set *uset);
+	__isl_give isl_basic_map *isl_basic_map_affine_hull(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_basic_map *isl_map_affine_hull(
+		__isl_take isl_map *map);
+	__isl_give isl_union_map *isl_union_map_affine_hull(
+		__isl_take isl_union_map *umap);
+
+In case of union sets and relations, the affine hull is computed
+per space.
+
+=item * Polyhedral hull
+
+	__isl_give isl_basic_set *isl_set_polyhedral_hull(
+		__isl_take isl_set *set);
+	__isl_give isl_basic_map *isl_map_polyhedral_hull(
+		__isl_take isl_map *map);
+	__isl_give isl_union_set *isl_union_set_polyhedral_hull(
+		__isl_take isl_union_set *uset);
+	__isl_give isl_union_map *isl_union_map_polyhedral_hull(
+		__isl_take isl_union_map *umap);
+
+These functions compute a single basic set or relation
+not involving any existentially quantified variables
+that contains the whole input set or relation.
+In case of union sets and relations, the polyhedral hull is computed
+per space.
+
+=item * Other approximations
+
+	#include <isl/set.h>
+	__isl_give isl_basic_set *
+	isl_basic_set_drop_constraints_involving_dims(
+		__isl_take isl_basic_set *bset,
+		enum isl_dim_type type,
+		unsigned first, unsigned n);
+	__isl_give isl_basic_set *
+	isl_basic_set_drop_constraints_not_involving_dims(
+		__isl_take isl_basic_set *bset,
+		enum isl_dim_type type,
+		unsigned first, unsigned n);
+	__isl_give isl_set *
+	isl_set_drop_constraints_involving_dims(
+		__isl_take isl_set *set,
+		enum isl_dim_type type,
+		unsigned first, unsigned n);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *
+	isl_basic_map_drop_constraints_involving_dims(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type,
+		unsigned first, unsigned n);
+	__isl_give isl_map *
+	isl_map_drop_constraints_involving_dims(
+		__isl_take isl_map *map,
+		enum isl_dim_type type,
+		unsigned first, unsigned n);
+
+These functions drop any constraints (not) involving the specified dimensions.
+Note that the result depends on the representation of the input.
+
+	#include <isl/polynomial.h>
+	__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_to_polynomial(
+		__isl_take isl_pw_qpolynomial *pwqp, int sign);
+	__isl_give isl_union_pw_qpolynomial *
+	isl_union_pw_qpolynomial_to_polynomial(
+		__isl_take isl_union_pw_qpolynomial *upwqp, int sign);
+
+Approximate each quasipolynomial by a polynomial.  If C<sign> is positive,
+the polynomial will be an overapproximation.  If C<sign> is negative,
+it will be an underapproximation.  If C<sign> is zero, the approximation
+will lie somewhere in between.
+
+=item * Feasibility
+
+	__isl_give isl_basic_set *isl_basic_set_sample(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_basic_set *isl_set_sample(
+		__isl_take isl_set *set);
+	__isl_give isl_basic_map *isl_basic_map_sample(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_basic_map *isl_map_sample(
+		__isl_take isl_map *map);
+
+If the input (basic) set or relation is non-empty, then return
+a singleton subset of the input.  Otherwise, return an empty set.
+
+=item * Optimization
+
+	#include <isl/ilp.h>
+	__isl_give isl_val *isl_basic_set_max_val(
+		__isl_keep isl_basic_set *bset,
+		__isl_keep isl_aff *obj);
+	__isl_give isl_val *isl_set_min_val(
+		__isl_keep isl_set *set,
+		__isl_keep isl_aff *obj);
+	__isl_give isl_val *isl_set_max_val(
+		__isl_keep isl_set *set,
+		__isl_keep isl_aff *obj);
+
+Compute the minimum or maximum of the integer affine expression C<obj>
+over the points in C<set>, returning the result in C<opt>.
+The result is C<NULL> in case of an error, the optimal value in case
+there is one, negative infinity or infinity if the problem is unbounded and
+NaN if the problem is empty.
+
+=item * Parametric optimization
+
+	__isl_give isl_pw_aff *isl_set_dim_min(
+		__isl_take isl_set *set, int pos);
+	__isl_give isl_pw_aff *isl_set_dim_max(
+		__isl_take isl_set *set, int pos);
+	__isl_give isl_pw_aff *isl_map_dim_max(
+		__isl_take isl_map *map, int pos);
+
+Compute the minimum or maximum of the given set or output dimension
+as a function of the parameters (and input dimensions), but independently
+of the other set or output dimensions.
+For lexicographic optimization, see L<"Lexicographic Optimization">.
+
+=item * Dual
+
+The following functions compute either the set of (rational) coefficient
+values of valid constraints for the given set or the set of (rational)
+values satisfying the constraints with coefficients from the given set.
+Internally, these two sets of functions perform essentially the
+same operations, except that the set of coefficients is assumed to
+be a cone, while the set of values may be any polyhedron.
+The current implementation is based on the Farkas lemma and
+Fourier-Motzkin elimination, but this may change or be made optional
+in future.  In particular, future implementations may use different
+dualization algorithms or skip the elimination step.
+
+	__isl_give isl_basic_set *isl_basic_set_coefficients(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_basic_set *isl_set_coefficients(
+		__isl_take isl_set *set);
+	__isl_give isl_union_set *isl_union_set_coefficients(
+		__isl_take isl_union_set *bset);
+	__isl_give isl_basic_set *isl_basic_set_solutions(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_basic_set *isl_set_solutions(
+		__isl_take isl_set *set);
+	__isl_give isl_union_set *isl_union_set_solutions(
+		__isl_take isl_union_set *bset);
+
+=item * Power
+
+	__isl_give isl_map *isl_map_fixed_power_val(
+		__isl_take isl_map *map,
+		__isl_take isl_val *exp);
+	__isl_give isl_union_map *
+	isl_union_map_fixed_power_val(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_val *exp);
+
+Compute the given power of C<map>, where C<exp> is assumed to be non-zero.
+If the exponent C<exp> is negative, then the -C<exp> th power of the inverse
+of C<map> is computed.
+
+	__isl_give isl_map *isl_map_power(__isl_take isl_map *map,
+		int *exact);
+	__isl_give isl_union_map *isl_union_map_power(
+		__isl_take isl_union_map *umap, int *exact);
+
+Compute a parametric representation for all positive powers I<k> of C<map>.
+The result maps I<k> to a nested relation corresponding to the
+I<k>th power of C<map>.
+The result may be an overapproximation.  If the result is known to be exact,
+then C<*exact> is set to C<1>.
+
+=item * Transitive closure
+
+	__isl_give isl_map *isl_map_transitive_closure(
+		__isl_take isl_map *map, int *exact);
+	__isl_give isl_union_map *isl_union_map_transitive_closure(
+		__isl_take isl_union_map *umap, int *exact);
+
+Compute the transitive closure of C<map>.
+The result may be an overapproximation.  If the result is known to be exact,
+then C<*exact> is set to C<1>.
+
+=item * Reaching path lengths
+
+	__isl_give isl_map *isl_map_reaching_path_lengths(
+		__isl_take isl_map *map, int *exact);
+
+Compute a relation that maps each element in the range of C<map>
+to the lengths of all paths composed of edges in C<map> that
+end up in the given element.
+The result may be an overapproximation.  If the result is known to be exact,
+then C<*exact> is set to C<1>.
+To compute the I<maximal> path length, the resulting relation
+should be postprocessed by C<isl_map_lexmax>.
+In particular, if the input relation is a dependence relation
+(mapping sources to sinks), then the maximal path length corresponds
+to the free schedule.
+Note, however, that C<isl_map_lexmax> expects the maximum to be
+finite, so if the path lengths are unbounded (possibly due to
+the overapproximation), then you will get an error message.
+
+=item * Wrapping
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_wrap(
+		__isl_take isl_space *space);
+	__isl_give isl_space *isl_space_unwrap(
+		__isl_take isl_space *space);
+
+	#include <isl/set.h>
+	__isl_give isl_basic_map *isl_basic_set_unwrap(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_map *isl_set_unwrap(
+		__isl_take isl_set *set);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_set *isl_basic_map_wrap(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_set *isl_map_wrap(
+		__isl_take isl_map *map);
+
+	#include <isl/union_set.h>
+	__isl_give isl_union_map *isl_union_set_unwrap(
+		__isl_take isl_union_set *uset);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_set *isl_union_map_wrap(
+		__isl_take isl_union_map *umap);
+
+The input to C<isl_space_unwrap> should
+be the space of a set, while that of
+C<isl_space_wrap> should be the space of a relation.
+Conversely, the output of C<isl_space_unwrap> is the space
+of a relation, while that of C<isl_space_wrap> is the space of a set.
+
+=item * Flattening
+
+Remove any internal structure of domain (and range) of the given
+set or relation.  If there is any such internal structure in the input,
+then the name of the space is also removed.
+
+	#include <isl/local_space.h>
+	__isl_give isl_local_space *
+	isl_local_space_flatten_domain(
+		__isl_take isl_local_space *ls);
+	__isl_give isl_local_space *
+	isl_local_space_flatten_range(
+		__isl_take isl_local_space *ls);
+
+	#include <isl/set.h>
+	__isl_give isl_basic_set *isl_basic_set_flatten(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_set *isl_set_flatten(
+		__isl_take isl_set *set);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *isl_basic_map_flatten_domain(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_basic_map *isl_basic_map_flatten_range(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_map *isl_map_flatten_range(
+		__isl_take isl_map *map);
+	__isl_give isl_map *isl_map_flatten_domain(
+		__isl_take isl_map *map);
+	__isl_give isl_basic_map *isl_basic_map_flatten(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_map *isl_map_flatten(
+		__isl_take isl_map *map);
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_flatten_range(
+		__isl_take isl_multi_val *mv);
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *isl_multi_aff_flatten_domain(
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_multi_aff *isl_multi_aff_flatten_range(
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_flatten_range(
+		__isl_take isl_multi_pw_aff *mpa);
+
+	#include <isl/map.h>
+	__isl_give isl_map *isl_set_flatten_map(
+		__isl_take isl_set *set);
+
+The function above constructs a relation
+that maps the input set to a flattened version of the set.
+
+=item * Lifting
+
+Lift the input set to a space with extra dimensions corresponding
+to the existentially quantified variables in the input.
+In particular, the result lives in a wrapped map where the domain
+is the original space and the range corresponds to the original
+existentially quantified variables.
+
+	#include <isl/set.h>
+	__isl_give isl_basic_set *isl_basic_set_lift(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_set *isl_set_lift(
+		__isl_take isl_set *set);
+	__isl_give isl_union_set *isl_union_set_lift(
+		__isl_take isl_union_set *uset);
+
+Given a local space that contains the existentially quantified
+variables of a set, a basic relation that, when applied to
+a basic set, has essentially the same effect as C<isl_basic_set_lift>,
+can be constructed using the following function.
+
+	#include <isl/local_space.h>
+	__isl_give isl_basic_map *isl_local_space_lifting(
+		__isl_take isl_local_space *ls);
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *isl_multi_aff_lift(
+		__isl_take isl_multi_aff *maff,
+		__isl_give isl_local_space **ls);
+
+If the C<ls> argument of C<isl_multi_aff_lift> is not C<NULL>,
+then it is assigned the local space that lies at the basis of
+the lifting applied.
+
+=item * Internal Product
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_zip(
+		__isl_take isl_space *space);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *isl_basic_map_zip(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_map *isl_map_zip(
+		__isl_take isl_map *map);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *isl_union_map_zip(
+		__isl_take isl_union_map *umap);
+
+Given a relation with nested relations for domain and range,
+interchange the range of the domain with the domain of the range.
+
+=item * Currying
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_curry(
+		__isl_take isl_space *space);
+	__isl_give isl_space *isl_space_uncurry(
+		__isl_take isl_space *space);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *isl_basic_map_curry(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_basic_map *isl_basic_map_uncurry(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_map *isl_map_curry(
+		__isl_take isl_map *map);
+	__isl_give isl_map *isl_map_uncurry(
+		__isl_take isl_map *map);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *isl_union_map_curry(
+		__isl_take isl_union_map *umap);
+	__isl_give isl_union_map *isl_union_map_uncurry(
+		__isl_take isl_union_map *umap);
+
+Given a relation with a nested relation for domain,
+the C<curry> functions
+move the range of the nested relation out of the domain
+and use it as the domain of a nested relation in the range,
+with the original range as range of this nested relation.
+The C<uncurry> functions perform the inverse operation.
+
+=item * Aligning parameters
+
+Change the order of the parameters of the given set, relation
+or function
+such that the first parameters match those of C<model>.
+This may involve the introduction of extra parameters.
+All parameters need to be named.
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_align_params(
+		__isl_take isl_space *space1,
+		__isl_take isl_space *space2)
+
+	#include <isl/set.h>
+	__isl_give isl_basic_set *isl_basic_set_align_params(
+		__isl_take isl_basic_set *bset,
+		__isl_take isl_space *model);
+	__isl_give isl_set *isl_set_align_params(
+		__isl_take isl_set *set,
+		__isl_take isl_space *model);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *isl_basic_map_align_params(
+		__isl_take isl_basic_map *bmap,
+		__isl_take isl_space *model);
+	__isl_give isl_map *isl_map_align_params(
+		__isl_take isl_map *map,
+		__isl_take isl_space *model);
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_align_params(
+		__isl_take isl_multi_val *mv,
+		__isl_take isl_space *model);
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_aff_align_params(
+		__isl_take isl_aff *aff,
+		__isl_take isl_space *model);
+	__isl_give isl_multi_aff *isl_multi_aff_align_params(
+		__isl_take isl_multi_aff *multi,
+		__isl_take isl_space *model);
+	__isl_give isl_pw_aff *isl_pw_aff_align_params(
+		__isl_take isl_pw_aff *pwaff,
+		__isl_take isl_space *model);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_align_params(
+		__isl_take isl_pw_multi_aff *pma,
+		__isl_take isl_space *model);
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_align_params(
+		__isl_take isl_union_pw_multi_aff *upma,
+		__isl_take isl_space *model);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_qpolynomial *isl_qpolynomial_align_params(
+		__isl_take isl_qpolynomial *qp,
+		__isl_take isl_space *model);
+
+=item * Unary Arithmethic Operations
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_aff_neg(
+		__isl_take isl_aff *aff);
+	__isl_give isl_pw_aff *isl_pw_aff_neg(
+		__isl_take isl_pw_aff *pwaff);
+	__isl_give isl_aff *isl_aff_ceil(
+		__isl_take isl_aff *aff);
+	__isl_give isl_pw_aff *isl_pw_aff_ceil(
+		__isl_take isl_pw_aff *pwaff);
+	__isl_give isl_aff *isl_aff_floor(
+		__isl_take isl_aff *aff);
+	__isl_give isl_multi_aff *isl_multi_aff_floor(
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_pw_aff *isl_pw_aff_floor(
+		__isl_take isl_pw_aff *pwaff);
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_aff *isl_pw_aff_list_min(
+		__isl_take isl_pw_aff_list *list);
+	__isl_give isl_pw_aff *isl_pw_aff_list_max(
+		__isl_take isl_pw_aff_list *list);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_qpolynomial *isl_qpolynomial_neg(
+		__isl_take isl_qpolynomial *qp);
+	__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_neg(
+		__isl_take isl_pw_qpolynomial *pwqp);
+	__isl_give isl_union_pw_qpolynomial *
+	isl_union_pw_qpolynomial_neg(
+		__isl_take isl_union_pw_qpolynomial *upwqp);
+	__isl_give isl_qpolynomial *isl_qpolynomial_pow(
+		__isl_take isl_qpolynomial *qp,
+		unsigned exponent);
+	__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_pow(
+		__isl_take isl_pw_qpolynomial *pwqp,
+		unsigned exponent);
+
+=item * Evaluation
+
+The following functions evaluate a function in a point.
+
+	#include <isl/polynomial.h>
+	__isl_give isl_val *isl_pw_qpolynomial_eval(
+		__isl_take isl_pw_qpolynomial *pwqp,
+		__isl_take isl_point *pnt);
+	__isl_give isl_val *isl_pw_qpolynomial_fold_eval(
+		__isl_take isl_pw_qpolynomial_fold *pwf,
+		__isl_take isl_point *pnt);
+	__isl_give isl_val *isl_union_pw_qpolynomial_eval(
+		__isl_take isl_union_pw_qpolynomial *upwqp,
+		__isl_take isl_point *pnt);
+	__isl_give isl_val *isl_union_pw_qpolynomial_fold_eval(
+		__isl_take isl_union_pw_qpolynomial_fold *upwf,
+		__isl_take isl_point *pnt);
+
+=item * Dimension manipulation
+
+It is usually not advisable to directly change the (input or output)
+space of a set or a relation as this removes the name and the internal
+structure of the space.  However, the functions below can be useful
+to add new parameters, assuming
+C<isl_set_align_params> and C<isl_map_align_params>
+are not sufficient.
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_add_dims(
+		__isl_take isl_space *space,
+		enum isl_dim_type type, unsigned n);
+	__isl_give isl_space *isl_space_insert_dims(
+		__isl_take isl_space *space,
+		enum isl_dim_type type, unsigned pos, unsigned n);
+	__isl_give isl_space *isl_space_drop_dims(
+		__isl_take isl_space *space,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_space *isl_space_move_dims(
+		__isl_take isl_space *space,
+		enum isl_dim_type dst_type, unsigned dst_pos,
+		enum isl_dim_type src_type, unsigned src_pos,
+		unsigned n);
+
+	#include <isl/local_space.h>
+	__isl_give isl_local_space *isl_local_space_add_dims(
+		__isl_take isl_local_space *ls,
+		enum isl_dim_type type, unsigned n);
+	__isl_give isl_local_space *isl_local_space_insert_dims(
+		__isl_take isl_local_space *ls,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_local_space *isl_local_space_drop_dims(
+		__isl_take isl_local_space *ls,
+		enum isl_dim_type type, unsigned first, unsigned n);
+
+	#include <isl/set.h>
+	__isl_give isl_basic_set *isl_basic_set_add_dims(
+		__isl_take isl_basic_set *bset,
+		enum isl_dim_type type, unsigned n);
+	__isl_give isl_set *isl_set_add_dims(
+		__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned n);
+	__isl_give isl_basic_set *isl_basic_set_insert_dims(
+		__isl_take isl_basic_set *bset,
+		enum isl_dim_type type, unsigned pos,
+		unsigned n);
+	__isl_give isl_set *isl_set_insert_dims(
+		__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned pos, unsigned n);
+	__isl_give isl_basic_set *isl_basic_set_move_dims(
+		__isl_take isl_basic_set *bset,
+		enum isl_dim_type dst_type, unsigned dst_pos,
+		enum isl_dim_type src_type, unsigned src_pos,
+		unsigned n);
+	__isl_give isl_set *isl_set_move_dims(
+		__isl_take isl_set *set,
+		enum isl_dim_type dst_type, unsigned dst_pos,
+		enum isl_dim_type src_type, unsigned src_pos,
+		unsigned n);
+
+	#include <isl/map.h>
+	__isl_give isl_map *isl_map_add_dims(
+		__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned n);
+	__isl_give isl_basic_map *isl_basic_map_insert_dims(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned pos,
+		unsigned n);
+	__isl_give isl_map *isl_map_insert_dims(
+		__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned pos, unsigned n);
+	__isl_give isl_basic_map *isl_basic_map_move_dims(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type dst_type, unsigned dst_pos,
+		enum isl_dim_type src_type, unsigned src_pos,
+		unsigned n);
+	__isl_give isl_map *isl_map_move_dims(
+		__isl_take isl_map *map,
+		enum isl_dim_type dst_type, unsigned dst_pos,
+		enum isl_dim_type src_type, unsigned src_pos,
+		unsigned n);
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_insert_dims(
+		__isl_take isl_multi_val *mv,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_multi_val *isl_multi_val_add_dims(
+		__isl_take isl_multi_val *mv,
+		enum isl_dim_type type, unsigned n);
+	__isl_give isl_multi_val *isl_multi_val_drop_dims(
+		__isl_take isl_multi_val *mv,
+		enum isl_dim_type type, unsigned first, unsigned n);
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_aff_insert_dims(
+		__isl_take isl_aff *aff,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_multi_aff *isl_multi_aff_insert_dims(
+		__isl_take isl_multi_aff *ma,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_pw_aff *isl_pw_aff_insert_dims(
+		__isl_take isl_pw_aff *pwaff,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_insert_dims(
+		__isl_take isl_multi_pw_aff *mpa,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_aff *isl_aff_add_dims(
+		__isl_take isl_aff *aff,
+		enum isl_dim_type type, unsigned n);
+	__isl_give isl_multi_aff *isl_multi_aff_add_dims(
+		__isl_take isl_multi_aff *ma,
+		enum isl_dim_type type, unsigned n);
+	__isl_give isl_pw_aff *isl_pw_aff_add_dims(
+		__isl_take isl_pw_aff *pwaff,
+		enum isl_dim_type type, unsigned n);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_add_dims(
+		__isl_take isl_multi_pw_aff *mpa,
+		enum isl_dim_type type, unsigned n);
+	__isl_give isl_aff *isl_aff_drop_dims(
+		__isl_take isl_aff *aff,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_multi_aff *isl_multi_aff_drop_dims(
+		__isl_take isl_multi_aff *maff,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_pw_aff *isl_pw_aff_drop_dims(
+		__isl_take isl_pw_aff *pwaff,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_drop_dims(
+		__isl_take isl_pw_multi_aff *pma,
+		enum isl_dim_type type, unsigned first, unsigned n);
+	__isl_give isl_aff *isl_aff_move_dims(
+		__isl_take isl_aff *aff,
+		enum isl_dim_type dst_type, unsigned dst_pos,
+		enum isl_dim_type src_type, unsigned src_pos,
+		unsigned n);
+	__isl_give isl_multi_aff *isl_multi_aff_move_dims(
+		__isl_take isl_multi_aff *ma,
+		enum isl_dim_type dst_type, unsigned dst_pos,
+		enum isl_dim_type src_type, unsigned src_pos,
+		unsigned n);
+	__isl_give isl_pw_aff *isl_pw_aff_move_dims(
+		__isl_take isl_pw_aff *pa,
+		enum isl_dim_type dst_type, unsigned dst_pos,
+		enum isl_dim_type src_type, unsigned src_pos,
+		unsigned n);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_move_dims(
+		__isl_take isl_multi_pw_aff *pma,
+		enum isl_dim_type dst_type, unsigned dst_pos,
+		enum isl_dim_type src_type, unsigned src_pos,
+		unsigned n);
+
+=back
+
+=head2 Binary Operations
+
+The two arguments of a binary operation not only need to live
+in the same C<isl_ctx>, they currently also need to have
+the same (number of) parameters.
+
+=head3 Basic Operations
+
+=over
+
+=item * Intersection
+
+	#include <isl/local_space.h>
+	__isl_give isl_local_space *isl_local_space_intersect(
+		__isl_take isl_local_space *ls1,
+		__isl_take isl_local_space *ls2);
+
+	#include <isl/set.h>
+	__isl_give isl_basic_set *isl_basic_set_intersect_params(
+		__isl_take isl_basic_set *bset1,
+		__isl_take isl_basic_set *bset2);
+	__isl_give isl_basic_set *isl_basic_set_intersect(
+		__isl_take isl_basic_set *bset1,
+		__isl_take isl_basic_set *bset2);
+	__isl_give isl_basic_set *isl_basic_set_list_intersect(
+		__isl_take struct isl_basic_set_list *list);
+	__isl_give isl_set *isl_set_intersect_params(
+		__isl_take isl_set *set,
+		__isl_take isl_set *params);
+	__isl_give isl_set *isl_set_intersect(
+		__isl_take isl_set *set1,
+		__isl_take isl_set *set2);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *isl_basic_map_intersect_domain(
+		__isl_take isl_basic_map *bmap,
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_basic_map *isl_basic_map_intersect_range(
+		__isl_take isl_basic_map *bmap,
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_basic_map *isl_basic_map_intersect(
+		__isl_take isl_basic_map *bmap1,
+		__isl_take isl_basic_map *bmap2);
+	__isl_give isl_map *isl_map_intersect_params(
+		__isl_take isl_map *map,
+		__isl_take isl_set *params);
+	__isl_give isl_map *isl_map_intersect_domain(
+		__isl_take isl_map *map,
+		__isl_take isl_set *set);
+	__isl_give isl_map *isl_map_intersect_range(
+		__isl_take isl_map *map,
+		__isl_take isl_set *set);
+	__isl_give isl_map *isl_map_intersect(
+		__isl_take isl_map *map1,
+		__isl_take isl_map *map2);
+
+	#include <isl/union_set.h>
+	__isl_give isl_union_set *isl_union_set_intersect_params(
+		__isl_take isl_union_set *uset,
+		__isl_take isl_set *set);
+	__isl_give isl_union_set *isl_union_set_intersect(
+		__isl_take isl_union_set *uset1,
+		__isl_take isl_union_set *uset2);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *isl_union_map_intersect_params(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_set *set);
+	__isl_give isl_union_map *isl_union_map_intersect_domain(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_union_set *uset);
+	__isl_give isl_union_map *isl_union_map_intersect_range(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_union_set *uset);
+	__isl_give isl_union_map *isl_union_map_intersect(
+		__isl_take isl_union_map *umap1,
+		__isl_take isl_union_map *umap2);
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_aff *isl_pw_aff_intersect_domain(
+		__isl_take isl_pw_aff *pa,
+		__isl_take isl_set *set);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_intersect_domain(
+		__isl_take isl_multi_pw_aff *mpa,
+		__isl_take isl_set *domain);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_intersect_domain(
+		__isl_take isl_pw_multi_aff *pma,
+		__isl_take isl_set *set);
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_intersect_domain(
+		__isl_take isl_union_pw_multi_aff *upma,
+		__isl_take isl_union_set *uset);
+	__isl_give isl_pw_aff *isl_pw_aff_intersect_params(
+		__isl_take isl_pw_aff *pa,
+		__isl_take isl_set *set);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_intersect_params(
+		__isl_take isl_multi_pw_aff *mpa,
+		__isl_take isl_set *set);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_intersect_params(
+		__isl_take isl_pw_multi_aff *pma,
+		__isl_take isl_set *set);
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_intersect_params(
+		__isl_take isl_union_pw_multi_aff *upma,
+		__isl_take isl_set *set);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_pw_qpolynomial *
+	isl_pw_qpolynomial_intersect_domain(
+		__isl_take isl_pw_qpolynomial *pwpq,
+		__isl_take isl_set *set);
+	__isl_give isl_union_pw_qpolynomial *
+	isl_union_pw_qpolynomial_intersect_domain(
+		__isl_take isl_union_pw_qpolynomial *upwpq,
+		__isl_take isl_union_set *uset);
+	__isl_give isl_union_pw_qpolynomial_fold *
+	isl_union_pw_qpolynomial_fold_intersect_domain(
+		__isl_take isl_union_pw_qpolynomial_fold *upwf,
+		__isl_take isl_union_set *uset);
+	__isl_give isl_pw_qpolynomial *
+	isl_pw_qpolynomial_intersect_params(
+		__isl_take isl_pw_qpolynomial *pwpq,
+		__isl_take isl_set *set);
+	__isl_give isl_pw_qpolynomial_fold *
+	isl_pw_qpolynomial_fold_intersect_params(
+		__isl_take isl_pw_qpolynomial_fold *pwf,
+		__isl_take isl_set *set);
+	__isl_give isl_union_pw_qpolynomial *
+	isl_union_pw_qpolynomial_intersect_params(
+		__isl_take isl_union_pw_qpolynomial *upwpq,
+		__isl_take isl_set *set);
+	__isl_give isl_union_pw_qpolynomial_fold *
+	isl_union_pw_qpolynomial_fold_intersect_params(
+		__isl_take isl_union_pw_qpolynomial_fold *upwf,
+		__isl_take isl_set *set);
+
+The second argument to the C<_params> functions needs to be
+a parametric (basic) set.  For the other functions, a parametric set
+for either argument is only allowed if the other argument is
+a parametric set as well.
+The list passed to C<isl_basic_set_list_intersect> needs to have
+at least one element and all elements need to live in the same space.
+
+=item * Union
+
+	__isl_give isl_set *isl_basic_set_union(
+		__isl_take isl_basic_set *bset1,
+		__isl_take isl_basic_set *bset2);
+	__isl_give isl_map *isl_basic_map_union(
+		__isl_take isl_basic_map *bmap1,
+		__isl_take isl_basic_map *bmap2);
+	__isl_give isl_set *isl_set_union(
+		__isl_take isl_set *set1,
+		__isl_take isl_set *set2);
+	__isl_give isl_map *isl_map_union(
+		__isl_take isl_map *map1,
+		__isl_take isl_map *map2);
+	__isl_give isl_union_set *isl_union_set_union(
+		__isl_take isl_union_set *uset1,
+		__isl_take isl_union_set *uset2);
+	__isl_give isl_union_map *isl_union_map_union(
+		__isl_take isl_union_map *umap1,
+		__isl_take isl_union_map *umap2);
+
+=item * Set difference
+
+	__isl_give isl_set *isl_set_subtract(
+		__isl_take isl_set *set1,
+		__isl_take isl_set *set2);
+	__isl_give isl_map *isl_map_subtract(
+		__isl_take isl_map *map1,
+		__isl_take isl_map *map2);
+	__isl_give isl_map *isl_map_subtract_domain(
+		__isl_take isl_map *map,
+		__isl_take isl_set *dom);
+	__isl_give isl_map *isl_map_subtract_range(
+		__isl_take isl_map *map,
+		__isl_take isl_set *dom);
+	__isl_give isl_union_set *isl_union_set_subtract(
+		__isl_take isl_union_set *uset1,
+		__isl_take isl_union_set *uset2);
+	__isl_give isl_union_map *isl_union_map_subtract(
+		__isl_take isl_union_map *umap1,
+		__isl_take isl_union_map *umap2);
+	__isl_give isl_union_map *isl_union_map_subtract_domain(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_union_set *dom);
+	__isl_give isl_union_map *isl_union_map_subtract_range(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_union_set *dom);
+
+=item * Application
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_join(
+		__isl_take isl_space *left,
+		__isl_take isl_space *right);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_set *isl_basic_set_apply(
+		__isl_take isl_basic_set *bset,
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_set *isl_set_apply(
+		__isl_take isl_set *set,
+		__isl_take isl_map *map);
+	__isl_give isl_union_set *isl_union_set_apply(
+		__isl_take isl_union_set *uset,
+		__isl_take isl_union_map *umap);
+	__isl_give isl_basic_map *isl_basic_map_apply_domain(
+		__isl_take isl_basic_map *bmap1,
+		__isl_take isl_basic_map *bmap2);
+	__isl_give isl_basic_map *isl_basic_map_apply_range(
+		__isl_take isl_basic_map *bmap1,
+		__isl_take isl_basic_map *bmap2);
+	__isl_give isl_map *isl_map_apply_domain(
+		__isl_take isl_map *map1,
+		__isl_take isl_map *map2);
+	__isl_give isl_map *isl_map_apply_range(
+		__isl_take isl_map *map1,
+		__isl_take isl_map *map2);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *isl_union_map_apply_domain(
+		__isl_take isl_union_map *umap1,
+		__isl_take isl_union_map *umap2);
+	__isl_give isl_union_map *isl_union_map_apply_range(
+		__isl_take isl_union_map *umap1,
+		__isl_take isl_union_map *umap2);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_pw_qpolynomial_fold *
+	isl_set_apply_pw_qpolynomial_fold(
+		__isl_take isl_set *set,
+		__isl_take isl_pw_qpolynomial_fold *pwf,
+		int *tight);
+	__isl_give isl_pw_qpolynomial_fold *
+	isl_map_apply_pw_qpolynomial_fold(
+		__isl_take isl_map *map,
+		__isl_take isl_pw_qpolynomial_fold *pwf,
+		int *tight);
+	__isl_give isl_union_pw_qpolynomial_fold *
+	isl_union_set_apply_union_pw_qpolynomial_fold(
+		__isl_take isl_union_set *uset,
+		__isl_take isl_union_pw_qpolynomial_fold *upwf,
+		int *tight);
+	__isl_give isl_union_pw_qpolynomial_fold *
+	isl_union_map_apply_union_pw_qpolynomial_fold(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_union_pw_qpolynomial_fold *upwf,
+		int *tight);
+
+The functions taking a map
+compose the given map with the given piecewise quasipolynomial reduction.
+That is, compute a bound (of the same type as C<pwf> or C<upwf> itself)
+over all elements in the intersection of the range of the map
+and the domain of the piecewise quasipolynomial reduction
+as a function of an element in the domain of the map.
+The functions taking a set compute a bound over all elements in the
+intersection of the set and the domain of the
+piecewise quasipolynomial reduction.
+
+=item * Preimage
+
+	#include <isl/set.h>
+	__isl_give isl_basic_set *
+	isl_basic_set_preimage_multi_aff(
+		__isl_take isl_basic_set *bset,
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_set *isl_set_preimage_multi_aff(
+		__isl_take isl_set *set,
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_set *isl_set_preimage_pw_multi_aff(
+		__isl_take isl_set *set,
+		__isl_take isl_pw_multi_aff *pma);
+	__isl_give isl_set *isl_set_preimage_multi_pw_aff(
+		__isl_take isl_set *set,
+		__isl_take isl_multi_pw_aff *mpa);
+
+	#include <isl/union_set.h>
+	__isl_give isl_union_set *
+	isl_union_set_preimage_multi_aff(
+		__isl_take isl_union_set *uset,
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_union_set *
+	isl_union_set_preimage_pw_multi_aff(
+		__isl_take isl_union_set *uset,
+		__isl_take isl_pw_multi_aff *pma);
+	__isl_give isl_union_set *
+	isl_union_set_preimage_union_pw_multi_aff(
+		__isl_take isl_union_set *uset,
+		__isl_take isl_union_pw_multi_aff *upma);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *
+	isl_basic_map_preimage_domain_multi_aff(
+		__isl_take isl_basic_map *bmap,
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_map *isl_map_preimage_domain_multi_aff(
+		__isl_take isl_map *map,
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_map *isl_map_preimage_range_multi_aff(
+		__isl_take isl_map *map,
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_map *
+	isl_map_preimage_domain_pw_multi_aff(
+		__isl_take isl_map *map,
+		__isl_take isl_pw_multi_aff *pma);
+	__isl_give isl_map *
+	isl_map_preimage_range_pw_multi_aff(
+		__isl_take isl_map *map,
+		__isl_take isl_pw_multi_aff *pma);
+	__isl_give isl_map *
+	isl_map_preimage_domain_multi_pw_aff(
+		__isl_take isl_map *map,
+		__isl_take isl_multi_pw_aff *mpa);
+	__isl_give isl_basic_map *
+	isl_basic_map_preimage_range_multi_aff(
+		__isl_take isl_basic_map *bmap,
+		__isl_take isl_multi_aff *ma);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *
+	isl_union_map_preimage_domain_multi_aff(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_union_map *
+	isl_union_map_preimage_range_multi_aff(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_union_map *
+	isl_union_map_preimage_domain_pw_multi_aff(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_pw_multi_aff *pma);
+	__isl_give isl_union_map *
+	isl_union_map_preimage_range_pw_multi_aff(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_pw_multi_aff *pma);
+	__isl_give isl_union_map *
+	isl_union_map_preimage_domain_union_pw_multi_aff(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_union_pw_multi_aff *upma);
+	__isl_give isl_union_map *
+	isl_union_map_preimage_range_union_pw_multi_aff(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_union_pw_multi_aff *upma);
+
+These functions compute the preimage of the given set or map domain/range under
+the given function.  In other words, the expression is plugged
+into the set description or into the domain/range of the map.
+
+=item * Pullback
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_aff_pullback_aff(
+		__isl_take isl_aff *aff1,
+		__isl_take isl_aff *aff2);
+	__isl_give isl_aff *isl_aff_pullback_multi_aff(
+		__isl_take isl_aff *aff,
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_pw_aff *isl_pw_aff_pullback_multi_aff(
+		__isl_take isl_pw_aff *pa,
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_pw_aff *isl_pw_aff_pullback_pw_multi_aff(
+		__isl_take isl_pw_aff *pa,
+		__isl_take isl_pw_multi_aff *pma);
+	__isl_give isl_pw_aff *isl_pw_aff_pullback_multi_pw_aff(
+		__isl_take isl_pw_aff *pa,
+		__isl_take isl_multi_pw_aff *mpa);
+	__isl_give isl_multi_aff *isl_multi_aff_pullback_multi_aff(
+		__isl_take isl_multi_aff *ma1,
+		__isl_take isl_multi_aff *ma2);
+	__isl_give isl_pw_multi_aff *
+	isl_pw_multi_aff_pullback_multi_aff(
+		__isl_take isl_pw_multi_aff *pma,
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_pullback_multi_aff(
+		__isl_take isl_multi_pw_aff *mpa,
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_pw_multi_aff *
+	isl_pw_multi_aff_pullback_pw_multi_aff(
+		__isl_take isl_pw_multi_aff *pma1,
+		__isl_take isl_pw_multi_aff *pma2);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_pullback_pw_multi_aff(
+		__isl_take isl_multi_pw_aff *mpa,
+		__isl_take isl_pw_multi_aff *pma);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_pullback_multi_pw_aff(
+		__isl_take isl_multi_pw_aff *mpa1,
+		__isl_take isl_multi_pw_aff *mpa2);
+
+These functions precompose the first expression by the second function.
+In other words, the second function is plugged
+into the first expression.
+
+=item * Locus
+
+	#include <isl/aff.h>
+	__isl_give isl_basic_set *isl_aff_le_basic_set(
+		__isl_take isl_aff *aff1,
+		__isl_take isl_aff *aff2);
+	__isl_give isl_basic_set *isl_aff_ge_basic_set(
+		__isl_take isl_aff *aff1,
+		__isl_take isl_aff *aff2);
+	__isl_give isl_set *isl_pw_aff_eq_set(
+		__isl_take isl_pw_aff *pwaff1,
+		__isl_take isl_pw_aff *pwaff2);
+	__isl_give isl_set *isl_pw_aff_ne_set(
+		__isl_take isl_pw_aff *pwaff1,
+		__isl_take isl_pw_aff *pwaff2);
+	__isl_give isl_set *isl_pw_aff_le_set(
+		__isl_take isl_pw_aff *pwaff1,
+		__isl_take isl_pw_aff *pwaff2);
+	__isl_give isl_set *isl_pw_aff_lt_set(
+		__isl_take isl_pw_aff *pwaff1,
+		__isl_take isl_pw_aff *pwaff2);
+	__isl_give isl_set *isl_pw_aff_ge_set(
+		__isl_take isl_pw_aff *pwaff1,
+		__isl_take isl_pw_aff *pwaff2);
+	__isl_give isl_set *isl_pw_aff_gt_set(
+		__isl_take isl_pw_aff *pwaff1,
+		__isl_take isl_pw_aff *pwaff2);
+
+	__isl_give isl_set *isl_multi_aff_lex_le_set(
+		__isl_take isl_multi_aff *ma1,
+		__isl_take isl_multi_aff *ma2);
+	__isl_give isl_set *isl_multi_aff_lex_ge_set(
+		__isl_take isl_multi_aff *ma1,
+		__isl_take isl_multi_aff *ma2);
+
+	__isl_give isl_set *isl_pw_aff_list_eq_set(
+		__isl_take isl_pw_aff_list *list1,
+		__isl_take isl_pw_aff_list *list2);
+	__isl_give isl_set *isl_pw_aff_list_ne_set(
+		__isl_take isl_pw_aff_list *list1,
+		__isl_take isl_pw_aff_list *list2);
+	__isl_give isl_set *isl_pw_aff_list_le_set(
+		__isl_take isl_pw_aff_list *list1,
+		__isl_take isl_pw_aff_list *list2);
+	__isl_give isl_set *isl_pw_aff_list_lt_set(
+		__isl_take isl_pw_aff_list *list1,
+		__isl_take isl_pw_aff_list *list2);
+	__isl_give isl_set *isl_pw_aff_list_ge_set(
+		__isl_take isl_pw_aff_list *list1,
+		__isl_take isl_pw_aff_list *list2);
+	__isl_give isl_set *isl_pw_aff_list_gt_set(
+		__isl_take isl_pw_aff_list *list1,
+		__isl_take isl_pw_aff_list *list2);
+
+The function C<isl_aff_ge_basic_set> returns a basic set
+containing those elements in the shared space
+of C<aff1> and C<aff2> where C<aff1> is greater than or equal to C<aff2>.
+The function C<isl_pw_aff_ge_set> returns a set
+containing those elements in the shared domain
+of C<pwaff1> and C<pwaff2> where C<pwaff1> is
+greater than or equal to C<pwaff2>.
+The function C<isl_multi_aff_lex_le_set> returns a set
+containing those elements in the shared domain space
+where C<ma1> is lexicographically smaller than or
+equal to C<ma2>.
+The functions operating on C<isl_pw_aff_list> apply the corresponding
+C<isl_pw_aff> function to each pair of elements in the two lists.
+
+=item * Cartesian Product
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_product(
+		__isl_take isl_space *space1,
+		__isl_take isl_space *space2);
+	__isl_give isl_space *isl_space_domain_product(
+		__isl_take isl_space *space1,
+		__isl_take isl_space *space2);
+	__isl_give isl_space *isl_space_range_product(
+		__isl_take isl_space *space1,
+		__isl_take isl_space *space2);
+
+The functions
+C<isl_space_product>, C<isl_space_domain_product>
+and C<isl_space_range_product> take pairs or relation spaces and
+produce a single relations space, where either the domain, the range
+or both domain and range are wrapped spaces of relations between
+the domains and/or ranges of the input spaces.
+If the product is only constructed over the domain or the range
+then the ranges or the domains of the inputs should be the same.
+The function C<isl_space_product> also accepts a pair of set spaces,
+in which case it returns a wrapped space of a relation between the
+two input spaces.
+
+	#include <isl/set.h>
+	__isl_give isl_set *isl_set_product(
+		__isl_take isl_set *set1,
+		__isl_take isl_set *set2);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *isl_basic_map_domain_product(
+		__isl_take isl_basic_map *bmap1,
+		__isl_take isl_basic_map *bmap2);
+	__isl_give isl_basic_map *isl_basic_map_range_product(
+		__isl_take isl_basic_map *bmap1,
+		__isl_take isl_basic_map *bmap2);
+	__isl_give isl_basic_map *isl_basic_map_product(
+		__isl_take isl_basic_map *bmap1,
+		__isl_take isl_basic_map *bmap2);
+	__isl_give isl_map *isl_map_domain_product(
+		__isl_take isl_map *map1,
+		__isl_take isl_map *map2);
+	__isl_give isl_map *isl_map_range_product(
+		__isl_take isl_map *map1,
+		__isl_take isl_map *map2);
+	__isl_give isl_map *isl_map_product(
+		__isl_take isl_map *map1,
+		__isl_take isl_map *map2);
+
+	#include <isl/union_set.h>
+	__isl_give isl_union_set *isl_union_set_product(
+		__isl_take isl_union_set *uset1,
+		__isl_take isl_union_set *uset2);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *isl_union_map_domain_product(
+		__isl_take isl_union_map *umap1,
+		__isl_take isl_union_map *umap2);
+	__isl_give isl_union_map *isl_union_map_range_product(
+		__isl_take isl_union_map *umap1,
+		__isl_take isl_union_map *umap2);
+	__isl_give isl_union_map *isl_union_map_product(
+		__isl_take isl_union_map *umap1,
+		__isl_take isl_union_map *umap2);
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_range_product(
+		__isl_take isl_multi_val *mv1,
+		__isl_take isl_multi_val *mv2);
+	__isl_give isl_multi_val *isl_multi_val_product(
+		__isl_take isl_multi_val *mv1,
+		__isl_take isl_multi_val *mv2);
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *isl_multi_aff_range_product(
+		__isl_take isl_multi_aff *ma1,
+		__isl_take isl_multi_aff *ma2);
+	__isl_give isl_multi_aff *isl_multi_aff_product(
+		__isl_take isl_multi_aff *ma1,
+		__isl_take isl_multi_aff *ma2);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_range_product(
+		__isl_take isl_multi_pw_aff *mpa1,
+		__isl_take isl_multi_pw_aff *mpa2);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_product(
+		__isl_take isl_multi_pw_aff *mpa1,
+		__isl_take isl_multi_pw_aff *mpa2);
+	__isl_give isl_pw_multi_aff *
+	isl_pw_multi_aff_range_product(
+		__isl_take isl_pw_multi_aff *pma1,
+		__isl_take isl_pw_multi_aff *pma2);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_product(
+		__isl_take isl_pw_multi_aff *pma1,
+		__isl_take isl_pw_multi_aff *pma2);
+
+The above functions compute the cross product of the given
+sets, relations or functions.  The domains and ranges of the results
+are wrapped maps between domains and ranges of the inputs.
+To obtain a ``flat'' product, use the following functions
+instead.
+
+	#include <isl/set.h>
+	__isl_give isl_basic_set *isl_basic_set_flat_product(
+		__isl_take isl_basic_set *bset1,
+		__isl_take isl_basic_set *bset2);
+	__isl_give isl_set *isl_set_flat_product(
+		__isl_take isl_set *set1,
+		__isl_take isl_set *set2);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *isl_basic_map_flat_range_product(
+		__isl_take isl_basic_map *bmap1,
+		__isl_take isl_basic_map *bmap2);
+	__isl_give isl_map *isl_map_flat_domain_product(
+		__isl_take isl_map *map1,
+		__isl_take isl_map *map2);
+	__isl_give isl_map *isl_map_flat_range_product(
+		__isl_take isl_map *map1,
+		__isl_take isl_map *map2);
+	__isl_give isl_basic_map *isl_basic_map_flat_product(
+		__isl_take isl_basic_map *bmap1,
+		__isl_take isl_basic_map *bmap2);
+	__isl_give isl_map *isl_map_flat_product(
+		__isl_take isl_map *map1,
+		__isl_take isl_map *map2);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *
+	isl_union_map_flat_range_product(
+		__isl_take isl_union_map *umap1,
+		__isl_take isl_union_map *umap2);
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_flat_range_product(
+		__isl_take isl_multi_val *mv1,
+		__isl_take isl_multi_aff *mv2);
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *isl_multi_aff_flat_range_product(
+		__isl_take isl_multi_aff *ma1,
+		__isl_take isl_multi_aff *ma2);
+	__isl_give isl_pw_multi_aff *
+	isl_pw_multi_aff_flat_range_product(
+		__isl_take isl_pw_multi_aff *pma1,
+		__isl_take isl_pw_multi_aff *pma2);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_flat_range_product(
+		__isl_take isl_multi_pw_aff *mpa1,
+		__isl_take isl_multi_pw_aff *mpa2);
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_flat_range_product(
+		__isl_take isl_union_pw_multi_aff *upma1,
+		__isl_take isl_union_pw_multi_aff *upma2);
+
+	#include <isl/space.h>
+	__isl_give isl_space *isl_space_factor_domain(
+		__isl_take isl_space *space);
+	__isl_give isl_space *isl_space_factor_range(
+		__isl_take isl_space *space);
+	__isl_give isl_space *isl_space_domain_factor_domain(
+		__isl_take isl_space *space);
+	__isl_give isl_space *isl_space_domain_factor_range(
+		__isl_take isl_space *space);
+	__isl_give isl_space *isl_space_range_factor_domain(
+		__isl_take isl_space *space);
+	__isl_give isl_space *isl_space_range_factor_range(
+		__isl_take isl_space *space);
+
+The functions C<isl_space_range_factor_domain> and
+C<isl_space_range_factor_range> extract the two arguments from
+the result of a call to C<isl_space_range_product>.
+
+The arguments of a call to C<isl_map_range_product> can be extracted
+from the result using the following functions.
+
+	#include <isl/map.h>
+	__isl_give isl_map *isl_map_factor_domain(
+		__isl_take isl_map *map);
+	__isl_give isl_map *isl_map_factor_range(
+		__isl_take isl_map *map);
+	__isl_give isl_map *isl_map_domain_factor_domain(
+		__isl_take isl_map *map);
+	__isl_give isl_map *isl_map_domain_factor_range(
+		__isl_take isl_map *map);
+	__isl_give isl_map *isl_map_range_factor_domain(
+		__isl_take isl_map *map);
+	__isl_give isl_map *isl_map_range_factor_range(
+		__isl_take isl_map *map);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *isl_union_map_factor_domain(
+		__isl_take isl_union_map *umap);
+	__isl_give isl_union_map *isl_union_map_factor_range(
+		__isl_take isl_union_map *umap);
+	__isl_give isl_union_map *
+	isl_union_map_domain_factor_domain(
+		__isl_take isl_union_map *umap);
+	__isl_give isl_union_map *
+	isl_union_map_domain_factor_range(
+		__isl_take isl_union_map *umap);
+	__isl_give isl_union_map *
+	isl_union_map_range_factor_range(
+		__isl_take isl_union_map *umap);
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *
+	isl_multi_val_range_factor_domain(
+		__isl_take isl_multi_val *mv);
+	__isl_give isl_multi_val *
+	isl_multi_val_range_factor_range(
+		__isl_take isl_multi_val *mv);
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *
+	isl_multi_aff_range_factor_domain(
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_multi_aff *
+	isl_multi_aff_range_factor_range(
+		__isl_take isl_multi_aff *ma);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_range_factor_domain(
+		__isl_take isl_multi_pw_aff *mpa);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_range_factor_range(
+		__isl_take isl_multi_pw_aff *mpa);
+
+The splice functions are a generalization of the flat product functions,
+where the second argument may be inserted at any position inside
+the first argument rather than being placed at the end.
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_range_splice(
+		__isl_take isl_multi_val *mv1, unsigned pos,
+		__isl_take isl_multi_val *mv2);
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *isl_multi_aff_range_splice(
+		__isl_take isl_multi_aff *ma1, unsigned pos,
+		__isl_take isl_multi_aff *ma2);
+	__isl_give isl_multi_aff *isl_multi_aff_splice(
+		__isl_take isl_multi_aff *ma1,
+		unsigned in_pos, unsigned out_pos,
+		__isl_take isl_multi_aff *ma2);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_range_splice(
+		__isl_take isl_multi_pw_aff *mpa1, unsigned pos,
+		__isl_take isl_multi_pw_aff *mpa2);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_splice(
+		__isl_take isl_multi_pw_aff *mpa1,
+		unsigned in_pos, unsigned out_pos,
+		__isl_take isl_multi_pw_aff *mpa2);
+
+=item * Simplification
+
+When applied to a set or relation,
+the gist operation returns a set or relation that has the
+same intersection with the context as the input set or relation.
+Any implicit equality in the intersection is made explicit in the result,
+while all inequalities that are redundant with respect to the intersection
+are removed.
+In case of union sets and relations, the gist operation is performed
+per space.
+
+When applied to a function,
+the gist operation applies the set gist operation to each of
+the cells in the domain of the input piecewise expression.
+The context is also exploited
+to simplify the expression associated to each cell.
+
+	#include <isl/set.h>
+	__isl_give isl_basic_set *isl_basic_set_gist(
+		__isl_take isl_basic_set *bset,
+		__isl_take isl_basic_set *context);
+	__isl_give isl_set *isl_set_gist(__isl_take isl_set *set,
+		__isl_take isl_set *context);
+	__isl_give isl_set *isl_set_gist_params(
+		__isl_take isl_set *set,
+		__isl_take isl_set *context);
+
+	#include <isl/map.h>
+	__isl_give isl_basic_map *isl_basic_map_gist(
+		__isl_take isl_basic_map *bmap,
+		__isl_take isl_basic_map *context);
+	__isl_give isl_map *isl_map_gist(__isl_take isl_map *map,
+		__isl_take isl_map *context);
+	__isl_give isl_map *isl_map_gist_params(
+		__isl_take isl_map *map,
+		__isl_take isl_set *context);
+	__isl_give isl_map *isl_map_gist_domain(
+		__isl_take isl_map *map,
+		__isl_take isl_set *context);
+	__isl_give isl_map *isl_map_gist_range(
+		__isl_take isl_map *map,
+		__isl_take isl_set *context);
+
+	#include <isl/union_set.h>
+	__isl_give isl_union_set *isl_union_set_gist(
+		__isl_take isl_union_set *uset,
+		__isl_take isl_union_set *context);
+	__isl_give isl_union_set *isl_union_set_gist_params(
+		__isl_take isl_union_set *uset,
+		__isl_take isl_set *set);
+
+	#include <isl/union_map.h>
+	__isl_give isl_union_map *isl_union_map_gist(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_union_map *context);
+	__isl_give isl_union_map *isl_union_map_gist_params(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_set *set);
+	__isl_give isl_union_map *isl_union_map_gist_domain(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_union_set *uset);
+	__isl_give isl_union_map *isl_union_map_gist_range(
+		__isl_take isl_union_map *umap,
+		__isl_take isl_union_set *uset);
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_aff_gist_params(
+		__isl_take isl_aff *aff,
+		__isl_take isl_set *context);
+	__isl_give isl_aff *isl_aff_gist(__isl_take isl_aff *aff,
+		__isl_take isl_set *context);
+	__isl_give isl_multi_aff *isl_multi_aff_gist_params(
+		__isl_take isl_multi_aff *maff,
+		__isl_take isl_set *context);
+	__isl_give isl_multi_aff *isl_multi_aff_gist(
+		__isl_take isl_multi_aff *maff,
+		__isl_take isl_set *context);
+	__isl_give isl_pw_aff *isl_pw_aff_gist_params(
+		__isl_take isl_pw_aff *pwaff,
+		__isl_take isl_set *context);
+	__isl_give isl_pw_aff *isl_pw_aff_gist(
+		__isl_take isl_pw_aff *pwaff,
+		__isl_take isl_set *context);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_gist_params(
+		__isl_take isl_pw_multi_aff *pma,
+		__isl_take isl_set *set);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_gist(
+		__isl_take isl_pw_multi_aff *pma,
+		__isl_take isl_set *set);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_gist_params(
+		__isl_take isl_multi_pw_aff *mpa,
+		__isl_take isl_set *set);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_gist(
+		__isl_take isl_multi_pw_aff *mpa,
+		__isl_take isl_set *set);
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_gist_params(
+		__isl_take isl_union_pw_multi_aff *upma,
+		__isl_take isl_set *context);
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_gist(
+		__isl_take isl_union_pw_multi_aff *upma,
+		__isl_take isl_union_set *context);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_qpolynomial *isl_qpolynomial_gist_params(
+		__isl_take isl_qpolynomial *qp,
+		__isl_take isl_set *context);
+	__isl_give isl_qpolynomial *isl_qpolynomial_gist(
+		__isl_take isl_qpolynomial *qp,
+		__isl_take isl_set *context);
+	__isl_give isl_qpolynomial_fold *
+	isl_qpolynomial_fold_gist_params(
+		__isl_take isl_qpolynomial_fold *fold,
+		__isl_take isl_set *context);
+	__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_gist(
+		__isl_take isl_qpolynomial_fold *fold,
+		__isl_take isl_set *context);
+	__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_gist_params(
+		__isl_take isl_pw_qpolynomial *pwqp,
+		__isl_take isl_set *context);
+	__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_gist(
+		__isl_take isl_pw_qpolynomial *pwqp,
+		__isl_take isl_set *context);
+	__isl_give isl_pw_qpolynomial_fold *
+	isl_pw_qpolynomial_fold_gist(
+		__isl_take isl_pw_qpolynomial_fold *pwf,
+		__isl_take isl_set *context);
+	__isl_give isl_pw_qpolynomial_fold *
+	isl_pw_qpolynomial_fold_gist_params(
+		__isl_take isl_pw_qpolynomial_fold *pwf,
+		__isl_take isl_set *context);
+	__isl_give isl_union_pw_qpolynomial *
+	isl_union_pw_qpolynomial_gist_params(
+		__isl_take isl_union_pw_qpolynomial *upwqp,
+		__isl_take isl_set *context);
+	__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_gist(
+		__isl_take isl_union_pw_qpolynomial *upwqp,
+		__isl_take isl_union_set *context);
+	__isl_give isl_union_pw_qpolynomial_fold *
+	isl_union_pw_qpolynomial_fold_gist(
+		__isl_take isl_union_pw_qpolynomial_fold *upwf,
+		__isl_take isl_union_set *context);
+	__isl_give isl_union_pw_qpolynomial_fold *
+	isl_union_pw_qpolynomial_fold_gist_params(
+		__isl_take isl_union_pw_qpolynomial_fold *upwf,
+		__isl_take isl_set *context);
+
+=item * Binary Arithmethic Operations
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_aff_add(
+		__isl_take isl_aff *aff1,
+		__isl_take isl_aff *aff2);
+	__isl_give isl_multi_aff *isl_multi_aff_add(
+		__isl_take isl_multi_aff *maff1,
+		__isl_take isl_multi_aff *maff2);
+	__isl_give isl_pw_aff *isl_pw_aff_add(
+		__isl_take isl_pw_aff *pwaff1,
+		__isl_take isl_pw_aff *pwaff2);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_add(
+		__isl_take isl_pw_multi_aff *pma1,
+		__isl_take isl_pw_multi_aff *pma2);
+	__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_add(
+		__isl_take isl_union_pw_multi_aff *upma1,
+		__isl_take isl_union_pw_multi_aff *upma2);
+	__isl_give isl_pw_aff *isl_pw_aff_min(
+		__isl_take isl_pw_aff *pwaff1,
+		__isl_take isl_pw_aff *pwaff2);
+	__isl_give isl_pw_aff *isl_pw_aff_max(
+		__isl_take isl_pw_aff *pwaff1,
+		__isl_take isl_pw_aff *pwaff2);
+	__isl_give isl_aff *isl_aff_sub(
+		__isl_take isl_aff *aff1,
+		__isl_take isl_aff *aff2);
+	__isl_give isl_multi_aff *isl_multi_aff_sub(
+		__isl_take isl_multi_aff *ma1,
+		__isl_take isl_multi_aff *ma2);
+	__isl_give isl_pw_aff *isl_pw_aff_sub(
+		__isl_take isl_pw_aff *pwaff1,
+		__isl_take isl_pw_aff *pwaff2);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_sub(
+		__isl_take isl_pw_multi_aff *pma1,
+		__isl_take isl_pw_multi_aff *pma2);
+	__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_sub(
+		__isl_take isl_union_pw_multi_aff *upma1,
+		__isl_take isl_union_pw_multi_aff *upma2);
+
+C<isl_aff_sub> subtracts the second argument from the first.
+
+	#include <isl/polynomial.h>
+	__isl_give isl_qpolynomial *isl_qpolynomial_add(
+		__isl_take isl_qpolynomial *qp1,
+		__isl_take isl_qpolynomial *qp2);
+	__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add(
+		__isl_take isl_pw_qpolynomial *pwqp1,
+		__isl_take isl_pw_qpolynomial *pwqp2);
+	__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add_disjoint(
+		__isl_take isl_pw_qpolynomial *pwqp1,
+		__isl_take isl_pw_qpolynomial *pwqp2);
+	__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_add(
+		__isl_take isl_pw_qpolynomial_fold *pwf1,
+		__isl_take isl_pw_qpolynomial_fold *pwf2);
+	__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_add(
+		__isl_take isl_union_pw_qpolynomial *upwqp1,
+		__isl_take isl_union_pw_qpolynomial *upwqp2);
+	__isl_give isl_qpolynomial *isl_qpolynomial_sub(
+		__isl_take isl_qpolynomial *qp1,
+		__isl_take isl_qpolynomial *qp2);
+	__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_sub(
+		__isl_take isl_pw_qpolynomial *pwqp1,
+		__isl_take isl_pw_qpolynomial *pwqp2);
+	__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_sub(
+		__isl_take isl_union_pw_qpolynomial *upwqp1,
+		__isl_take isl_union_pw_qpolynomial *upwqp2);
+	__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_fold(
+		__isl_take isl_pw_qpolynomial_fold *pwf1,
+		__isl_take isl_pw_qpolynomial_fold *pwf2);
+	__isl_give isl_union_pw_qpolynomial_fold *
+	isl_union_pw_qpolynomial_fold_fold(
+		__isl_take isl_union_pw_qpolynomial_fold *upwf1,
+		__isl_take isl_union_pw_qpolynomial_fold *upwf2);
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_aff *isl_pw_aff_union_add(
+		__isl_take isl_pw_aff *pwaff1,
+		__isl_take isl_pw_aff *pwaff2);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_add(
+		__isl_take isl_pw_multi_aff *pma1,
+		__isl_take isl_pw_multi_aff *pma2);
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_union_add(
+		__isl_take isl_union_pw_multi_aff *upma1,
+		__isl_take isl_union_pw_multi_aff *upma2);
+	__isl_give isl_pw_aff *isl_pw_aff_union_min(
+		__isl_take isl_pw_aff *pwaff1,
+		__isl_take isl_pw_aff *pwaff2);
+	__isl_give isl_pw_aff *isl_pw_aff_union_max(
+		__isl_take isl_pw_aff *pwaff1,
+		__isl_take isl_pw_aff *pwaff2);
+
+The function C<isl_pw_aff_union_max> computes a piecewise quasi-affine
+expression with a domain that is the union of those of C<pwaff1> and
+C<pwaff2> and such that on each cell, the quasi-affine expression is
+the maximum of those of C<pwaff1> and C<pwaff2>.  If only one of
+C<pwaff1> or C<pwaff2> is defined on a given cell, then the
+associated expression is the defined one.
+This in contrast to the C<isl_pw_aff_max> function, which is
+only defined on the shared definition domain of the arguments.
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_add_val(
+		__isl_take isl_multi_val *mv,
+		__isl_take isl_val *v);
+	__isl_give isl_multi_val *isl_multi_val_mod_val(
+		__isl_take isl_multi_val *mv,
+		__isl_take isl_val *v);
+	__isl_give isl_multi_val *isl_multi_val_scale_val(
+		__isl_take isl_multi_val *mv,
+		__isl_take isl_val *v);
+	__isl_give isl_multi_val *isl_multi_val_scale_down_val(
+		__isl_take isl_multi_val *mv,
+		__isl_take isl_val *v);
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_aff_mod_val(__isl_take isl_aff *aff,
+		__isl_take isl_val *mod);
+	__isl_give isl_pw_aff *isl_pw_aff_mod_val(
+		__isl_take isl_pw_aff *pa,
+		__isl_take isl_val *mod);
+	__isl_give isl_aff *isl_aff_scale_val(__isl_take isl_aff *aff,
+		__isl_take isl_val *v);
+	__isl_give isl_multi_aff *isl_multi_aff_scale_val(
+		__isl_take isl_multi_aff *ma,
+		__isl_take isl_val *v);
+	__isl_give isl_pw_aff *isl_pw_aff_scale_val(
+		__isl_take isl_pw_aff *pa, __isl_take isl_val *v);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_scale_val(
+		__isl_take isl_multi_pw_aff *mpa,
+		__isl_take isl_val *v);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_scale_val(
+		__isl_take isl_pw_multi_aff *pma,
+		__isl_take isl_val *v);
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_scale_val(
+		__isl_take isl_union_pw_multi_aff *upma,
+		__isl_take isl_val *val);
+	__isl_give isl_aff *isl_aff_scale_down_ui(
+		__isl_take isl_aff *aff, unsigned f);
+	__isl_give isl_aff *isl_aff_scale_down_val(
+		__isl_take isl_aff *aff, __isl_take isl_val *v);
+	__isl_give isl_multi_aff *isl_multi_aff_scale_down_val(
+		__isl_take isl_multi_aff *ma,
+		__isl_take isl_val *v);
+	__isl_give isl_pw_aff *isl_pw_aff_scale_down_val(
+		__isl_take isl_pw_aff *pa,
+		__isl_take isl_val *f);
+	__isl_give isl_multi_pw_aff *isl_multi_pw_aff_scale_down_val(
+		__isl_take isl_multi_pw_aff *mpa,
+		__isl_take isl_val *v);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_scale_down_val(
+		__isl_take isl_pw_multi_aff *pma,
+		__isl_take isl_val *v);
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_scale_down_val(
+		__isl_take isl_union_pw_multi_aff *upma,
+		__isl_take isl_val *val);
+
+	#include <isl/polynomial.h>
+	__isl_give isl_qpolynomial *isl_qpolynomial_scale_val(
+		__isl_take isl_qpolynomial *qp,
+		__isl_take isl_val *v);
+	__isl_give isl_qpolynomial_fold *
+	isl_qpolynomial_fold_scale_val(
+		__isl_take isl_qpolynomial_fold *fold,
+		__isl_take isl_val *v);
+	__isl_give isl_pw_qpolynomial *
+	isl_pw_qpolynomial_scale_val(
+		__isl_take isl_pw_qpolynomial *pwqp,
+		__isl_take isl_val *v);
+	__isl_give isl_pw_qpolynomial_fold *
+	isl_pw_qpolynomial_fold_scale_val(
+		__isl_take isl_pw_qpolynomial_fold *pwf,
+		__isl_take isl_val *v);
+	__isl_give isl_union_pw_qpolynomial *
+	isl_union_pw_qpolynomial_scale_val(
+		__isl_take isl_union_pw_qpolynomial *upwqp,
+		__isl_take isl_val *v);
+	__isl_give isl_union_pw_qpolynomial_fold *
+	isl_union_pw_qpolynomial_fold_scale_val(
+		__isl_take isl_union_pw_qpolynomial_fold *upwf,
+		__isl_take isl_val *v);
+	__isl_give isl_qpolynomial *
+	isl_qpolynomial_scale_down_val(
+		__isl_take isl_qpolynomial *qp,
+		__isl_take isl_val *v);
+	__isl_give isl_qpolynomial_fold *
+	isl_qpolynomial_fold_scale_down_val(
+		__isl_take isl_qpolynomial_fold *fold,
+		__isl_take isl_val *v);
+	__isl_give isl_pw_qpolynomial *
+	isl_pw_qpolynomial_scale_down_val(
+		__isl_take isl_pw_qpolynomial *pwqp,
+		__isl_take isl_val *v);
+	__isl_give isl_pw_qpolynomial_fold *
+	isl_pw_qpolynomial_fold_scale_down_val(
+		__isl_take isl_pw_qpolynomial_fold *pwf,
+		__isl_take isl_val *v);
+	__isl_give isl_union_pw_qpolynomial *
+	isl_union_pw_qpolynomial_scale_down_val(
+		__isl_take isl_union_pw_qpolynomial *upwqp,
+		__isl_take isl_val *v);
+	__isl_give isl_union_pw_qpolynomial_fold *
+	isl_union_pw_qpolynomial_fold_scale_down_val(
+		__isl_take isl_union_pw_qpolynomial_fold *upwf,
+		__isl_take isl_val *v);
+
+	#include <isl/val.h>
+	__isl_give isl_multi_val *isl_multi_val_scale_multi_val(
+		__isl_take isl_multi_val *mv1,
+		__isl_take isl_multi_val *mv2);
+	__isl_give isl_multi_val *
+	isl_multi_val_scale_down_multi_val(
+		__isl_take isl_multi_val *mv1,
+		__isl_take isl_multi_val *mv2);
+
+	#include <isl/aff.h>
+	__isl_give isl_multi_aff *isl_multi_aff_scale_multi_val(
+		__isl_take isl_multi_aff *ma,
+		__isl_take isl_multi_val *mv);
+	__isl_give isl_pw_multi_aff *
+	isl_pw_multi_aff_scale_multi_val(
+		__isl_take isl_pw_multi_aff *pma,
+		__isl_take isl_multi_val *mv);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_scale_multi_val(
+		__isl_take isl_multi_pw_aff *mpa,
+		__isl_take isl_multi_val *mv);
+	__isl_give isl_union_pw_multi_aff *
+	isl_union_pw_multi_aff_scale_multi_val(
+		__isl_take isl_union_pw_multi_aff *upma,
+		__isl_take isl_multi_val *mv);
+	__isl_give isl_multi_aff *
+	isl_multi_aff_scale_down_multi_val(
+		__isl_take isl_multi_aff *ma,
+		__isl_take isl_multi_val *mv);
+	__isl_give isl_multi_pw_aff *
+	isl_multi_pw_aff_scale_down_multi_val(
+		__isl_take isl_multi_pw_aff *mpa,
+		__isl_take isl_multi_val *mv);
+
+C<isl_multi_aff_scale_multi_val> scales the elements of C<ma>
+by the corresponding elements of C<mv>.
+
+	#include <isl/aff.h>
+	__isl_give isl_aff *isl_aff_mul(
+		__isl_take isl_aff *aff1,
+		__isl_take isl_aff *aff2);
+	__isl_give isl_aff *isl_aff_div(
+		__isl_take isl_aff *aff1,
+		__isl_take isl_aff *aff2);
+	__isl_give isl_pw_aff *isl_pw_aff_mul(
+		__isl_take isl_pw_aff *pwaff1,
+		__isl_take isl_pw_aff *pwaff2);
+	__isl_give isl_pw_aff *isl_pw_aff_div(
+		__isl_take isl_pw_aff *pa1,
+		__isl_take isl_pw_aff *pa2);
+	__isl_give isl_pw_aff *isl_pw_aff_tdiv_q(
+		__isl_take isl_pw_aff *pa1,
+		__isl_take isl_pw_aff *pa2);
+	__isl_give isl_pw_aff *isl_pw_aff_tdiv_r(
+		__isl_take isl_pw_aff *pa1,
+		__isl_take isl_pw_aff *pa2);
+
+When multiplying two affine expressions, at least one of the two needs
+to be a constant.  Similarly, when dividing an affine expression by another,
+the second expression needs to be a constant.
+C<isl_pw_aff_tdiv_q> computes the quotient of an integer division with
+rounding towards zero.  C<isl_pw_aff_tdiv_r> computes the corresponding
+remainder.
+
+	#include <isl/polynomial.h>
+	__isl_give isl_qpolynomial *isl_qpolynomial_mul(
+		__isl_take isl_qpolynomial *qp1,
+		__isl_take isl_qpolynomial *qp2);
+	__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_mul(
+		__isl_take isl_pw_qpolynomial *pwqp1,
+		__isl_take isl_pw_qpolynomial *pwqp2);
+	__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_mul(
+		__isl_take isl_union_pw_qpolynomial *upwqp1,
+		__isl_take isl_union_pw_qpolynomial *upwqp2);
+
+=back
+
+=head3 Lexicographic Optimization
+
+Given a (basic) set C<set> (or C<bset>) and a zero-dimensional domain C<dom>,
+the following functions
+compute a set that contains the lexicographic minimum or maximum
+of the elements in C<set> (or C<bset>) for those values of the parameters
+that satisfy C<dom>.
+If C<empty> is not C<NULL>, then C<*empty> is assigned a set
+that contains the parameter values in C<dom> for which C<set> (or C<bset>)
+has no elements.
+In other words, the union of the parameter values
+for which the result is non-empty and of C<*empty>
+is equal to C<dom>.
+
+	#include <isl/set.h>
+	__isl_give isl_set *isl_basic_set_partial_lexmin(
+		__isl_take isl_basic_set *bset,
+		__isl_take isl_basic_set *dom,
+		__isl_give isl_set **empty);
+	__isl_give isl_set *isl_basic_set_partial_lexmax(
+		__isl_take isl_basic_set *bset,
+		__isl_take isl_basic_set *dom,
+		__isl_give isl_set **empty);
+	__isl_give isl_set *isl_set_partial_lexmin(
+		__isl_take isl_set *set, __isl_take isl_set *dom,
+		__isl_give isl_set **empty);
+	__isl_give isl_set *isl_set_partial_lexmax(
+		__isl_take isl_set *set, __isl_take isl_set *dom,
+		__isl_give isl_set **empty);
+
+Given a (basic) set C<set> (or C<bset>), the following functions simply
+return a set containing the lexicographic minimum or maximum
+of the elements in C<set> (or C<bset>).
+In case of union sets, the optimum is computed per space.
+
+	#include <isl/set.h>
+	__isl_give isl_set *isl_basic_set_lexmin(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_set *isl_basic_set_lexmax(
+		__isl_take isl_basic_set *bset);
+	__isl_give isl_set *isl_set_lexmin(
+		__isl_take isl_set *set);
+	__isl_give isl_set *isl_set_lexmax(
+		__isl_take isl_set *set);
+	__isl_give isl_union_set *isl_union_set_lexmin(
+		__isl_take isl_union_set *uset);
+	__isl_give isl_union_set *isl_union_set_lexmax(
+		__isl_take isl_union_set *uset);
+
+Given a (basic) relation C<map> (or C<bmap>) and a domain C<dom>,
+the following functions
+compute a relation that maps each element of C<dom>
+to the single lexicographic minimum or maximum
+of the elements that are associated to that same
+element in C<map> (or C<bmap>).
+If C<empty> is not C<NULL>, then C<*empty> is assigned a set
+that contains the elements in C<dom> that do not map
+to any elements in C<map> (or C<bmap>).
+In other words, the union of the domain of the result and of C<*empty>
+is equal to C<dom>.
+
+	#include <isl/map.h>
+	__isl_give isl_map *isl_basic_map_partial_lexmax(
+		__isl_take isl_basic_map *bmap,
+		__isl_take isl_basic_set *dom,
+		__isl_give isl_set **empty);
+	__isl_give isl_map *isl_basic_map_partial_lexmin(
+		__isl_take isl_basic_map *bmap,
+		__isl_take isl_basic_set *dom,
+		__isl_give isl_set **empty);
+	__isl_give isl_map *isl_map_partial_lexmax(
+		__isl_take isl_map *map, __isl_take isl_set *dom,
+		__isl_give isl_set **empty);
+	__isl_give isl_map *isl_map_partial_lexmin(
+		__isl_take isl_map *map, __isl_take isl_set *dom,
+		__isl_give isl_set **empty);
+
+Given a (basic) map C<map> (or C<bmap>), the following functions simply
+return a map mapping each element in the domain of
+C<map> (or C<bmap>) to the lexicographic minimum or maximum
+of all elements associated to that element.
+In case of union relations, the optimum is computed per space.
+
+	#include <isl/map.h>
+	__isl_give isl_map *isl_basic_map_lexmin(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_map *isl_basic_map_lexmax(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_map *isl_map_lexmin(
+		__isl_take isl_map *map);
+	__isl_give isl_map *isl_map_lexmax(
+		__isl_take isl_map *map);
+	__isl_give isl_union_map *isl_union_map_lexmin(
+		__isl_take isl_union_map *umap);
+	__isl_give isl_union_map *isl_union_map_lexmax(
+		__isl_take isl_union_map *umap);
+
+The following functions return their result in the form of
+a piecewise multi-affine expression,
+but are otherwise equivalent to the corresponding functions
+returning a basic set or relation.
+
+	#include <isl/set.h>
+	__isl_give isl_pw_multi_aff *
+	isl_basic_set_partial_lexmin_pw_multi_aff(
+		__isl_take isl_basic_set *bset,
+		__isl_take isl_basic_set *dom,
+		__isl_give isl_set **empty);
+	__isl_give isl_pw_multi_aff *
+	isl_basic_set_partial_lexmax_pw_multi_aff(
+		__isl_take isl_basic_set *bset,
+		__isl_take isl_basic_set *dom,
+		__isl_give isl_set **empty);
+	__isl_give isl_pw_multi_aff *isl_set_lexmin_pw_multi_aff(
+		__isl_take isl_set *set);
+	__isl_give isl_pw_multi_aff *isl_set_lexmax_pw_multi_aff(
+		__isl_take isl_set *set);
+
+	#include <isl/map.h>
+	__isl_give isl_pw_multi_aff *
+	isl_basic_map_lexmin_pw_multi_aff(
+		__isl_take isl_basic_map *bmap);
+	__isl_give isl_pw_multi_aff *
+	isl_basic_map_partial_lexmin_pw_multi_aff(
+		__isl_take isl_basic_map *bmap,
+		__isl_take isl_basic_set *dom,
+		__isl_give isl_set **empty);
+	__isl_give isl_pw_multi_aff *
+	isl_basic_map_partial_lexmax_pw_multi_aff(
+		__isl_take isl_basic_map *bmap,
+		__isl_take isl_basic_set *dom,
+		__isl_give isl_set **empty);
+	__isl_give isl_pw_multi_aff *isl_map_lexmin_pw_multi_aff(
+		__isl_take isl_map *map);
+	__isl_give isl_pw_multi_aff *isl_map_lexmax_pw_multi_aff(
+		__isl_take isl_map *map);
+
+The following functions return the lexicographic minimum or maximum
+on the shared domain of the inputs and the single defined function
+on those parts of the domain where only a single function is defined.
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_lexmin(
+		__isl_take isl_pw_multi_aff *pma1,
+		__isl_take isl_pw_multi_aff *pma2);
+	__isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_lexmax(
+		__isl_take isl_pw_multi_aff *pma1,
+		__isl_take isl_pw_multi_aff *pma2);
+
+=head2 Ternary Operations
+
+	#include <isl/aff.h>
+	__isl_give isl_pw_aff *isl_pw_aff_cond(
+		__isl_take isl_pw_aff *cond,
+		__isl_take isl_pw_aff *pwaff_true,
+		__isl_take isl_pw_aff *pwaff_false);
+
+The function C<isl_pw_aff_cond> performs a conditional operator
+and returns an expression that is equal to C<pwaff_true>
+for elements where C<cond> is non-zero and equal to C<pwaff_false> for elements
+where C<cond> is zero.
+
+=head2 Lists
+
+Lists are defined over several element types, including
+C<isl_val>, C<isl_id>, C<isl_aff>, C<isl_pw_aff>, C<isl_constraint>,
+C<isl_basic_set>, C<isl_set>, C<isl_ast_expr> and C<isl_ast_node>.
+Here we take lists of C<isl_set>s as an example.
+Lists can be created, copied, modified and freed using the following functions.
+
+	#include <isl/set.h>
+	__isl_give isl_set_list *isl_set_list_from_set(
+		__isl_take isl_set *el);
+	__isl_give isl_set_list *isl_set_list_alloc(
+		isl_ctx *ctx, int n);
+	__isl_give isl_set_list *isl_set_list_copy(
+		__isl_keep isl_set_list *list);
+	__isl_give isl_set_list *isl_set_list_insert(
+		__isl_take isl_set_list *list, unsigned pos,
+		__isl_take isl_set *el);
+	__isl_give isl_set_list *isl_set_list_add(
+		__isl_take isl_set_list *list,
+		__isl_take isl_set *el);
+	__isl_give isl_set_list *isl_set_list_drop(
+		__isl_take isl_set_list *list,
+		unsigned first, unsigned n);
+	__isl_give isl_set_list *isl_set_list_set_set(
+		__isl_take isl_set_list *list, int index,
+		__isl_take isl_set *set);
+	__isl_give isl_set_list *isl_set_list_concat(
+		__isl_take isl_set_list *list1,
+		__isl_take isl_set_list *list2);
+	__isl_give isl_set_list *isl_set_list_sort(
+		__isl_take isl_set_list *list,
+		int (*cmp)(__isl_keep isl_set *a,
+			__isl_keep isl_set *b, void *user),
+		void *user);
+	__isl_null isl_set_list *isl_set_list_free(
+		__isl_take isl_set_list *list);
+
+C<isl_set_list_alloc> creates an empty list with a capacity for
+C<n> elements.  C<isl_set_list_from_set> creates a list with a single
+element.
+
+Lists can be inspected using the following functions.
+
+	#include <isl/set.h>
+	int isl_set_list_n_set(__isl_keep isl_set_list *list);
+	__isl_give isl_set *isl_set_list_get_set(
+		__isl_keep isl_set_list *list, int index);
+	int isl_set_list_foreach(__isl_keep isl_set_list *list,
+		int (*fn)(__isl_take isl_set *el, void *user),
+		void *user);
+	int isl_set_list_foreach_scc(__isl_keep isl_set_list *list,
+		int (*follows)(__isl_keep isl_set *a,
+			__isl_keep isl_set *b, void *user),
+		void *follows_user
+		int (*fn)(__isl_take isl_set *el, void *user),
+		void *fn_user);
+
+The function C<isl_set_list_foreach_scc> calls C<fn> on each of the
+strongly connected components of the graph with as vertices the elements
+of C<list> and a directed edge from vertex C<b> to vertex C<a>
+iff C<follows(a, b)> returns C<1>.  The callbacks C<follows> and C<fn>
+should return C<-1> on error.
+
+Lists can be printed using
+
+	#include <isl/set.h>
+	__isl_give isl_printer *isl_printer_print_set_list(
+		__isl_take isl_printer *p,
+		__isl_keep isl_set_list *list);
+
+=head2 Associative arrays
+
+Associative arrays map isl objects of a specific type to isl objects
+of some (other) specific type.  They are defined for several pairs
+of types, including (C<isl_map>, C<isl_basic_set>),
+(C<isl_id>, C<isl_ast_expr>) and.
+(C<isl_id>, C<isl_pw_aff>).
+Here, we take associative arrays that map C<isl_id>s to C<isl_ast_expr>s
+as an example.
+
+Associative arrays can be created, copied and freed using
+the following functions.
+
+	#include <isl/id_to_ast_expr.h>
+	__isl_give id_to_ast_expr *isl_id_to_ast_expr_alloc(
+		isl_ctx *ctx, int min_size);
+	__isl_give id_to_ast_expr *isl_id_to_ast_expr_copy(
+		__isl_keep id_to_ast_expr *id2expr);
+	__isl_null id_to_ast_expr *isl_id_to_ast_expr_free(
+		__isl_take id_to_ast_expr *id2expr);
+
+The C<min_size> argument to C<isl_id_to_ast_expr_alloc> can be used
+to specify the expected size of the associative array.
+The associative array will be grown automatically as needed.
+
+Associative arrays can be inspected using the following functions.
+
+	#include <isl/id_to_ast_expr.h>
+	int isl_id_to_ast_expr_has(
+		__isl_keep id_to_ast_expr *id2expr,
+		__isl_keep isl_id *key);
+	__isl_give isl_ast_expr *isl_id_to_ast_expr_get(
+		__isl_keep id_to_ast_expr *id2expr,
+		__isl_take isl_id *key);
+	int isl_id_to_ast_expr_foreach(
+		__isl_keep id_to_ast_expr *id2expr,
+		int (*fn)(__isl_take isl_id *key,
+			__isl_take isl_ast_expr *val, void *user),
+		void *user);
+
+They can be modified using the following function.
+
+	#include <isl/id_to_ast_expr.h>
+	__isl_give id_to_ast_expr *isl_id_to_ast_expr_set(
+		__isl_take id_to_ast_expr *id2expr,
+		__isl_take isl_id *key,
+		__isl_take isl_ast_expr *val);
+	__isl_give id_to_ast_expr *isl_id_to_ast_expr_drop(
+		__isl_take id_to_ast_expr *id2expr,
+		__isl_take isl_id *key);
+
+Associative arrays can be printed using the following function.
+
+	#include <isl/id_to_ast_expr.h>
+	__isl_give isl_printer *isl_printer_print_id_to_ast_expr(
+		__isl_take isl_printer *p,
+		__isl_keep id_to_ast_expr *id2expr);
+
+=head2 Vectors
+
+Vectors can be created, copied and freed using the following functions.
+
+	#include <isl/vec.h>
+	__isl_give isl_vec *isl_vec_alloc(isl_ctx *ctx,
+		unsigned size);
+	__isl_give isl_vec *isl_vec_copy(__isl_keep isl_vec *vec);
+	__isl_null isl_vec *isl_vec_free(__isl_take isl_vec *vec);
+
+Note that the elements of a newly created vector may have arbitrary values.
+The elements can be changed and inspected using the following functions.
+
+	int isl_vec_size(__isl_keep isl_vec *vec);
+	__isl_give isl_val *isl_vec_get_element_val(
+		__isl_keep isl_vec *vec, int pos);
+	__isl_give isl_vec *isl_vec_set_element_si(
+		__isl_take isl_vec *vec, int pos, int v);
+	__isl_give isl_vec *isl_vec_set_element_val(
+		__isl_take isl_vec *vec, int pos,
+		__isl_take isl_val *v);
+	__isl_give isl_vec *isl_vec_set_si(__isl_take isl_vec *vec,
+		int v);
+	__isl_give isl_vec *isl_vec_set_val(
+		__isl_take isl_vec *vec, __isl_take isl_val *v);
+	int isl_vec_cmp_element(__isl_keep isl_vec *vec1,
+		__isl_keep isl_vec *vec2, int pos);
+
+C<isl_vec_get_element> will return a negative value if anything went wrong.
+In that case, the value of C<*v> is undefined.
+
+The following function can be used to concatenate two vectors.
+
+	__isl_give isl_vec *isl_vec_concat(__isl_take isl_vec *vec1,
+		__isl_take isl_vec *vec2);
+
+=head2 Matrices
+
+Matrices can be created, copied and freed using the following functions.
+
+	#include <isl/mat.h>
+	__isl_give isl_mat *isl_mat_alloc(isl_ctx *ctx,
+		unsigned n_row, unsigned n_col);
+	__isl_give isl_mat *isl_mat_copy(__isl_keep isl_mat *mat);
+	__isl_null isl_mat *isl_mat_free(__isl_take isl_mat *mat);
+
+Note that the elements of a newly created matrix may have arbitrary values.
+The elements can be changed and inspected using the following functions.
+
+	int isl_mat_rows(__isl_keep isl_mat *mat);
+	int isl_mat_cols(__isl_keep isl_mat *mat);
+	__isl_give isl_val *isl_mat_get_element_val(
+		__isl_keep isl_mat *mat, int row, int col);
+	__isl_give isl_mat *isl_mat_set_element_si(__isl_take isl_mat *mat,
+		int row, int col, int v);
+	__isl_give isl_mat *isl_mat_set_element_val(
+		__isl_take isl_mat *mat, int row, int col,
+		__isl_take isl_val *v);
+
+C<isl_mat_get_element> will return a negative value if anything went wrong.
+In that case, the value of C<*v> is undefined.
+
+The following function can be used to compute the (right) inverse
+of a matrix, i.e., a matrix such that the product of the original
+and the inverse (in that order) is a multiple of the identity matrix.
+The input matrix is assumed to be of full row-rank.
+
+	__isl_give isl_mat *isl_mat_right_inverse(__isl_take isl_mat *mat);
+
+The following function can be used to compute the (right) kernel
+(or null space) of a matrix, i.e., a matrix such that the product of
+the original and the kernel (in that order) is the zero matrix.
+
+	__isl_give isl_mat *isl_mat_right_kernel(__isl_take isl_mat *mat);
+
+=head2 Bounds on Piecewise Quasipolynomials and Piecewise Quasipolynomial Reductions
+
+The following functions determine
+an upper or lower bound on a quasipolynomial over its domain.
+
+	__isl_give isl_pw_qpolynomial_fold *
+	isl_pw_qpolynomial_bound(
+		__isl_take isl_pw_qpolynomial *pwqp,
+		enum isl_fold type, int *tight);
+
+	__isl_give isl_union_pw_qpolynomial_fold *
+	isl_union_pw_qpolynomial_bound(
+		__isl_take isl_union_pw_qpolynomial *upwqp,
+		enum isl_fold type, int *tight);
+
+The C<type> argument may be either C<isl_fold_min> or C<isl_fold_max>.
+If C<tight> is not C<NULL>, then C<*tight> is set to C<1>
+is the returned bound is known be tight, i.e., for each value
+of the parameters there is at least
+one element in the domain that reaches the bound.
+If the domain of C<pwqp> is not wrapping, then the bound is computed
+over all elements in that domain and the result has a purely parametric
+domain.  If the domain of C<pwqp> is wrapping, then the bound is
+computed over the range of the wrapped relation.  The domain of the
+wrapped relation becomes the domain of the result.
+
+=head2 Parametric Vertex Enumeration
+
+The parametric vertex enumeration described in this section
+is mainly intended to be used internally and by the C<barvinok>
+library.
+
+	#include <isl/vertices.h>
+	__isl_give isl_vertices *isl_basic_set_compute_vertices(
+		__isl_keep isl_basic_set *bset);
+
+The function C<isl_basic_set_compute_vertices> performs the
+actual computation of the parametric vertices and the chamber
+decomposition and store the result in an C<isl_vertices> object.
+This information can be queried by either iterating over all
+the vertices or iterating over all the chambers or cells
+and then iterating over all vertices that are active on the chamber.
+
+	int isl_vertices_foreach_vertex(
+		__isl_keep isl_vertices *vertices,
+		int (*fn)(__isl_take isl_vertex *vertex, void *user),
+		void *user);
+
+	int isl_vertices_foreach_cell(
+		__isl_keep isl_vertices *vertices,
+		int (*fn)(__isl_take isl_cell *cell, void *user),
+		void *user);
+	int isl_cell_foreach_vertex(__isl_keep isl_cell *cell,
+		int (*fn)(__isl_take isl_vertex *vertex, void *user),
+		void *user);
+
+Other operations that can be performed on an C<isl_vertices> object are
+the following.
+
+	int isl_vertices_get_n_vertices(
+		__isl_keep isl_vertices *vertices);
+	void isl_vertices_free(__isl_take isl_vertices *vertices);
+
+Vertices can be inspected and destroyed using the following functions.
+
+	int isl_vertex_get_id(__isl_keep isl_vertex *vertex);
+	__isl_give isl_basic_set *isl_vertex_get_domain(
+		__isl_keep isl_vertex *vertex);
+	__isl_give isl_multi_aff *isl_vertex_get_expr(
+		__isl_keep isl_vertex *vertex);
+	void isl_vertex_free(__isl_take isl_vertex *vertex);
+
+C<isl_vertex_get_expr> returns a multiple quasi-affine expression
+describing the vertex in terms of the parameters,
+while C<isl_vertex_get_domain> returns the activity domain
+of the vertex.
+
+Chambers can be inspected and destroyed using the following functions.
+
+	__isl_give isl_basic_set *isl_cell_get_domain(
+		__isl_keep isl_cell *cell);
+	void isl_cell_free(__isl_take isl_cell *cell);
+
+=head1 Polyhedral Compilation Library
+
+This section collects functionality in C<isl> that has been specifically
+designed for use during polyhedral compilation.
+
+=head2 Dependence Analysis
+
+C<isl> contains specialized functionality for performing
+array dataflow analysis.  That is, given a I<sink> access relation
+and a collection of possible I<source> access relations,
+C<isl> can compute relations that describe
+for each iteration of the sink access, which iteration
+of which of the source access relations was the last
+to access the same data element before the given iteration
+of the sink access.
+The resulting dependence relations map source iterations
+to the corresponding sink iterations.
+To compute standard flow dependences, the sink should be
+a read, while the sources should be writes.
+If any of the source accesses are marked as being I<may>
+accesses, then there will be a dependence from the last
+I<must> access B<and> from any I<may> access that follows
+this last I<must> access.
+In particular, if I<all> sources are I<may> accesses,
+then memory based dependence analysis is performed.
+If, on the other hand, all sources are I<must> accesses,
+then value based dependence analysis is performed.
+
+	#include <isl/flow.h>
+
+	typedef int (*isl_access_level_before)(void *first, void *second);
+
+	__isl_give isl_access_info *isl_access_info_alloc(
+		__isl_take isl_map *sink,
+		void *sink_user, isl_access_level_before fn,
+		int max_source);
+	__isl_give isl_access_info *isl_access_info_add_source(
+		__isl_take isl_access_info *acc,
+		__isl_take isl_map *source, int must,
+		void *source_user);
+	__isl_null isl_access_info *isl_access_info_free(
+		__isl_take isl_access_info *acc);
+
+	__isl_give isl_flow *isl_access_info_compute_flow(
+		__isl_take isl_access_info *acc);
+
+	int isl_flow_foreach(__isl_keep isl_flow *deps,
+		int (*fn)(__isl_take isl_map *dep, int must,
+			  void *dep_user, void *user),
+		void *user);
+	__isl_give isl_map *isl_flow_get_no_source(
+		__isl_keep isl_flow *deps, int must);
+	void isl_flow_free(__isl_take isl_flow *deps);
+
+The function C<isl_access_info_compute_flow> performs the actual
+dependence analysis.  The other functions are used to construct
+the input for this function or to read off the output.
+
+The input is collected in an C<isl_access_info>, which can
+be created through a call to C<isl_access_info_alloc>.
+The arguments to this functions are the sink access relation
+C<sink>, a token C<sink_user> used to identify the sink
+access to the user, a callback function for specifying the
+relative order of source and sink accesses, and the number
+of source access relations that will be added.
+The callback function has type C<int (*)(void *first, void *second)>.
+The function is called with two user supplied tokens identifying
+either a source or the sink and it should return the shared nesting
+level and the relative order of the two accesses.
+In particular, let I<n> be the number of loops shared by
+the two accesses.  If C<first> precedes C<second> textually,
+then the function should return I<2 * n + 1>; otherwise,
+it should return I<2 * n>.
+The sources can be added to the C<isl_access_info> by performing
+(at most) C<max_source> calls to C<isl_access_info_add_source>.
+C<must> indicates whether the source is a I<must> access
+or a I<may> access.  Note that a multi-valued access relation
+should only be marked I<must> if every iteration in the domain
+of the relation accesses I<all> elements in its image.
+The C<source_user> token is again used to identify
+the source access.  The range of the source access relation
+C<source> should have the same dimension as the range
+of the sink access relation.
+The C<isl_access_info_free> function should usually not be
+called explicitly, because it is called implicitly by
+C<isl_access_info_compute_flow>.
+
+The result of the dependence analysis is collected in an
+C<isl_flow>.  There may be elements of
+the sink access for which no preceding source access could be
+found or for which all preceding sources are I<may> accesses.
+The relations containing these elements can be obtained through
+calls to C<isl_flow_get_no_source>, the first with C<must> set
+and the second with C<must> unset.
+In the case of standard flow dependence analysis,
+with the sink a read and the sources I<must> writes,
+the first relation corresponds to the reads from uninitialized
+array elements and the second relation is empty.
+The actual flow dependences can be extracted using
+C<isl_flow_foreach>.  This function will call the user-specified
+callback function C<fn> for each B<non-empty> dependence between
+a source and the sink.  The callback function is called
+with four arguments, the actual flow dependence relation
+mapping source iterations to sink iterations, a boolean that
+indicates whether it is a I<must> or I<may> dependence, a token
+identifying the source and an additional C<void *> with value
+equal to the third argument of the C<isl_flow_foreach> call.
+A dependence is marked I<must> if it originates from a I<must>
+source and if it is not followed by any I<may> sources.
+
+After finishing with an C<isl_flow>, the user should call
+C<isl_flow_free> to free all associated memory.
+
+A higher-level interface to dependence analysis is provided
+by the following function.
+
+	#include <isl/flow.h>
+
+	int isl_union_map_compute_flow(__isl_take isl_union_map *sink,
+		__isl_take isl_union_map *must_source,
+		__isl_take isl_union_map *may_source,
+		__isl_take isl_union_map *schedule,
+		__isl_give isl_union_map **must_dep,
+		__isl_give isl_union_map **may_dep,
+		__isl_give isl_union_map **must_no_source,
+		__isl_give isl_union_map **may_no_source);
+
+The arrays are identified by the tuple names of the ranges
+of the accesses.  The iteration domains by the tuple names
+of the domains of the accesses and of the schedule.
+The relative order of the iteration domains is given by the
+schedule.  The relations returned through C<must_no_source>
+and C<may_no_source> are subsets of C<sink>.
+Any of C<must_dep>, C<may_dep>, C<must_no_source>
+or C<may_no_source> may be C<NULL>, but a C<NULL> value for
+any of the other arguments is treated as an error.
+
+=head3 Interaction with Dependence Analysis
+
+During the dependence analysis, we frequently need to perform
+the following operation.  Given a relation between sink iterations
+and potential source iterations from a particular source domain,
+what is the last potential source iteration corresponding to each
+sink iteration.  It can sometimes be convenient to adjust
+the set of potential source iterations before or after each such operation.
+The prototypical example is fuzzy array dataflow analysis,
+where we need to analyze if, based on data-dependent constraints,
+the sink iteration can ever be executed without one or more of
+the corresponding potential source iterations being executed.
+If so, we can introduce extra parameters and select an unknown
+but fixed source iteration from the potential source iterations.
+To be able to perform such manipulations, C<isl> provides the following
+function.
+
+	#include <isl/flow.h>
+
+	typedef __isl_give isl_restriction *(*isl_access_restrict)(
+		__isl_keep isl_map *source_map,
+		__isl_keep isl_set *sink, void *source_user,
+		void *user);
+	__isl_give isl_access_info *isl_access_info_set_restrict(
+		__isl_take isl_access_info *acc,
+		isl_access_restrict fn, void *user);
+
+The function C<isl_access_info_set_restrict> should be called
+before calling C<isl_access_info_compute_flow> and registers a callback function
+that will be called any time C<isl> is about to compute the last
+potential source.  The first argument is the (reverse) proto-dependence,
+mapping sink iterations to potential source iterations.
+The second argument represents the sink iterations for which
+we want to compute the last source iteration.
+The third argument is the token corresponding to the source
+and the final argument is the token passed to C<isl_access_info_set_restrict>.
+The callback is expected to return a restriction on either the input or
+the output of the operation computing the last potential source.
+If the input needs to be restricted then restrictions are needed
+for both the source and the sink iterations.  The sink iterations
+and the potential source iterations will be intersected with these sets.
+If the output needs to be restricted then only a restriction on the source
+iterations is required.
+If any error occurs, the callback should return C<NULL>.
+An C<isl_restriction> object can be created, freed and inspected
+using the following functions.
+
+	#include <isl/flow.h>
+
+	__isl_give isl_restriction *isl_restriction_input(
+		__isl_take isl_set *source_restr,
+		__isl_take isl_set *sink_restr);
+	__isl_give isl_restriction *isl_restriction_output(
+		__isl_take isl_set *source_restr);
+	__isl_give isl_restriction *isl_restriction_none(
+		__isl_take isl_map *source_map);
+	__isl_give isl_restriction *isl_restriction_empty(
+		__isl_take isl_map *source_map);
+	__isl_null isl_restriction *isl_restriction_free(
+		__isl_take isl_restriction *restr);
+
+C<isl_restriction_none> and C<isl_restriction_empty> are special
+cases of C<isl_restriction_input>.  C<isl_restriction_none>
+is essentially equivalent to
+
+	isl_restriction_input(isl_set_universe(
+	    isl_space_range(isl_map_get_space(source_map))),
+			    isl_set_universe(
+	    isl_space_domain(isl_map_get_space(source_map))));
+
+whereas C<isl_restriction_empty> is essentially equivalent to
+
+	isl_restriction_input(isl_set_empty(
+	    isl_space_range(isl_map_get_space(source_map))),
+			    isl_set_universe(
+	    isl_space_domain(isl_map_get_space(source_map))));
+
+=head2 Scheduling
+
+B<The functionality described in this section is fairly new
+and may be subject to change.>
+
+	#include <isl/schedule.h>
+	__isl_give isl_schedule *
+	isl_schedule_constraints_compute_schedule(
+		__isl_take isl_schedule_constraints *sc);
+	__isl_null isl_schedule *isl_schedule_free(
+		__isl_take isl_schedule *sched);
+
+The function C<isl_schedule_constraints_compute_schedule> can be
+used to compute a schedule that satisfies the given schedule constraints.
+These schedule constraints include the iteration domain for which
+a schedule should be computed and dependences between pairs of
+iterations.  In particular, these dependences include
+I<validity> dependences and I<proximity> dependences.
+By default, the algorithm used to construct the schedule is similar
+to that of C<Pluto>.
+Alternatively, Feautrier's multi-dimensional scheduling algorithm can
+be selected.
+The generated schedule respects all validity dependences.
+That is, all dependence distances over these dependences in the
+scheduled space are lexicographically positive.
+The default algorithm tries to ensure that the dependence distances
+over coincidence constraints are zero and to minimize the
+dependence distances over proximity dependences.
+Moreover, it tries to obtain sequences (bands) of schedule dimensions
+for groups of domains where the dependence distances over validity
+dependences have only non-negative values.
+When using Feautrier's algorithm, the coincidence and proximity constraints
+are only taken into account during the extension to a
+full-dimensional schedule.
+
+An C<isl_schedule_constraints> object can be constructed
+and manipulated using the following functions.
+
+	#include <isl/schedule.h>
+	__isl_give isl_schedule_constraints *
+	isl_schedule_constraints_copy(
+		__isl_keep isl_schedule_constraints *sc);
+	__isl_give isl_schedule_constraints *
+	isl_schedule_constraints_on_domain(
+		__isl_take isl_union_set *domain);
+	__isl_give isl_schedule_constraints *
+	isl_schedule_constraints_set_validity(
+		__isl_take isl_schedule_constraints *sc,
+		__isl_take isl_union_map *validity);
+	__isl_give isl_schedule_constraints *
+	isl_schedule_constraints_set_coincidence(
+		__isl_take isl_schedule_constraints *sc,
+		__isl_take isl_union_map *coincidence);
+	__isl_give isl_schedule_constraints *
+	isl_schedule_constraints_set_proximity(
+		__isl_take isl_schedule_constraints *sc,
+		__isl_take isl_union_map *proximity);
+	__isl_give isl_schedule_constraints *
+	isl_schedule_constraints_set_conditional_validity(
+		__isl_take isl_schedule_constraints *sc,
+		__isl_take isl_union_map *condition,
+		__isl_take isl_union_map *validity);
+	__isl_null isl_schedule_constraints *
+	isl_schedule_constraints_free(
+		__isl_take isl_schedule_constraints *sc);
+
+The initial C<isl_schedule_constraints> object created by
+C<isl_schedule_constraints_on_domain> does not impose any constraints.
+That is, it has an empty set of dependences.
+The function C<isl_schedule_constraints_set_validity> replaces the
+validity dependences, mapping domain elements I<i> to domain
+elements that should be scheduled after I<i>.
+The function C<isl_schedule_constraints_set_coincidence> replaces the
+coincidence dependences, mapping domain elements I<i> to domain
+elements that should be scheduled together with I<I>, if possible.
+The function C<isl_schedule_constraints_set_proximity> replaces the
+proximity dependences, mapping domain elements I<i> to domain
+elements that should be scheduled either before I<I>
+or as early as possible after I<i>.
+
+The function C<isl_schedule_constraints_set_conditional_validity>
+replaces the conditional validity constraints.
+A conditional validity constraint is only imposed when any of the corresponding
+conditions is satisfied, i.e., when any of them is non-zero.
+That is, the scheduler ensures that within each band if the dependence
+distances over the condition constraints are not all zero
+then all corresponding conditional validity constraints are respected.
+A conditional validity constraint corresponds to a condition
+if the two are adjacent, i.e., if the domain of one relation intersect
+the range of the other relation.
+The typical use case of conditional validity constraints is
+to allow order constraints between live ranges to be violated
+as long as the live ranges themselves are local to the band.
+To allow more fine-grained control over which conditions correspond
+to which conditional validity constraints, the domains and ranges
+of these relations may include I<tags>.  That is, the domains and
+ranges of those relation may themselves be wrapped relations
+where the iteration domain appears in the domain of those wrapped relations
+and the range of the wrapped relations can be arbitrarily chosen
+by the user.  Conditions and conditional validity constraints are only
+considered adjacent to each other if the entire wrapped relation matches.
+In particular, a relation with a tag will never be considered adjacent
+to a relation without a tag.
+
+The following function computes a schedule directly from
+an iteration domain and validity and proximity dependences
+and is implemented in terms of the functions described above.
+The use of C<isl_union_set_compute_schedule> is discouraged.
+
+	#include <isl/schedule.h>
+	__isl_give isl_schedule *isl_union_set_compute_schedule(
+		__isl_take isl_union_set *domain,
+		__isl_take isl_union_map *validity,
+		__isl_take isl_union_map *proximity);
+
+A mapping from the domains to the scheduled space can be obtained
+from an C<isl_schedule> using the following function.
+
+	__isl_give isl_union_map *isl_schedule_get_map(
+		__isl_keep isl_schedule *sched);
+
+A representation of the schedule can be printed using
+	 
+	__isl_give isl_printer *isl_printer_print_schedule(
+		__isl_take isl_printer *p,
+		__isl_keep isl_schedule *schedule);
+
+A representation of the schedule as a forest of bands can be obtained
+using the following function.
+
+	__isl_give isl_band_list *isl_schedule_get_band_forest(
+		__isl_keep isl_schedule *schedule);
+
+The individual bands can be visited in depth-first post-order
+using the following function.
+
+	#include <isl/schedule.h>
+	int isl_schedule_foreach_band(
+		__isl_keep isl_schedule *sched,
+		int (*fn)(__isl_keep isl_band *band, void *user),
+		void *user);
+
+The list can be manipulated as explained in L<"Lists">.
+The bands inside the list can be copied and freed using the following
+functions.
+
+	#include <isl/band.h>
+	__isl_give isl_band *isl_band_copy(
+		__isl_keep isl_band *band);
+	__isl_null isl_band *isl_band_free(
+		__isl_take isl_band *band);
+
+Each band contains zero or more scheduling dimensions.
+These are referred to as the members of the band.
+The section of the schedule that corresponds to the band is
+referred to as the partial schedule of the band.
+For those nodes that participate in a band, the outer scheduling
+dimensions form the prefix schedule, while the inner scheduling
+dimensions form the suffix schedule.
+That is, if we take a cut of the band forest, then the union of
+the concatenations of the prefix, partial and suffix schedules of
+each band in the cut is equal to the entire schedule (modulo
+some possible padding at the end with zero scheduling dimensions).
+The properties of a band can be inspected using the following functions.
+
+	#include <isl/band.h>
+	int isl_band_has_children(__isl_keep isl_band *band);
+	__isl_give isl_band_list *isl_band_get_children(
+		__isl_keep isl_band *band);
+
+	__isl_give isl_union_map *isl_band_get_prefix_schedule(
+		__isl_keep isl_band *band);
+	__isl_give isl_union_map *isl_band_get_partial_schedule(
+		__isl_keep isl_band *band);
+	__isl_give isl_union_map *isl_band_get_suffix_schedule(
+		__isl_keep isl_band *band);
+
+	int isl_band_n_member(__isl_keep isl_band *band);
+	int isl_band_member_is_coincident(
+		__isl_keep isl_band *band, int pos);
+
+	int isl_band_list_foreach_band(
+		__isl_keep isl_band_list *list,
+		int (*fn)(__isl_keep isl_band *band, void *user),
+		void *user);
+
+Note that a scheduling dimension is considered to be ``coincident''
+if it satisfies the coincidence constraints within its band.
+That is, if the dependence distances of the coincidence
+constraints are all zero in that direction (for fixed
+iterations of outer bands).
+Like C<isl_schedule_foreach_band>,
+the function C<isl_band_list_foreach_band> calls C<fn> on the bands
+in depth-first post-order.
+
+A band can be tiled using the following function.
+
+	#include <isl/band.h>
+	int isl_band_tile(__isl_keep isl_band *band,
+		__isl_take isl_vec *sizes);
+
+	int isl_options_set_tile_scale_tile_loops(isl_ctx *ctx,
+		int val);
+	int isl_options_get_tile_scale_tile_loops(isl_ctx *ctx);
+	int isl_options_set_tile_shift_point_loops(isl_ctx *ctx,
+		int val);
+	int isl_options_get_tile_shift_point_loops(isl_ctx *ctx);
+
+The C<isl_band_tile> function tiles the band using the given tile sizes
+inside its schedule.
+A new child band is created to represent the point loops and it is
+inserted between the modified band and its children.
+The C<tile_scale_tile_loops> option specifies whether the tile
+loops iterators should be scaled by the tile sizes.
+If the C<tile_shift_point_loops> option is set, then the point loops
+are shifted to start at zero.
+
+A band can be split into two nested bands using the following function.
+
+	int isl_band_split(__isl_keep isl_band *band, int pos);
+
+The resulting outer band contains the first C<pos> dimensions of C<band>
+while the inner band contains the remaining dimensions.
+
+A representation of the band can be printed using
+
+	#include <isl/band.h>
+	__isl_give isl_printer *isl_printer_print_band(
+		__isl_take isl_printer *p,
+		__isl_keep isl_band *band);
+
+=head3 Options
+
+	#include <isl/schedule.h>
+	int isl_options_set_schedule_max_coefficient(
+		isl_ctx *ctx, int val);
+	int isl_options_get_schedule_max_coefficient(
+		isl_ctx *ctx);
+	int isl_options_set_schedule_max_constant_term(
+		isl_ctx *ctx, int val);
+	int isl_options_get_schedule_max_constant_term(
+		isl_ctx *ctx);
+	int isl_options_set_schedule_fuse(isl_ctx *ctx, int val);
+	int isl_options_get_schedule_fuse(isl_ctx *ctx);
+	int isl_options_set_schedule_maximize_band_depth(
+		isl_ctx *ctx, int val);
+	int isl_options_get_schedule_maximize_band_depth(
+		isl_ctx *ctx);
+	int isl_options_set_schedule_outer_coincidence(
+		isl_ctx *ctx, int val);
+	int isl_options_get_schedule_outer_coincidence(
+		isl_ctx *ctx);
+	int isl_options_set_schedule_split_scaled(
+		isl_ctx *ctx, int val);
+	int isl_options_get_schedule_split_scaled(
+		isl_ctx *ctx);
+	int isl_options_set_schedule_algorithm(
+		isl_ctx *ctx, int val);
+	int isl_options_get_schedule_algorithm(
+		isl_ctx *ctx);
+	int isl_options_set_schedule_separate_components(
+		isl_ctx *ctx, int val);
+	int isl_options_get_schedule_separate_components(
+		isl_ctx *ctx);
+
+=over
+
+=item * schedule_max_coefficient
+
+This option enforces that the coefficients for variable and parameter
+dimensions in the calculated schedule are not larger than the specified value.
+This option can significantly increase the speed of the scheduling calculation
+and may also prevent fusing of unrelated dimensions. A value of -1 means that
+this option does not introduce bounds on the variable or parameter
+coefficients.
+
+=item * schedule_max_constant_term
+
+This option enforces that the constant coefficients in the calculated schedule
+are not larger than the maximal constant term. This option can significantly
+increase the speed of the scheduling calculation and may also prevent fusing of
+unrelated dimensions. A value of -1 means that this option does not introduce
+bounds on the constant coefficients.
+
+=item * schedule_fuse
+
+This option controls the level of fusion.
+If this option is set to C<ISL_SCHEDULE_FUSE_MIN>, then loops in the
+resulting schedule will be distributed as much as possible.
+If this option is set to C<ISL_SCHEDULE_FUSE_MAX>, then C<isl> will
+try to fuse loops in the resulting schedule.
+
+=item * schedule_maximize_band_depth
+
+If this option is set, we do not split bands at the point
+where we detect splitting is necessary. Instead, we
+backtrack and split bands as early as possible. This
+reduces the number of splits and maximizes the width of
+the bands. Wider bands give more possibilities for tiling.
+Note that if the C<schedule_fuse> option is set to C<ISL_SCHEDULE_FUSE_MIN>,
+then bands will be split as early as possible, even if there is no need.
+The C<schedule_maximize_band_depth> option therefore has no effect in this case.
+
+=item * schedule_outer_coincidence
+
+If this option is set, then we try to construct schedules
+where the outermost scheduling dimension in each band
+satisfies the coincidence constraints.
+
+=item * schedule_split_scaled
+
+If this option is set, then we try to construct schedules in which the
+constant term is split off from the linear part if the linear parts of
+the scheduling rows for all nodes in the graphs have a common non-trivial
+divisor.
+The constant term is then placed in a separate band and the linear
+part is reduced.
+
+=item * schedule_algorithm
+
+Selects the scheduling algorithm to be used.
+Available scheduling algorithms are C<ISL_SCHEDULE_ALGORITHM_ISL>
+and C<ISL_SCHEDULE_ALGORITHM_FEAUTRIER>.
+
+=item * schedule_separate_components
+
+If at any point the dependence graph contains any (weakly connected) components,
+then these components are scheduled separately.
+If this option is not set, then some iterations of the domains
+in these components may be scheduled together.
+If this option is set, then the components are given consecutive
+schedules.
+
+=back
+
+=head2 AST Generation
+
+This section describes the C<isl> functionality for generating
+ASTs that visit all the elements
+in a domain in an order specified by a schedule.
+In particular, given a C<isl_union_map>, an AST is generated
+that visits all the elements in the domain of the C<isl_union_map>
+according to the lexicographic order of the corresponding image
+element(s).  If the range of the C<isl_union_map> consists of
+elements in more than one space, then each of these spaces is handled
+separately in an arbitrary order.
+It should be noted that the image elements only specify the I<order>
+in which the corresponding domain elements should be visited.
+No direct relation between the image elements and the loop iterators
+in the generated AST should be assumed.
+
+Each AST is generated within a build.  The initial build
+simply specifies the constraints on the parameters (if any)
+and can be created, inspected, copied and freed using the following functions.
+
+	#include <isl/ast_build.h>
+	__isl_give isl_ast_build *isl_ast_build_from_context(
+		__isl_take isl_set *set);
+	__isl_give isl_ast_build *isl_ast_build_copy(
+		__isl_keep isl_ast_build *build);
+	__isl_null isl_ast_build *isl_ast_build_free(
+		__isl_take isl_ast_build *build);
+
+The C<set> argument is usually a parameter set with zero or more parameters.
+More C<isl_ast_build> functions are described in L</"Nested AST Generation">
+and L</"Fine-grained Control over AST Generation">.
+Finally, the AST itself can be constructed using the following
+function.
+
+	#include <isl/ast_build.h>
+	__isl_give isl_ast_node *isl_ast_build_ast_from_schedule(
+		__isl_keep isl_ast_build *build,
+		__isl_take isl_union_map *schedule);
+
+=head3 Inspecting the AST
+
+The basic properties of an AST node can be obtained as follows.
+
+	#include <isl/ast.h>
+	enum isl_ast_node_type isl_ast_node_get_type(
+		__isl_keep isl_ast_node *node);
+
+The type of an AST node is one of
+C<isl_ast_node_for>,
+C<isl_ast_node_if>,
+C<isl_ast_node_block> or
+C<isl_ast_node_user>.
+An C<isl_ast_node_for> represents a for node.
+An C<isl_ast_node_if> represents an if node.
+An C<isl_ast_node_block> represents a compound node.
+An C<isl_ast_node_user> represents an expression statement.
+An expression statement typically corresponds to a domain element, i.e.,
+one of the elements that is visited by the AST.
+
+Each type of node has its own additional properties.
+
+	#include <isl/ast.h>
+	__isl_give isl_ast_expr *isl_ast_node_for_get_iterator(
+		__isl_keep isl_ast_node *node);
+	__isl_give isl_ast_expr *isl_ast_node_for_get_init(
+		__isl_keep isl_ast_node *node);
+	__isl_give isl_ast_expr *isl_ast_node_for_get_cond(
+		__isl_keep isl_ast_node *node);
+	__isl_give isl_ast_expr *isl_ast_node_for_get_inc(
+		__isl_keep isl_ast_node *node);
+	__isl_give isl_ast_node *isl_ast_node_for_get_body(
+		__isl_keep isl_ast_node *node);
+	int isl_ast_node_for_is_degenerate(
+		__isl_keep isl_ast_node *node);
+
+An C<isl_ast_for> is considered degenerate if it is known to execute
+exactly once.
+
+	#include <isl/ast.h>
+	__isl_give isl_ast_expr *isl_ast_node_if_get_cond(
+		__isl_keep isl_ast_node *node);
+	__isl_give isl_ast_node *isl_ast_node_if_get_then(
+		__isl_keep isl_ast_node *node);
+	int isl_ast_node_if_has_else(
+		__isl_keep isl_ast_node *node);
+	__isl_give isl_ast_node *isl_ast_node_if_get_else(
+		__isl_keep isl_ast_node *node);
+
+	__isl_give isl_ast_node_list *
+	isl_ast_node_block_get_children(
+		__isl_keep isl_ast_node *node);
+
+	__isl_give isl_ast_expr *isl_ast_node_user_get_expr(
+		__isl_keep isl_ast_node *node);
+
+Each of the returned C<isl_ast_expr>s can in turn be inspected using
+the following functions.
+
+	#include <isl/ast.h>
+	enum isl_ast_expr_type isl_ast_expr_get_type(
+		__isl_keep isl_ast_expr *expr);
+
+The type of an AST expression is one of
+C<isl_ast_expr_op>,
+C<isl_ast_expr_id> or
+C<isl_ast_expr_int>.
+An C<isl_ast_expr_op> represents the result of an operation.
+An C<isl_ast_expr_id> represents an identifier.
+An C<isl_ast_expr_int> represents an integer value.
+
+Each type of expression has its own additional properties.
+
+	#include <isl/ast.h>
+	enum isl_ast_op_type isl_ast_expr_get_op_type(
+		__isl_keep isl_ast_expr *expr);
+	int isl_ast_expr_get_op_n_arg(__isl_keep isl_ast_expr *expr);
+	__isl_give isl_ast_expr *isl_ast_expr_get_op_arg(
+		__isl_keep isl_ast_expr *expr, int pos);
+	int isl_ast_node_foreach_ast_op_type(
+		__isl_keep isl_ast_node *node,
+		int (*fn)(enum isl_ast_op_type type, void *user),
+		void *user);
+
+C<isl_ast_expr_get_op_type> returns the type of the operation
+performed.  C<isl_ast_expr_get_op_n_arg> returns the number of
+arguments.  C<isl_ast_expr_get_op_arg> returns the specified
+argument.
+C<isl_ast_node_foreach_ast_op_type> calls C<fn> for each distinct
+C<isl_ast_op_type> that appears in C<node>.
+The operation type is one of the following.
+
+=over
+
+=item C<isl_ast_op_and>
+
+Logical I<and> of two arguments.
+Both arguments can be evaluated.
+
+=item C<isl_ast_op_and_then>
+
+Logical I<and> of two arguments.
+The second argument can only be evaluated if the first evaluates to true.
+
+=item C<isl_ast_op_or>
+
+Logical I<or> of two arguments.
+Both arguments can be evaluated.
+
+=item C<isl_ast_op_or_else>
+
+Logical I<or> of two arguments.
+The second argument can only be evaluated if the first evaluates to false.
+
+=item C<isl_ast_op_max>
+
+Maximum of two or more arguments.
+
+=item C<isl_ast_op_min>
+
+Minimum of two or more arguments.
+
+=item C<isl_ast_op_minus>
+
+Change sign.
+
+=item C<isl_ast_op_add>
+
+Sum of two arguments.
+
+=item C<isl_ast_op_sub>
+
+Difference of two arguments.
+
+=item C<isl_ast_op_mul>
+
+Product of two arguments.
+
+=item C<isl_ast_op_div>
+
+Exact division.  That is, the result is known to be an integer.
+
+=item C<isl_ast_op_fdiv_q>
+
+Result of integer division, rounded towards negative
+infinity.
+
+=item C<isl_ast_op_pdiv_q>
+
+Result of integer division, where dividend is known to be non-negative.
+
+=item C<isl_ast_op_pdiv_r>
+
+Remainder of integer division, where dividend is known to be non-negative.
+
+=item C<isl_ast_op_zdiv_r>
+
+Equal to zero iff the remainder on integer division is zero.
+
+=item C<isl_ast_op_cond>
+
+Conditional operator defined on three arguments.
+If the first argument evaluates to true, then the result
+is equal to the second argument.  Otherwise, the result
+is equal to the third argument.
+The second and third argument may only be evaluated if
+the first argument evaluates to true and false, respectively.
+Corresponds to C<a ? b : c> in C.
+
+=item C<isl_ast_op_select>
+
+Conditional operator defined on three arguments.
+If the first argument evaluates to true, then the result
+is equal to the second argument.  Otherwise, the result
+is equal to the third argument.
+The second and third argument may be evaluated independently
+of the value of the first argument.
+Corresponds to C<a * b + (1 - a) * c> in C.
+
+=item C<isl_ast_op_eq>
+
+Equality relation.
+
+=item C<isl_ast_op_le>
+
+Less than or equal relation.
+
+=item C<isl_ast_op_lt>
+
+Less than relation.
+
+=item C<isl_ast_op_ge>
+
+Greater than or equal relation.
+
+=item C<isl_ast_op_gt>
+
+Greater than relation.
+
+=item C<isl_ast_op_call>
+
+A function call.
+The number of arguments of the C<isl_ast_expr> is one more than
+the number of arguments in the function call, the first argument
+representing the function being called.
+
+=item C<isl_ast_op_access>
+
+An array access.
+The number of arguments of the C<isl_ast_expr> is one more than
+the number of index expressions in the array access, the first argument
+representing the array being accessed.
+
+=item C<isl_ast_op_member>
+
+A member access.
+This operation has two arguments, a structure and the name of
+the member of the structure being accessed.
+
+=back
+
+	#include <isl/ast.h>
+	__isl_give isl_id *isl_ast_expr_get_id(
+		__isl_keep isl_ast_expr *expr);
+
+Return the identifier represented by the AST expression.
+
+	#include <isl/ast.h>
+	__isl_give isl_val *isl_ast_expr_get_val(
+		__isl_keep isl_ast_expr *expr);
+
+Return the integer represented by the AST expression.
+
+=head3 Properties of ASTs
+
+	#include <isl/ast.h>
+	int isl_ast_expr_is_equal(__isl_keep isl_ast_expr *expr1,
+		__isl_keep isl_ast_expr *expr2);
+
+Check if two C<isl_ast_expr>s are equal to each other.
+
+=head3 Manipulating and printing the AST
+
+AST nodes can be copied and freed using the following functions.
+
+	#include <isl/ast.h>
+	__isl_give isl_ast_node *isl_ast_node_copy(
+		__isl_keep isl_ast_node *node);
+	__isl_null isl_ast_node *isl_ast_node_free(
+		__isl_take isl_ast_node *node);
+
+AST expressions can be copied and freed using the following functions.
+
+	#include <isl/ast.h>
+	__isl_give isl_ast_expr *isl_ast_expr_copy(
+		__isl_keep isl_ast_expr *expr);
+	__isl_null isl_ast_expr *isl_ast_expr_free(
+		__isl_take isl_ast_expr *expr);
+
+New AST expressions can be created either directly or within
+the context of an C<isl_ast_build>.
+
+	#include <isl/ast.h>
+	__isl_give isl_ast_expr *isl_ast_expr_from_val(
+		__isl_take isl_val *v);
+	__isl_give isl_ast_expr *isl_ast_expr_from_id(
+		__isl_take isl_id *id);
+	__isl_give isl_ast_expr *isl_ast_expr_neg(
+		__isl_take isl_ast_expr *expr);
+	__isl_give isl_ast_expr *isl_ast_expr_address_of(
+		__isl_take isl_ast_expr *expr);
+	__isl_give isl_ast_expr *isl_ast_expr_add(
+		__isl_take isl_ast_expr *expr1,
+		__isl_take isl_ast_expr *expr2);
+	__isl_give isl_ast_expr *isl_ast_expr_sub(
+		__isl_take isl_ast_expr *expr1,
+		__isl_take isl_ast_expr *expr2);
+	__isl_give isl_ast_expr *isl_ast_expr_mul(
+		__isl_take isl_ast_expr *expr1,
+		__isl_take isl_ast_expr *expr2);
+	__isl_give isl_ast_expr *isl_ast_expr_div(
+		__isl_take isl_ast_expr *expr1,
+		__isl_take isl_ast_expr *expr2);
+	__isl_give isl_ast_expr *isl_ast_expr_and(
+		__isl_take isl_ast_expr *expr1,
+		__isl_take isl_ast_expr *expr2)
+	__isl_give isl_ast_expr *isl_ast_expr_or(
+		__isl_take isl_ast_expr *expr1,
+		__isl_take isl_ast_expr *expr2)
+	__isl_give isl_ast_expr *isl_ast_expr_eq(
+		__isl_take isl_ast_expr *expr1,
+		__isl_take isl_ast_expr *expr2);
+	__isl_give isl_ast_expr *isl_ast_expr_le(
+		__isl_take isl_ast_expr *expr1,
+		__isl_take isl_ast_expr *expr2);
+	__isl_give isl_ast_expr *isl_ast_expr_lt(
+		__isl_take isl_ast_expr *expr1,
+		__isl_take isl_ast_expr *expr2);
+	__isl_give isl_ast_expr *isl_ast_expr_ge(
+		__isl_take isl_ast_expr *expr1,
+		__isl_take isl_ast_expr *expr2);
+	__isl_give isl_ast_expr *isl_ast_expr_gt(
+		__isl_take isl_ast_expr *expr1,
+		__isl_take isl_ast_expr *expr2);
+	__isl_give isl_ast_expr *isl_ast_expr_access(
+		__isl_take isl_ast_expr *array,
+		__isl_take isl_ast_expr_list *indices);
+
+The function C<isl_ast_expr_address_of> can be applied to an
+C<isl_ast_expr> of type C<isl_ast_op_access> only. It is meant
+to represent the address of the C<isl_ast_expr_access>.
+
+	#include <isl/ast_build.h>
+	__isl_give isl_ast_expr *isl_ast_build_expr_from_pw_aff(
+		__isl_keep isl_ast_build *build,
+		__isl_take isl_pw_aff *pa);
+	__isl_give isl_ast_expr *
+	isl_ast_build_access_from_pw_multi_aff(
+		__isl_keep isl_ast_build *build,
+		__isl_take isl_pw_multi_aff *pma);
+	__isl_give isl_ast_expr *
+	isl_ast_build_access_from_multi_pw_aff(
+		__isl_keep isl_ast_build *build,
+		__isl_take isl_multi_pw_aff *mpa);
+	__isl_give isl_ast_expr *
+	isl_ast_build_call_from_pw_multi_aff(
+		__isl_keep isl_ast_build *build,
+		__isl_take isl_pw_multi_aff *pma);
+	__isl_give isl_ast_expr *
+	isl_ast_build_call_from_multi_pw_aff(
+		__isl_keep isl_ast_build *build,
+		__isl_take isl_multi_pw_aff *mpa);
+
+The domains of C<pa>, C<mpa> and C<pma> should correspond
+to the schedule space of C<build>.
+The tuple id of C<mpa> or C<pma> is used as the array being accessed or
+the function being called.
+If the accessed space is a nested relation, then it is taken
+to represent an access of the member specified by the range
+of this nested relation of the structure specified by the domain
+of the nested relation.
+
+The following functions can be used to modify an C<isl_ast_expr>.
+
+	#include <isl/ast.h>
+	__isl_give isl_ast_expr *isl_ast_expr_set_op_arg(
+		__isl_take isl_ast_expr *expr, int pos,
+		__isl_take isl_ast_expr *arg);
+
+Replace the argument of C<expr> at position C<pos> by C<arg>.
+
+	#include <isl/ast.h>
+	__isl_give isl_ast_expr *isl_ast_expr_substitute_ids(
+		__isl_take isl_ast_expr *expr,
+		__isl_take isl_id_to_ast_expr *id2expr);
+
+The function C<isl_ast_expr_substitute_ids> replaces the
+subexpressions of C<expr> of type C<isl_ast_expr_id>
+by the corresponding expression in C<id2expr>, if there is any.
+
+
+User specified data can be attached to an C<isl_ast_node> and obtained
+from the same C<isl_ast_node> using the following functions.
+
+	#include <isl/ast.h>
+	__isl_give isl_ast_node *isl_ast_node_set_annotation(
+		__isl_take isl_ast_node *node,
+		__isl_take isl_id *annotation);
+	__isl_give isl_id *isl_ast_node_get_annotation(
+		__isl_keep isl_ast_node *node);
+
+Basic printing can be performed using the following functions.
+
+	#include <isl/ast.h>
+	__isl_give isl_printer *isl_printer_print_ast_expr(
+		__isl_take isl_printer *p,
+		__isl_keep isl_ast_expr *expr);
+	__isl_give isl_printer *isl_printer_print_ast_node(
+		__isl_take isl_printer *p,
+		__isl_keep isl_ast_node *node);
+	__isl_give char *isl_ast_expr_to_str(
+		__isl_keep isl_ast_expr *expr);
+
+More advanced printing can be performed using the following functions.
+
+	#include <isl/ast.h>
+	__isl_give isl_printer *isl_ast_op_type_print_macro(
+		enum isl_ast_op_type type,
+		__isl_take isl_printer *p);
+	__isl_give isl_printer *isl_ast_node_print_macros(
+		__isl_keep isl_ast_node *node,
+		__isl_take isl_printer *p);
+	__isl_give isl_printer *isl_ast_node_print(
+		__isl_keep isl_ast_node *node,
+		__isl_take isl_printer *p,
+		__isl_take isl_ast_print_options *options);
+	__isl_give isl_printer *isl_ast_node_for_print(
+		__isl_keep isl_ast_node *node,
+		__isl_take isl_printer *p,
+		__isl_take isl_ast_print_options *options);
+	__isl_give isl_printer *isl_ast_node_if_print(
+		__isl_keep isl_ast_node *node,
+		__isl_take isl_printer *p,
+		__isl_take isl_ast_print_options *options);
+
+While printing an C<isl_ast_node> in C<ISL_FORMAT_C>,
+C<isl> may print out an AST that makes use of macros such
+as C<floord>, C<min> and C<max>.
+C<isl_ast_op_type_print_macro> prints out the macro
+corresponding to a specific C<isl_ast_op_type>.
+C<isl_ast_node_print_macros> scans the C<isl_ast_node>
+for expressions where these macros would be used and prints
+out the required macro definitions.
+Essentially, C<isl_ast_node_print_macros> calls
+C<isl_ast_node_foreach_ast_op_type> with C<isl_ast_op_type_print_macro>
+as function argument.
+C<isl_ast_node_print>, C<isl_ast_node_for_print> and
+C<isl_ast_node_if_print> print an C<isl_ast_node>
+in C<ISL_FORMAT_C>, but allow for some extra control
+through an C<isl_ast_print_options> object.
+This object can be created using the following functions.
+
+	#include <isl/ast.h>
+	__isl_give isl_ast_print_options *
+	isl_ast_print_options_alloc(isl_ctx *ctx);
+	__isl_give isl_ast_print_options *
+	isl_ast_print_options_copy(
+		__isl_keep isl_ast_print_options *options);
+	__isl_null isl_ast_print_options *
+	isl_ast_print_options_free(
+		__isl_take isl_ast_print_options *options);
+
+	__isl_give isl_ast_print_options *
+	isl_ast_print_options_set_print_user(
+		__isl_take isl_ast_print_options *options,
+		__isl_give isl_printer *(*print_user)(
+			__isl_take isl_printer *p,
+			__isl_take isl_ast_print_options *options,
+			__isl_keep isl_ast_node *node, void *user),
+		void *user);
+	__isl_give isl_ast_print_options *
+	isl_ast_print_options_set_print_for(
+		__isl_take isl_ast_print_options *options,
+		__isl_give isl_printer *(*print_for)(
+			__isl_take isl_printer *p,
+			__isl_take isl_ast_print_options *options,
+			__isl_keep isl_ast_node *node, void *user),
+		void *user);
+
+The callback set by C<isl_ast_print_options_set_print_user>
+is called whenever a node of type C<isl_ast_node_user> needs to
+be printed.
+The callback set by C<isl_ast_print_options_set_print_for>
+is called whenever a node of type C<isl_ast_node_for> needs to
+be printed.
+Note that C<isl_ast_node_for_print> will I<not> call the
+callback set by C<isl_ast_print_options_set_print_for> on the node
+on which C<isl_ast_node_for_print> is called, but only on nested
+nodes of type C<isl_ast_node_for>.  It is therefore safe to
+call C<isl_ast_node_for_print> from within the callback set by
+C<isl_ast_print_options_set_print_for>.
+
+The following option determines the type to be used for iterators
+while printing the AST.
+
+	int isl_options_set_ast_iterator_type(
+		isl_ctx *ctx, const char *val);
+	const char *isl_options_get_ast_iterator_type(
+		isl_ctx *ctx);
+
+The AST printer only prints body nodes as blocks if these
+blocks cannot be safely omitted.
+For example, a C<for> node with one body node will not be
+surrounded with braces in C<ISL_FORMAT_C>.
+A block will always be printed by setting the following option.
+
+	int isl_options_set_ast_always_print_block(isl_ctx *ctx,
+		int val);
+	int isl_options_get_ast_always_print_block(isl_ctx *ctx);
+
+=head3 Options
+
+	#include <isl/ast_build.h>
+	int isl_options_set_ast_build_atomic_upper_bound(
+		isl_ctx *ctx, int val);
+	int isl_options_get_ast_build_atomic_upper_bound(
+		isl_ctx *ctx);
+	int isl_options_set_ast_build_prefer_pdiv(isl_ctx *ctx,
+		int val);
+	int isl_options_get_ast_build_prefer_pdiv(isl_ctx *ctx);
+	int isl_options_set_ast_build_exploit_nested_bounds(
+		isl_ctx *ctx, int val);
+	int isl_options_get_ast_build_exploit_nested_bounds(
+		isl_ctx *ctx);
+	int isl_options_set_ast_build_group_coscheduled(
+		isl_ctx *ctx, int val);
+	int isl_options_get_ast_build_group_coscheduled(
+		isl_ctx *ctx);
+	int isl_options_set_ast_build_scale_strides(
+		isl_ctx *ctx, int val);
+	int isl_options_get_ast_build_scale_strides(
+		isl_ctx *ctx);
+	int isl_options_set_ast_build_allow_else(isl_ctx *ctx,
+		int val);
+	int isl_options_get_ast_build_allow_else(isl_ctx *ctx);
+	int isl_options_set_ast_build_allow_or(isl_ctx *ctx,
+		int val);
+	int isl_options_get_ast_build_allow_or(isl_ctx *ctx);
+
+=over
+
+=item * ast_build_atomic_upper_bound
+
+Generate loop upper bounds that consist of the current loop iterator,
+an operator and an expression not involving the iterator.
+If this option is not set, then the current loop iterator may appear
+several times in the upper bound.
+For example, when this option is turned off, AST generation
+for the schedule
+
+	[n] -> { A[i] -> [i] : 0 <= i <= 100, n }
+
+produces
+
+	for (int c0 = 0; c0 <= 100 && n >= c0; c0 += 1)
+	  A(c0);
+
+When the option is turned on, the following AST is generated
+
+	for (int c0 = 0; c0 <= min(100, n); c0 += 1)
+	  A(c0);
+
+=item * ast_build_prefer_pdiv
+
+If this option is turned off, then the AST generation will
+produce ASTs that may only contain C<isl_ast_op_fdiv_q>
+operators, but no C<isl_ast_op_pdiv_q> or
+C<isl_ast_op_pdiv_r> operators.
+If this options is turned on, then C<isl> will try to convert
+some of the C<isl_ast_op_fdiv_q> operators to (expressions containing)
+C<isl_ast_op_pdiv_q> or C<isl_ast_op_pdiv_r> operators.
+
+=item * ast_build_exploit_nested_bounds
+
+Simplify conditions based on bounds of nested for loops.
+In particular, remove conditions that are implied by the fact
+that one or more nested loops have at least one iteration,
+meaning that the upper bound is at least as large as the lower bound.
+For example, when this option is turned off, AST generation
+for the schedule
+
+	[N,M] -> { A[i,j] -> [i,j] : 0 <= i <= N and
+					0 <= j <= M }
+
+produces
+
+	if (M >= 0)
+	  for (int c0 = 0; c0 <= N; c0 += 1)
+	    for (int c1 = 0; c1 <= M; c1 += 1)
+	      A(c0, c1);
+
+When the option is turned on, the following AST is generated
+
+	for (int c0 = 0; c0 <= N; c0 += 1)
+	  for (int c1 = 0; c1 <= M; c1 += 1)
+	    A(c0, c1);
+
+=item * ast_build_group_coscheduled
+
+If two domain elements are assigned the same schedule point, then
+they may be executed in any order and they may even appear in different
+loops.  If this options is set, then the AST generator will make
+sure that coscheduled domain elements do not appear in separate parts
+of the AST.  This is useful in case of nested AST generation
+if the outer AST generation is given only part of a schedule
+and the inner AST generation should handle the domains that are
+coscheduled by this initial part of the schedule together.
+For example if an AST is generated for a schedule
+
+	{ A[i] -> [0]; B[i] -> [0] }
+
+then the C<isl_ast_build_set_create_leaf> callback described
+below may get called twice, once for each domain.
+Setting this option ensures that the callback is only called once
+on both domains together.
+
+=item * ast_build_separation_bounds
+
+This option specifies which bounds to use during separation.
+If this option is set to C<ISL_AST_BUILD_SEPARATION_BOUNDS_IMPLICIT>
+then all (possibly implicit) bounds on the current dimension will
+be used during separation.
+If this option is set to C<ISL_AST_BUILD_SEPARATION_BOUNDS_EXPLICIT>
+then only those bounds that are explicitly available will
+be used during separation.
+
+=item * ast_build_scale_strides
+
+This option specifies whether the AST generator is allowed
+to scale down iterators of strided loops.
+
+=item * ast_build_allow_else
+
+This option specifies whether the AST generator is allowed
+to construct if statements with else branches.
+
+=item * ast_build_allow_or
+
+This option specifies whether the AST generator is allowed
+to construct if conditions with disjunctions.
+
+=back
+
+=head3 Fine-grained Control over AST Generation
+
+Besides specifying the constraints on the parameters,
+an C<isl_ast_build> object can be used to control
+various aspects of the AST generation process.
+The most prominent way of control is through ``options'',
+which can be set using the following function.
+
+	#include <isl/ast_build.h>
+	__isl_give isl_ast_build *
+	isl_ast_build_set_options(
+		__isl_take isl_ast_build *control,
+		__isl_take isl_union_map *options);
+
+The options are encoded in an C<isl_union_map>.
+The domain of this union relation refers to the schedule domain,
+i.e., the range of the schedule passed to C<isl_ast_build_ast_from_schedule>.
+In the case of nested AST generation (see L</"Nested AST Generation">),
+the domain of C<options> should refer to the extra piece of the schedule.
+That is, it should be equal to the range of the wrapped relation in the
+range of the schedule.
+The range of the options can consist of elements in one or more spaces,
+the names of which determine the effect of the option.
+The values of the range typically also refer to the schedule dimension
+to which the option applies.  In case of nested AST generation
+(see L</"Nested AST Generation">), these values refer to the position
+of the schedule dimension within the innermost AST generation.
+The constraints on the domain elements of
+the option should only refer to this dimension and earlier dimensions.
+We consider the following spaces.
+
+=over
+
+=item C<separation_class>
+
+This space is a wrapped relation between two one dimensional spaces.
+The input space represents the schedule dimension to which the option
+applies and the output space represents the separation class.
+While constructing a loop corresponding to the specified schedule
+dimension(s), the AST generator will try to generate separate loops
+for domain elements that are assigned different classes.
+If only some of the elements are assigned a class, then those elements
+that are not assigned any class will be treated as belonging to a class
+that is separate from the explicitly assigned classes.
+The typical use case for this option is to separate full tiles from
+partial tiles.
+The other options, described below, are applied after the separation
+into classes.
+
+As an example, consider the separation into full and partial tiles
+of a tiling of a triangular domain.
+Take, for example, the domain
+
+	{ A[i,j] : 0 <= i,j and i + j <= 100 }
+
+and a tiling into tiles of 10 by 10.  The input to the AST generator
+is then the schedule
+
+	{ A[i,j] -> [([i/10]),[j/10],i,j] : 0 <= i,j and
+						i + j <= 100 }
+
+Without any options, the following AST is generated
+
+	for (int c0 = 0; c0 <= 10; c0 += 1)
+	  for (int c1 = 0; c1 <= -c0 + 10; c1 += 1)
+	    for (int c2 = 10 * c0;
+		 c2 <= min(-10 * c1 + 100, 10 * c0 + 9);
+		 c2 += 1)
+	      for (int c3 = 10 * c1;
+		   c3 <= min(10 * c1 + 9, -c2 + 100);
+		   c3 += 1)
+		A(c2, c3);
+
+Separation into full and partial tiles can be obtained by assigning
+a class, say C<0>, to the full tiles.  The full tiles are represented by those
+values of the first and second schedule dimensions for which there are
+values of the third and fourth dimensions to cover an entire tile.
+That is, we need to specify the following option
+
+	{ [a,b,c,d] -> separation_class[[0]->[0]] :
+		exists b': 0 <= 10a,10b' and
+			   10a+9+10b'+9 <= 100;
+	  [a,b,c,d] -> separation_class[[1]->[0]] :
+		0 <= 10a,10b and 10a+9+10b+9 <= 100 }
+
+which simplifies to
+
+	{ [a, b, c, d] -> separation_class[[1] -> [0]] :
+		a >= 0 and b >= 0 and b <= 8 - a;
+	  [a, b, c, d] -> separation_class[[0] -> [0]] :
+		a >= 0 and a <= 8 }
+
+With this option, the generated AST is as follows
+
+	{
+	  for (int c0 = 0; c0 <= 8; c0 += 1) {
+	    for (int c1 = 0; c1 <= -c0 + 8; c1 += 1)
+	      for (int c2 = 10 * c0;
+		   c2 <= 10 * c0 + 9; c2 += 1)
+		for (int c3 = 10 * c1;
+		     c3 <= 10 * c1 + 9; c3 += 1)
+		  A(c2, c3);
+	    for (int c1 = -c0 + 9; c1 <= -c0 + 10; c1 += 1)
+	      for (int c2 = 10 * c0;
+		   c2 <= min(-10 * c1 + 100, 10 * c0 + 9);
+		   c2 += 1)
+		for (int c3 = 10 * c1;
+		     c3 <= min(-c2 + 100, 10 * c1 + 9);
+		     c3 += 1)
+		  A(c2, c3);
+	  }
+	  for (int c0 = 9; c0 <= 10; c0 += 1)
+	    for (int c1 = 0; c1 <= -c0 + 10; c1 += 1)
+	      for (int c2 = 10 * c0;
+		   c2 <= min(-10 * c1 + 100, 10 * c0 + 9);
+		   c2 += 1)
+		for (int c3 = 10 * c1;
+		     c3 <= min(10 * c1 + 9, -c2 + 100);
+		     c3 += 1)
+		  A(c2, c3);
+	}
+
+=item C<separate>
+
+This is a single-dimensional space representing the schedule dimension(s)
+to which ``separation'' should be applied.  Separation tries to split
+a loop into several pieces if this can avoid the generation of guards
+inside the loop.
+See also the C<atomic> option.
+
+=item C<atomic>
+
+This is a single-dimensional space representing the schedule dimension(s)
+for which the domains should be considered ``atomic''.  That is, the
+AST generator will make sure that any given domain space will only appear
+in a single loop at the specified level.
+
+Consider the following schedule
+
+	{ a[i] -> [i] : 0 <= i < 10;
+	  b[i] -> [i+1] : 0 <= i < 10 }
+
+If the following option is specified
+
+	{ [i] -> separate[x] }
+
+then the following AST will be generated
+
+	{
+	  a(0);
+	  for (int c0 = 1; c0 <= 9; c0 += 1) {
+	    a(c0);
+	    b(c0 - 1);
+	  }
+	  b(9);
+	}
+
+If, on the other hand, the following option is specified
+
+	{ [i] -> atomic[x] }
+
+then the following AST will be generated
+
+	for (int c0 = 0; c0 <= 10; c0 += 1) {
+	  if (c0 <= 9)
+	    a(c0);
+	  if (c0 >= 1)
+	    b(c0 - 1);
+	}
+
+If neither C<atomic> nor C<separate> is specified, then the AST generator
+may produce either of these two results or some intermediate form.
+
+=item C<unroll>
+
+This is a single-dimensional space representing the schedule dimension(s)
+that should be I<completely> unrolled.
+To obtain a partial unrolling, the user should apply an additional
+strip-mining to the schedule and fully unroll the inner loop.
+
+=back
+
+Additional control is available through the following functions.
+
+	#include <isl/ast_build.h>
+	__isl_give isl_ast_build *
+	isl_ast_build_set_iterators(
+		__isl_take isl_ast_build *control,
+		__isl_take isl_id_list *iterators);
+
+The function C<isl_ast_build_set_iterators> allows the user to
+specify a list of iterator C<isl_id>s to be used as iterators.
+If the input schedule is injective, then
+the number of elements in this list should be as large as the dimension
+of the schedule space, but no direct correspondence should be assumed
+between dimensions and elements.
+If the input schedule is not injective, then an additional number
+of C<isl_id>s equal to the largest dimension of the input domains
+may be required.
+If the number of provided C<isl_id>s is insufficient, then additional
+names are automatically generated.
+
+	#include <isl/ast_build.h>
+	__isl_give isl_ast_build *
+	isl_ast_build_set_create_leaf(
+		__isl_take isl_ast_build *control,
+		__isl_give isl_ast_node *(*fn)(
+			__isl_take isl_ast_build *build,
+			void *user), void *user);
+
+The
+C<isl_ast_build_set_create_leaf> function allows for the
+specification of a callback that should be called whenever the AST
+generator arrives at an element of the schedule domain.
+The callback should return an AST node that should be inserted
+at the corresponding position of the AST.  The default action (when
+the callback is not set) is to continue generating parts of the AST to scan
+all the domain elements associated to the schedule domain element
+and to insert user nodes, ``calling'' the domain element, for each of them.
+The C<build> argument contains the current state of the C<isl_ast_build>.
+To ease nested AST generation (see L</"Nested AST Generation">),
+all control information that is
+specific to the current AST generation such as the options and
+the callbacks has been removed from this C<isl_ast_build>.
+The callback would typically return the result of a nested
+AST generation or a
+user defined node created using the following function.
+
+	#include <isl/ast.h>
+	__isl_give isl_ast_node *isl_ast_node_alloc_user(
+		__isl_take isl_ast_expr *expr);
+
+	#include <isl/ast_build.h>
+	__isl_give isl_ast_build *
+	isl_ast_build_set_at_each_domain(
+		__isl_take isl_ast_build *build,
+		__isl_give isl_ast_node *(*fn)(
+			__isl_take isl_ast_node *node,
+			__isl_keep isl_ast_build *build,
+			void *user), void *user);
+	__isl_give isl_ast_build *
+	isl_ast_build_set_before_each_for(
+		__isl_take isl_ast_build *build,
+		__isl_give isl_id *(*fn)(
+			__isl_keep isl_ast_build *build,
+			void *user), void *user);
+	__isl_give isl_ast_build *
+	isl_ast_build_set_after_each_for(
+		__isl_take isl_ast_build *build,
+		__isl_give isl_ast_node *(*fn)(
+			__isl_take isl_ast_node *node,
+			__isl_keep isl_ast_build *build,
+			void *user), void *user);
+
+The callback set by C<isl_ast_build_set_at_each_domain> will
+be called for each domain AST node.
+The callbacks set by C<isl_ast_build_set_before_each_for>
+and C<isl_ast_build_set_after_each_for> will be called
+for each for AST node.  The first will be called in depth-first
+pre-order, while the second will be called in depth-first post-order.
+Since C<isl_ast_build_set_before_each_for> is called before the for
+node is actually constructed, it is only passed an C<isl_ast_build>.
+The returned C<isl_id> will be added as an annotation (using
+C<isl_ast_node_set_annotation>) to the constructed for node.
+In particular, if the user has also specified an C<after_each_for>
+callback, then the annotation can be retrieved from the node passed to
+that callback using C<isl_ast_node_get_annotation>.
+All callbacks should C<NULL> on failure.
+The given C<isl_ast_build> can be used to create new
+C<isl_ast_expr> objects using C<isl_ast_build_expr_from_pw_aff>
+or C<isl_ast_build_call_from_pw_multi_aff>.
+
+=head3 Nested AST Generation
+
+C<isl> allows the user to create an AST within the context
+of another AST.  These nested ASTs are created using the
+same C<isl_ast_build_ast_from_schedule> function that is used to create the
+outer AST.  The C<build> argument should be an C<isl_ast_build>
+passed to a callback set by
+C<isl_ast_build_set_create_leaf>.
+The space of the range of the C<schedule> argument should refer
+to this build.  In particular, the space should be a wrapped
+relation and the domain of this wrapped relation should be the
+same as that of the range of the schedule returned by
+C<isl_ast_build_get_schedule> below.
+In practice, the new schedule is typically
+created by calling C<isl_union_map_range_product> on the old schedule
+and some extra piece of the schedule.
+The space of the schedule domain is also available from
+the C<isl_ast_build>.
+
+	#include <isl/ast_build.h>
+	__isl_give isl_union_map *isl_ast_build_get_schedule(
+		__isl_keep isl_ast_build *build);
+	__isl_give isl_space *isl_ast_build_get_schedule_space(
+		__isl_keep isl_ast_build *build);
+	__isl_give isl_ast_build *isl_ast_build_restrict(
+		__isl_take isl_ast_build *build,
+		__isl_take isl_set *set);
+
+The C<isl_ast_build_get_schedule> function returns a (partial)
+schedule for the domains elements for which part of the AST still needs to
+be generated in the current build.
+In particular, the domain elements are mapped to those iterations of the loops
+enclosing the current point of the AST generation inside which
+the domain elements are executed.
+No direct correspondence between
+the input schedule and this schedule should be assumed.
+The space obtained from C<isl_ast_build_get_schedule_space> can be used
+to create a set for C<isl_ast_build_restrict> to intersect
+with the current build.  In particular, the set passed to
+C<isl_ast_build_restrict> can have additional parameters.
+The ids of the set dimensions in the space returned by
+C<isl_ast_build_get_schedule_space> correspond to the
+iterators of the already generated loops.
+The user should not rely on the ids of the output dimensions
+of the relations in the union relation returned by
+C<isl_ast_build_get_schedule> having any particular value.
+
+=head1 Applications
+
+Although C<isl> is mainly meant to be used as a library,
+it also contains some basic applications that use some
+of the functionality of C<isl>.
+The input may be specified in either the L<isl format>
+or the L<PolyLib format>.
+
+=head2 C<isl_polyhedron_sample>
+
+C<isl_polyhedron_sample> takes a polyhedron as input and prints
+an integer element of the polyhedron, if there is any.
+The first column in the output is the denominator and is always
+equal to 1.  If the polyhedron contains no integer points,
+then a vector of length zero is printed.
+
+=head2 C<isl_pip>
+
+C<isl_pip> takes the same input as the C<example> program
+from the C<piplib> distribution, i.e., a set of constraints
+on the parameters, a line containing only -1 and finally a set
+of constraints on a parametric polyhedron.
+The coefficients of the parameters appear in the last columns
+(but before the final constant column).
+The output is the lexicographic minimum of the parametric polyhedron.
+As C<isl> currently does not have its own output format, the output
+is just a dump of the internal state.
+
+=head2 C<isl_polyhedron_minimize>
+
+C<isl_polyhedron_minimize> computes the minimum of some linear
+or affine objective function over the integer points in a polyhedron.
+If an affine objective function
+is given, then the constant should appear in the last column.
+
+=head2 C<isl_polytope_scan>
+
+Given a polytope, C<isl_polytope_scan> prints
+all integer points in the polytope.
+
+=head2 C<isl_codegen>
+
+Given a schedule, a context set and an options relation,
+C<isl_codegen> prints out an AST that scans the domain elements
+of the schedule in the order of their image(s) taking into account
+the constraints in the context set.
diff --git a/polly/lib/External/isl/gitversion.h b/polly/lib/External/isl/gitversion.h
new file mode 100644
index 00000000000..3e4cb24fff3
--- /dev/null
+++ b/polly/lib/External/isl/gitversion.h
@@ -0,0 +1 @@
+#define GIT_HEAD_ID "UNKNOWN"
diff --git a/polly/lib/External/isl/imath/ChangeLog b/polly/lib/External/isl/imath/ChangeLog
new file mode 100644
index 00000000000..5222475d084
--- /dev/null
+++ b/polly/lib/External/isl/imath/ChangeLog
@@ -0,0 +1,377 @@
+1.0.1
+	First released version.
+
+1.0.2
+	Fixed a bug in mp_int_div() which would yield incorrect quotients
+	when the divisor was very close in value to a prefix of the
+	dividend.  This is now fixed, and there are regression tests in
+	the tests directory.
+
+	Added recursive multiplication and squaring (Karatsuba-Ofman) for
+	large input values.  Integrated these with the existing code for
+	exponentiation, too.  See the code for s_kmul() and s_ksqr() in
+	imath.c.  Tests added and verified against GNU bc.
+
+	Added documentation on mp_get_multiply_threshold() and the reason
+	why it exists.
+
+1.0.3
+	Fixed a couple of bugs in pi.c that were causing incorrect values
+	to be computed for > 30 digits or so.  Added a pi-computation test
+	to the default test suite (make test), checked against a static
+	file computed by bc (set scale=1024, compute 4 * atan(1)).  Added
+	command line option to specify output radix for pi.
+
+	Cleaned up a sign-related bug in mp_int_gcd(), which would cause
+	the sign of gcd(0, x) to be incorrect when x < 0.  Test cases
+	added for future regression.
+
+	Fixed a bug in s_reduce() which would give incorrect results for
+	powers of 2 in certain circumstances.  Added tests to drive this
+	case for future regression.
+
+	Added mp_int_exptmod_evalue() and mp_int_exptmod_bvalue() to make
+	it easier to work with small bases and small exponents.
+
+	Set default recursive multiplication threshold to 50 digits, since
+	this seems to work best for the platforms I've tested so far.
+
+	Added iprime.h and iprime.c to the distribution.
+
+1.0.4
+	Added `findsizes.pl' to the distribution.
+
+	Revised the type declarations in imath.h to use 32/64 bit
+	operations where the "long long" type is supported.
+
+	Fixed a sign-related bug in mp_int_invmod().
+
+	Fixed several small bugs related to shifting which affect the use
+	of 32-bit digits.  Many architectures cannot shift by 32 bits at a
+	time (e.g., MIPS), so I split each of these cases into two shifts
+	of half the size, which should scale properly for both the smaller
+	and larger cases.
+
+	Fixed several arithmetic issues with 32-bit digits that arose due
+	to missing type-casts on the right-hand sides of assignments.
+
+	Fixed s_print() and s_print_buf() to handle the sizes of digits
+	transparently.
+
+1.0.5
+	Updated the Makefile to include the _GNU_SOURCE macro.  For many
+	GCC systems, this is necessary to get the correct definition of
+	the ULLONG_MAX macro in <limits.h>.  Also, you may now build with
+	the make option DEBUG=Y to enable debugging, e.g.:
+
+	make DEBUG=Y imtest
+
+	By default, the Makefile builds with the optimizer enabled.
+
+	Cleaned up the definitions triggered by USE_LONG_LONG in imath.h,
+	and added an #error instruction in case the build is unable to
+	find a definition of ULLONG_MAX or ULONG_LONG_MAX in <limits.h>.
+	Also added the mp_int_to_unsigned(), mp_int_read_unsigned(), and
+	mp_int_unsigned_len() prototypes.
+
+	Fixed a bug in s_qmul() [imath.c:2493] that would grow the value
+	being multiplied even if there was room in the existing digits to
+	hold the result.  This was driving an (apparent) bug in the more
+	general mp_int_read_binary() routine.  Added the routines
+	mentioned in the previous paragraph, and factored some common
+	code out into a static s_tobin().
+
+	Added reset_registers() to imdrover.{h,c}.  Added new test
+	driver functions test_to_uns() and test_read_uns().  Renamed
+	test_read_bin to test_read_binary().
+
+	Silenced a sign-related warning in pi.c (related to printf).
+
+	Added many new test vectors to tests/conv.t, including the
+	original bug proof-of-concept from Tom Wu, and a series of new
+	tests for the unsigned conversion routines.
+
+	Updated `doc.txt' to reflect the changes described above.
+
+1.0.6
+	Updated copyright notices, added LICENSE file explaining the
+	license I am using.  This is basically the BSD license, so
+	you should have no trouble incorporating this code into other
+	open source projects.
+
+	No new functionality in this release.
+
+1.0.7
+	The mp_int_invmod(a, m, c) function would compute a negative value
+	for c when given a < 0.  I added some code to insure that the value
+	returned is always the least non-negative member of the congruence
+	class, if the inverse exists.  A test for this was added to invmod.t.
+
+1.0.8
+	Fixed a small buffer-overrun in s_qmul().  Because it only
+	allocates an extra digit if it absolutely has to, the test for
+	whether it needs to carry a shift out into the "spare" digit had
+	to be written carefully; I missed a subtlety, which is now
+	fixed. Along the way, I fixed a minor performance-related bug in
+	the same routine.
+
+	Added mp_int_error_string(), which converts mp_result values
+	into descriptive strings.  These are statically allocated, so
+	you don't have to free them.
+
+	This version also adds an "examples" subdirectory.  Currently,
+	there is only one program there, but I will add more examples as
+	time permits me.  You have to read the source to understand them
+	anyway, so I won't explain them here.
+
+1.1.0
+	Added imrat.h and imrat.c, containing routines for rational number
+	arithmetic at arbitrary precision.  Added support to the test driver,
+	in imath.c and included various tests in the tests/ subdirectory.
+
+	Fixed a sign-of-zero bug in mp_int_mul().  Tests added to mul.t to
+	regress this fix.
+
+1.1.2
+	Fixed a bug with leading zeroes after the decimal point in the
+	mp_rat_read_decimal() function (imrat.c).  Along the way, I also
+	found a sign-related bug, in which -0.5 would be treated as if it
+	were positive, because the sign of zero is implicitly positive,
+	and the denominator is treated as unsigned always.
+
+	Thanks to Eric Silva for pointing out the leading zeroes bug.
+	The solution isn't the most efficient possible.
+
+1.1.3
+	Rewrote mp_int_to_decimal() to support new rounding modes.  The
+	modes are documented in doc.txt.  Some of the code sucked anyway,
+	so I rewrote pretty much the entire function.
+
+	Added new rounding mode constants.
+
+1.1.4
+	Added mixed rational/integer operations:
+	  mp_rat_add_int, mp_rat_sub_int, mp_rat_mul_int, mp_rat_div_int
+	Added rational exponentiation (with integer exponents):
+	  mp_rat_expt
+
+	Tests for same were added to the tests/ subdirectory.
+
+1.1.5
+	Added mp_rat_read_cdecimal() and mp_rat_read_ustring()
+	Updated the input.c example.
+
+1.1.6
+	Fixed a bug in mp_int_read_cstring() which would read the string
+	"-0" with incorrect sign (MP_NEG instead of MP_ZPOS).  This would
+	violate an invariant that zero is always signed with positives.
+
+	Added some tests to tests/neg.t to catch this case.
+
+1.1.7
+	Fixed a bug in s_udiv(), internal to imath.c, which caused
+	division to fail in some corner cases masked by the use of long
+	long as a word type.  As a result, s_udiv() has now been wholly
+	rewritten.  I also fixed a few lingering buffer-length errors in
+	s_kmul(), and added a "const" qualifier to the input buffers for
+	the mp_int_read_string() and mp_int_read_cstring() functions,
+	and their analogs in imrat.c.
+
+1.1.8
+	Added mp_int_alloc() and mp_int_free().
+
+1.1.9
+	Added mp_rat_alloc() and mp_rat_free().  Fixed a couple of minor
+	bugs in the doc.txt file.  Added mp_int_sqrt() to imath.{h,c} and
+	doc.txt.
+
+1.2
+	Dropped bugfix component of revision number.  Fixed rsakey.c
+	example program to be complete and work faster.
+
+1.3
+	Replaced findsizes.pl with findsizes.py.  Fixed two bugs in the
+	rsakey tool that were leading to incorrect output.
+
+1.4
+	Fixed a bug in mp_int_alloc(), it was not returning NULL when out
+	of memory, but rather failing in assert() instead.  Also, updated
+	the documentation to have better language about the return values
+	in various error conditions.
+
+1.5
+	Changed the API for rational rounding.  Removed the two functions
+	mp_rat_set_rounding() and mp_rat_get_rounding(), along with the
+	round_output global variable.  Redefined the MP_ROUND_* constants
+	as an enumeration type (mp_round_mode).  Added a new parameter to
+	the mp_rat_to_decimal() function to accept a rounding mode.  Unit
+	tests and doc.txt updated suitably.
+
+	This release also incorporates a small patch submitted by Jonathan
+	Shapiro to support compilation in C++.
+
+1.6
+	Defined default_precision and multiply_threshold to be constant
+	and static.  If IMATH_TEST is defined at compile time, these are
+	made global, and can be modified by the caller (the imtimer tool
+	makes use of this ability, for example).
+
+	Implemented a single-digit optimization suggested by J. Shapiro.
+	Documentation updated.
+
+1.7
+	Fixed a subtle casting problem in the use of the ctype macros that
+	would permit negative signed character values to produce illogical
+	behaviour in some configurations (e.g., NetBSD).  Removed a dead
+	"return" statement.
+
+	Added the -pedantic switch for gcc, to get more aggressive
+	warnings; to permit the nonstandard "long long" type to still be
+	used, I also added -Wno-long-long when building with long long
+	enabled (the standard configuration).
+
+	Fixed a bug found by the Samba team running Valgrind on the
+	Heimdal project, and reported by Love Hörnquist Âstrand:  One of
+	the intermediate values used during modular exponentiation could
+	be overflowed during recursive multiplication.  Fixed by taking a
+	more conservative approach to buffer sizing.
+
+	Added a "contrib" subdirectory, whose first entry is a Makefile
+	to build IMath with the MSVC++ "nmake" program, contributed by
+	Matus Horvath.
+
+1.8
+	Fixed a bug in s_udiv() affecting the computation of quotient
+	digits.  Thanks to Love Âstrand for isolating this bug.  Also in
+	this release, defining USELLONG=Y or USELLONG=N on the command
+	line for make will switch support for the "long long" data type on
+	or off without having to edit the Makefile.  The default is still
+	to permit use of "long long", even though the type is not standard
+	ANSI C90.
+
+1.9
+	Increased the number of small primes used for primality testing to
+	100 from 32.  Removed an unwanted #define from imath.c, left over
+	from testing; added "static" to the declaration of the s_embar()
+	internal function since it is not used outside imath.c.  Reduced
+	the quantity of feedback generated by rsakey.c during the prime
+	finding stage of key generation.
+
+1.10
+	All primes less than 1000 are now used in iprime.c for preliminary
+	testing of prime candidates.  Removed declaration of s_pad() from
+	rsakey.c example.  Added imcalc.c example.
+
+	Beginning with this release, defining the DEBUG preprocessor macro
+	when compiling imath.c causes all the normally-static helper
+	functions to be exported.  This makes it easier to troubleshoot
+	bugs in the back end functions without manually editing the source
+	till you have found where the bug actually is.
+
+	Fixed a memory leak in the test driver (imtest.c) where the input
+	buffers allocated for test specs were not released before being
+	released.  No impact on the core routines, but nevertheless not a
+	good thing.
+
+	Fixed several uninitialized memory reads and one subtle read past
+	the end of a buffer in s_kmul(), found during a run of Purify.
+	Thanks to Love Hörnquist Âstrand for finding this one, and
+	providing a good test case so I could isolate the problem.  Also
+	fixed a buglet in s_kmul(), in which free() was being called
+	instead of s_free(), which would break if you provided a custom
+	version of s_alloc() and s_free() for your application.
+
+1.11
+	Those functions which take int parameters to supply one or more of
+	the arithmetic values of the function have been converted to use a
+	typedef "mp_small".  This is defined in imath.h, along with some
+	supporting macros.
+
+	Added mp_int_to_uint() and mp_int_lcm() in imath.{h,c}, based on a
+	patch contributed by Hal Finkel.  Added LCM tests as as well as
+	some more GCD tests in tests/lcm.t and tests/gcd.t
+
+	Also at Hal Finkel's request, added mp_int_root() to compute the
+	integer nth root, i.e., \lfloor a^{1/b}\rfloor; replaced the old
+	mp_int_sqrt() function with a call to mp_int_root() via a macro.
+	The new implementation is probably slightly less efficient for
+	square roots, but more general.  Added tests/root.t and moved the
+	sqrt tests there, also.
+
+1.12
+	Added a new global constant MP_MINERR which is the value of the
+	smallest error code defined by IMath itself.  This can be used by
+	clients who wish to define and use additional error codes, so that
+	those codes will not conflict with the existing set.
+
+	Extended the imcalc example to include memory.
+
+	Fixed a bug in mp_int_add() in which -1 + 1 = -0 (the sign of zero
+	was recorded incorrectly).  Added tests to the regression suite
+	for this fix.
+
+1.13
+	Cosmetic change -- updated all the files with my new web address.
+
+	Fixed a buglet caught by Love Hörnquist Âstrand using the LLVM
+	static checker tools, in which a mp_int_copy() failure would be
+	silently ignored and cause an extra copy to be generated.
+
+	Fixed a bug in the testing suite while building on MinGW.  The pi
+	generation tests compare to static files and these tests fail if
+	CR/LF is output instead of just LF.  The test script now strips
+	all CR and LF from the output and compares to files lacking them.
+	Reported by Chris Cole <cjcole@gmail.com>.
+
+1.14
+	Instead of using the preprocessor to delete "static", the static
+	definitions in imath.c now use an explicit STATIC macro, that is
+	made null when DEBUG is defined.  This avoids a subtle problem
+	with static variables defined inside functions (although no bugs
+	actually arose from it).
+
+	Fixed a bug in s_udiv() while building on MinGW.  When building
+	with short type digits, the routine was incorrectly discarding
+	overflow when computing the next quotient digit.
+	Reported by Paul DeMarco <pdemarco@ppg.com>.
+
+1.15
+	Fixed a bug in the definition of MP_DIGIT_MAX that caused errors
+	when IMath is built under 64-bit Linux.  Reported by
+	Klaus Stengel <klaus.stengel@informatik.stud.uni-erlangen.de>.
+
+	Unpacked the macro definitions in imath.c a bit, to make them more
+	readable.
+
+	Added mp_int_expt_full() by request of Andrea Barberio
+	<insomniac@slackware.it>.
+
+1.16
+	Fixed a bug in mp_int_to_uint() which was causing incorrect MP_RANGE
+	errors during small integer conversion.
+	Reported by Andrea Barberio <insomniac@slackware.it>
+
+	Added mp_int_compare_uvalue().
+	Added some new testing hooks in imtest.c, new unit tests.
+
+	Made some code style changes that do not affect functionality.
+
+1.17
+	Fixed a bug in mp_int_swap() where mpz_t structures using their single
+	field as storage would not get swapped correctly.
+	Reported by Andres Navarro <canavarro82@gmail.com>
+
+	Added regression test for this and some hooks for future
+	regressions in the tests/test.sh script.
+
+1.18
+	Made mp_int_rat() use mp_int_init() to initialize numerator and
+	denominator instead of mp_int_init_size().
+	Some minor code cleanup inside the testing code (imdrover.c).
+
+	Fixed an off-by-one bug in s_udiv() which could cause the quotient
+	guessing loop to spin.  Reported by Andres Navarro.  Added
+	triggering example to div.t as a regression test.
+
+1.19
+	Fix signedness error in compile.  Reported by Paweł Sikora.
diff --git a/polly/lib/External/isl/imath/LICENSE b/polly/lib/External/isl/imath/LICENSE
new file mode 100644
index 00000000000..087fb612583
--- /dev/null
+++ b/polly/lib/External/isl/imath/LICENSE
@@ -0,0 +1,20 @@
+IMath is Copyright © 2002-2009 Michael J. Fromberger
+You may use it subject to the following Licensing Terms:
+
+Permission is hereby granted, free of charge, to any person obtaining a copy of
+this software and associated documentation files (the "Software"), to deal in
+the Software without restriction, including without limitation the rights to
+use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies
+of the Software, and to permit persons to whom the Software is furnished to do
+so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.
diff --git a/polly/lib/External/isl/imath/README.md b/polly/lib/External/isl/imath/README.md
new file mode 100644
index 00000000000..637467177d0
--- /dev/null
+++ b/polly/lib/External/isl/imath/README.md
@@ -0,0 +1,106 @@
+IMath
+=====
+
+Arbitrary precision integer and rational arithmetic library.
+
+IMath is an open-source ANSI C arbitrary precision integer and rational
+arithmetic library.
+
+IMath is copyright &copy; 2002-2009 Michael J. Fromberger.
+
+> Permission is hereby granted, free of charge, to any person obtaining a copy
+> of this software and associated documentation files (the "Software"), to deal
+> in the Software without restriction, including without limitation the rights
+> to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+> copies of the Software, and to permit persons to whom the Software is
+> furnished to do so, subject to the following conditions:
+>
+> The above copyright notice and this permission notice shall be included in
+> all copies or substantial portions of the Software.
+>
+> THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+> IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+> FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+> AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+> LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+> OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+> SOFTWARE.
+
+
+About IMath
+-----------
+
+IMath is a library written in portable ANSI C that allows you to perform
+arithmetic on integers and rational numbers of arbitrary precision.  While many
+programming languages, including Java, Perl, and Python provide arbitrary
+precision numbers as a standard library or language feature, C does not.
+
+IMath was designed to be small, self-contained, easy to understand and use, and
+as portable as possible across various platforms.  The API is simple, and the
+code should be comparatively easy to modify or extend.  Simplicity and
+portability are useful goals for some applications&#8212;however, IMath does
+not attempt to break performance records.  If you need the fastest possible
+implementation, you might consider some other libraries, such as GNU MP (GMP),
+MIRACL, or the bignum library from OpenSSL.
+
+Programming with IMath
+----------------------
+
+Detailed descriptions of the IMath API can be found in `doc.txt`.  However, the
+following is a brief synopsis of how to get started with some simple tasks.
+
+To do basic integer arithmetic, you must declare variables of type `mpz_t` in
+your program, and call the functions defined in `imath.h` to operate on them.
+Here is a simple example that reads one base-10 integer from the command line,
+multiplies it by another (fixed) value, and prints the result to the standard
+output in base-10 notation:
+
+    #include <stdio.h>
+    #include <stdlib.h>
+    #include "imath.h"
+
+    int main(int argc, char *argv[])
+    {
+      mpz_t  a, b;
+      char  *buf;
+      int    len;
+
+      if(argc < 2) {
+        fprintf(stderr, "Usage: testprogram <integer>\n");
+        return 1;
+      }
+
+      /* Initialize a new zero-valued mpz_t structure */
+      mp_int_init(&a);
+
+      /* Initialize a new mpz_t with a small integer value */
+      mp_int_init_value(&b, 25101);
+
+      /* Read a string value in the specified radix */
+      mp_int_read_string(&a, 10, argv[1]);
+
+      /* Multiply the two together... */
+      mp_int_mul(&a, &b, &a);
+
+      /* Print out the result */
+      len = mp_int_string_len(&a, 10);
+      buf = calloc(len, sizeof(*buf));
+      mp_int_to_string(&a, 10, buf, len);
+      printf("result = %s\n", buf);
+      free(buf);
+
+      /* Release memory occupied by mpz_t structures when finished */
+      mp_int_clear(&b);
+      mp_int_clear(&a);
+
+      return 0;
+    }
+
+This simple example program does not do any error checking, but all the IMath
+API functions return an `mp_result` value which can be used to detect various
+problems like range errors, running out of memory, and undefined results.
+
+The IMath API also supports operations on arbitrary precision rational numbers.
+The functions for creating and manipulating rational values (type `mpq_t`) are
+defined in `imrat.h`, so that you need only include them in your project if you
+wish to.
diff --git a/polly/lib/External/isl/imath/bintest.c b/polly/lib/External/isl/imath/bintest.c
new file mode 100644
index 00000000000..bec34945da2
--- /dev/null
+++ b/polly/lib/External/isl/imath/bintest.c
@@ -0,0 +1,106 @@
+/*
+  Name:     bintest.c
+  Purpose:  Test driver for binary input/output formats from IMath.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Copyright (C) 2002-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+
+#include "imath.h"
+
+int main(int argc, char *argv[])
+{
+  unsigned char buf[512];
+  mpz_t  v, w;
+  mp_result res;
+  int len;
+
+  if(argc < 2 || argv[1][0] == '\0') {
+    fprintf(stderr, "Usage: bintest <value>\n");
+    return 1;
+  }
+
+  mp_int_init(&v);
+  mp_int_init(&w);
+  res = mp_int_read_string(&v, 10, argv[1]);
+  printf("Result code from mp_int_read_string() = %d (%s)\n",
+	 res, mp_error_string(res));
+
+  len = mp_int_binary_len(&v);
+  printf("%d bytes needed to write this value in 2's complement.\n", len);
+
+  res = mp_int_to_binary(&v, buf, sizeof(buf));
+  printf("Result code from mp_int_to_binary() = %d (%s)\n",
+	 res, mp_error_string(res));
+  if(res == MP_OK) {
+    int ix;
+
+    for(ix = 0; ix < (len - 1); ++ix)
+      printf("%d.", buf[ix]);
+
+    printf("%d\n", buf[ix]);
+  } else {
+    return 1;
+  }
+
+  /* Try converting back... */
+  res = mp_int_read_binary(&w, buf, len);
+  printf("Result code from mp_int_read_binary() = %d (%s)\n",
+	 res, mp_error_string(res));
+  if(res == MP_OK) {
+    mp_int_to_string(&w, 10, (char *) buf, sizeof(buf));
+
+    printf("[%s]\n\n", buf);
+  }
+
+  len = mp_int_unsigned_len(&v);
+  printf("%d bytes needed to write this value as unsigned.\n", len);
+
+  res = mp_int_to_unsigned(&v, buf, sizeof(buf));
+  printf("Result code from mp_int_to_unsigned() = %d\n", res);
+  if(res == MP_OK) {
+    int ix;
+
+    for(ix = 0; ix < (len - 1); ++ix)
+      printf("%d.", buf[ix]);
+
+    printf("%d\n", buf[ix]);
+  } else {
+    return 1;
+  }
+
+  res = mp_int_read_unsigned(&w, buf, len);
+  printf("Result code from mp_int_read_unsigned() = %d (%s)\n",
+	 res, mp_error_string(res));
+  if(res == MP_OK) {
+    mp_int_to_string(&w, 10, (char *) buf, sizeof(buf));
+
+    printf("[%s]\n\n", buf);
+  }
+
+  mp_int_clear(&v);
+  mp_int_clear(&w);
+  return 0;
+}
diff --git a/polly/lib/External/isl/imath/bug-swap.c b/polly/lib/External/isl/imath/bug-swap.c
new file mode 100644
index 00000000000..c9adca9f928
--- /dev/null
+++ b/polly/lib/External/isl/imath/bug-swap.c
@@ -0,0 +1,25 @@
+/* Regression test for mp_int_swap() bug on self-stored values. */
+#include <stdio.h>
+#include "imath.h"
+
+int main(int argc, char* argv[]) {
+  mpz_t a, b;
+  int result;
+
+  mp_int_init_value(&a, 1);
+  mp_int_init_value(&b, 16);
+
+  mp_int_swap(&a, &b);
+  result = (a.digits == &(a.single) &&
+	    b.digits == &(b.single) &&
+	    a.digits[0] == 16 &&
+	    b.digits[0] == 1);
+
+  printf("REGRESSION: mp_int_swap() on self-stored values: %s\n",
+	 result ? "OK" : "FAILED");
+
+  mp_int_clear(&b);
+  mp_int_clear(&a);
+  return !result;
+}
+   
diff --git a/polly/lib/External/isl/imath/contrib/Makefile.msvc b/polly/lib/External/isl/imath/contrib/Makefile.msvc
new file mode 100644
index 00000000000..0eb24fb352a
--- /dev/null
+++ b/polly/lib/External/isl/imath/contrib/Makefile.msvc
@@ -0,0 +1,84 @@
+##
+## Name:     Makefile.msvc
+## Purpose:  Makefile for IMath library and associated tools
+##           for Microsoft Visual Studio 2005
+## Author:   Matus Horvath <matus.horvath@nextra.sk>
+##
+## Copyright (C) 2006 Matus Horvath.  Permission has been granted to use,
+## modify, and redistribute this file according to the terms of the IMath
+## license.
+##
+## Usage: nmake /f Makefile.msvc
+##
+
+# --- begin configuration section ---
+
+## Settings for Microsoft Windows systems using nmake.
+## To build with debugging, add DEBUG=Y on the "nmake" command line.
+CC=cl.exe
+LD=link.exe
+CFLAGS=$(CFLAGS) -nologo -I. -D_CRT_SECURE_NO_DEPRECATE $(DCFLAGS)
+LDFLAGS=$(LDFLAGS) -nologo $(DLDFLAGS)
+LIBS=$(DLIBS)
+
+!if "$(DEBUG)" == "Y"
+DCFLAGS=-ZI -Od -DDEBUG=1 -DTRACEABLE_FREE=1
+DLDFLAGS=-DEBUG
+#DLIBS=-lefence
+!else
+DCFLAGS=-O2 -Ob2
+DLDFLAGS=
+#DLIBS=
+!endif
+
+## Visual Studio C/C++ 2005 compiler supports "long long" 64-bit type.  
+CFLAGS=$(CFLAGS) -DUSE_LONG_LONG
+
+# --- end of configuration section ---
+TARGETS=imtest.exe pi.exe bintest.exe findprime.exe
+HDRS=imath.h imrat.h iprime.h imdrover.h rsamath.h
+SRCS=$(HDRS:.h=.c) $(TARGETS:.exe=.c)
+OBJS=$(SRCS:.c=.obj)
+EXAMPLES=input.exe basecvt.exe rounding.exe
+
+.c.obj:
+	$(CC) $(CFLAGS) -c $<
+
+all: objs examples test
+
+objs: $(OBJS)
+
+# Because Visual Studio does not permit Unix shell syntax, you will
+# have to run the tests manually once the "test" target is built.
+test: imtest.exe pi.exe
+#	@ echo ""
+#	@ echo "Running tests, you should not see any 'FAILED' lines here."
+#	@ echo "If you do, please see doc.txt for how to report a bug."
+#	@ echo ""
+#	(cd tests && ./test.sh)
+
+$(EXAMPLES): imath.obj imrat.obj iprime.obj examples/$*.obj
+	@move $*.obj examples/$*.obj
+	$(LD) $(LDFLAGS) -out:$@ $** $(LIBS)
+
+examples: $(EXAMPLES)
+
+imtest.exe: imtest.obj imath.obj imrat.obj imdrover.obj
+	$(LD) $(LDFLAGS) -out:$@ $** $(LIBS)
+
+pi.exe: pi.obj imath.obj
+	$(LD) $(LDFLAGS) -out:$@ $** $(LIBS)
+
+findprime.exe: findprime.obj imath.obj iprime.obj
+	$(LD) $(LDFLAGS) -out:$@ $** $(LIBS)
+
+rtest.exe: rtest.obj imath.obj rsamath.obj
+	$(LD) $(LDFLAGS) -out:$@ $** $(LIBS)
+
+bintest.exe: imath.obj bintest.obj
+	$(LD) $(LDFLAGS) -out:$@ $** $(LIBS)
+
+clean:
+	del /q /f *.exe *.obj examples\*.obj
+
+# End of Makefile.msvc
diff --git a/polly/lib/External/isl/imath/contrib/README b/polly/lib/External/isl/imath/contrib/README
new file mode 100644
index 00000000000..b699944123b
--- /dev/null
+++ b/polly/lib/External/isl/imath/contrib/README
@@ -0,0 +1,3 @@
+The files in this directory have been contributed to the IMath project
+by other authors.
+
diff --git a/polly/lib/External/isl/imath/divide.txt b/polly/lib/External/isl/imath/divide.txt
new file mode 100644
index 00000000000..41092b3a606
--- /dev/null
+++ b/polly/lib/External/isl/imath/divide.txt
@@ -0,0 +1,43 @@
+Algorithm for Arbitrary Precision Unsigned Division
+
+Inputs:
+  A = [a1 ... an]       -- array representation
+  B = [b1 ... bm]
+  ai, bi \in {0,..radix}
+
+Preconditions:
+  n >= m
+
+Outputs:
+  Q, R satisfying A = QB + R with 0 <= R < B.
+
+let Q = [], j = n, i = 1, skip = 0 in 
+  while j > 0 do
+    if B > A[j,..j + i + 1] then
+      decrement j
+      increment i
+      increment skip
+      if skip > 1 then
+        Q.append(0)
+    else
+      let pfx = A[j] in
+        if pfx < bm and j > 1 then
+          pfx := pfx * radix + A[j - 1]
+        
+        q := pfx div bm
+        if q >= radix then
+          if q > radix then
+            q = radix
+          else 
+            q := 1
+
+        let t = B * q in
+          while t > A[j,..j + i + 1] do
+            t := t - B
+            decrement q
+        
+        Q.append(q)
+        skip := 0
+  end    
+  
+  result Q, A
diff --git a/polly/lib/External/isl/imath/doc.txt b/polly/lib/External/isl/imath/doc.txt
new file mode 100644
index 00000000000..12f7ee62e55
--- /dev/null
+++ b/polly/lib/External/isl/imath/doc.txt
@@ -0,0 +1,820 @@
+Name:     doc.txt
+Purpose:  User documentation for IMath library
+Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+--- Installation:
+
+1. Edit Makefile to select compiler and options.  The default uses gcc.  If you
+   are building under MacOS 10.1.x, you will need to change CC to 'cc' instead
+   of 'gcc', but otherwise you can use the default GCC settings.
+ 
+   By default, the Makefile assumes you can use the "long long" integer type,
+   even though it is not standard in ANSI C90.  You can control this by setting
+   the USELLONG variable in your make command line.  Setting USELLONG=Y enables
+   "long long"; USELLONG=N disables it.  The default is Y.
+
+   If the built-in types do not work for your particular platform, you can try
+   to find suitable integer type sizes by running the Python program
+   `findsizes.py'.  This requires a Python interpreter, version 2.4.x or later.
+
+2. Type 'make' or 'make test' to build the test driver and run the unit tests.
+   None of these should fail.  If they do, see below for how you can report
+   bugs.
+
+   To build with debugging enabled (and optimization disabled), run make
+   DEBUG=Y.  This sets the preprocessor macro DEBUG to 1, and several other
+   things (see Makefile for details).
+
+To use the library in your code, include "imath.h" wherever you intend to use
+the library's routines.  The integer library is just a single source file, so
+you can compile it into your project in whatever way makes sense.  If you wish
+to use rational arithmetic, you will also need to include "imrat.h".
+
+--- Background:
+
+The basic types defined by the imath library are mpz_t, an arbitrary precision
+signed integer, and mpq_t, an arbitrary precision signed rational number.  The
+type mp_int is a pointer to an mpz_t, and mp_rat is a pointer to an mpq_t.
+
+Most of the functions in the imath library return a value of type mp_result.
+This is a signed integer type which can be used to convey status information
+and also return small values.  Any negative value is considered to be a status
+message.  The following constants are defined for processing these:
+
+ MP_OK        operation successful, all is well (= 0)
+ MP_FALSE     boolean false (= MP_OK)
+ MP_TRUE      boolean true
+ MP_MEMORY    out of memory
+ MP_RANGE     parameter out of range
+ MP_UNDEF     result is undefined (e.g., division by zero)
+ MP_TRUNC     output value was truncated
+ MP_BADARG    an invalid parameter was passed
+
+If you obtain a zero or negative value of an mp_result, you can use the
+mp_int_error_string() routine to obtain a pointer to a brief human readable
+string describing the error.  These strings are statically allocated, so they
+need not be freed by the caller; the same strings are re-used from call to
+call.
+
+Unless otherwise noted, it is legal to use the same parameter for both inputs
+and output with most of the functions in this library.  For example, you can
+add a number to itself and replace the original by writing:
+
+  mp_int_add(a, a, a);  /* a = a + a */
+
+Any cases in which this is not legal will be noted in the function summaries
+below (if you discover that this is not so, please report it as a bug; I will
+fix either the function or the documentation :)
+
+--- The IMath API
+
+Each of the API functions is documented here.  The general format of the
+entries is:
+
+return_type function_name(parameters ...)
+            English description.
+
+Unless otherwise noted, any API function that returns mp_result may be expected
+to return MP_OK, MP_BADARG, or MP_MEMORY.  Other return values should be
+documented in the description.  Please let me know if you discover this is not
+the case.
+
+The following macros are defined in "imath.h", to define the sizes of the
+various data types used in the library:
+
+ MP_DIGIT_BIT    : the number of bits in a single mpz_t digit.
+ MP_WORD_BIT     : the number of bits in a mpz_t word.
+ MP_SMALL_MIN    : the minimum value representable by an mp_small.
+ MP_SMALL_MAX    : the maximum value representable by an mp_small.
+ MP_USMALL_MIN   : the minimum value representable by an mp_usmall.
+ MP_USMALL_MAX   : the maximum value representable by an mp_usmall.
+ MP_MIN_RADIX    : the minimum radix accepted for base conversion.
+ MP_MAX_RADIX    : the maximum radix accepted for base conversion.
+
+An mp_int must be initialized before use.  This may be accomplished using the
+following functions:
+
+mp_result mp_int_init(mp_int z);
+          Initializes z with a default precision, sets the value to zero.  This
+          function cannot fail unless z is NULL.
+
+mp_int    mp_int_alloc();
+          Dynamically allocates an mpz_t, initializes it to the value zero, and
+          returns a pointer to it.  Returns NULL in case of error (the only
+          error possible is MP_MEMORY in this case).
+
+mp_result mp_int_init_size(mp_int z, mp_size prec);
+          Initializes z with at least prec digits of storage, sets the value to
+          zero.  If prec is zero, the default size is used, defined in imath.h
+          as MP_DEFAULT_PREC.  The size is rounded up to the nearest word
+          boundary.
+
+mp_result mp_int_init_copy(mp_int z, mp_int old);
+          Initializes z to be a copy of an already-initialized mp_int in old.
+          They do not share storage.
+
+mp_result mp_int_init_value(mp_int z, mp_small value);
+          Initializes z with default precision and sets its value to the value
+          of the supplied integer.
+
+To copy one mp_int to another, use:
+
+mp_result mp_int_copy(mp_int a, mp_int c);
+          Copies the value of a into c.  Does not allocate new memory unless a
+          has more significant digits than c has room for.
+
+When you are finished with an mp_int, you must free the memory it uses:
+
+void      mp_int_clear(mp_int z);
+          Releases the storage used by z.
+
+void      mp_int_free(mp_int z);
+          Releases the storage used by z, and frees the mpz_t structure z
+          points to.  This should only be used for values allocated by
+          mp_int_alloc().
+
+To set an mp_int, which has already been initialized, to a small integer value,
+use the following:
+
+mp_result mp_int_set_value(mp_int z, mp_small value);
+          Sets the value of z to the value of the supplied integer.
+
+By default, an mp_int is initialized with a certain minimum amount of storage
+for digits.  This storage is expanded automatically as needed.
+
+- The arithmetic functions are:
+
+int       mp_int_is_odd(mp_int z);
+          Returns true if z is an odd integer (z = 1 (mod 2))
+          [currently implemented as a macro]
+
+int       mp_int_is_even(mp_int z);
+          Returns true if z is an even integer (z = 0 (mod 2))
+          [currently implemented as a macro]
+
+void      mp_int_zero(mp_int z);
+          Sets z to zero.
+
+mp_result mp_int_abs(mp_int a, mp_int c);
+          Sets c to the absolute value of a. 
+          If a < 0, c = -a, else c = a.
+
+mp_result mp_int_neg(mp_int a, mp_int c);
+          Sets c to be the additive inverse of a, c = -a.
+
+mp_result mp_int_add(mp_int a, mp_int b, mp_int c);
+          Computes c = a + b.
+
+mp_result mp_int_add_value(mp_int a, mp_small value, mp_int c);
+          Computes c = a + value, where value is a small integer.
+
+mp_result mp_int_sub(mp_int a, mp_int b, mp_int c);
+          Computes c = a - b.
+
+mp_result mp_int_sub_value(mp_int a, mp_small value, mp_int c);
+          Computes c = a - value, where value is a small integer.
+
+mp_result mp_int_mul(mp_int a, mp_int b, mp_int c);
+          Computes c = a * b
+
+mp_result mp_int_mul_value(mp_int a, mp_small value, mp_int c);
+          Computes c = a * value, where value is a small integer.
+
+mp_result mp_int_mul_pow2(mp_int a, mp_small p2, mp_int c);
+          Computes c = a * 2^p2, where p2 >= 0.
+
+mp_result mp_int_sqr(mp_int a, mp_int c);
+          Computes c = a * a.  Faster than using mp_int_mul(a, a, c).
+
+mp_result mp_int_root(mp_int a, mp_small b, mp_int c);
+          Computes c = floor(a^{1/b}).  Returns MP_UNDEF if the root is
+          undefined, i.e., if a < 0 and b is even.  Uses Newton's method to
+          obtain the root.
+
+mp_result mp_int_sqrt(mp_int a, mp_int c);
+          Computes c = floor(sqrt(a)) if a >= 0.  Returns MP_UNDEF if
+          a < 0.
+          [currently implemented as a macro]
+
+mp_result mp_int_div(mp_int a, mp_int b, mp_int q, mp_int r);
+          Computes q, r such that a = bq + r and 0 <= r < b.
+
+          Pass NULL for q or r if you don't need its value.  Detects and
+          handles division by powers of two in an efficient manner.  Returns
+          MP_UNDEF if b = 0.  If both q and r point to the same non-NULL
+          location, their values on output will be arbitrary (usually
+          incorrect).
+
+mp_result mp_int_div_value(mp_int a, mp_small v, mp_int q, mp_small *r);
+          Computes q, r such that a = qv + r and 0 <= r < v, where v is a small
+          integer.  Pass NULL for q or r if you don't need its value.
+
+mp_result mp_int_div_pow2(mp_int a, mp_small p2, mp_int q, mp_int r);
+          Computes q, r such that a = q * 2^p2 + r.  This is a special case for
+          division by powers of two that is much more efficient than using the
+          regular division algorithm.  Note that mp_int_div() will
+          automatically handle this case if b = 2^k for some k >= 0;
+          mp_int_div_pow2() is for when you have only the exponent, not the
+          expanded value.
+
+mp_result mp_int_mod(mp_int a, mp_int m, mp_int c);
+          Computes the least non-negative residue of a (mod m), and assigns the
+          result to c.
+
+mp_result mp_int_mod_value(mp_int a, mp_int value, mp_small *r);
+          Computes the least non-negative residue of a (mod value), where value
+          is a small integer, and assigns the result to r.
+
+mp_result mp_int_expt(mp_int a, mp_small b, mp_int c);
+          Raises a to the b power, and assigns the result to c.  It is an error
+          if b < 0.
+
+mp_result mp_int_expt_value(mp_small a, mp_small b, mp_int c);
+          Raises a to the b power, and assigns the result to c.  It is an error
+          if b < 0.
+
+mp_result mp_int_expt_full(mp_int a, mp_int b, mp_int c);
+          Raises a to the b power, and assigns the result to c.  It is an error
+          if b < 0.
+
+- Comparison functions:
+
+Unless otherwise specified, comparison between values x and y returns a value <
+0 if x is less than y, = 0 if x is equal to y, and > 0 if x is greater than y.
+
+int       mp_int_compare(mp_int a, mp_int b);
+          Signed comparison of a and b.
+
+int       mp_int_compare_unsigned(mp_int a, mp_int b);
+          Unsigned (magnitude) comparison of a and b.  The signs of a and b are
+          not modified.
+
+int       mp_int_compare_zero(mp_int z);
+          Compare z to zero.
+
+int       mp_int_compare_value(mp_int z, mp_small value);
+int       mp_int_compare_uvalue(mp_int z, mp_usmall value);
+          Compare z to small signed (value) or unsigned (uvalue) integer value.
+
+int       mp_int_divisible_value(mp_int a, mp_small v);
+          Returns true (nonzero) if a is divisible by small integer v,
+          otherwise false (zero)
+
+int       mp_int_is_pow2(mp_int z);
+          Returns k >= 0 such that z = 2^k, if such a k exists; otherwise a
+          value < 0 is returned.
+
+- Other useful functions:
+
+mp_result mp_int_exptmod(mp_int a, mp_int b, mp_int m, mp_int c);
+          Efficiently computes c = a^b (mod m).
+          Returns MP_UNDEF if m = 0; returns MP_RANGE if b < 0.
+
+mp_result mp_int_exptmod_evalue(mp_int a, mp_small v, mp_int m, mp_int c);
+          Efficiently computes c = a^v (mod m).
+mp_result mp_int_exptmod_bvalue(mp_small v, mp_int b, mp_int m, mp_int c);
+          Efficiently computes c = v^b (mod m).
+
+          Note: These routines use Barrett's algorithm for modular reduction.
+          It is widely held (probably correctly) that using Peter Montgomery's
+          multiplication algorithm would make this operation faster; but that
+          algorithm has the restriction that a and m must be coprime, so I have
+          not implemented it here.
+
+mp_result mp_int_exptmod_known(mp_int a, mp_int b, mp_int m, mp_int mu, 
+                               mp_int c);
+          Efficiently computes c = a^b (mod m), given a precomputed reduction
+          constant mu, as defined for Barrett's modular reduction algorithm.
+          Returns MP_UNDEF if m = 0; returns MP_RANGE if b < 0.
+
+mp_result mp_int_redux_const(mp_int m, mp_int c);
+          Computes reduction constant mu for Barrett reduction by modulus m,
+          stores the result in c.
+
+mp_result mp_int_invmod(mp_int a, mp_int m, mp_int c);
+          Computes the modular inverse of a (mod m), if it exists, and assigns
+          the result to c.  Returns the least non-negative representative of
+          the congruence class (mod m) containing this inverse.  Returns
+          MP_UNDEF if the inverse does not exist; returns MP_RANGE if a = 0 or
+          m <= 0.
+
+mp_result mp_int_gcd(mp_int a, mp_int b, mp_int c);
+          Compute the greatest common divisor if a and b, and assign the result
+          to c.  Returns MP_UNDEF if the GCD is not defined (e.g., if a = 0 and
+          b = 0).
+
+mp_result mp_int_egcd(mp_int a, mp_int b, mp_int c, mp_int x, mp_int y);
+          Compute the greatest common divisor of a and b, and assign the result
+          to c.  Also computes x and y satisfying Bezout's identity, namely (a,
+          b) = ax + by.  Returns MP_UNDEF if the GCD is not defined (e.g., if a
+          = b = 0).
+
+mp_result mp_int_lcm(mp_int a, mp_int b, mp_int c)
+          Compute the least common multiple of a and b, and assign the result
+          to c.  Returns MP_UNDEF if the LCM is not defined (e.g., if a = 0 and
+          b = 0).
+
+- Conversion of values:
+
+mp_result mp_int_to_int(mp_int z, mp_small *out);
+          Convert z to an int type, if it is representable as such.  Returns
+          MP_RANGE if z cannot be represented as an value of type mp_small.  If
+          out is NULL no value is stored, but the return value will still be
+          correct.
+
+mp_result mp_int_to_uint(mp_int z, mp_usmall *out);
+          Convert z to an unsigned int type, if it is representable as such.
+          Returns MP_RANGE if z cannot be represented as a value of type
+          mp_usmall.  If out is NULL no value is stored, but the return value
+          will still be correct.
+
+mp_result mp_int_to_string(mp_int z, mp_size radix, char *str, int limit);
+          Convert z to a zero-terminated string of characters in the given
+          radix, writing at most 'limit' characters including the terminating
+          NUL value.  A leading '-' is used to indicate a negative value.
+
+          Returns MP_RANGE if radix < MP_MIN_RADIX or radix > MP_MAX_RADIX.
+          Returns MP_TRUNC if limit is too small to write out all of z.
+
+mp_result mp_int_string_len(mp_int z, mp_size radix);
+          Return the minimum number of characters required to represent z as a
+          zero-terminated string of characters in the given radix. May
+          over-estimate (but generally will not).
+
+          Returns MP_RANGE if radix < MP_MIN_RADIX or radix > MP_MAX_RADIX.
+
+mp_result mp_int_read_string(mp_int z, mp_size radix, const char *str);
+mp_result mp_int_read_cstring(mp_int z, mp_size radix, const char *str, 
+                              char **end);
+          Read a string of ASCII digits in the specified radix from the
+          zero-terminated string provided, and assign z to the corresponding
+          value.  For radices greater than 10, the ASCII letters 'A' .. 'Z' or
+          'a' .. 'z' are used.  Letters are interpreted without respect to
+          case.
+
+          Leading whitespace is ignored, and a leading '+' or '-' is
+          interpreted as a sign flag.  Processing stops when ASCII NUL or any
+          character which is out of range for a digit in the given radix is
+          encountered.
+
+          If the whole string was processed, MP_OK is returned; otherwise,
+          MP_TRUNC is returned.
+
+          Returns MP_RANGE if radix < MP_MIN_RADIX or radix > MP_MAX_RADIX.
+
+          With mp_int_read_cstring(), if end is not NULL, the target pointer is
+          set to point to the first unconsumed character of the input string
+          (the NUL byte, if the whole string was consumed).  This emulates the
+          behavior of the standard C strtol() function.
+
+mp_result mp_int_count_bits(mp_int z);
+          Returns the number of significant bits in z.
+
+mp_result mp_int_to_binary(mp_int z, unsigned char *buf, int limit);
+          Convert z to 2's complement binary, writing at most 'limit' bytes
+          into the given buffer.  Returns MP_TRUNC if the buffer limit was too
+          small to contain the whole value.  If this occurs, the contents of
+          buf will be effectively garbage, as the function uses the buffer as
+          scratch space.
+
+          The binary representation of z is in base-256 with digits ordered
+          from most significant to least significant (network byte ordering).
+          The high-order bit of the first byte is set for negative values,
+          clear for non-negative values.
+
+          As a result, non-negative values will be padded with a leading zero
+          byte if the high-order byte of the base-256 magnitude is set.  This
+          extra byte is accounted for by the mp_int_binary_len() function
+          described below.
+
+mp_result mp_int_read_binary(mp_int z, unsigned char *buf, int len);
+          Read a 2's complement binary value into z, where the length of the
+          buffer is given as 'len'.  The contents of 'buf' may be overwritten
+          during processing, although they will be restored when the function
+          returns.
+
+mp_result mp_int_binary_len(mp_int z);
+          Return the number of bytes required to represent z in 2's complement
+          binary.
+
+mp_result mp_int_to_unsigned(mp_int z, unsigned char *buf, int limit);
+          Convert |z| to unsigned binary, writing at most 'limit' bytes into
+          the given buffer.  The sign of z is ignored, but z is not modified.
+          Returns MP_TRUNC if the buffer limit was too small to contain the
+          whole value.  If this occurs, the contents of buf will be effectively
+          garbage, as the function uses the buffer as scratch space during
+          conversion.
+
+          The binary representation of z is in base-256 with digits ordered
+          from most significant to least significant (network byte ordering).
+
+mp_result mp_int_read_unsigned(mp_int z, unsigned char *buf, int len);
+          Read an unsigned binary value into z, where the length of the buffer
+          is given as 'len'.  The contents of 'buf' will not be modified during
+          processing.
+
+mp_result mp_int_unsigned_len(mp_int z);
+          Return the number of bytes required to represent z as an unsigned
+          binary value using mp_int_to_unsigned().
+
+- Other functions:
+
+Ordinarily, integer multiplication and squaring are done using the simple
+quadratic "schoolbook" algorithm.  However, for sufficiently large values,
+there is a more efficient algorithm usually attributed to Karatsuba and Ofman
+that is usually faster.  See Knuth Vol. 2 for more details about how this
+algorithm works.
+
+The breakpoint between the "normal" and the recursive algorithm is controlled
+by a static constant "multiply_threshold" defined in imath.c, which contains
+the number of significant digits below which the standard algorithm should be
+used.  This is initialized to the value of the compile-time constant
+MP_MULT_THRESH from imath.h.  If you wish to be able to modify this value at
+runtime, compile imath.c with IMATH_TEST defined true in the preprocessor, and
+declare
+
+  extern mp_size multiply_threshold;
+
+When IMATH_TEST is defined, this variable is defined as a mutable global, and
+can be changed.  Otherwise, it is defined as an immutable static constant.  The
+`imtimer' program and the `findthreshold.py' script (Python) can help you find
+a suitable value for MP_MULT_THRESH for your particular platform.
+
+const char *mp_int_error_string(mp_result res);
+          Return a pointer to a brief string describing 'res'.  These strings
+          are defined as a constant array in `imath.c', if you wish to change
+          them for your application.
+
+--- Rational Arithmetic:
+
+mp_result mp_rat_init(mp_rat r);
+          Initialize a new zero-valued rational number in r.
+
+mp_result mp_rat_reduce(mp_rat r);
+          Reduce a rational to canonical form.
+          Removes common factors from the numerator and denomenator and makes
+          the denomenator positive.
+
+mp_result mp_rat_init_size(mp_rat r, mp_size n_prec, mp_size d_prec);
+          As mp_rat_init(), but specifies the number of long digits of
+          precision for numerator (n_prec) and denominator (d_prec).  Use this
+          if you wish to preallocate storage for operations of known output
+          size.
+
+mp_result mp_rat_init_copy(mp_rat r, mp_rat old);
+          As mp_rat_init(), but initializes a copy of an existing rational
+          value.
+
+mp_result mp_rat_set_value(mp_rat r, mp_small numer, mp_small denom);
+          Set the value of the given rational to a ratio specified as ordinary
+          signed integers (denom != 0).  Returns MP_UNDEF if denom = 0.
+
+void      mp_rat_clear(mp_rat r);
+          Release the memory occupied by the given rational number.
+
+mp_result mp_rat_numer(mp_rat r, mp_int z);
+          Extract the numerator of r as an mp_int, and store it in z.
+
+mp_result mp_rat_denom(mp_rat r, mp_int z);
+          Extract the denominator of r as an mp_int, and store it in z.
+
+mp_sign   mp_rat_sign(mp_rat r);
+          Return the sign of the rational number.  Note that an mpq_t
+          is always stored so that the sign of the numerator is the
+          correct sign of the whole value.
+
+mp_result mp_rat_copy(mp_rat a, mp_rat c);
+          Copy a to c.  Avoids unnecessary allocations.
+
+void      mp_rat_zero(mp_rat r);
+          Set r to have the value zero (canonical with denominator 1).
+
+mp_result mp_rat_abs(mp_rat a, mp_rat c);
+          Set c to the absolute value of a.
+
+mp_result mp_rat_neg(mp_rat a, mp_rat c);
+          Set c to the negative (additive inverse) of a.
+
+mp_result mp_rat_recip(mp_rat a, mp_rat c);
+          Take the reciprocal of a and store it in c, if defined.
+          Returns MP_UNDEF if a/c = 0.
+
+mp_result mp_rat_add(mp_rat a, mp_rat b, mp_rat c);
+          Add a + b and store the result in c.
+
+mp_result mp_rat_sub(mp_rat a, mp_rat b, mp_rat c);
+          Subtract a - b and store the result in c.
+
+mp_result mp_rat_mul(mp_rat a, mp_rat b, mp_rat c);
+          Multiply a * b and store the result in c.
+
+mp_result mp_rat_div(mp_rat a, mp_rat b, mp_rat c);
+          Divide a / b, if possible, and store the result in c.
+          Returns MP_UNDEF if b = 0.
+
+mp_result mp_rat_add_int(mp_rat a, mp_int b, mp_rat c);
+          Add a + b and store the result in c.  Note:  b is an integer.
+
+mp_result mp_rat_sub_int(mp_rat a, mp_int b, mp_rat c);
+          Subtract a - b and store the result in c.  Note:  b is an integer.
+
+mp_result mp_rat_mul_int(mp_rat a, mp_int b, mp_rat c);
+          Multiply a * b and store the result in c.  Note:  b is an integer.
+
+mp_result mp_rat_div_int(mp_rat a, mp_int b, mp_rat c);
+          Divide a / b, if possible, and store the result in c.
+          Note:  b is an integer.
+          Returns MP_UNDEF if b = 0.
+
+mp_result mp_rat_expt(mp_rat a, mp_small b, mp_rat c);
+          Raise a to the b power, where b >= 0, and store the result in c.
+
+int       mp_rat_compare(mp_rat a, mp_rat b);
+          Full signed comparison of rational values.
+
+int       mp_rat_compare_unsigned(mp_rat a, mp_rat b);
+          Compare the absolute values of a and b.
+
+int       mp_rat_compare_zero(mp_rat r);
+          Compare r to zero.
+
+int       mp_rat_compare_value(mp_rat r, mp_small n, mp_small d);
+          Compare r to the ratio n/d.
+
+int       mp_rat_is_integer(mp_rat r);
+          Returns true if r can be represented by an integer (i.e.,
+          its denominator is one).
+
+mp_result mp_rat_to_ints(mp_rat r, mp_small *num, mp_small *den);
+          If it is possible to do so, extract the numerator and the denominator
+          of r as regular (signed) integers.  Returns MP_RANGE if either cannot
+          be so represented.
+
+mp_result mp_rat_to_string(mp_rat r, mp_size radix, char *str, int limit);
+          Convert the value of r to a string of the format "n/d" with n and d
+          in the specified radix, writing no more than "limit" bytes of data to
+          the given output buffer.  Includes sign flag.
+
+mp_result mp_rat_to_decimal(mp_rat r, mp_size radix, mp_size prec,
+                            mp_round_mode rmode, char *str, int limit);
+          Convert the value of r to a string in decimal-point notation with the
+          specified radix, writing no more than "limit" bytes of data to the
+          given output buffer.  Generates "prec" digits of precision.
+
+          Values numbers may be rounded when they are being converted for
+          output as a decimal value.  There are four rounding modes currently
+          supported:
+
+          MP_ROUND_DOWN
+            Truncates the value toward zero.  
+            Example:  12.009 to 2dp becomes 12.00
+
+          MP_ROUND_UP
+            Rounds the value away from zero:
+            Example:  12.001 to 2dp becomes 12.01, but
+                      12.000 to 2dp remains 12.00
+
+          MP_ROUND_HALF_DOWN
+             Rounds the value to nearest digit, half goes toward zero.
+             Example:  12.005 to 2dp becomes 12.00, but
+                       12.006 to 2dp becomes 12.01
+
+          MP_ROUND_HALF_UP
+             Rounds the value to nearest digit, half rounds upward.
+             Example:  12.005 to 2dp becomes 12.01, but
+                       12.004 to 2dp becomes 12.00
+
+mp_result mp_rat_string_len(mp_rat r, mp_size radix);
+          Return the length of buffer required to convert r using the
+          mp_rat_to_string() function.  May over-estimate slightly.
+
+mp_result mp_rat_decimal_len(mp_rat r, mp_size radix, mp_size prec);
+          Return the length of buffer required to convert r using the
+          mp_rat_to_decimal() function.  May over-estimate slightly.
+
+mp_result mp_rat_read_string(mp_rat r, mp_size radix, char *str);
+          Read a zero-terminated string in the format "n/d" (including sign
+          flag), and replace the value of r with it.
+
+mp_result mp_rat_read_cstring(mp_rat r, mp_size radix, char *str, char **end);
+          Like mp_rat_read_string(), but with a similar interface to the
+          strtoul() library function.  Used as the back end for the
+          mp_rat_read_string() function.  Returns MP_UNDEF if the denominator
+          read has value zero.
+
+mp_result mp_rat_read_ustring(mp_rat r, mp_size radix, char *str, char **end);
+          A "universal" reader.  Capable of reading plain integers, rational
+          number written in a/b notation, and decimal values in z.f format.
+          The end parameter works as for mp_int_read_cstring().
+
+mp_result mp_rat_read_decimal(mp_rat r, mp_size radix, char *str);
+          A wrapper around mp_rat_read_cdecimal(), which discards the resulting
+          end pointer.
+
+mp_result mp_rat_read_cdecimal(mp_rat r, mp_size radix, char *str, char **end);
+          Read a zero-terminated string in the format "z.f" (including sign
+          flag), and replace r with its value.  If end is not NULL, a pointer
+          to the first unconsumed character of the string is returned.
+
+--- Representation Details
+
+    NOTE:  You do not need to read this section to use IMath.  This is provided
+           for the benefit of developers wishing to extend or modify the
+           internals of the library.
+
+IMath uses a signed magnitude representation for arbitrary precision integers.
+The magnitude is represented as an array of radix-R digits in increasing order
+of significance; the value of R is chosen to be half the size of the largest
+available unsigned integer type, so typically 16 or 32 bits.  Digits are
+represented as mp_digit, which must be an unsigned integral type.
+
+Digit arrays are allocated using malloc(3) and realloc(3).  Because this can be
+an expensive operation, the library takes pains to avoid allocation as much as
+possible.  For this reason, the mpz_t structure distinguishes between how many
+digits are allocated and how many digits are actually consumed by the
+representation.  The fields of an mpz_t are:
+
+       mp_digit    single;  /* single-digit value (see note) */
+       mp_digit   *digits;  /* array of digits               */
+       mp_size     alloc;   /* how many digits are allocated */
+       mp_size     used;    /* how many digits are in use    */
+       mp_sign     sign;    /* the sign of the value         */
+
+The elements of "digits" at indices less than "used" are the significant
+figures of the value; the elements at indices greater than or equal to "used"
+are undefined (and may contain garbage).  At all times, "used" must be at least
+1 and at most "alloc".
+
+To avoid interaction with the memory allocator, single-digit values are stored
+directly in the mpz_t structure, in the .single field.  The semantics of access
+are the same as the more general case.
+
+The number of digits allocated for an mpz_t is referred to in the library
+documentation as its "precision".  Operations that affect an mpz_t cause
+precision to increase as needed.  In any case, all allocations are measured in
+digits, and rounded up to the nearest mp_word boundary.  There is a default
+minimum precision stored as a static constant default_precision (imath.c); its
+value is set to MP_DEFAULT_PREC (imath.h).  If the preprocessor symbol
+IMATH_TEST is defined, this value becomes global and modifiable.
+
+Note that the allocated size of an mpz_t can only grow; the library never
+reallocates in order to decrease the size.  A simple way to do so explicitly is
+to use mp_int_init_copy(), as in:
+
+  mpz_t big, new;
+
+  /* ... */
+  mp_int_init_copy(&new, &big);
+  mp_int_swap(&new, &big);
+  mp_int_clear(&new);
+
+The value of "sign" is 0 for positive values and zero, 1 for negative values.
+Constants MP_ZPOS and MP_NEG are defined for these; no other sign values are
+used.
+
+If you are adding to this library, you should be careful to preserve the
+convention that inputs and outputs can overlap, as described above.  So, for
+example, mp_int_add(a, a, a) is legal.  Often, this means you must maintain one
+or more temporary mpz_t structures for intermediate values.  The private macros
+SETUP(E, C) and TEMP(K) can be used to enforce a conventional structure like
+this:
+
+  {
+    mpz_t     temp[NUM_TEMPS];  /* declare how many you need here */
+    int       last = 0;         /* number of in-use temps         */
+    mp_result res;              /* used for checking results      */
+    ...
+
+    /* Initialization phase */ 
+    SETUP(mp_int_init(TEMP(0)), last);
+    SETUP(mp_int_init_copy(TEMP(1), x), last);
+    ...
+    SETUP(mp_int_init_value(TEMP(7), 3), last);
+
+    /* Work phase */
+    ...
+
+  CLEANUP:
+    while(--last >= 0)
+      mp_int_clear(TEMP(last));
+
+    return res;
+  }
+
+The names "temp" and "res" are fixed -- the SETUP and TEMP macros assume they
+exist.  TEMP(k) returns a pointer to the kth entry of temp.  This structure
+insures that even if a failure occurs during the "initialization phase", no
+memory is leaked.
+
+"Small" integer values are represented by the types mp_small and mp_usmall,
+which are mapped to appropriately-sized types on the host system.  The default
+for mp_small is "long" and the default for mp_usmall is "unsigned long".  You
+may change these, provided you insure that mp_small is signed and mp_usmall is
+unsigned.  You will also need to adjust the size macros:
+
+  MP_SMALL_MIN, MP_SMALL_MAX
+  MP_USMALL_MIN, MP_USMALL_MAX
+
+... which are defined in <imath.h>, if you change these.
+
+Rational numbers are represented using a pair of arbitrary precision integers,
+with the convention that the sign of the numerator is the sign of the rational
+value, and that the result of any rational operation is always represented in
+lowest terms.  The canonical representation for rational zero is 0/1.  See
+"imrat.h".
+
+--- Testing and Reporting of Bugs:
+
+Test vectors are included in the 'tests' subdirectory of the imath
+distribution.  When you run 'make test', it builds the imtest program and runs
+all available test vectors.  If any tests fail, you will get a line like this:
+
+  x    y    FAILED      v
+
+Here, x is the line number of the test which failed, y is index of the test
+within the file, and v is the value(s) actually computed.  The name of the file
+is printed at the beginning of each test, so you can find out what test vector
+failed by executing the following (with x, y, and v replaced by the above
+values, and where "foo.t" is the name of the test file that was being processed
+at the time):
+
+   % tail +x tests/foo.t | head -1
+
+None of the tests should fail [but see Note 2]; if any do, it probably
+indicates a bug in the library (or at the very least, some assumption I made
+which I shouldn't have).  Please send a bug report to the address below, which
+includes the FAILED test line above, as well as the output of the above 'tail'
+command (so I know what inputs caused the failure).
+
+If you build with the preprocessor symbol DEBUG defined as a positive integer,
+you will have access to several things:
+
+ 1. The static functions defined in imath.c are made globally visible so that
+    you can call them from a test driver.
+
+ 2. The s_print() and s_print_buf() routines are defined; these make it easier
+    to dump the contents of an mpz_t to text.
+
+ 3. If DEBUG > 1, the digit allocators (s_alloc, s_realloc) fill all new
+    buffers with the value 0xDEADBEEF, or as much of it as will fit in a digit,
+    so that you can more easily catch uninitialized reads in the debugger.
+
+--- Notes:
+
+1. You can generally use the same variables for both input and output.  One
+   exception is that you may not use the same variable for both the quotient
+   and the remainder of mp_int_div().
+
+2. Many of the tests for this library were written under the assumption that
+   the mp_small type is 32 bits or more.  If you compile with a smaller type,
+   you may see MP_RANGE errors in some of the tests that otherwise pass (due to
+   conversion failures).  Also, the pi generator (pi.c) will not work correctly
+   if mp_small is too short, as its algorithm for arc tangent is fairly
+   simple-minded.
+
+--- Contacts:
+
+The IMath library was written by 
+Michael J. Fromberger <http://spinning-yarns.org/michael/>
+
+If you discover any bugs or testing failures, please send e-mail to
+the following address:
+
+    <michael.j.fromberger@gmail.com>
+
+Please be sure to include, with any bug report, a complete description of what
+goes wrong, and if possible, a test vector for imtest or a minimal test program
+that will demonstrate the bug on your system.  Please also let me know what
+hardware, operating system, and compiler you're using.
+
+--- Acknowledgements:
+
+The algorithms used in this library came from Vol. 2 of Donald Knuth's "The Art
+of Computer Programming" (Seminumerical Algorithms).  Thanks to Nelson Bolyard,
+Bryan Olson, Tom St. Denis, Tushar Udeshi, and Eric Silva for excellent
+feedback on earlier versions of this code.  Special thanks to Jonathan Shapiro
+for some very helpful design advice, as well as feedback and some clever ideas
+for improving performance in some common use cases.
+
+--- License and Disclaimers:
+
+IMath is Copyright 2002-2009 Michael J. Fromberger
+You may use it subject to the following Licensing Terms:
+
+Permission is hereby granted, free of charge, to any person obtaining a copy of
+this software and associated documentation files (the "Software"), to deal in
+the Software without restriction, including without limitation the rights to
+use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies
+of the Software, and to permit persons to whom the Software is furnished to do
+so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.
diff --git a/polly/lib/External/isl/imath/examples/basecvt.c b/polly/lib/External/isl/imath/examples/basecvt.c
new file mode 100644
index 00000000000..c067be904f1
--- /dev/null
+++ b/polly/lib/External/isl/imath/examples/basecvt.c
@@ -0,0 +1,121 @@
+/*
+  Name:     basecvt.c
+  Purpose:  Convert integers and rationals from one base to another.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+  Info:     $Id: basecvt.c 635 2008-01-08 18:19:40Z sting $
+
+  Copyright (C) 2004-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+#include <errno.h>
+
+#include "imath.h"
+#include "imrat.h"
+
+int main(int argc, char *argv[])
+{
+  mp_size   in_rdx, out_rdx;
+  mpq_t     value;
+  mp_result res;
+  int       ix;
+
+  if(argc < 4) {
+    fprintf(stderr, "Usage: basecvt <ibase> <obase> <values>+\n");
+    return 1;
+  }
+
+  in_rdx  = atoi(argv[1]);
+  out_rdx = atoi(argv[2]);
+
+  if(in_rdx < MP_MIN_RADIX || in_rdx > MP_MAX_RADIX) {
+    fprintf(stderr, 
+	    "basecvt: input radix %u not allowed (minimum %u, "
+	    "maximum %u)\n", in_rdx, MP_MIN_RADIX, MP_MAX_RADIX);
+    return 3;
+  }
+  if(out_rdx < MP_MIN_RADIX || out_rdx > MP_MAX_RADIX) {
+    fprintf(stderr, 
+	    "basecvt: output radix %u not allowed (minimum %u, "
+	    "maximum %u)\n", out_rdx, MP_MIN_RADIX, MP_MAX_RADIX);
+    return 3;
+  }
+  
+  if((res = mp_rat_init(&value)) != MP_OK) {
+    fprintf(stderr, "basecvt: out of memory\n");
+    return 2;
+  }
+
+  for(ix = 3; ix < argc; ++ix) {
+    char *buf, *endp = NULL;
+    mp_result   len;
+    int         is_int;
+
+    if((res = mp_rat_read_ustring(&value, in_rdx, argv[ix], &endp)) != MP_OK &&
+       res != MP_TRUNC) {
+      fprintf(stderr, "basecvt:  error reading argument %d: %s\n", ix,
+	      mp_error_string(res));
+      break;
+    }
+    else if(*endp != '\0') {
+      fprintf(stderr, "basecvt:  argument %d contains '%s' not in base %u\n",
+	      ix, endp, in_rdx);
+      continue;
+    }
+
+    if((is_int = mp_rat_is_integer(&value)) != 0) {
+      len = mp_int_string_len(MP_NUMER_P(&value), out_rdx);
+    }
+    else {
+      len = mp_rat_string_len(&value, out_rdx);
+    }
+    
+    if((buf = malloc(len)) == NULL) {
+      fprintf(stderr, "basecvt:  out of memory\n");
+      break;
+    }
+    
+    if(is_int) {
+      res = mp_int_to_string(MP_NUMER_P(&value), out_rdx, buf, len);
+    }
+    else {
+      res = mp_rat_to_string(&value, out_rdx, buf, len);
+    }
+
+    if(res != MP_OK) {
+      fprintf(stderr, "basecvt:  error converting argument %d: %s\n", ix,
+	      mp_error_string(res));
+      free(buf);
+      break;
+    }
+   
+    printf("%s\n", buf);
+    free(buf);
+  }
+  
+  mp_rat_clear(&value);
+
+  return (res != MP_OK);
+}
+
+/* Here there be dragons */
diff --git a/polly/lib/External/isl/imath/examples/findprime.c b/polly/lib/External/isl/imath/examples/findprime.c
new file mode 100644
index 00000000000..070f14194d9
--- /dev/null
+++ b/polly/lib/External/isl/imath/examples/findprime.c
@@ -0,0 +1,72 @@
+/*
+  Name:     findprime.c
+  Purpose:  Find probable primes.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+  Info:     $Id: findprime.c 635 2008-01-08 18:19:40Z sting $
+
+  Copyright (C) 2002-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Notes:
+  Find the first prime number in sequence starting from the given value.
+  Demonstrates the use of mp_int_find_prime().
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#include <stdio.h>
+
+#include "iprime.h"
+
+int main(int argc, char *argv[])
+{
+  char   buf[4096];
+  mpz_t  seed;
+  mp_result res;
+
+  if(argc < 2) {
+    fprintf(stderr, "Usage: %s <start-value>\n", argv[0]);
+    return 1;
+  }
+
+  mp_int_init(&seed);
+  if((res = mp_int_read_string(&seed, 10, argv[1])) != MP_OK) {
+    fprintf(stderr, "%s: error reading `%s': %d\n", 
+	    argv[0], argv[1], res);
+    return 2;
+  }
+
+  if(mp_int_compare_value(&seed, 131) <= 0) {
+    fprintf(stderr, "%s: please enter a start value > 131\n",
+	    argv[0]);
+    return 1;
+  }
+
+  if((res = mp_int_find_prime(&seed)) != MP_TRUE) {
+    fprintf(stderr, "%s: error finding prime: %d\n",
+	    argv[0], res);
+    return 2;
+  }
+
+  mp_int_to_string(&seed, 10, buf, sizeof(buf));
+  printf("=> %s\n", buf);
+
+  mp_int_clear(&seed);
+
+  return 0;
+}
diff --git a/polly/lib/External/isl/imath/examples/imcalc.c b/polly/lib/External/isl/imath/examples/imcalc.c
new file mode 100644
index 00000000000..97f0dc5affb
--- /dev/null
+++ b/polly/lib/External/isl/imath/examples/imcalc.c
@@ -0,0 +1,1471 @@
+/*
+  Name:     imcalc.c
+  Purpose:  Simple RPN calculator based on IMath library.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  This is a very simplistic RPN calculator that will let you test the features
+  of the IMath built-in functions.
+
+  Copyright (C) 2002-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+#include <limits.h>
+#include <ctype.h>
+#include <errno.h>
+#include <assert.h>
+
+#include <unistd.h>
+
+#include "imath.h"
+#include "imrat.h"
+#include "iprime.h"
+
+/* A cstate_t represents a stack of operands; numeric operands are pushed on
+   the stack, and commands cause them to be consumed in various ways.
+ */
+typedef struct {
+  /* Operand stack    */
+  mp_int    *elts;
+  mp_size    alloc;   /* number of slots available */
+  mp_size    used;    /* number of slots free      */
+
+  /* Named variables  */
+  mp_int    *mem;     /* named memory slots        */
+  char     **names;   /* names of memory slots     */
+  mp_size    mslots;  /* number of memory slots    */
+  mp_size    mused;   /* number of used memories   */
+
+  /* I/O components   */
+  FILE      *ifp;     /* input file handle         */
+  char      *ibuf;    /* input scratch buffer      */
+  int        buflen;  /* size of scratch buffer    */
+} cstate_t;
+
+/* {{{ State function prototypes */
+
+static mp_result state_init(cstate_t *sp, mp_size n_elts);
+static void      state_clear(cstate_t *sp);
+static void      stack_flush(cstate_t *sp);
+static mp_result stack_push(cstate_t *sp, mp_int elt);
+static mp_result stack_pop(cstate_t *sp);
+static mp_result mem_insert(cstate_t *sp, const char *name, mp_int value);
+static mp_result mem_recall(cstate_t *sp, const char *name, mp_int value);
+static mp_result mem_clear(cstate_t *sp);
+
+/* }}} */
+
+/* {{{ Calculation function prototypes */
+
+typedef mp_result (*op_func)(cstate_t *);
+
+static mp_result cf_abs(cstate_t *sp);
+static mp_result cf_neg(cstate_t *sp);
+static mp_result cf_add(cstate_t *sp);
+static mp_result cf_sub(cstate_t *sp);
+static mp_result cf_mul(cstate_t *sp);
+static mp_result cf_divmod(cstate_t *sp);
+static mp_result cf_div(cstate_t *sp);
+static mp_result cf_mod(cstate_t *sp);
+static mp_result cf_expt(cstate_t *sp);
+static mp_result cf_exptmod(cstate_t *sp);
+static mp_result cf_square(cstate_t *sp);
+static mp_result cf_invmod(cstate_t *sp);
+static mp_result cf_gcd(cstate_t *sp);
+static mp_result cf_xgcd(cstate_t *sp);
+static mp_result cf_sqrt(cstate_t *sp);
+static mp_result cf_root(cstate_t *sp);
+static mp_result cf_cmplt(cstate_t *sp);
+static mp_result cf_cmpgt(cstate_t *sp);
+static mp_result cf_cmple(cstate_t *sp);
+static mp_result cf_cmpge(cstate_t *sp);
+static mp_result cf_cmpeq(cstate_t *sp);
+static mp_result cf_cmpne(cstate_t *sp);
+static mp_result cf_inc(cstate_t *sp);
+static mp_result cf_dec(cstate_t *sp);
+static mp_result cf_fact(cstate_t *sp);
+static mp_result cf_pprint(cstate_t *sp);
+static mp_result cf_print(cstate_t *sp);
+static mp_result cf_pstack(cstate_t *sp);
+static mp_result cf_clstk(cstate_t *sp);
+static mp_result cf_pop(cstate_t *sp);
+static mp_result cf_dup(cstate_t *sp);
+static mp_result cf_copy(cstate_t *sp);
+static mp_result cf_swap(cstate_t *sp);
+static mp_result cf_rot(cstate_t *sp);
+static mp_result cf_pick(cstate_t *sp);
+static mp_result cf_setr(cstate_t *sp);
+static mp_result cf_setbin(cstate_t *sp);
+static mp_result cf_help(cstate_t *sp);
+static mp_result cf_store(cstate_t *sp);
+static mp_result cf_recall(cstate_t *sp);
+static mp_result cf_cmem(cstate_t *sp);
+static mp_result cf_pmem(cstate_t *sp);
+static mp_result cf_qrecall(cstate_t *sp);
+
+/* }}} */
+
+/* {{{ Built-in operator records, g_ops[] */
+
+typedef struct {
+  char    *name;         /* The name of the operator.           */
+  int      stack_size;   /* Number of stack arguments required. */
+  op_func  handler;      /* Function implementing operation.    */
+  char    *descript;     /* Human-readable description.         */
+} calcop_t;
+
+static calcop_t g_ops[] = {
+  { "abs",   1, cf_abs,     "x -- |x|" },
+  { "neg",   1, cf_neg,     "x -- (-x)" },
+  { "+",     2, cf_add,     "x y -- (x+y)" },
+  { "add",   2, cf_add,     "x y -- (x+y)" },
+  { "-",     2, cf_sub,     "x y -- (x-y)" },
+  { "sub",   2, cf_sub,     "x y -- (x-y)" },
+  { "*",     2, cf_mul,     "x y -- (x*y)" },
+  { "mul",   2, cf_mul,     "x y -- (x*y)" },
+  { "/",     2, cf_divmod,  "x y -- q r ; x = yq + r, 0 <= r < y" },
+  { "//",    2, cf_div,     "x y -- (x div y)" },
+  { "div",   2, cf_div,     "x y -- (x div y)" },
+  { "%",     2, cf_mod,     "x y -- (x mod y)" },
+  { "mod",   2, cf_mod,     "x y -- (x mod y)" },
+  { "^",     2, cf_expt,    "x y -- (x^y)" },
+  { "expt",  2, cf_expt,    "x y -- (x^y)" },
+  { "^^",    3, cf_exptmod, "x y m -- (x^y mod m)" },
+  { "emod",  3, cf_exptmod, "x y m -- (x^y mod m)" },
+  { "sqr",   1, cf_square,  "x -- (x*x)" },
+  { "inv",   2, cf_invmod,  "x m -- (1/x mod m)" },
+  { "gcd",   2, cf_gcd,     "x y -- gcd(x, y)" },
+  { "xgcd",  2, cf_xgcd,    "x y -- g u v ; g = ux + vy" },
+  { "sqrt",  1, cf_sqrt,    "x -- floor(sqrt(x))" },
+  { "root",  2, cf_root,    "x y -- floor(x^{1/y}) ; y > 0" },
+  { "<",     2, cf_cmplt,   "x y -- (x<y)" },
+  { ">",     2, cf_cmpgt,   "x y -- (x>y)" }, 
+  { "<=",    2, cf_cmple,   "x y -- (x<=y)" },
+  { ">=",    2, cf_cmpge,   "x y -- (x>=y)" },
+  { "=",     2, cf_cmpeq,   "x y -- (x=y)" },
+  { "<>",    2, cf_cmpne,   "x y -- (x<>y)" },
+  { "inc",   1, cf_inc,     "x -- (x+1)" },
+  { "dec",   1, cf_dec,     "x -- (x-1)" },
+  { "!",     1, cf_fact,    "x -- x!" },
+  { "fact",  1, cf_fact,    "x -- x!" },
+
+  { ".",     1, cf_pprint,  "x -- ; print x in current output mode" },
+  { ";",     1, cf_print,   "x -- x ; print x in current output mode" },
+  { "?",     0, cf_pstack,  "-- ; print stack" },
+  { "cls",   0, cf_clstk,   "... -- ; clear stack" },
+  { "$",     1, cf_pop,     "x --" },
+  { "drop",  1, cf_pop,     "x --" },
+  { "dup",   1, cf_dup,     "x -- x x" },
+  { "copy",  2, cf_copy,    "vn ... v1 v0 n -- vn ... v0 vn ... v0" },
+  { "swap",  2, cf_swap,    "x y -- y x" },
+  { "rot",   3, cf_rot,     "a b c -- b c a" },
+  { "pick",  2, cf_pick,    "... v2 v1 v0 n -- ... v2 v1 v0 vn" },
+
+  { ">>",    1, cf_store,   "x -- ; save in named variable" },
+  { "<<",    0, cf_recall,  "-- x ; recall from named variable" },
+  { "clm",   0, cf_cmem,    "-- ; clear memory" },
+  { "??",    0, cf_pmem,    "-- ; print memory" },
+
+  { "out",   1, cf_setr,    "r -- ; set output radix to r" },
+  { "bin",   0, cf_setbin,  "-- ; set output format to binary" },
+  { "help",  0, cf_help,    "-- ; print help message" },
+
+  /* This is the end-marker, but it is also used to catch implicit
+     variable lookups from memory.
+   */
+  { NULL,    0, cf_qrecall, "-- x ; recall from named variable" }, 
+};
+
+/* }}} */
+
+#define BUFFER_SIZE  16384 /* max. length of input values, in chars */
+
+/* Token types from the primitive lexical analyzer */
+typedef enum { t_eof, t_symbol, t_number, t_error } token_t;
+
+/* {{{ Helper functions */
+
+static token_t   next_token(FILE *ifp, char *buf, int size);
+static mp_result read_number(char *buf, mp_int *out);
+static int       find_command(cstate_t *ops);
+static void      print_value(mp_int v);
+static mp_result run_file(FILE *ifp, cstate_t *op_state);
+
+/* }}} */
+
+/* Error code used internally to signal input problems. */
+static mp_result MP_INPUT;
+
+static int g_output_radix = 10;  /* output radix */
+static FILE *g_output_file = NULL;
+
+int main(int argc, char *argv[]) 
+{
+  extern char *optarg;
+  extern int   optind;
+
+  int   opt, errs = 0;
+  FILE *ifp;
+
+  cstate_t op_state;
+  mp_result res;
+
+  MP_INPUT = MP_MINERR - 1;
+
+  g_output_file = stdout;
+  while((opt = getopt(argc, argv, "ho:")) != EOF) {
+    switch(opt) {
+    case 'h':
+      fprintf(stderr,
+	      "Usage: imcalc [-h] [-o <output>] input*\n\n"
+	      "Options:\n"
+	      "  -h          : display this help message.\n"
+	      "  -o <output> : send output to file.\n\n"
+
+	      "If no input files are given, the standard input is read.  The\n"
+	      "special file name \"-\" is interpreted to mean the standard input.\n"
+	      "Output goes to standard output unless \"-o\" is used.\n\n");
+      return 0;
+
+    case 'o':
+      if((g_output_file = fopen(optarg, "wt")) == NULL) {
+	fprintf(stderr, "Unable to open \"%s\" for writing: %s\n",
+		optarg, strerror(errno));
+	return 1;
+      }
+      break;
+
+    default:
+      fprintf(stderr, 
+	      "Usage: imcalc [-h] [-o <output>] input*\n"
+	      "       [use \"imcalc -h\" to get help]\n\n");
+      return 1;
+    }
+  }
+
+  if((res = state_init(&op_state, 1)) != MP_OK) {
+    fprintf(stderr, "Error: state_init: %s\n", 
+	    mp_error_string(res));
+    return 1;
+  }
+
+  if(optind < argc) {
+    int ix;
+
+    for(ix = optind; ix < argc; ++ix) {
+      if(strcmp(argv[ix], "-") == 0) 
+	ifp = stdin;
+      else if((ifp = fopen(argv[optind], "rt")) == NULL) {
+	fprintf(stderr, "Unable to open \"%s\" for reading: %s\n",
+		argv[optind], strerror(errno));
+	return 1;
+      }
+
+      if(run_file(ifp, &op_state) != MP_OK)
+	++errs;
+    }
+
+    state_clear(&op_state);
+    return errs > 0;
+  }
+  else {
+    int rv = 1 - (run_file(stdin, &op_state) == MP_OK);
+    state_clear(&op_state);    
+    return rv;
+  }
+}
+
+/* {{{ next_token(*ifp, *buf, size) */
+
+static token_t next_token(FILE *ifp, char *buf, int size)
+{
+  int ch, pos = 0;
+  token_t res;
+
+  assert(buf != NULL && size > 0);
+  
+  while((ch = fgetc(ifp)) != EOF && isspace(ch))
+    /* empty */;
+  
+  if(ch == EOF) {
+    buf[0] = '\0';
+    return t_eof;
+  }
+
+  if(ch == '-') {
+    int next = fgetc(ifp);
+    if(next == EOF || !isdigit(next))
+      res = t_symbol;
+    else
+      res = t_number;
+    ungetc(next, ifp);
+  }
+  else if(isdigit(ch) || ch == '#')
+    res = t_number;
+  else
+    res = t_symbol;
+
+  buf[pos++] = ch;
+  while((ch = fgetc(ifp)) != EOF) {
+    if((res == t_number && ispunct(ch) && ch != '-') ||
+       (res == t_symbol && isdigit(ch)) ||
+       isspace(ch)) {
+      ungetc(ch, ifp);
+      break;
+    }
+    else if(pos + 1 >= size) {
+      res = t_error;
+      break;
+    }
+    buf[pos++] = ch;
+  }
+
+  buf[pos] = '\0';
+  return res;
+}
+
+/* }}} */
+
+/* {{{ read_number(*buf, *out) */
+
+static mp_result read_number(char *buf, mp_int *out)
+{
+  int radix = 10, pos = 0;
+  mp_result res;
+  mp_int    value;
+
+  assert(buf != NULL && out != NULL);
+
+  if(buf[pos] == '#') {
+    switch(buf[1]) {
+    case 'b': case 'B':
+      radix = 2;
+      break;
+    case 'd': case 'D':
+      radix = 10;
+      break;
+    case 'o': case 'O':
+      radix = 8;
+      break;
+    case 'x': case 'X':
+      radix = 16;
+      break;
+    default:
+      return MP_BADARG;
+    }
+
+    pos += 2;
+  }
+
+  if((value = mp_int_alloc()) == NULL) {
+    *out = NULL; 
+    return MP_MEMORY;
+  }
+  
+  if((res = mp_int_read_string(value, radix, buf + pos)) != MP_OK) {
+    mp_int_free(value);
+    *out = NULL;
+    return res;
+  }
+  
+  *out = value;
+  return res;
+}
+
+/* }}} */
+
+/* {{{ find_command(*buf) */
+
+static int       find_command(cstate_t *op)
+{
+  int ix, jx;
+  char *buf = op->ibuf;
+
+  /* First, try to find the command by name */
+  for(ix = 0; g_ops[ix].name != NULL; ++ix) {
+    if(strcasecmp(buf, g_ops[ix].name) == 0)
+      return ix;
+  }
+
+  /* If we don't find the command, try a variable lookup */
+  for(jx = 0; jx < op->mused; ++jx) {
+    if(strcmp(buf, op->names[jx]) == 0) 
+      return ix; /* sentinel */
+  }
+
+  /* If variable lookup fails, report command not found */
+  return -1;
+}
+
+/* }}} */
+
+/* {{{ print_value(v) */
+
+static void      print_value(mp_int v)
+{
+  if(g_output_radix == 0) {
+    mp_result len = mp_int_binary_len(v);
+    unsigned char *buf = malloc(len);
+    int ix;
+
+    if(buf != NULL) {
+      mp_int_to_binary(v, buf, len);
+      for(ix = 0; ix < len - 1; ++ix) {
+	fprintf(g_output_file, "%02x.", buf[ix]);
+      }
+      fprintf(g_output_file, "%02x\n", buf[ix]);
+      free(buf);
+    }
+    else 
+      fprintf(g_output_file, "<insufficient memory to print>\n");
+  }
+  else {
+    mp_result len = mp_int_string_len(v, g_output_radix);
+    char *buf = malloc(len);
+
+    if(buf != NULL) {
+      mp_int_to_string(v, g_output_radix, buf, len);
+      fputs(buf, g_output_file);
+      fputc('\n', g_output_file);
+      free(buf);
+    }
+    else
+      fprintf(g_output_file, "<insufficient memory to print>\n");
+  }
+}
+
+/* }}} */
+
+/* {{{ run_file(*ifp, *op_state) */
+
+static mp_result run_file(FILE *ifp, cstate_t *op_state)
+{
+  mp_result  res = MP_OK;
+  token_t    next;
+
+  op_state->ifp = ifp;
+  while((next = next_token(ifp, op_state->ibuf, op_state->buflen)) != t_eof) {
+    mp_int value = NULL;
+    int    cpos;
+
+    switch(next) {
+    case t_number:
+      if((res = read_number(op_state->ibuf, &value)) != MP_OK) 
+	fprintf(stderr, "error: invalid number syntax: %s\n", op_state->ibuf);
+      else if((res = stack_push(op_state, value)) != MP_OK)
+	goto EXIT;
+      break;
+    case t_symbol:
+      if((cpos = find_command(op_state)) < 0) 
+	fprintf(stderr, "error: command not understood: %s\n", op_state->ibuf);
+      else if(op_state->used < g_ops[cpos].stack_size) {
+	fprintf(stderr, "error: not enough arguments (have %d, want %d)\n",
+		op_state->used, g_ops[cpos].stack_size);
+      }
+      else if((res = (g_ops[cpos].handler)(op_state)) != MP_OK) {
+	if(res == MP_INPUT) 
+	  fprintf(stderr, "error: incorrect input format\n");
+	else 
+	  fprintf(stderr, "error: %s\n", mp_error_string(res));
+      }
+      break;
+    default:
+      fprintf(stderr, "error: invalid input token: %s\n", op_state->ibuf);
+      res = MP_BADARG;
+      goto EXIT;
+    }
+  }
+
+ EXIT:
+  return res;
+}
+
+/* }}} */
+
+/* {{{ state_init(*sp, n_elts) */
+
+static mp_result state_init(cstate_t *sp, mp_size n_elts)
+{
+  int ix;
+
+  assert(sp != NULL && n_elts > 0);
+
+  if((sp->elts = malloc(n_elts * sizeof(*(sp->elts)))) == NULL)
+    return MP_MEMORY;
+  if((sp->mem = malloc(n_elts * sizeof(*(sp->mem)))) == NULL) {
+    free(sp->elts);
+    return MP_MEMORY;
+  }
+  if((sp->names = malloc(n_elts * sizeof(*(sp->names)))) == NULL) {
+    free(sp->mem);
+    free(sp->elts);
+    return MP_MEMORY;
+  }
+  if((sp->ibuf = malloc(BUFFER_SIZE * sizeof(char))) == NULL) {
+    free(sp->names);
+    free(sp->mem);
+    free(sp->elts);
+    return MP_MEMORY;
+  }
+  
+  for(ix = 0; ix < n_elts; ++ix) {
+    sp->elts[ix]  = NULL;
+    sp->mem[ix]   = NULL;
+    sp->names[ix] = NULL;
+  }
+
+  sp->alloc  = n_elts;
+  sp->used   = 0;
+  sp->mslots = n_elts;
+  sp->mused  = 0;
+  sp->buflen = BUFFER_SIZE;
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ state_clear(*sp) */
+
+static void      state_clear(cstate_t *sp)
+{
+  assert(sp != NULL);
+
+  if(sp->elts != NULL) {
+    int ix;
+
+    for(ix = 0; ix < sp->used; ++ix) {
+      mp_int_clear(sp->elts[ix]);
+      sp->elts[ix] = NULL; 
+    }
+
+    free(sp->elts);
+    sp->elts = NULL;
+    sp->alloc = 0;
+    sp->used = 0;
+  }
+  if(sp->mem != NULL) {
+    int ix;
+
+    for(ix = 0; ix < sp->mused; ++ix) {
+      mp_int_free(sp->mem[ix]);
+      sp->mem[ix] = NULL;
+      free(sp->names[ix]);
+      sp->names[ix] = NULL;
+    }
+
+    free(sp->mem);
+    sp->mem = NULL;
+    free(sp->names);
+    sp->names = NULL;
+    
+    sp->mslots = 0;
+    sp->mused  = 0;
+  }
+  if(sp->ibuf != NULL) {
+    free(sp->ibuf);
+    sp->buflen = 0;
+  }
+  if(sp->ifp != NULL) {
+    fclose(sp->ifp);
+    sp->ifp = NULL;
+  }
+}
+
+/* }}} */
+
+/* {{{ stack_flush(*sp) */
+
+static void      stack_flush(cstate_t *sp)
+{
+  int ix;
+
+  assert(sp != NULL && sp->elts != NULL);
+  
+  for(ix = 0; ix < sp->used; ++ix) {
+    mp_int_clear(sp->elts[ix]);
+    sp->elts[ix] = NULL;
+  }
+
+  sp->used = 0;
+}
+
+/* }}} */
+
+/* {{{ stack_push(*sp, elt) */
+
+static mp_result stack_push(cstate_t *sp, mp_int elt)
+{
+  if(sp->used >= sp->alloc) {
+    mp_size nsize = 2 * sp->alloc;
+    mp_int *tmp;
+    int ix;
+    
+    if((tmp = malloc(nsize * sizeof(*(sp->elts)))) == NULL)
+      return MP_MEMORY;
+
+    for(ix = 0; ix < sp->used; ++ix)
+      tmp[ix] = sp->elts[ix];
+
+    free(sp->elts);
+    sp->elts = tmp;
+    sp->alloc = nsize;
+  }
+
+  sp->elts[sp->used++] = elt;
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ stack_pop(*sp) */
+
+static mp_result stack_pop(cstate_t *sp)
+{
+  assert(sp != NULL && sp->elts != NULL);
+
+  if(sp->used == 0)
+    return MP_UNDEF;
+
+  sp->used -= 1;
+  mp_int_clear(sp->elts[sp->used]);
+  sp->elts[sp->used] = NULL;
+  
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mem_insert(*sp, *name, value) */
+
+static mp_result mem_insert(cstate_t *sp, const char *name, mp_int value)
+{
+  int ix;
+
+  for(ix = 0; ix < sp->mused; ++ix) {
+    if(strcmp(name, sp->names[ix]) == 0)
+      break;
+  }
+
+  /* Two cases:
+     ix < sp->mused   ==> replacing existing entry.
+     otherwise        ==> adding new entry, may need to grow dictionary.
+   */
+  if(ix < sp->mused) { 
+    mp_int_free(sp->mem[ix]); /* fall through to the end */
+  } 
+  else {
+    if(sp->mused >= sp->mslots) {
+      mp_size nsize = 2 * sp->mslots;
+      mp_int  *tz;
+      char   **tc;
+      int      jx;
+
+      if((tz = malloc(nsize * sizeof(*(sp->mem)))) == NULL)
+	return MP_MEMORY;
+      if((tc = malloc(nsize * sizeof(*(sp->names)))) == NULL) {
+	free(tz);
+	return MP_MEMORY;
+      }
+      
+      for(jx = 0; jx < sp->mused; ++jx) {
+	tz[jx] = sp->mem[jx];
+	tc[jx] = sp->names[jx];
+      }
+      
+      free(sp->mem);
+      sp->mem = tz;
+      free(sp->names);
+      sp->names = tc;
+      
+      sp->mslots = nsize;
+    }
+
+    sp->mused += 1;
+    sp->names[ix] = malloc(1 + strlen(name));
+    strcpy(sp->names[ix], name);
+  }
+
+  sp->mem[ix] = mp_int_alloc();
+  return mp_int_copy(value, sp->mem[ix]);
+}
+
+/* }}} */
+
+/* {{{ mem_recall(*sp, *name, value) */
+
+static mp_result mem_recall(cstate_t *sp, const char *name, mp_int value)
+{
+  int ix;
+
+  for(ix = 0; ix < sp->mused; ++ix) {
+    if(strcmp(name, sp->names[ix]) == 0) {
+      return mp_int_copy(sp->mem[ix], value);
+    }
+  }
+
+  return MP_UNDEF; /* not found */
+}
+
+/* }}} */
+
+/* {{{ mem_clear(*sp) */
+
+static mp_result mem_clear(cstate_t *sp)
+{
+  int ix;
+
+  for(ix = 0; ix < sp->mused; ++ix) {
+    mp_int_free(sp->mem[ix]);
+    free(sp->names[ix]);
+  }
+  sp->mused = 0;
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ cf_abs(sp) */
+
+static mp_result cf_abs(cstate_t *sp)
+{
+  mp_int a = sp->elts[sp->used - 1];
+
+  return mp_int_abs(a, a);
+}
+
+/* }}} */
+
+/* {{{ cf_neg(sp) */
+
+static mp_result cf_neg(cstate_t *sp)
+{
+  mp_int a = sp->elts[sp->used - 1];
+
+  return mp_int_neg(a, a);
+}
+
+/* }}} */
+
+/* {{{ cf_add(sp) */
+
+static mp_result cf_add(cstate_t *sp)
+{
+  mp_int b = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  mp_result res = mp_int_add(a, b, a);
+
+  if(res == MP_OK)
+    stack_pop(sp);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_sub(sp) */
+
+static mp_result cf_sub(cstate_t *sp)
+{
+  mp_int b = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  mp_result res = mp_int_sub(a, b, a);
+
+  if(res == MP_OK)
+    stack_pop(sp);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_mul(sp) */
+
+static mp_result cf_mul(cstate_t *sp)
+{
+  mp_int b = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  mp_result res = mp_int_mul(a, b, a);
+
+  if(res == MP_OK)
+    stack_pop(sp);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_divmod(sp) */
+
+static mp_result cf_divmod(cstate_t *sp)
+{
+  mp_int b = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  
+  return mp_int_div(a, b, a, b);
+}
+
+/* }}} */
+
+/* {{{ cf_div(sp) */
+
+static mp_result cf_div(cstate_t *sp)
+{
+  mp_int b = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  mp_result res = mp_int_div(a, b, a, NULL);
+
+  if(res == MP_OK)
+    stack_pop(sp);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_mod(sp) */
+
+static mp_result cf_mod(cstate_t *sp)
+{
+  mp_int b = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  mp_result res = mp_int_mod(a, b, a);
+
+  if(res == MP_OK)
+    stack_pop(sp);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_expt(sp) */
+
+static mp_result cf_expt(cstate_t *sp)
+{
+  mp_int b = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  mp_result res;
+  mp_small  bval;
+
+  if((res = mp_int_to_int(b, &bval)) != MP_OK)
+    return res;
+
+  stack_pop(sp);
+  return mp_int_expt(a, bval, a);
+}
+
+/* }}} */
+
+/* {{{ cf_exptmod(sp) */
+
+static mp_result cf_exptmod(cstate_t *sp)
+{
+  mp_int m = sp->elts[sp->used - 1];
+  mp_int b = sp->elts[sp->used - 2];
+  mp_int a = sp->elts[sp->used - 3];
+  mp_result res = mp_int_exptmod(a, b, m, a);
+
+  if(res == MP_OK) {
+    stack_pop(sp);
+    stack_pop(sp);
+  }
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_square(sp) */
+
+static mp_result cf_square(cstate_t *sp)
+{
+  mp_int a = sp->elts[sp->used - 1];
+
+  return mp_int_sqr(a, a);
+}
+
+/* }}} */
+
+/* {{{ cf_invmod(sp) */
+
+static mp_result cf_invmod(cstate_t *sp)
+{
+  mp_int m = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  mp_result res = mp_int_invmod(a, m, a);
+
+  stack_pop(sp);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_gcd(sp) */
+
+static mp_result cf_gcd(cstate_t *sp)
+{
+  mp_int b = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  mp_result res = mp_int_gcd(a, b, a);
+
+  if(res == MP_OK)
+    stack_pop(sp);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_xgcd(sp) */
+
+static mp_result cf_xgcd(cstate_t *sp)
+{
+  mp_int b = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  mp_int t;
+  mp_result res;
+
+  if((t = mp_int_alloc()) == NULL) 
+    return MP_MEMORY;
+  if((res = mp_int_egcd(a, b, a, b, t)) != MP_OK) {
+    mp_int_free(t);
+    return res;
+  }
+
+  if((res = stack_push(sp, t)) != MP_OK)
+    mp_int_free(t);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_sqrt(sp) */
+
+static mp_result cf_sqrt(cstate_t *sp)
+{
+  mp_int a = sp->elts[sp->used - 1];
+
+  return mp_int_sqrt(a, a);
+}
+
+/* }}} */
+
+/* {{{ cf_root(sp) */
+
+static mp_result cf_root(cstate_t *sp)
+{
+  mp_int a = sp->elts[sp->used - 2];
+  mp_int bp = sp->elts[sp->used - 1];
+  mp_small b;
+  mp_result res;
+  
+  if((res = mp_int_to_int(bp, &b)) != MP_OK)
+    return res;
+
+  stack_pop(sp);
+  return mp_int_root(a, b, a);
+}
+
+/* }}} */
+
+/* {{{ cf_cmplt(sp) */
+
+static mp_result cf_cmplt(cstate_t *sp)
+{
+  mp_int b = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  mp_result res;
+
+  res = mp_int_set_value(a, (mp_int_compare(a, b) < 0));
+  stack_pop(sp);
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_cmpgt(sp) */
+
+static mp_result cf_cmpgt(cstate_t *sp)
+{
+  mp_int b = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  mp_result res;
+
+  res = mp_int_set_value(a, (mp_int_compare(a, b) > 0));
+  stack_pop(sp);
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_cmple(sp) */
+
+static mp_result cf_cmple(cstate_t *sp)
+{
+  mp_int b = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  mp_result res;
+
+  res = mp_int_set_value(a, (mp_int_compare(a, b) <= 0));
+  stack_pop(sp);
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_cmpge(sp) */
+
+static mp_result cf_cmpge(cstate_t *sp)
+{
+  mp_int b = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  mp_result res;
+
+  res = mp_int_set_value(a, (mp_int_compare(a, b) >= 0));
+  stack_pop(sp);
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_cmpeq(sp) */
+
+static mp_result cf_cmpeq(cstate_t *sp)
+{
+  mp_int b = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  mp_result res;
+
+  res = mp_int_set_value(a, (mp_int_compare(a, b) == 0));
+  stack_pop(sp);
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_cmpne(sp) */
+
+static mp_result cf_cmpne(cstate_t *sp)
+{
+  mp_int b = sp->elts[sp->used - 1];
+  mp_int a = sp->elts[sp->used - 2];
+  mp_result res;
+
+  res = mp_int_set_value(a, (mp_int_compare(a, b) != 0));
+  stack_pop(sp);
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_inc(sp) */
+
+static mp_result cf_inc(cstate_t *sp)
+{
+  mp_int a = sp->elts[sp->used - 1];
+
+  return mp_int_add_value(a, 1, a);
+}
+
+/* }}} */
+
+/* {{{ cf_dec(sp) */
+
+static mp_result cf_dec(cstate_t *sp)
+{
+  mp_int a = sp->elts[sp->used - 1];
+
+  return mp_int_sub_value(a, 1, a);
+}
+
+/* }}} */
+
+/* {{{ cf_fact(sp) */
+
+static mp_result cf_fact(cstate_t *sp)
+{
+  mpz_t  tmp;
+  mp_int x = sp->elts[sp->used - 1];
+  mp_result res = MP_OK;
+
+  if (mp_int_compare_zero(x) < 0)
+    return MP_UNDEF;
+
+  (void) mp_int_init_value(&tmp, 1);
+
+  while (mp_int_compare_value(x, 1) > 0) {
+    if ((res = mp_int_mul(&tmp, x, &tmp)) != MP_OK)
+      goto CLEANUP;
+    if ((res = mp_int_sub_value(x, 1, x)) != MP_OK)
+      goto CLEANUP;
+  }
+
+  res = mp_int_copy(&tmp, x);
+
+ CLEANUP:
+  mp_int_clear(&tmp);
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_pprint(sp) */
+
+static mp_result cf_pprint(cstate_t *sp)
+{
+  print_value(sp->elts[sp->used - 1]);
+  stack_pop(sp);
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ cf_print(sp) */
+
+static mp_result cf_print(cstate_t *sp)
+{
+  print_value(sp->elts[sp->used - 1]);
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ cf_pstack(sp) */
+
+static mp_result cf_pstack(cstate_t *sp)
+{
+  int ix;
+
+  if(sp->used == 0) {
+    fprintf(g_output_file, "<stack empty>\n");
+  }
+  else {
+    for(ix = 0; ix < sp->used; ++ix) {
+      fprintf(g_output_file, "%2d: ", ix);
+      print_value(sp->elts[sp->used - 1 - ix]);
+    }
+  }
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ cf_clstk(sp) */
+
+static mp_result cf_clstk(cstate_t *sp)
+{
+  stack_flush(sp);
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ cf_pop(sp) */
+
+static mp_result cf_pop(cstate_t *sp)
+{
+  return stack_pop(sp);
+}
+
+/* }}} */
+
+/* {{{ cf_dup(sp) */
+
+static mp_result cf_dup(cstate_t *sp)
+{
+  mp_int cp = mp_int_alloc();
+  mp_result res;
+
+  if(cp == NULL)
+    return MP_MEMORY;
+
+  if((res = mp_int_copy(sp->elts[sp->used - 1], cp)) != MP_OK) {
+    mp_int_free(cp);
+    return res;
+  }
+
+  if((res = stack_push(sp, cp)) != MP_OK) 
+    mp_int_free(cp);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ cf_copy(sp) */
+
+static mp_result cf_copy(cstate_t *sp)
+{
+  mp_int n = sp->elts[sp->used - 1];
+  mp_result res;
+  mp_small  ncopy;
+  int       ix;
+
+  if((res = mp_int_to_int(n, &ncopy)) != MP_OK)
+    return res;
+
+  if(ncopy < 1 || ncopy >= sp->used)
+    return MP_RANGE;
+  
+  stack_pop(sp);
+  
+  for(ix = 0; ix < ncopy; ++ix) {
+    mp_int old = sp->elts[sp->used - ncopy];
+    mp_int new = mp_int_alloc();
+
+    if(new == NULL)
+      return MP_MEMORY;
+
+    if((res = mp_int_copy(old, new)) != MP_OK) {
+      mp_int_free(new);
+      return res;
+    }
+    if((res = stack_push(sp, new)) != MP_OK)
+      return res;
+  }
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ cf_swap(sp) */
+
+static mp_result cf_swap(cstate_t *sp)
+{
+  mp_int t = sp->elts[sp->used - 1];
+  
+  sp->elts[sp->used - 1] = sp->elts[sp->used - 2];
+  sp->elts[sp->used - 2] = t;
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ cf_rot(sp) */
+
+static mp_result cf_rot(cstate_t *sp)
+{
+  mp_int t = sp->elts[sp->used - 3];
+
+  sp->elts[sp->used - 3] = sp->elts[sp->used - 2];
+  sp->elts[sp->used - 2] = sp->elts[sp->used - 1];
+  sp->elts[sp->used - 1] = t;
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ cf_pick(sp) */
+
+static mp_result cf_pick(cstate_t *sp)
+{
+  mp_int n = sp->elts[sp->used - 1];
+  mp_result res;
+  mp_small  pos = 0;
+
+  if((res = mp_int_to_int(n, &pos)) != MP_OK)
+    return res;
+
+  if(pos < 0 || pos >= sp->used - 1)
+    return MP_RANGE;
+
+  return mp_int_copy(sp->elts[sp->used - 2 - pos], n);
+}
+
+/* }}} */
+
+/* {{{ cf_setr(sp) */
+
+static mp_result cf_setr(cstate_t *sp)
+{
+  mp_int a = sp->elts[sp->used - 1];
+  mp_result res;
+  mp_small  rdx = 0;
+
+  if((res = mp_int_to_int(a, &rdx)) != MP_OK)
+    return res;
+
+  if(rdx < MP_MIN_RADIX || rdx > MP_MAX_RADIX)
+    return MP_RANGE;
+
+  g_output_radix = rdx;
+  stack_pop(sp);
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ cf_setbin(sp) */
+
+static mp_result cf_setbin(cstate_t *sp)
+{
+  g_output_radix = 0;
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ cf_help(sp) */
+
+static mp_result cf_help(cstate_t *sp)
+{
+  int ix, maxlen = 10; /* minimum width */
+
+  for(ix = 0; g_ops[ix].name != NULL; ++ix) {
+    int len = strlen(g_ops[ix].name);
+
+    if(len > maxlen)
+      maxlen = len;
+  }
+
+  fprintf(stderr, "Operators understood:\n");
+  for(ix = 0; g_ops[ix].name != NULL; ++ix) {
+    int len = strlen(g_ops[ix].name);
+
+    fputs(g_ops[ix].name, stderr);
+    while(len++ <=  maxlen)
+      fputc(' ', stderr);
+
+    fprintf(stderr, "%s\n", g_ops[ix].descript);
+  }
+  fputc('\n', stderr);
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ cf_store(*sp) */
+
+static mp_result cf_store(cstate_t *sp)
+{
+  mp_result res;
+
+  if(next_token(sp->ifp, sp->ibuf, sp->buflen) != t_symbol)
+    return MP_INPUT;
+
+  if((res = mem_insert(sp, sp->ibuf, sp->elts[sp->used - 1])) != MP_OK)
+    return res;
+
+  return stack_pop(sp);
+}
+
+/* }}} */
+
+/* {{{ cf_recall(*sp) */
+
+static mp_result cf_recall(cstate_t *sp)
+{
+  mp_result res;
+  mp_int    val;
+
+  if(next_token(sp->ifp, sp->ibuf, sp->buflen) != t_symbol)
+    return MP_INPUT;
+
+  if((val = mp_int_alloc()) == NULL)
+    return MP_MEMORY;
+  if((res = mem_recall(sp, sp->ibuf, val)) != MP_OK) {
+    mp_int_free(val);
+    return res;
+  }
+
+  return stack_push(sp, val);
+}
+
+/* }}} */
+
+/* {{{ cf_cmem(*sp) */
+
+static mp_result cf_cmem(cstate_t *sp)
+{
+  return mem_clear(sp);
+}
+
+/* }}} */
+
+/* {{{ cf_pmem(*sp) */
+
+static mp_result cf_pmem(cstate_t *sp)
+{
+  int ix, max_len = 0;
+
+  if(sp->mused == 0) {
+    fprintf(g_output_file, "<memory empty>\n");
+    return MP_OK;
+  }
+
+  for(ix = 0; ix < sp->mused; ++ix) {
+    int ln = strlen(sp->names[ix]);
+
+    if(ln > max_len)
+      max_len = ln;
+  }
+
+  max_len += 1; /* allow for a padding space */
+
+  for(ix = 0; ix < sp->mused; ++ix) {
+    int ln = strlen(sp->names[ix]);
+
+    fprintf(g_output_file, "%s:", sp->names[ix]);
+
+    while(ln++ < max_len) 
+      fputc(' ', g_output_file);
+
+    print_value(sp->mem[ix]);
+  }
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ cf_qrecall(*sp) */
+
+static mp_result cf_qrecall(cstate_t *sp)
+{
+  mp_result res;
+  mp_int    val;
+
+  if((val = mp_int_alloc()) == NULL)
+    return MP_MEMORY;
+
+  if((res = mem_recall(sp, sp->ibuf, val)) != MP_OK) {
+    mp_int_free(val); 
+    return res;
+  }
+
+  return stack_push(sp, val);
+}
+
+/* }}} */
+
+/* Here there be dragons */
diff --git a/polly/lib/External/isl/imath/examples/input.c b/polly/lib/External/isl/imath/examples/input.c
new file mode 100644
index 00000000000..6997c3e9321
--- /dev/null
+++ b/polly/lib/External/isl/imath/examples/input.c
@@ -0,0 +1,120 @@
+/*
+  input.c
+
+  Demonstrates basic input and output of arbitrary precision integers using
+  IMath.
+
+  by Michael J. Fromberger <http://spinning-yarns.org/michael/>
+  Copyright (C) 2003-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+
+#include "imrat.h"
+
+int main(int argc, char *argv[])
+{
+  mp_size   radix = 10;  /* Default output radix */
+  mpq_t     value;
+  mp_result res;
+  char      *endp;
+
+  if(argc < 2) {
+    fprintf(stderr, "Usage: input <value> [output-base]\n");
+    return 1;
+  }
+  if(argc > 2) {
+    if((radix = atoi(argv[2])) < MP_MIN_RADIX ||
+       (radix > MP_MAX_RADIX)) {
+      fprintf(stderr, "Error:  Specified radix is out of range (%d)\n",
+	      radix);
+      return 1;
+    }
+  }
+
+  /* Initialize a new value, initially zero; illustrates how to check
+     for errors (e.g., out of memory) and display a message.  */
+  if((res = mp_rat_init(&value)) != MP_OK) {
+    fprintf(stderr, "Error in mp_rat_init(): %s\n", 
+	    mp_error_string(res));
+    return 1;
+  }
+
+  /* Read value in base 10 */
+  if((res = mp_rat_read_ustring(&value, 0, argv[1], &endp)) != MP_OK) {
+    fprintf(stderr, "Error in mp_rat_read_ustring(): %s\n",
+	    mp_error_string(res));
+    
+    if(res == MP_TRUNC)
+      fprintf(stderr, " -- remaining input is: %s\n", endp);
+    
+    mp_rat_clear(&value);
+    return 1;
+  }
+  
+  printf("Here is your value in base %d\n", radix);
+  {
+    mp_result buf_size, res;
+    char *obuf;
+
+    if(mp_rat_is_integer(&value)) {
+      /* Allocate a buffer big enough to hold the given value, including
+	 sign and zero terminator. */
+      buf_size = mp_int_string_len(MP_NUMER_P(&value), radix);
+      obuf = malloc(buf_size);
+
+      /* Convert the value to a string in the desired radix. */
+      if((res = mp_int_to_string(MP_NUMER_P(&value), radix, 
+				 obuf, buf_size)) != MP_OK) {
+	fprintf(stderr, "Converstion to base %d failed: %s\n",
+		radix, mp_error_string(res));
+	mp_rat_clear(&value);
+	return 1;
+      }
+    } 
+    else {
+      /* Allocate a buffer big enough to hold the given value, including
+	 sign and zero terminator. */
+      buf_size = mp_rat_string_len(&value, radix);
+      obuf = malloc(buf_size);
+
+      /* Convert the value to a string in the desired radix. */
+      if((res = mp_rat_to_string(&value, radix, obuf, buf_size)) != MP_OK) {
+	fprintf(stderr, "Conversion to base %d failed: %s\n",
+		radix, mp_error_string(res));
+	mp_rat_clear(&value);
+	return 1;
+      }
+    }
+    fputs(obuf, stdout);
+    fputc('\n', stdout);
+    free(obuf);
+  }
+
+  /* When you are done with a value, it must be "cleared" to release
+     the memory it occupies */
+  mp_rat_clear(&value);
+  return 0;
+}
+
+/* Here there be dragons */
diff --git a/polly/lib/External/isl/imath/examples/randprime.c b/polly/lib/External/isl/imath/examples/randprime.c
new file mode 100644
index 00000000000..84ccdc98c98
--- /dev/null
+++ b/polly/lib/External/isl/imath/examples/randprime.c
@@ -0,0 +1,261 @@
+/*
+  Name:     randprime.c
+  Purpose:  Generate a probable prime at random.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Usage:  randprime [-s] <bits> [<outfile>]
+
+  Generate a randomly-chosen probable prime having <bits> significant bits, and
+  write it to the specified output file or to the standard output.  If the "-s"
+  option is given, a prime p is chosen such that (p - 1) / 2 is also prime.
+
+  A prime is obtained by reading random bits from /dev/random, setting the
+  low-order bit, and testing for primality.  If the first candidate is not
+  prime, successive odd candidates are tried until a probable prime is found.
+
+  Copyright (C) 2002-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+#include <errno.h>
+#include <limits.h>
+
+#include <unistd.h>
+
+#include "imath.h"
+#include "iprime.h"
+
+/* Load the specified buffer with random bytes */
+int randomize(unsigned char *buf, size_t len);
+
+/* Overwrite the specified value with n_bits random bits */
+mp_result mp_int_randomize(mp_int a, mp_size n_bits);
+
+/* Find a prime starting from the given odd seed */
+mp_result find_prime(mp_int seed, FILE *fb);
+mp_result find_strong_prime(mp_int seed, FILE *fb);
+
+typedef mp_result (*find_f)(mp_int, FILE *);
+
+int main(int argc, char *argv[])
+{
+  int       opt, modbits;
+  FILE     *ofp = stdout;
+  mp_result res;
+  find_f    find_func = find_prime;
+  char      tag = 'p';
+  mpz_t     value;
+
+  /* Process command-line arguments */
+  while((opt = getopt(argc, argv, "s")) != EOF) {
+    switch(opt) {
+    case 's':
+      find_func = find_strong_prime;
+      tag = 'P';
+      break;
+    default:
+      fprintf(stderr, "Usage: randprime [-s] <bits> [<outfile>]\n");
+      return 1;
+    }
+  }
+  
+  if(optind >= argc) {
+    fprintf(stderr, "Error:  You must specify the number of significant bits.\n");
+    fprintf(stderr, "Usage: randprime [-s] <bits> [<outfile>]\n");
+    return 1;
+  }
+  modbits = (int) strtol(argv[optind++], NULL, 0);
+  if(modbits < CHAR_BIT) {
+    fprintf(stderr, "Error:  Invalid value for number of significant bits.\n");
+    return 1;
+  }
+  if(modbits % 2 == 1)
+    ++modbits;
+
+  /* Check if output file is specified */
+  if(optind < argc) {
+    if((ofp = fopen(argv[optind], "wt")) == NULL) {
+      fprintf(stderr, "Error:  Unable to open output file for writing.\n"
+	      " - Filename: %s\n"
+	      " - Error:    %s\n", argv[optind], strerror(errno));
+      return 1;
+    }
+  }
+  
+  mp_int_init(&value);
+  if ((res = mp_int_randomize(&value, modbits - 1)) != MP_OK) {
+    fprintf(stderr, "Error:  Unable to generate random start value.\n"
+	    " - %s (%d)\n", mp_error_string(res), res);
+    goto EXIT;
+  }
+  fprintf(stderr, "%c: ", tag);
+  find_func(&value, stderr);
+  fputc('\n', stderr);
+
+  /* Write the completed value to the specified output file */
+  {
+    int len;
+    char *obuf;
+
+    len = mp_int_string_len(&value, 10);
+    obuf = malloc(len);
+    mp_int_to_string(&value, 10, obuf, len);
+    fputs(obuf, ofp);
+    fputc('\n', ofp);
+
+    free(obuf);
+  }
+
+ EXIT:
+  fclose(ofp);
+  mp_int_clear(&value);
+  return 0;
+}
+
+/* {{{ randomize(buf, len) */
+
+int randomize(unsigned char *buf, size_t len)
+{
+  FILE *rnd = fopen("/dev/random", "rb");
+  size_t nr;
+
+  if(rnd == NULL)
+    return -1;
+  
+  nr = fread(buf, sizeof(*buf), len, rnd);
+  fclose(rnd);
+  
+  return (int) nr;
+}
+
+/* }}} */
+
+/* {{{ mp_int_randomize(a, n_bits) */
+
+mp_result mp_int_randomize(mp_int a, mp_size n_bits)
+{
+  mp_size n_bytes = (n_bits + CHAR_BIT - 1) / CHAR_BIT;
+  unsigned char *buf;
+  mp_result res = MP_OK;
+  
+  if((buf = malloc(n_bytes)) == NULL)
+    return MP_MEMORY;
+  
+  if(randomize(buf, n_bytes) != n_bytes) {
+    res = MP_TRUNC;
+    goto CLEANUP;
+  }
+
+  /* Clear bits beyond the number requested */
+  if(n_bits % CHAR_BIT != 0) {
+    unsigned char b_mask = (1 << (n_bits % CHAR_BIT)) - 1;
+    unsigned char t_mask = (1 << (n_bits % CHAR_BIT)) >> 1;
+
+    buf[0] &= b_mask;
+    buf[0] |= t_mask;
+  }
+
+  /* Set low-order bit to insure value is odd */
+  buf[n_bytes - 1] |= 1;
+
+  res = mp_int_read_unsigned(a, buf, n_bytes);
+
+ CLEANUP:
+  memset(buf, 0, n_bytes);
+  free(buf);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ find_prime(seed, fb) */
+
+mp_result find_prime(mp_int seed, FILE *fb)
+{
+  mp_result res;
+  int       count = 0;
+
+  if(mp_int_is_even(seed))
+    if((res = mp_int_add_value(seed, 1, seed)) != MP_OK)
+      return res;
+  
+  while((res = mp_int_is_prime(seed)) == MP_FALSE) {
+    ++count;
+
+    if(fb != NULL && (count % 50) == 0)
+      fputc('.', fb);
+
+    if((res = mp_int_add_value(seed, 2, seed)) != MP_OK)
+      return res;
+  }
+
+  if(res == MP_TRUE && fb != NULL)
+    fputc('+', fb);
+  
+  return res;
+}
+
+/* }}} */
+
+/* {{{ find_strong_prime(seed, fb) */
+
+mp_result find_strong_prime(mp_int seed, FILE *fb)
+{
+  mp_result res;
+  mpz_t     t;
+
+  mp_int_init(&t);
+  for(;;) {
+    if ((res = find_prime(seed, fb)) != MP_TRUE)
+      break;
+    if ((res = mp_int_copy(seed, &t)) != MP_OK)
+      break;
+
+    if ((res = mp_int_mul_pow2(&t, 1, &t)) != MP_OK ||
+	(res = mp_int_add_value(&t, 1, &t)) != MP_OK) 
+      break;
+
+    if ((res = mp_int_is_prime(&t)) == MP_TRUE) {
+      if (fb != NULL)
+	fputc('!', fb);
+
+      res = mp_int_copy(&t, seed);
+      break;
+    }
+    else if (res != MP_FALSE) 
+      break;
+
+    if (fb != NULL)
+      fputc('x', fb);
+    if ((res = mp_int_add_value(seed, 2, seed)) != MP_OK)
+      break;
+  }
+
+  mp_int_clear(&t);
+  return res;
+}
+
+/* }}} */
+
+/* Here there be dragons */
diff --git a/polly/lib/External/isl/imath/examples/rounding.c b/polly/lib/External/isl/imath/examples/rounding.c
new file mode 100644
index 00000000000..674b2bd39a4
--- /dev/null
+++ b/polly/lib/External/isl/imath/examples/rounding.c
@@ -0,0 +1,83 @@
+/*
+  Name:     rounding.c
+  Purpose:  Demonstrates rounding modes.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Bugs:  The rounding mode can only be specified by value, not name.
+
+  Copyright (C) 2002-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+#include <limits.h>
+
+#include "imath.h"
+#include "imrat.h"
+
+int main(int argc, char *argv[])
+{
+  mp_result  mode, len, res = 0;
+  mp_size    prec, radix;
+  mpq_t      value;
+  char      *buf;
+
+  if(argc < 5) {
+    fprintf(stderr, "Usage: rounding <mode> <precision> <radix> <value>\n");
+    return 1;
+  }
+
+  if((res = mp_rat_init(&value)) != MP_OK) {
+    fprintf(stderr, "Error initializing: %s\n", mp_error_string(res));
+    return 2;
+  }
+
+  mode = atoi(argv[1]);
+  prec = atoi(argv[2]);
+  radix = atoi(argv[3]);
+  
+  printf("Rounding mode:   %d\n"
+	 "Precision:       %u digits\n"
+	 "Radix:           %u\n"
+	 "Input string:    \"%s\"\n", mode, prec, radix, argv[4]);
+  
+  if((res = mp_rat_read_decimal(&value, radix, argv[4])) != MP_OK) {
+    fprintf(stderr, "Error reading input string: %s\n", 
+	    mp_error_string(res));
+    goto CLEANUP;
+  }
+
+  len = mp_rat_decimal_len(&value, radix, prec);
+  buf = malloc(len);
+
+  if((res = mp_rat_to_decimal(&value, radix, prec, mode, buf, len)) != MP_OK)
+    fprintf(stderr, "Error converting output: %s\n",
+	    mp_error_string(res));
+
+  printf("Result string:   \"%s\"\n", buf);
+  free(buf);
+
+ CLEANUP:
+  mp_rat_clear(&value);
+  return res;
+}
+
+/* Here there be dragons */
diff --git a/polly/lib/External/isl/imath/examples/rsakey.c b/polly/lib/External/isl/imath/examples/rsakey.c
new file mode 100644
index 00000000000..34190fa414f
--- /dev/null
+++ b/polly/lib/External/isl/imath/examples/rsakey.c
@@ -0,0 +1,325 @@
+/*
+  Name:     rsakey.c
+  Purpose:  Generate keys for the RSA cryptosystem.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Usage:  rsakey [-e <expt>] <modbits> [<outfile>]
+
+  Generates an RSA key pair with a modulus having <modbits> significant bits,
+  and writes it to the specified output file, or to the standard output.  The
+  -e option allows the user to specify an encryption exponent; otherwise, an
+  encryption exponent is chosen at random.
+  
+  Primes p and q are obtained by reading random bits from /dev/random, setting
+  the low-order bit, and testing for primality.  If the first candidate is not
+  prime, successive odd candidates are tried until a probable prime is found.
+
+  Copyright (C) 2002-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+#include <errno.h>
+#include <limits.h>
+
+#include <unistd.h>
+
+#include "imath.h"
+#include "iprime.h"
+
+typedef struct {
+  mpz_t   p;
+  mpz_t   q;
+  mpz_t   n;
+  mpz_t   e;
+  mpz_t   d;
+} rsa_key;
+
+/* Load the specified buffer with random bytes */
+int randomize(unsigned char *buf, size_t len);
+
+/* Overwrite the specified value with n_bits random bits */
+mp_result mp_int_randomize(mp_int a, mp_size n_bits);
+
+/* Find a prime starting from the given odd seed */
+mp_result find_prime(mp_int seed, FILE *fb);
+
+/* Initialize/destroy an rsa_key structure */
+mp_result rsa_key_init(rsa_key *kp);
+void      rsa_key_clear(rsa_key *kp);
+void      rsa_key_write(rsa_key *kp, FILE *ofp);
+
+int main(int argc, char *argv[])
+{
+  int       opt, modbits;
+  FILE     *ofp = stdout;
+  char     *expt = NULL;
+  rsa_key   the_key;
+  mp_result res;
+
+  /* Process command-line arguments */
+  while((opt = getopt(argc, argv, "e:")) != EOF) {
+    switch(opt) {
+    case 'e':
+      expt = optarg;
+      break;
+    default:
+      fprintf(stderr, "Usage: rsakey [-e <expt>] <modbits> [<outfile>]\n");
+      return 1;
+    }
+  }
+  
+  if(optind >= argc) {
+    fprintf(stderr, "Error:  You must specify the number of modulus bits.\n");
+    fprintf(stderr, "Usage: rsakey [-e <expt>] <modbits> [<outfile>]\n");
+    return 1;
+  }
+  modbits = (int) strtol(argv[optind++], NULL, 0);
+  if(modbits < CHAR_BIT) {
+    fprintf(stderr, "Error:  Invalid value for number of modulus bits.\n");
+    return 1;
+  }
+  if(modbits % 2 == 1)
+    ++modbits;
+
+  /* Check if output file is specified */
+  if(optind < argc) {
+    if((ofp = fopen(argv[optind], "wt")) == NULL) {
+      fprintf(stderr, "Error:  Unable to open output file for writing.\n"
+	      " - Filename: %s\n"
+	      " - Error:    %s\n", argv[optind], strerror(errno));
+      return 1;
+    }
+  }
+  
+  if((res = rsa_key_init(&the_key)) != MP_OK) {
+    fprintf(stderr, "Error initializing RSA key structure:\n"
+	    " - %s (%d)\n", mp_error_string(res), res);
+    return 1;
+  }
+
+  /* If specified, try to load the key exponent */
+  if(expt != NULL) {
+    if((res = mp_int_read_string(&(the_key.e), 10, expt)) != MP_OK) {
+      fprintf(stderr, "Error:  Invalid value for encryption exponent.\n"
+	      " - %s (%d)\n", mp_error_string(res), res);
+      goto EXIT;
+    }
+  }
+
+  if((res = mp_int_randomize(&(the_key.p), (modbits / 2))) != MP_OK) {
+    fprintf(stderr, "Error:  Unable to randomize first prime.\n"
+	    " - %s (%d)\n", mp_error_string(res), res);
+    goto EXIT;
+  }
+  fprintf(stderr, "p: ");
+  find_prime(&(the_key.p), stderr);
+
+  if((res = mp_int_randomize(&(the_key.q), (modbits / 2))) != MP_OK) {
+    fprintf(stderr, "Error:  Unable to randomize second prime.\n"
+	    " - %s (%d)\n", mp_error_string(res), res);
+    goto EXIT;
+  }
+  fprintf(stderr, "\nq: ");
+  find_prime(&(the_key.q), stderr);
+  fputc('\n', stderr);
+
+  /* Temporarily, the key's "n" field will be (p - 1) * (q - 1) for
+     purposes of computing the decryption exponent.
+   */
+  mp_int_mul(&(the_key.p), &(the_key.q), &(the_key.n));
+  mp_int_sub(&(the_key.n), &(the_key.p), &(the_key.n));
+  mp_int_sub(&(the_key.n), &(the_key.q), &(the_key.n));
+  mp_int_add_value(&(the_key.n), 1, &(the_key.n));
+
+  if(expt == NULL &&
+     (res = mp_int_randomize(&(the_key.e), (modbits / 2))) != MP_OK) {
+    fprintf(stderr, "Error:  Unable to randomize encryption exponent.\n"
+	    " - %s (%d)\n", mp_error_string(res), res);
+    goto EXIT;
+  }
+  while((res = mp_int_invmod(&(the_key.e), &(the_key.n), 
+			     &(the_key.d))) != MP_OK) {
+    if(expt != NULL) {
+      fprintf(stderr, "Error:  Unable to compute decryption exponent.\n"
+	      " - %s (%d)\n", mp_error_string(res), res);
+      goto EXIT;
+    }
+    if((res = mp_int_randomize(&(the_key.e), (modbits / 2))) != MP_OK) {
+      fprintf(stderr, "Error:  Unable to re-randomize encryption exponent.\n"
+	      " - %s (%d)\n", mp_error_string(res), res);
+      goto EXIT;
+    }
+  }
+
+  /* Recompute the real modulus, now that exponents are done. */
+  mp_int_mul(&(the_key.p), &(the_key.q), &(the_key.n));
+
+  /* Write completed key to the specified output file */
+  rsa_key_write(&the_key, ofp);
+
+ EXIT:
+  fclose(ofp);
+  rsa_key_clear(&the_key);
+  return 0;
+}
+
+/* {{{ randomize(buf, len) */
+
+int randomize(unsigned char *buf, size_t len)
+{
+  FILE *rnd = fopen("/dev/random", "rb");
+  size_t nr;
+
+  if(rnd == NULL)
+    return -1;
+  
+  nr = fread(buf, sizeof(*buf), len, rnd);
+  fclose(rnd);
+  
+  return (int) nr;
+}
+
+/* }}} */
+
+/* {{{ mp_int_randomize(a, n_bits) */
+
+mp_result mp_int_randomize(mp_int a, mp_size n_bits)
+{
+  mp_size n_bytes = (n_bits + CHAR_BIT - 1) / CHAR_BIT;
+  unsigned char *buf;
+  mp_result res = MP_OK;
+  
+  if((buf = malloc(n_bytes)) == NULL)
+    return MP_MEMORY;
+  
+  if(randomize(buf, n_bytes) != n_bytes) {
+    res = MP_TRUNC;
+    goto CLEANUP;
+  }
+
+  /* Clear bits beyond the number requested */
+  if(n_bits % CHAR_BIT != 0) {
+    unsigned char b_mask = (1 << (n_bits % CHAR_BIT)) - 1;
+    unsigned char t_mask = (1 << (n_bits % CHAR_BIT)) >> 1;
+
+    buf[0] &= b_mask;
+    buf[0] |= t_mask;
+  }
+
+  /* Set low-order bit to insure value is odd */
+  buf[n_bytes - 1] |= 1;
+
+  res = mp_int_read_unsigned(a, buf, n_bytes);
+
+ CLEANUP:
+  memset(buf, 0, n_bytes);
+  free(buf);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ find_prime(seed, fb) */
+
+mp_result find_prime(mp_int seed, FILE *fb)
+{
+  mp_result res;
+  int       count = 0;
+
+  if(mp_int_is_even(seed))
+    if((res = mp_int_add_value(seed, 1, seed)) != MP_OK)
+      return res;
+  
+  while((res = mp_int_is_prime(seed)) == MP_FALSE) {
+    ++count;
+
+    if(fb != NULL && (count % 50) == 0)
+      fputc('.', fb);
+
+    if((res = mp_int_add_value(seed, 2, seed)) != MP_OK)
+      return res;
+  }
+
+  if(res == MP_TRUE && fb != NULL)
+    fputc('+', fb);
+  
+  return res;
+}
+
+/* }}} */
+
+/* {{{ rsa_key_init(kp) */
+
+mp_result rsa_key_init(rsa_key *kp)
+{
+  mp_int_init(&(kp->p));
+  mp_int_init(&(kp->q));
+  mp_int_init(&(kp->n));
+  mp_int_init(&(kp->e));
+  mp_int_init(&(kp->d));
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ rsa_key_clear(kp) */
+
+void      rsa_key_clear(rsa_key *kp)
+{
+  mp_int_clear(&(kp->p));
+  mp_int_clear(&(kp->q));
+  mp_int_clear(&(kp->n));
+  mp_int_clear(&(kp->e));
+  mp_int_clear(&(kp->d));  
+}
+
+/* }}} */
+
+/* {{{ rsa_key_write(kp, ofp) */
+
+void      rsa_key_write(rsa_key *kp, FILE *ofp)
+{
+  int   len;
+  char *obuf;
+
+  len = mp_int_string_len(&(kp->n), 10);
+  obuf = malloc(len);
+  mp_int_to_string(&(kp->p), 10, obuf, len);
+  fprintf(ofp, "p = %s\n", obuf);
+  mp_int_to_string(&(kp->q), 10, obuf, len);
+  fprintf(ofp, "q = %s\n", obuf);
+  mp_int_to_string(&(kp->e), 10, obuf, len);
+  fprintf(ofp, "e = %s\n", obuf);
+  mp_int_to_string(&(kp->d), 10, obuf, len);
+  fprintf(ofp, "d = %s\n", obuf);
+  mp_int_to_string(&(kp->n), 10, obuf, len);
+  fprintf(ofp, "n = %s\n", obuf);
+
+  free(obuf);
+}
+
+/* }}} */
+
+/* Here there be dragons */
diff --git a/polly/lib/External/isl/imath/findsizes.py b/polly/lib/External/isl/imath/findsizes.py
new file mode 100755
index 00000000000..fc270b38761
--- /dev/null
+++ b/polly/lib/External/isl/imath/findsizes.py
@@ -0,0 +1,174 @@
+#!/usr/bin/env python
+##
+## Name:     findsizes.py
+## Purpose:  Find acceptable digit and word types for IMath.
+## Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+##
+##  Copyright (C) 2002-2008 Michael J. Fromberger, All Rights Reserved.
+##
+##  Permission is hereby granted, free of charge, to any person obtaining a
+##  copy of this software and associated documentation files (the "Software"),
+##  to deal in the Software without restriction, including without limitation
+##  the rights to use, copy, modify, merge, publish, distribute, sublicense,
+##  and/or sell copies of the Software, and to permit persons to whom the
+##  Software is furnished to do so, subject to the following conditions:
+##
+##  The above copyright notice and this permission notice shall be included in
+##  all copies or substantial portions of the Software.
+##
+##  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+##  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+##  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL
+##  THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+##  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
+##  FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
+##  DEALINGS IN THE SOFTWARE.
+##
+import getopt, os, re, subprocess, sys, tempfile
+
+# These are the type names to test for suitability.  If your compiler
+# does not support "long long", e.g., it is strict ANSI C90, then you
+# should remove "long long" from this list.
+try_types = [ "unsigned %s" % s for s in
+              ("char", "short", "int", "long", "long long") ]
+
+def main(args):
+    """Scan the Makefile to find appropriate compiler settings, and then
+    compile a test program to emit the sizes of the various types that are
+    considered candidates.  The -L (--nolonglong) command line option disables
+    the use of the"long long" type, which is not standard ANSI C90; by default,
+    "long long" is considered.
+    """
+    try:
+        opts, args = getopt.getopt(args, 'L', ('nolonglong',))
+    except getopt.GetoptError, e:
+        print >> sys.stderr, "Usage: findsizes.py [-L/--nolonglong]"
+        sys.exit(1)
+
+    for opt, arg in opts:
+        if opt in ('-L', '--nolonglong'):
+            try:
+                try_types.pop(try_types.index("unsigned long long"))
+            except ValueError: pass
+    
+    vars = get_make_info()
+    sizes = get_type_sizes(try_types, vars.get('CC', 'cc'),
+                           vars.get('CFLAGS', '').split())
+    
+    stypes = sorted(sizes.keys(), key = lambda k: sizes[k], reverse = True)
+    word_type = stypes[0]
+    for t in stypes[1:]:
+        if sizes[t] <= sizes[word_type] / 2:
+            digit_type = t
+            break
+    else:
+        print >> sys.stderr, "Unable to find a compatible digit type."
+        sys.exit(1)
+
+    print "typedef %-20s mp_word;   /* %d bytes */\n" \
+          "typedef %-20s mp_digit;  /* %d bytes */" % \
+          (word_type, sizes[word_type], digit_type, sizes[digit_type])
+
+def get_type_sizes(types, cc, cflags = ()):
+    """Return a dictionary mapping the names of the specified types to
+    their sizes in bytes, based on the output of the C compiler whose
+    path and arguments are given.
+    """
+    fd, tpath = tempfile.mkstemp(suffix = '.c')
+    fp = os.fdopen(fd, 'r+')
+    fp.seek(0)
+    
+    fp.write("#include <stdio.h>\n\nint main(void)\n{\n")
+    for t in types:
+        fp.write('  printf("%%lu\\t%%s\\n", (unsigned long) sizeof(%s), '
+                 '\"%s\");\n' % (t, t))
+    fp.write('\n  return 0;\n}\n')
+    fp.close()
+
+    print >> sys.stderr, \
+          "Compiler:  %s\n" \
+          "Flags:     %s\n" \
+          "Source:    %s\n" % (cc, ' '.join(cflags), tpath)
+    
+    cmd = [cc] + list(cflags) + [tpath]
+    if subprocess.call(cmd) <> 0:
+        raise ValueError("Error while running '%s'" % ' '.join(cmd))
+
+    os.unlink(tpath)
+    if not os.path.isfile('a.out'):
+        raise ValueError("No executable a.out found")
+
+    result = subprocess.Popen(['./a.out'],
+                              stdout = subprocess.PIPE).communicate()[0]
+
+    out = {}
+    for line in result.split('\n'):
+        if line.strip() == '':
+            continue
+        size, type = re.split(r'\s+', line, 1)
+        out[type] = int(size)
+    
+    os.unlink("a.out")
+    return out
+
+def sub_make_vars(input, vars):
+    """Perform Make style variable substitution in the given input
+    string, using vars as a dictionary of variables to substitute.
+    """
+    def frep(m):
+        try:
+            return vars[m.group(1).strip()]
+        except KeyError:
+            return ' '
+    
+    expr = re.compile(r'\$\((\s*\w+\s*)\)')
+    out = input
+    while True:
+        next = expr.sub(frep, out)
+        if next == out:
+            break
+        out = next
+    
+    return out
+
+def get_make_info(target = None, makefile = None, makepath = "make",
+                  defs = ()):
+    """Extract a listing of all of the variables defined by Make.
+    Returns a dictionary mapping variable names to their string
+    values.
+
+    Optional arguments:
+    target   -- the name of the target to request that Make build.
+    makefile -- the path to the Makefile.
+    makepath -- the path to the make executable.
+    defs     -- a sequence of strings, additional make arguments.
+    """
+    cmd = [makepath]
+    if defs:
+        cmd.extend(defs)
+    cmd.extend(("-p", "-n"))
+    if makefile is not None:
+        cmd.extend(["-f", makefile])
+    if target is not None:
+        cmd.append(target)
+    
+    output = subprocess.Popen(cmd,
+                              stdout = subprocess.PIPE,
+                              stderr = subprocess.PIPE).communicate()[0]
+    vrule = re.compile(r'^(\w+)\s*=\s*(.*)$')
+    
+    vars = {}
+    for line in output.split('\n'):
+        m = vrule.match(line)
+        if m:
+            vars[m.group(1)] = m.group(2)
+
+    for key, val in vars.iteritems():
+        vars[key] = sub_make_vars(val, vars)
+    
+    return vars
+
+if __name__ == "__main__":
+    main(sys.argv[1:])
+
+# Here there be dragons
diff --git a/polly/lib/External/isl/imath/findthreshold.py b/polly/lib/External/isl/imath/findthreshold.py
new file mode 100755
index 00000000000..daf3de6580b
--- /dev/null
+++ b/polly/lib/External/isl/imath/findthreshold.py
@@ -0,0 +1,89 @@
+#!/usr/bin/env python
+##
+## Name:     findthreshold.py
+## Purpose:  Find a good threshold for recursive multiplication.
+## Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+## Info:     $Id: findthreshold.py 456 2006-08-16 14:24:06Z sting $
+##
+## This tool computes some timing statistics to help you select a suitable
+## recursive multiplication breakpoint.  It uses the imtimer tool to run a
+## series of tests varying precision and breakpoint, and prints out a summary
+## of the "best" values for each category.  Each summary line contains the
+## following fields, tab-separated:
+##
+## prec     -- the precision of the operands (in digits).
+## thresh   -- the threshold for recursive multiplication (digits).
+## trec     -- total time using recursive algorithm (sec).
+## tnorm    -- total time without recursive algorithm (sec).
+## ratio    -- speedup (ratio of tnorm/trec).
+##
+## You are responsible for reading and interpreting the resulting table to
+## obtain a useful value for your workload.  Change the value of MP_MULT_THRESH
+## in imath.h once you have a satisfactory result.
+##
+import math, os, random, sys, time
+
+def get_timing_stats(num_tests, precision, threshold, seed = None):
+    """Obtain timing statistics for multiplication.
+
+    num_tests      -- number of tests to run.
+    precision      -- number of digits per operand.
+    threshold      -- threshold in digits for recursive multiply.
+    seed           -- random seed; if None, the clock is used.
+
+    Returns a tuple of (seed, bits, time) where seed is the random seed used,
+    bits is the number of bits per operand, and time is a float giving the
+    total time taken for the test run.
+    """
+    if seed is None:
+        seed = int(time.time())
+    
+    line = os.popen('./imtimer -mn -p %d -t %d -s %d %d' %
+                    (precision, threshold, seed, num_tests), 'r').readline()
+    
+    count, prec, bits, thresh, status = line.strip().split('\t')
+    kind, total, unit = status.split()
+
+    return seed, int(bits), float(total)
+
+def check_binary(name):
+    if not os.path.exists(name):
+        os.system('make %s' % name)
+        if not os.path.exists(name):
+            raise ValueError("Unable to build %s" % name)
+    elif not os.path.isfile(name):
+        raise ValueError("Path exists with wrong type")
+
+def compute_stats():
+    check_binary('imtimer')
+    seed = int(time.time())
+
+    print >> sys.stderr, "Computing timer statistics (this may take a while)"
+    stats = {}
+    for prec in (32, 40, 64, 80, 128, 150, 256, 384, 512, 600, 768, 1024):
+        sys.stderr.write('%-4d ' % prec)
+        stats[prec] = (None, 1000000., 0.)
+    
+        for thresh in xrange(16, 65, 2):
+            s, b, t = get_timing_stats(1000, prec, thresh, seed)
+            sp, bp, tp = get_timing_stats(1000, prec, prec + 1, seed)
+            
+            if t < stats[prec][1]:
+                stats[prec] = (thresh, t, tp)
+                sys.stderr.write('+')
+            else:
+                sys.stderr.write('.')
+        sys.stderr.write('\n')
+
+    return list((p, h, t, tp) for p, (h, t, tp) in stats.iteritems())
+
+if __name__ == "__main__":
+    stats = compute_stats()
+    stats.sort(key = lambda s: s[3]/s[2])
+    for prec, thresh, trec, tnorm in stats:
+        print "%d\t%d\t%.3f\t%.3f\t%.4f" % (prec, thresh, trec, tnorm,
+                                            tnorm / trec)
+
+    print
+
+# Here there be dragons
diff --git a/polly/lib/External/isl/imath/gmp_compat.c b/polly/lib/External/isl/imath/gmp_compat.c
new file mode 100644
index 00000000000..5d89b3097da
--- /dev/null
+++ b/polly/lib/External/isl/imath/gmp_compat.c
@@ -0,0 +1,861 @@
+/*
+  Name:     gmp_compat.c
+  Purpose:  Provide GMP compatiable routines for imath library
+  Author:   David Peixotto
+
+  Copyright (c) 2012 Qualcomm Innovation Center, Inc. All rights reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+#include "gmp_compat.h"
+#include <stdlib.h>
+#include <assert.h>
+#include <ctype.h>
+#include <string.h>
+
+#ifdef  NDEBUG
+#define CHECK(res) (res)
+#else
+#define CHECK(res) assert(((res) == MP_OK) && "expected MP_OK")
+#endif
+
+/*************************************************************************
+ *
+ * Functions with direct translations
+ *
+ *************************************************************************/
+/* gmp: mpq_clear */
+void GMPQAPI(clear)(mp_rat x) {
+  mp_rat_clear(x);
+}
+
+/* gmp: mpq_cmp */
+int GMPQAPI(cmp)(mp_rat op1, mp_rat op2) {
+  return mp_rat_compare(op1, op2);
+}
+
+/* gmp: mpq_init */
+void GMPQAPI(init)(mp_rat x) {
+  CHECK(mp_rat_init(x));
+}
+
+/* gmp: mpq_mul */
+void GMPQAPI(mul)(mp_rat product, mp_rat multiplier, mp_rat multiplicand) {
+  CHECK(mp_rat_mul(multiplier, multiplicand, product));
+}
+
+/* gmp: mpq_set*/
+void GMPQAPI(set)(mp_rat rop, mp_rat op) {
+  CHECK(mp_rat_copy(op, rop));
+}
+
+/* gmp: mpz_abs */
+void GMPZAPI(abs)(mp_int rop, mp_int op) {
+  CHECK(mp_int_abs(op, rop));
+}
+
+/* gmp: mpz_add */
+void GMPZAPI(add)(mp_int rop, mp_int op1, mp_int op2) {
+  CHECK(mp_int_add(op1, op2, rop));
+}
+
+/* gmp: mpz_clear */
+void GMPZAPI(clear)(mp_int x) {
+  mp_int_clear(x);
+}
+
+/* gmp: mpz_cmp_si */
+int GMPZAPI(cmp_si)(mp_int op1, long op2) {
+  return mp_int_compare_value(op1, op2);
+}
+
+/* gmp: mpz_cmpabs */
+int GMPZAPI(cmpabs)(mp_int op1, mp_int op2) {
+  return mp_int_compare_unsigned(op1, op2);
+}
+
+/* gmp: mpz_cmp */
+int GMPZAPI(cmp)(mp_int op1, mp_int op2) {
+  return mp_int_compare(op1, op2);
+}
+
+/* gmp: mpz_init */
+void GMPZAPI(init)(mp_int x) {
+  CHECK(mp_int_init(x));
+}
+
+/* gmp: mpz_mul */
+void GMPZAPI(mul)(mp_int rop, mp_int op1, mp_int op2) {
+  CHECK(mp_int_mul(op1, op2, rop));
+}
+
+/* gmp: mpz_neg */
+void GMPZAPI(neg)(mp_int rop, mp_int op) {
+  CHECK(mp_int_neg(op, rop));
+}
+
+/* gmp: mpz_set_si */
+void GMPZAPI(set_si)(mp_int rop, long op) {
+  CHECK(mp_int_set_value(rop, op));
+}
+
+/* gmp: mpz_set */
+void GMPZAPI(set)(mp_int rop, mp_int op) {
+  CHECK(mp_int_copy(op, rop));
+}
+
+/* gmp: mpz_sub */
+void GMPZAPI(sub)(mp_int rop, mp_int op1, mp_int op2) {
+  CHECK(mp_int_sub(op1, op2, rop));
+}
+
+/* gmp: mpz_swap */
+void GMPZAPI(swap)(mp_int rop1, mp_int rop2) {
+  mp_int_swap(rop1, rop2);
+}
+
+/* gmp: mpq_sgn */
+int GMPQAPI(sgn)(mp_rat op) {
+  return mp_rat_compare_zero(op);
+}
+
+/* gmp: mpz_sgn */
+int GMPZAPI(sgn)(mp_int op) {
+  return mp_int_compare_zero(op);
+}
+
+/* gmp: mpq_set_ui */
+void GMPQAPI(set_ui)(mp_rat rop, unsigned long op1, unsigned long op2) {
+  CHECK(mp_rat_set_uvalue(rop, op1, op2));
+}
+
+/* gmp: mpz_set_ui */
+void GMPZAPI(set_ui)(mp_int rop, unsigned long op) {
+  CHECK(mp_int_set_uvalue(rop, op));
+}
+
+/* gmp: mpq_den_ref */
+mp_int GMPQAPI(denref)(mp_rat op) {
+  return mp_rat_denom_ref(op);
+}
+
+/* gmp: mpq_num_ref */
+mp_int GMPQAPI(numref)(mp_rat op) {
+  return mp_rat_numer_ref(op);
+}
+
+/* gmp: mpq_canonicalize */
+void GMPQAPI(canonicalize)(mp_rat op) {
+  CHECK(mp_rat_reduce(op));
+}
+
+/*************************************************************************
+ *
+ * Functions that can be implemented as a combination of imath functions
+ *
+ *************************************************************************/
+/* gmp: mpz_addmul */
+/* gmp: rop = rop + (op1 * op2) */
+void GMPZAPI(addmul)(mp_int rop, mp_int op1, mp_int op2) {
+  mpz_t tempz;
+  mp_int temp = &tempz;
+  mp_int_init(temp);
+
+  CHECK(mp_int_mul(op1, op2, temp));
+  CHECK(mp_int_add(rop, temp, rop));
+  mp_int_clear(temp);
+}
+
+/* gmp: mpz_divexact */
+/* gmp: only produces correct results when d divides n */
+void GMPZAPI(divexact)(mp_int q, mp_int n, mp_int d) {
+  CHECK(mp_int_div(n, d, q, NULL));
+}
+
+/* gmp: mpz_divisible_p */
+/* gmp: return 1 if d divides n, 0 otherwise */
+/* gmp: 0 is considered to divide 0*/
+int GMPZAPI(divisible_p)(mp_int n, mp_int d) {
+  /* variables to hold remainder */
+  mpz_t rz;
+  mp_int r = &rz;
+  int r_is_zero;
+
+  /* check for n = 0, d = 0 */
+  int n_is_zero = mp_int_compare_zero(n) == 0;
+  int d_is_zero = mp_int_compare_zero(d) == 0;
+  if (n_is_zero && d_is_zero)
+    return 1;
+
+  /* return true if remainder is 0 */
+  CHECK(mp_int_init(r));
+  CHECK(mp_int_div(n, d, NULL, r));
+  r_is_zero = mp_int_compare_zero(r) == 0;
+  mp_int_clear(r);
+
+  return r_is_zero;
+}
+
+/* gmp: mpz_submul */
+/* gmp: rop = rop - (op1 * op2) */
+void GMPZAPI(submul)(mp_int rop, mp_int op1, mp_int op2) {
+  mpz_t tempz;
+  mp_int temp = &tempz;
+  mp_int_init(temp);
+
+  CHECK(mp_int_mul(op1, op2, temp));
+  CHECK(mp_int_sub(rop, temp, rop));
+
+  mp_int_clear(temp);
+}
+
+/* gmp: mpz_add_ui */
+void GMPZAPI(add_ui)(mp_int rop, mp_int op1, unsigned long op2) {
+  mpz_t tempz;
+  mp_int temp = &tempz;
+  CHECK(mp_int_init_uvalue(temp, op2));
+
+  CHECK(mp_int_add(op1, temp, rop));
+
+  mp_int_clear(temp);
+}
+
+/* gmp: mpz_divexact_ui */
+/* gmp: only produces correct results when d divides n */
+void GMPZAPI(divexact_ui)(mp_int q, mp_int n, unsigned long d) {
+  mpz_t tempz;
+  mp_int temp = &tempz;
+  CHECK(mp_int_init_uvalue(temp, d));
+
+  CHECK(mp_int_div(n, temp, q, NULL));
+
+  mp_int_clear(temp);
+}
+
+/* gmp: mpz_mul_ui */
+void GMPZAPI(mul_ui)(mp_int rop, mp_int op1, unsigned long op2) {
+  mpz_t tempz;
+  mp_int temp = &tempz;
+  CHECK(mp_int_init_uvalue(temp, op2));
+
+  CHECK(mp_int_mul(op1, temp, rop));
+
+  mp_int_clear(temp);
+}
+
+/* gmp: mpz_pow_ui */
+/* gmp: 0^0 = 1 */
+void GMPZAPI(pow_ui)(mp_int rop, mp_int base, unsigned long exp) {
+  mpz_t tempz;
+  mp_int temp = &tempz;
+
+  /* check for 0^0 */
+  if (exp == 0 && mp_int_compare_zero(base) == 0) {
+    CHECK(mp_int_set_value(rop, 1));
+    return;
+  }
+
+  /* rop = base^exp */
+  CHECK(mp_int_init_uvalue(temp, exp));
+  CHECK(mp_int_expt_full(base, temp, rop));
+  mp_int_clear(temp);
+}
+
+/* gmp: mpz_sub_ui */
+void GMPZAPI(sub_ui)(mp_int rop, mp_int op1, unsigned long op2) {
+  mpz_t tempz;
+  mp_int temp = &tempz;
+  CHECK(mp_int_init_uvalue(temp, op2));
+
+  CHECK(mp_int_sub(op1, temp, rop));
+
+  mp_int_clear(temp);
+}
+
+/*************************************************************************
+ *
+ * Functions with different behavior in corner cases
+ *
+ *************************************************************************/
+
+/* gmp: mpz_gcd */
+void GMPZAPI(gcd)(mp_int rop, mp_int op1, mp_int op2) {
+  int op1_is_zero = mp_int_compare_zero(op1) == 0;
+  int op2_is_zero = mp_int_compare_zero(op2) == 0;
+
+  if (op1_is_zero && op2_is_zero) {
+    mp_int_zero(rop);
+    return;
+  }
+
+  CHECK(mp_int_gcd(op1, op2, rop));
+}
+
+/* gmp: mpz_get_str */
+char* GMPZAPI(get_str)(char *str, int radix, mp_int op) {
+  int i, r, len;
+
+  /* Support negative radix like gmp */
+  r = radix;
+  if (r < 0)
+    r = -r;
+
+  /* Compute the length of the string needed to hold the int */
+  len = mp_int_string_len(op, r);
+  if (str == NULL) {
+    str = malloc(len);
+  }
+
+  /* Convert to string using imath function */
+  CHECK(mp_int_to_string(op, r, str, len));
+
+  /* Change case to match gmp */
+  for (i = 0; i < len; i++)
+    if (radix < 0)
+      str[i] = toupper(str[i]);
+    else
+      str[i] = tolower(str[i]);
+  return str;
+}
+
+/* gmp: mpq_get_str */
+char* GMPQAPI(get_str)(char *str, int radix, mp_rat op) {
+  int i, r, len;
+
+  /* Only print numerator if it is a whole number */
+  if (mp_int_compare_value(mp_rat_denom_ref(op), 1) == 0)
+    return GMPZAPI(get_str)(str, radix, mp_rat_numer_ref(op));
+
+  /* Support negative radix like gmp */
+  r = radix;
+  if (r < 0)
+    r = -r;
+
+  /* Compute the length of the string needed to hold the int */
+  len = mp_rat_string_len(op, r);
+  if (str == NULL) {
+    str = malloc(len);
+  }
+
+  /* Convert to string using imath function */
+  CHECK(mp_rat_to_string(op, r, str, len));
+
+  /* Change case to match gmp */
+  for (i = 0; i < len; i++)
+    if (radix < 0)
+      str[i] = toupper(str[i]);
+    else
+      str[i] = tolower(str[i]);
+
+  return str;
+}
+
+/* gmp: mpz_set_str */
+int GMPZAPI(set_str)(mp_int rop, char *str, int base) {
+  mp_result res = mp_int_read_string(rop, base, str);
+  return ((res == MP_OK) ? 0 : -1);
+}
+
+/* gmp: mpq_set_str */
+int GMPQAPI(set_str)(mp_rat rop, char *s, int base) {
+  char *slash;
+  char *str;
+  mp_result resN;
+  mp_result resD;
+  int res = 0;
+
+  /* Copy string to temporary storage so we can modify it below */
+  str = malloc(strlen(s)+1);
+  strcpy(str, s);
+
+  /* Properly format the string as an int by terminating at the / */
+  slash = strchr(str, '/');
+  if (slash)
+    *slash = '\0';
+
+  /* Parse numerator */
+  resN = mp_int_read_string(mp_rat_numer_ref(rop), base, str);
+
+  /* Parse denomenator if given or set to 1 if not */
+  if (slash)
+    resD = mp_int_read_string(mp_rat_denom_ref(rop), base, slash+1);
+  else
+    resD = mp_int_set_uvalue(mp_rat_denom_ref(rop), 1);
+
+  /* Return failure if either parse failed */
+  if (resN != MP_OK || resD != MP_OK)
+    res = -1;
+
+  free(str);
+  return res;
+}
+
+static unsigned long get_long_bits(mp_int op) {
+  /* Deal with integer that does not fit into unsigned long. We want to grab
+   * the least significant digits that will fit into the long.  Read the digits
+   * into the long starting at the most significant digit that fits into a
+   * long. The long is shifted over by MP_DIGIT_BIT before each digit is added.
+   * The shift is decomposed into two steps to follow the patten used in the
+   * rest of the imath library. The two step shift is used to accomedate
+   * architectures that don't deal well with 32-bit shifts. */
+  mp_size num_digits_in_long = sizeof(unsigned long) / sizeof(mp_digit);
+  mp_digit *digits = MP_DIGITS(op);
+  unsigned long out = 0;
+  int i;
+
+  for (i = num_digits_in_long - 1; i >= 0; i--) {
+    out <<= (MP_DIGIT_BIT/2);
+    out <<= (MP_DIGIT_BIT/2);
+    out  |= digits[i];
+  }
+
+  return out;
+}
+
+/* gmp: mpz_get_ui */
+unsigned long GMPZAPI(get_ui)(mp_int op) {
+  unsigned long out;
+
+  /* Try a standard conversion that fits into an unsigned long */
+  mp_result res = mp_int_to_uint(op, &out);
+  if (res == MP_OK)
+    return out;
+
+  /* Abort the try if we don't have a range error in the conversion.
+   * The range error indicates that the value cannot fit into a long. */
+  CHECK(res == MP_RANGE ? MP_OK : MP_RANGE);
+  if (res != MP_RANGE)
+    return 0;
+
+  return get_long_bits(op);
+}
+
+/* gmp: mpz_get_si */
+long GMPZAPI(get_si)(mp_int op) {
+  long out;
+  unsigned long uout;
+  int long_msb;
+
+  /* Try a standard conversion that fits into a long */
+  mp_result res = mp_int_to_int(op, &out);
+  if (res == MP_OK)
+    return out;
+
+  /* Abort the try if we don't have a range error in the conversion.
+   * The range error indicates that the value cannot fit into a long. */
+  CHECK(res == MP_RANGE ? MP_OK : MP_RANGE);
+  if (res != MP_RANGE)
+    return 0;
+
+  /* get least significant bits into an unsigned long */
+  uout = get_long_bits(op);
+
+  /* clear the top bit */
+  long_msb = (sizeof(unsigned long) * 8) - 1;
+  uout &= (~(1UL << long_msb));
+
+  /* convert to negative if needed based on sign of op */
+  if (MP_SIGN(op) == MP_NEG)
+    uout = 0 - uout;
+
+  out = (long) uout;
+  return out;
+}
+
+/* gmp: mpz_lcm */
+void GMPZAPI(lcm)(mp_int rop, mp_int op1, mp_int op2) {
+  int op1_is_zero = mp_int_compare_zero(op1) == 0;
+  int op2_is_zero = mp_int_compare_zero(op2) == 0;
+
+  if (op1_is_zero || op2_is_zero) {
+    mp_int_zero(rop);
+    return;
+  }
+
+  CHECK(mp_int_lcm(op1, op2, rop));
+  CHECK(mp_int_abs(rop, rop));
+}
+
+/* gmp: mpz_mul_2exp */
+/* gmp: allow big values for op2 when op1 == 0 */
+void GMPZAPI(mul_2exp)(mp_int rop, mp_int op1, unsigned long op2) {
+  if (mp_int_compare_zero(op1) == 0)
+    mp_int_zero(rop);
+  else
+    CHECK(mp_int_mul_pow2(op1, op2, rop));
+}
+
+/*************************************************************************
+ *
+ * Functions needing expanded functionality
+ *
+ *************************************************************************/
+/* [Note]Overview of division implementation
+
+    All division operations (N / D) compute q and r such that
+
+      N = q * D + r, with 0 <= abs(r) < abs(d)
+
+    The q and r values are not uniquely specified by N and D. To specify which q
+    and r values should be used, GMP implements three different rounding modes
+    for integer division:
+
+      ceiling  - round q twords +infinity, r has opposite sign as d
+      floor    - round q twords -infinity, r has same sign as d
+      truncate - round q twords zero,      r has same sign as n
+
+    The imath library only supports truncate as a rounding mode. We need to
+    implement the other rounding modes in terms of truncating division. We first
+    perform the division in trucate mode and then adjust q accordingly. Once we
+    know q, we can easily compute the correct r according the the formula above
+    by computing:
+
+      r = N - q * D
+
+    The main task is to compute q. We can compute the correct q from a trucated
+    version as follows.
+
+    For ceiling rounding mode, if q is less than 0 then the truncated rounding
+    mode is the same as the ceiling rounding mode.  If q is greater than zero
+    then we need to round q up by one because the truncated version was rounded
+    down to zero. If q equals zero then check to see if the result of the
+    divison is positive. A positive result needs to increment q to one.
+
+    For floor rounding mode, if q is greater than 0 then the trucated rounding
+    mode is the same as the floor rounding mode. If q is less than zero then we
+    need to round q down by one because the trucated mode rounded q up by one
+    twords zero. If q is zero then we need to check to see if the result of the
+    division is negative. A negative result needs to decrement q to negative
+    one.
+ */
+
+/* gmp: mpz_cdiv_q */
+void GMPZAPI(cdiv_q)(mp_int q, mp_int n, mp_int d) {
+  mpz_t rz;
+  mp_int r = &rz;
+  int qsign, rsign, nsign, dsign;
+  CHECK(mp_int_init(r));
+
+  /* save signs before division because q can alias with n or d */
+  nsign = mp_int_compare_zero(n);
+  dsign = mp_int_compare_zero(d);
+
+  /* truncating division */
+  CHECK(mp_int_div(n, d, q, r));
+
+  /* see: [Note]Overview of division implementation */
+  qsign = mp_int_compare_zero(q);
+  rsign = mp_int_compare_zero(r);
+  if (qsign > 0) {    /* q > 0 */
+    if (rsign != 0) { /* r != 0 */
+      CHECK(mp_int_add_value(q, 1, q));
+    }
+  }
+  else if (qsign == 0) { /* q == 0 */
+    if (rsign != 0) {    /* r != 0 */
+      if ((nsign > 0 && dsign > 0) || (nsign < 0 && dsign < 0)) {
+        CHECK(mp_int_set_value(q, 1));
+      }
+    }
+  }
+  mp_int_clear(r);
+}
+
+/* gmp: mpz_fdiv_q */
+void GMPZAPI(fdiv_q)(mp_int q, mp_int n, mp_int d) {
+  mpz_t rz;
+  mp_int r = &rz;
+  int qsign, rsign, nsign, dsign;
+  CHECK(mp_int_init(r));
+
+  /* save signs before division because q can alias with n or d */
+  nsign = mp_int_compare_zero(n);
+  dsign = mp_int_compare_zero(d);
+
+  /* truncating division */
+  CHECK(mp_int_div(n, d, q, r));
+
+  /* see: [Note]Overview of division implementation */
+  qsign = mp_int_compare_zero(q);
+  rsign = mp_int_compare_zero(r);
+  if (qsign < 0) {    /* q  < 0 */
+    if (rsign != 0) { /* r != 0 */
+      CHECK(mp_int_sub_value(q, 1, q));
+    }
+  }
+  else if (qsign == 0) { /* q == 0 */
+    if (rsign != 0) {    /* r != 0 */
+      if ((nsign < 0 && dsign > 0) || (nsign > 0 && dsign < 0)) {
+        CHECK(mp_int_set_value(q, -1));
+      }
+    }
+  }
+  mp_int_clear(r);
+}
+
+/* gmp: mpz_fdiv_r */
+void GMPZAPI(fdiv_r)(mp_int r, mp_int n, mp_int d) {
+  mpz_t qz;
+  mpz_t tempz;
+  mpz_t orig_dz;
+  mpz_t orig_nz;
+  mp_int q = &qz;
+  mp_int temp = &tempz;
+  mp_int orig_d = &orig_dz;
+  mp_int orig_n = &orig_nz;
+  CHECK(mp_int_init(q));
+  CHECK(mp_int_init(temp));
+  /* Make a copy of n in case n and d in case they overlap with q */
+  CHECK(mp_int_init_copy(orig_d, d));
+  CHECK(mp_int_init_copy(orig_n, n));
+
+  /* floor division */
+  GMPZAPI(fdiv_q)(q, n, d);
+
+  /* see: [Note]Overview of division implementation */
+  /* n = q * d + r  ==>  r = n - q * d */
+  mp_int_mul(q, orig_d, temp);
+  mp_int_sub(orig_n, temp, r);
+
+  mp_int_clear(q);
+  mp_int_clear(temp);
+  mp_int_clear(orig_d);
+  mp_int_clear(orig_n);
+}
+
+/* gmp: mpz_tdiv_q */
+void GMPZAPI(tdiv_q)(mp_int q, mp_int n, mp_int d) {
+  /* truncating division*/
+  CHECK(mp_int_div(n, d, q, NULL));
+}
+
+/* gmp: mpz_fdiv_q_ui */
+unsigned long GMPZAPI(fdiv_q_ui)(mp_int q, mp_int n, unsigned long d) {
+  mpz_t tempz;
+  mp_int temp = &tempz;
+  mpz_t rz;
+  mp_int r = &rz;
+  mpz_t orig_nz;
+  mp_int orig_n = &orig_nz;
+  unsigned long rl;
+  CHECK(mp_int_init_uvalue(temp, d));
+  CHECK(mp_int_init(r));
+  /* Make a copy of n in case n and q overlap */
+  CHECK(mp_int_init_copy(orig_n, n));
+
+  /* use floor division mode to compute q and r */
+  GMPZAPI(fdiv_q)(q, n, temp);
+  GMPZAPI(fdiv_r)(r, orig_n, temp);
+  CHECK(mp_int_to_uint(r, &rl));
+
+  mp_int_clear(temp);
+  mp_int_clear(r);
+  mp_int_clear(orig_n);
+
+  return rl;
+}
+
+/* gmp: mpz_export */
+void* GMPZAPI(export)(void *rop, size_t *countp, int order, size_t size, int endian, size_t nails, mp_int op) {
+  int i;
+  int num_used_bytes;
+  size_t num_words, num_missing_bytes;
+  unsigned char* dst;
+  unsigned char* src;
+
+  /* We do not have a complete implementation. Assert to ensure our
+   * restrictions are in place, We do not support big endian output, but do not
+   * check that native endian is little endian. */
+  assert(nails  == 0 && "Do not support non-full words");
+  assert((endian == 0 || endian == -1) && "Do not support big endian");
+
+  /* The gmp API requires that order must be -1 or 1.
+     Not sure how gmp behaves when order is not 1 or -1, so force all non-one
+     values to -1 for now. */
+  if (order != 1) order = -1;
+
+  /* Test for zero */
+  if (mp_int_compare_zero(op) == 0) {
+    if (countp)
+      *countp = 0;
+    return rop;
+  }
+
+  /* Calculate how many words we need */
+  num_used_bytes  = mp_int_unsigned_len(op);
+  num_words       = (num_used_bytes + (size-1)) / size; /* ceil division */
+  assert(num_used_bytes > 0);
+
+  /* Check to see if we will have missing bytes in the last word.
+
+     Missing bytes can only occur when the size of words we output is
+     greater than the size of words used internally by imath. The number of
+     missing bytes is the number of bytes needed to fill out the last word. If
+     this number is greater than the size of a single mp_digit, then we need to
+     pad the word with extra zeros. Otherwise, the missing bytes can be filled
+     directly from the zeros in the last digit in the number.
+   */
+  num_missing_bytes   = (size * num_words) - num_used_bytes;
+  assert(num_missing_bytes < size);
+
+  /* Allocate space for the result if needed */
+  if (rop == NULL) {
+    rop = malloc(num_words * size);
+  }
+
+  /* Initialize dst and src pointers */
+  dst = (unsigned char *)rop;
+  src = (unsigned char *)MP_DIGITS(op);
+
+  /* Most significant word first */
+  if (order == 1) {
+    size_t words_written = 0;
+    src += (num_words-1) * size;
+
+    /* Handle write of first word specially */
+    for (i = 0; i < size - num_missing_bytes; i++)
+      dst[i] = src[i];
+    for (; i < size; i++)
+      dst[i] = 0;
+    dst += size;
+    src -= size;
+    words_written++;
+
+    for (; words_written < num_words; words_written++) {
+      for (i = 0; i < size; i++)
+        dst[i] = src[i];
+      dst += size;
+      src -= size;
+    }
+  }
+  /* Least significant word first */
+  else {
+    size_t words_written = 0;
+    for (; words_written < num_words - 1; words_written++) {
+      for (i = 0; i < size; i++)
+        dst[i] = src[i];
+      dst += size;
+      src += size;
+    }
+
+    /* Handle write of last word specially */
+    for (i = 0; i < size - num_missing_bytes; i++)
+      dst[i] = src[i];
+
+    for (; i < size; i++)
+      dst[i] = 0;
+  }
+
+  if (countp)
+    *countp = num_words;
+  return rop;
+}
+
+/* gmp: mpz_import */
+void GMPZAPI(import)(mp_int rop, size_t count, int order, size_t size, int endian, size_t nails, const void* op) {
+  mpz_t tmpz;
+  mp_int tmp = &tmpz;
+  size_t total_size;
+  size_t num_digits;
+  const char *src;
+  char *dst;
+  int i;
+  if (count == 0 || op == NULL)
+    return;
+
+  /* We do not have a complete implementation. Assert to ensure our
+   * restrictions are in place, We do not support big endian output, but do not
+   * check that native endian is little endian. */
+  assert(nails  == 0 && "Do not support non-full words");
+  assert((endian == 0 || endian == -1) && "Do not support big endian");
+
+  /* Compute number of needed digits by ceil division */
+  total_size = count * size;
+  num_digits = (total_size + sizeof(mp_digit) - 1) / sizeof(mp_digit);
+
+  /* Init temporary */
+  mp_int_init_size(tmp, num_digits);
+  for (i = 0; i < num_digits; i++)
+    tmp->digits[i] = 0;
+
+  /* Copy bytes */
+  src = (const char *) op;
+  dst = (char *)MP_DIGITS(tmp);
+
+  /* Most significant word is first */
+  if (order == 1) {
+    size_t word;
+    dst += (count - 1) * size;
+    for (word = 0; word < count; word++) {
+      for (i = 0; i < size; i++)
+        dst[i] = src[i];
+      dst -= size;
+      src += size;
+    }
+  }
+  /* Least significant word is first */
+  else {
+    size_t word;
+    for (word = 0; word < count; word++) {
+      for (i = 0; i < size; i++)
+        dst[i] = src[i];
+      dst += size;
+      src += size;
+    }
+  }
+  MP_USED(tmp) = num_digits;
+
+  /* Remove leading zeros from number */
+  {
+    mp_size uz_   = MP_USED(tmp);
+    mp_digit *dz_ = MP_DIGITS(tmp) + uz_ -1;
+    while (uz_ > 1 && (*dz_-- == 0))
+      --uz_;
+    MP_USED(tmp) = uz_;
+  }
+
+  /* Copy to destination */
+  mp_int_copy(tmp, rop);
+}
+
+/* gmp: mpz_sizeinbase */
+size_t GMPZAPI(sizeinbase)(mp_int op, int base) {
+  mp_result res;
+  size_t size;
+
+  /* If op == 0, return 1 */
+  if (mp_int_compare_zero(op) == 0)
+    return 1;
+
+  /* Compute string length in base */
+  res = mp_int_string_len(op, base);
+  CHECK((res > 0) == MP_OK);
+
+  /* Now adjust the final size by getting rid of string artifacts */
+  size = res;
+
+  /* subtract one for the null terminator */
+  size -= 1;
+
+  /* subtract one for the negative sign */
+  if (mp_int_compare_zero(op) < 0)
+    size -= 1;
+
+  return size;
+}
diff --git a/polly/lib/External/isl/imath/gmp_compat.h b/polly/lib/External/isl/imath/gmp_compat.h
new file mode 100644
index 00000000000..949e377e979
--- /dev/null
+++ b/polly/lib/External/isl/imath/gmp_compat.h
@@ -0,0 +1,229 @@
+/*
+  Name:     gmp_compat.h
+  Purpose:  Provide GMP compatiable routines for imath library
+  Author:   David Peixotto
+
+  Copyright (c) 2012 Qualcomm Innovation Center, Inc. All rights reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#ifndef IMATH_GMP_COMPAT_H_
+#define IMATH_GMP_COMPAT_H_
+#include "imath.h"
+#include "imrat.h"
+#include <stddef.h>
+
+#define GMPZAPI(fun) impz_ ## fun
+#define GMPQAPI(fun) impq_ ## fun
+
+#ifdef __cplusplus
+extern "C" {
+#endif
+/*************************************************************************
+ *
+ * Functions with direct translations
+ *
+ *************************************************************************/
+/* gmp: mpq_clear */
+void GMPQAPI(clear)(mp_rat x);
+
+/* gmp: mpq_cmp */
+int GMPQAPI(cmp)(mp_rat op1, mp_rat op2);
+
+/* gmp: mpq_init */
+void GMPQAPI(init)(mp_rat x);
+
+/* gmp: mpq_mul */
+void GMPQAPI(mul)(mp_rat product, mp_rat multiplier, mp_rat multiplicand);
+
+/* gmp: mpq_set */
+void GMPQAPI(set)(mp_rat rop, mp_rat op);
+
+/* gmp: mpz_abs */
+void GMPZAPI(abs)(mp_int rop, mp_int op);
+
+/* gmp: mpz_add */
+void GMPZAPI(add)(mp_int rop, mp_int op1, mp_int op2);
+
+/* gmp: mpz_clear */
+void GMPZAPI(clear)(mp_int x);
+
+/* gmp: mpz_cmp_si */
+int GMPZAPI(cmp_si)(mp_int op1, long op2);
+
+/* gmp: mpz_cmpabs */
+int GMPZAPI(cmpabs)(mp_int op1, mp_int op2);
+
+/* gmp: mpz_cmp */
+int GMPZAPI(cmp)(mp_int op1, mp_int op2);
+
+/* gmp: mpz_init */
+void GMPZAPI(init)(mp_int x);
+
+/* gmp: mpz_mul */
+void GMPZAPI(mul)(mp_int rop, mp_int op1, mp_int op2);
+
+/* gmp: mpz_neg */
+void GMPZAPI(neg)(mp_int rop, mp_int op);
+
+/* gmp: mpz_set_si */
+void GMPZAPI(set_si)(mp_int rop, long op);
+
+/* gmp: mpz_set */
+void GMPZAPI(set)(mp_int rop, mp_int op);
+
+/* gmp: mpz_sub */
+void GMPZAPI(sub)(mp_int rop, mp_int op1, mp_int op2);
+
+/* gmp: mpz_swap */
+void GMPZAPI(swap)(mp_int rop1, mp_int rop2);
+
+/* gmp: mpq_sgn */
+int GMPQAPI(sgn)(mp_rat op);
+
+/* gmp: mpz_sgn */
+int GMPZAPI(sgn)(mp_int op);
+
+/* gmp: mpq_set_ui */
+void GMPQAPI(set_ui)(mp_rat rop, unsigned long op1, unsigned long op2);
+
+/* gmp: mpz_set_ui */
+void GMPZAPI(set_ui)(mp_int rop, unsigned long op);
+
+/* gmp: mpq_den_ref */
+mp_int GMPQAPI(denref)(mp_rat op);
+
+/* gmp: mpq_num_ref */
+mp_int GMPQAPI(numref)(mp_rat op);
+
+/* gmp: mpq_canonicalize */
+void GMPQAPI(canonicalize)(mp_rat op);
+
+/*************************************************************************
+ *
+ * Functions that can be implemented as a combination of imath functions
+ *
+ *************************************************************************/
+/* gmp: mpz_addmul */
+void GMPZAPI(addmul)(mp_int rop, mp_int op1, mp_int op2);
+
+/* gmp: mpz_divexact */
+void GMPZAPI(divexact)(mp_int q, mp_int n, mp_int d);
+
+/* gmp: mpz_divisible_p */
+int GMPZAPI(divisible_p)(mp_int n, mp_int d);
+
+/* gmp: mpz_submul */
+void GMPZAPI(submul)(mp_int rop, mp_int op1, mp_int op2);
+
+/* gmp: mpz_add_ui */
+void GMPZAPI(add_ui)(mp_int rop, mp_int op1, unsigned long op2);
+
+/* gmp: mpz_divexact_ui */
+void GMPZAPI(divexact_ui)(mp_int q, mp_int n, unsigned long d);
+
+/* gmp: mpz_mul_ui */
+void GMPZAPI(mul_ui)(mp_int rop, mp_int op1, unsigned long op2);
+
+/* gmp: mpz_pow_ui */
+void GMPZAPI(pow_ui)(mp_int rop, mp_int base, unsigned long exp);
+
+/* gmp: mpz_sub_ui */
+void GMPZAPI(sub_ui)(mp_int rop, mp_int op1, unsigned long op2);
+
+/* gmp: mpz_fdiv_q_ui */
+unsigned long GMPZAPI(fdiv_q_ui)(mp_int q, mp_int n, unsigned long d);
+
+/* gmp: mpz_sizeinbase */
+size_t GMPZAPI(sizeinbase)(mp_int op, int base);
+
+/*************************************************************************
+ *
+ * Functions with different behavior in corner cases
+ *
+ *************************************************************************/
+/* gmp: mpz_gcd */
+/* gmp: When op1 = 0 and op2 = 0, return 0.*/
+void GMPZAPI(gcd)(mp_int rop, mp_int op1, mp_int op2);
+
+/* gmp: mpz_get_str */
+/* gmp: If str is NULL then allocate space using the default allocator. */
+char* GMPZAPI(get_str)(char *str, int radix, mp_int op);
+
+/* gmp: mpq_get_str */
+/* gmp: If str is NULL then allocate space using the default allocator. */
+/* gmp: If value is a whole number do not print denomenator. */
+/* TODO: Need to handle 0 values better. GMP prints 0/4 instead of 0.*/
+char* GMPQAPI(get_str)(char *str, int radix, mp_rat op);
+
+/* gmp: mpz_set_str */
+/* gmp: Allow and ignore spaces in string. */
+int GMPZAPI(set_str)(mp_int rop, char *str, int base);
+
+/* gmp: mpq_set_str */
+int GMPQAPI(set_str)(mp_rat rop, char *str, int base);
+
+/* gmp: mpz_get_ui */
+/* gmp: Return least significant bits if value is too big for a long. */
+unsigned long GMPZAPI(get_ui)(mp_int op);
+
+/* gmp: mpz_get_si */
+/* gmp: Return least significant bits if value is too bit for a long. */
+/* gmp: If value is too big for long, return the least significant
+        (8*sizeof(long)-1) bits from the op and set the sign bit according to
+        the sign of the op. */
+long GMPZAPI(get_si)(mp_int op);
+
+/* gmp: mpz_lcm */
+/* gmp: When op1 = 0 or op2 = 0, return 0.*/
+/* gmp: The resutl of lcm(a,b) is always positive. */
+void GMPZAPI(lcm)(mp_int rop, mp_int op1, mp_int op2);
+
+/* gmp: mpz_mul_2exp */
+/* gmp: allow big values for op2 when op1 == 0 */
+void GMPZAPI(mul_2exp)(mp_int rop, mp_int op1, unsigned long op2);
+
+/*************************************************************************
+ *
+ * Functions needing expanded functionality
+ *
+ *************************************************************************/
+/* gmp: mpz_cdiv_q */
+void GMPZAPI(cdiv_q)(mp_int q, mp_int n, mp_int d);
+
+/* gmp: mpz_fdiv_q */
+void GMPZAPI(fdiv_q)(mp_int q, mp_int n, mp_int d);
+
+/* gmp: mpz_fdiv_r */
+void GMPZAPI(fdiv_r)(mp_int r, mp_int n, mp_int d);
+
+/* gmp: mpz_tdiv_q */
+void GMPZAPI(tdiv_q)(mp_int q, mp_int n, mp_int d);
+
+/* gmp: mpz_export */
+void* GMPZAPI(export)(void *rop, size_t *countp, int order, size_t size, int endian, size_t nails, mp_int op);
+
+/* gmp: mpz_import */
+void GMPZAPI(import)(mp_int rop, size_t count, int order, size_t size, int endian, size_t nails, const void* op);
+
+#ifdef __cplusplus
+}
+#endif
+#endif /* end IMATH_GMP_COMPAT_H_ */
diff --git a/polly/lib/External/isl/imath/imath.c b/polly/lib/External/isl/imath/imath.c
new file mode 100644
index 00000000000..306678ec963
--- /dev/null
+++ b/polly/lib/External/isl/imath/imath.c
@@ -0,0 +1,3668 @@
+/*
+  Name:     imath.c
+  Purpose:  Arbitrary precision integer arithmetic routines.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Copyright (C) 2002-2007 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#include "imath.h"
+
+#if DEBUG
+#include <stdio.h>
+#endif
+
+#include <stdlib.h>
+#include <string.h>
+#include <ctype.h>
+
+#include <assert.h>
+
+#if DEBUG
+#define STATIC /* public */
+#else
+#define STATIC static
+#endif
+
+/* {{{ Constants */
+
+const mp_result MP_OK     = 0;  /* no error, all is well  */
+const mp_result MP_FALSE  = 0;  /* boolean false          */
+const mp_result MP_TRUE   = -1; /* boolean true           */
+const mp_result MP_MEMORY = -2; /* out of memory          */
+const mp_result MP_RANGE  = -3; /* argument out of range  */
+const mp_result MP_UNDEF  = -4; /* result undefined       */
+const mp_result MP_TRUNC  = -5; /* output truncated       */
+const mp_result MP_BADARG = -6; /* invalid null argument  */
+const mp_result MP_MINERR = -6;
+
+const mp_sign   MP_NEG  = 1;    /* value is strictly negative */
+const mp_sign   MP_ZPOS = 0;    /* value is non-negative      */
+
+STATIC const char *s_unknown_err = "unknown result code";
+STATIC const char *s_error_msg[] = {
+  "error code 0",
+  "boolean true",
+  "out of memory",
+  "argument out of range",
+  "result undefined",
+  "output truncated",
+  "invalid argument",
+  NULL
+};
+
+/* }}} */
+
+/* Argument checking macros
+   Use CHECK() where a return value is required; NRCHECK() elsewhere */
+#define CHECK(TEST)   assert(TEST)
+#define NRCHECK(TEST) assert(TEST)
+
+/* {{{ Logarithm table for computing output sizes */
+
+/* The ith entry of this table gives the value of log_i(2).
+
+   An integer value n requires ceil(log_i(n)) digits to be represented
+   in base i.  Since it is easy to compute lg(n), by counting bits, we
+   can compute log_i(n) = lg(n) * log_i(2).
+
+   The use of this table eliminates a dependency upon linkage against
+   the standard math libraries.
+
+   If MP_MAX_RADIX is increased, this table should be expanded too.
+ */
+STATIC const double s_log2[] = {
+   0.000000000, 0.000000000, 1.000000000, 0.630929754, 	/* (D)(D) 2  3 */
+   0.500000000, 0.430676558, 0.386852807, 0.356207187, 	/*  4  5  6  7 */
+   0.333333333, 0.315464877, 0.301029996, 0.289064826, 	/*  8  9 10 11 */
+   0.278942946, 0.270238154, 0.262649535, 0.255958025, 	/* 12 13 14 15 */
+   0.250000000, 0.244650542, 0.239812467, 0.235408913, 	/* 16 17 18 19 */
+   0.231378213, 0.227670249, 0.224243824, 0.221064729, 	/* 20 21 22 23 */
+   0.218104292, 0.215338279, 0.212746054, 0.210309918, 	/* 24 25 26 27 */
+   0.208014598, 0.205846832, 0.203795047, 0.201849087, 	/* 28 29 30 31 */
+   0.200000000, 0.198239863, 0.196561632, 0.194959022, 	/* 32 33 34 35 */
+   0.193426404,                                         /* 36          */
+};
+
+/* }}} */
+/* {{{ Various macros */
+
+/* Return the number of digits needed to represent a static value */
+#define MP_VALUE_DIGITS(V) \
+((sizeof(V)+(sizeof(mp_digit)-1))/sizeof(mp_digit))
+
+/* Round precision P to nearest word boundary */
+#define ROUND_PREC(P) ((mp_size)(2*(((P)+1)/2)))
+
+/* Set array P of S digits to zero */
+#define ZERO(P, S) \
+do{ \
+  mp_size i__ = (S) * sizeof(mp_digit); \
+  mp_digit *p__ = (P); \
+  memset(p__, 0, i__); \
+} while(0)
+
+/* Copy S digits from array P to array Q */
+#define COPY(P, Q, S) \
+do{ \
+  mp_size i__ = (S) * sizeof(mp_digit); \
+  mp_digit *p__ = (P), *q__ = (Q); \
+  memcpy(q__, p__, i__); \
+} while(0)
+
+/* Reverse N elements of type T in array A */
+#define REV(T, A, N) \
+do{ \
+  T *u_ = (A), *v_ = u_ + (N) - 1; \
+  while (u_ < v_) { \
+    T xch = *u_; \
+    *u_++ = *v_; \
+    *v_-- = xch; \
+  } \
+} while(0)
+
+#define CLAMP(Z) \
+do{ \
+  mp_int z_ = (Z); \
+  mp_size uz_ = MP_USED(z_); \
+  mp_digit *dz_ = MP_DIGITS(z_) + uz_ -1; \
+  while (uz_ > 1 && (*dz_-- == 0)) \
+    --uz_; \
+  MP_USED(z_) = uz_; \
+} while(0)
+
+/* Select min/max.  Do not provide expressions for which multiple
+   evaluation would be problematic, e.g. x++ */
+#define MIN(A, B) ((B)<(A)?(B):(A))
+#define MAX(A, B) ((B)>(A)?(B):(A))
+
+/* Exchange lvalues A and B of type T, e.g.
+   SWAP(int, x, y) where x and y are variables of type int. */
+#define SWAP(T, A, B) \
+do{ \
+  T t_ = (A); \
+  A = (B); \
+  B = t_; \
+} while(0)
+
+/* Used to set up and access simple temp stacks within functions. */
+#define DECLARE_TEMP(N) \
+  mpz_t temp[(N)]; \
+  int last__ = 0
+#define CLEANUP_TEMP() \
+ CLEANUP: \
+  while (--last__ >= 0) \
+    mp_int_clear(TEMP(last__))
+#define TEMP(K) (temp + (K))
+#define LAST_TEMP() TEMP(last__)
+#define SETUP(E) \
+do{ \
+  if ((res = (E)) != MP_OK) \
+    goto CLEANUP; \
+  ++(last__); \
+} while(0)
+
+/* Compare value to zero. */
+#define CMPZ(Z) \
+(((Z)->used==1&&(Z)->digits[0]==0)?0:((Z)->sign==MP_NEG)?-1:1)
+
+/* Multiply X by Y into Z, ignoring signs.  Requires that Z have
+   enough storage preallocated to hold the result. */
+#define UMUL(X, Y, Z) \
+do{ \
+  mp_size ua_ = MP_USED(X), ub_ = MP_USED(Y); \
+  mp_size o_ = ua_ + ub_; \
+  ZERO(MP_DIGITS(Z), o_); \
+  (void) s_kmul(MP_DIGITS(X), MP_DIGITS(Y), MP_DIGITS(Z), ua_, ub_); \
+  MP_USED(Z) = o_; \
+  CLAMP(Z); \
+} while(0)
+
+/* Square X into Z.  Requires that Z have enough storage to hold the
+   result. */
+#define USQR(X, Z) \
+do{ \
+  mp_size ua_ = MP_USED(X), o_ = ua_ + ua_; \
+  ZERO(MP_DIGITS(Z), o_); \
+  (void) s_ksqr(MP_DIGITS(X), MP_DIGITS(Z), ua_); \
+  MP_USED(Z) = o_; \
+  CLAMP(Z); \
+} while(0)
+
+#define UPPER_HALF(W)           ((mp_word)((W) >> MP_DIGIT_BIT))
+#define LOWER_HALF(W)           ((mp_digit)(W))
+#define HIGH_BIT_SET(W)         ((W) >> (MP_WORD_BIT - 1))
+#define ADD_WILL_OVERFLOW(W, V) ((MP_WORD_MAX - (V)) < (W))
+
+/* }}} */
+/* {{{ Default configuration settings */
+
+/* Default number of digits allocated to a new mp_int */
+#if IMATH_TEST
+mp_size default_precision = MP_DEFAULT_PREC;
+#else
+STATIC const mp_size default_precision = MP_DEFAULT_PREC;
+#endif
+
+/* Minimum number of digits to invoke recursive multiply */
+#if IMATH_TEST
+mp_size multiply_threshold = MP_MULT_THRESH;
+#else
+STATIC const mp_size multiply_threshold = MP_MULT_THRESH;
+#endif
+
+/* }}} */
+
+/* Allocate a buffer of (at least) num digits, or return
+   NULL if that couldn't be done.  */
+STATIC mp_digit *s_alloc(mp_size num);
+
+/* Release a buffer of digits allocated by s_alloc(). */
+STATIC void s_free(void *ptr);
+
+/* Insure that z has at least min digits allocated, resizing if
+   necessary.  Returns true if successful, false if out of memory. */
+STATIC int  s_pad(mp_int z, mp_size min);
+
+/* Fill in a "fake" mp_int on the stack with a given value */
+STATIC void      s_fake(mp_int z, mp_small value, mp_digit vbuf[]);
+STATIC void      s_ufake(mp_int z, mp_usmall value, mp_digit vbuf[]);
+
+/* Compare two runs of digits of given length, returns <0, 0, >0 */
+STATIC int       s_cdig(mp_digit *da, mp_digit *db, mp_size len);
+
+/* Pack the unsigned digits of v into array t */
+STATIC int       s_uvpack(mp_usmall v, mp_digit t[]);
+
+/* Compare magnitudes of a and b, returns <0, 0, >0 */
+STATIC int       s_ucmp(mp_int a, mp_int b);
+
+/* Compare magnitudes of a and v, returns <0, 0, >0 */
+STATIC int       s_vcmp(mp_int a, mp_small v);
+STATIC int       s_uvcmp(mp_int a, mp_usmall uv);
+
+/* Unsigned magnitude addition; assumes dc is big enough.
+   Carry out is returned (no memory allocated). */
+STATIC mp_digit  s_uadd(mp_digit *da, mp_digit *db, mp_digit *dc,
+		        mp_size size_a, mp_size size_b);
+
+/* Unsigned magnitude subtraction.  Assumes dc is big enough. */
+STATIC void      s_usub(mp_digit *da, mp_digit *db, mp_digit *dc,
+		        mp_size size_a, mp_size size_b);
+
+/* Unsigned recursive multiplication.  Assumes dc is big enough. */
+STATIC int       s_kmul(mp_digit *da, mp_digit *db, mp_digit *dc,
+			mp_size size_a, mp_size size_b);
+
+/* Unsigned magnitude multiplication.  Assumes dc is big enough. */
+STATIC void      s_umul(mp_digit *da, mp_digit *db, mp_digit *dc,
+			mp_size size_a, mp_size size_b);
+
+/* Unsigned recursive squaring.  Assumes dc is big enough. */
+STATIC int       s_ksqr(mp_digit *da, mp_digit *dc, mp_size size_a);
+
+/* Unsigned magnitude squaring.  Assumes dc is big enough. */
+STATIC void      s_usqr(mp_digit *da, mp_digit *dc, mp_size size_a);
+
+/* Single digit addition.  Assumes a is big enough. */
+STATIC void      s_dadd(mp_int a, mp_digit b);
+
+/* Single digit multiplication.  Assumes a is big enough. */
+STATIC void      s_dmul(mp_int a, mp_digit b);
+
+/* Single digit multiplication on buffers; assumes dc is big enough. */
+STATIC void      s_dbmul(mp_digit *da, mp_digit b, mp_digit *dc,
+			 mp_size size_a);
+
+/* Single digit division.  Replaces a with the quotient,
+   returns the remainder.  */
+STATIC mp_digit  s_ddiv(mp_int a, mp_digit b);
+
+/* Quick division by a power of 2, replaces z (no allocation) */
+STATIC void      s_qdiv(mp_int z, mp_size p2);
+
+/* Quick remainder by a power of 2, replaces z (no allocation) */
+STATIC void      s_qmod(mp_int z, mp_size p2);
+
+/* Quick multiplication by a power of 2, replaces z.
+   Allocates if necessary; returns false in case this fails. */
+STATIC int       s_qmul(mp_int z, mp_size p2);
+
+/* Quick subtraction from a power of 2, replaces z.
+   Allocates if necessary; returns false in case this fails. */
+STATIC int       s_qsub(mp_int z, mp_size p2);
+
+/* Return maximum k such that 2^k divides z. */
+STATIC int       s_dp2k(mp_int z);
+
+/* Return k >= 0 such that z = 2^k, or -1 if there is no such k. */
+STATIC int       s_isp2(mp_int z);
+
+/* Set z to 2^k.  May allocate; returns false in case this fails. */
+STATIC int       s_2expt(mp_int z, mp_small k);
+
+/* Normalize a and b for division, returns normalization constant */
+STATIC int       s_norm(mp_int a, mp_int b);
+
+/* Compute constant mu for Barrett reduction, given modulus m, result
+   replaces z, m is untouched. */
+STATIC mp_result s_brmu(mp_int z, mp_int m);
+
+/* Reduce a modulo m, using Barrett's algorithm. */
+STATIC int       s_reduce(mp_int x, mp_int m, mp_int mu, mp_int q1, mp_int q2);
+
+/* Modular exponentiation, using Barrett reduction */
+STATIC mp_result s_embar(mp_int a, mp_int b, mp_int m, mp_int mu, mp_int c);
+
+/* Unsigned magnitude division.  Assumes |a| > |b|.  Allocates temporaries;
+   overwrites a with quotient, b with remainder. */
+STATIC mp_result s_udiv_knuth(mp_int a, mp_int b);
+
+/* Compute the number of digits in radix r required to represent the given
+   value.  Does not account for sign flags, terminators, etc. */
+STATIC int       s_outlen(mp_int z, mp_size r);
+
+/* Guess how many digits of precision will be needed to represent a radix r
+   value of the specified number of digits.  Returns a value guaranteed to be
+   no smaller than the actual number required. */
+STATIC mp_size   s_inlen(int len, mp_size r);
+
+/* Convert a character to a digit value in radix r, or
+   -1 if out of range */
+STATIC int       s_ch2val(char c, int r);
+
+/* Convert a digit value to a character */
+STATIC char      s_val2ch(int v, int caps);
+
+/* Take 2's complement of a buffer in place */
+STATIC void      s_2comp(unsigned char *buf, int len);
+
+/* Convert a value to binary, ignoring sign.  On input, *limpos is the bound on
+   how many bytes should be written to buf; on output, *limpos is set to the
+   number of bytes actually written. */
+STATIC mp_result s_tobin(mp_int z, unsigned char *buf, int *limpos, int pad);
+
+#if DEBUG
+/* Dump a representation of the mp_int to standard output */
+void      s_print(char *tag, mp_int z);
+void      s_print_buf(char *tag, mp_digit *buf, mp_size num);
+#endif
+
+/* {{{ mp_int_init(z) */
+
+mp_result mp_int_init(mp_int z)
+{
+  if (z == NULL)
+    return MP_BADARG;
+
+  z->single = 0;
+  z->digits = &(z->single);
+  z->alloc  = 1;
+  z->used   = 1;
+  z->sign   = MP_ZPOS;
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_int_alloc() */
+
+mp_int    mp_int_alloc(void)
+{
+  mp_int out = malloc(sizeof(mpz_t));
+
+  if (out != NULL)
+    mp_int_init(out);
+
+  return out;
+}
+
+/* }}} */
+
+/* {{{ mp_int_init_size(z, prec) */
+
+mp_result mp_int_init_size(mp_int z, mp_size prec)
+{
+  CHECK(z != NULL);
+
+  if (prec == 0)
+    prec = default_precision;
+  else if (prec == 1)
+    return mp_int_init(z);
+  else
+    prec = (mp_size) ROUND_PREC(prec);
+
+  if ((MP_DIGITS(z) = s_alloc(prec)) == NULL)
+    return MP_MEMORY;
+
+  z->digits[0] = 0;
+  MP_USED(z) = 1;
+  MP_ALLOC(z) = prec;
+  MP_SIGN(z) = MP_ZPOS;
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_int_init_copy(z, old) */
+
+mp_result mp_int_init_copy(mp_int z, mp_int old)
+{
+  mp_result res;
+  mp_size uold;
+
+  CHECK(z != NULL && old != NULL);
+
+  uold = MP_USED(old);
+  if (uold == 1) {
+    mp_int_init(z);
+  }
+  else {
+    mp_size target = MAX(uold, default_precision);
+
+    if ((res = mp_int_init_size(z, target)) != MP_OK)
+      return res;
+  }
+
+  MP_USED(z) = uold;
+  MP_SIGN(z) = MP_SIGN(old);
+  COPY(MP_DIGITS(old), MP_DIGITS(z), uold);
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_int_init_value(z, value) */
+
+mp_result mp_int_init_value(mp_int z, mp_small value)
+{
+  mpz_t    vtmp;
+  mp_digit vbuf[MP_VALUE_DIGITS(value)];
+
+  s_fake(&vtmp, value, vbuf);
+  return mp_int_init_copy(z, &vtmp);
+}
+
+/* }}} */
+
+/* {{{ mp_int_init_uvalue(z, uvalue) */
+
+mp_result mp_int_init_uvalue(mp_int z, mp_usmall uvalue)
+{
+  mpz_t    vtmp;
+  mp_digit vbuf[MP_VALUE_DIGITS(uvalue)];
+
+  s_ufake(&vtmp, uvalue, vbuf);
+  return mp_int_init_copy(z, &vtmp);
+}
+
+/* }}} */
+
+/* {{{ mp_int_set_value(z, value) */
+
+mp_result  mp_int_set_value(mp_int z, mp_small value)
+{
+  mpz_t    vtmp;
+  mp_digit vbuf[MP_VALUE_DIGITS(value)];
+
+  s_fake(&vtmp, value, vbuf);
+  return mp_int_copy(&vtmp, z);
+}
+
+/* }}} */
+
+/* {{{ mp_int_set_uvalue(z, value) */
+
+mp_result  mp_int_set_uvalue(mp_int z, mp_usmall uvalue)
+{
+  mpz_t    vtmp;
+  mp_digit vbuf[MP_VALUE_DIGITS(uvalue)];
+
+  s_ufake(&vtmp, uvalue, vbuf);
+  return mp_int_copy(&vtmp, z);
+}
+
+/* }}} */
+
+/* {{{ mp_int_clear(z) */
+
+void      mp_int_clear(mp_int z)
+{
+  if (z == NULL)
+    return;
+
+  if (MP_DIGITS(z) != NULL) {
+    if (MP_DIGITS(z) != &(z->single))
+      s_free(MP_DIGITS(z));
+
+    MP_DIGITS(z) = NULL;
+  }
+}
+
+/* }}} */
+
+/* {{{ mp_int_free(z) */
+
+void      mp_int_free(mp_int z)
+{
+  NRCHECK(z != NULL);
+
+  mp_int_clear(z);
+  free(z); /* note: NOT s_free() */
+}
+
+/* }}} */
+
+/* {{{ mp_int_copy(a, c) */
+
+mp_result mp_int_copy(mp_int a, mp_int c)
+{
+  CHECK(a != NULL && c != NULL);
+
+  if (a != c) {
+    mp_size ua = MP_USED(a);
+    mp_digit *da, *dc;
+
+    if (!s_pad(c, ua))
+      return MP_MEMORY;
+
+    da = MP_DIGITS(a); dc = MP_DIGITS(c);
+    COPY(da, dc, ua);
+
+    MP_USED(c) = ua;
+    MP_SIGN(c) = MP_SIGN(a);
+  }
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_int_swap(a, c) */
+
+void      mp_int_swap(mp_int a, mp_int c)
+{
+  if (a != c) {
+    mpz_t tmp = *a;
+
+    *a = *c;
+    *c = tmp;
+
+    if (MP_DIGITS(a) == &(c->single))
+      MP_DIGITS(a) = &(a->single);
+    if (MP_DIGITS(c) == &(a->single))
+      MP_DIGITS(c) = &(c->single);
+  }
+}
+
+/* }}} */
+
+/* {{{ mp_int_zero(z) */
+
+void      mp_int_zero(mp_int z)
+{
+  NRCHECK(z != NULL);
+
+  z->digits[0] = 0;
+  MP_USED(z) = 1;
+  MP_SIGN(z) = MP_ZPOS;
+}
+
+/* }}} */
+
+/* {{{ mp_int_abs(a, c) */
+
+mp_result mp_int_abs(mp_int a, mp_int c)
+{
+  mp_result res;
+
+  CHECK(a != NULL && c != NULL);
+
+  if ((res = mp_int_copy(a, c)) != MP_OK)
+    return res;
+
+  MP_SIGN(c) = MP_ZPOS;
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_int_neg(a, c) */
+
+mp_result mp_int_neg(mp_int a, mp_int c)
+{
+  mp_result res;
+
+  CHECK(a != NULL && c != NULL);
+
+  if ((res = mp_int_copy(a, c)) != MP_OK)
+    return res;
+
+  if (CMPZ(c) != 0)
+    MP_SIGN(c) = 1 - MP_SIGN(a);
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_int_add(a, b, c) */
+
+mp_result mp_int_add(mp_int a, mp_int b, mp_int c)
+{
+  mp_size ua, ub, uc, max;
+
+  CHECK(a != NULL && b != NULL && c != NULL);
+
+  ua = MP_USED(a); ub = MP_USED(b); uc = MP_USED(c);
+  max = MAX(ua, ub);
+
+  if (MP_SIGN(a) == MP_SIGN(b)) {
+    /* Same sign -- add magnitudes, preserve sign of addends */
+    mp_digit carry;
+
+    if (!s_pad(c, max))
+      return MP_MEMORY;
+
+    carry = s_uadd(MP_DIGITS(a), MP_DIGITS(b), MP_DIGITS(c), ua, ub);
+    uc = max;
+
+    if (carry) {
+      if (!s_pad(c, max + 1))
+	return MP_MEMORY;
+
+      c->digits[max] = carry;
+      ++uc;
+    }
+
+    MP_USED(c) = uc;
+    MP_SIGN(c) = MP_SIGN(a);
+
+  }
+  else {
+    /* Different signs -- subtract magnitudes, preserve sign of greater */
+    mp_int  x, y;
+    int     cmp = s_ucmp(a, b); /* magnitude comparision, sign ignored */
+
+    /* Set x to max(a, b), y to min(a, b) to simplify later code.
+       A special case yields zero for equal magnitudes.
+    */
+    if (cmp == 0) {
+      mp_int_zero(c);
+      return MP_OK;
+    }
+    else if (cmp < 0) {
+      x = b; y = a;
+    }
+    else {
+      x = a; y = b;
+    }
+
+    if (!s_pad(c, MP_USED(x)))
+      return MP_MEMORY;
+
+    /* Subtract smaller from larger */
+    s_usub(MP_DIGITS(x), MP_DIGITS(y), MP_DIGITS(c), MP_USED(x), MP_USED(y));
+    MP_USED(c) = MP_USED(x);
+    CLAMP(c);
+
+    /* Give result the sign of the larger */
+    MP_SIGN(c) = MP_SIGN(x);
+  }
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_int_add_value(a, value, c) */
+
+mp_result mp_int_add_value(mp_int a, mp_small value, mp_int c)
+{
+  mpz_t    vtmp;
+  mp_digit vbuf[MP_VALUE_DIGITS(value)];
+
+  s_fake(&vtmp, value, vbuf);
+
+  return mp_int_add(a, &vtmp, c);
+}
+
+/* }}} */
+
+/* {{{ mp_int_sub(a, b, c) */
+
+mp_result mp_int_sub(mp_int a, mp_int b, mp_int c)
+{
+  mp_size ua, ub, uc, max;
+
+  CHECK(a != NULL && b != NULL && c != NULL);
+
+  ua = MP_USED(a); ub = MP_USED(b); uc = MP_USED(c);
+  max = MAX(ua, ub);
+
+  if (MP_SIGN(a) != MP_SIGN(b)) {
+    /* Different signs -- add magnitudes and keep sign of a */
+    mp_digit carry;
+
+    if (!s_pad(c, max))
+      return MP_MEMORY;
+
+    carry = s_uadd(MP_DIGITS(a), MP_DIGITS(b), MP_DIGITS(c), ua, ub);
+    uc = max;
+
+    if (carry) {
+      if (!s_pad(c, max + 1))
+	return MP_MEMORY;
+
+      c->digits[max] = carry;
+      ++uc;
+    }
+
+    MP_USED(c) = uc;
+    MP_SIGN(c) = MP_SIGN(a);
+
+  }
+  else {
+    /* Same signs -- subtract magnitudes */
+    mp_int  x, y;
+    mp_sign osign;
+    int     cmp = s_ucmp(a, b);
+
+    if (!s_pad(c, max))
+      return MP_MEMORY;
+
+    if (cmp >= 0) {
+      x = a; y = b; osign = MP_ZPOS;
+    }
+    else {
+      x = b; y = a; osign = MP_NEG;
+    }
+
+    if (MP_SIGN(a) == MP_NEG && cmp != 0)
+      osign = 1 - osign;
+
+    s_usub(MP_DIGITS(x), MP_DIGITS(y), MP_DIGITS(c), MP_USED(x), MP_USED(y));
+    MP_USED(c) = MP_USED(x);
+    CLAMP(c);
+
+    MP_SIGN(c) = osign;
+  }
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_int_sub_value(a, value, c) */
+
+mp_result mp_int_sub_value(mp_int a, mp_small value, mp_int c)
+{
+  mpz_t    vtmp;
+  mp_digit vbuf[MP_VALUE_DIGITS(value)];
+
+  s_fake(&vtmp, value, vbuf);
+
+  return mp_int_sub(a, &vtmp, c);
+}
+
+/* }}} */
+
+/* {{{ mp_int_mul(a, b, c) */
+
+mp_result mp_int_mul(mp_int a, mp_int b, mp_int c)
+{
+  mp_digit *out;
+  mp_size   osize, ua, ub, p = 0;
+  mp_sign   osign;
+
+  CHECK(a != NULL && b != NULL && c != NULL);
+
+  /* If either input is zero, we can shortcut multiplication */
+  if (mp_int_compare_zero(a) == 0 || mp_int_compare_zero(b) == 0) {
+    mp_int_zero(c);
+    return MP_OK;
+  }
+
+  /* Output is positive if inputs have same sign, otherwise negative */
+  osign = (MP_SIGN(a) == MP_SIGN(b)) ? MP_ZPOS : MP_NEG;
+
+  /* If the output is not identical to any of the inputs, we'll write the
+     results directly; otherwise, allocate a temporary space. */
+  ua = MP_USED(a); ub = MP_USED(b);
+  osize = MAX(ua, ub);
+  osize = 4 * ((osize + 1) / 2);
+
+  if (c == a || c == b) {
+    p = ROUND_PREC(osize);
+    p = MAX(p, default_precision);
+
+    if ((out = s_alloc(p)) == NULL)
+      return MP_MEMORY;
+  }
+  else {
+    if (!s_pad(c, osize))
+      return MP_MEMORY;
+
+    out = MP_DIGITS(c);
+  }
+  ZERO(out, osize);
+
+  if (!s_kmul(MP_DIGITS(a), MP_DIGITS(b), out, ua, ub))
+    return MP_MEMORY;
+
+  /* If we allocated a new buffer, get rid of whatever memory c was already
+     using, and fix up its fields to reflect that.
+   */
+  if (out != MP_DIGITS(c)) {
+    if ((void *) MP_DIGITS(c) != (void *) c)
+      s_free(MP_DIGITS(c));
+    MP_DIGITS(c) = out;
+    MP_ALLOC(c) = p;
+  }
+
+  MP_USED(c) = osize; /* might not be true, but we'll fix it ... */
+  CLAMP(c);           /* ... right here */
+  MP_SIGN(c) = osign;
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_int_mul_value(a, value, c) */
+
+mp_result mp_int_mul_value(mp_int a, mp_small value, mp_int c)
+{
+  mpz_t    vtmp;
+  mp_digit vbuf[MP_VALUE_DIGITS(value)];
+
+  s_fake(&vtmp, value, vbuf);
+
+  return mp_int_mul(a, &vtmp, c);
+}
+
+/* }}} */
+
+/* {{{ mp_int_mul_pow2(a, p2, c) */
+
+mp_result mp_int_mul_pow2(mp_int a, mp_small p2, mp_int c)
+{
+  mp_result res;
+  CHECK(a != NULL && c != NULL && p2 >= 0);
+
+  if ((res = mp_int_copy(a, c)) != MP_OK)
+    return res;
+
+  if (s_qmul(c, (mp_size) p2))
+    return MP_OK;
+  else
+    return MP_MEMORY;
+}
+
+/* }}} */
+
+/* {{{ mp_int_sqr(a, c) */
+
+mp_result mp_int_sqr(mp_int a, mp_int c)
+{
+  mp_digit *out;
+  mp_size   osize, p = 0;
+
+  CHECK(a != NULL && c != NULL);
+
+  /* Get a temporary buffer big enough to hold the result */
+  osize = (mp_size) 4 * ((MP_USED(a) + 1) / 2);
+  if (a == c) {
+    p = ROUND_PREC(osize);
+    p = MAX(p, default_precision);
+
+    if ((out = s_alloc(p)) == NULL)
+      return MP_MEMORY;
+  }
+  else {
+    if (!s_pad(c, osize))
+      return MP_MEMORY;
+
+    out = MP_DIGITS(c);
+  }
+  ZERO(out, osize);
+
+  s_ksqr(MP_DIGITS(a), out, MP_USED(a));
+
+  /* Get rid of whatever memory c was already using, and fix up its fields to
+     reflect the new digit array it's using
+   */
+  if (out != MP_DIGITS(c)) {
+    if ((void *) MP_DIGITS(c) != (void *) c)
+      s_free(MP_DIGITS(c));
+    MP_DIGITS(c) = out;
+    MP_ALLOC(c) = p;
+  }
+
+  MP_USED(c) = osize; /* might not be true, but we'll fix it ... */
+  CLAMP(c);           /* ... right here */
+  MP_SIGN(c) = MP_ZPOS;
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_int_div(a, b, q, r) */
+
+mp_result mp_int_div(mp_int a, mp_int b, mp_int q, mp_int r)
+{
+  int cmp, lg;
+  mp_result res = MP_OK;
+  mp_int qout, rout;
+  mp_sign sa = MP_SIGN(a), sb = MP_SIGN(b);
+  DECLARE_TEMP(2);
+
+  CHECK(a != NULL && b != NULL && q != r);
+
+  if (CMPZ(b) == 0)
+    return MP_UNDEF;
+  else if ((cmp = s_ucmp(a, b)) < 0) {
+    /* If |a| < |b|, no division is required:
+       q = 0, r = a
+     */
+    if (r && (res = mp_int_copy(a, r)) != MP_OK)
+      return res;
+
+    if (q)
+      mp_int_zero(q);
+
+    return MP_OK;
+  }
+  else if (cmp == 0) {
+    /* If |a| = |b|, no division is required:
+       q = 1 or -1, r = 0
+     */
+    if (r)
+      mp_int_zero(r);
+
+    if (q) {
+      mp_int_zero(q);
+      q->digits[0] = 1;
+
+      if (sa != sb)
+	MP_SIGN(q) = MP_NEG;
+    }
+
+    return MP_OK;
+  }
+
+  /* When |a| > |b|, real division is required.  We need someplace to store
+     quotient and remainder, but q and r are allowed to be NULL or to overlap
+     with the inputs.
+   */
+  if ((lg = s_isp2(b)) < 0) {
+    if (q && b != q) {
+      if ((res = mp_int_copy(a, q)) != MP_OK)
+	goto CLEANUP;
+      else
+	qout = q;
+    }
+    else {
+      qout = LAST_TEMP();
+      SETUP(mp_int_init_copy(LAST_TEMP(), a));
+    }
+
+    if (r && a != r) {
+      if ((res = mp_int_copy(b, r)) != MP_OK)
+	goto CLEANUP;
+      else
+	rout = r;
+    }
+    else {
+      rout = LAST_TEMP();
+      SETUP(mp_int_init_copy(LAST_TEMP(), b));
+    }
+
+    if ((res = s_udiv_knuth(qout, rout)) != MP_OK) goto CLEANUP;
+  }
+  else {
+    if (q && (res = mp_int_copy(a, q)) != MP_OK) goto CLEANUP;
+    if (r && (res = mp_int_copy(a, r)) != MP_OK) goto CLEANUP;
+
+    if (q) s_qdiv(q, (mp_size) lg); qout = q;
+    if (r) s_qmod(r, (mp_size) lg); rout = r;
+  }
+
+  /* Recompute signs for output */
+  if (rout) {
+    MP_SIGN(rout) = sa;
+    if (CMPZ(rout) == 0)
+      MP_SIGN(rout) = MP_ZPOS;
+  }
+  if (qout) {
+    MP_SIGN(qout) = (sa == sb) ? MP_ZPOS : MP_NEG;
+    if (CMPZ(qout) == 0)
+      MP_SIGN(qout) = MP_ZPOS;
+  }
+
+  if (q && (res = mp_int_copy(qout, q)) != MP_OK) goto CLEANUP;
+  if (r && (res = mp_int_copy(rout, r)) != MP_OK) goto CLEANUP;
+
+  CLEANUP_TEMP();
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_int_mod(a, m, c) */
+
+mp_result mp_int_mod(mp_int a, mp_int m, mp_int c)
+{
+  mp_result res;
+  mpz_t     tmp;
+  mp_int    out;
+
+  if (m == c) {
+    mp_int_init(&tmp);
+    out = &tmp;
+  }
+  else {
+    out = c;
+  }
+
+  if ((res = mp_int_div(a, m, NULL, out)) != MP_OK)
+    goto CLEANUP;
+
+  if (CMPZ(out) < 0)
+    res = mp_int_add(out, m, c);
+  else
+    res = mp_int_copy(out, c);
+
+ CLEANUP:
+  if (out != c)
+    mp_int_clear(&tmp);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_int_div_value(a, value, q, r) */
+
+mp_result mp_int_div_value(mp_int a, mp_small value, mp_int q, mp_small *r)
+{
+  mpz_t     vtmp, rtmp;
+  mp_digit  vbuf[MP_VALUE_DIGITS(value)];
+  mp_result res;
+
+  mp_int_init(&rtmp);
+  s_fake(&vtmp, value, vbuf);
+
+  if ((res = mp_int_div(a, &vtmp, q, &rtmp)) != MP_OK)
+    goto CLEANUP;
+
+  if (r)
+    (void) mp_int_to_int(&rtmp, r); /* can't fail */
+
+ CLEANUP:
+  mp_int_clear(&rtmp);
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_int_div_pow2(a, p2, q, r) */
+
+mp_result mp_int_div_pow2(mp_int a, mp_small p2, mp_int q, mp_int r)
+{
+  mp_result res = MP_OK;
+
+  CHECK(a != NULL && p2 >= 0 && q != r);
+
+  if (q != NULL && (res = mp_int_copy(a, q)) == MP_OK)
+    s_qdiv(q, (mp_size) p2);
+
+  if (res == MP_OK && r != NULL && (res = mp_int_copy(a, r)) == MP_OK)
+    s_qmod(r, (mp_size) p2);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_int_expt(a, b, c) */
+
+mp_result mp_int_expt(mp_int a, mp_small b, mp_int c)
+{
+  mpz_t t;
+  mp_result res;
+  unsigned int v = abs(b);
+
+  CHECK(c != NULL);
+  if (b < 0)
+    return MP_RANGE;
+
+  if ((res = mp_int_init_copy(&t, a)) != MP_OK)
+    return res;
+
+  (void) mp_int_set_value(c, 1);
+  while (v != 0) {
+    if (v & 1) {
+      if ((res = mp_int_mul(c, &t, c)) != MP_OK)
+	goto CLEANUP;
+    }
+
+    v >>= 1;
+    if (v == 0) break;
+
+    if ((res = mp_int_sqr(&t, &t)) != MP_OK)
+      goto CLEANUP;
+  }
+
+ CLEANUP:
+  mp_int_clear(&t);
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_int_expt_value(a, b, c) */
+
+mp_result mp_int_expt_value(mp_small a, mp_small b, mp_int c)
+{
+  mpz_t     t;
+  mp_result res;
+  unsigned int v = abs(b);
+
+  CHECK(c != NULL);
+  if (b < 0)
+    return MP_RANGE;
+
+  if ((res = mp_int_init_value(&t, a)) != MP_OK)
+    return res;
+
+  (void) mp_int_set_value(c, 1);
+  while (v != 0) {
+    if (v & 1) {
+      if ((res = mp_int_mul(c, &t, c)) != MP_OK)
+	goto CLEANUP;
+    }
+
+    v >>= 1;
+    if (v == 0) break;
+
+    if ((res = mp_int_sqr(&t, &t)) != MP_OK)
+      goto CLEANUP;
+  }
+
+ CLEANUP:
+  mp_int_clear(&t);
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_int_expt_full(a, b, c) */
+
+mp_result mp_int_expt_full(mp_int a, mp_int b, mp_int c)
+{
+  mpz_t t;
+  mp_result res;
+  unsigned ix, jx;
+
+  CHECK(a != NULL && b != NULL && c != NULL);
+  if (MP_SIGN(b) == MP_NEG)
+    return MP_RANGE;
+
+  if ((res = mp_int_init_copy(&t, a)) != MP_OK)
+    return res;
+
+  (void) mp_int_set_value(c, 1);
+  for (ix = 0; ix < MP_USED(b); ++ix) {
+    mp_digit d = b->digits[ix];
+
+    for (jx = 0; jx < MP_DIGIT_BIT; ++jx) {
+      if (d & 1) {
+	if ((res = mp_int_mul(c, &t, c)) != MP_OK)
+	  goto CLEANUP;
+      }
+
+      d >>= 1;
+      if (d == 0 && ix + 1 == MP_USED(b))
+	break;
+      if ((res = mp_int_sqr(&t, &t)) != MP_OK)
+	goto CLEANUP;
+    }
+  }
+
+ CLEANUP:
+  mp_int_clear(&t);
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_int_compare(a, b) */
+
+int       mp_int_compare(mp_int a, mp_int b)
+{
+  mp_sign sa;
+
+  CHECK(a != NULL && b != NULL);
+
+  sa = MP_SIGN(a);
+  if (sa == MP_SIGN(b)) {
+    int cmp = s_ucmp(a, b);
+
+    /* If they're both zero or positive, the normal comparison applies; if both
+       negative, the sense is reversed. */
+    if (sa == MP_ZPOS)
+      return cmp;
+    else
+      return -cmp;
+
+  }
+  else {
+    if (sa == MP_ZPOS)
+      return 1;
+    else
+      return -1;
+  }
+}
+
+/* }}} */
+
+/* {{{ mp_int_compare_unsigned(a, b) */
+
+int       mp_int_compare_unsigned(mp_int a, mp_int b)
+{
+  NRCHECK(a != NULL && b != NULL);
+
+  return s_ucmp(a, b);
+}
+
+/* }}} */
+
+/* {{{ mp_int_compare_zero(z) */
+
+int       mp_int_compare_zero(mp_int z)
+{
+  NRCHECK(z != NULL);
+
+  if (MP_USED(z) == 1 && z->digits[0] == 0)
+    return 0;
+  else if (MP_SIGN(z) == MP_ZPOS)
+    return 1;
+  else
+    return -1;
+}
+
+/* }}} */
+
+/* {{{ mp_int_compare_value(z, value) */
+
+int       mp_int_compare_value(mp_int z, mp_small value)
+{
+  mp_sign vsign = (value < 0) ? MP_NEG : MP_ZPOS;
+  int cmp;
+
+  CHECK(z != NULL);
+
+  if (vsign == MP_SIGN(z)) {
+    cmp = s_vcmp(z, value);
+
+    return (vsign == MP_ZPOS) ? cmp : -cmp;
+  }
+  else {
+    return (value < 0) ? 1 : -1;
+  }
+}
+
+/* }}} */
+
+/* {{{ mp_int_compare_uvalue(z, uv) */
+
+int       mp_int_compare_uvalue(mp_int z, mp_usmall uv)
+{
+  CHECK(z != NULL);
+
+  if (MP_SIGN(z) == MP_NEG)
+    return -1;
+  else
+    return s_uvcmp(z, uv);
+}
+
+/* }}} */
+
+/* {{{ mp_int_exptmod(a, b, m, c) */
+
+mp_result mp_int_exptmod(mp_int a, mp_int b, mp_int m, mp_int c)
+{
+  mp_result res;
+  mp_size um;
+  mp_int s;
+  DECLARE_TEMP(3);
+
+  CHECK(a != NULL && b != NULL && c != NULL && m != NULL);
+
+  /* Zero moduli and negative exponents are not considered. */
+  if (CMPZ(m) == 0)
+    return MP_UNDEF;
+  if (CMPZ(b) < 0)
+    return MP_RANGE;
+
+  um = MP_USED(m);
+  SETUP(mp_int_init_size(TEMP(0), 2 * um));
+  SETUP(mp_int_init_size(TEMP(1), 2 * um));
+
+  if (c == b || c == m) {
+    SETUP(mp_int_init_size(TEMP(2), 2 * um));
+    s = TEMP(2);
+  }
+  else {
+    s = c;
+  }
+
+  if ((res = mp_int_mod(a, m, TEMP(0))) != MP_OK) goto CLEANUP;
+
+  if ((res = s_brmu(TEMP(1), m)) != MP_OK) goto CLEANUP;
+
+  if ((res = s_embar(TEMP(0), b, m, TEMP(1), s)) != MP_OK)
+    goto CLEANUP;
+
+  res = mp_int_copy(s, c);
+
+  CLEANUP_TEMP();
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_int_exptmod_evalue(a, value, m, c) */
+
+mp_result mp_int_exptmod_evalue(mp_int a, mp_small value, mp_int m, mp_int c)
+{
+  mpz_t vtmp;
+  mp_digit vbuf[MP_VALUE_DIGITS(value)];
+
+  s_fake(&vtmp, value, vbuf);
+
+  return mp_int_exptmod(a, &vtmp, m, c);
+}
+
+/* }}} */
+
+/* {{{ mp_int_exptmod_bvalue(v, b, m, c) */
+
+mp_result mp_int_exptmod_bvalue(mp_small value, mp_int b,
+				mp_int m, mp_int c)
+{
+  mpz_t vtmp;
+  mp_digit vbuf[MP_VALUE_DIGITS(value)];
+
+  s_fake(&vtmp, value, vbuf);
+
+  return mp_int_exptmod(&vtmp, b, m, c);
+}
+
+/* }}} */
+
+/* {{{ mp_int_exptmod_known(a, b, m, mu, c) */
+
+mp_result mp_int_exptmod_known(mp_int a, mp_int b, mp_int m, mp_int mu, mp_int c)
+{
+  mp_result res;
+  mp_size um;
+  mp_int s;
+  DECLARE_TEMP(2);
+
+  CHECK(a && b && m && c);
+
+  /* Zero moduli and negative exponents are not considered. */
+  if (CMPZ(m) == 0)
+    return MP_UNDEF;
+  if (CMPZ(b) < 0)
+    return MP_RANGE;
+
+  um = MP_USED(m);
+  SETUP(mp_int_init_size(TEMP(0), 2 * um));
+
+  if (c == b || c == m) {
+    SETUP(mp_int_init_size(TEMP(1), 2 * um));
+    s = TEMP(1);
+  }
+  else {
+    s = c;
+  }
+
+  if ((res = mp_int_mod(a, m, TEMP(0))) != MP_OK) goto CLEANUP;
+
+  if ((res = s_embar(TEMP(0), b, m, mu, s)) != MP_OK)
+    goto CLEANUP;
+
+  res = mp_int_copy(s, c);
+
+  CLEANUP_TEMP();
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_int_redux_const(m, c) */
+
+mp_result mp_int_redux_const(mp_int m, mp_int c)
+{
+  CHECK(m != NULL && c != NULL && m != c);
+
+  return s_brmu(c, m);
+}
+
+/* }}} */
+
+/* {{{ mp_int_invmod(a, m, c) */
+
+mp_result mp_int_invmod(mp_int a, mp_int m, mp_int c)
+{
+  mp_result res;
+  mp_sign sa;
+  DECLARE_TEMP(2);
+
+  CHECK(a != NULL && m != NULL && c != NULL);
+
+  if (CMPZ(a) == 0 || CMPZ(m) <= 0)
+    return MP_RANGE;
+
+  sa = MP_SIGN(a); /* need this for the result later */
+
+  for (last__ = 0; last__ < 2; ++last__)
+    mp_int_init(LAST_TEMP());
+
+  if ((res = mp_int_egcd(a, m, TEMP(0), TEMP(1), NULL)) != MP_OK)
+    goto CLEANUP;
+
+  if (mp_int_compare_value(TEMP(0), 1) != 0) {
+    res = MP_UNDEF;
+    goto CLEANUP;
+  }
+
+  /* It is first necessary to constrain the value to the proper range */
+  if ((res = mp_int_mod(TEMP(1), m, TEMP(1))) != MP_OK)
+    goto CLEANUP;
+
+  /* Now, if 'a' was originally negative, the value we have is actually the
+     magnitude of the negative representative; to get the positive value we
+     have to subtract from the modulus.  Otherwise, the value is okay as it
+     stands.
+   */
+  if (sa == MP_NEG)
+    res = mp_int_sub(m, TEMP(1), c);
+  else
+    res = mp_int_copy(TEMP(1), c);
+
+  CLEANUP_TEMP();
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_int_gcd(a, b, c) */
+
+/* Binary GCD algorithm due to Josef Stein, 1961 */
+mp_result mp_int_gcd(mp_int a, mp_int b, mp_int c)
+{
+  int ca, cb, k = 0;
+  mpz_t u, v, t;
+  mp_result res;
+
+  CHECK(a != NULL && b != NULL && c != NULL);
+
+  ca = CMPZ(a);
+  cb = CMPZ(b);
+  if (ca == 0 && cb == 0)
+    return MP_UNDEF;
+  else if (ca == 0)
+    return mp_int_abs(b, c);
+  else if (cb == 0)
+    return mp_int_abs(a, c);
+
+  mp_int_init(&t);
+  if ((res = mp_int_init_copy(&u, a)) != MP_OK)
+    goto U;
+  if ((res = mp_int_init_copy(&v, b)) != MP_OK)
+    goto V;
+
+  MP_SIGN(&u) = MP_ZPOS; MP_SIGN(&v) = MP_ZPOS;
+
+  { /* Divide out common factors of 2 from u and v */
+    int div2_u = s_dp2k(&u), div2_v = s_dp2k(&v);
+
+    k = MIN(div2_u, div2_v);
+    s_qdiv(&u, (mp_size) k);
+    s_qdiv(&v, (mp_size) k);
+  }
+
+  if (mp_int_is_odd(&u)) {
+    if ((res = mp_int_neg(&v, &t)) != MP_OK)
+      goto CLEANUP;
+  }
+  else {
+    if ((res = mp_int_copy(&u, &t)) != MP_OK)
+      goto CLEANUP;
+  }
+
+  for (;;) {
+    s_qdiv(&t, s_dp2k(&t));
+
+    if (CMPZ(&t) > 0) {
+      if ((res = mp_int_copy(&t, &u)) != MP_OK)
+	goto CLEANUP;
+    }
+    else {
+      if ((res = mp_int_neg(&t, &v)) != MP_OK)
+	goto CLEANUP;
+    }
+
+    if ((res = mp_int_sub(&u, &v, &t)) != MP_OK)
+      goto CLEANUP;
+
+    if (CMPZ(&t) == 0)
+      break;
+  }
+
+  if ((res = mp_int_abs(&u, c)) != MP_OK)
+    goto CLEANUP;
+  if (!s_qmul(c, (mp_size) k))
+    res = MP_MEMORY;
+
+ CLEANUP:
+  mp_int_clear(&v);
+ V: mp_int_clear(&u);
+ U: mp_int_clear(&t);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_int_egcd(a, b, c, x, y) */
+
+/* This is the binary GCD algorithm again, but this time we keep track of the
+   elementary matrix operations as we go, so we can get values x and y
+   satisfying c = ax + by.
+ */
+mp_result mp_int_egcd(mp_int a, mp_int b, mp_int c,
+		      mp_int x, mp_int y)
+{
+  int k, ca, cb;
+  mp_result res;
+  DECLARE_TEMP(8);
+
+  CHECK(a != NULL && b != NULL && c != NULL &&
+	(x != NULL || y != NULL));
+
+  ca = CMPZ(a);
+  cb = CMPZ(b);
+  if (ca == 0 && cb == 0)
+    return MP_UNDEF;
+  else if (ca == 0) {
+    if ((res = mp_int_abs(b, c)) != MP_OK) return res;
+    mp_int_zero(x); (void) mp_int_set_value(y, 1); return MP_OK;
+  }
+  else if (cb == 0) {
+    if ((res = mp_int_abs(a, c)) != MP_OK) return res;
+    (void) mp_int_set_value(x, 1); mp_int_zero(y); return MP_OK;
+  }
+
+  /* Initialize temporaries:
+     A:0, B:1, C:2, D:3, u:4, v:5, ou:6, ov:7 */
+  for (last__ = 0; last__ < 4; ++last__)
+    mp_int_init(LAST_TEMP());
+  TEMP(0)->digits[0] = 1;
+  TEMP(3)->digits[0] = 1;
+
+  SETUP(mp_int_init_copy(TEMP(4), a));
+  SETUP(mp_int_init_copy(TEMP(5), b));
+
+  /* We will work with absolute values here */
+  MP_SIGN(TEMP(4)) = MP_ZPOS;
+  MP_SIGN(TEMP(5)) = MP_ZPOS;
+
+  { /* Divide out common factors of 2 from u and v */
+    int  div2_u = s_dp2k(TEMP(4)), div2_v = s_dp2k(TEMP(5));
+
+    k = MIN(div2_u, div2_v);
+    s_qdiv(TEMP(4), k);
+    s_qdiv(TEMP(5), k);
+  }
+
+  SETUP(mp_int_init_copy(TEMP(6), TEMP(4)));
+  SETUP(mp_int_init_copy(TEMP(7), TEMP(5)));
+
+  for (;;) {
+    while (mp_int_is_even(TEMP(4))) {
+      s_qdiv(TEMP(4), 1);
+
+      if (mp_int_is_odd(TEMP(0)) || mp_int_is_odd(TEMP(1))) {
+	if ((res = mp_int_add(TEMP(0), TEMP(7), TEMP(0))) != MP_OK)
+	  goto CLEANUP;
+	if ((res = mp_int_sub(TEMP(1), TEMP(6), TEMP(1))) != MP_OK)
+	  goto CLEANUP;
+      }
+
+      s_qdiv(TEMP(0), 1);
+      s_qdiv(TEMP(1), 1);
+    }
+
+    while (mp_int_is_even(TEMP(5))) {
+      s_qdiv(TEMP(5), 1);
+
+      if (mp_int_is_odd(TEMP(2)) || mp_int_is_odd(TEMP(3))) {
+	if ((res = mp_int_add(TEMP(2), TEMP(7), TEMP(2))) != MP_OK)
+	  goto CLEANUP;
+	if ((res = mp_int_sub(TEMP(3), TEMP(6), TEMP(3))) != MP_OK)
+	  goto CLEANUP;
+      }
+
+      s_qdiv(TEMP(2), 1);
+      s_qdiv(TEMP(3), 1);
+    }
+
+    if (mp_int_compare(TEMP(4), TEMP(5)) >= 0) {
+      if ((res = mp_int_sub(TEMP(4), TEMP(5), TEMP(4))) != MP_OK) goto CLEANUP;
+      if ((res = mp_int_sub(TEMP(0), TEMP(2), TEMP(0))) != MP_OK) goto CLEANUP;
+      if ((res = mp_int_sub(TEMP(1), TEMP(3), TEMP(1))) != MP_OK) goto CLEANUP;
+    }
+    else {
+      if ((res = mp_int_sub(TEMP(5), TEMP(4), TEMP(5))) != MP_OK) goto CLEANUP;
+      if ((res = mp_int_sub(TEMP(2), TEMP(0), TEMP(2))) != MP_OK) goto CLEANUP;
+      if ((res = mp_int_sub(TEMP(3), TEMP(1), TEMP(3))) != MP_OK) goto CLEANUP;
+    }
+
+    if (CMPZ(TEMP(4)) == 0) {
+      if (x && (res = mp_int_copy(TEMP(2), x)) != MP_OK) goto CLEANUP;
+      if (y && (res = mp_int_copy(TEMP(3), y)) != MP_OK) goto CLEANUP;
+      if (c) {
+	if (!s_qmul(TEMP(5), k)) {
+	  res = MP_MEMORY;
+	  goto CLEANUP;
+	}
+
+	res = mp_int_copy(TEMP(5), c);
+      }
+
+      break;
+    }
+  }
+
+  CLEANUP_TEMP();
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_int_lcm(a, b, c) */
+
+mp_result mp_int_lcm(mp_int a, mp_int b, mp_int c)
+{
+  mpz_t lcm;
+  mp_result res;
+
+  CHECK(a != NULL && b != NULL && c != NULL);
+
+  /* Since a * b = gcd(a, b) * lcm(a, b), we can compute
+     lcm(a, b) = (a / gcd(a, b)) * b.
+
+     This formulation insures everything works even if the input
+     variables share space.
+   */
+  if ((res = mp_int_init(&lcm)) != MP_OK)
+    return res;
+  if ((res = mp_int_gcd(a, b, &lcm)) != MP_OK)
+    goto CLEANUP;
+  if ((res = mp_int_div(a, &lcm, &lcm, NULL)) != MP_OK)
+    goto CLEANUP;
+  if ((res = mp_int_mul(&lcm, b, &lcm)) != MP_OK)
+    goto CLEANUP;
+
+  res = mp_int_copy(&lcm, c);
+
+  CLEANUP:
+    mp_int_clear(&lcm);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_int_divisible_value(a, v) */
+
+int       mp_int_divisible_value(mp_int a, mp_small v)
+{
+  mp_small rem = 0;
+
+  if (mp_int_div_value(a, v, NULL, &rem) != MP_OK)
+    return 0;
+
+  return rem == 0;
+}
+
+/* }}} */
+
+/* {{{ mp_int_is_pow2(z) */
+
+int       mp_int_is_pow2(mp_int z)
+{
+  CHECK(z != NULL);
+
+  return s_isp2(z);
+}
+
+/* }}} */
+
+/* {{{ mp_int_root(a, b, c) */
+
+/* Implementation of Newton's root finding method, based loosely on a patch
+   contributed by Hal Finkel <half@halssoftware.com>
+   modified by M. J. Fromberger.
+ */
+mp_result mp_int_root(mp_int a, mp_small b, mp_int c)
+{
+  mp_result res = MP_OK;
+  int flips = 0;
+  DECLARE_TEMP(5);
+
+  CHECK(a != NULL && c != NULL && b > 0);
+
+  if (b == 1) {
+    return mp_int_copy(a, c);
+  }
+  if (MP_SIGN(a) == MP_NEG) {
+    if (b % 2 == 0)
+      return MP_UNDEF; /* root does not exist for negative a with even b */
+    else
+      flips = 1;
+  }
+
+  SETUP(mp_int_init_copy(LAST_TEMP(), a));
+  SETUP(mp_int_init_copy(LAST_TEMP(), a));
+  SETUP(mp_int_init(LAST_TEMP()));
+  SETUP(mp_int_init(LAST_TEMP()));
+  SETUP(mp_int_init(LAST_TEMP()));
+
+  (void) mp_int_abs(TEMP(0), TEMP(0));
+  (void) mp_int_abs(TEMP(1), TEMP(1));
+
+  for (;;) {
+    if ((res = mp_int_expt(TEMP(1), b, TEMP(2))) != MP_OK)
+      goto CLEANUP;
+
+    if (mp_int_compare_unsigned(TEMP(2), TEMP(0)) <= 0)
+      break;
+
+    if ((res = mp_int_sub(TEMP(2), TEMP(0), TEMP(2))) != MP_OK)
+      goto CLEANUP;
+    if ((res = mp_int_expt(TEMP(1), b - 1, TEMP(3))) != MP_OK)
+      goto CLEANUP;
+    if ((res = mp_int_mul_value(TEMP(3), b, TEMP(3))) != MP_OK)
+      goto CLEANUP;
+    if ((res = mp_int_div(TEMP(2), TEMP(3), TEMP(4), NULL)) != MP_OK)
+      goto CLEANUP;
+    if ((res = mp_int_sub(TEMP(1), TEMP(4), TEMP(4))) != MP_OK)
+      goto CLEANUP;
+
+    if (mp_int_compare_unsigned(TEMP(1), TEMP(4)) == 0) {
+      if ((res = mp_int_sub_value(TEMP(4), 1, TEMP(4))) != MP_OK)
+	goto CLEANUP;
+    }
+    if ((res = mp_int_copy(TEMP(4), TEMP(1))) != MP_OK)
+      goto CLEANUP;
+  }
+
+  if ((res = mp_int_copy(TEMP(1), c)) != MP_OK)
+    goto CLEANUP;
+
+  /* If the original value of a was negative, flip the output sign. */
+  if (flips)
+    (void) mp_int_neg(c, c); /* cannot fail */
+
+  CLEANUP_TEMP();
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_int_to_int(z, *out) */
+
+mp_result mp_int_to_int(mp_int z, mp_small *out)
+{
+  mp_usmall uv = 0;
+  mp_size   uz;
+  mp_digit *dz;
+  mp_sign   sz;
+
+  CHECK(z != NULL);
+
+  /* Make sure the value is representable as a small integer */
+  sz = MP_SIGN(z);
+  if ((sz == MP_ZPOS && mp_int_compare_value(z, MP_SMALL_MAX) > 0) ||
+      mp_int_compare_value(z, MP_SMALL_MIN) < 0)
+    return MP_RANGE;
+
+  uz = MP_USED(z);
+  dz = MP_DIGITS(z) + uz - 1;
+
+  while (uz > 0) {
+    uv <<= MP_DIGIT_BIT/2;
+    uv = (uv << (MP_DIGIT_BIT/2)) | *dz--;
+    --uz;
+  }
+
+  if (out)
+    *out = (sz == MP_NEG) ? -(mp_small)uv : (mp_small)uv;
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_int_to_uint(z, *out) */
+
+mp_result mp_int_to_uint(mp_int z, mp_usmall *out)
+{
+  mp_usmall uv = 0;
+  mp_size   uz;
+  mp_digit *dz;
+  mp_sign   sz;
+  
+  CHECK(z != NULL);
+
+  /* Make sure the value is representable as an unsigned small integer */
+  sz = MP_SIGN(z);
+  if (sz == MP_NEG || mp_int_compare_uvalue(z, MP_USMALL_MAX) > 0)
+    return MP_RANGE;
+     
+  uz = MP_USED(z);
+  dz = MP_DIGITS(z) + uz - 1;
+  
+  while (uz > 0) {
+    uv <<= MP_DIGIT_BIT/2;
+    uv = (uv << (MP_DIGIT_BIT/2)) | *dz--;
+    --uz;
+  }
+  
+  if (out)
+    *out = uv;
+  
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_int_to_string(z, radix, str, limit) */
+
+mp_result mp_int_to_string(mp_int z, mp_size radix,
+			   char *str, int limit)
+{
+  mp_result res;
+  int       cmp = 0;
+
+  CHECK(z != NULL && str != NULL && limit >= 2);
+
+  if (radix < MP_MIN_RADIX || radix > MP_MAX_RADIX)
+    return MP_RANGE;
+
+  if (CMPZ(z) == 0) {
+    *str++ = s_val2ch(0, 1);
+  }
+  else {
+    mpz_t tmp;
+    char  *h, *t;
+
+    if ((res = mp_int_init_copy(&tmp, z)) != MP_OK)
+      return res;
+
+    if (MP_SIGN(z) == MP_NEG) {
+      *str++ = '-';
+      --limit;
+    }
+    h = str;
+
+    /* Generate digits in reverse order until finished or limit reached */
+    for (/* */; limit > 0; --limit) {
+      mp_digit d;
+
+      if ((cmp = CMPZ(&tmp)) == 0)
+	break;
+
+      d = s_ddiv(&tmp, (mp_digit)radix);
+      *str++ = s_val2ch(d, 1);
+    }
+    t = str - 1;
+
+    /* Put digits back in correct output order */
+    while (h < t) {
+      char tc = *h;
+      *h++ = *t;
+      *t-- = tc;
+    }
+
+    mp_int_clear(&tmp);
+  }
+
+  *str = '\0';
+  if (cmp == 0)
+    return MP_OK;
+  else
+    return MP_TRUNC;
+}
+
+/* }}} */
+
+/* {{{ mp_int_string_len(z, radix) */
+
+mp_result mp_int_string_len(mp_int z, mp_size radix)
+{
+  int  len;
+
+  CHECK(z != NULL);
+
+  if (radix < MP_MIN_RADIX || radix > MP_MAX_RADIX)
+    return MP_RANGE;
+
+  len = s_outlen(z, radix) + 1; /* for terminator */
+
+  /* Allow for sign marker on negatives */
+  if (MP_SIGN(z) == MP_NEG)
+    len += 1;
+
+  return len;
+}
+
+/* }}} */
+
+/* {{{ mp_int_read_string(z, radix, *str) */
+
+/* Read zero-terminated string into z */
+mp_result mp_int_read_string(mp_int z, mp_size radix, const char *str)
+{
+  return mp_int_read_cstring(z, radix, str, NULL);
+}
+
+/* }}} */
+
+/* {{{ mp_int_read_cstring(z, radix, *str, **end) */
+
+mp_result mp_int_read_cstring(mp_int z, mp_size radix, const char *str, char **end)
+{
+  int ch;
+
+  CHECK(z != NULL && str != NULL);
+
+  if (radix < MP_MIN_RADIX || radix > MP_MAX_RADIX)
+    return MP_RANGE;
+
+  /* Skip leading whitespace */
+  while (isspace((int)*str))
+    ++str;
+
+  /* Handle leading sign tag (+/-, positive default) */
+  switch (*str) {
+  case '-':
+    MP_SIGN(z) = MP_NEG;
+    ++str;
+    break;
+  case '+':
+    ++str; /* fallthrough */
+  default:
+    MP_SIGN(z) = MP_ZPOS;
+    break;
+  }
+
+  /* Skip leading zeroes */
+  while ((ch = s_ch2val(*str, radix)) == 0)
+    ++str;
+
+  /* Make sure there is enough space for the value */
+  if (!s_pad(z, s_inlen(strlen(str), radix)))
+    return MP_MEMORY;
+
+  MP_USED(z) = 1; z->digits[0] = 0;
+
+  while (*str != '\0' && ((ch = s_ch2val(*str, radix)) >= 0)) {
+    s_dmul(z, (mp_digit)radix);
+    s_dadd(z, (mp_digit)ch);
+    ++str;
+  }
+
+  CLAMP(z);
+
+  /* Override sign for zero, even if negative specified. */
+  if (CMPZ(z) == 0)
+    MP_SIGN(z) = MP_ZPOS;
+
+  if (end != NULL)
+    *end = (char *)str;
+
+  /* Return a truncation error if the string has unprocessed characters
+     remaining, so the caller can tell if the whole string was done */
+  if (*str != '\0')
+    return MP_TRUNC;
+  else
+    return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_int_count_bits(z) */
+
+mp_result mp_int_count_bits(mp_int z)
+{
+  mp_size  nbits = 0, uz;
+  mp_digit d;
+
+  CHECK(z != NULL);
+
+  uz = MP_USED(z);
+  if (uz == 1 && z->digits[0] == 0)
+    return 1;
+
+  --uz;
+  nbits = uz * MP_DIGIT_BIT;
+  d = z->digits[uz];
+
+  while (d != 0) {
+    d >>= 1;
+    ++nbits;
+  }
+
+  return nbits;
+}
+
+/* }}} */
+
+/* {{{ mp_int_to_binary(z, buf, limit) */
+
+mp_result mp_int_to_binary(mp_int z, unsigned char *buf, int limit)
+{
+  static const int PAD_FOR_2C = 1;
+
+  mp_result res;
+  int limpos = limit;
+
+  CHECK(z != NULL && buf != NULL);
+
+  res = s_tobin(z, buf, &limpos, PAD_FOR_2C);
+
+  if (MP_SIGN(z) == MP_NEG)
+    s_2comp(buf, limpos);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_int_read_binary(z, buf, len) */
+
+mp_result mp_int_read_binary(mp_int z, unsigned char *buf, int len)
+{
+  mp_size need, i;
+  unsigned char *tmp;
+  mp_digit *dz;
+
+  CHECK(z != NULL && buf != NULL && len > 0);
+
+  /* Figure out how many digits are needed to represent this value */
+  need = ((len * CHAR_BIT) + (MP_DIGIT_BIT - 1)) / MP_DIGIT_BIT;
+  if (!s_pad(z, need))
+    return MP_MEMORY;
+
+  mp_int_zero(z);
+
+  /* If the high-order bit is set, take the 2's complement before reading the
+     value (it will be restored afterward) */
+  if (buf[0] >> (CHAR_BIT - 1)) {
+    MP_SIGN(z) = MP_NEG;
+    s_2comp(buf, len);
+  }
+
+  dz = MP_DIGITS(z);
+  for (tmp = buf, i = len; i > 0; --i, ++tmp) {
+    s_qmul(z, (mp_size) CHAR_BIT);
+    *dz |= *tmp;
+  }
+
+  /* Restore 2's complement if we took it before */
+  if (MP_SIGN(z) == MP_NEG)
+    s_2comp(buf, len);
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_int_binary_len(z) */
+
+mp_result mp_int_binary_len(mp_int z)
+{
+  mp_result  res = mp_int_count_bits(z);
+  int        bytes = mp_int_unsigned_len(z);
+
+  if (res <= 0)
+    return res;
+
+  bytes = (res + (CHAR_BIT - 1)) / CHAR_BIT;
+
+  /* If the highest-order bit falls exactly on a byte boundary, we need to pad
+     with an extra byte so that the sign will be read correctly when reading it
+     back in. */
+  if (bytes * CHAR_BIT == res)
+    ++bytes;
+
+  return bytes;
+}
+
+/* }}} */
+
+/* {{{ mp_int_to_unsigned(z, buf, limit) */
+
+mp_result mp_int_to_unsigned(mp_int z, unsigned char *buf, int limit)
+{
+  static const int NO_PADDING = 0;
+
+  CHECK(z != NULL && buf != NULL);
+
+  return s_tobin(z, buf, &limit, NO_PADDING);
+}
+
+/* }}} */
+
+/* {{{ mp_int_read_unsigned(z, buf, len) */
+
+mp_result mp_int_read_unsigned(mp_int z, unsigned char *buf, int len)
+{
+  mp_size need, i;
+  unsigned char *tmp;
+
+  CHECK(z != NULL && buf != NULL && len > 0);
+
+  /* Figure out how many digits are needed to represent this value */
+  need = ((len * CHAR_BIT) + (MP_DIGIT_BIT - 1)) / MP_DIGIT_BIT;
+  if (!s_pad(z, need))
+    return MP_MEMORY;
+
+  mp_int_zero(z);
+
+  for (tmp = buf, i = len; i > 0; --i, ++tmp) {
+    (void) s_qmul(z, CHAR_BIT);
+    *MP_DIGITS(z) |= *tmp;
+  }
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_int_unsigned_len(z) */
+
+mp_result mp_int_unsigned_len(mp_int z)
+{
+  mp_result  res = mp_int_count_bits(z);
+  int        bytes;
+
+  if (res <= 0)
+    return res;
+
+  bytes = (res + (CHAR_BIT - 1)) / CHAR_BIT;
+
+  return bytes;
+}
+
+/* }}} */
+
+/* {{{ mp_error_string(res) */
+
+const char *mp_error_string(mp_result res)
+{
+  int ix;
+  if (res > 0)
+    return s_unknown_err;
+
+  res = -res;
+  for (ix = 0; ix < res && s_error_msg[ix] != NULL; ++ix)
+    ;
+
+  if (s_error_msg[ix] != NULL)
+    return s_error_msg[ix];
+  else
+    return s_unknown_err;
+}
+
+/* }}} */
+
+/*------------------------------------------------------------------------*/
+/* Private functions for internal use.  These make assumptions.           */
+
+/* {{{ s_alloc(num) */
+
+STATIC mp_digit *s_alloc(mp_size num)
+{
+  mp_digit *out = malloc(num * sizeof(mp_digit));
+
+  assert(out != NULL); /* for debugging */
+#if DEBUG > 1
+  {
+    mp_digit v = (mp_digit) 0xdeadbeef;
+    int      ix;
+
+    for (ix = 0; ix < num; ++ix)
+      out[ix] = v;
+  }
+#endif
+
+  return out;
+}
+
+/* }}} */
+
+/* {{{ s_realloc(old, osize, nsize) */
+
+STATIC mp_digit *s_realloc(mp_digit *old, mp_size osize, mp_size nsize)
+{
+#if DEBUG > 1
+  mp_digit *new = s_alloc(nsize);
+  int       ix;
+
+  for (ix = 0; ix < nsize; ++ix)
+    new[ix] = (mp_digit) 0xdeadbeef;
+
+  memcpy(new, old, osize * sizeof(mp_digit));
+#else
+  mp_digit *new = realloc(old, nsize * sizeof(mp_digit));
+
+  assert(new != NULL); /* for debugging */
+#endif
+  return new;
+}
+
+/* }}} */
+
+/* {{{ s_free(ptr) */
+
+STATIC void s_free(void *ptr)
+{
+  free(ptr);
+}
+
+/* }}} */
+
+/* {{{ s_pad(z, min) */
+
+STATIC int      s_pad(mp_int z, mp_size min)
+{
+  if (MP_ALLOC(z) < min) {
+    mp_size nsize = ROUND_PREC(min);
+    mp_digit *tmp;
+
+    if ((void *)z->digits == (void *)z) {
+      if ((tmp = s_alloc(nsize)) == NULL)
+        return 0;
+
+      COPY(MP_DIGITS(z), tmp, MP_USED(z));
+    }
+    else if ((tmp = s_realloc(MP_DIGITS(z), MP_ALLOC(z), nsize)) == NULL)
+      return 0;
+
+    MP_DIGITS(z) = tmp;
+    MP_ALLOC(z) = nsize;
+  }
+
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ s_fake(z, value, vbuf[]) */
+
+/* Note: This will not work correctly when value == MP_SMALL_MIN */
+STATIC void      s_fake(mp_int z, mp_small value, mp_digit vbuf[])
+{
+  mp_usmall uv = (mp_usmall) (value < 0) ? -value : value;
+  s_ufake(z, uv, vbuf);
+  if (value < 0)
+    z->sign = MP_NEG;
+}
+
+/* }}} */
+
+/* {{{ s_ufake(z, value, vbuf[]) */
+
+STATIC void      s_ufake(mp_int z, mp_usmall value, mp_digit vbuf[])
+{
+  mp_size ndig = (mp_size) s_uvpack(value, vbuf);
+
+  z->used = ndig;
+  z->alloc = MP_VALUE_DIGITS(value);
+  z->sign = MP_ZPOS;
+  z->digits = vbuf;
+}
+
+/* }}} */
+
+/* {{{ s_cdig(da, db, len) */
+
+STATIC int      s_cdig(mp_digit *da, mp_digit *db, mp_size len)
+{
+  mp_digit *dat = da + len - 1, *dbt = db + len - 1;
+
+  for (/* */; len != 0; --len, --dat, --dbt) {
+    if (*dat > *dbt)
+      return 1;
+    else if (*dat < *dbt)
+      return -1;
+  }
+
+  return 0;
+}
+
+/* }}} */
+
+/* {{{ s_uvpack(uv, t[]) */
+
+STATIC int       s_uvpack(mp_usmall uv, mp_digit t[])
+{
+  int ndig = 0;
+
+  if (uv == 0)
+    t[ndig++] = 0;
+  else {
+    while (uv != 0) {
+      t[ndig++] = (mp_digit) uv;
+      uv >>= MP_DIGIT_BIT/2;
+      uv >>= MP_DIGIT_BIT/2;
+    }
+  }
+
+  return ndig;
+}
+
+/* }}} */
+
+/* {{{ s_ucmp(a, b) */
+
+STATIC int      s_ucmp(mp_int a, mp_int b)
+{
+  mp_size  ua = MP_USED(a), ub = MP_USED(b);
+
+  if (ua > ub)
+    return 1;
+  else if (ub > ua)
+    return -1;
+  else
+    return s_cdig(MP_DIGITS(a), MP_DIGITS(b), ua);
+}
+
+/* }}} */
+
+/* {{{ s_vcmp(a, v) */
+
+STATIC int      s_vcmp(mp_int a, mp_small v)
+{
+  mp_usmall uv = (mp_usmall) (v < 0) ? -v : v;
+  return s_uvcmp(a, uv);
+}
+
+/* }}} */
+
+/* {{{ s_uvcmp(a, v) */
+
+STATIC int      s_uvcmp(mp_int a, mp_usmall uv)
+{
+  mpz_t vtmp;
+  mp_digit vdig[MP_VALUE_DIGITS(uv)];
+
+  s_ufake(&vtmp, uv, vdig);
+  return s_ucmp(a, &vtmp);
+}
+
+/* }}} */
+
+/* {{{ s_uadd(da, db, dc, size_a, size_b) */
+
+STATIC mp_digit s_uadd(mp_digit *da, mp_digit *db, mp_digit *dc,
+		       mp_size size_a, mp_size size_b)
+{
+  mp_size pos;
+  mp_word w = 0;
+
+  /* Insure that da is the longer of the two to simplify later code */
+  if (size_b > size_a) {
+    SWAP(mp_digit *, da, db);
+    SWAP(mp_size, size_a, size_b);
+  }
+
+  /* Add corresponding digits until the shorter number runs out */
+  for (pos = 0; pos < size_b; ++pos, ++da, ++db, ++dc) {
+    w = w + (mp_word) *da + (mp_word) *db;
+    *dc = LOWER_HALF(w);
+    w = UPPER_HALF(w);
+  }
+
+  /* Propagate carries as far as necessary */
+  for (/* */; pos < size_a; ++pos, ++da, ++dc) {
+    w = w + *da;
+
+    *dc = LOWER_HALF(w);
+    w = UPPER_HALF(w);
+  }
+
+  /* Return carry out */
+  return (mp_digit)w;
+}
+
+/* }}} */
+
+/* {{{ s_usub(da, db, dc, size_a, size_b) */
+
+STATIC void     s_usub(mp_digit *da, mp_digit *db, mp_digit *dc,
+		       mp_size size_a, mp_size size_b)
+{
+  mp_size pos;
+  mp_word w = 0;
+
+  /* We assume that |a| >= |b| so this should definitely hold */
+  assert(size_a >= size_b);
+
+  /* Subtract corresponding digits and propagate borrow */
+  for (pos = 0; pos < size_b; ++pos, ++da, ++db, ++dc) {
+    w = ((mp_word)MP_DIGIT_MAX + 1 +  /* MP_RADIX */
+	 (mp_word)*da) - w - (mp_word)*db;
+
+    *dc = LOWER_HALF(w);
+    w = (UPPER_HALF(w) == 0);
+  }
+
+  /* Finish the subtraction for remaining upper digits of da */
+  for (/* */; pos < size_a; ++pos, ++da, ++dc) {
+    w = ((mp_word)MP_DIGIT_MAX + 1 +  /* MP_RADIX */
+	 (mp_word)*da) - w;
+
+    *dc = LOWER_HALF(w);
+    w = (UPPER_HALF(w) == 0);
+  }
+
+  /* If there is a borrow out at the end, it violates the precondition */
+  assert(w == 0);
+}
+
+/* }}} */
+
+/* {{{ s_kmul(da, db, dc, size_a, size_b) */
+
+STATIC int       s_kmul(mp_digit *da, mp_digit *db, mp_digit *dc,
+			mp_size size_a, mp_size size_b)
+{
+  mp_size  bot_size;
+
+  /* Make sure b is the smaller of the two input values */
+  if (size_b > size_a) {
+    SWAP(mp_digit *, da, db);
+    SWAP(mp_size, size_a, size_b);
+  }
+
+  /* Insure that the bottom is the larger half in an odd-length split; the code
+     below relies on this being true.
+   */
+  bot_size = (size_a + 1) / 2;
+
+  /* If the values are big enough to bother with recursion, use the Karatsuba
+     algorithm to compute the product; otherwise use the normal multiplication
+     algorithm
+   */
+  if (multiply_threshold &&
+      size_a >= multiply_threshold &&
+      size_b > bot_size) {
+
+    mp_digit *t1, *t2, *t3, carry;
+
+    mp_digit *a_top = da + bot_size;
+    mp_digit *b_top = db + bot_size;
+
+    mp_size  at_size = size_a - bot_size;
+    mp_size  bt_size = size_b - bot_size;
+    mp_size  buf_size = 2 * bot_size;
+
+    /* Do a single allocation for all three temporary buffers needed; each
+       buffer must be big enough to hold the product of two bottom halves, and
+       one buffer needs space for the completed product; twice the space is
+       plenty.
+     */
+    if ((t1 = s_alloc(4 * buf_size)) == NULL) return 0;
+    t2 = t1 + buf_size;
+    t3 = t2 + buf_size;
+    ZERO(t1, 4 * buf_size);
+
+    /* t1 and t2 are initially used as temporaries to compute the inner product
+       (a1 + a0)(b1 + b0) = a1b1 + a1b0 + a0b1 + a0b0
+     */
+    carry = s_uadd(da, a_top, t1, bot_size, at_size);      /* t1 = a1 + a0 */
+    t1[bot_size] = carry;
+
+    carry = s_uadd(db, b_top, t2, bot_size, bt_size);      /* t2 = b1 + b0 */
+    t2[bot_size] = carry;
+
+    (void) s_kmul(t1, t2, t3, bot_size + 1, bot_size + 1); /* t3 = t1 * t2 */
+
+    /* Now we'll get t1 = a0b0 and t2 = a1b1, and subtract them out so that
+       we're left with only the pieces we want:  t3 = a1b0 + a0b1
+     */
+    ZERO(t1, buf_size);
+    ZERO(t2, buf_size);
+    (void) s_kmul(da, db, t1, bot_size, bot_size);     /* t1 = a0 * b0 */
+    (void) s_kmul(a_top, b_top, t2, at_size, bt_size); /* t2 = a1 * b1 */
+
+    /* Subtract out t1 and t2 to get the inner product */
+    s_usub(t3, t1, t3, buf_size + 2, buf_size);
+    s_usub(t3, t2, t3, buf_size + 2, buf_size);
+
+    /* Assemble the output value */
+    COPY(t1, dc, buf_size);
+    carry = s_uadd(t3, dc + bot_size, dc + bot_size,
+		   buf_size + 1, buf_size);
+    assert(carry == 0);
+
+    carry = s_uadd(t2, dc + 2*bot_size, dc + 2*bot_size,
+		   buf_size, buf_size);
+    assert(carry == 0);
+
+    s_free(t1); /* note t2 and t3 are just internal pointers to t1 */
+  }
+  else {
+    s_umul(da, db, dc, size_a, size_b);
+  }
+
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ s_umul(da, db, dc, size_a, size_b) */
+
+STATIC void     s_umul(mp_digit *da, mp_digit *db, mp_digit *dc,
+		       mp_size size_a, mp_size size_b)
+{
+  mp_size a, b;
+  mp_word w;
+
+  for (a = 0; a < size_a; ++a, ++dc, ++da) {
+    mp_digit *dct = dc;
+    mp_digit *dbt = db;
+
+    if (*da == 0)
+      continue;
+
+    w = 0;
+    for (b = 0; b < size_b; ++b, ++dbt, ++dct) {
+      w = (mp_word)*da * (mp_word)*dbt + w + (mp_word)*dct;
+
+      *dct = LOWER_HALF(w);
+      w = UPPER_HALF(w);
+    }
+
+    *dct = (mp_digit)w;
+  }
+}
+
+/* }}} */
+
+/* {{{ s_ksqr(da, dc, size_a) */
+
+STATIC int       s_ksqr(mp_digit *da, mp_digit *dc, mp_size size_a)
+{
+  if (multiply_threshold && size_a > multiply_threshold) {
+    mp_size  bot_size = (size_a + 1) / 2;
+    mp_digit *a_top = da + bot_size;
+    mp_digit *t1, *t2, *t3, carry;
+    mp_size  at_size = size_a - bot_size;
+    mp_size  buf_size = 2 * bot_size;
+
+    if ((t1 = s_alloc(4 * buf_size)) == NULL) return 0;
+    t2 = t1 + buf_size;
+    t3 = t2 + buf_size;
+    ZERO(t1, 4 * buf_size);
+
+    (void) s_ksqr(da, t1, bot_size);    /* t1 = a0 ^ 2 */
+    (void) s_ksqr(a_top, t2, at_size);  /* t2 = a1 ^ 2 */
+
+    (void) s_kmul(da, a_top, t3, bot_size, at_size);  /* t3 = a0 * a1 */
+
+    /* Quick multiply t3 by 2, shifting left (can't overflow) */
+    {
+      int i, top = bot_size + at_size;
+      mp_word w, save = 0;
+
+      for (i = 0; i < top; ++i) {
+	w = t3[i];
+	w = (w << 1) | save;
+	t3[i] = LOWER_HALF(w);
+	save = UPPER_HALF(w);
+      }
+      t3[i] = LOWER_HALF(save);
+    }
+
+    /* Assemble the output value */
+    COPY(t1, dc, 2 * bot_size);
+    carry = s_uadd(t3, dc + bot_size, dc + bot_size,
+		   buf_size + 1, buf_size);
+    assert(carry == 0);
+
+    carry = s_uadd(t2, dc + 2*bot_size, dc + 2*bot_size,
+		   buf_size, buf_size);
+    assert(carry == 0);
+
+    s_free(t1); /* note that t2 and t2 are internal pointers only */
+
+  } 
+  else {
+    s_usqr(da, dc, size_a);
+  }
+
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ s_usqr(da, dc, size_a) */
+
+STATIC void      s_usqr(mp_digit *da, mp_digit *dc, mp_size size_a)
+{
+  mp_size i, j;
+  mp_word w;
+
+  for (i = 0; i < size_a; ++i, dc += 2, ++da) {
+    mp_digit  *dct = dc, *dat = da;
+
+    if (*da == 0)
+      continue;
+
+    /* Take care of the first digit, no rollover */
+    w = (mp_word)*dat * (mp_word)*dat + (mp_word)*dct;
+    *dct = LOWER_HALF(w);
+    w = UPPER_HALF(w);
+    ++dat; ++dct;
+
+    for (j = i + 1; j < size_a; ++j, ++dat, ++dct) {
+      mp_word  t = (mp_word)*da * (mp_word)*dat;
+      mp_word  u = w + (mp_word)*dct, ov = 0;
+
+      /* Check if doubling t will overflow a word */
+      if (HIGH_BIT_SET(t))
+	ov = 1;
+
+      w = t + t;
+
+      /* Check if adding u to w will overflow a word */
+      if (ADD_WILL_OVERFLOW(w, u))
+	ov = 1;
+
+      w += u;
+
+      *dct = LOWER_HALF(w);
+      w = UPPER_HALF(w);
+      if (ov) {
+	w += MP_DIGIT_MAX; /* MP_RADIX */
+	++w;
+      }
+    }
+
+    w = w + *dct;
+    *dct = (mp_digit)w;
+    while ((w = UPPER_HALF(w)) != 0) {
+      ++dct; w = w + *dct;
+      *dct = LOWER_HALF(w);
+    }
+
+    assert(w == 0);
+  }
+}
+
+/* }}} */
+
+/* {{{ s_dadd(a, b) */
+
+STATIC void      s_dadd(mp_int a, mp_digit b)
+{
+  mp_word w = 0;
+  mp_digit *da = MP_DIGITS(a);
+  mp_size ua = MP_USED(a);
+
+  w = (mp_word)*da + b;
+  *da++ = LOWER_HALF(w);
+  w = UPPER_HALF(w);
+
+  for (ua -= 1; ua > 0; --ua, ++da) {
+    w = (mp_word)*da + w;
+
+    *da = LOWER_HALF(w);
+    w = UPPER_HALF(w);
+  }
+
+  if (w) {
+    *da = (mp_digit)w;
+    MP_USED(a) += 1;
+  }
+}
+
+/* }}} */
+
+/* {{{ s_dmul(a, b) */
+
+STATIC void      s_dmul(mp_int a, mp_digit b)
+{
+  mp_word w = 0;
+  mp_digit *da = MP_DIGITS(a);
+  mp_size ua = MP_USED(a);
+
+  while (ua > 0) {
+    w = (mp_word)*da * b + w;
+    *da++ = LOWER_HALF(w);
+    w = UPPER_HALF(w);
+    --ua;
+  }
+
+  if (w) {
+    *da = (mp_digit)w;
+    MP_USED(a) += 1;
+  }
+}
+
+/* }}} */
+
+/* {{{ s_dbmul(da, b, dc, size_a) */
+
+STATIC void      s_dbmul(mp_digit *da, mp_digit b, mp_digit *dc, mp_size size_a)
+{
+  mp_word  w = 0;
+
+  while (size_a > 0) {
+    w = (mp_word)*da++ * (mp_word)b + w;
+
+    *dc++ = LOWER_HALF(w);
+    w = UPPER_HALF(w);
+    --size_a;
+  }
+
+  if (w)
+    *dc = LOWER_HALF(w);
+}
+
+/* }}} */
+
+/* {{{ s_ddiv(da, d, dc, size_a) */
+
+STATIC mp_digit  s_ddiv(mp_int a, mp_digit b)
+{
+  mp_word w = 0, qdigit;
+  mp_size ua = MP_USED(a);
+  mp_digit *da = MP_DIGITS(a) + ua - 1;
+
+  for (/* */; ua > 0; --ua, --da) {
+    w = (w << MP_DIGIT_BIT) | *da;
+
+    if (w >= b) {
+      qdigit = w / b;
+      w = w % b;
+    }
+    else {
+      qdigit = 0;
+    }
+
+    *da = (mp_digit)qdigit;
+  }
+
+  CLAMP(a);
+  return (mp_digit)w;
+}
+
+/* }}} */
+
+/* {{{ s_qdiv(z, p2) */
+
+STATIC void     s_qdiv(mp_int z, mp_size p2)
+{
+  mp_size ndig = p2 / MP_DIGIT_BIT, nbits = p2 % MP_DIGIT_BIT;
+  mp_size uz = MP_USED(z);
+
+  if (ndig) {
+    mp_size  mark;
+    mp_digit *to, *from;
+
+    if (ndig >= uz) {
+      mp_int_zero(z);
+      return;
+    }
+
+    to = MP_DIGITS(z); from = to + ndig;
+
+    for (mark = ndig; mark < uz; ++mark)
+      *to++ = *from++;
+
+    MP_USED(z) = uz - ndig;
+  }
+
+  if (nbits) {
+    mp_digit d = 0, *dz, save;
+    mp_size  up = MP_DIGIT_BIT - nbits;
+
+    uz = MP_USED(z);
+    dz = MP_DIGITS(z) + uz - 1;
+
+    for (/* */; uz > 0; --uz, --dz) {
+      save = *dz;
+
+      *dz = (*dz >> nbits) | (d << up);
+      d = save;
+    }
+
+    CLAMP(z);
+  }
+
+  if (MP_USED(z) == 1 && z->digits[0] == 0)
+    MP_SIGN(z) = MP_ZPOS;
+}
+
+/* }}} */
+
+/* {{{ s_qmod(z, p2) */
+
+STATIC void     s_qmod(mp_int z, mp_size p2)
+{
+  mp_size start = p2 / MP_DIGIT_BIT + 1, rest = p2 % MP_DIGIT_BIT;
+  mp_size uz = MP_USED(z);
+  mp_digit mask = (1 << rest) - 1;
+
+  if (start <= uz) {
+    MP_USED(z) = start;
+    z->digits[start - 1] &= mask;
+    CLAMP(z);
+  }
+}
+
+/* }}} */
+
+/* {{{ s_qmul(z, p2) */
+
+STATIC int      s_qmul(mp_int z, mp_size p2)
+{
+  mp_size   uz, need, rest, extra, i;
+  mp_digit *from, *to, d;
+
+  if (p2 == 0)
+    return 1;
+
+  uz = MP_USED(z); 
+  need = p2 / MP_DIGIT_BIT; rest = p2 % MP_DIGIT_BIT;
+
+  /* Figure out if we need an extra digit at the top end; this occurs if the
+     topmost `rest' bits of the high-order digit of z are not zero, meaning
+     they will be shifted off the end if not preserved */
+  extra = 0;
+  if (rest != 0) {
+    mp_digit *dz = MP_DIGITS(z) + uz - 1;
+
+    if ((*dz >> (MP_DIGIT_BIT - rest)) != 0)
+      extra = 1;
+  }
+
+  if (!s_pad(z, uz + need + extra))
+    return 0;
+
+  /* If we need to shift by whole digits, do that in one pass, then
+     to back and shift by partial digits.
+   */
+  if (need > 0) {
+    from = MP_DIGITS(z) + uz - 1;
+    to = from + need;
+
+    for (i = 0; i < uz; ++i)
+      *to-- = *from--;
+
+    ZERO(MP_DIGITS(z), need);
+    uz += need;
+  }
+
+  if (rest) {
+    d = 0;
+    for (i = need, from = MP_DIGITS(z) + need; i < uz; ++i, ++from) {
+      mp_digit save = *from;
+      
+      *from = (*from << rest) | (d >> (MP_DIGIT_BIT - rest));
+      d = save;
+    }
+
+    d >>= (MP_DIGIT_BIT - rest);
+    if (d != 0) {
+      *from = d;
+      uz += extra;
+    }
+  }
+
+  MP_USED(z) = uz;
+  CLAMP(z);
+
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ s_qsub(z, p2) */
+
+/* Compute z = 2^p2 - |z|; requires that 2^p2 >= |z|
+   The sign of the result is always zero/positive.
+ */
+STATIC int       s_qsub(mp_int z, mp_size p2)
+{
+  mp_digit hi = (1 << (p2 % MP_DIGIT_BIT)), *zp;
+  mp_size  tdig = (p2 / MP_DIGIT_BIT), pos;
+  mp_word  w = 0;
+
+  if (!s_pad(z, tdig + 1))
+    return 0;
+
+  for (pos = 0, zp = MP_DIGITS(z); pos < tdig; ++pos, ++zp) {
+    w = ((mp_word) MP_DIGIT_MAX + 1) - w - (mp_word)*zp;
+
+    *zp = LOWER_HALF(w);
+    w = UPPER_HALF(w) ? 0 : 1;
+  }
+
+  w = ((mp_word) MP_DIGIT_MAX + 1 + hi) - w - (mp_word)*zp;
+  *zp = LOWER_HALF(w);
+
+  assert(UPPER_HALF(w) != 0); /* no borrow out should be possible */
+
+  MP_SIGN(z) = MP_ZPOS;
+  CLAMP(z);
+
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ s_dp2k(z) */
+
+STATIC int      s_dp2k(mp_int z)
+{
+  int       k = 0;
+  mp_digit *dp = MP_DIGITS(z), d;
+
+  if (MP_USED(z) == 1 && *dp == 0)
+    return 1;
+
+  while (*dp == 0) {
+    k += MP_DIGIT_BIT;
+    ++dp;
+  }
+
+  d = *dp;
+  while ((d & 1) == 0) {
+    d >>= 1;
+    ++k;
+  }
+
+  return k;
+}
+
+/* }}} */
+
+/* {{{ s_isp2(z) */
+
+STATIC int       s_isp2(mp_int z)
+{
+  mp_size uz = MP_USED(z), k = 0;
+  mp_digit *dz = MP_DIGITS(z), d;
+
+  while (uz > 1) {
+    if (*dz++ != 0)
+      return -1;
+    k += MP_DIGIT_BIT;
+    --uz;
+  }
+
+  d = *dz;
+  while (d > 1) {
+    if (d & 1)
+      return -1;
+    ++k; d >>= 1;
+  }
+
+  return (int) k;
+}
+
+/* }}} */
+
+/* {{{ s_2expt(z, k) */
+
+STATIC int       s_2expt(mp_int z, mp_small k)
+{
+  mp_size  ndig, rest;
+  mp_digit *dz;
+
+  ndig = (k + MP_DIGIT_BIT) / MP_DIGIT_BIT;
+  rest = k % MP_DIGIT_BIT;
+
+  if (!s_pad(z, ndig))
+    return 0;
+
+  dz = MP_DIGITS(z);
+  ZERO(dz, ndig);
+  *(dz + ndig - 1) = (1 << rest);
+  MP_USED(z) = ndig;
+
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ s_norm(a, b) */
+
+STATIC int      s_norm(mp_int a, mp_int b)
+{
+  mp_digit d = b->digits[MP_USED(b) - 1];
+  int k = 0;
+
+  while (d < (mp_digit) (1 << (MP_DIGIT_BIT - 1))) { /* d < (MP_RADIX / 2) */
+    d <<= 1;
+    ++k;
+  }
+
+  /* These multiplications can't fail */
+  if (k != 0) {
+    (void) s_qmul(a, (mp_size) k);
+    (void) s_qmul(b, (mp_size) k);
+  }
+
+  return k;
+}
+
+/* }}} */
+
+/* {{{ s_brmu(z, m) */
+
+STATIC mp_result s_brmu(mp_int z, mp_int m)
+{
+  mp_size um = MP_USED(m) * 2;
+
+  if (!s_pad(z, um))
+    return MP_MEMORY;
+
+  s_2expt(z, MP_DIGIT_BIT * um);
+  return mp_int_div(z, m, z, NULL);
+}
+
+/* }}} */
+
+/* {{{ s_reduce(x, m, mu, q1, q2) */
+
+STATIC int       s_reduce(mp_int x, mp_int m, mp_int mu, mp_int q1, mp_int q2)
+{
+  mp_size   um = MP_USED(m), umb_p1, umb_m1;
+
+  umb_p1 = (um + 1) * MP_DIGIT_BIT;
+  umb_m1 = (um - 1) * MP_DIGIT_BIT;
+
+  if (mp_int_copy(x, q1) != MP_OK)
+    return 0;
+
+  /* Compute q2 = floor((floor(x / b^(k-1)) * mu) / b^(k+1)) */
+  s_qdiv(q1, umb_m1);
+  UMUL(q1, mu, q2);
+  s_qdiv(q2, umb_p1);
+
+  /* Set x = x mod b^(k+1) */
+  s_qmod(x, umb_p1);
+
+  /* Now, q is a guess for the quotient a / m.
+     Compute x - q * m mod b^(k+1), replacing x.  This may be off
+     by a factor of 2m, but no more than that.
+   */
+  UMUL(q2, m, q1);
+  s_qmod(q1, umb_p1);
+  (void) mp_int_sub(x, q1, x); /* can't fail */
+
+  /* The result may be < 0; if it is, add b^(k+1) to pin it in the proper
+     range. */
+  if ((CMPZ(x) < 0) && !s_qsub(x, umb_p1))
+    return 0;
+
+  /* If x > m, we need to back it off until it is in range.  This will be
+     required at most twice.  */
+  if (mp_int_compare(x, m) >= 0) {
+    (void) mp_int_sub(x, m, x);
+    if (mp_int_compare(x, m) >= 0)
+      (void) mp_int_sub(x, m, x);
+  }
+
+  /* At this point, x has been properly reduced. */
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ s_embar(a, b, m, mu, c) */
+
+/* Perform modular exponentiation using Barrett's method, where mu is the
+   reduction constant for m.  Assumes a < m, b > 0. */
+STATIC mp_result s_embar(mp_int a, mp_int b, mp_int m, mp_int mu, mp_int c)
+{
+  mp_digit  *db, *dbt, umu, d;
+  mp_result res;
+  DECLARE_TEMP(3);
+
+  umu = MP_USED(mu); db = MP_DIGITS(b); dbt = db + MP_USED(b) - 1;
+
+  while (last__ < 3) {
+    SETUP(mp_int_init_size(LAST_TEMP(), 4 * umu));
+    ZERO(MP_DIGITS(TEMP(last__ - 1)), MP_ALLOC(TEMP(last__ - 1)));
+  }
+
+  (void) mp_int_set_value(c, 1);
+
+  /* Take care of low-order digits */
+  while (db < dbt) {
+    int      i;
+
+    for (d = *db, i = MP_DIGIT_BIT; i > 0; --i, d >>= 1) {
+      if (d & 1) {
+	/* The use of a second temporary avoids allocation */
+	UMUL(c, a, TEMP(0));
+	if (!s_reduce(TEMP(0), m, mu, TEMP(1), TEMP(2))) {
+	  res = MP_MEMORY; goto CLEANUP;
+	}
+	mp_int_copy(TEMP(0), c);
+      }
+
+
+      USQR(a, TEMP(0));
+      assert(MP_SIGN(TEMP(0)) == MP_ZPOS);
+      if (!s_reduce(TEMP(0), m, mu, TEMP(1), TEMP(2))) {
+	res = MP_MEMORY; goto CLEANUP;
+      }
+      assert(MP_SIGN(TEMP(0)) == MP_ZPOS);
+      mp_int_copy(TEMP(0), a);
+    }
+
+    ++db;
+  }
+
+  /* Take care of highest-order digit */
+  d = *dbt;
+  for (;;) {
+    if (d & 1) {
+      UMUL(c, a, TEMP(0));
+      if (!s_reduce(TEMP(0), m, mu, TEMP(1), TEMP(2))) {
+	res = MP_MEMORY; goto CLEANUP;
+      }
+      mp_int_copy(TEMP(0), c);
+    }
+
+    d >>= 1;
+    if (!d) break;
+
+    USQR(a, TEMP(0));
+    if (!s_reduce(TEMP(0), m, mu, TEMP(1), TEMP(2))) {
+      res = MP_MEMORY; goto CLEANUP;
+    }
+    (void) mp_int_copy(TEMP(0), a);
+  }
+
+  CLEANUP_TEMP();
+  return res;
+}
+
+/* }}} */
+
+#if 0
+/* {{{ s_udiv(a, b) */
+/* The s_udiv function produces incorrect results. For example, with test
+     div:11141460315522012760862883825:48318382095:0,230584300062375935
+   commenting out the function for now and using s_udiv_knuth instead.
+   STATIC mp_result s_udiv(mp_int a, mp_int b);
+*/
+
+/* Precondition:  a >= b and b > 0
+   Postcondition: a' = a / b, b' = a % b
+ */
+STATIC mp_result s_udiv(mp_int a, mp_int b)
+{
+  mpz_t q, r, t;
+  mp_size ua, ub, qpos = 0;
+  mp_digit *da, btop;
+  mp_result res = MP_OK;
+  int k, skip = 0;
+
+  /* Force signs to positive */
+  MP_SIGN(a) = MP_ZPOS;
+  MP_SIGN(b) = MP_ZPOS;
+
+  /* Normalize, per Knuth */
+  k = s_norm(a, b);
+
+  ua = MP_USED(a); ub = MP_USED(b); btop = b->digits[ub - 1];
+  if ((res = mp_int_init_size(&q, ua)) != MP_OK) return res;
+  if ((res = mp_int_init_size(&t, ua + 1)) != MP_OK) goto CLEANUP;
+
+  da = MP_DIGITS(a);
+  r.digits = da + ua - 1;  /* The contents of r are shared with a */
+  r.used   = 1;
+  r.sign   = MP_ZPOS;
+  r.alloc  = MP_ALLOC(a);
+  ZERO(t.digits, t.alloc);
+
+  /* Solve for quotient digits, store in q.digits in reverse order */
+  while (r.digits >= da) {
+    assert(qpos <= q.alloc);
+
+    if (s_ucmp(b, &r) > 0) {
+      r.digits -= 1;
+      r.used += 1;
+
+      if (++skip > 1 && qpos > 0)
+	q.digits[qpos++] = 0;
+
+      CLAMP(&r);
+    }
+    else {
+      mp_word  pfx = r.digits[r.used - 1];
+      mp_word  qdigit;
+
+      if (r.used > 1 && pfx < btop) {
+	pfx <<= MP_DIGIT_BIT / 2;
+	pfx <<= MP_DIGIT_BIT / 2;
+	pfx |= r.digits[r.used - 2];
+      }
+
+      qdigit = pfx / btop;
+      if (qdigit > MP_DIGIT_MAX) {
+	qdigit = MP_DIGIT_MAX;
+      }
+
+      s_dbmul(MP_DIGITS(b), (mp_digit) qdigit, t.digits, ub);
+      t.used = ub + 1; CLAMP(&t);
+      while (s_ucmp(&t, &r) > 0) {
+	--qdigit;
+	(void) mp_int_sub(&t, b, &t); /* cannot fail */
+      }
+
+      s_usub(r.digits, t.digits, r.digits, r.used, t.used);
+      CLAMP(&r);
+
+      q.digits[qpos++] = (mp_digit) qdigit;
+      ZERO(t.digits, t.used);
+      skip = 0;
+    }
+  }
+
+  /* Put quotient digits in the correct order, and discard extra zeroes */
+  q.used = qpos;
+  REV(mp_digit, q.digits, qpos);
+  CLAMP(&q);
+
+  /* Denormalize the remainder */
+  CLAMP(a);
+  if (k != 0)
+    s_qdiv(a, k);
+
+  mp_int_copy(a, b);  /* ok:  0 <= r < b */
+  mp_int_copy(&q, a); /* ok:  q <= a     */
+
+  mp_int_clear(&t);
+ CLEANUP:
+  mp_int_clear(&q);
+  return res;
+}
+
+/* }}} */
+#endif
+
+/* Division of nonnegative integers
+
+   This function implements division algorithm for unsigned multi-precision
+   integers. The algorithm is based on Algorithm D from Knuth's "The Art of
+   Computer Programming", 3rd ed. 1998, pg 272-273.
+
+   We diverge from Knuth's algorithm in that we do not perform the subtraction
+   from the remainder until we have determined that we have the correct
+   quotient digit. This makes our algorithm less efficient that Knuth because
+   we might have to perform multiple multiplication and comparison steps before
+   the subtraction. The advantage is that it is easy to implement and ensure
+   correctness without worrying about underflow from the subtraction.
+
+   inputs: u   a n+m digit integer in base b (b is 2^MP_DIGIT_BIT)
+           v   a n   digit integer in base b (b is 2^MP_DIGIT_BIT)
+           n >= 1
+           m >= 0
+  outputs: u / v stored in u
+           u % v stored in v
+ */
+STATIC mp_result s_udiv_knuth(mp_int u, mp_int v) {
+  mpz_t q, r, t;
+  mp_result
+  res = MP_OK;
+  int k,j;
+  mp_size m,n;
+
+  /* Force signs to positive */
+  MP_SIGN(u) = MP_ZPOS;
+  MP_SIGN(v) = MP_ZPOS;
+
+  /* Use simple division algorithm when v is only one digit long */
+  if (MP_USED(v) == 1) {
+    mp_digit d, rem;
+    d   = v->digits[0];
+    rem = s_ddiv(u, d);
+    mp_int_set_value(v, rem);
+    return MP_OK;
+  }
+
+  /************************************************************/
+  /* Algorithm D */
+  /************************************************************/
+  /* The n and m variables are defined as used by Knuth.
+     u is an n digit number with digits u_{n-1}..u_0.
+     v is an n+m digit number with digits from v_{m+n-1}..v_0.
+     We require that n > 1 and m >= 0 */
+  n = MP_USED(v);
+  m = MP_USED(u) - n;
+  assert(n >  1);
+  assert(m >= 0);
+
+  /************************************************************/
+  /* D1: Normalize.
+     The normalization step provides the necessary condition for Theorem B,
+     which states that the quotient estimate for q_j, call it qhat
+
+       qhat = u_{j+n}u_{j+n-1} / v_{n-1}
+
+     is bounded by
+
+      qhat - 2 <= q_j <= qhat.
+
+     That is, qhat is always greater than the actual quotient digit q,
+     and it is never more than two larger than the actual quotient digit.  */
+  k = s_norm(u, v);
+
+  /* Extend size of u by one if needed.
+
+     The algorithm begins with a value of u that has one more digit of input.
+     The normalization step sets u_{m+n}..u_0 = 2^k * u_{m+n-1}..u_0. If the
+     multiplication did not increase the number of digits of u, we need to add
+     a leading zero here.
+   */
+  if (k == 0 || MP_USED(u) != m + n + 1) {
+    if (!s_pad(u, m+n+1))
+      return MP_MEMORY;
+    u->digits[m+n] = 0;
+    u->used = m+n+1;
+  }
+
+  /* Add a leading 0 to v.
+
+     The multiplication in step D4 multiplies qhat * 0v_{n-1}..v_0.  We need to
+     add the leading zero to v here to ensure that the multiplication will
+     produce the full n+1 digit result.  */
+  if (!s_pad(v, n+1)) return MP_MEMORY; v->digits[n] = 0;
+
+  /* Initialize temporary variables q and t.
+     q allocates space for m+1 digits to store the quotient digits
+     t allocates space for n+1 digits to hold the result of q_j*v */
+  if ((res = mp_int_init_size(&q, m + 1)) != MP_OK) return res;
+  if ((res = mp_int_init_size(&t, n + 1)) != MP_OK) goto CLEANUP;
+
+  /************************************************************/
+  /* D2: Initialize j */
+  j = m;
+  r.digits = MP_DIGITS(u) + j;  /* The contents of r are shared with u */
+  r.used   = n + 1;
+  r.sign   = MP_ZPOS;
+  r.alloc  = MP_ALLOC(u);
+  ZERO(t.digits, t.alloc);
+
+  /* Calculate the m+1 digits of the quotient result */
+  for (; j >= 0; j--) {
+    /************************************************************/
+    /* D3: Calculate q' */
+    /* r->digits is aligned to position j of the number u */
+    mp_word pfx, qhat;
+    pfx   = r.digits[n];
+    pfx <<= MP_DIGIT_BIT / 2;
+    pfx <<= MP_DIGIT_BIT / 2;
+    pfx |= r.digits[n-1]; /* pfx = u_{j+n}{j+n-1} */
+
+    qhat = pfx / v->digits[n-1];
+    /* Check to see if qhat > b, and decrease qhat if so.
+       Theorem B guarantess that qhat is at most 2 larger than the
+       actual value, so it is possible that qhat is greater than
+       the maximum value that will fit in a digit */
+    if (qhat > MP_DIGIT_MAX)
+      qhat = MP_DIGIT_MAX;
+
+    /************************************************************/
+    /* D4,D5,D6: Multiply qhat * v and test for a correct value of q
+
+       We proceed a bit different than the way described by Knuth. This way is
+       simpler but less efficent. Instead of doing the multiply and subtract
+       then checking for underflow, we first do the multiply of qhat * v and
+       see if it is larger than the current remainder r. If it is larger, we
+       decrease qhat by one and try again. We may need to decrease qhat one
+       more time before we get a value that is smaller than r.
+
+       This way is less efficent than Knuth becuase we do more multiplies, but
+       we do not need to worry about underflow this way.  */
+    /* t = qhat * v */
+    s_dbmul(MP_DIGITS(v), (mp_digit) qhat, t.digits, n+1); t.used = n + 1;
+    CLAMP(&t);
+
+    /* Clamp r for the comparison. Comparisons do not like leading zeros. */
+    CLAMP(&r);
+    if (s_ucmp(&t, &r) > 0) {   /* would the remainder be negative? */
+      qhat -= 1;   /* try a smaller q */
+      s_dbmul(MP_DIGITS(v), (mp_digit) qhat, t.digits, n+1);
+      t.used = n + 1; CLAMP(&t);
+      if (s_ucmp(&t, &r) > 0) { /* would the remainder be negative? */
+        assert(qhat > 0);
+        qhat -= 1; /* try a smaller q */
+        s_dbmul(MP_DIGITS(v), (mp_digit) qhat, t.digits, n+1);
+        t.used = n + 1; CLAMP(&t);
+      }
+      assert(s_ucmp(&t, &r) <=  0 && "The mathematics failed us.");
+    }
+    /* Unclamp r. The D algorithm expects r = u_{j+n}..u_j to always be n+1
+       digits long. */
+    r.used = n + 1;
+
+    /************************************************************/
+    /* D4: Multiply and subtract */
+    /* note: The multiply was completed above so we only need to subtract here.
+     **/
+    s_usub(r.digits, t.digits, r.digits, r.used, t.used);
+
+    /************************************************************/
+    /* D5: Test remainder */
+    /* note: Not needed because we always check that qhat is the correct value
+     *       before performing the subtract. */
+    q.digits[j] = qhat;
+
+    /************************************************************/
+    /* D6: Add back */
+    /* note: Not needed because we always check that qhat is the correct value
+     *       before performing the subtract. */
+
+    /************************************************************/
+    /* D7: Loop on j */
+    r.digits--;
+    ZERO(t.digits, t.alloc);
+  }
+
+  /* Get rid of leading zeros in q */
+  q.used = m + 1;
+  CLAMP(&q);
+
+  /* Denormalize the remainder */
+  CLAMP(u); /* use u here because the r.digits pointer is off-by-one */
+  if (k != 0)
+    s_qdiv(u, k);
+
+  mp_int_copy(u, v);  /* ok:  0 <= r < v */
+  mp_int_copy(&q, u); /* ok:  q <= u     */
+
+  mp_int_clear(&t);
+ CLEANUP:
+  mp_int_clear(&q);
+  return res;
+}
+
+/* {{{ s_outlen(z, r) */
+
+STATIC int       s_outlen(mp_int z, mp_size r)
+{
+  mp_result bits;
+  double raw;
+
+  assert(r >= MP_MIN_RADIX && r <= MP_MAX_RADIX);
+
+  bits = mp_int_count_bits(z);
+  raw = (double)bits * s_log2[r];
+
+  return (int)(raw + 0.999999);
+}
+
+/* }}} */
+
+/* {{{ s_inlen(len, r) */
+
+STATIC mp_size   s_inlen(int len, mp_size r)
+{
+  double  raw = (double)len / s_log2[r];
+  mp_size bits = (mp_size)(raw + 0.5);
+
+  return (mp_size)((bits + (MP_DIGIT_BIT - 1)) / MP_DIGIT_BIT) + 1;
+}
+
+/* }}} */
+
+/* {{{ s_ch2val(c, r) */
+
+STATIC int       s_ch2val(char c, int r)
+{
+  int out;
+
+  if (isdigit((unsigned char) c))
+    out = c - '0';
+  else if (r > 10 && isalpha((unsigned char) c))
+    out = toupper(c) - 'A' + 10;
+  else
+    return -1;
+
+  return (out >= r) ? -1 : out;
+}
+
+/* }}} */
+
+/* {{{ s_val2ch(v, caps) */
+
+STATIC char      s_val2ch(int v, int caps)
+{
+  assert(v >= 0);
+
+  if (v < 10)
+    return v + '0';
+  else {
+    char out = (v - 10) + 'a';
+
+    if (caps)
+      return toupper(out);
+    else
+      return out;
+  }
+}
+
+/* }}} */
+
+/* {{{ s_2comp(buf, len) */
+
+STATIC void      s_2comp(unsigned char *buf, int len)
+{
+  int i;
+  unsigned short s = 1;
+
+  for (i = len - 1; i >= 0; --i) {
+    unsigned char c = ~buf[i];
+
+    s = c + s;
+    c = s & UCHAR_MAX;
+    s >>= CHAR_BIT;
+
+    buf[i] = c;
+  }
+
+  /* last carry out is ignored */
+}
+
+/* }}} */
+
+/* {{{ s_tobin(z, buf, *limpos) */
+
+STATIC mp_result s_tobin(mp_int z, unsigned char *buf, int *limpos, int pad)
+{
+  mp_size uz;
+  mp_digit *dz;
+  int pos = 0, limit = *limpos;
+
+  uz = MP_USED(z); dz = MP_DIGITS(z);
+  while (uz > 0 && pos < limit) {
+    mp_digit d = *dz++;
+    int i;
+
+    for (i = sizeof(mp_digit); i > 0 && pos < limit; --i) {
+      buf[pos++] = (unsigned char)d;
+      d >>= CHAR_BIT;
+
+      /* Don't write leading zeroes */
+      if (d == 0 && uz == 1)
+	i = 0; /* exit loop without signaling truncation */
+    }
+
+    /* Detect truncation (loop exited with pos >= limit) */
+    if (i > 0) break;
+
+    --uz;
+  }
+
+  if (pad != 0 && (buf[pos - 1] >> (CHAR_BIT - 1))) {
+    if (pos < limit)
+      buf[pos++] = 0;
+    else
+      uz = 1;
+  }
+
+  /* Digits are in reverse order, fix that */
+  REV(unsigned char, buf, pos);
+
+  /* Return the number of bytes actually written */
+  *limpos = pos;
+
+  return (uz == 0) ? MP_OK : MP_TRUNC;
+}
+
+/* }}} */
+
+/* {{{ s_print(tag, z) */
+
+#if DEBUG
+void      s_print(char *tag, mp_int z)
+{
+  int  i;
+
+  fprintf(stderr, "%s: %c ", tag,
+	  (MP_SIGN(z) == MP_NEG) ? '-' : '+');
+
+  for (i = MP_USED(z) - 1; i >= 0; --i)
+    fprintf(stderr, "%0*X", (int)(MP_DIGIT_BIT / 4), z->digits[i]);
+
+  fputc('\n', stderr);
+
+}
+
+void      s_print_buf(char *tag, mp_digit *buf, mp_size num)
+{
+  int i;
+
+  fprintf(stderr, "%s: ", tag);
+
+  for (i = num - 1; i >= 0; --i)
+    fprintf(stderr, "%0*X", (int)(MP_DIGIT_BIT / 4), buf[i]);
+
+  fputc('\n', stderr);
+}
+#endif
+
+/* }}} */
+
+/* Here there be dragons */
diff --git a/polly/lib/External/isl/imath/imath.h b/polly/lib/External/isl/imath/imath.h
new file mode 100644
index 00000000000..a9f5e408a48
--- /dev/null
+++ b/polly/lib/External/isl/imath/imath.h
@@ -0,0 +1,232 @@
+/*
+  Name:     imath.h
+  Purpose:  Arbitrary precision integer arithmetic routines.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Copyright (C) 2002-2007 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#ifndef IMATH_H_
+#define IMATH_H_
+
+#include <stdint.h>
+#include <limits.h>
+
+#ifdef __cplusplus
+extern "C" {
+#endif
+
+typedef unsigned char      mp_sign;
+typedef unsigned int       mp_size;
+typedef int                mp_result;
+typedef long               mp_small;  /* must be a signed type */
+typedef unsigned long      mp_usmall; /* must be an unsigned type */
+
+/* Force building with uint64_t so that the library builds consistently
+ * whether we build from the makefile or by embedding imath in another project.
+ */
+#undef  USE_64BIT_WORDS
+#define USE_64BIT_WORDS
+#ifdef  USE_64BIT_WORDS
+typedef uint32_t           mp_digit;
+typedef uint64_t           mp_word;
+#else
+typedef uint16_t           mp_digit;
+typedef uint32_t           mp_word;
+#endif
+
+typedef struct mpz {
+  mp_digit    single;
+  mp_digit   *digits;
+  mp_size     alloc;
+  mp_size     used;
+  mp_sign     sign;
+} mpz_t, *mp_int;
+
+#define MP_DIGITS(Z) ((Z)->digits)
+#define MP_ALLOC(Z)  ((Z)->alloc)
+#define MP_USED(Z)   ((Z)->used)
+#define MP_SIGN(Z)   ((Z)->sign)
+
+extern const mp_result MP_OK;
+extern const mp_result MP_FALSE;
+extern const mp_result MP_TRUE;
+extern const mp_result MP_MEMORY;
+extern const mp_result MP_RANGE;
+extern const mp_result MP_UNDEF;
+extern const mp_result MP_TRUNC;
+extern const mp_result MP_BADARG;
+extern const mp_result MP_MINERR;
+
+#define MP_DIGIT_BIT    (sizeof(mp_digit) * CHAR_BIT)
+#define MP_WORD_BIT     (sizeof(mp_word) * CHAR_BIT)
+#define MP_SMALL_MIN    LONG_MIN
+#define MP_SMALL_MAX    LONG_MAX
+#define MP_USMALL_MIN   ULONG_MIN
+#define MP_USMALL_MAX   ULONG_MAX
+
+#ifdef USE_64BIT_WORDS
+#  define MP_DIGIT_MAX   (UINT32_MAX * UINT64_C(1))
+#  define MP_WORD_MAX    (UINT64_MAX)
+#else
+#  define MP_DIGIT_MAX   (UINT16_MAX * 1UL)
+#  define MP_WORD_MAX    (UINT32_MAX * 1UL)
+#endif
+
+#define MP_MIN_RADIX    2
+#define MP_MAX_RADIX    36
+
+/* Values with fewer than this many significant digits use the standard
+   multiplication algorithm; otherwise, a recursive algorithm is used.  
+   Choose a value to suit your platform.
+ */
+#define MP_MULT_THRESH  22
+
+#define MP_DEFAULT_PREC 8   /* default memory allocation, in digits */
+
+extern const mp_sign   MP_NEG;
+extern const mp_sign   MP_ZPOS;
+
+#define mp_int_is_odd(Z)  ((Z)->digits[0] & 1)
+#define mp_int_is_even(Z) !((Z)->digits[0] & 1)
+
+mp_result mp_int_init(mp_int z);
+mp_int    mp_int_alloc(void);
+mp_result mp_int_init_size(mp_int z, mp_size prec);
+mp_result mp_int_init_copy(mp_int z, mp_int old);
+mp_result mp_int_init_value(mp_int z, mp_small value);
+mp_result mp_int_init_uvalue(mp_int z, mp_usmall uvalue);
+mp_result mp_int_set_value(mp_int z, mp_small value);
+mp_result mp_int_set_uvalue(mp_int z, mp_usmall uvalue);
+void      mp_int_clear(mp_int z);
+void      mp_int_free(mp_int z);
+
+mp_result mp_int_copy(mp_int a, mp_int c);           /* c = a     */
+void      mp_int_swap(mp_int a, mp_int c);           /* swap a, c */
+void      mp_int_zero(mp_int z);                     /* z = 0     */
+mp_result mp_int_abs(mp_int a, mp_int c);            /* c = |a|   */
+mp_result mp_int_neg(mp_int a, mp_int c);            /* c = -a    */
+mp_result mp_int_add(mp_int a, mp_int b, mp_int c);  /* c = a + b */
+mp_result mp_int_add_value(mp_int a, mp_small value, mp_int c);
+mp_result mp_int_sub(mp_int a, mp_int b, mp_int c);  /* c = a - b */
+mp_result mp_int_sub_value(mp_int a, mp_small value, mp_int c);
+mp_result mp_int_mul(mp_int a, mp_int b, mp_int c);  /* c = a * b */
+mp_result mp_int_mul_value(mp_int a, mp_small value, mp_int c);
+mp_result mp_int_mul_pow2(mp_int a, mp_small p2, mp_int c);
+mp_result mp_int_sqr(mp_int a, mp_int c);            /* c = a * a */
+mp_result mp_int_div(mp_int a, mp_int b,             /* q = a / b */
+		     mp_int q, mp_int r);            /* r = a % b */
+mp_result mp_int_div_value(mp_int a, mp_small value, /* q = a / value */
+			   mp_int q, mp_small *r);   /* r = a % value */
+mp_result mp_int_div_pow2(mp_int a, mp_small p2,     /* q = a / 2^p2  */
+			  mp_int q, mp_int r);       /* r = q % 2^p2  */
+mp_result mp_int_mod(mp_int a, mp_int m, mp_int c);  /* c = a % m */
+#define   mp_int_mod_value(A, V, R) mp_int_div_value((A), (V), 0, (R))
+mp_result mp_int_expt(mp_int a, mp_small b, mp_int c);         /* c = a^b */
+mp_result mp_int_expt_value(mp_small a, mp_small b, mp_int c); /* c = a^b */
+mp_result mp_int_expt_full(mp_int a, mp_int b, mp_int c);      /* c = a^b */
+
+int       mp_int_compare(mp_int a, mp_int b);          /* a <=> b     */
+int       mp_int_compare_unsigned(mp_int a, mp_int b); /* |a| <=> |b| */
+int       mp_int_compare_zero(mp_int z);                 /* a <=> 0  */
+int       mp_int_compare_value(mp_int z, mp_small v);    /* a <=> v  */
+int       mp_int_compare_uvalue(mp_int z, mp_usmall uv); /* a <=> uv */
+
+/* Returns true if v|a, false otherwise (including errors) */
+int       mp_int_divisible_value(mp_int a, mp_small v);
+
+/* Returns k >= 0 such that z = 2^k, if one exists; otherwise < 0 */
+int       mp_int_is_pow2(mp_int z);
+
+mp_result mp_int_exptmod(mp_int a, mp_int b, mp_int m,
+			 mp_int c);                    /* c = a^b (mod m) */
+mp_result mp_int_exptmod_evalue(mp_int a, mp_small value,
+				mp_int m, mp_int c);   /* c = a^v (mod m) */
+mp_result mp_int_exptmod_bvalue(mp_small value, mp_int b,
+				mp_int m, mp_int c);   /* c = v^b (mod m) */
+mp_result mp_int_exptmod_known(mp_int a, mp_int b,
+			       mp_int m, mp_int mu,
+			       mp_int c);              /* c = a^b (mod m) */
+mp_result mp_int_redux_const(mp_int m, mp_int c);
+
+mp_result mp_int_invmod(mp_int a, mp_int m, mp_int c); /* c = 1/a (mod m) */
+
+mp_result mp_int_gcd(mp_int a, mp_int b, mp_int c);    /* c = gcd(a, b)   */
+
+mp_result mp_int_egcd(mp_int a, mp_int b, mp_int c,    /* c = gcd(a, b)   */
+		      mp_int x, mp_int y);             /* c = ax + by     */
+
+mp_result mp_int_lcm(mp_int a, mp_int b, mp_int c);    /* c = lcm(a, b)   */
+
+mp_result mp_int_root(mp_int a, mp_small b, mp_int c); /* c = floor(a^{1/b}) */
+#define   mp_int_sqrt(a, c) mp_int_root(a, 2, c)       /* c = floor(sqrt(a)) */
+
+/* Convert to a small int, if representable; else MP_RANGE */
+mp_result mp_int_to_int(mp_int z, mp_small *out);
+mp_result mp_int_to_uint(mp_int z, mp_usmall *out);
+
+/* Convert to nul-terminated string with the specified radix, writing at
+   most limit characters including the nul terminator  */
+mp_result mp_int_to_string(mp_int z, mp_size radix,
+			   char *str, int limit);
+
+/* Return the number of characters required to represent
+   z in the given radix.  May over-estimate. */
+mp_result mp_int_string_len(mp_int z, mp_size radix);
+
+/* Read zero-terminated string into z */
+mp_result mp_int_read_string(mp_int z, mp_size radix, const char *str);
+mp_result mp_int_read_cstring(mp_int z, mp_size radix, const char *str,
+			      char **end);
+
+/* Return the number of significant bits in z */
+mp_result mp_int_count_bits(mp_int z);
+
+/* Convert z to two's complement binary, writing at most limit bytes */
+mp_result mp_int_to_binary(mp_int z, unsigned char *buf, int limit);
+
+/* Read a two's complement binary value into z from the given buffer */
+mp_result mp_int_read_binary(mp_int z, unsigned char *buf, int len);
+
+/* Return the number of bytes required to represent z in binary. */
+mp_result mp_int_binary_len(mp_int z);
+
+/* Convert z to unsigned binary, writing at most limit bytes */
+mp_result mp_int_to_unsigned(mp_int z, unsigned char *buf, int limit);
+
+/* Read an unsigned binary value into z from the given buffer */
+mp_result mp_int_read_unsigned(mp_int z, unsigned char *buf, int len);
+
+/* Return the number of bytes required to represent z as unsigned output */
+mp_result mp_int_unsigned_len(mp_int z);
+
+/* Return a statically allocated string describing error code res */
+const char *mp_error_string(mp_result res);
+
+#if DEBUG
+void      s_print(char *tag, mp_int z);
+void      s_print_buf(char *tag, mp_digit *buf, mp_size num);
+#endif
+
+#ifdef __cplusplus
+}
+#endif
+#endif /* end IMATH_H_ */
diff --git a/polly/lib/External/isl/imath/imdrover.c b/polly/lib/External/isl/imath/imdrover.c
new file mode 100644
index 00000000000..be0de16cc8f
--- /dev/null
+++ b/polly/lib/External/isl/imath/imdrover.c
@@ -0,0 +1,1780 @@
+/*
+  Name:     imdrover.c
+  Purpose:  Keeper of the hordes of testing code.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Copyright (C) 2002-2007 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#include <stdlib.h>
+#include <string.h>
+#include <limits.h>
+#include <assert.h>
+
+#include "imdrover.h"
+#include "imath.h"
+#include "imrat.h"
+
+/* Globals visible from outside this file */
+mp_result imath_errno;
+char     *imath_errmsg;
+
+/* Set imath_errno and return failure from a test. */
+#define FAIL(E) return (imath_errno = (E), 0)
+
+/* Check that an expression X yields the expected mp_result value V. */
+#define VCHECK(X, V)         \
+do {                         \
+  mp_result res_;            \
+  if ((res_ = (X)) != (V)) { \
+    FAIL(res_);              \
+  }                          \
+} while(0)
+#define CHECK(X) VCHECK(X, MP_OK)
+#define ECHECK(X) VCHECK(X, expect)
+#define ACHECK(X)    \
+do {                 \
+  if (!(X)) {        \
+    FAIL(MP_BADARG); \
+  }                  \
+} while(0)
+
+#define OUTPUT_LIMIT 2048
+#define NUM_REGS     16
+#define OTHER_ERROR  -1024
+
+static char g_output[OUTPUT_LIMIT];
+static mpz_t g_zreg[NUM_REGS];
+static mpq_t g_qreg[NUM_REGS];
+static unsigned char g_bin1[OUTPUT_LIMIT];
+static unsigned char g_bin2[OUTPUT_LIMIT];
+
+extern void trim_line(char *line); /* borrowed from imtest.c */
+
+/* Read in a string with radix tags */
+static mp_result read_int_value(mp_int z, char *str);
+static mp_result read_rat_value(mp_rat q, char *str);
+
+/* Read in a string with radix tags, as a long (not an mp_int) */
+static int read_long(long *z, char *str);
+
+/* Parse the input and output values and fill in pointers to the
+   registers containing them.  Returns true if all is well, false
+   in case of error.  Caller allocates in/out to correct sizes. */
+static int parse_int_values(testspec_t* t, mp_int *in, mp_int *out,
+			    mp_result *rval);
+static int parse_rat_values(testspec_t* t, mp_rat *in, mp_rat *out,
+			    mp_result *rval);
+
+/* Parse a result code name and return the corresponding result
+   code */
+static int parse_result_code(char *str, mp_result *code);
+
+/* Read in a dot-delimited binary sequence to the given buffer, and
+   return the number of bytes read.  Returns < 0 in case of a syntax
+   error.  Records no more than limit bytes. */
+static int parse_binary(char *str, unsigned char *buf, int limit);
+
+/* Clean up registers (called from atexit()) */
+static void done_testing(void);
+
+/*------------------------------------------------------------------------*/
+/* Utility subroutines for writing tests (explained above)                */
+
+/* {{{ read_int_value(z, str) */
+
+static mp_result read_int_value(mp_int z, char *str)
+{
+  int radix = 10;
+
+  if(*str == '#') {
+    ++str;
+    switch(*str) {
+    case 'x': case 'X':
+      radix = 16;
+      break;
+    case 'd': case 'D':
+      radix = 10;
+      break;
+    case 'o': case 'O':
+      radix = 8;
+      break;
+    case 'b': case 'B':
+      radix = 2;
+      break;
+    default:
+      return MP_RANGE;
+    }
+    ++str;
+  }
+
+  return mp_int_read_string(z, radix, str);
+}
+
+/* }}} */
+
+/* {{{ read_rat_value(q, str) */
+
+static mp_result read_rat_value(mp_rat q, char *str)
+{
+  int radix = 10;
+
+  if(*str == '#') {
+    ++str;
+    switch(*str) {
+    case 'x': case 'X':
+      radix = 16;
+      break;
+    case 'd': case 'D':
+      radix = 10;
+      break;
+    case 'o': case 'O':
+      radix = 8;
+      break;
+    case 'b': case 'B':
+      radix = 2;
+      break;
+    default:
+      return MP_RANGE;
+    }
+    ++str;
+  }
+
+  if(*str == '@')
+    return mp_rat_read_decimal(q, radix, str + 1);
+  else
+    return mp_rat_read_string(q, radix, str);
+}
+
+/* }}} */
+
+/* {{{ read_long(z, str) */
+
+static int read_long(long *z, char *str)
+{
+  char *end;
+  int radix = 10;
+
+  if (*str == '#') {
+    ++str;
+    switch (*str) {
+    case 'x': case 'X':
+      radix = 16;
+      break;
+    case 'd': case 'D':
+      radix = 10;
+      break;
+    case 'o': case 'O':
+      radix = 8;
+      break;
+    case 'b': case 'B':
+      radix = 2;
+      break;
+    default:
+      return 0;
+    }
+    ++str;
+  }
+
+  *z = strtol(str, &end, radix);
+  return (end != str && *end == '\0');
+}
+
+/* }}} */
+
+/* {{{ parse_int_values(t, in, out) */
+
+static int parse_int_values(testspec_t* t, mp_int* in, mp_int* out,
+			    mp_result* rval)
+{
+  int i, pos = 0;
+  char *str;
+
+  if (rval != NULL)
+    *rval = MP_OK;  /* default */
+
+  if (in != NULL) {
+    for (i = 0; i < t->num_inputs; ++i) {
+      str = t->input[i];
+
+      trim_line(str);
+
+      if (*str == '=') {
+	int k = abs(atoi(str + 1)) - 1;
+
+	if (k < 0 || k >= i) {
+	  fprintf(stderr, "Line %d: Invalid input back-reference [%s]\n",
+		  t->line, str);
+	  return 0;
+	}
+
+	in[i] = in[k];
+      } 
+      else {
+	mp_int reg = g_zreg + pos++; /* grab next free register */
+
+	if (read_int_value(reg, str) != MP_OK) {
+	  fprintf(stderr, "Line %d: Invalid input value [%s]\n",
+		  t->line, str);
+	  return 0;
+	}
+
+	in[i] = reg;
+      }
+    }
+  }
+
+  for (i = 0; i < t->num_outputs; ++i) {
+    mp_int reg = g_zreg + pos++;
+
+    str = t->output[i];
+
+    trim_line(str);
+
+    if (strcmp(str, "?") == 0)
+      mp_int_zero(reg);
+    else if (*str == '$') {
+      mp_result code;
+
+      if (!parse_result_code(str, &code)) {
+	fprintf(stderr, "Line %d: Invalid result code [%s]\n",
+		t->line, str);
+	return 0;
+      }
+      else if(rval == NULL) {
+	fprintf(stderr, "Line %d: Result code not permitted here [%s]\n",
+		t->line, str);
+	return 0;
+      }
+      else
+	*rval = code;
+
+      /* Provide a dummy value for the corresponding output */
+      mp_int_zero(reg);
+    }
+    else if (out != NULL && read_int_value(reg, str) != MP_OK) {
+      fprintf(stderr, "Line %d: Invalid output value [%s]\n",
+	      t->line, str);
+      return 0;
+    }
+
+    if (out != NULL)
+      out[i] = reg;
+  }
+
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ parse_rat_values(t, in, out) */
+
+static int parse_rat_values(testspec_t* t, mp_rat *in, mp_rat *out,
+			    mp_result *rval)
+{
+  int i, pos = 0;
+  char *str;
+
+  if (rval != NULL)
+    *rval = MP_OK;  /* default */
+
+  if (in != NULL) {
+    for (i = 0; i < t->num_inputs; ++i) {
+      str = t->input[i];
+
+      trim_line(str);
+
+      if (*str == '=') {
+	int k = abs(atoi(str + 1)) - 1;
+
+	if (k < 0 || k >= i) {
+	  fprintf(stderr, "Line %d: Invalid input back-reference [%s]\n",
+		  t->line, str);
+	  return 0;
+	}
+
+	in[i] = in[k];
+      } 
+      else {
+	mp_rat reg = g_qreg + pos++; /* grab next free register */
+
+	if (read_rat_value(reg, str) != MP_OK) {
+	  fprintf(stderr, "Line %d: Invalid input value [%s]\n",
+		  t->line, str);
+	  return 0;
+	}
+
+	in[i] = reg;
+      }
+    }
+  }
+
+  for (i = 0; i < t->num_outputs; ++i) {
+    mp_rat reg = g_qreg + pos++;
+
+    str = t->output[i];
+
+    trim_line(str);
+
+    if (strcmp(str, "?") == 0)
+      mp_rat_zero(reg);
+    else if (*str == '$') {
+      mp_result code;
+
+      if (!parse_result_code(str, &code)) {
+	fprintf(stderr, "Line %d: Invalid result code [%s]\n",
+		t->line, str);
+	return 0;
+      }
+      else if (rval == NULL) {
+	fprintf(stderr, "Line %d: Result code not permitted here [%s]\n",
+		t->line, str);
+	return 0;
+      }
+      else
+	*rval = code;
+
+      /* Provide a dummy value for the corresponding output */
+      mp_rat_zero(reg);
+    }
+    else if (out != NULL && read_rat_value(reg, str) != MP_OK) {
+      fprintf(stderr, "Line %d: Invalid output value [%s]\n",
+	      t->line, str);
+      return 0;
+    }
+
+    if (out != NULL)
+      out[i] = reg;
+  }
+
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ parse_result_code(str, *code) */
+
+static int parse_result_code(char *str, mp_result *code)
+{
+  if (str[0] == '$') {
+    if (str[1] == '#') {
+      long v;
+
+      if (!read_long(&v, str + 2))
+	return 0;
+
+      *code = (mp_result) v;
+    }
+    else if (strcmp(str + 1, "MP_OK") == 0 ||
+	    strcmp(str + 1, "MP_FALSE") == 0)
+      *code = MP_OK;
+    else if (strcmp(str + 1, "MP_TRUE") == 0)
+      *code = MP_TRUE;
+    else if (strcmp(str + 1, "MP_MEMORY") == 0)
+      *code = MP_MEMORY;
+    else if (strcmp(str + 1, "MP_RANGE") == 0)
+      *code = MP_RANGE;
+    else if (strcmp(str + 1, "MP_UNDEF") == 0)
+      *code = MP_UNDEF;
+    else if (strcmp(str + 1, "MP_TRUNC") == 0)
+      *code = MP_TRUNC;
+    else if (strcmp(str + 1, "MP_ROUND_UP") == 0)
+      *code = MP_ROUND_UP;
+    else if (strcmp(str + 1, "MP_ROUND_DOWN") == 0)
+      *code = MP_ROUND_DOWN;
+    else if (strcmp(str + 1, "MP_ROUND_HALF_UP") == 0)
+      *code = MP_ROUND_HALF_UP;
+    else if (strcmp(str + 1, "MP_ROUND_HALF_DOWN") == 0)
+      *code = MP_ROUND_HALF_DOWN;
+    else
+      return 0;
+  }
+
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ parse_binary(str, buf, limit) */
+
+static int parse_binary(char *str, unsigned char *buf, int limit)
+{
+  int pos = 0;
+  char *tok;
+
+  trim_line(str);
+
+  for (tok = strtok(str, ".");
+       tok != NULL && pos < limit;
+       tok = strtok(NULL, ".")) {
+    long v;
+
+    if (!read_long(&v, tok) || v > UCHAR_MAX || v < 0)
+      return -1;
+
+    buf[pos++] = (unsigned char)v;
+  }
+
+  return pos;
+}
+
+/* }}} */
+
+/* {{{ done_testing() */
+
+static void done_testing(void)
+{
+  int i;
+
+  for (i = 0; i < NUM_REGS; ++i) {
+    mp_int_clear(g_zreg + i);
+    mp_rat_clear(g_qreg + i);
+  }
+}
+
+/* }}} */
+
+/*------------------------------------------------------------------------*/
+/* Global functions visible to callers outside this file                  */
+
+/* {{{ init_testing() */
+
+void init_testing(void)
+{
+  static int is_done = 0;
+  int i;
+
+  if (is_done)
+    return;
+
+  for (i = 0; i < NUM_REGS; ++i) {
+    assert(mp_int_init(g_zreg + i) == MP_OK);
+    assert(mp_rat_init(g_qreg + i) == MP_OK);
+  }
+
+  imath_errmsg = g_output;
+
+  assert(atexit(done_testing) == 0);
+  is_done = 1;
+}
+
+/* }}} */
+
+/* {{{ reset_registers() */
+
+void reset_registers(void)
+{
+  int i;
+
+  for (i = 0; i < NUM_REGS; ++i) {
+    mp_int_zero(g_zreg + i);
+    mp_rat_zero(g_qreg + i);
+  }
+}
+
+/* }}} */
+
+int test_init(testspec_t* t, FILE* ofp)
+{
+  mp_int in[2], out[1];
+  mp_small   v;
+  mp_usmall uv;
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+
+  if (strcmp(t->code, "initu") == 0) {
+    CHECK(mp_int_to_uint(in[1], &uv));
+    ECHECK(mp_int_init_uvalue(in[0], uv));
+  }
+  else { /* initv */
+    CHECK(mp_int_to_int(in[1], &v));
+    ECHECK(mp_int_init_value(in[0], v));
+  }
+
+
+  if (expect == MP_OK && mp_int_compare(in[0], out[0]) != 0) {
+    mp_int_to_string(in[0], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+
+  return 1;
+}
+
+int test_set(testspec_t* t, FILE* ofp)
+{
+  mp_int in[2], out[1];
+  mp_small   v;
+  mp_usmall uv;
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+
+  if (strcmp(t->code, "setu") == 0) {
+    CHECK(mp_int_to_uint(in[1], &uv));
+    ECHECK(mp_int_set_uvalue(in[0], uv));
+  }
+  else { /* setv */
+    CHECK(mp_int_to_int(in[1], &v));
+    ECHECK(mp_int_set_value(in[0], v));
+  }
+
+  if (expect == MP_OK && mp_int_compare(in[0], out[0]) != 0) {
+    mp_int_to_string(in[0], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+
+  return 1;
+}
+
+/* {{{ test_neg(t, ofp) */
+
+int test_neg(testspec_t* t, FILE* ofp)
+{
+  mp_int in[2], out[1];
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  ECHECK(mp_int_neg(in[0], in[1]));
+
+  if (expect == MP_OK && mp_int_compare(in[1], out[0]) != 0) {
+    mp_int_to_string(in[1], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_abs(t, ofp) */
+
+int test_abs(testspec_t* t, FILE* ofp)
+{
+  mp_int in[2], out[1];
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  ECHECK(mp_int_abs(in[0], in[1]));
+
+  if (expect == MP_OK && mp_int_compare(in[1], out[0]) != 0) {
+    mp_int_to_string(in[1], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_add(t, ofp) */
+
+int test_add(testspec_t* t, FILE* ofp)
+{
+  mp_int in[3], out[1];
+  mp_small v;
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+
+  if (strcmp(t->code, "addv") == 0) {
+    CHECK(mp_int_to_int(in[1], &v));
+    ECHECK(mp_int_add_value(in[0], v, in[2]));
+  } 
+  else {
+    ECHECK(mp_int_add(in[0], in[1], in[2]));
+  }
+
+  if (expect == MP_OK && mp_int_compare(in[2], out[0]) != 0) {
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_sub(t, ofp) */
+
+int test_sub(testspec_t* t, FILE* ofp)
+{
+  mp_int in[3], out[1];
+  mp_small v;
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+
+  if (strcmp(t->code, "subv") == 0) {
+    CHECK(mp_int_to_int(in[1], &v));
+    ECHECK(mp_int_sub_value(in[0], v, in[2]));
+  } 
+  else {
+    ECHECK(mp_int_sub(in[0], in[1], in[2]));
+  }
+
+  if (expect == MP_OK && mp_int_compare(in[2], out[0]) != 0) {
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_mul(t, ofp) */
+
+int test_mul(testspec_t* t, FILE* ofp)
+{
+  mp_int in[3], out[1];
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  ECHECK(mp_int_mul(in[0], in[1], in[2]));
+
+  if(expect == MP_OK && mp_int_compare(in[2], out[0]) != 0) {
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_mulp2(t, ofp) */
+
+int test_mulp2(testspec_t* t, FILE* ofp)
+{
+  mp_int in[3], out[1];
+  mp_result expect;
+  mp_small p2;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  CHECK(mp_int_to_int(in[1], &p2));
+  ECHECK(mp_int_mul_pow2(in[0], p2, in[2]));
+
+  if(expect == MP_OK && mp_int_compare(in[2], out[0]) != 0) {
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_mulv(t, ofp) */
+
+int test_mulv(testspec_t* t, FILE* ofp)
+{
+  mp_int in[3], out[1];
+  mp_result expect;
+  mp_small v;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  CHECK(mp_int_to_int(in[1], &v));
+  ECHECK(mp_int_mul_value(in[0], v, in[2]));
+
+  if(expect == MP_OK && mp_int_compare(in[2], out[0]) != 0) {
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_sqr(t, ofp) */
+
+int test_sqr(testspec_t* t, FILE* ofp)
+{
+  mp_int in[2], out[1];
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  ECHECK(mp_int_sqr(in[0], in[1]));
+
+  if(expect == MP_OK && mp_int_compare(in[1], out[0]) != 0) {
+    mp_int_to_string(in[1], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_div(t, ofp) */
+
+int test_div(testspec_t* t, FILE* ofp)
+{
+  mp_int in[4], out[2];
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  ECHECK(mp_int_div(in[0], in[1], in[2], in[3]));
+
+  if (expect == MP_OK &&
+      ((mp_int_compare(in[2], out[0]) != 0) ||
+       (mp_int_compare(in[3], out[1]) != 0))) {
+    int len;
+    char *str;
+
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    str = g_output + (len = strlen(g_output));
+    *str++ = ',';
+    mp_int_to_string(in[3], 10, str, OUTPUT_LIMIT - (len + 1));
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_divp2(t, ofp) */
+
+int test_divp2(testspec_t* t, FILE* ofp)
+{
+  mp_int in[4], out[2];
+  mp_result expect;
+  mp_small p2;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  CHECK(mp_int_to_int(in[1], &p2));
+  ECHECK(mp_int_div_pow2(in[0], p2, in[2], in[3]));
+
+  if (expect == MP_OK &&
+      ((mp_int_compare(in[2], out[0]) != 0) ||
+       (mp_int_compare(in[3], out[1]) != 0))) {
+    int len;
+    char *str;
+
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    str = g_output + (len = strlen(g_output));
+    *str++ = ',';
+    mp_int_to_string(in[3], 10, str, OUTPUT_LIMIT - (len + 1));
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_divv(t, ofp) */
+
+int test_divv(testspec_t* t, FILE* ofp)
+{
+  mp_int in[3], out[2];
+  mp_result expect;
+  mp_small v, rem, orem;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  CHECK(mp_int_to_int(in[1], &v));
+  CHECK(mp_int_to_int(out[1], &orem));
+  ECHECK(mp_int_div_value(in[0], v, in[2], &rem));
+
+  if (expect == MP_OK &&
+      ((mp_int_compare(in[2], out[0]) != 0) || (rem != orem))) {
+    char *str;
+
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    str = g_output + strlen(g_output);
+    *str++ = ',';
+    sprintf(str, "%ld", rem);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_expt(t, ofp) */
+
+int test_expt(testspec_t* t, FILE* ofp)
+{
+  mp_int in[3], out[1];
+  mp_result expect;
+  mp_small  pow;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  CHECK(mp_int_to_int(in[1], &pow));
+  ECHECK(mp_int_expt(in[0], pow, in[2]));
+
+  if (expect == MP_OK && mp_int_compare(in[2], out[0]) != 0) {
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_exptv(t, ofp) */
+
+int test_exptv(testspec_t* t, FILE* ofp)
+{
+  mp_int in[3], out[1];
+  mp_result expect;
+  mp_small a, b;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  CHECK(mp_int_to_int(in[0], &a));
+  CHECK(mp_int_to_int(in[1], &b));
+  ECHECK(mp_int_expt_value(a, b, in[2]));
+
+  if (expect == MP_OK && mp_int_compare(in[2], out[0]) != 0) {
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_exptf(t, ofp) */
+
+int test_exptf(testspec_t* t, FILE* ofp)
+{
+  mp_int in[3], out[1];
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  ECHECK(mp_int_expt_full(in[0], in[1], in[2]));
+
+  if (expect == MP_OK && mp_int_compare(in[2], out[0]) != 0) {
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_mod(t, ofp) */
+
+int test_mod(testspec_t* t, FILE* ofp)
+{
+  mp_int in[3], out[1];
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  ECHECK(mp_int_mod(in[0], in[1], in[2]));
+
+  if (expect == MP_OK && mp_int_compare(in[2], out[0]) != 0) {
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_gcd(t, ofp) */
+
+int test_gcd(testspec_t* t, FILE* ofp)
+{
+  mp_int in[3], out[1];
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  ECHECK(mp_int_gcd(in[0], in[1], in[2]));
+
+  if (expect == MP_OK && mp_int_compare(in[2], out[0]) != 0) {
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_egcd(t, ofp) */
+
+int test_egcd(testspec_t* t, FILE* ofp)
+{
+  mp_int in[5], out[3], t1 = g_zreg + 8, t2 = g_zreg + 9;
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  ECHECK(mp_int_egcd(in[0], in[1], in[2], in[3], in[4]));
+
+  /* If we got an error we expected, return success immediately */
+  if(expect != MP_OK)
+    return 1;
+
+  if ((mp_int_compare(in[2], out[0]) != 0) ||
+      (mp_int_compare(in[3], out[1]) != 0) ||
+      (mp_int_compare(in[4], out[2]) != 0)) {
+    int len, len2;
+    char *str;
+
+    /* Failure might occur because the tester computed x and y in a different
+       way than we did.  Verify that the results are correct before reporting
+       an error. */
+    mp_int_mul(in[3], in[0], t1);
+    mp_int_mul(in[4], in[1], t2);
+    mp_int_add(t1, t2, t2);
+    if (mp_int_compare(t2, in[2]) == 0)
+      return 1;
+
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    str = g_output + (len = strlen(g_output));
+    *str++ = ',';
+    mp_int_to_string(in[3], 10, str, OUTPUT_LIMIT - (len + 1));
+    str = str + (len2 = strlen(str));
+    *str++ = ',';
+    mp_int_to_string(in[4], 10, str, OUTPUT_LIMIT - (len + len2 + 2));
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_lcm(t, ofp) */
+
+int test_lcm(testspec_t* t, FILE* ofp)
+{
+  mp_int in[3], out[1];
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  ECHECK(mp_int_lcm(in[0], in[1], in[2]));
+
+  if (expect == MP_OK && mp_int_compare(in[2], out[0]) != 0) {
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_sqrt(t, ofp) */
+
+int test_sqrt(testspec_t* t, FILE* ofp)
+{
+  mp_int in[2], out[1];
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  ECHECK(mp_int_sqrt(in[0], in[1]));
+
+  if (expect == MP_OK && mp_int_compare(in[1], out[0]) != 0) {
+    mp_int_to_string(in[1], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_root(t, ofp) */
+
+int test_root(testspec_t* t, FILE* ofp)
+{
+  mp_int in[3], out[1];
+  mp_small v;
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  CHECK(mp_int_to_int(in[1], &v));
+  ECHECK(mp_int_root(in[0], v, in[2]));
+
+  if (expect == MP_OK && mp_int_compare(in[2], out[0]) != 0) {
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_invmod(t, ofp) */
+
+int test_invmod(testspec_t* t, FILE* ofp)
+{
+  mp_int in[3], out[1];
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  ECHECK(mp_int_invmod(in[0], in[1], in[2]));
+
+  if (expect == MP_OK && mp_int_compare(in[2], out[0]) != 0) {
+    mp_int_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_exptmod(t, ofp) */
+
+int test_exptmod(testspec_t* t, FILE* ofp)
+{
+  mp_int in[4], out[1];
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  ECHECK(mp_int_exptmod(in[0], in[1], in[2], in[3]));
+
+  if (expect == MP_OK && mp_int_compare(in[3], out[0]) != 0) {
+    mp_int_to_string(in[3], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_exptmod_ev(t, ofp) */
+
+int test_exptmod_ev(testspec_t* t, FILE* ofp)
+{
+  mp_int in[4], out[1];
+  mp_result expect;
+  mp_small v;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  CHECK(mp_int_to_int(in[1], &v));
+  ECHECK(mp_int_exptmod_evalue(in[0], v, in[2], in[3]));
+
+  if (expect == MP_OK && mp_int_compare(in[3], out[0]) != 0) {
+    mp_int_to_string(in[3], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_exptmod_bv(t, ofp) */
+
+int test_exptmod_bv(testspec_t* t, FILE* ofp)
+{
+  mp_int in[4], out[1];
+  mp_result expect;
+  mp_small v;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  CHECK(mp_int_to_int(in[0], &v));
+  ECHECK(mp_int_exptmod_bvalue(v, in[1], in[2], in[3]));
+
+  if (expect == MP_OK && mp_int_compare(in[3], out[0]) != 0) {
+    mp_int_to_string(in[3], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_comp(t, ofp) */
+
+int test_comp(testspec_t* t, FILE* ofp)
+{
+  mp_int in[2];
+  mp_result res, expect;
+
+  ACHECK(parse_int_values(t, in, NULL, &expect));
+
+  if ((res = mp_int_compare(in[0], in[1])) != expect) {
+    sprintf(g_output, "Incorrect comparison result (want %d, got %d)", 
+	    expect, res);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_ucomp(t, ofp) */
+
+int test_ucomp(testspec_t* t, FILE* ofp)
+{
+  mp_int in[2];
+  mp_result res, expect;
+
+  ACHECK(parse_int_values(t, in, NULL, &expect));
+
+  if ((res = mp_int_compare_unsigned(in[0], in[1])) != expect) {
+    sprintf(g_output, "Incorrect comparison result (want %d, got %d)", 
+	    expect, res);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_zcomp(t, ofp) */
+
+int test_zcomp(testspec_t* t, FILE* ofp)
+{
+  mp_int in[1];
+  mp_result res, expect;
+
+  ACHECK(parse_int_values(t, in, NULL, &expect));
+
+  if ((res = mp_int_compare_zero(in[0])) != expect) {
+    sprintf(g_output, "Incorrect comparison result (want %d, got %d)", 
+	    expect, res);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_vcomp(t, ofp) */
+
+int test_vcomp(testspec_t* t, FILE* ofp)
+{
+  mp_int in[2];
+  mp_result res, expect;
+  mp_small v;
+
+  ACHECK(parse_int_values(t, in, NULL, &expect));
+
+  v = atoi(t->input[1]);
+  if ((res = mp_int_compare_value(in[0], v)) != expect) {
+    sprintf(g_output, "Incorrect comparison result (want %d, got %d)", 
+	    expect, res);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_uvcomp(t, ofp) */
+
+int test_uvcomp(testspec_t* t, FILE* ofp)
+{
+  mp_int in[2];
+  mp_result res, expect;
+  mp_usmall v;
+
+  ACHECK(parse_int_values(t, in, NULL, &expect));
+
+  v = strtoul(t->input[1], NULL, 0);
+  if ((res = mp_int_compare_uvalue(in[0], v)) != expect) {
+    sprintf(g_output, "Incorrect comparison result (want %d, got %d)",
+	    expect, res);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_tostr(t, ofp) */
+
+int test_tostr(testspec_t* t, FILE* ofp)
+{
+  mp_int in[2];
+  mp_small radix;
+  mp_result len;
+
+  ACHECK(parse_int_values(t, in, NULL, NULL));
+  ACHECK(mp_int_to_int(in[1], &radix) == MP_OK);
+
+  if (radix < MP_MIN_RADIX || radix > MP_MAX_RADIX)
+    FAIL(MP_RANGE);
+
+  trim_line(t->output[0]);
+  len = mp_int_string_len(in[0], radix);
+
+  CHECK(mp_int_to_string(in[0], radix, g_output, len));
+
+  if(strcmp(t->output[0], g_output) != 0)
+    FAIL(OTHER_ERROR);
+
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_tobin(t, ofp) */
+
+int test_tobin(testspec_t* t, FILE* ofp)
+{
+  mp_int in[1];
+  int test_len, out_len;
+
+  ACHECK(parse_int_values(t, in, NULL, NULL));
+
+  trim_line(t->output[0]);
+  if((out_len = parse_binary(t->output[0], g_bin1, sizeof(g_bin1))) < 0)
+    FAIL(MP_BADARG);
+
+  if((test_len = mp_int_binary_len(in[0])) != out_len) {
+    sprintf(g_output, "Output lengths do not match (want %d, got %d)",
+	    test_len, out_len);
+    FAIL(OTHER_ERROR);
+  }
+
+  CHECK(mp_int_to_binary(in[0], g_bin2, sizeof(g_bin2)));
+
+  if (memcmp(g_bin1, g_bin2, test_len) != 0) {
+    int pos = 0, i;
+
+    for(i = 0; i < test_len - 1; ++i)
+      pos += sprintf(g_output + pos, "%d.", g_bin2[i]);
+
+    sprintf(g_output + pos, "%d", g_bin2[i]);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_to_int(t, ofp) */
+
+int test_to_int(testspec_t* t, FILE* ofp)
+{
+  mp_int in[1], out[1];
+  mp_small v;
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  ECHECK(mp_int_to_int(in[0], &v));
+
+  if (expect == MP_OK && mp_int_compare_value(out[0], v) != 0) {
+    sprintf(g_output, "Incorrect value (got %ld)", v);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_to_uint(t, ofp) */
+
+int test_to_uint(testspec_t* t, FILE* ofp)
+{
+  mp_int in[1], out[1];
+  mp_usmall v;
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+  ECHECK(mp_int_to_uint(in[0], &v));
+
+  if (expect == MP_OK && mp_int_compare_uvalue(out[0], v) != 0) {
+    sprintf(g_output, "Incorrect value (got %lu)", v);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_read_binary(t, ofp) */
+
+int test_read_binary(testspec_t* t, FILE* ofp)
+{
+  mp_int out[1], in = g_zreg + 1;
+  int in_len;
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, NULL, out, &expect));
+
+  trim_line(t->input[0]);
+  if((in_len = parse_binary(t->input[0], g_bin1, sizeof(g_bin1))) < 0)
+    FAIL(MP_BADARG);
+
+  ECHECK(mp_int_read_binary(in, g_bin1, in_len));
+
+  if (expect == MP_OK && mp_int_compare(in, out[0]) != 0) {
+    mp_int_to_string(in, 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_to_uns(t, ofp) */
+
+int test_to_uns(testspec_t* t, FILE* ofp)
+{
+  mp_int in[1];
+  int test_len, out_len;
+
+  ACHECK(parse_int_values(t, in, NULL, NULL));
+
+  trim_line(t->output[0]);
+  if((out_len = parse_binary(t->output[0], g_bin1, sizeof(g_bin1))) < 0)
+    FAIL(MP_BADARG);
+
+  if((test_len = mp_int_unsigned_len(in[0])) != out_len) {
+    sprintf(g_output, "Output lengths do not match (want %d, got %d)",
+	    test_len, out_len);
+    FAIL(OTHER_ERROR);
+  }
+
+  CHECK(mp_int_to_unsigned(in[0], g_bin2, sizeof(g_bin2)));
+
+  if (memcmp(g_bin1, g_bin2, test_len) != 0) {
+    int pos = 0, i;
+
+    for(i = 0; i < test_len - 1; ++i)
+      pos += sprintf(g_output + pos, "%d.", g_bin2[i]);
+
+    sprintf(g_output + pos, "%d", g_bin2[i]);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_read_uns(t, ofp) */
+
+int test_read_uns(testspec_t* t, FILE* ofp)
+{
+  mp_int out[1], in = g_zreg + 1;
+  int in_len;
+  mp_result expect;
+
+  ACHECK(parse_int_values(t, NULL, out, &expect));
+
+  trim_line(t->input[0]);
+  if((in_len = parse_binary(t->input[0], g_bin1, sizeof(g_bin1))) < 0)
+    FAIL(MP_BADARG);
+
+  ECHECK(mp_int_read_unsigned(in, g_bin1, in_len));
+
+  if (expect == MP_OK && mp_int_compare(in, out[0]) != 0) {
+    mp_int_to_string(in, 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_meta(t, ofp) */
+
+int test_meta(testspec_t* t, FILE* ofp)
+{
+  mp_int *in = NULL, *out = NULL;
+  int i, j;
+  mp_result expect;
+
+  if (t->num_inputs > 0)
+    in = calloc(t->num_inputs, sizeof(mp_int));
+  if (t->num_outputs > 0)
+    out = calloc(t->num_outputs, sizeof(mp_int));
+
+  ACHECK(parse_int_values(t, in, out, &expect));
+
+  fprintf(ofp, "Test '%s' defined at line %d\n",
+	  t->code, t->line);
+  fprintf(ofp, "Expected result: %d\n", expect);
+  fprintf(ofp, "Input values: %d\n", t->num_inputs);
+  for (i = 0; i < t->num_inputs; ++i) {
+    mp_int_to_string(in[i], 10, g_output, OUTPUT_LIMIT);
+
+    fprintf(ofp, " %2d.) %s", i + 1, g_output);
+
+    for (j = i - 1; j >= 0; --j)
+      if (in[j] == in[i]) {
+	fprintf(ofp, " (=> %d)", j + 1);
+	break;
+      }
+
+    fputc('\n', ofp);
+  }
+  fprintf(ofp, "Output values: %d\n", t->num_outputs);
+  for (i = 0; i < t->num_outputs; ++i) {
+    mp_int_to_string(out[i], 10, g_output, OUTPUT_LIMIT);
+
+    fprintf(ofp, " %2d.) %s\n", i + 1, g_output);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_qneg(t, ofp) */
+
+int test_qneg(testspec_t* t, FILE* ofp)
+{
+  mp_rat in[2], out[1];
+  mp_result expect;
+
+  ACHECK(parse_rat_values(t, in, out, &expect));
+  ECHECK(mp_rat_neg(in[0], in[1]));
+
+  if (expect == MP_OK && mp_rat_compare(in[1], out[0]) != 0) {
+    mp_rat_to_string(in[1], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_qrecip(t, ofp) */
+
+int test_qrecip(testspec_t* t, FILE* ofp)
+{
+  mp_rat in[2], out[1];
+  mp_result expect;
+
+  ACHECK(parse_rat_values(t, in, out, &expect));
+  ECHECK(mp_rat_recip(in[0], in[1]));
+
+  if (expect == MP_OK  && mp_rat_compare(in[1], out[0]) != 0) {
+    mp_rat_to_string(in[1], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_qabs(t, ofp) */
+
+int test_qabs(testspec_t* t, FILE* ofp)
+{
+  mp_rat in[2], out[1];
+  mp_result expect;
+
+  ACHECK(parse_rat_values(t, in, out, &expect));
+  ECHECK(mp_rat_abs(in[0], in[1]));
+
+  if (expect == MP_OK && mp_rat_compare(in[1], out[0]) != 0) {
+    mp_rat_to_string(in[1], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_qadd(t, ofp) */
+
+int test_qadd(testspec_t* t, FILE* ofp)
+{
+  mp_rat in[3], out[1];
+  mp_result expect;
+
+  ACHECK(parse_rat_values(t, in, out, &expect));
+  ECHECK(mp_rat_add(in[0], in[1], in[2]));
+
+  if (expect == MP_OK && mp_rat_compare(in[2], out[0]) != 0) {
+    mp_rat_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_qsub(t, ofp) */
+
+int test_qsub(testspec_t* t, FILE* ofp)
+{
+  mp_rat in[3], out[1];
+  mp_result expect;
+
+  ACHECK(parse_rat_values(t, in, out, &expect));
+  ECHECK(mp_rat_sub(in[0], in[1], in[2]));
+
+  if (expect == MP_OK && mp_rat_compare(in[2], out[0]) != 0) {
+    mp_rat_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_qmul(t, ofp) */
+
+int test_qmul(testspec_t* t, FILE* ofp)
+{
+  mp_rat in[3], out[1];
+  mp_result expect;
+
+  ACHECK(parse_rat_values(t, in, out, &expect));
+  ECHECK(mp_rat_mul(in[0], in[1], in[2]));
+
+  if (expect == MP_OK && mp_rat_compare(in[2], out[0]) != 0) {
+    mp_rat_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_qdiv(t, ofp) */
+
+int test_qdiv(testspec_t* t, FILE* ofp)
+{
+  mp_rat in[3], out[1];
+  mp_result expect;
+
+  ACHECK(parse_rat_values(t, in, out, &expect));
+  ECHECK(mp_rat_div(in[0], in[1], in[2]));
+
+  if (expect == MP_OK && mp_rat_compare(in[2], out[0]) != 0) {
+    mp_rat_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_qaddz(t, ofp) */
+
+int test_qaddz(testspec_t* t, FILE* ofp)
+{
+  mp_rat in[3], out[1];
+  mp_result expect;
+
+  ACHECK(parse_rat_values(t, in, out, &expect));
+
+  if (!mp_rat_is_integer(in[1])) {
+    fprintf(stderr, "Line %d: Second argument must be an integer (test_qaddz)\n",
+	    t->line);
+    FAIL(MP_BADARG);
+  }
+
+  ECHECK(mp_rat_add_int(in[0], MP_NUMER_P(in[1]), in[2]));
+
+  if (expect == MP_OK && mp_rat_compare(in[2], out[0]) != 0) {
+    mp_rat_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_qsubz(t, ofp) */
+
+int test_qsubz(testspec_t* t, FILE* ofp)
+{
+  mp_rat in[3], out[1];
+  mp_result expect;
+
+  ACHECK(parse_rat_values(t, in, out, &expect));
+
+  if (!mp_rat_is_integer(in[1])) {
+    fprintf(stderr, "Line %d: Second argument must be an integer (test_qsubz)\n",
+	    t->line);
+    FAIL(MP_BADARG);
+  }
+
+  ECHECK(mp_rat_sub_int(in[0], MP_NUMER_P(in[1]), in[2]));
+
+  if (expect == MP_OK && mp_rat_compare(in[2], out[0]) != 0) {
+    mp_rat_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_qmulz(t, ofp) */
+
+int test_qmulz(testspec_t* t, FILE* ofp)
+{
+  mp_rat in[3], out[1];
+  mp_result expect;
+
+  ACHECK(parse_rat_values(t, in, out, &expect));
+
+  if (!mp_rat_is_integer(in[1])) {
+    fprintf(stderr, "Line %d: Second argument must be an integer (test_qmulz)\n",
+	    t->line);
+    FAIL(MP_BADARG);
+  }
+
+  ECHECK(mp_rat_mul_int(in[0], MP_NUMER_P(in[1]), in[2]));
+
+  if (expect == MP_OK && mp_rat_compare(in[2], out[0]) != 0) {
+    mp_rat_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_qdivz(t, ofp) */
+
+int test_qdivz(testspec_t* t, FILE* ofp)
+{
+  mp_rat in[3], out[1];
+  mp_result expect;
+
+  ACHECK(parse_rat_values(t, in, out, &expect));
+
+  if (!mp_rat_is_integer(in[1])) {
+    fprintf(stderr, "Line %d: Second argument must be an integer (test_qdivz)\n",
+	    t->line);
+    FAIL(MP_BADARG);
+  }
+
+  ECHECK(mp_rat_div_int(in[0], MP_NUMER_P(in[1]), in[2]));
+
+  if (expect == MP_OK && mp_rat_compare(in[2], out[0]) != 0) {
+    mp_rat_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_qexpt(t, ofp) */
+
+int test_qexpt(testspec_t* t, FILE* ofp)
+{
+  mp_rat in[3], out[1];
+  mp_result expect;
+  mp_small power;
+
+  ACHECK(parse_rat_values(t, in, out, &expect));
+
+  if (!mp_rat_is_integer(in[1])) {
+    fprintf(stderr, "Line %d: Second argument must be an integer (test_qexpt)\n",
+	    t->line);
+    FAIL(MP_BADARG);
+  }
+
+  CHECK(mp_int_to_int(MP_NUMER_P(in[1]), &power));
+  ECHECK(mp_rat_expt(in[0], power, in[2]));
+
+  if (expect == MP_OK && mp_rat_compare(in[2], out[0]) != 0) {
+    mp_rat_to_string(in[2], 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_qtostr(t, ofp) */
+
+int test_qtostr(testspec_t* t, FILE* ofp)
+{
+  mp_rat in[2];
+  long radix;
+  mp_result len;
+
+  ACHECK(parse_rat_values(t, in, NULL, NULL));
+  trim_line(t->input[1]);
+  ACHECK(read_long(&radix, t->input[1]));
+
+  if (radix < MP_MIN_RADIX || radix > MP_MAX_RADIX) {
+    fprintf(stderr, "Line %d: Radix %ld out of range\n",
+	    t->line, radix);
+    FAIL(MP_RANGE);
+  }
+
+  trim_line(t->output[0]);
+  len = mp_rat_string_len(in[0], radix);
+
+  CHECK(mp_rat_to_string(in[0], radix, g_output, len));
+
+  if (strcmp(t->output[0], g_output) != 0)
+    FAIL(OTHER_ERROR);
+
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_qtodec(t, ofp) */
+
+int test_qtodec(testspec_t* t, FILE* ofp)
+{
+  mp_rat in[4];
+  long radix, prec, m;
+  mp_round_mode rmode;
+  mp_result res, expect = MP_OK, len;
+
+  ACHECK(parse_rat_values(t, in, NULL, NULL));
+
+  if (t->output[0][0] == '$' && !parse_result_code(t->output[0], &expect)) {
+    fprintf(stderr, "Line %d: Invalid result code [%s]\n",
+	    t->line, t->output[0]);
+    FAIL(OTHER_ERROR);
+  }
+
+  trim_line(t->input[1]);
+  trim_line(t->input[2]);
+  trim_line(t->input[3]);
+  ACHECK(read_long(&radix, t->input[1]));
+  ACHECK(read_long(&prec, t->input[2]));
+  ACHECK(read_long(&m, t->input[3]));
+  rmode = (mp_round_mode) m;
+
+  if (prec < 0) {
+    fprintf(stderr, "Line %d: Precision %ld out of range\n",
+	    t->line, prec);
+    FAIL(MP_RANGE);
+  }
+
+  trim_line(t->output[0]);
+  len = mp_rat_decimal_len(in[0], radix, prec);
+  ECHECK((res = mp_rat_to_decimal(in[0], radix, prec, rmode, g_output, len)));
+
+  if (res == MP_OK && strcmp(t->output[0], g_output) != 0)
+    FAIL(OTHER_ERROR);
+
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ test_qrdec(t, ofp) */
+
+int test_qrdec(testspec_t* t, FILE* ofp)
+{
+  mp_rat out[1], reg = g_qreg + 1;
+  long radix;
+  mp_result expect;
+
+  ACHECK(parse_rat_values(t, NULL, out, &expect));
+  trim_line(t->input[1]);
+  ACHECK(read_long(&radix, t->input[1]));
+
+  ECHECK(mp_rat_read_decimal(reg, radix, t->input[0]));
+  if (expect == MP_OK &&
+      mp_rat_compare(reg, out[0]) != 0) {
+    mp_rat_to_string(reg, 10, g_output, OUTPUT_LIMIT);
+    FAIL(OTHER_ERROR);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* Here there be dragons */
diff --git a/polly/lib/External/isl/imath/imdrover.h b/polly/lib/External/isl/imath/imdrover.h
new file mode 100644
index 00000000000..df74d2b6700
--- /dev/null
+++ b/polly/lib/External/isl/imath/imdrover.h
@@ -0,0 +1,106 @@
+/*
+  Name:     imdrover.h
+  Purpose:  Keeper of the hordes of testing code.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Copyright (C) 2002-2007 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#ifndef IMDROVER_H_
+#define IMDROVER_H_
+
+#include <stdio.h>
+
+typedef struct {
+  int    line;
+  char  *code;
+  int    num_inputs;
+  char **input;
+  int    num_outputs;
+  char **output;
+} testspec_t;
+
+typedef int (*test_f)(testspec_t *, FILE *);
+
+/* Call this once at the outset to set up test registers */
+void init_testing(void);
+void reset_registers(void);
+
+/* Integer tests, and general */
+int test_init(testspec_t* t, FILE* ofp);
+int test_set(testspec_t* t, FILE* ofp);
+int test_neg(testspec_t* t, FILE* ofp);
+int test_abs(testspec_t* t, FILE* ofp);
+int test_add(testspec_t* t, FILE* ofp);
+int test_sub(testspec_t* t, FILE* ofp);
+int test_mul(testspec_t* t, FILE* ofp);
+int test_mulp2(testspec_t* t, FILE* ofp);
+int test_mulv(testspec_t* t, FILE* ofp);
+int test_sqr(testspec_t* t, FILE* ofp);
+int test_div(testspec_t* t, FILE* ofp);
+int test_divp2(testspec_t* t, FILE* ofp);
+int test_divv(testspec_t* t, FILE* ofp);
+int test_expt(testspec_t* t, FILE* ofp);
+int test_exptv(testspec_t* t, FILE* ofp);
+int test_exptf(testspec_t* t, FILE* ofp);
+int test_mod(testspec_t* t, FILE* ofp);
+int test_gcd(testspec_t* t, FILE* ofp);
+int test_egcd(testspec_t* t, FILE* ofp);
+int test_lcm(testspec_t* t, FILE* ofp);
+int test_sqrt(testspec_t* t, FILE* ofp);
+int test_root(testspec_t* t, FILE* ofp);
+int test_invmod(testspec_t* t, FILE* ofp);
+int test_exptmod(testspec_t* t, FILE* ofp);
+int test_exptmod_ev(testspec_t* t, FILE* ofp);
+int test_exptmod_bv(testspec_t* t, FILE* ofp);
+int test_comp(testspec_t* t, FILE* ofp);
+int test_ucomp(testspec_t* t, FILE* ofp);
+int test_zcomp(testspec_t* t, FILE* ofp);
+int test_vcomp(testspec_t* t, FILE* ofp);
+int test_uvcomp(testspec_t* t, FILE* ofp);
+int test_tostr(testspec_t* t, FILE* ofp);
+int test_tobin(testspec_t* t, FILE* ofp);
+int test_to_int(testspec_t* t, FILE* ofp);
+int test_to_uint(testspec_t* t, FILE* ofp);
+int test_read_binary(testspec_t* t, FILE* ofp);
+int test_to_uns(testspec_t* t, FILE* ofp);
+int test_read_uns(testspec_t* t, FILE* ofp);
+int test_meta(testspec_t* t, FILE* ofp);
+
+/* Rational tests */
+int test_qneg(testspec_t* t, FILE* ofp);
+int test_qrecip(testspec_t* t, FILE* ofp);
+int test_qabs(testspec_t* t, FILE* ofp);
+int test_qadd(testspec_t* t, FILE* ofp);
+int test_qsub(testspec_t* t, FILE* ofp);
+int test_qmul(testspec_t* t, FILE* ofp);
+int test_qdiv(testspec_t* t, FILE* ofp);
+int test_qdiv(testspec_t* t, FILE* ofp);
+int test_qaddz(testspec_t* t, FILE* ofp);
+int test_qsubz(testspec_t* t, FILE* ofp);
+int test_qmulz(testspec_t* t, FILE* ofp);
+int test_qdivz(testspec_t* t, FILE* ofp);
+int test_qexpt(testspec_t* t, FILE* ofp);
+int test_qtostr(testspec_t* t, FILE* ofp);
+int test_qtodec(testspec_t* t, FILE* ofp);
+int test_qrdec(testspec_t* t, FILE* ofp);
+
+#endif /* IMDROVER_H_ */
diff --git a/polly/lib/External/isl/imath/imrat.c b/polly/lib/External/isl/imath/imrat.c
new file mode 100644
index 00000000000..c6e9cde003e
--- /dev/null
+++ b/polly/lib/External/isl/imath/imrat.c
@@ -0,0 +1,1141 @@
+/*
+  Name:     imrat.c
+  Purpose:  Arbitrary precision rational arithmetic routines.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Copyright (C) 2002-2007 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#include "imrat.h"
+#include <stdlib.h>
+#include <string.h>
+#include <ctype.h>
+#include <assert.h>
+
+/* {{{ Useful macros */
+
+#define TEMP(K) (temp + (K))
+#define SETUP(E, C) \
+do{if((res = (E)) != MP_OK) goto CLEANUP; ++(C);}while(0)
+
+/* Argument checking:
+   Use CHECK() where a return value is required; NRCHECK() elsewhere */
+#define CHECK(TEST)   assert(TEST)
+#define NRCHECK(TEST) assert(TEST)
+
+/* }}} */
+
+/* Reduce the given rational, in place, to lowest terms and canonical form.
+   Zero is represented as 0/1, one as 1/1.  Signs are adjusted so that the sign
+   of the numerator is definitive. */
+static mp_result s_rat_reduce(mp_rat r);
+
+/* Common code for addition and subtraction operations on rationals. */
+static mp_result s_rat_combine(mp_rat a, mp_rat b, mp_rat c,
+			       mp_result (*comb_f)(mp_int, mp_int, mp_int));
+
+/* {{{ mp_rat_init(r) */
+
+mp_result mp_rat_init(mp_rat r)
+{
+  mp_result res;
+
+  if ((res = mp_int_init(MP_NUMER_P(r))) != MP_OK)
+    return res;
+  if ((res = mp_int_init(MP_DENOM_P(r))) != MP_OK) {
+    mp_int_clear(MP_NUMER_P(r));
+    return res;
+  }
+
+  return mp_int_set_value(MP_DENOM_P(r), 1);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_alloc() */
+
+mp_rat mp_rat_alloc(void)
+{
+  mp_rat out = malloc(sizeof(*out));
+
+  if (out != NULL) {
+    if (mp_rat_init(out) != MP_OK) {
+      free(out);
+      return NULL;
+    }
+  }
+
+  return out;
+}
+
+/* }}} */
+
+/* {{{ mp_rat_reduce(r) */
+
+mp_result mp_rat_reduce(mp_rat r) {
+  return s_rat_reduce(r);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_init_size(r, n_prec, d_prec) */
+
+mp_result mp_rat_init_size(mp_rat r, mp_size n_prec, mp_size d_prec)
+{
+  mp_result res;
+
+  if ((res = mp_int_init_size(MP_NUMER_P(r), n_prec)) != MP_OK)
+    return res;
+  if ((res = mp_int_init_size(MP_DENOM_P(r), d_prec)) != MP_OK) {
+    mp_int_clear(MP_NUMER_P(r));
+    return res;
+  }
+  
+  return mp_int_set_value(MP_DENOM_P(r), 1);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_init_copy(r, old) */
+
+mp_result mp_rat_init_copy(mp_rat r, mp_rat old)
+{
+  mp_result res;
+
+  if ((res = mp_int_init_copy(MP_NUMER_P(r), MP_NUMER_P(old))) != MP_OK)
+    return res;
+  if ((res = mp_int_init_copy(MP_DENOM_P(r), MP_DENOM_P(old))) != MP_OK) 
+    mp_int_clear(MP_NUMER_P(r));
+  
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_rat_set_value(r, numer, denom) */
+
+mp_result mp_rat_set_value(mp_rat r, mp_small numer, mp_small denom)
+{
+  mp_result res;
+
+  if (denom == 0)
+    return MP_UNDEF;
+
+  if ((res = mp_int_set_value(MP_NUMER_P(r), numer)) != MP_OK)
+    return res;
+  if ((res = mp_int_set_value(MP_DENOM_P(r), denom)) != MP_OK)
+    return res;
+
+  return s_rat_reduce(r);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_set_uvalue(r, numer, denom) */
+
+mp_result mp_rat_set_uvalue(mp_rat r, mp_usmall numer, mp_usmall denom)
+{
+  mp_result res;
+
+  if (denom == 0)
+    return MP_UNDEF;
+
+  if ((res = mp_int_set_uvalue(MP_NUMER_P(r), numer)) != MP_OK)
+    return res;
+  if ((res = mp_int_set_uvalue(MP_DENOM_P(r), denom)) != MP_OK)
+    return res;
+
+  return s_rat_reduce(r);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_clear(r) */
+
+void      mp_rat_clear(mp_rat r)
+{
+  mp_int_clear(MP_NUMER_P(r));
+  mp_int_clear(MP_DENOM_P(r));
+
+}
+
+/* }}} */
+
+/* {{{ mp_rat_free(r) */
+
+void      mp_rat_free(mp_rat r)
+{
+  NRCHECK(r != NULL);
+  
+  if (r->num.digits != NULL)
+    mp_rat_clear(r);
+
+  free(r);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_numer(r, z) */
+
+mp_result mp_rat_numer(mp_rat r, mp_int z)
+{
+  return mp_int_copy(MP_NUMER_P(r), z);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_numer_ref(r) */
+
+mp_int mp_rat_numer_ref(mp_rat r)
+{
+  return MP_NUMER_P(r);
+}
+
+/* }}} */
+
+
+/* {{{ mp_rat_denom(r, z) */
+
+mp_result mp_rat_denom(mp_rat r, mp_int z)
+{
+  return mp_int_copy(MP_DENOM_P(r), z);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_denom_ref(r) */
+
+mp_int    mp_rat_denom_ref(mp_rat r)
+{
+  return MP_DENOM_P(r);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_sign(r) */
+
+mp_sign   mp_rat_sign(mp_rat r)
+{
+  return MP_SIGN(MP_NUMER_P(r));
+}
+
+/* }}} */
+
+/* {{{ mp_rat_copy(a, c) */
+
+mp_result mp_rat_copy(mp_rat a, mp_rat c)
+{
+  mp_result res;
+
+  if ((res = mp_int_copy(MP_NUMER_P(a), MP_NUMER_P(c))) != MP_OK)
+    return res;
+  
+  res = mp_int_copy(MP_DENOM_P(a), MP_DENOM_P(c));
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_rat_zero(r) */
+
+void      mp_rat_zero(mp_rat r)
+{
+  mp_int_zero(MP_NUMER_P(r));
+  mp_int_set_value(MP_DENOM_P(r), 1);
+  
+}
+
+/* }}} */
+
+/* {{{ mp_rat_abs(a, c) */
+
+mp_result mp_rat_abs(mp_rat a, mp_rat c)
+{
+  mp_result res;
+
+  if ((res = mp_int_abs(MP_NUMER_P(a), MP_NUMER_P(c))) != MP_OK)
+    return res;
+  
+  res = mp_int_abs(MP_DENOM_P(a), MP_DENOM_P(c));
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_rat_neg(a, c) */
+
+mp_result mp_rat_neg(mp_rat a, mp_rat c)
+{
+  mp_result res;
+
+  if ((res = mp_int_neg(MP_NUMER_P(a), MP_NUMER_P(c))) != MP_OK)
+    return res;
+
+  res = mp_int_copy(MP_DENOM_P(a), MP_DENOM_P(c));
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_rat_recip(a, c) */
+
+mp_result mp_rat_recip(mp_rat a, mp_rat c)
+{
+  mp_result res;
+
+  if (mp_rat_compare_zero(a) == 0)
+    return MP_UNDEF;
+
+  if ((res = mp_rat_copy(a, c)) != MP_OK)
+    return res;
+
+  mp_int_swap(MP_NUMER_P(c), MP_DENOM_P(c));
+
+  /* Restore the signs of the swapped elements */
+  {
+    mp_sign tmp = MP_SIGN(MP_NUMER_P(c));
+
+    MP_SIGN(MP_NUMER_P(c)) = MP_SIGN(MP_DENOM_P(c));
+    MP_SIGN(MP_DENOM_P(c)) = tmp;
+  }
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ mp_rat_add(a, b, c) */
+
+mp_result mp_rat_add(mp_rat a, mp_rat b, mp_rat c)
+{
+  return s_rat_combine(a, b, c, mp_int_add);
+
+}
+
+/* }}} */
+
+/* {{{ mp_rat_sub(a, b, c) */
+
+mp_result mp_rat_sub(mp_rat a, mp_rat b, mp_rat c)
+{
+  return s_rat_combine(a, b, c, mp_int_sub);
+
+}
+
+/* }}} */
+
+/* {{{ mp_rat_mul(a, b, c) */
+
+mp_result mp_rat_mul(mp_rat a, mp_rat b, mp_rat c)
+{
+  mp_result res;
+
+  if ((res = mp_int_mul(MP_NUMER_P(a), MP_NUMER_P(b), MP_NUMER_P(c))) != MP_OK)
+    return res;
+
+  if (mp_int_compare_zero(MP_NUMER_P(c)) != 0) {
+    if ((res = mp_int_mul(MP_DENOM_P(a), MP_DENOM_P(b), MP_DENOM_P(c))) != MP_OK)
+      return res;
+  }
+
+  return s_rat_reduce(c);
+}
+
+/* }}} */
+
+/* {{{ mp_int_div(a, b, c) */
+
+mp_result mp_rat_div(mp_rat a, mp_rat b, mp_rat c)
+{
+  mp_result res = MP_OK;
+
+  if (mp_rat_compare_zero(b) == 0)
+    return MP_UNDEF;
+
+  if (c == a || c == b) {
+    mpz_t tmp;
+
+    if ((res = mp_int_init(&tmp)) != MP_OK) return res;
+    if ((res = mp_int_mul(MP_NUMER_P(a), MP_DENOM_P(b), &tmp)) != MP_OK) 
+      goto CLEANUP;
+    if ((res = mp_int_mul(MP_DENOM_P(a), MP_NUMER_P(b), MP_DENOM_P(c))) != MP_OK)
+      goto CLEANUP;
+    res = mp_int_copy(&tmp, MP_NUMER_P(c));
+
+  CLEANUP:
+    mp_int_clear(&tmp);
+  }
+  else {
+    if ((res = mp_int_mul(MP_NUMER_P(a), MP_DENOM_P(b), MP_NUMER_P(c))) != MP_OK)
+      return res;
+    if ((res = mp_int_mul(MP_DENOM_P(a), MP_NUMER_P(b), MP_DENOM_P(c))) != MP_OK)
+      return res;
+  }
+
+  if (res != MP_OK)
+    return res;
+  else
+    return s_rat_reduce(c);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_add_int(a, b, c) */
+
+mp_result mp_rat_add_int(mp_rat a, mp_int b, mp_rat c)
+{
+  mpz_t tmp;
+  mp_result res;
+
+  if ((res = mp_int_init_copy(&tmp, b)) != MP_OK)
+    return res;
+
+  if ((res = mp_int_mul(&tmp, MP_DENOM_P(a), &tmp)) != MP_OK)
+    goto CLEANUP;
+
+  if ((res = mp_rat_copy(a, c)) != MP_OK)
+    goto CLEANUP;
+
+  if ((res = mp_int_add(MP_NUMER_P(c), &tmp, MP_NUMER_P(c))) != MP_OK)
+    goto CLEANUP;
+
+  res = s_rat_reduce(c);
+
+ CLEANUP:
+  mp_int_clear(&tmp);
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_rat_sub_int(a, b, c) */
+
+mp_result mp_rat_sub_int(mp_rat a, mp_int b, mp_rat c)
+{
+  mpz_t tmp;
+  mp_result res;
+
+  if ((res = mp_int_init_copy(&tmp, b)) != MP_OK)
+    return res;
+
+  if ((res = mp_int_mul(&tmp, MP_DENOM_P(a), &tmp)) != MP_OK)
+    goto CLEANUP;
+
+  if ((res = mp_rat_copy(a, c)) != MP_OK)
+    goto CLEANUP;
+
+  if ((res = mp_int_sub(MP_NUMER_P(c), &tmp, MP_NUMER_P(c))) != MP_OK)
+    goto CLEANUP;
+
+  res = s_rat_reduce(c);
+
+ CLEANUP:
+  mp_int_clear(&tmp);
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_rat_mul_int(a, b, c) */
+
+mp_result mp_rat_mul_int(mp_rat a, mp_int b, mp_rat c)
+{
+  mp_result res;
+
+  if ((res = mp_rat_copy(a, c)) != MP_OK)
+    return res;
+
+  if ((res = mp_int_mul(MP_NUMER_P(c), b, MP_NUMER_P(c))) != MP_OK)
+    return res;
+
+  return s_rat_reduce(c);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_div_int(a, b, c) */
+
+mp_result mp_rat_div_int(mp_rat a, mp_int b, mp_rat c)
+{
+  mp_result res;
+
+  if (mp_int_compare_zero(b) == 0)
+    return MP_UNDEF;
+
+  if ((res = mp_rat_copy(a, c)) != MP_OK)
+    return res;
+
+  if ((res = mp_int_mul(MP_DENOM_P(c), b, MP_DENOM_P(c))) != MP_OK)
+    return res;
+
+  return s_rat_reduce(c);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_expt(a, b, c) */
+
+mp_result mp_rat_expt(mp_rat a, mp_small b, mp_rat c)
+{
+  mp_result  res;
+
+  /* Special cases for easy powers. */
+  if (b == 0)
+    return mp_rat_set_value(c, 1, 1);
+  else if(b == 1)
+    return mp_rat_copy(a, c);
+
+  /* Since rationals are always stored in lowest terms, it is not necessary to
+     reduce again when raising to an integer power. */
+  if ((res = mp_int_expt(MP_NUMER_P(a), b, MP_NUMER_P(c))) != MP_OK)
+    return res;
+
+  return mp_int_expt(MP_DENOM_P(a), b, MP_DENOM_P(c));
+}
+
+/* }}} */
+
+/* {{{ mp_rat_compare(a, b) */
+
+int       mp_rat_compare(mp_rat a, mp_rat b)
+{
+  /* Quick check for opposite signs.  Works because the sign of the numerator
+     is always definitive. */
+  if (MP_SIGN(MP_NUMER_P(a)) != MP_SIGN(MP_NUMER_P(b))) {
+    if (MP_SIGN(MP_NUMER_P(a)) == MP_ZPOS)
+      return 1;
+    else
+      return -1;
+  }
+  else {
+    /* Compare absolute magnitudes; if both are positive, the answer stands,
+       otherwise it needs to be reflected about zero. */
+    int cmp = mp_rat_compare_unsigned(a, b);
+
+    if (MP_SIGN(MP_NUMER_P(a)) == MP_ZPOS)
+      return cmp;
+    else
+      return -cmp;
+  }
+}
+
+/* }}} */
+
+/* {{{ mp_rat_compare_unsigned(a, b) */
+
+int       mp_rat_compare_unsigned(mp_rat a, mp_rat b)
+{
+  /* If the denominators are equal, we can quickly compare numerators without
+     multiplying.  Otherwise, we actually have to do some work. */
+  if (mp_int_compare_unsigned(MP_DENOM_P(a), MP_DENOM_P(b)) == 0)
+    return mp_int_compare_unsigned(MP_NUMER_P(a), MP_NUMER_P(b));
+
+  else {
+    mpz_t  temp[2];
+    mp_result res;
+    int  cmp = INT_MAX, last = 0;
+
+    /* t0 = num(a) * den(b), t1 = num(b) * den(a) */
+    SETUP(mp_int_init_copy(TEMP(last), MP_NUMER_P(a)), last);
+    SETUP(mp_int_init_copy(TEMP(last), MP_NUMER_P(b)), last);
+
+    if ((res = mp_int_mul(TEMP(0), MP_DENOM_P(b), TEMP(0))) != MP_OK ||
+	(res = mp_int_mul(TEMP(1), MP_DENOM_P(a), TEMP(1))) != MP_OK)
+      goto CLEANUP;
+    
+    cmp = mp_int_compare_unsigned(TEMP(0), TEMP(1));
+    
+  CLEANUP:
+    while (--last >= 0)
+      mp_int_clear(TEMP(last));
+
+    return cmp;
+  }
+}
+
+/* }}} */
+
+/* {{{ mp_rat_compare_zero(r) */
+
+int       mp_rat_compare_zero(mp_rat r)
+{
+  return mp_int_compare_zero(MP_NUMER_P(r));
+}
+
+/* }}} */
+
+/* {{{ mp_rat_compare_value(r, n, d) */
+
+int       mp_rat_compare_value(mp_rat r, mp_small n, mp_small d)
+{
+  mpq_t tmp;
+  mp_result res;
+  int  out = INT_MAX;
+
+  if ((res = mp_rat_init(&tmp)) != MP_OK)
+    return out;
+  if ((res = mp_rat_set_value(&tmp, n, d)) != MP_OK)
+    goto CLEANUP;
+  
+  out = mp_rat_compare(r, &tmp);
+  
+ CLEANUP:
+  mp_rat_clear(&tmp);
+  return out;
+}
+
+/* }}} */
+
+/* {{{ mp_rat_is_integer(r) */
+
+int       mp_rat_is_integer(mp_rat r)
+{
+  return (mp_int_compare_value(MP_DENOM_P(r), 1) == 0);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_to_ints(r, *num, *den) */
+
+mp_result mp_rat_to_ints(mp_rat r, mp_small *num, mp_small *den)
+{
+  mp_result res;
+
+  if ((res = mp_int_to_int(MP_NUMER_P(r), num)) != MP_OK)
+    return res;
+
+  res = mp_int_to_int(MP_DENOM_P(r), den);
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_rat_to_string(r, radix, *str, limit) */
+
+mp_result mp_rat_to_string(mp_rat r, mp_size radix, char *str, int limit)
+{
+  char *start;
+  int   len;
+  mp_result res;
+
+  /* Write the numerator.  The sign of the rational number is written by the
+     underlying integer implementation. */
+  if ((res = mp_int_to_string(MP_NUMER_P(r), radix, str, limit)) != MP_OK)
+    return res;
+
+  /* If the value is zero, don't bother writing any denominator */
+  if (mp_int_compare_zero(MP_NUMER_P(r)) == 0)
+    return MP_OK;
+  
+  /* Locate the end of the numerator, and make sure we are not going to exceed
+     the limit by writing a slash. */
+  len = strlen(str);
+  start = str + len;
+  limit -= len;
+  if(limit == 0)
+    return MP_TRUNC;
+
+  *start++ = '/';
+  limit -= 1;
+  
+  res = mp_int_to_string(MP_DENOM_P(r), radix, start, limit);
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_rat_to_decimal(r, radix, prec, *str, limit) */
+mp_result mp_rat_to_decimal(mp_rat r, mp_size radix, mp_size prec,
+                            mp_round_mode round, char *str, int limit)
+{
+  mpz_t temp[3];
+  mp_result res;
+  char *start = str;
+  int len, lead_0, left = limit, last = 0;
+    
+  SETUP(mp_int_init_copy(TEMP(last), MP_NUMER_P(r)), last);
+  SETUP(mp_int_init(TEMP(last)), last);
+  SETUP(mp_int_init(TEMP(last)), last);
+
+  /* Get the unsigned integer part by dividing denominator into the absolute
+     value of the numerator. */
+  mp_int_abs(TEMP(0), TEMP(0));
+  if ((res = mp_int_div(TEMP(0), MP_DENOM_P(r), TEMP(0), TEMP(1))) != MP_OK)
+    goto CLEANUP;
+
+  /* Now:  T0 = integer portion, unsigned;
+           T1 = remainder, from which fractional part is computed. */
+
+  /* Count up leading zeroes after the radix point. */
+  for (lead_0 = 0; lead_0 < prec && mp_int_compare(TEMP(1), MP_DENOM_P(r)) < 0; 
+      ++lead_0) {
+    if ((res = mp_int_mul_value(TEMP(1), radix, TEMP(1))) != MP_OK)
+      goto CLEANUP;
+  }
+
+  /* Multiply remainder by a power of the radix sufficient to get the right
+     number of significant figures. */
+  if (prec > lead_0) {
+    if ((res = mp_int_expt_value(radix, prec - lead_0, TEMP(2))) != MP_OK)
+      goto CLEANUP;
+    if ((res = mp_int_mul(TEMP(1), TEMP(2), TEMP(1))) != MP_OK)
+      goto CLEANUP;
+  }
+  if ((res = mp_int_div(TEMP(1), MP_DENOM_P(r), TEMP(1), TEMP(2))) != MP_OK)
+    goto CLEANUP;
+
+  /* Now:  T1 = significant digits of fractional part;
+           T2 = leftovers, to use for rounding. 
+
+     At this point, what we do depends on the rounding mode.  The default is
+     MP_ROUND_DOWN, for which everything is as it should be already.
+  */
+  switch (round) {
+    int cmp;
+
+  case MP_ROUND_UP:
+    if (mp_int_compare_zero(TEMP(2)) != 0) {
+      if (prec == 0)
+	res = mp_int_add_value(TEMP(0), 1, TEMP(0));
+      else
+	res = mp_int_add_value(TEMP(1), 1, TEMP(1));
+    }
+    break;
+
+  case MP_ROUND_HALF_UP:
+  case MP_ROUND_HALF_DOWN:
+    if ((res = mp_int_mul_pow2(TEMP(2), 1, TEMP(2))) != MP_OK)
+      goto CLEANUP;
+
+    cmp = mp_int_compare(TEMP(2), MP_DENOM_P(r));    
+
+    if (round == MP_ROUND_HALF_UP)
+      cmp += 1;
+
+    if (cmp > 0) {
+      if (prec == 0)
+	res = mp_int_add_value(TEMP(0), 1, TEMP(0));
+      else
+	res = mp_int_add_value(TEMP(1), 1, TEMP(1));
+    }
+    break;
+    
+  case MP_ROUND_DOWN:
+    break;  /* No action required */
+
+  default: 
+    return MP_BADARG; /* Invalid rounding specifier */
+  }
+
+  /* The sign of the output should be the sign of the numerator, but if all the
+     displayed digits will be zero due to the precision, a negative shouldn't
+     be shown. */
+  if (MP_SIGN(MP_NUMER_P(r)) == MP_NEG &&
+      (mp_int_compare_zero(TEMP(0)) != 0 ||
+       mp_int_compare_zero(TEMP(1)) != 0)) {
+    *start++ = '-';
+    left -= 1;
+  }
+
+  if ((res = mp_int_to_string(TEMP(0), radix, start, left)) != MP_OK)
+    goto CLEANUP;
+  
+  len = strlen(start);
+  start += len;
+  left -= len;
+  
+  if (prec == 0) 
+    goto CLEANUP;
+  
+  *start++ = '.';
+  left -= 1;
+  
+  if (left < prec + 1) {
+    res = MP_TRUNC;
+    goto CLEANUP;
+  }
+
+  memset(start, '0', lead_0 - 1);
+  left -= lead_0;
+  start += lead_0 - 1;
+
+  res = mp_int_to_string(TEMP(1), radix, start, left);
+
+ CLEANUP:
+  while (--last >= 0)
+    mp_int_clear(TEMP(last));
+  
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_rat_string_len(r, radix) */
+
+mp_result mp_rat_string_len(mp_rat r, mp_size radix)
+{
+  mp_result n_len, d_len = 0;
+
+  n_len = mp_int_string_len(MP_NUMER_P(r), radix);
+
+  if (mp_int_compare_zero(MP_NUMER_P(r)) != 0)
+    d_len = mp_int_string_len(MP_DENOM_P(r), radix);
+
+  /* Though simplistic, this formula is correct.  Space for the sign flag is
+     included in n_len, and the space for the NUL that is counted in n_len
+     counts for the separator here.  The space for the NUL counted in d_len
+     counts for the final terminator here. */
+
+  return n_len + d_len;
+
+}
+
+/* }}} */
+
+/* {{{ mp_rat_decimal_len(r, radix, prec) */
+
+mp_result mp_rat_decimal_len(mp_rat r, mp_size radix, mp_size prec)
+{
+  int  z_len, f_len;
+
+  z_len = mp_int_string_len(MP_NUMER_P(r), radix);
+  
+  if (prec == 0)
+    f_len = 1; /* terminator only */
+  else
+    f_len = 1 + prec + 1; /* decimal point, digits, terminator */
+  
+  return z_len + f_len;
+}
+
+/* }}} */
+
+/* {{{ mp_rat_read_string(r, radix, *str) */
+
+mp_result mp_rat_read_string(mp_rat r, mp_size radix, const char *str)
+{
+  return mp_rat_read_cstring(r, radix, str, NULL);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_read_cstring(r, radix, *str, **end) */
+
+mp_result mp_rat_read_cstring(mp_rat r, mp_size radix, const char *str, 
+			      char **end)
+{
+  mp_result res;
+  char *endp;
+
+  if ((res = mp_int_read_cstring(MP_NUMER_P(r), radix, str, &endp)) != MP_OK &&
+      (res != MP_TRUNC))
+    return res;
+
+  /* Skip whitespace between numerator and (possible) separator */
+  while (isspace((unsigned char) *endp))
+    ++endp;
+  
+  /* If there is no separator, we will stop reading at this point. */
+  if (*endp != '/') {
+    mp_int_set_value(MP_DENOM_P(r), 1);
+    if (end != NULL)
+      *end = endp;
+    return res;
+  }
+  
+  ++endp; /* skip separator */
+  if ((res = mp_int_read_cstring(MP_DENOM_P(r), radix, endp, end)) != MP_OK)
+    return res;
+  
+  /* Make sure the value is well-defined */
+  if (mp_int_compare_zero(MP_DENOM_P(r)) == 0)
+    return MP_UNDEF;
+
+  /* Reduce to lowest terms */
+  return s_rat_reduce(r);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_read_ustring(r, radix, *str, **end) */
+
+/* Read a string and figure out what format it's in.  The radix may be supplied
+   as zero to use "default" behaviour.
+
+   This function will accept either a/b notation or decimal notation.
+ */
+mp_result mp_rat_read_ustring(mp_rat r, mp_size radix, const char *str, 
+			      char **end)
+{
+  char      *endp;
+  mp_result  res;
+
+  if (radix == 0)
+    radix = 10;  /* default to decimal input */
+
+  if ((res = mp_rat_read_cstring(r, radix, str, &endp)) != MP_OK) {
+    if (res == MP_TRUNC) {
+      if (*endp == '.')
+	res = mp_rat_read_cdecimal(r, radix, str, &endp);
+    }
+    else
+      return res;
+  }
+
+  if (end != NULL)
+    *end = endp;
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ mp_rat_read_decimal(r, radix, *str) */
+
+mp_result mp_rat_read_decimal(mp_rat r, mp_size radix, const char *str)
+{
+  return mp_rat_read_cdecimal(r, radix, str, NULL);
+}
+
+/* }}} */
+
+/* {{{ mp_rat_read_cdecimal(r, radix, *str, **end) */
+
+mp_result mp_rat_read_cdecimal(mp_rat r, mp_size radix, const char *str, 
+			       char **end)
+{
+  mp_result res;
+  mp_sign   osign;
+  char *endp;
+
+  while (isspace((unsigned char) *str))
+    ++str;
+  
+  switch (*str) {
+  case '-':
+    osign = MP_NEG;
+    break;
+  default:
+    osign = MP_ZPOS;
+  }
+  
+  if ((res = mp_int_read_cstring(MP_NUMER_P(r), radix, str, &endp)) != MP_OK &&
+     (res != MP_TRUNC))
+    return res;
+
+  /* This needs to be here. */
+  (void) mp_int_set_value(MP_DENOM_P(r), 1);
+
+  if (*endp != '.') {
+    if (end != NULL)
+      *end = endp;
+    return res;
+  }
+
+  /* If the character following the decimal point is whitespace or a sign flag,
+     we will consider this a truncated value.  This special case is because
+     mp_int_read_string() will consider whitespace or sign flags to be valid
+     starting characters for a value, and we do not want them following the
+     decimal point.
+
+     Once we have done this check, it is safe to read in the value of the
+     fractional piece as a regular old integer.
+  */
+  ++endp;
+  if (*endp == '\0') {
+    if (end != NULL)
+      *end = endp;
+    return MP_OK;
+  }
+  else if(isspace((unsigned char) *endp) || *endp == '-' || *endp == '+') {
+    return MP_TRUNC;
+  }
+  else {
+    mpz_t  frac;
+    mp_result save_res;
+    char  *save = endp;
+    int    num_lz = 0;
+
+    /* Make a temporary to hold the part after the decimal point. */
+    if ((res = mp_int_init(&frac)) != MP_OK)
+      return res;
+    
+    if ((res = mp_int_read_cstring(&frac, radix, endp, &endp)) != MP_OK &&
+       (res != MP_TRUNC))
+      goto CLEANUP;
+
+    /* Save this response for later. */
+    save_res = res;
+
+    if (mp_int_compare_zero(&frac) == 0)
+      goto FINISHED;
+
+    /* Discard trailing zeroes (somewhat inefficiently) */
+    while (mp_int_divisible_value(&frac, radix))
+      if ((res = mp_int_div_value(&frac, radix, &frac, NULL)) != MP_OK)
+	goto CLEANUP;
+    
+    /* Count leading zeros after the decimal point */
+    while (save[num_lz] == '0')
+      ++num_lz;
+
+    /* Find the least power of the radix that is at least as large as the
+       significant value of the fractional part, ignoring leading zeroes.  */
+    (void) mp_int_set_value(MP_DENOM_P(r), radix); 
+    
+    while (mp_int_compare(MP_DENOM_P(r), &frac) < 0) {
+      if ((res = mp_int_mul_value(MP_DENOM_P(r), radix, MP_DENOM_P(r))) != MP_OK)
+	goto CLEANUP;
+    }
+    
+    /* Also shift by enough to account for leading zeroes */
+    while (num_lz > 0) {
+      if ((res = mp_int_mul_value(MP_DENOM_P(r), radix, MP_DENOM_P(r))) != MP_OK)
+	goto CLEANUP;
+
+      --num_lz;
+    }
+
+    /* Having found this power, shift the numerator leftward that many, digits,
+       and add the nonzero significant digits of the fractional part to get the
+       result. */
+    if ((res = mp_int_mul(MP_NUMER_P(r), MP_DENOM_P(r), MP_NUMER_P(r))) != MP_OK)
+      goto CLEANUP;
+    
+    { /* This addition needs to be unsigned. */
+      MP_SIGN(MP_NUMER_P(r)) = MP_ZPOS;
+      if ((res = mp_int_add(MP_NUMER_P(r), &frac, MP_NUMER_P(r))) != MP_OK)
+	goto CLEANUP;
+
+      MP_SIGN(MP_NUMER_P(r)) = osign;
+    }
+    if ((res = s_rat_reduce(r)) != MP_OK)
+      goto CLEANUP;
+
+    /* At this point, what we return depends on whether reading the fractional
+       part was truncated or not.  That information is saved from when we
+       called mp_int_read_string() above. */
+  FINISHED:
+    res = save_res;
+    if (end != NULL)
+      *end = endp;
+
+  CLEANUP:
+    mp_int_clear(&frac);
+
+    return res;
+  }
+}
+
+/* }}} */
+
+/* Private functions for internal use.  Make unchecked assumptions about format
+   and validity of inputs. */
+
+/* {{{ s_rat_reduce(r) */
+
+static mp_result s_rat_reduce(mp_rat r)
+{
+  mpz_t gcd;
+  mp_result res = MP_OK;
+
+  if (mp_int_compare_zero(MP_NUMER_P(r)) == 0) {
+    mp_int_set_value(MP_DENOM_P(r), 1);
+    return MP_OK;
+  }
+
+  /* If the greatest common divisor of the numerator and denominator is greater
+     than 1, divide it out. */
+  if ((res = mp_int_init(&gcd)) != MP_OK)
+    return res;
+
+  if ((res = mp_int_gcd(MP_NUMER_P(r), MP_DENOM_P(r), &gcd)) != MP_OK)
+    goto CLEANUP;
+
+  if (mp_int_compare_value(&gcd, 1) != 0) {
+    if ((res = mp_int_div(MP_NUMER_P(r), &gcd, MP_NUMER_P(r), NULL)) != MP_OK)
+      goto CLEANUP;
+    if ((res = mp_int_div(MP_DENOM_P(r), &gcd, MP_DENOM_P(r), NULL)) != MP_OK)
+      goto CLEANUP;
+  }
+
+  /* Fix up the signs of numerator and denominator */
+  if (MP_SIGN(MP_NUMER_P(r)) == MP_SIGN(MP_DENOM_P(r)))
+    MP_SIGN(MP_NUMER_P(r)) = MP_SIGN(MP_DENOM_P(r)) = MP_ZPOS;
+  else {
+    MP_SIGN(MP_NUMER_P(r)) = MP_NEG;
+    MP_SIGN(MP_DENOM_P(r)) = MP_ZPOS;
+  }
+
+ CLEANUP:
+  mp_int_clear(&gcd);
+
+  return res;
+}
+
+/* }}} */
+
+/* {{{ s_rat_combine(a, b, c, comb_f) */
+
+static mp_result s_rat_combine(mp_rat a, mp_rat b, mp_rat c, 
+			       mp_result (*comb_f)(mp_int, mp_int, mp_int))
+{
+  mp_result res;
+
+  /* Shortcut when denominators are already common */
+  if (mp_int_compare(MP_DENOM_P(a), MP_DENOM_P(b)) == 0) {
+    if ((res = (comb_f)(MP_NUMER_P(a), MP_NUMER_P(b), MP_NUMER_P(c))) != MP_OK)
+      return res;
+    if ((res = mp_int_copy(MP_DENOM_P(a), MP_DENOM_P(c))) != MP_OK)
+      return res;
+    
+    return s_rat_reduce(c);
+  }
+  else {
+    mpz_t  temp[2];
+    int    last = 0;
+
+    SETUP(mp_int_init_copy(TEMP(last), MP_NUMER_P(a)), last);
+    SETUP(mp_int_init_copy(TEMP(last), MP_NUMER_P(b)), last);
+    
+    if ((res = mp_int_mul(TEMP(0), MP_DENOM_P(b), TEMP(0))) != MP_OK)
+      goto CLEANUP;
+    if ((res = mp_int_mul(TEMP(1), MP_DENOM_P(a), TEMP(1))) != MP_OK)
+      goto CLEANUP;
+    if ((res = (comb_f)(TEMP(0), TEMP(1), MP_NUMER_P(c))) != MP_OK)
+      goto CLEANUP;
+
+    res = mp_int_mul(MP_DENOM_P(a), MP_DENOM_P(b), MP_DENOM_P(c));
+
+  CLEANUP:
+    while (--last >= 0) 
+      mp_int_clear(TEMP(last));
+
+    if (res == MP_OK)
+      return s_rat_reduce(c);
+    else
+      return res;
+  }
+}
+
+/* }}} */
+
+/* Here there be dragons */
diff --git a/polly/lib/External/isl/imath/imrat.h b/polly/lib/External/isl/imath/imrat.h
new file mode 100644
index 00000000000..a3678a23922
--- /dev/null
+++ b/polly/lib/External/isl/imath/imrat.h
@@ -0,0 +1,124 @@
+/*
+  Name:     imrat.h
+  Purpose:  Arbitrary precision rational arithmetic routines.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Copyright (C) 2002-2007 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#ifndef IMRAT_H_
+#define IMRAT_H_
+
+#include "imath.h"
+
+#ifdef __cplusplus
+extern "C" {
+#endif
+
+typedef struct mpq {
+  mpz_t   num;    /* Numerator         */
+  mpz_t   den;    /* Denominator, <> 0 */
+} mpq_t, *mp_rat;
+
+#define MP_NUMER_P(Q)  (&((Q)->num)) /* Pointer to numerator   */
+#define MP_DENOM_P(Q)  (&((Q)->den)) /* Pointer to denominator */
+
+/* Rounding constants */
+typedef enum {
+  MP_ROUND_DOWN,
+  MP_ROUND_HALF_UP,
+  MP_ROUND_UP,
+  MP_ROUND_HALF_DOWN
+} mp_round_mode;
+
+mp_result mp_rat_init(mp_rat r);
+mp_rat    mp_rat_alloc(void);
+mp_result mp_rat_reduce(mp_rat r);
+mp_result mp_rat_init_size(mp_rat r, mp_size n_prec, mp_size d_prec);
+mp_result mp_rat_init_copy(mp_rat r, mp_rat old);
+mp_result mp_rat_set_value(mp_rat r, mp_small numer, mp_small denom);
+mp_result mp_rat_set_uvalue(mp_rat r, mp_usmall numer, mp_usmall denom);
+void      mp_rat_clear(mp_rat r);
+void      mp_rat_free(mp_rat r);
+mp_result mp_rat_numer(mp_rat r, mp_int z);             /* z = num(r)  */
+mp_int    mp_rat_numer_ref(mp_rat r);                   /* &num(r)     */
+mp_result mp_rat_denom(mp_rat r, mp_int z);             /* z = den(r)  */
+mp_int    mp_rat_denom_ref(mp_rat r);                   /* &den(r)     */
+mp_sign   mp_rat_sign(mp_rat r);
+
+mp_result mp_rat_copy(mp_rat a, mp_rat c);              /* c = a       */
+void      mp_rat_zero(mp_rat r);                        /* r = 0       */
+mp_result mp_rat_abs(mp_rat a, mp_rat c);               /* c = |a|     */
+mp_result mp_rat_neg(mp_rat a, mp_rat c);               /* c = -a      */
+mp_result mp_rat_recip(mp_rat a, mp_rat c);             /* c = 1 / a   */
+mp_result mp_rat_add(mp_rat a, mp_rat b, mp_rat c);     /* c = a + b   */
+mp_result mp_rat_sub(mp_rat a, mp_rat b, mp_rat c);     /* c = a - b   */
+mp_result mp_rat_mul(mp_rat a, mp_rat b, mp_rat c);     /* c = a * b   */
+mp_result mp_rat_div(mp_rat a, mp_rat b, mp_rat c);     /* c = a / b   */
+
+mp_result mp_rat_add_int(mp_rat a, mp_int b, mp_rat c); /* c = a + b   */
+mp_result mp_rat_sub_int(mp_rat a, mp_int b, mp_rat c); /* c = a - b   */
+mp_result mp_rat_mul_int(mp_rat a, mp_int b, mp_rat c); /* c = a * b   */
+mp_result mp_rat_div_int(mp_rat a, mp_int b, mp_rat c); /* c = a / b   */
+mp_result mp_rat_expt(mp_rat a, mp_small b, mp_rat c);  /* c = a ^ b   */
+
+int       mp_rat_compare(mp_rat a, mp_rat b);           /* a <=> b     */
+int       mp_rat_compare_unsigned(mp_rat a, mp_rat b);  /* |a| <=> |b| */
+int       mp_rat_compare_zero(mp_rat r);                /* r <=> 0     */
+int       mp_rat_compare_value(mp_rat r, mp_small n, mp_small d); /* r <=> n/d */
+int       mp_rat_is_integer(mp_rat r);
+
+/* Convert to integers, if representable (returns MP_RANGE if not). */
+mp_result mp_rat_to_ints(mp_rat r, mp_small *num, mp_small *den);
+
+/* Convert to nul-terminated string with the specified radix, writing
+   at most limit characters including the nul terminator. */
+mp_result mp_rat_to_string(mp_rat r, mp_size radix, char *str, int limit);
+
+/* Convert to decimal format in the specified radix and precision,
+   writing at most limit characters including a nul terminator. */
+mp_result mp_rat_to_decimal(mp_rat r, mp_size radix, mp_size prec,
+                            mp_round_mode round, char *str, int limit);
+
+/* Return the number of characters required to represent r in the given
+   radix.  May over-estimate. */
+mp_result mp_rat_string_len(mp_rat r, mp_size radix);
+
+/* Return the number of characters required to represent r in decimal
+   format with the given radix and precision.  May over-estimate. */
+mp_result mp_rat_decimal_len(mp_rat r, mp_size radix, mp_size prec);
+
+/* Read zero-terminated string into r */
+mp_result mp_rat_read_string(mp_rat r, mp_size radix, const char *str);
+mp_result mp_rat_read_cstring(mp_rat r, mp_size radix, const char *str,
+			      char **end);
+mp_result mp_rat_read_ustring(mp_rat r, mp_size radix, const char *str,
+			      char **end);
+
+/* Read zero-terminated string in decimal format into r */
+mp_result mp_rat_read_decimal(mp_rat r, mp_size radix, const char *str);
+mp_result mp_rat_read_cdecimal(mp_rat r, mp_size radix, const char *str,
+			       char **end);
+
+#ifdef __cplusplus
+}
+#endif
+#endif /* IMRAT_H_ */
diff --git a/polly/lib/External/isl/imath/imtest.c b/polly/lib/External/isl/imath/imtest.c
new file mode 100644
index 00000000000..804034f51e6
--- /dev/null
+++ b/polly/lib/External/isl/imath/imtest.c
@@ -0,0 +1,474 @@
+/*
+  Name:     imtest.c
+  Purpose:  Test driver for imath library.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Copyright (C) 2002-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+
+  Reads tests from input files or standard input, and runs them.  Tests have
+  the form:
+
+  code:inputs:outputs
+
+  The 'code' is a string identifying the test to be performed.  The inputs and
+  outputs are comma-separated sequences of values.  The format of each input
+  is:
+
+    1005    number in decimal notation (signs ok)
+    #x-C0E  number in hexadecimal notation
+    #b1011  number in binary notation
+    #o37750 number in octal notation
+    =k      use register k for this input
+
+  For rational tests, the following syntax is also legal:
+    @5.33   use decimal notation (for rationals only)
+            may be combined with radix notation, e.g. #x@A0.5C
+
+  Each output is a string representing the value to which the corresponding
+  result is compared in order to pass the test.  By default, tests are expected
+  to succeed (i.e., return MP_OK).  To specify an alternate return value, use
+  the notation $RESULT, where RESULT is the name of an error (e.g., MP_MEMORY,
+  MP_UNDEF, etc.) or a numeric result denoted $#number (e.g., $#-5).
+
+  Results are written to standard output in the following formats:
+
+  line<tab>number<tab>result<eoln>
+  line<tab>number<tab>result<tab>message<eoln>
+
+  The number is the number of the test in the input (starting from 1).  The
+  result is a textual description of the result code returned by the operation
+  being tested.
+
+  Note:  There is currently a fixed limit on the length of lines by this test
+  ----   driver.  You can increase it if you wish, but the code doesn't check;
+         lines over the length are truncated (split).
+ */
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+#include <limits.h>
+#include <ctype.h>
+#include <time.h>
+#include <errno.h>
+#include <assert.h>
+
+#include "imath.h"
+#include "imdrover.h"
+
+#ifdef LINE_MAX
+#undef LINE_MAX
+#endif
+
+#define LINE_MAX 4096
+
+typedef struct {
+  char  *code;
+  int    num_inputs;
+  int    num_outputs;
+  test_f call;
+} test_t;
+
+test_t g_tests[] = {                  /* What it does...    */
+  { "initu",   2,  1, test_init},     /* r0 = uv(r1)        */
+  { "initv",   2,  1, test_init},     /* r0 = v(r1)         */
+  { "setu",    2,  1, test_set},      /* r0 = uv(r1)        */
+  { "setv",    2,  1, test_set},      /* r0 = v(r1)         */
+  { "neg",     2,  1, test_neg },     /* r1 = -r0           */
+  { "abs",     2,  1, test_abs },     /* r1 = |r0|          */
+  { "add",     3,  1, test_add },     /* r3 = r1 + r2       */
+  { "addv",    3,  1, test_add },     /* r3 = r1 + v(r2)    */
+  { "sub",     3,  1, test_sub },     /* r3 = r1 - r2       */
+  { "subv",    3,  1, test_sub },     /* r3 = r1 - v(r2)    */
+  { "mul",     3,  1, test_mul },     /* r3 = r1 * r2       */
+  { "mulp2",   3,  1, test_mulp2 },   /* r3 = r1 * 2^v(r2)  */
+  { "mulv",    3,  1, test_mulv },    /* r3 = r1 * v(r2)    */
+  { "sqr",     2,  1, test_sqr },     /* r2 = r1 * r1       */
+  { "div",     4,  2, test_div },     /* r2 = r1 / r2, r3 = r1 % r2         */
+  { "divp2",   4,  2, test_divp2 },   /* r2 = r1 / 2^v(r2),r3 = r1 % 2^v(r2)*/
+  { "divv",    3,  2, test_divv },    /* r2 = r1 / v(r2), r3 = r1 % v(r2)   */
+  { "expt",    3,  1, test_expt },    /* r3 = r1 ^ v(r2) */
+  { "exptv",   3,  1, test_exptv },   /* r3 = v(r1) ^ v(r2) */
+  { "exptf",   3,  1, test_exptf },   /* r3 = r1 ^ r2 */
+  { "mod",     3,  1, test_mod },     /* r3 = r1 % r2       */
+  { "gcd",     3,  1, test_gcd },     /* r3 = gcd(r1, r2)   */
+  { "egcd",    5,  3, test_egcd },    /* r3 = gcd(r1, r2) = r1*r4 + r2*r5   */
+  { "lcm",     3,  1, test_lcm },     /* r3 = lcm(r1, r2)   */
+  { "sqrt",    2,  1, test_sqrt },    /* r2 = sqrt(r1)      */
+  { "root",    3,  1, test_root },    /* r3 = r1^(1/v(r2))  */
+  { "invmod",  3,  1, test_invmod },  /* r3 = r1^-1 mod r2  */
+  { "emod",    4,  1, test_exptmod }, /* r4 = r1^r2 mod r3  */
+  { "emodev",  4,  1, test_exptmod_ev }, /* r4 = r1^v(r2) mod r3  */
+  { "emodbv",  4,  1, test_exptmod_bv }, /* r4 = v(r1)^r2 mod r3  */
+  { "cmp",     2,  1, test_comp },       /* rtn = compare(r1, r2) */
+  { "cmpu",    2,  1, test_ucomp },      /* rtn = compare(|r1|, |r2|) */
+  { "cmpz",    1,  1, test_zcomp },      /* rtn = compare(r1, 0)      */
+  { "cmpv",    2,  1, test_vcomp },      /* rtn = compare(r1, v(r2))  */
+  { "cmpuv",   2,  1, test_uvcomp },     /* rtn = compare(r1, v(r2))  */
+  { "tostr",   2,  1, test_tostr },      /* r1: value, r2: radix, o1: result */
+  { "tobin",   1,  1, test_tobin },      /* r1: value, o1: result binary     */
+  { "readbin", 1,  1, test_read_binary },/* r1: 2's comp, o1: result value   */
+  { "to-uns",  1,  1, test_to_uns },     /* r1: value, o1: result binary     */
+  { "readuns", 1,  1, test_read_uns },   /* r1: unsigned, o1: result value   */
+  { "to-int",  1,  1, test_to_int },     /* r1: value, o1: result */
+  { "to-uint", 1,  1, test_to_uint },    /* r1: value, o1: result */
+  { "meta",   -1, -1, test_meta },
+  { "qneg",    2,  1, test_qneg },       /* r2 = -r1        */
+  { "qrecip",  2,  1, test_qrecip },     /* r2 = 1 / r1     */
+  { "qabs",    2,  1, test_qabs },       /* r2 = |r1|       */
+  { "qadd",    3,  1, test_qadd },       /* r3 = r1 + r2    */
+  { "qsub",    3,  1, test_qsub },       /* r3 = r1 - r2    */
+  { "qmul",    3,  1, test_qmul },       /* r3 = r1 * r2    */
+  { "qdiv",    3,  1, test_qdiv },       /* r3 = r1 / r2    */
+  { "qaddz",   3,  1, test_qaddz },      /* r3 = r1 + r2    */
+  { "qsubz",   3,  1, test_qsubz },      /* r3 = r1 - r2    */
+  { "qmulz",   3,  1, test_qmulz },      /* r3 = r1 * r2    */
+  { "qdivz",   3,  1, test_qdivz },      /* r3 = r1 / r2    */
+  { "qexpt",   3,  1, test_qexpt },      /* r3 = r1 ^ v(r2) */
+  { "qtostr",  2,  1, test_qtostr },     /* r1: value, r2: radix; o1: result */
+  { "qtodec",  4,  1, test_qtodec },     /* r1: val, r2: rdx, r3: prec,
+                                            r4: rounding mode; o1: res */
+  { "qrdec",   2,  1, test_qrdec },   /* r1: dec, r2: rdx; o1: result value  */
+  { NULL, 0, 0, NULL } /* end of list marker */
+};
+
+char      g_line[LINE_MAX];
+
+extern mp_result imath_errno;
+extern char *imath_errmsg;
+
+const char *g_imath_strerr[] = {
+  "MP_OK",    "MP_TRUE",  "MP_MEMORY",  "MP_RANGE",
+  "MP_UNDEF", "MP_TRUNC", "MP_BADARG"
+};
+
+int  process_file(FILE *ifp, FILE *ofp);
+int  read_line(FILE *ifp, char *line, int limit);
+void trim_line(char *line);
+int  is_blank(char *line);
+int  parse_line(char *line, testspec_t *t);
+int  count_fields(char *line, int delim);
+void parse_fields(char *line, int delim, char **start);
+int  run_test(int test_num, testspec_t *t, FILE *ofp);
+void free_test(testspec_t *t);
+int  find_test(char *code, test_t *info);
+char *error_string(mp_result res);
+
+int main(int argc, char *argv[])
+{
+  int exit_status = 0;
+
+  init_testing();
+
+  if (argc == 1) {
+    process_file(stdin, stdout);
+  }
+  else {
+    FILE *ifp;
+    int i;
+
+    for (i = 1; i < argc; ++i) {
+      if (strcmp(argv[i], "-") == 0) {
+	ifp = stdin;
+	printf("# [stdin]\n");
+      }
+      else if ((ifp = fopen(argv[i], "r")) == NULL) {
+	fprintf(stderr, "Cannot open '%s': %s\n",
+		argv[i], strerror(errno));
+	return 1;
+      }
+      else {
+	printf("# %s\n", argv[i]);
+      }
+
+      if (process_file(ifp, stdout) != 0)
+	exit_status = 1;
+
+      fclose(ifp);
+    }
+  }
+  return exit_status;
+}
+
+/* {{{ process_file(ifp, ofp) */
+
+int process_file(FILE *ifp, FILE *ofp)
+{
+  int        res, line_num, test_num = 0, num_failed = 0, num_bogus = 0;
+  clock_t    start, finish;
+  testspec_t t;
+
+  start = clock();
+  while ((line_num = read_line(ifp, g_line, LINE_MAX)) != 0) {
+    if (parse_line(g_line, &t)) {
+      t.line = line_num;
+      if ((res = run_test(++test_num, &t, ofp)) < 0)
+	++num_bogus;
+      else if (!res)
+	++num_failed;
+
+      free_test(&t);
+    }
+    else {
+      fprintf(stderr, "Line %d: Incorrect input syntax.\n", line_num);
+    }
+  }
+  finish = clock();
+
+  fprintf(ofp, "# %d tests: %d passed, %d failed, %d errors. (%.2f seconds)\n",
+	  test_num, (test_num - num_failed - num_bogus), num_failed, num_bogus,
+	  ((double)(finish - start) / CLOCKS_PER_SEC));
+
+  return num_failed;
+}
+
+/* }}} */
+
+/* {{{ read_line(ifp, line, limit) */
+
+int  read_line(FILE *ifp, char *line, int limit)
+{
+  static FILE *current_fp = NULL;
+  static int   current_line = 0;
+
+  if (ifp != current_fp) {
+    current_fp = ifp;
+    current_line = 0;
+  }
+
+  do {
+    if (fgets(line, limit, ifp) == NULL)
+      return 0;
+
+    ++current_line;
+  } while (is_blank(line));
+
+  trim_line(line);
+  return current_line;
+}
+
+/* }}} */
+
+/* {{{ trim_line(line) */
+
+void trim_line(char *line)
+{
+  int len;
+  char *fnw = line;
+
+  /* Remove leading whitespace */
+  while (isspace((unsigned char) *fnw))
+    ++fnw;
+
+  len = strlen(fnw);
+  memmove(line, fnw, len);
+
+  /* Remove trailing whitespace (including linefeeds) */
+  fnw = line + len - 1;
+  while (fnw >= line && isspace((unsigned char) *fnw))
+    *fnw-- = '\0';
+}
+
+/* }}} */
+
+/* {{{ is_blank(line) */
+
+int  is_blank(char *line)
+{
+  while (*line && *line != '#' && isspace((unsigned char) *line))
+    ++line;
+
+  return *line == '\0' || *line == '#';
+}
+
+/* }}} */
+
+/* {{{ parse_line(line, t) */
+
+int  parse_line(char *line, testspec_t *t)
+{
+  char *code_brk, *in_brk;
+  int   num_fields;
+
+  if ((code_brk = strchr(line, ':')) == NULL)
+    return 0;
+  if ((in_brk = strchr(code_brk + 1, ':')) == NULL)
+    return 0;
+
+  *code_brk = '\0';
+  t->code = line;
+  *in_brk = '\0';
+
+  num_fields = count_fields(code_brk + 1, ',');
+  t->num_inputs = num_fields;
+
+  num_fields = count_fields(in_brk + 1, ',');
+  t->num_outputs = num_fields;
+
+  if (t->num_inputs > 0) {
+    t->input = calloc(t->num_inputs, sizeof(char *));
+    parse_fields(code_brk + 1, ',', t->input);
+  }
+  if (t->num_outputs > 0) {
+    t->output = calloc(t->num_outputs, sizeof(char *));
+    parse_fields(in_brk + 1, ',', t->output);
+  }
+  return 1;
+}
+
+/* }}} */
+
+/* {{{ count_fields(line, delim) */
+
+int  count_fields(char *line, int delim)
+{
+  int count = 1;
+
+  if (*line == '\0')
+    return 0;
+
+  while (*line) {
+    if (*line == (char)delim && *(line+1) != '\0')
+      ++count;
+    ++line;
+  }
+  return count;
+}
+
+/* }}} */
+
+/* {{{ parse_fields(line, delim, start) */
+
+void parse_fields(char *line, int delim, char **start)
+{
+  int pos = 0;
+
+  start[pos++] = line;
+  while ((line = strchr(line, delim)) != NULL) {
+    *line++ = '\0';
+    start[pos++] = line;
+  }
+}
+
+/* }}} */
+
+/* {{{ run_test(test_num, t, ofp) */
+
+int  run_test(int test_num, testspec_t *t, FILE *ofp)
+{
+  test_t info;
+
+  /* Look up and reality check test parameters */
+  if (find_test(t->code, &info) < 0) {
+    fprintf(stderr, "Line %d: Test code '%s' is unknown.\n",
+	    t->line, t->code);
+    return -1;
+  }
+  else {
+    int errs = 0;
+
+    if (info.num_inputs >= 0 && t->num_inputs != info.num_inputs) {
+      fprintf(stderr,
+	      "Line %d: Wrong number of inputs to %s (want %d, have %d)\n",
+	      t->line, t->code, info.num_inputs, t->num_inputs);
+      ++errs;
+    }
+    if (info.num_outputs >= 0 && t->num_outputs != info.num_outputs) {
+      fprintf(stderr,
+	      "Line %d: Wrong number of outputs to %s (want %d, have %d)\n",
+	      t->line, t->code, info.num_outputs, t->num_outputs);
+      ++errs;
+    }
+    if (errs) {
+      fprintf(stderr, "Line %d: %d error(s), skipping this test.\n",
+	      t->line, errs);
+      return -1;
+    }
+  }
+
+  /* If return value is true, just print a generic OK message;
+     otherwise, it is assumed that imath_errno has been set to
+     a value indicating the problem. */
+  if ((info.call)(t, ofp)) {
+    fprintf(ofp, "%d\t%d\tOK\n", t->line, test_num);
+    return 1;
+  }
+  else if(imath_errno >= MP_BADARG) {
+    fprintf(ofp, "%d\t%d\t%s\n",
+	    t->line, test_num, error_string(imath_errno));
+  }
+  else {
+    fprintf(ofp, "%d\t%d\tFAILED\t%s\n",
+	    t->line, test_num, imath_errmsg);
+  }
+  return 0;
+}
+
+/* }}} */
+
+/* {{{ find_test(code, info) */
+
+int  find_test(char *code, test_t *info)
+{
+  int i = 0;
+
+  while (g_tests[i].code != NULL) {
+    if (strcmp(g_tests[i].code, code) == 0) {
+      *info = g_tests[i];
+      return i;
+    }
+    ++i;
+  }
+  return -1;
+}
+
+/* }}} */
+
+/* {{{ free_test(*t) */
+
+void free_test(testspec_t *t)
+{
+  assert(t != NULL);
+
+  if (t->input != NULL) {
+    free(t->input);
+    t->input = NULL;
+  }
+  if (t->output != NULL) {
+    free(t->output);
+    t->output = NULL;
+  }
+}
+
+/* }}} */
+
+/* {{{ error_string(res) */
+
+char *error_string(mp_result res)
+{
+  int v = abs(res);
+
+  return (char *)g_imath_strerr[v];
+}
+
+/* }}} */
+
+/* Here there be dragons */
diff --git a/polly/lib/External/isl/imath/imtimer.c b/polly/lib/External/isl/imath/imtimer.c
new file mode 100644
index 00000000000..f5a0ecf5641
--- /dev/null
+++ b/polly/lib/External/isl/imath/imtimer.c
@@ -0,0 +1,259 @@
+/*
+  Name:     imtimer.c
+  Purpose:  Timing tests for the imath library.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Copyright (C) 2002-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+#include <limits.h>
+#include <time.h>
+
+#include <unistd.h>
+
+#include "imath.h"
+
+extern mp_size multiply_threshold; /* global in imath.c */
+extern mp_size default_precision;  /* global in imath.c */
+
+double clocks_to_seconds(clock_t start, clock_t end);
+double get_multiply_time(int nt, int prec);
+double get_exptmod_time(int nt, int prec);
+mp_int alloc_values(int nt, int prec);
+void   randomize_values(mp_int values, int nt, int prec);
+void   release_values(mp_int values, int nt);
+void   mp_int_random(mp_int z, int prec);
+
+const int g_mul_factor = 1000;
+
+int main(int argc, char *argv[])
+{
+  int do_mul = 0, do_exp = 0, do_header = 1;
+  int num_tests, precision = default_precision, opt;
+  unsigned int seed = (unsigned int) time(NULL);
+
+  while ((opt = getopt(argc, argv, "ehmnp:s:t:")) != EOF) {
+    switch (opt) {
+    case 'e':
+      do_exp = 1;
+      break;
+    case 'm':
+      do_mul = 1;
+      break;
+    case 'n':
+      do_header = 0;
+      break;
+    case 'p':
+      precision = atoi(optarg);
+      break;
+    case 's':
+      seed = atoi(optarg);
+      break;
+    case 't':
+      multiply_threshold = (mp_size) atoi(optarg);
+      break;
+    default:
+      fprintf(stderr, "Usage:  imtimer [options] <num-tests>\n\n"
+              "Options understood:\n"
+              " -e        -- test modular exponentiation speed.\n"
+              " -h        -- display this help message.\n"
+              " -m        -- test multiplication speed.\n"
+              " -n        -- no header line.\n"
+              " -p <dig>  -- use values with <dig> digits.\n"
+              " -s <rnd>  -- set random seed to <rnd>.\n"
+              " -t <dig>  -- set recursion threshold to <dig> digits.\n\n");
+      return (opt != 'h');
+    }
+  }
+
+  if (optind >= argc) {
+    fprintf(stderr, "Usage:  imtimer [options] <num-tests>\n"
+            "[use \"imtimer -h\" for help with options]\n\n");
+    return 1;
+  }
+  else
+    num_tests = atoi(argv[optind]);
+
+  srand(seed);
+
+  if (num_tests <= 0) {
+    fprintf(stderr, "You must request at least one test.\n");
+    return 1;
+  }
+  if (precision <= 0) {
+    fprintf(stderr, "Precision must be positive.\n");
+    return 1;
+  } 
+  
+  if (do_header)
+    printf("NUM\tPREC\tBITS\tREC\tRESULT\n");
+  printf("%d\t%d\t%d\t%u", num_tests, precision,
+         (int) (precision * MP_DIGIT_BIT), 
+         multiply_threshold);
+
+  if (do_mul) {
+    double m_time = get_multiply_time(num_tests, precision);
+    
+    printf("\tMUL %.3f %.3f", m_time, m_time / num_tests);
+  }
+
+  if (do_exp) {
+    double e_time = get_exptmod_time(num_tests, precision);
+    
+    printf("\tEXP %.3f %.3f", e_time, e_time / num_tests);
+  }
+  fputc('\n', stdout);
+  fflush(stdout);
+
+  return 0;
+}
+
+/* {{{ clocks_to_seconds(start, end) */
+
+double clocks_to_seconds(clock_t start, clock_t end)
+{
+  return (double)(end - start) / CLOCKS_PER_SEC;
+}
+
+/* }}} */
+
+/* {{{ alloc_values(nt, prec) */
+
+mp_int alloc_values(int nt, int prec)
+{
+  mp_int out = malloc(nt * sizeof(mpz_t));
+  int i;
+
+  if (out == NULL)
+    return NULL;
+
+  for (i = 0; i < nt; ++i) {
+    if (mp_int_init_size(out + i, prec) != MP_OK) {
+      while (--i >= 0)
+	mp_int_clear(out + i);
+      return NULL;
+    }
+  }
+  
+  return out;
+}
+
+/* }}} */
+
+/* {{{ randomize_values(values, nt, prec) */
+
+void   randomize_values(mp_int values, int nt, int prec)
+{
+  int i;
+  
+  for (i = 0; i < nt; ++i)
+    mp_int_random(values + i, prec);
+}
+
+/* }}} */
+
+/* {{{ release_values(values, nt) */
+
+void   release_values(mp_int values, int nt)
+{
+  int i;
+
+  for (i = 0; i < nt; ++i)
+    mp_int_clear(values + i);
+
+  free(values);
+}
+
+/* }}} */
+
+/* {{{ get_multiply_time(nt, prec) */
+
+double get_multiply_time(int nt, int prec)
+{
+  clock_t start, end;
+  mp_int values;
+  int i;
+
+  if ((values = alloc_values(3, prec)) == NULL)
+    return 0.0;
+  randomize_values(values, 2, prec);
+
+  start = clock(); 
+  for (i = 0; i < nt; ++i)
+    mp_int_mul(values, values + 1, values + 2);
+  end = clock();
+
+  release_values(values, 3);
+
+  return clocks_to_seconds(start, end);
+}
+
+/* }}} */
+
+/* {{{ get_exptmod_time(nt, prec) */
+
+double get_exptmod_time(int nt, int prec)
+{
+  clock_t start, end;
+  mp_int values;
+  int i;
+
+  if ((values = alloc_values(4, prec)) == NULL)
+    return 0.0;
+  randomize_values(values, 3, prec);
+
+  start = clock(); 
+  for (i = 0; i < nt; ++i)
+    mp_int_exptmod(values, values + 1, values + 2, values + 3);
+  end = clock();
+
+  release_values(values, 4);
+
+  return clocks_to_seconds(start, end);
+}
+
+/* }}} */
+
+/* {{{ mp_int_random(z, prec) */
+
+void   mp_int_random(mp_int z, int prec)
+{
+  int i;
+
+  if (prec > MP_ALLOC(z))
+    prec = MP_ALLOC(z);
+
+  for (i = 0; i < prec; ++i) {
+    mp_digit d = 0;
+    int j;
+    
+    for (j = 0; j < sizeof(d); ++j)
+      d = (d << CHAR_BIT) | (rand() & UCHAR_MAX);
+    
+    z->digits[i] = d;
+  }
+  MP_USED(z) = prec;
+}
+
+/* }}} */
diff --git a/polly/lib/External/isl/imath/iprime.c b/polly/lib/External/isl/imath/iprime.c
new file mode 100644
index 00000000000..864eadbac94
--- /dev/null
+++ b/polly/lib/External/isl/imath/iprime.c
@@ -0,0 +1,116 @@
+/*
+  Name:     iprime.c
+  Purpose:  Pseudoprimality testing routines
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Copyright (C) 2002-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#include "iprime.h"
+#include <stdlib.h>
+
+static int s_ptab_size = 168;
+static int s_ptab[] = {
+  2,   3,   5,   7,   11,  13,  17,  19,  23,  29,  31,  37,  41,  43,
+  47,  53,  59,  61,  67,  71,  73,  79,  83,  89,  97,  101, 103, 107,
+  109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181,
+  191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263,
+  269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349,
+  353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433,
+  439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521,
+  523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613,
+  617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701,
+  709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809,
+  811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887,
+  907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997
+};
+
+/* {{{ mp_int_is_prime(z) */
+
+/* Test whether z is likely to be prime:
+   MP_TRUE  means it is probably prime
+   MP_FALSE means it is definitely composite
+ */
+mp_result mp_int_is_prime(mp_int z)
+{
+  int       i;
+  mp_small  rem;
+  mp_result res;
+
+  /* First check for divisibility by small primes; this eliminates a large
+     number of composite candidates quickly
+   */
+  for(i = 0; i < s_ptab_size; ++i) {
+    if((res = mp_int_div_value(z, s_ptab[i], NULL, &rem)) != MP_OK)
+      return res;
+
+    if(rem == 0)
+      return MP_FALSE;
+  }
+
+  /* Now try Fermat's test for several prime witnesses (since we now know from
+     the above that z is not a multiple of any of them)
+   */
+  {
+    mpz_t  tmp;
+
+    if((res = mp_int_init(&tmp)) != MP_OK) return res;
+
+    for(i = 0; i < 10 && i < s_ptab_size; ++i) {
+      if((res = mp_int_exptmod_bvalue(s_ptab[i], z, z, &tmp)) != MP_OK)
+	return res;
+
+      if(mp_int_compare_value(&tmp, s_ptab[i]) != 0) {
+	mp_int_clear(&tmp);
+	return MP_FALSE;
+      }
+    }
+
+    mp_int_clear(&tmp);
+  }
+
+  return MP_TRUE;
+}
+
+/* }}} */
+
+/* {{{ mp_int_find_prime(z) */
+
+/* Find the first apparent prime in ascending order from z */
+mp_result mp_int_find_prime(mp_int z)
+{
+  mp_result  res;
+
+  if(mp_int_is_even(z) && ((res = mp_int_add_value(z, 1, z)) != MP_OK))
+    return res;
+
+  while((res = mp_int_is_prime(z)) == MP_FALSE) {
+    if((res = mp_int_add_value(z, 2, z)) != MP_OK)
+      break;
+
+  }
+
+  return res;
+}
+
+/* }}} */
+
+/* Here there be dragons */
diff --git a/polly/lib/External/isl/imath/iprime.h b/polly/lib/External/isl/imath/iprime.h
new file mode 100644
index 00000000000..03ea8f9d3cd
--- /dev/null
+++ b/polly/lib/External/isl/imath/iprime.h
@@ -0,0 +1,48 @@
+/*
+  Name:     iprime.h
+  Purpose:  Pseudoprimality testing routines
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Copyright (C) 2002-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#ifndef IPRIME_H_
+#define IPRIME_H_
+
+#include "imath.h"
+
+#ifdef __cplusplus
+extern "C" {
+#endif
+
+/* Test whether z is likely to be prime
+   MP_YES means it is probably prime
+   MP_NO  means it is definitely composite
+ */
+mp_result mp_int_is_prime(mp_int z);
+
+/* Find the first apparent prime in ascending order from z */
+mp_result mp_int_find_prime(mp_int z);
+
+#ifdef __cplusplus
+}
+#endif
+#endif /* IPRIME_H_ */
diff --git a/polly/lib/External/isl/imath/pi.c b/polly/lib/External/isl/imath/pi.c
new file mode 100644
index 00000000000..e1d46487b30
--- /dev/null
+++ b/polly/lib/External/isl/imath/pi.c
@@ -0,0 +1,177 @@
+/*
+  Name:     pi.c
+  Purpose:  Computes digits of the physical constant pi.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Copyright (C) 2002-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Notes:
+  Uses Machin's formula, which should be suitable for a few thousand digits.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+#include <time.h>
+
+#include "imath.h"
+
+int g_radix = 10; /* use this radix for output */
+
+mp_result arctan(mp_small radix, mp_small mul, mp_small x, 
+		 mp_small prec, mp_int sum);
+
+char g_buf[4096];
+
+int main(int argc, char *argv[]) 
+{
+  mp_result  res;
+  mpz_t      sum1, sum2;
+  int        ndigits, out = 0;
+  clock_t    start, end;
+
+  if(argc < 2) {
+    fprintf(stderr, "Usage: %s <num-digits> [<radix>]\n", argv[0]);
+    return 1;
+  }
+
+  if((ndigits = abs(atoi(argv[1]))) == 0) {
+    fprintf(stderr, "%s: you must request at least 1 digit\n", argv[0]);
+    return 1;
+  } else if(ndigits > MP_DIGIT_MAX) {
+    fprintf(stderr, "%s: you may request at most %u digits\n", 
+	    argv[0], (unsigned int)MP_DIGIT_MAX);
+    return 1;
+  }
+
+  if(argc > 2) {
+    int radix = atoi(argv[2]);
+
+    if(radix < MP_MIN_RADIX || radix > MP_MAX_RADIX) {
+      fprintf(stderr, "%s: you may only specify a radix between %d and %d\n",
+	      argv[0], MP_MIN_RADIX, MP_MAX_RADIX);
+      return 1;
+    }
+    g_radix = radix;
+  }
+
+  mp_int_init(&sum1); mp_int_init(&sum2);
+  start = clock();
+
+  /* sum1 = 16 * arctan(1/5) */
+  if((res = arctan(g_radix, 16, 5, ndigits, &sum1)) != MP_OK) {
+    fprintf(stderr, "%s: error computing arctan: %d\n", argv[0], res);
+    out = 1; goto CLEANUP;
+  }
+
+  /* sum2 = 4 * arctan(1/239) */
+  if((res = arctan(g_radix, 4, 239, ndigits, &sum2)) != MP_OK) {
+    fprintf(stderr, "%s: error computing arctan: %d\n", argv[0], res);
+    out = 1; goto CLEANUP;
+  }
+
+  /* pi = sum1 - sum2 */
+  if((res = mp_int_sub(&sum1, &sum2, &sum1)) != MP_OK) {
+    fprintf(stderr, "%s: error computing pi: %d\n", argv[0], res);
+    out = 1; goto CLEANUP;
+  }
+  end = clock();
+
+  mp_int_to_string(&sum1, g_radix, g_buf, sizeof(g_buf));
+  printf("%c.%s\n", g_buf[0], g_buf + 1);
+
+  fprintf(stderr, "Computation took %.2f sec.\n",
+	  (double)(end - start) / CLOCKS_PER_SEC);
+
+ CLEANUP:
+  mp_int_clear(&sum1);
+  mp_int_clear(&sum2);
+
+  return out;
+}
+
+/* {{{ arctan(radix, mul, x, prec, sum) */
+
+/*
+  Compute mul * atan(1/x) to prec digits of precision, and store the
+  result in sum.
+
+  Computes atan(1/x) using the formula:
+
+               1     1      1      1  
+  atan(1/x) = --- - ---- + ---- - ---- + ...
+               x    3x^3   5x^5   7x^7
+  
+ */
+mp_result arctan(mp_small radix, mp_small mul, mp_small x, 
+		 mp_small prec, mp_int sum)
+{
+  mpz_t   t, v;
+  mp_result res;
+  mp_small  rem, sign = 1, coeff = 1;
+
+  mp_int_init(&t);
+  mp_int_init(&v);
+  ++prec;
+
+  /* Compute mul * radix^prec * x
+     The initial multiplication by x saves a special case in the loop for
+     the first term of the series.
+   */
+  if((res = mp_int_expt_value(radix, prec, &t)) != MP_OK ||
+     (res = mp_int_mul_value(&t, mul, &t)) != MP_OK ||
+     (res = mp_int_mul_value(&t, x, &t)) != MP_OK)
+    goto CLEANUP;
+
+  x *= x; /* assumes x <= sqrt(MP_SMALL_MAX) */
+  mp_int_zero(sum);
+
+  do {
+    if((res = mp_int_div_value(&t, x, &t, &rem)) != MP_OK)
+      goto CLEANUP;
+
+    if((res = mp_int_div_value(&t, coeff, &v, &rem)) != MP_OK)
+      goto CLEANUP;
+
+    /* Add or subtract the result depending on the current sign (1 = add) */
+    if(sign > 0) 
+      res = mp_int_add(sum, &v, sum);
+    else
+      res = mp_int_sub(sum, &v, sum);
+
+    if(res != MP_OK) goto CLEANUP;
+    sign = -sign;
+    coeff += 2;
+
+  } while(mp_int_compare_zero(&t) != 0);
+
+  res = mp_int_div_value(sum, radix, sum, NULL);
+
+ CLEANUP:
+  mp_int_clear(&v);
+  mp_int_clear(&t);
+  
+  return res;
+}
+
+/* }}} */
+
+/* Here there be dragons */
diff --git a/polly/lib/External/isl/imath/rsamath.c b/polly/lib/External/isl/imath/rsamath.c
new file mode 100644
index 00000000000..ff9aa1898bb
--- /dev/null
+++ b/polly/lib/External/isl/imath/rsamath.c
@@ -0,0 +1,203 @@
+/*
+  Name:     rsamath.c
+  Purpose:  Implements part of PKCS#1, v. 2.1, June 14, 2002 (RSA Labs)
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Copyright (C) 2002-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#include "rsamath.h"
+
+#include <string.h>
+#include <limits.h>
+
+static mp_result s_rsa_transform(mp_int msg, mp_int exp, 
+				 mp_int mod, mp_int out);
+
+/* {{{ rsa_i2osp(z, out, len) */
+
+/* Convert integer to octet string, per PKCS#1 v.2.1 */
+mp_result rsa_i2osp(mp_int z, unsigned char *out, int len)
+{
+  int  excess_len = mp_int_binary_len(z);
+
+  if(excess_len < len)
+    return MP_RANGE;
+
+  memset(out, 0, len);
+
+  excess_len -= len;
+  mp_int_to_binary(z, out + excess_len, len);
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ rsa_os2ip(z, in, len) */
+
+/* Convert octet string to integer, per PKCS#1 v.2.1 */
+mp_result rsa_os2ip(mp_int z, unsigned char *in, int len)
+{
+  return mp_int_read_binary(z, in, len);
+}
+
+/* }}} */
+
+/* {{{ rsa_rsaep(msg, exp, mod, cipher) */
+
+/* Primitive RSA encryption operation */
+mp_result rsa_rsaep(mp_int msg, mp_int exp, mp_int mod, mp_int cipher)
+{
+  return s_rsa_transform(msg, exp, mod, cipher);
+}
+
+/* }}} */
+
+/* {{{ rsa_rsadp(cipher, exp, mod, msg) */
+
+/* Primitive RSA decryption operation */
+mp_result rsa_rsadp(mp_int cipher, mp_int exp, mp_int mod, mp_int msg)
+{
+  return s_rsa_transform(cipher, exp, mod, msg);
+}
+
+/* }}} */
+
+/* {{{ rsa_rsasp(msg, exp, mod, signature) */
+
+/* Primitive RSA signing operation */
+mp_result rsa_rsasp(mp_int msg, mp_int exp, mp_int mod, mp_int signature)
+{
+  return s_rsa_transform(msg, exp, mod, signature);
+}
+
+/* }}} */
+
+/* {{{ rsa_rsavp(signature, exp, mod, msg) */
+
+/* Primitive RSA verification operation */
+mp_result rsa_rsavp(mp_int signature, mp_int exp, mp_int mod, mp_int msg)
+{
+  return s_rsa_transform(signature, exp, mod, msg);
+}
+
+/* }}} */
+
+/* {{{ rsa_max_message_len(mod) */
+
+/* Compute the maximum length in bytes a message can have using PKCS#1
+   v.1.5 encoding with the given modulus */
+int       rsa_max_message_len(mp_int mod)
+{
+  int  num_bits = mp_int_count_bits(mod);
+  int  num_bytes = num_bits / CHAR_BIT;
+
+  if(num_bytes < 11)
+    return 0; /* at least eleven bytes are required for padding */
+  else
+    return num_bytes - 11;
+}
+
+/* }}} */
+
+/* {{{ rsa_pkcs1v15_encode(buf, msg_len, buf_len, tag, filler) */
+
+mp_result rsa_pkcs1v15_encode(unsigned char *buf, int msg_len, 
+			      int buf_len, int tag, random_f filler)
+{
+  int  pad_len, msg_start;
+
+  /* Make sure there is enough space for the encoded output */
+  if(msg_len > (buf_len - 11))
+    return MP_RANGE;
+
+  msg_start = buf_len - msg_len;
+  pad_len = msg_start - 3;
+
+  /* Move message to top of buffer -- these might overlap, so we rely
+     on the semantics of memmove() here */
+  memmove(buf + msg_start, buf, msg_len);
+
+  /* Set initial bytes as required by the specification */
+  buf[0] = 0x00;
+  buf[1] = (unsigned char)tag;
+
+  /* Fill with random padding.  We'll just assume the filler function
+     does the right thing and only writes the requested number of
+     nonzero bytes */
+  (filler)(buf + 2, pad_len);
+
+  /* Write separator between pad and message body */
+  buf[msg_start - 1] = 0x00;
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ rsa_pkcs1v15_decode(buf, buf_len, *msg_len) */
+
+mp_result rsa_pkcs1v15_decode(unsigned char *buf, int buf_len, 
+			      int tag, int *msg_len)
+{
+  int  pad_len = 0, data_len, data_start, i;
+
+  /* Make sure the buffer is syntactically valid */
+  if(buf_len < 11 || buf[0] != 0x00 || buf[1] != (unsigned char)tag)
+    return MP_UNDEF;
+
+  /* Figure out how many bytes of random padding there are */
+  i = 2;
+  while(buf[i++] != '\0')
+    ++pad_len;
+
+  data_start = i;
+  data_len = buf_len - data_start;
+
+  /* Shift the message to the front of the buffer */
+  memmove(buf, buf + data_start, data_len);
+
+  /* Zero out the rest of the buffer */
+  memset(buf + data_len, 0, pad_len + 3);
+
+  *msg_len = data_len;
+
+  return MP_OK;
+}
+
+/* }}} */
+
+/* {{{ s_rsa_transform(msg, exp, mod, out) */
+
+static mp_result s_rsa_transform(mp_int msg, mp_int exp, 
+				 mp_int mod, mp_int out)
+{
+  if(mp_int_compare_zero(msg) < 0 ||
+     mp_int_compare(msg, mod) >= 0)
+    return MP_RANGE;
+
+  return mp_int_exptmod(msg, exp, mod, out);
+}
+
+/* }}} */
+
+/* Here there be dragons */
diff --git a/polly/lib/External/isl/imath/rsamath.h b/polly/lib/External/isl/imath/rsamath.h
new file mode 100644
index 00000000000..01ebedebaaf
--- /dev/null
+++ b/polly/lib/External/isl/imath/rsamath.h
@@ -0,0 +1,94 @@
+/*
+  Name:     rsamath.h
+  Purpose:  Implements part of PKCS#1, v. 2.1, June 14, 2002 (RSA Labs)
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+
+  Copyright (C) 2002-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#ifndef RSAMATH_H_
+#define RSAMATH_H_
+
+#include "imath.h"
+
+#ifdef __cplusplus
+extern "C" {
+#endif
+
+/* Function to fill a buffer with nonzero random bytes */
+typedef void (*random_f)(unsigned char *, int);
+
+/* Convert integer to octet string, per PKCS#1 v.2.1 */
+mp_result rsa_i2osp(mp_int z, unsigned char *out, int len);
+
+/* Convert octet string to integer, per PKCS#1 v.2.1 */
+mp_result rsa_os2ip(mp_int z, unsigned char *in, int len);
+
+/* The following operations assume that you have converted your keys
+   and message data into mp_int values somehow.                      */
+
+/* Primitive RSA encryption operation */
+mp_result rsa_rsaep(mp_int msg, mp_int exp, mp_int mod, mp_int cipher);
+
+/* Primitive RSA decryption operation */
+mp_result rsa_rsadp(mp_int cipher, mp_int exp, mp_int mod, mp_int msg);
+
+/* Primitive RSA signing operation */
+mp_result rsa_rsasp(mp_int msg, mp_int exp, mp_int mod, mp_int signature);
+
+/* Primitive RSA verification operation */
+mp_result rsa_rsavp(mp_int signature, mp_int exp, mp_int mod, mp_int msg);
+
+/* Compute the maximum length in bytes a message can have using PKCS#1
+   v.1.5 encoding with the given modulus */
+int       rsa_max_message_len(mp_int mod);
+
+/* Encode a raw message per PKCS#1 v.1.5
+   buf      - the buffer containing the message
+   msg_len  - the length in bytes of the message
+   buf_len  - the size in bytes of the buffer
+   tag      - the message tag (nonzero byte)
+   filler   - function to generate pseudorandom nonzero padding
+
+   On input, the message is in the first msg_len bytes of the buffer;
+   on output, the contents of the buffer are replaced by the padded
+   message.  If there is not enough room, MP_RANGE is returned.
+ */
+mp_result rsa_pkcs1v15_encode(unsigned char *buf, int msg_len, 
+			      int buf_len, int tag, random_f filler);
+
+/* Decode a PKCS#1 v.1.5 message back to its raw form 
+   buf      - the buffer containing the encoded message
+   buf_len  - the length in bytes of the buffer
+   tag      - the expected message tag (nonzero byte)
+   msg_len  - on output, receives the length of the message content
+   
+   On output, the message is packed into the first msg_len bytes of
+   the buffer, and the rest of the buffer is zeroed.  If the buffer is
+   not of the correct form, MP_UNDEF is returned and msg_len is undefined.
+ */
+mp_result rsa_pkcs1v15_decode(unsigned char *buf, int buf_len, 
+			      int tag, int *msg_len);
+
+#ifdef __cplusplus
+}
+#endif
+#endif /* end RSAMATH_H_ */
diff --git a/polly/lib/External/isl/imath/rtest.c b/polly/lib/External/isl/imath/rtest.c
new file mode 100644
index 00000000000..12c55207296
--- /dev/null
+++ b/polly/lib/External/isl/imath/rtest.c
@@ -0,0 +1,124 @@
+/* 
+  Name:     rtest.c
+  Purpose:  Test routines for RSA implementation.
+  Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+  
+  Copyright (C) 2002-2008 Michael J. Fromberger, All Rights Reserved.
+
+  Permission is hereby granted, free of charge, to any person obtaining a copy
+  of this software and associated documentation files (the "Software"), to deal
+  in the Software without restriction, including without limitation the rights
+  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+  copies of the Software, and to permit persons to whom the Software is
+  furnished to do so, subject to the following conditions:
+
+  The above copyright notice and this permission notice shall be included in
+  all copies or substantial portions of the Software.
+
+  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
+  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+  SOFTWARE.
+ */
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+#include <limits.h>
+#include <time.h>
+
+#include "rsamath.h"
+
+void random_fill(unsigned char *buf, int len);
+void print_buf(unsigned char *buf, int len, int brk, FILE *ofp);
+
+int main(int argc, char *argv[])
+{
+  int buf_len, msg_len, i;
+  unsigned char *buf;
+  mp_result      res;
+
+  if(argc < 3) {
+    fprintf(stderr, "Usage: %s <bufsize> <msglen>\n", argv[0]);
+    return 1;
+  }
+
+  srand((unsigned int)time(NULL));
+
+  if((buf_len = atoi(argv[1])) <= 0) {
+    fprintf(stderr, "Buffer length must be positive, not %d\n", buf_len);
+    return 2;
+  }
+  if((msg_len = atoi(argv[2])) <= 0) {
+    fprintf(stderr, "Message length must be positive, not %d\n", msg_len);
+    return 2;
+  }
+  if(msg_len > buf_len) 
+    msg_len = buf_len;
+
+  buf = calloc(buf_len, sizeof(*buf));
+  for(i = 0; i < msg_len; ++i)
+    buf[i] = i + 1;
+
+  printf("Buffer size:  %d bytes\n"
+	 "Message len:  %d bytes\n\n", buf_len, msg_len);
+
+  printf("Message:\n");
+  print_buf(buf, msg_len, 16, stdout);
+  fputc('\n', stdout);
+
+  if((res = rsa_pkcs1v15_encode(buf, msg_len, buf_len, 2, random_fill)) != MP_OK) {
+    printf("Error from encoding function: %d\n", res);
+    free(buf);
+    return 1;
+  }
+  printf("Encoded message:\n");
+  print_buf(buf, buf_len, 16, stdout);
+  fputc('\n', stdout);
+
+  msg_len = -1;  /* make decoder fill this in */
+  if((res = rsa_pkcs1v15_decode(buf, buf_len, 2, &msg_len)) != MP_OK) {
+    printf("Error from decoding function: %d\n", res);
+    free(buf);
+    return 1;
+  }
+  printf("Decoded message (%d bytes):\n", msg_len);
+  print_buf(buf, msg_len, 16, stdout);
+  fputc('\n', stdout);
+
+  free(buf);
+  return 0;
+}
+
+void random_fill(unsigned char *buf, int len)
+{
+  int i;
+
+  for(i = 0; i < len; ++i) {
+    unsigned char c = 0; 
+
+    while(c == 0) 
+      c = (unsigned char)rand();
+
+    buf[i] = c;
+  }
+}
+
+void print_buf(unsigned char *buf, int len, int brk, FILE *ofp)
+{
+  int i;
+
+  for(i = 0; i < len; ++i) {
+    fprintf(ofp, "%02X", buf[i]);
+
+    if((i + 1) % brk == 0)
+      fputc('\n', ofp);
+    else
+      fputc(' ', ofp);
+  }
+  if(i % brk)
+    fputc('\n', ofp);
+}
diff --git a/polly/lib/External/isl/imath/tests/add.t b/polly/lib/External/isl/imath/tests/add.t
new file mode 100644
index 00000000000..546aa673884
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/add.t
@@ -0,0 +1,848 @@
+# Addition tests
+
+add:0,0,0:0
+add:0,0,=1:0
+add:1,=1,=1:2
+add:1,-1,0:0
+add:-1,1,0:0
+add:-1,1,=1:0
+
+add:103427990038,909510006269847,0:909613434259885
+add:128593002,-9007199254740992,0:-9007199126147990
+add:-65537,70000,0:4463
+add:-1000000,-6543210,0:-7543210
+add:#x29C932F25E6CEF8046FA,#x-29C932F25E6CEF8046FA,0:0
+add:2,#x7FFFFFFFF,0:#x800000001
+add:95031018,=1,=1:190062036
+add:#x-176E7F8018D6BA4B550AB36AAAFDC78A9,#x186DAD2485A41B18AAF5EF17922D7E5F1F,=1:#x16F6C52C8416AF73F5A543E0E77DA1E676
+add:-628443782411312,-301652218193016,=2:-930096000604328
+add:13731535,499104,0:14230639
+add:46496,0,0:46496
+add:85658949,0,0:85658949
+add:3,86618070,0:86618073
+add:-364,0,0:-364
+add:916376,71094297,0:72010673
+add:-76,-48988,0:-49064
+add:7,98029,293800191852111:98036
+add:213,0,0:213
+add:116992,19352187,=1:19469179
+add:-54334706,-2638935,0:-56973641
+add:4210,4441,0:8651
+add:2470,-1654142,0:-1651672
+add:-32608,76896,0:44288
+add:598200,65918574,0:66516774
+add:-524863149,95573,0:-524767576
+add:4,6856995,0:6856999
+add:86710327,16209386,0:102919713
+add:31,-24,0:7
+add:5895523,9915789381,0:9921684904
+add:41557511,310172,0:41867683
+add:-6714,923,0:-5791
+add:466232157,987,0:466233144
+add:5919,-63,0:5856
+add:929994,66871,0:996865
+add:23,391,0:414
+add:751,-25255,0:-24504
+add:-282292,-1608,0:-283900
+add:0,-4119448990,0:-4119448990
+add:69248,-64165,0:5083
+add:816106,580,0:816686
+add:15171,219806,0:234977
+add:945180,-585066,0:360114
+add:27052,6664636892,0:6664663944
+add:-481622,-578,0:-482200
+add:926,-81,0:845
+add:-4084,62941238721,=1:62941234637
+add:888598,6601,0:895199
+add:-61,143796,0:143735
+add:408006264,51,0:408006315
+add:-582073,-6752,0:-588825
+add:88194084,595557600,0:683751684
+add:4963532385,72932,0:4963605317
+add:37370180,708428344,0:745798524
+add:638718578,-94054,0:638624524
+add:12603892606,194259,0:12604086865
+add:9812049,8843056,0:18655105
+add:-156093,-3718156523,0:-3718312616
+add:12264,949557843298,0:949557855562
+add:338742,9139,0:347881
+add:48848,-6268099,0:-6219251
+add:3653,981782890584,0:981782894237
+add:608763646,38233135026,0:38841898672
+add:-17063,-4811,0:-21874
+add:63244008,-380818980,0:-317574972
+add:624559862947,21088,0:624559884035
+add:-268795562,74755169,0:-194040393
+add:368827930,56987680,0:425815610
+add:142,-838643130506,0:-838643130364
+add:848909,137303147,0:138152056
+add:6682221270,4495632518460,0:4502314739730
+add:163999538727,754668,0:164000293395
+add:570098317,1230642588,0:1800740905
+add:814302609847,295465,0:814302905312
+add:61148411120,462527703,0:61610938823
+add:979075368979,64243429,0:979139612408
+add:8280601099859,431312,0:8280601531171
+add:29662,3920844058249,0:3920844087911
+add:381765363,6285,0:381771648
+add:6038301700102,167589011428,0:6205890711530
+add:-6720,50309,0:43589
+add:92070025383,8403955,0:92078429338
+add:9057100,9331,0:9066431
+add:46997620390,2556790331006,0:2603787951396
+add:9853238961850,760016661497,0:10613255623347
+add:642786326083,4994197649,0:647780523732
+add:60934395,192791945290,0:192852879685
+add:1304866806,9628566351206,0:9629871218012
+add:-537871392295,1691988267,0:-536179404028
+add:73595697641,63790,0:73595761431
+add:87798124384590,168308169,0:87798292692759
+add:77155772837617,993653875,0:77156766491492
+add:691109994727,83820903066409,0:84512013061136
+add:2012749349577,1194,0:2012749350771
+add:3909606,1258809350,0:1262718956
+add:5961027295,-191014,0:5960836281
+add:-9155782013,84622,0:-9155697391
+add:-7211667,-624806193439,0:-624813405106
+add:1754076,931310,0:2685386
+add:53288031910288,1866366148,0:53289898276436
+add:767503182,-78473238204,0:-77705735022
+add:-89763935864,-14028568280297,0:-14118332216161
+add:454616605,67243118209653,0:67243572826258
+add:674985,745910,0:1420895
+add:9182866281585,-7492901984,0:9175373379601
+add:8427973625,2948084578162,0:2956512551787
+add:89366530,96670928089,0:96760294619
+add:62255464298256,1694823,0:62255465993079
+add:5268676160308,-93499502334390,0:-88230826174082
+add:7165346,-12994330060,0:-12987164714
+add:73330752719,904668539586,0:977999292305
+add:8007349,88617667386,0:88625674735
+add:951867792,807329,0:952675121
+add:645715109491016,58609052382,0:645773718543398
+add:2037750169321,85113205892,0:2122863375213
+add:6585250,76715488825566,0:76715495410816
+add:4585299,4140224,0:8725523
+add:20194080,7047624,0:27241704
+add:786913195,5197776759303,0:5198563672498
+add:751261814658086,65539188,0:751261880197274
+add:-6325172443258,2171993016413249,0:2165667843969991
+add:3947284565,-78726628505,0:-74779343940
+add:-71699817835206,40239308204,0:-71659578527002
+add:53023714,-82553833053558,0:-82553780029844
+add:2808131187169,890250230608190,0:893058361795359
+add:4783795579,-953360022706,0:-948576227127
+add:22326520,59606609543578,0:59606631870098
+add:37110988189,2635629530529,0:2672740518718
+add:5415787,65557975919,0:65563391706
+add:687495904,-3296642739961554,0:-3296642052465650
+add:72930835789,43382168425705304,0:43382241356541093
+add:-995003864534,61722341747,0:-933281522787
+add:74285863538462,4701124896071184,0:4775410759609646
+add:-7259432400758987,917110091938734,0:-6342322308820253
+add:83640964028613,28436773,0:83640992465386
+add:12798300129,42764089830,0:55562389959
+add:907689161228474,9984041726746,0:917673202955220
+add:88648450941,-92224182312398621,0:-92224093663947680
+add:3423736794,992566059450,0:995989796244
+add:26095203,84516699542499712,0:84516699568594915
+add:-664955461457,34154013,0:-664921307444
+add:17408598155,57614703283466656,0:57614720692064811
+add:734135582143613,8386599520,0:734143968743133
+add:75433328059753089,3411724102,0:75433331471477191
+add:-5780056339,370823068636155,0:370817288579816
+add:747133022687630,743183409891560,0:1490316432579190
+add:1351023965,562216806411,0:563567830376
+add:8923051674,5453380084794,0:5462303136468
+add:244571930,-51111880145,0:-50867308215
+add:-41136584383913485,2817501078,0:-41136581566412407
+add:-8079768063523410,441741295522855156,0:433661527459331746
+add:599579537585,1505732756139,0:2105312293724
+add:6728375739,634744472,0:7363120211
+add:163686368,10797029513119,0:10797193199487
+add:38258778569482,81136715091,0:38339915284573
+add:89311393409893347,6487334114,0:89311399897227461
+add:4413952309911,-335091846,0:4413617218065
+add:-8490298221703,906197882389719833,0:906189392091498130
+add:547446627581960679,4111395072,0:547446631693355751
+add:-147199257725737,358585065185151457,0:358437865927425720
+add:733104781218,1305401890,0:734410183108
+add:66789232493812823,152353542633642764,0:219142775127455587
+add:91447917877257,890149399042107596,0:890240846959984853
+add:67840659340,45844535119015296,0:45844602959674636
+add:31351681435877221,-442068622969770457,0:-410716941533893236
+add:8505522087730720965,162001047045127445,0:8667523134775848410
+add:90408836865703,91766097115,0:90500602962818
+add:8278725919,-6565975065136635,0:-6565966786410716
+add:778195404262806210,9985183887639111776,0:10763379291901917986
+add:6696810212664216,312304914759219,0:7009115127423435
+add:3557735824865,584593215289536,0:588150951114401
+add:-49412044681,29956396329782239,0:29956346917737558
+add:11923449620,7765655187,0:19689104807
+add:1724176067,-67142223866,0:-65418047799
+add:525722978624,57023088377443306,0:57023614100421930
+add:726936921764427,3077076587081089,0:3804013508845516
+add:1790333591126,-91682335574328992,0:-91680545240737866
+add:-1112879092907469823,1867817571215,0:-1112877225089898608
+add:-8476115918476,3323398534004,0:-5152717384472
+add:83059391773,-65273298074046515729,0:-65273297990987123956
+add:2211388311050688,-245146603838894173,0:-242935215527843485
+add:585851683392336,3975804417513631370,0:3976390269197023706
+add:139186663640090,15082756961,0:139201746397051
+add:-48189710801086,-33546258184494787728,0:-33546306374205588814
+add:-100779556817445,-1522038785339961,0:-1622818342157406
+add:4885512159295669,480294778789866559,0:485180290949162228
+add:-557882003905193311,20195805293900,0:-557861808099899411
+add:154553862612072,9109218823125727072,0:9109373376988339144
+add:1483847515783086714,78233016786768531473,0:79716864302551618187
+add:632462956668123,-2951738540674650223,0:-2951106077717982100
+add:-60943119399848141,74036520715371,0:-60869082879132770
+add:-7905328383508507808,-207334403231,0:-7905328590842911039
+add:-376183968178,-862976187364,0:-1239160155542
+add:918327139068056021,-685922087142,0:918326453145968879
+add:-9846497109055054870,6775447579440637,0:-9839721661475614233
+add:-84073521083215,71309976692297262369,0:71309892618776179154
+add:-373011272994268095955,4157479098221637,0:-373007115515169874318
+add:49271774069573022576,378613526495506818344,0:427885300565079840920
+add:-5376376953931357562,7508833770175817,0:-5368868120161181745
+add:-23778714258598355551,-7185005392000393507,0:-30963719650598749058
+add:801031622749,53000130414840,0:53801162037589
+add:-970929952514676169,64017648311761173049,0:63046718359246496880
+add:444961381148474,-368409783275279249,0:-367964821894130775
+add:39999239040508219266,38054215870372108,0:40037293256378591374
+add:-712118755181107,21931036897383483,0:21218918142202376
+add:-646755675748528169,609844920243318603442,0:609198164567570075273
+add:171013028884891673,11967223547227113703,0:12138236576112005376
+add:90483783429450285931,8758299165450368514304,0:8848782948879818800235
+add:54722600417344222188,6655544009320,0:54722607072888231508
+add:7059355949487975,15428043300387,0:7074783992788362
+add:63718799432141668051,-1777818839779748621,0:61940980592361919430
+add:-811965656585153,-92999475442666083,0:-93811441099251236
+add:-480943344691207768,38637964026264,0:-480904706727181504
+add:87346474969320,11671605416687563,0:11758951891656883
+add:-144193490776783,320317203586930,0:176123712810147
+add:637224429856480,90100814693140,0:727325244549620
+add:1117448757980,20116238466569324081,0:20116239584018082061
+add:2134662653354031124,57535080216036041491,0:59669742869390072615
+add:-699305697439351543,-1322846377107844464831,0:-1323545682805283816374
+add:-287368527913803,559967333864139,0:272598805950336
+add:-4524131005300321178256,648090525452193270713,0:-3876040479848127907543
+add:983197813618615290581,-555223968315003163,0:982642589650300287418
+add:47595018158554,554354123612822084175,0:554354171207840242729
+add:76492304934567739056067,-17570682458452839171696,0:58921622476114899884371
+add:605579840932458751,5118917191828845481095,0:5119522771669777939846
+add:-77276783972774282464,97578221184517253974,0:20301437211742971510
+add:-55779468870779406,3628329199924803907,0:3572549731054024501
+add:-6810396240935929656,45581601665257956989667,0:45574791269017021060011
+add:-643395735022469,95994342321649,0:-547401392700820
+add:59033016804296,90010200392865311,0:90069233409669607
+add:412262412571713009,44652839425446643811,0:45065101838018356820
+add:-996242836885840433689,9932141967639296363784,0:8935899130753455930095
+add:-187591416585869366840,4868774783363997420,0:-182722641802505369420
+add:786124638558627932,120825949339625517416,0:121612073978184145348
+add:683056931297141410,13435555169749,0:683070366852311159
+add:78591895387046935416982,-3863271218201597391326,0:74728624168845338025656
+add:-581639737241844,59732569151616229,0:59150929414374385
+add:63839107438860260,-57936899988162199911626,0:-57936836149054761051366
+add:90241348579070434,4025779803051070287,0:4116021151630140721
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+add:46052697297460594549965,-2523434189144367108,0:46050173863271450182857
+add:-908472766655304200,503910242160770655137,0:503001769394115350937
+add:35664754770405781576262,-52328200879455304,0:35664702442204902120958
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+add:962044518204007,36941484559411981897006,0:36941485521456500101013
+add:1778825689878630,1752252372363744,0:3531078062242374
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+add:-8416111605414512356,-71387336643361802215,0:-79803448248776314571
+add:-4364114737364980,5713751339100810207300,0:5713746974986072842320
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+add:780056864215481439157498,-7015580969512485,0:780056857199900469645013
+add:978681982436757227400,2684547366513014,0:978684666984123740414
+add:-1128116293278100017914825,5548079193241448,0:-1128116287730020824673377
+add:69069382482494805,-40368676133739445253204,0:-40368607064356962758399
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+add:5107392493993300358729,-8580635176929229687,0:5098811858816371129042
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+add:289209848265652909988,896527174701053128584035,0:896816384549318781494023
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+add:-1998078850553780901,-491302474398583183312,0:-493300553249136964213
+add:48578611214917968888328,-9724795578642180430323005,0:-9676216967427262461434677
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+add:90799712490109616993113540668118622608569993361558,6473804353031813836768680654473327017068712061,0:90806186294462648806950309348773095935587062073619
+add:8558267182043409385092694859321311049409595764196,-43855190592575763264725873745536703978426263630380326,0:-43846632325393719855340781050677382667376854034616130
+add:8862769280704424674067234121859535881853068385985015416,3128430671333004136364095742988724990646419068,0:8862769283832855345400238258223631624841793376631434484
+add:378208270494674096813484033949342736267924012919568644,1144519071359286777414333143799173474364390418550794089,0:1522727341853960874227817177748516210632314431470362733
+add:-6579674113733012078327119937146015198044119737,-3734798125808966311088975985514063521776488722551148716,0:-3734798132388640424821988063841183458922503920595268453
+add:2960711245597899800032622343542149905437203154361936820,499809669955373495757639913113238315793000498691,0:2960711745407569755406118101182063018675518947362435511
+add:25821180677407112730506355642526955953730655437874901,220829252352340433327709631917215008695896671223582832,0:246650433029747546058215987559741964649627326661457733
+add:4105370173243521757766753923407753132483088841573916094,-5859495212721416724385219267491089611350684535452912,0:4099510678030800341042368704140262042871738157038463182
+add:5163869845851925892481681950908088101067549260179722231,4902983733534376916005427640372005698206345908473,0:5163874748835659426858597956335728473073247466525630704
+add:8939484182366010380381840561998154807909027117547207,-5791890006411319753232065830229236658118816108248,0:8933692292359599060628608496167925571250908301438959
+add:701510238238697450550654709447628676278557462178180,78660480845787107751570298546903383688737254959620217,0:79361991084025805202120953256351012365015812421798397
+add:341765117982065537129466755082499645994576852735,692520757209429295113115688021272259739538890867946,0:692862522327411360650245154776354759385533467720681
+add:-4895783578731845919967474094433303503864596778179935462,-8364042240998784278401769408809806799945229261776648,0:-4904147620972844704245875863842113310664542007441712110
+add:14238409419953914857984647802372979092302755219238310,6371574430505907367749465495358382099864633219840163,0:20609983850459822225734113297731361192167388439078473
+add:819145829642159547700859583648855896951539123580831,361506805586624128214007976586393021628930076090363940,0:362325951416266287761708836170041877525881615213944771
+add:1877636417667934516336941709697932194927216313530003266,3110415340804941864303432168123355167418335131760417702,0:4988051758472876380640373877821287362345551445290420968
+add:7672066575928913691752679142441515012172611183558245718,371851147532253430639711983046051123330781047618685,0:7672438427076445945183318854424561063295941964605864403
+add:95512312207249366298896076611182343964081859466,-97124870343231010626703079092608549509450053912,0:-1612558135981644327807002481426205545368194446
+add:-5806422968663023549900750663638875296207673594658363702,90516128085557874002777142664275230359248090059,0:-5806422878146895464342876660861732631932443235410273643
+add:-1527157822675106197481254438080912892158041984128532354,725563661143607175528703551625121106037288811283,0:-1527157097111445053874078909377361267036935946839721071
+add:-42827691392859896582177202183556705231114395686423,45128827735882217281406911116739020388874185279388551,0:45086000044489357384824733914555463683643070883702128
+add:46043838505488963276375641613705710661972295575962167,28467164488355628577488094369127349235531802379663224,0:74511002993844591853863735982833059897504097955625391
+add:-38951950148008162832054098288607961881664530927148,8335283480464728668637429037127284541356281150509317,0:8296331530316720505805374938838676579474616619582169
+add:622249316814675692027128671081941223678839381254582193,7261720220517111354905851547139726385938166469124332501,0:7883969537331787046932980218221667609617005850378914694
+add:3674907036214085782411042221746956364413334313796,1921363713435768167432104932165249283578036436190717,0:1925038620471982253214515974386996239942449770504513
+add:7188482591275581867180190647387989483887113319876542897,6602680320762178166851882906088739930967430231717025380,0:13791162912037760034032073553476729414854543551593568277
+add:-1122742189773381273084251248143689561591037547534016,-968674210476153943757292285354448302755574664218668317,0:-969796952665927325030376536602591992317165701766202333
+add:-5630591882567573704954413450307476327113066874903314334,39824636924677667925199181345729096573735144801160,0:-5630552057930649027286488251126130598016493139758513174
+add:209583289465308712087487686838240265261210109472315151239,81969909236232824125903325576417695025146872644272065,0:209665259374544944911613590163816682956235256344959423304
+add:647747950632636576869808198783011688046450148232434870552,916898592764747488235431320315097102338752371550,0:647747951549535169634555687018443008361547250571187242102
+add:28135712883527960051641471442637534495815689268599701727,-38552244671238077902324238695360695260926661808213,0:28135674331283288813563569118398839135120428341937893514
+add:-831441344128176770720666876352870008663528096084544984455,-2692062828036480019863362944971350891750264323444185775,0:-834133406956213250740530239297841359555278360407989170230
+add:794701017798083397494303069423796491577530373372626,-72102207786198755767165144812992747956259847426428,0:722598810011884641727137924610803743621270525946198
+add:197485890851875072994741135376825653896503519991,138163469337810386780302197638192920802358090884502167,0:138163666823701238655375192379328297628011987388022158
+add:937892167666694076995649413871805692261114284540745180898,484506468570589158974739913192421910069689875922723630197,0:1422398636237283235970389327064227602330804160463468811095
+add:49513458769560636111865879434482526113781325925566233,9100294802196263191040735829862469546124284819589,0:49522559064362832375056920170312388583327450210385822
+add:6804265638701156312303584072108871897818849901582,-11345730118278439960458447085533973659139348981794,0:-4541464479577283648154863013425101761320499080212
+add:51279593637893634015629961349071462173442991832147136305,-630963561638139460898375258178326027272464749013574827807,0:-579683968000245826882745296829254565099021757181427691502
+add:-171297863190088468833128267336456231676696349106086723,25562345013839243089120015450478579141692354834498920234,0:25391047150649154620286887183142122910015658485392833511
+add:236820721337287547681130447452461282950000944868027,7910172708500900617141147757503793584328456508673025,0:8146993429838188164822278204956254867278457453541052
+add:6265195034976332988146228918664074473854894060855990323,7606251428765818252926521082377358210182682774550344917,0:13871446463742151241072750001041432684037576835406335240
+add:89597374568972918585799186543628138088356766678901,1813314149334378741507569909010462456324603409087645773,0:1813403746708947714426155708197006084462691765854324674
+add:195306949022209072289234099720202676610869290157465206,-994716438467019815849178678443452892613653602203942,0:194312232583742052473384921041759223718255636555261264
+add:-939205563958045162651332219711896963000039406190357,563202907700832072617064560363622275558376679989258989,0:562263702136874027454413228143910378595376640583068632
+add:80098286952047593891272622966959373457660113936107,5904947277434951020763542155046693897001992702425164,0:5985045564386998614654814778013653270459652816361271
+add:19798014000926405624727315383918920042183182364693,6487262615903993794908286483828887670955698777423148134,0:6487282413917994721313911211144271589875740960605512827
+add:84945814099981001691936754290990954291541653727223572796,-456805979472350610125202046336073888213752286874218,0:84945357294001529341326629088944618217653439974936698578
+add:2512089645159345096564730857111312670343564829071686,2455481799353124733391499423535890463012321535291607772379,0:2455484311442769892736595988266747574324991878856436844065
+add:73301381174453517580526931086388280775135813526768,4626165037419596821133153230043681255350276120564978,0:4699466418594050338713680161130069536125411934091746
+add:792820818959899903274588755045896284554931997668781703281,2595874074799206526967637468037895862567437806098280407872,0:3388694893759106430242226223083792147122369803767062111153
+add:31597081840709827246680247566853001879220752914725613176,2390964620594830956235541402980808255816375199839045,0:31599472805330422077636483108255982687476569289925452221
+add:31597081840709827246680247566853001879220752914725613176,=1,=1:63194163681419654493360495133706003758441505829451226352
+add:2037822438092847430389261539441146424774943894590,39916000644658200588230707520252700806573106896546,0:41953823082751048018619969059693847231348050791136
+
+# Small integer addition tests
+
+addv:51776377,-46479,0:51729898
+addv:7494538123,-48092,0:7494490031
+addv:5,20101,=2:20106
+addv:626,15974,0:16600
+addv:927,14290,=1:15217
+addv:71398252,45970,=1:71444222
+addv:93063,60606,=1:153669
+addv:-4895,63134,0:58239
+addv:8033997,6091,0:8040088
+addv:92778437607,15749,=2:92778453356
+addv:807578710629,-16317,=1:807578694312
+addv:6722,34150,0:40872
+addv:484482964826,45791,0:484483010617
+addv:63333,19921,0:83254
+addv:757593394565,58611,0:757593453176
+addv:5848315192249,20055,=2:5848315212304
+addv:21778,35258,0:57036
+addv:639687980,-1547,0:639686433
+addv:-8993430803,-39481,0:-8993470284
+addv:-825483,40649,=2:-784834
+addv:375685401,51109,=1:375736510
+addv:134791583451863,4578,=2:134791583456441
+addv:53129115309,51316,0:53129166625
+addv:471178234304,3338,0:471178237642
+addv:-7636956216875810,23137,=1:-7636956216852673
+addv:-2312413443812,31719,=2:-2312413412093
+addv:5200691919,55510,0:5200747429
+addv:5715872742,43764,=1:5715916506
+addv:5568388095431485,6290,0:5568388095437775
+addv:76084126643,54578,0:76084181221
+addv:96531391779281,46438,0:96531391825719
+addv:-828922693,15465,=2:-828907228
+addv:769843302686441996,27790,0:769843302686469786
+addv:1414000736454,15931,0:1414000752385
+addv:606227414766,517,0:606227415283
+addv:717100323961,499,=2:717100324460
+addv:-85406334226225355,-50533,=1:-85406334226275888
+addv:-3753831486,-36277,0:-3753867763
+addv:-58465715745893828,12779,=1:-58465715745881049
+addv:2855399155447860,5468,0:2855399155453328
+addv:4552049505281749,-33297,0:4552049505248452
+addv:87837072754930,40967,=2:87837072795897
+addv:-42693214607695890,-30492,0:-42693214607726382
+addv:70742576302309,-20599,0:70742576281710
+addv:118600972756003,10194,0:118600972766197
+addv:-882812251166,55167,0:-882812195999
+addv:-416381834108529,5090,0:-416381834103439
+addv:2956620172683281,-15742,0:2956620172667539
+addv:2711117679184010399584,9500,=2:2711117679184010409084
+addv:1356315685039,40425,=1:1356315725464
+addv:100388100074891247,28468,0:100388100074919715
+addv:735453579551845,-55120,=2:735453579496725
+addv:9573520208716968,57962,=1:9573520208774930
+addv:22464333496911435917,45263,=2:22464333496911481180
+addv:2318330134061664645524,35817,=1:2318330134061664681341
+addv:40085116522469,35577,0:40085116558046
+addv:-9824195637526213,46625,=2:-9824195637479588
+addv:1387219546869173,16428,0:1387219546885601
+addv:519326948996459554946,16498,0:519326948996459571444
+addv:901089575144273153684,48818,0:901089575144273202502
+addv:-7267047498859484256669593,10213,=1:-7267047498859484256659380
+addv:2746871124496359442,58189,0:2746871124496417631
+addv:-1505797511965887167260747,10886,0:-1505797511965887167249861
+addv:816815491488878842,49097,=1:816815491488927939
+addv:16917424131349552780253,38836,=2:16917424131349552819089
+addv:957454833834708924361,39916,=1:957454833834708964277
+addv:202026277214860503572,-38223,=2:202026277214860465349
+addv:968746861113481671,1030,0:968746861113482701
+addv:895889431267844251,44333,=2:895889431267888584
+addv:-3357720693387242036833,6150,0:-3357720693387242030683
+addv:659201417242721199959654444,44813,0:659201417242721199959699257
+addv:-400853063392189225718219,-1010,0:-400853063392189225719229
+addv:7218879194561993107952676021,-44154,0:7218879194561993107952631867
+addv:4721568551140507782,28030,=1:4721568551140535812
+addv:17034283717006469465522672,-11569,=1:17034283717006469465511103
+addv:41795594273804830816397,-7668,=1:41795594273804830808729
diff --git a/polly/lib/External/isl/imath/tests/bigmul.t b/polly/lib/External/isl/imath/tests/bigmul.t
new file mode 100644
index 00000000000..72974f35827
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/bigmul.t
@@ -0,0 +1,10 @@
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diff --git a/polly/lib/External/isl/imath/tests/bigsqr.t b/polly/lib/External/isl/imath/tests/bigsqr.t
new file mode 100644
index 00000000000..67aaa506290
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/bigsqr.t
@@ -0,0 +1,11 @@
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diff --git a/polly/lib/External/isl/imath/tests/compare.t b/polly/lib/External/isl/imath/tests/compare.t
new file mode 100644
index 00000000000..98a11343aa5
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/compare.t
@@ -0,0 +1,31 @@
+# Test ordinary signed comparisons
+cmp:0,0:$#0
+cmp:0,1:$#-1
+cmp:1,0:$#1
+cmp:-1,1:$#-1
+cmp:1,-1:$#1
+cmp:123456789012345678901234567890,123456789012345678901234567891:$#-1
+
+# Test unsigned comparisons
+cmpu:0,0:$#0
+cmpu:0,1:$#-1
+cmpu:1,0:$#1
+cmpu:-1,1:$#0
+cmpu:1,-1:$#0
+cmpu:-25,15:$#1
+cmpu:#x-ffffffffffffffff,#xfffffffffffffffe:$#1
+
+# Test zero comparisons
+cmpz:0:$#0
+cmpz:-25:$#-1
+cmpz:105:$#1
+
+# Test small-value comparisons
+cmpv:0,0:$#0
+cmpv:0,1:$#-1
+cmpv:1,0:$#1
+cmpv:-1,1:$#-1
+cmpv:1,-1:$#1
+cmpv:499,108:$#1
+cmpv:499,499:$#0
+cmpv:499,-1024:$#1
diff --git a/polly/lib/External/isl/imath/tests/conv.t b/polly/lib/External/isl/imath/tests/conv.t
new file mode 100644
index 00000000000..23bedfa86ba
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/conv.t
@@ -0,0 +1,228 @@
+# Output conversion tests
+
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,2:1111011101001010001010000111011010100001010000110010011010011000100100100011101100000111011001111101101000011001110111001111001111010111000101111001010111101110
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,3:22012212212000122111011022212212002120111111102001120012102211211202200102022110202002000211111221202
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,4:33131022022013122201100302122120210203230013121331220121313033033113011321113232
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,5:404041130042310320100141302000203430214122130002340212132414134210033
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,6:44515230120451152500101352430105520150025145320010504454125502
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,7:644641136612541136016610100564613624243140151310023515322
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,8:173512120732412062323044435407317550316717172705712756
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,9:265785018434285762514442046172754680368422060744852
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,10:1411774500397290569709059837552310354075408897518
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,11:184064268501499311A17746095910428222A241708032A
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,12:47706011B225950B02BB45602AA039893118A85950892
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,13:1A188C826B982353CB58422563AC602B783101671A86
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,14:105957B358B89B018958908A9114BC3DDC410B77982
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,15:CB7B3387E23452178846C55DD9D70C7CA9AEA78E8
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,16:F74A2876A1432698923B0767DA19DCF3D71795EE
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,17:17BF7C3673B76D7G7A5GA836277296F806E7453A
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,18:2EBG8HH3HFA6185D6H0596AH96G24C966DD3HG2
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,19:6G3HGBFEG8I3F25EAF61B904EIA40CFDH2124F
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,20:10AHC3D29EBHDF3HD97905CG0JA8061855C3FI
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,21:3BA5A55J5K699B2D09C38A4B237CH51IHA132
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,22:EDEA90DJ0B5CB3FGG1C8587FEB99D3C143CA
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,23:31M26JI1BBD56K3I028MML4EEDMAJK60LGLE
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,24:GGG5M3142FKKG82EJ28111D70EMHC241E4E
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,25:4446F4D5H10982023N297BF0DKBBHLLJB0I
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,26:12E9DEEOBMKAKEP0IM284MIP7FO1O521M46
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,27:85NN0HD48NN2FDDB1F5BMMKIB8CK20MDPK
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,28:2D882A7A0O0JPCJ4APDRIB77IABAKDGJP2
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,29:MFMCI0R7S27AAA3O3L2S8K44HKA7O02CN
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,30:7IGQS73FFSHC50NNH44B6PTTNLC3M6H78
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,31:2KLUB3U9850CSN6ANIDNIF1LB29MJ43LH
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,32:UT52GTL18CJ9H4HR0TJTK6ESUFBHF5FE
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,33:BTVL87QQBMUGF8PFWU4W3VU7U922QTMW
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,34:4OG10HW0MSWJBIDEE2PDH24GA7RIHIAA
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,35:1W8W9AX2DRUX48GXOLMK0PE42H0FEUWN
+tostr:#xF74A2876A1432698923B0767DA19DCF3D71795EE,36:SVWI84VBH069WR15W1U2VTK06USY8Z2
+
+# Conversion to binary
+
+tobin:0:0
+tobin:-1:255
+tobin:1:1
+tobin:97:97
+tobin:8492:33.44
+tobin:-69393016:251.221.37.136
+tobin:-218193:252.171.175
+tobin:662844:10.29.60
+tobin:8241131209300:7.126.201.178.194.84
+tobin:705223363410:0.164.50.150.139.82
+tobin:-28632557663:249.85.93.215.161
+tobin:-553325346410701486:248.82.49.135.113.248.93.82
+tobin:-907482337424:255.44.181.214.111.112
+tobin:4770047168390927:16.242.85.39.8.167.15
+tobin:977875907836789049:13.146.28.245.9.6.65.57
+tobin:3572115845746903:12.176.210.8.21.16.215
+tobin:28949534667733:26.84.86.186.243.213
+tobin:-183125616657544851234:246.18.159.29.228.28.174.208.222
+tobin:827162476353576735:11.122.171.227.198.150.95.31
+tobin:77184512995655260:1.18.54.231.215.68.126.92
+tobin:115255556962215997764:6.63.125.254.104.50.175.49.68
+tobin:54297489021590339048:2.241.135.110.149.175.144.125.232
+tobin:393928929335369918394864191:1.69.217.178.12.173.1.47.149.43.18.63
+tobin:-75550878255305302924368236946:255.11.225.199.59.30.217.134.166.120.200.174.110
+tobin:35784643040276670536818634:29.153.177.98.228.47.31.93.65.31.202
+tobin:-5876364928874055124010543482048:181.212.112.219.201.219.43.162.130.153.141.251.64
+tobin:555590330388726236230755046204154:27.100.137.45.209.201.206.212.69.137.217.165.34.250
+tobin:2123372635933612270094308954044:26.204.252.173.123.198.84.97.94.88.53.79.188
+tobin:373404212231777284568356947010238:18.105.6.21.0.246.144.164.13.115.91.8.162.190
+tobin:355087294263155005473388626149338730:68.99.39.192.10.100.204.157.215.128.79.45.43.110.106
+tobin:9075406350072985013229496898492710997:6.211.220.20.76.13.2.68.169.142.198.197.165.106.64.85
+tobin:8633675426061757629757850232155550:1.169.172.77.223.131.135.137.86.71.95.134.109.237.158
+tobin:-217191451486096670046812323372237867852:255.92.154.114.175.151.49.93.18.105.200.58.156.24.75.180.180
+tobin:26829035080908738042076334435395170:5.42.198.7.200.110.201.64.35.201.25.125.64.230.98
+tobin:16360606577002719669450329100985058711:12.78.240.46.160.248.51.247.149.232.48.220.205.240.69.151
+tobin:26026773591715017125540836707619882:5.3.56.17.26.95.176.214.116.53.154.110.71.84.42
+tobin:-315068060596719455443436752409094193:195.81.241.253.255.182.14.126.183.113.140.138.49.151.207
+tobin:1404578023533398209741079145277644921444663:16.31.175.15.121.161.137.196.26.125.240.115.59.220.1.85.25.55
+tobin:713575197074178787527618959331047471995825:8.49.2.79.246.51.45.90.242.199.86.95.47.64.104.90.91.177
+tobin:83871430502261490413450673851124769661:63.25.12.201.105.63.89.102.174.73.208.109.149.109.99.125
+tobin:1369887131891462493813741667459688172443624:15.185.188.136.198.40.60.46.152.22.161.146.105.88.210.253.87.232
+tobin:829432057166494966641315010911028795638578:9.133.123.83.146.186.254.230.112.155.225.249.127.219.248.43.131.50
+tobin:312089922808936140630879643949120224211369:3.149.38.92.196.150.155.232.60.223.247.126.50.176.14.127.177.169
+tobin:152333180111442438605972376130506110244327899219:26.174.219.37.168.212.28.162.39.139.47.209.114.56.212.157.2.33.192.83
+tobin:-2899223787613074281110944570982644571450999:222.183.242.113.50.20.39.238.192.239.78.170.102.70.102.231.209.137
+tobin:444765267932710530788173695233378407323077034767:77.231.246.225.189.253.236.180.85.80.95.147.220.91.211.171.195.205.195.15
+tobin:2928186048102542874310326072002539929893828809270937:7.211.139.211.153.76.242.8.164.101.41.243.188.255.255.239.194.98.21.205.38.153
+tobin:-826647521214480264223642441769570761615607819637146:253.202.98.131.34.175.212.21.119.210.126.69.61.176.197.180.182.201.116.60.110.102
+tobin:8400624623087855337736820815095169248114548022:1.120.178.112.248.52.41.234.187.122.100.10.161.143.20.186.83.68.157.54
+tobin:-14981001549924280586667180618266982428182337700429:245.191.228.168.55.2.155.157.248.12.207.194.168.98.220.91.66.110.69.189.179
+tobin:710160715229320560332911086106093153060357460087262479:7.106.23.173.245.150.209.228.252.222.62.180.93.143.150.200.182.76.119.140.239.205.15
+tobin:73883264784349702632503413711514594728450092072160898:0.197.120.251.47.105.95.210.233.70.183.8.43.3.136.2.36.106.51.96.215.206.130
+
+# Conversion from binary
+
+readbin:0:0
+readbin:255:-1
+readbin:1:1
+readbin:97:97
+readbin:33.44:8492
+readbin:251.221.37.136:-69393016
+readbin:252.171.175:-218193
+readbin:10.29.60:662844
+readbin:7.126.201.178.194.84:8241131209300
+readbin:0.164.50.150.139.82:705223363410
+readbin:249.85.93.215.161:-28632557663
+readbin:248.82.49.135.113.248.93.82:-553325346410701486
+readbin:255.44.181.214.111.112:-907482337424
+readbin:16.242.85.39.8.167.15:4770047168390927
+readbin:13.146.28.245.9.6.65.57:977875907836789049
+readbin:12.176.210.8.21.16.215:3572115845746903
+readbin:26.84.86.186.243.213:28949534667733
+readbin:246.18.159.29.228.28.174.208.222:-183125616657544851234
+readbin:11.122.171.227.198.150.95.31:827162476353576735
+readbin:1.18.54.231.215.68.126.92:77184512995655260
+readbin:6.63.125.254.104.50.175.49.68:115255556962215997764
+readbin:2.241.135.110.149.175.144.125.232:54297489021590339048
+readbin:1.69.217.178.12.173.1.47.149.43.18.63:393928929335369918394864191
+readbin:255.11.225.199.59.30.217.134.166.120.200.174.110:-75550878255305302924368236946
+readbin:29.153.177.98.228.47.31.93.65.31.202:35784643040276670536818634
+readbin:181.212.112.219.201.219.43.162.130.153.141.251.64:-5876364928874055124010543482048
+readbin:27.100.137.45.209.201.206.212.69.137.217.165.34.250:555590330388726236230755046204154
+readbin:26.204.252.173.123.198.84.97.94.88.53.79.188:2123372635933612270094308954044
+readbin:18.105.6.21.0.246.144.164.13.115.91.8.162.190:373404212231777284568356947010238
+readbin:68.99.39.192.10.100.204.157.215.128.79.45.43.110.106:355087294263155005473388626149338730
+readbin:6.211.220.20.76.13.2.68.169.142.198.197.165.106.64.85:9075406350072985013229496898492710997
+readbin:1.169.172.77.223.131.135.137.86.71.95.134.109.237.158:8633675426061757629757850232155550
+readbin:255.92.154.114.175.151.49.93.18.105.200.58.156.24.75.180.180:-217191451486096670046812323372237867852
+readbin:5.42.198.7.200.110.201.64.35.201.25.125.64.230.98:26829035080908738042076334435395170
+readbin:12.78.240.46.160.248.51.247.149.232.48.220.205.240.69.151:16360606577002719669450329100985058711
+readbin:5.3.56.17.26.95.176.214.116.53.154.110.71.84.42:26026773591715017125540836707619882
+readbin:195.81.241.253.255.182.14.126.183.113.140.138.49.151.207:-315068060596719455443436752409094193
+readbin:16.31.175.15.121.161.137.196.26.125.240.115.59.220.1.85.25.55:1404578023533398209741079145277644921444663
+readbin:8.49.2.79.246.51.45.90.242.199.86.95.47.64.104.90.91.177:713575197074178787527618959331047471995825
+readbin:63.25.12.201.105.63.89.102.174.73.208.109.149.109.99.125:83871430502261490413450673851124769661
+readbin:15.185.188.136.198.40.60.46.152.22.161.146.105.88.210.253.87.232:1369887131891462493813741667459688172443624
+readbin:9.133.123.83.146.186.254.230.112.155.225.249.127.219.248.43.131.50:829432057166494966641315010911028795638578
+readbin:3.149.38.92.196.150.155.232.60.223.247.126.50.176.14.127.177.169:312089922808936140630879643949120224211369
+readbin:26.174.219.37.168.212.28.162.39.139.47.209.114.56.212.157.2.33.192.83:152333180111442438605972376130506110244327899219
+readbin:222.183.242.113.50.20.39.238.192.239.78.170.102.70.102.231.209.137:-2899223787613074281110944570982644571450999
+readbin:77.231.246.225.189.253.236.180.85.80.95.147.220.91.211.171.195.205.195.15:444765267932710530788173695233378407323077034767
+readbin:7.211.139.211.153.76.242.8.164.101.41.243.188.255.255.239.194.98.21.205.38.153:2928186048102542874310326072002539929893828809270937
+readbin:253.202.98.131.34.175.212.21.119.210.126.69.61.176.197.180.182.201.116.60.110.102:-826647521214480264223642441769570761615607819637146
+readbin:1.120.178.112.248.52.41.234.187.122.100.10.161.143.20.186.83.68.157.54:8400624623087855337736820815095169248114548022
+readbin:245.191.228.168.55.2.155.157.248.12.207.194.168.98.220.91.66.110.69.189.179:-14981001549924280586667180618266982428182337700429
+readbin:7.106.23.173.245.150.209.228.252.222.62.180.93.143.150.200.182.76.119.140.239.205.15:710160715229320560332911086106093153060357460087262479
+readbin:0.197.120.251.47.105.95.210.233.70.183.8.43.3.136.2.36.106.51.96.215.206.130:73883264784349702632503413711514594728450092072160898
+#
+# -- From Tom Wu, triggered a bug in s_qmul():
+#
+readbin:22.200.146.12.243.10.174.111.79.92.242.37.59.156.205.75.163.223.107.103.20.203.19.168.35.69.241.146.93.87.91.210:10305260085868809781027103920001537650251216529133196585936939957967423757266
+
+# Conversion to unsigned
+
+to-uns:19:19
+to-uns:732849:11.46.177
+to-uns:2456939301:146.113.235.37
+to-uns:65221819301662:59.81.163.94.255.30
+to-uns:-844378241131209300:11.183.213.137.94.177.226.84
+to-uns:8570522336341023286325:1.208.155.227.240.84.37.160.100.53
+to-uns:57663995533253464107014863:47.178.211.118.41.71.211.74.100.90.207
+to-uns:290748233742495477004716839092:3.171.117.77.85.127.49.10.213.243.253.144.180
+to-uns:7769778759078367890498335721158457:1.127.20.101.49.86.0.169.68.4.237.33.216.119.57
+to-uns:46903702894953466773316183125616657544:35.73.83.24.98.4.211.124.18.69.32.88.95.142.132.136
+to-uns:-851234028271624763535767352077184512995655:9.197.141.78.121.20.132.41.143.246.59.33.126.2.92.102.177.71
+to-uns:-2609311525555696221599776451542974890215903390:117.1.109.101.2.62.220.206.134.57.198.156.134.252.135.230.114.144.158
+to-uns:48673939289293353699183948641914875550878255305302:33.77.214.236.147.20.12.74.7.55.12.207.229.43.143.81.121.102.216.82.86
+to-uns:924368236946243578464304027667053681863478587636492887:9.166.158.105.220.232.187.149.229.98.114.203.143.151.179.13.117.242.141.170.94.170.87
+to-uns:4055124010543482048995555903303887262362307550462041543621:165.97.120.89.181.168.202.95.163.0.0.6.89.126.141.209.204.168.88.165.194.166.83.197
+to-uns:23372635933612270094308954044673734042122317772845683569470102:14.139.121.201.25.49.14.230.89.139.204.223.141.232.201.189.74.36.63.100.115.152.44.110.42.150
+to-uns:382935508729426315500547338862614933873079907540635007298501322949:3.162.221.90.36.235.220.148.206.72.220.121.230.47.59.194.10.29.105.64.171.5.133.101.247.195.168.197
+to-uns:6898492710997158633675426061757629757850232155550392171914514860966700:255.225.26.217.121.147.85.123.62.227.155.115.151.17.14.226.20.151.216.70.129.251.123.127.200.253.229.131.44
+to-uns:46812323372237867852342682903508090873804207633443539517026163606065770027:26.126.173.147.234.225.62.206.132.172.56.142.212.124.86.139.56.123.128.83.59.140.190.156.37.116.71.8.107.238.43
+to-uns:196694503291009850587117226026773591715017125540836707619882023150680605967194:1.178.221.36.198.34.91.147.187.41.80.248.32.21.173.116.21.184.114.128.226.148.105.231.211.137.139.144.142.36.59.195.90
+to-uns:-5544343675240909419388140457802353339820974107914527764492144466356713575197074178:187.9.226.242.27.68.215.247.155.152.75.195.109.94.123.200.36.80.1.19.58.240.99.181.158.175.120.210.195.130.202.54.99.2
+to-uns:78752761895933104747199582531838714305022614904134506738511247696618513698871318914624:40.137.215.13.233.153.110.152.75.205.157.188.162.180.175.192.136.147.0.49.169.41.111.20.20.6.222.4.94.232.86.88.35.227.166.64
+to-uns:938137416674596881724436242382943205716649496664131501091102879563857843031208992280893614:7.94.95.163.78.130.176.52.11.67.110.168.191.141.240.164.45.130.231.131.248.48.191.162.48.249.20.181.82.141.159.150.82.213.105.159.88.174
+to-uns:-630879643949120224211369280152333180111442438605972376130506110244327899219912899223787613074:19.91.70.129.236.212.139.74.185.0.218.23.135.137.133.202.56.181.112.58.173.172.130.179.124.221.173.55.223.123.126.130.88.154.176.35.3.43.146
+to-uns:28111094457098264457145099945444765267932710530788173695233378407323077034767182928186048102542874:13.41.35.240.41.168.225.145.107.18.87.253.43.231.202.150.209.215.163.157.127.252.45.194.199.62.14.60.97.225.213.204.55.37.185.133.79.239.24.46.26
+
+# Conversion from unsigned
+
+readuns:19:19
+readuns:11.46.177:732849
+readuns:146.113.235.37:2456939301
+readuns:59.81.163.94.255.30:65221819301662
+readuns:11.183.213.137.94.177.226.84:844378241131209300
+readuns:1.208.155.227.240.84.37.160.100.53:8570522336341023286325
+readuns:47.178.211.118.41.71.211.74.100.90.207:57663995533253464107014863
+readuns:3.171.117.77.85.127.49.10.213.243.253.144.180:290748233742495477004716839092
+readuns:1.127.20.101.49.86.0.169.68.4.237.33.216.119.57:7769778759078367890498335721158457
+readuns:35.73.83.24.98.4.211.124.18.69.32.88.95.142.132.136:46903702894953466773316183125616657544
+readuns:9.197.141.78.121.20.132.41.143.246.59.33.126.2.92.102.177.71:851234028271624763535767352077184512995655
+readuns:117.1.109.101.2.62.220.206.134.57.198.156.134.252.135.230.114.144.158:2609311525555696221599776451542974890215903390
+readuns:33.77.214.236.147.20.12.74.7.55.12.207.229.43.143.81.121.102.216.82.86:48673939289293353699183948641914875550878255305302
+readuns:9.166.158.105.220.232.187.149.229.98.114.203.143.151.179.13.117.242.141.170.94.170.87:924368236946243578464304027667053681863478587636492887
+readuns:165.97.120.89.181.168.202.95.163.0.0.6.89.126.141.209.204.168.88.165.194.166.83.197:4055124010543482048995555903303887262362307550462041543621
+readuns:14.139.121.201.25.49.14.230.89.139.204.223.141.232.201.189.74.36.63.100.115.152.44.110.42.150:23372635933612270094308954044673734042122317772845683569470102
+readuns:3.162.221.90.36.235.220.148.206.72.220.121.230.47.59.194.10.29.105.64.171.5.133.101.247.195.168.197:382935508729426315500547338862614933873079907540635007298501322949
+readuns:255.225.26.217.121.147.85.123.62.227.155.115.151.17.14.226.20.151.216.70.129.251.123.127.200.253.229.131.44:6898492710997158633675426061757629757850232155550392171914514860966700
+readuns:26.126.173.147.234.225.62.206.132.172.56.142.212.124.86.139.56.123.128.83.59.140.190.156.37.116.71.8.107.238.43:46812323372237867852342682903508090873804207633443539517026163606065770027
+readuns:1.178.221.36.198.34.91.147.187.41.80.248.32.21.173.116.21.184.114.128.226.148.105.231.211.137.139.144.142.36.59.195.90:196694503291009850587117226026773591715017125540836707619882023150680605967194
+readuns:187.9.226.242.27.68.215.247.155.152.75.195.109.94.123.200.36.80.1.19.58.240.99.181.158.175.120.210.195.130.202.54.99.2:5544343675240909419388140457802353339820974107914527764492144466356713575197074178
+readuns:40.137.215.13.233.153.110.152.75.205.157.188.162.180.175.192.136.147.0.49.169.41.111.20.20.6.222.4.94.232.86.88.35.227.166.64:78752761895933104747199582531838714305022614904134506738511247696618513698871318914624
+readuns:7.94.95.163.78.130.176.52.11.67.110.168.191.141.240.164.45.130.231.131.248.48.191.162.48.249.20.181.82.141.159.150.82.213.105.159.88.174:938137416674596881724436242382943205716649496664131501091102879563857843031208992280893614
+readuns:19.91.70.129.236.212.139.74.185.0.218.23.135.137.133.202.56.181.112.58.173.172.130.179.124.221.173.55.223.123.126.130.88.154.176.35.3.43.146:630879643949120224211369280152333180111442438605972376130506110244327899219912899223787613074
+readuns:13.41.35.240.41.168.225.145.107.18.87.253.43.231.202.150.209.215.163.157.127.252.45.194.199.62.14.60.97.225.213.204.55.37.185.133.79.239.24.46.26:28111094457098264457145099945444765267932710530788173695233378407323077034767182928186048102542874
+
+# Conversion to small signed integers
+to-int:0:0
+to-int:399392010000012093849283940182938492839481928349823984:$MP_RANGE
+to-int:-29381082039841029384018230948203801985928309481203984:$MP_RANGE
+to-int:-25:-25
+to-int:101:101
+to-int:-1235:-1235
+
+# Conversion to small signed integers
+to-uint:0:0
+to-uint:1:1
+to-uint:10:10
+to-uint:-15:$MP_RANGE
+to-uint:101:101
+to-uint:100000000000000000000000000000000000000000000000000:$MP_RANGE
+
+# Tests to trigger off-by-one bug in computing number of digits needed for string
+tostr:99999999999999999999999999999,10:99999999999999999999999999999
+tostr:9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999,10:9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
+tostr:832136388736902947561088644572239329432528812529007215254079045034263648726187965907662527121488680217097287175564480588901672671661161,10:832136388736902947561088644572239329432528812529007215254079045034263648726187965907662527121488680217097287175564480588901672671661161
diff --git a/polly/lib/External/isl/imath/tests/div.t b/polly/lib/External/isl/imath/tests/div.t
new file mode 100644
index 00000000000..36a4831d408
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/div.t
@@ -0,0 +1,963 @@
+# Division tests
+
+# Tests for error conditions
+
+div:102233,0,=1,1:$MP_UNDEF,?
+div:0,=1,583922,-12345:$MP_UNDEF,?
+
+# General division tests, verified using GNU bc
+
+div:0,847887705,0,0:0,0
+div:80000000000000000000,25,0,0:3200000000000000000,0
+div:65000,25,0,0:2600,0
+div:800000000000000,25,0,0:32000000000000,0
+div:-4,58,0,0:0,-4
+div:36869,818935,0,0:0,36869
+div:364,921,0,0:0,364
+div:5169,898,0,0:5,679
+div:#x100000000,#xE424,=1,0:#x11F42,#xD2B8
+div:-6144,15855207,0,0:0,-6144
+div:680441,-8247,0,0:-82,4187
+div:-31218529,577983165,0,0:0,-31218529
+div:3,6280,0,0:0,3
+div:-2,39790,0,0:0,-2
+div:-310,611711,0,0:0,-310
+div:46573596,6038351,0,0:7,4305139
+div:4525534,39110,0,0:115,27884
+div:5938,7789378,0,0:0,5938
+div:19732849245693930165221819301662844,18446744073709551616,=1,=2:1069719901075515,6029073732495380604
+div:9,456578633,=2,=1:0,9
+div:538090499,1858,0,0:289607,693
+div:572,3804550,0,0:0,572
+div:288850,21164,0,0:13,13718
+div:-5190106,34726,0,0:-149,-15932
+div:7795,72811,0,0:0,7795
+div:781579692,-5,0,0:-156315938,2
+div:781579692,-5,=1,=2:-156315938,2
+div:781579692,-5,=2,=1:-156315938,2
+div:453020143,-3473628,0,0:-130,1448503
+div:8,853320290,0,0:0,8
+div:852316479,2925,0,0:291390,729
+div:8,5,0,0:1,3
+div:859848,9,0,0:95538,6
+div:70,48,0,0:1,22
+div:253385147,659538174,0,0:0,253385147
+div:-455686878,87344,0,0:-5217,-13230
+div:28271,635389,0,0:0,28271
+div:-86,2692,0,0:0,-86
+div:8591158476,73,0,0:117687102,30
+div:9082,275389,0,0:0,9082
+div:-582874,74264,0,0:-7,-63026
+div:237,4104,0,0:0,237
+div:7076150589,92653,0,0:76372,55673
+div:28665011462,-1272,0,0:-22535386,470
+div:-8056787359,8082209,0,0:-996,-6907195
+div:53192266,900107,0,0:59,85953
+div:91513,740,0,0:123,493
+div:-98721211,8015090320,0,0:0,-98721211
+div:11450,-36952886107,0,0:0,11450
+div:13925193,949882,0,0:14,626845
+div:683469315,-73092992,0,0:-9,25632387
+div:1164719,42872,0,0:27,7175
+div:-83594243092,3591081825,0,0:-23,-999361117
+div:87138285,266592788,0,0:0,87138285
+div:936091,704216,0,0:1,231875
+div:142238814011,709885571028,0,0:0,142238814011
+div:580,-60286,0,0:0,580
+div:5794931,-5726480,0,0:-1,68451
+div:88764678929,-2091175751,0,0:-42,935297387
+div:950181713208,-13449,0,0:-70650733,5091
+div:97846262,387939152,0,0:0,97846262
+div:32321004,4640815041,0,0:0,32321004
+div:9059,24970299,0,0:0,9059
+div:-223020,974872432,0,0:0,-223020
+div:628713585932,70889408675,0,0:8,61598316532
+div:5404031582279,-4343762403,0,0:-1244,391152947
+div:-27687,92528708054,0,0:0,-27687
+div:71252,-444744751915,0,0:0,71252
+div:2821624059,830276221,0,0:3,330795396
+div:77957,604211270,0,0:0,77957
+div:4336867764,377604023491,0,0:0,4336867764
+div:-1183882,537011618,0,0:0,-1183882
+div:7224227137658,26159,0,0:276166028,11206
+div:40420610021,29600662602,0,0:1,10819947419
+div:-53857,7190456320,0,0:0,-53857
+div:9365,85703,0,0:0,9365
+div:84454441,7874100414786,0,0:0,84454441
+div:-692,-92802,0,0:0,-692
+div:-31682292914,2103996,0,0:-15058,-321146
+div:675174791229,617791,0,0:1092885,274194
+div:151503,-7352433026033,0,0:0,151503
+div:-5675649814790,-45077421077,0,0:125,-40972180165
+div:-259966,4211178684,0,0:0,-259966
+div:99243,8777962733,0,0:0,99243
+div:-603163384555,4331787866,0,0:-139,-1044871181
+div:602983,3355548,0,0:0,602983
+div:-6945091776175,64113609219877,0,0:0,-6945091776175
+div:637048330,39484763123,0,0:0,637048330
+div:-268774166,54251789102520,0,0:0,-268774166
+div:96621785,-44159202070842,0,0:0,96621785
+div:259095,798144185517,0,0:0,259095
+div:673478104612,6115944997,0,0:110,724154942
+div:64865142,-7529340145,0,0:0,64865142
+div:62901084202908,923068964,0,0:68143,395789056
+div:857311552,30222,0,0:28367,4078
+div:6646483,1292987,0,0:5,181548
+div:855267457921,77070194891856,0,0:0,855267457921
+div:930905,-429874001253,0,0:0,930905
+div:-82732,973702689,0,0:0,-82732
+div:7498877,33402368,0,0:0,7498877
+div:2891132,3009602159727,0,0:0,2891132
+div:-16580361,45469862,0,0:0,-16580361
+div:28811879,1156690,0,0:24,1051319
+div:681237481,42215447,0,0:16,5790329
+div:952119,65641,0,0:14,33145
+div:669100,32909317724335,0,0:0,669100
+div:406354444955,4811582310504,0,0:0,406354444955
+div:-87218901,5441737753,0,0:0,-87218901
+div:262693,279060,0,0:0,262693
+div:950539838,98609240829243,0,0:0,950539838
+div:8061719120,-28536222,0,0:-282,14504516
+div:-3551675,11288323481405,0,0:0,-3551675
+div:3133528691,1315285,0,0:2382,519821
+div:1588170591121,9023526203,0,0:176,29979393
+div:1709754910,343280090631449,0,0:0,1709754910
+div:7694474,6311674489334,0,0:0,7694474
+div:285071476,771758808364462,0,0:0,285071476
+div:4473571504184,5027949393624517,0,0:0,4473571504184
+div:-314089112111113,-8919383,0,0:35214219,-5804236
+div:919605168565,-857045700084,0,0:-1,62559468481
+div:-2803909388714,46792205688194,0,0:0,-2803909388714
+div:-864476,-809506582,0,0:0,-864476
+div:157477674603,932718332239208,0,0:0,157477674603
+div:-86938062350,44113358240184,0,0:0,-86938062350
+div:7332602805,35651734,0,0:205,23997335
+div:40677613,7075290162,0,0:0,40677613
+div:6636106412767060,-388484912,0,0:-17082018,153254644
+div:9453362014,-644867786149524,0,0:0,9453362014
+div:123163311944389,-16324640549,0,0:-7544,10223642733
+div:55603403145688,-51970044763,0,0:-1069,47425294041
+div:-54436610546,-9254821627256041,0,0:0,-54436610546
+div:34541223746384523,408005019,0,0:84658820,283766943
+div:23462284247353823,2661546,0,0:8815284142,350291
+div:417392442148,-30391736401961,0,0:0,417392442148
+div:-718458863,-73668989433156990,0,0:0,-718458863
+div:-1215789,25426915315,0,0:0,-1215789
+div:8299846204145462,9023932210324,0,0:919,6852502857706
+div:-82615892630,85109462984880568,0,0:0,-82615892630
+div:828237348372,822759132224411,0,0:0,828237348372
+div:35003657,287087229590,0,0:0,35003657
+div:741329413510689,600380266222,0,0:1234,460164992741
+div:-454795890210,66807881739,0,0:-6,-53948599776
+div:442416942,-79456387864,0,0:0,442416942
+div:7067406608460,507872176187,0,0:13,465068318029
+div:57674203676661022,3315186364,0,0:17396971,2642557578
+div:-6677233842,881012314,0,0:-7,-510147644
+div:828372724537918527,2204184463,0,0:375818239,1222097870
+div:-97445457356928,3845827480,0,0:-25337,-3726496168
+div:644919922824665106,692147849,0,0:931766130,174110736
+div:-54599915326211,-274657876971100,0,0:0,-54599915326211
+div:3507375128,373578606,0,0:9,145167674
+div:19124901,485472050038835999,0,0:0,19124901
+div:687842561,49925524361,0,0:0,687842561
+div:193224391137090311,84335903540,0,0:2291128,41131297191
+div:388138056,66626944,0,0:5,55003336
+div:-656523898932,84101085851135799,0,0:0,-656523898932
+div:86004003567986,713613577148692,0,0:0,86004003567986
+div:7992270343,87209595913499289,0,0:0,7992270343
+div:-890969253231921,739402045213270955,0,0:0,-890969253231921
+div:6345032414365,821532347711110478,0,0:0,6345032414365
+div:-2988926571013999683,766437559276027,0,0:-3899,-586527396770410
+div:-15464671727416040,-618822744180643,0,0:24,-612925867080608
+div:21542413221581,-12802725163,0,0:-1682,8229497415
+div:210193665939,3964410139763307,0,0:0,210193665939
+div:-79273337971,91480286016,0,0:0,-79273337971
+div:6859567649816379680,-89417752889,0,0:-76713710,85837971490
+div:4244108501419250,1773608945178,0,0:2392,1635904553474
+div:-7873175456557395,-541450782136,0,0:14540,-481084299955
+div:2898452653,8240117642629,0,0:0,2898452653
+div:-8585567176400751363,39056741283,0,0:-219822926,-27546697305
+div:4923064817310182,488568656518880796,0,0:0,4923064817310182
+div:3946741285053501,-8922249541453011,0,0:0,3946741285053501
+div:438914382542168,98348892578,0,0:4462,81623859132
+div:257824992496,-9594074905801512,0,0:0,257824992496
+div:6213773111135227489,251684062660,0,0:24688782,155248147369
+div:-851789213354610296,395154960128827139,0,0:-2,-61479293096956018
+div:234318209260,31123149038286353070,0,0:0,234318209260
+div:243854015291982,52511298700,0,0:4643,44055427882
+div:7259566528156,6524807428024562671,0,0:0,7259566528156
+div:-887488809078966765,845297673812,0,0:-1049912,-637771662221
+div:65325904373281379,40360887138954,0,0:1618,21988982453807
+div:85548880813,9240276773237567918,0,0:0,85548880813
+div:7737568342993115224,69017119565996596,0,0:112,7650951601496472
+div:-2862875578129,87167074447,0,0:-32,-73529195825
+div:48121669474016423,812759701293155,0,0:59,168847097720278
+div:-82120995972822350,9793327794599,0,0:-8385,-3942415109735
+div:96669753363716639,3137138094734,0,0:30814,1980112583163
+div:44554701178616,52377515472783262637,0,0:0,44554701178616
+div:12903174971697,577437337389558244582,0,0:0,12903174971697
+div:38970472084822,-12255774904287,0,0:-3,2203147371961
+div:-371784186042,36063372236212364134,0,0:0,-371784186042
+div:-5423240799687893943,69598779690234855,0,0:-77,-64134763539810108
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+div:1690204685535914155988172971158586449220783297513957809,9929613337808118536737016690304412293482744573038,0,0:170218,5762400891834901871464168349997448737481780575525
+
+# Division by powers of 2
+
+divp2:6650822,18,0,0:25,97222
+divp2:-22,3,0,0:-2,-6
+divp2:626665862,11,0,0:305989,390
+divp2:382,7,0,0:2,126
+divp2:-8554783,11,0,0:-4177,-287
+divp2:80613086,5,0,0:2519158,30
+divp2:2476669303,30,0,0:2,329185655
+divp2:234993203,12,0,0:57371,1587
+divp2:78175,1,0,0:39087,1
+divp2:1092697,3,0,0:136587,1
+divp2:-51601200,8,0,0:-201567,-48
+divp2:-133,8,0,0:0,-133
+divp2:67494772097,28,0,0:251,117472641
+divp2:81070756,20,0,0:77,330404
+divp2:597,10,0,0:0,597
+divp2:-6061,14,0,0:0,-6061
+divp2:914736,17,0,0:6,128304
+divp2:5146181168,8,0,0:20102270,48
+divp2:908169,16,0,0:13,56201
+divp2:99131,5,0,0:3097,27
+divp2:828749,17,0,0:6,42317
+divp2:57934632020,24,0,0:3453,2905172
+divp2:374659,20,0,0:0,374659
+divp2:881062175551,18,0,0:3360985,123711
+divp2:8978255,9,0,0:17535,335
+divp2:-306454,22,0,0:0,-306454
+divp2:455535314,22,0,0:108,2550482
+divp2:-7013481097,8,0,0:-27396410,-137
+divp2:1902116776574,7,0,0:14860287316,126
+divp2:57980384,6,0,0:905943,32
+divp2:-6429952,10,0,0:-6279,-256
+divp2:-121080553817,25,0,0:-3608,-16163161
+divp2:9486785,18,0,0:36,49601
+divp2:-264976349260,22,0,0:-63175,-1194060
+divp2:-9957437441589,16,0,0:-151938437,-34357
+divp2:42280023,10,0,0:41289,87
+divp2:9445756856,10,0,0:9224371,952
+divp2:542833427595245,43,0,0:61,6271753240557
+divp2:-1989040985179,37,0,0:-14,-64895636571
+divp2:-842046065662,21,0,0:-401518,-1788926
+divp2:-36486783654,37,0,0:0,-36486783654
+divp2:12161238,12,0,0:2969,214
+divp2:522437544246,12,0,0:127548228,2358
+divp2:5771713949080,24,0,0:344020,16100760
+divp2:-192975675937641,21,0,0:-92017972,-1921897
+divp2:51399227509,40,0,0:0,51399227509
+divp2:76781489061746141,15,0,0:2343185090995,21981
+divp2:624421287579358777,15,0,0:19055825426616,5689
+divp2:585792449987,17,0,0:4469241,93635
+divp2:64111034807,16,0,0:978256,49591
+divp2:-4872049514614819145,60,0,0:-4,-260363496187431241
+divp2:-513726738058624,27,0,0:-3827562,-62639488
+divp2:-5129414645415250,35,0,0:-149285,-21103148370
+divp2:9777073958585027,20,0,0:9324144323,950979
+divp2:4917705257,31,0,0:2,622737961
+divp2:9309157350108809,25,0,0:277434508,16969353
+divp2:549889891480003,40,0,0:500,134077592003
+divp2:43259632605223,33,0,0:5036,721999911
+divp2:80760535636025,12,0,0:19716927645,2105
+divp2:95148114869248768947,22,0,0:22685078351318,2276275
+divp2:72784357307196492,23,0,0:8676571525,9292
+divp2:8695487295473310914,48,0,0:30892,162314927725762
+divp2:-999894977194,21,0,0:-476787,-166570
+divp2:422873004052896,16,0,0:6452529969,4512
+divp2:5850059500167,33,0,0:681,314043015
+divp2:279461298693250805,55,0,0:7,27259719560503029
+divp2:-2391765116812271696,39,0,0:-4350595,-221690208336
+divp2:93848076004932,44,0,0:5,5887145782852
+divp2:54463987988137594412,48,0,0:193494,268844485922348
+divp2:9735669349328114,31,0,0:4533524,691512562
+divp2:-96573804848256945392306,69,0,0:-163,-355587759787924163250
+divp2:693910684713664114,18,0,0:2647059191565,48754
+divp2:56085146568929935461,53,0,0:6226,6324008912519269
+divp2:7090107398039299,30,0,0:6603177,81864451
+divp2:767239571830013898753,66,0,0:10,29369808881631834113
+divp2:-74636360117262549362461,43,0,0:-8485171760,-6890174916381
+divp2:5461169750136459,50,0,0:4,957570122765963
+divp2:976702847245116276,37,0,0:7106448,71221928820
+divp2:3617864287401198,48,0,0:12,240164566873326
+divp2:-1309978694518086461,40,0,0:-1191418,-749976460093
+divp2:-2386749120888691269,33,0,0:-277854167,-244046405
+divp2:7266512965529689,33,0,0:845933,3826315353
+divp2:-6961513563355648932180971,48,0,0:-24732264461,-193888863384555
+divp2:5143351773441402886258,50,0,0:4568214,56404232132722
+divp2:900799510679250396,62,0,0:0,900799510679250396
+divp2:30978095034050773575947556,48,0,0:110056301970,184099523155236
+divp2:398942302365209775391228,57,0,0:2768218,44660240185115132
+divp2:914903526125867075664,27,0,0:6816562459810,77563984
+divp2:2551094792632393428431439,87,0,0:0,2551094792632393428431439
+divp2:6909886286515846175008,60,0,0:5993,427709407012247840
+divp2:461670131377821667,44,0,0:26242,15985200256995
+divp2:7884527998899506514865191,44,0,0:448183527561,3748608715815
+divp2:-17784618887403951072,44,0,0:-1010938,-9512034128864
+divp2:-2928855265612995457576576,88,0,0:0,-2928855265612995457576576
+divp2:8468445340017772238,58,0,0:29,109764431618131662
+divp2:97699128177500917404004263,94,0,0:0,97699128177500917404004263
+divp2:-901210903909447624384117,51,0,0:-400218038,-507572120280693
+divp2:-9099254471589743272646814275,80,0,0:-7526,-878753170044103808133699
+divp2:88420842262358445100031098,32,0,0:20587081616362194,3379223674
+divp2:5920604485769478876616913,30,0,0:5513992612966782,320526545
+
+# Division by small integers
+
+divv:-5721818303,-31144,0:183721,-11479
+divv:100359289911028,16,0:6272455619439,4
+divv:-80498620,-41490,0:1940,-8020
+divv:37386622,55859,0:669,16951
+divv:501578,15096,0:33,3410
+divv:6823,11067,0:0,6823
+divv:-56695604,31395,0:-1805,-27629
+divv:396,1627,0:0,396
+divv:82237128,16423,0:5007,7167
+divv:274600467,39390,0:6971,12777
+divv:258805,22852,0:11,7433
+divv:5139922232,23459,0:219102,8414
+divv:8755497783395,46470,0:188411830,43295
+divv:56370361453114,52292,0:1077992072,24090
+divv:81119249096,17631,0:4600944,5432
+divv:3925148017708459,17729,0:221397034108,7727
+divv:-289130964,59901,0:-4826,-48738
+divv:-2394107507926474,13709,0:-174637647379,-7763
+divv:-50229865870,26540,0:-1892609,-23010
+divv:3694598237147,64566,0:57222040,2507
+divv:-158417511658676756,-64615,0:2451714178730,-37806
+divv:4290308067505999,15193,0:282387156421,1746
+divv:560784439224,18032,0:31099403,4328
+divv:9207709447044,43308,0:212609897,27768
+divv:42833310641849859264,57815,0:740868470844069,10029
+divv:-95089845960500,-61243,0:1552664728,-23596
+divv:80687877739837,-30796,0:-2620076559,28873
+divv:87851714436662969875,61349,0:1431999126907740,28615
+divv:-5469457537,11848,0:-461635,-6057
+divv:-87902672172139257996,27307,0:-3219052703414481,-25329
+divv:-694576880064214,41179,0:-16867259527,-1881
+divv:47595892116434828,47204,0:1008302095509,27992
+divv:-505437926464614447947,63863,0:-7914409383596361,-45404
+divv:-747214784268033231559289,38881,0:-19217992959749832348,-36701
+divv:40648086806090824914,-26501,0:-1533832187694457,19957
+divv:23860923193973296,51883,0:459898679605,27081
+divv:-54927711987466879,-11937,0:4601467034218,-6613
+divv:4311756940677447464744993405,57924,0:74438176587898754656877,50057
+divv:6015889311543917179186719299,-50635,0:-118808913035329656940588,45919
+divv:3915173119649887691276,20797,0:188256629304702009,10103
+divv:209840208083311316090,36847,0:5694906181868573,6759
+divv:80000000000000000000,25,0:3200000000000000000,0
+divv:65000,25,0:2600,0
+divv:800000000000000,25,0:32000000000000,0
+
+# More regular division tests, all with remainder zero
+
+div:2546905999569583642767,14957376976449,0,0:170277583,0
+div:8820890915810513585008,1290654714992,0,0:6834431249,0
+div:32722399167479650872,485171756,0,0:67444979562,0
+div:281565136283209685892638788,662476076955569,0,0:425019326852,0
+div:378946105625441540737303,730177609549,0,0:518977986547,0
+div:4582336996784637790714331883462,524659399010705751,0,0:8733927201962,0
+div:30908196512568974558050436710,558351619821782,0,0:55356150882905,0
+div:22323179841972640636983704753,52956211127329649,0,0:421540351297,0
+div:52861155259463695471936251160,9468000366853,0,0:5583138277489720,0
+div:63260535208765360919938678377,42122224781389,0,0:1501832715082893,0
+div:28984143265639802151626094051438,400082299653446,0,0:72445452574997853,0
+div:3317964166851176108015612343691026488,4043377664512227983,0,0:820592198441457736,0
+div:66344183633702866132053676311141052606577,91178778320288608847,0,0:727627468320008374591,0
+div:5695550794829300030546270882518324992076491,700880200540761512031,0,0:8126282908883599504661,0
+div:39037424763225371397599506194643545880,7823304879274443640,0,0:4989889230399751117,0
+div:436302766535175847931361924986715811355695802824125,539874776595118725233603625,0,0:808155493551392334022517,0
+div:346745970223883677972239586841121624456285168,91526846077189641641348,0,0:3788461911289433710716,0
+div:52724986995800116311347732059133990047851510404883234,568154076534742483223872266,0,0:92800508125151146767566749,0
+div:2377848542111935708145523356022435793438514852230023651,78161062689778162572883876143153,0,0:30422418276855244470867,0
+div:232771550187355002750375300477927538332648185231317422242,385169380058229884085331084789,0,0:604335552717520311627548378,0
+
+# Bug fix regression tests
+div:#xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF,#xFFFFFFFFFFFFFFFFFFFFFFFC7,=1,=2:#x100000000000000000000000390000000000000000000000CB1000000000000000000002D369,#xA11260
+
+div:#xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF,#xFFFFFFFFFFFFFFFFFFFFFFFC7,0,0:#x100000000000000000000000390000000000000000000000CB1000000000000000000002D369,#xA11260
+
+div:#xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001,65532,=1,=2:#x100040010004001000400100040010004001000400100040010004001000400100040010004001000400100040010004000fe03f80fe03f80fe03f80fe03f80fe03f80fe03f80fe03f80fe03f80fe03f80fe03f80fe03f80fe03f80fe03f80fe03f80,#xfe01
+
+div:#x0144E8D32EFB430AB38966D6CF217276A50D882C9BB1956E47F0C11F809819482F6671C7CDD70EBFD40A67E96E8728B1FB26092168F13A2689FEB0B86D689F12949DDF22AF0A030FDF2A27BB4CE09998ADFDD014B507C487D2BB6C71122AF93F13E99F22A7C5F2E728A50C7761551C4E9322502DB31D03BDED78A9301C50D57AEA88C1FEE4EAC7F0B38002C0F86D434B7D8311D4C410A0CECD6B81D40E8289F31045CC6F6C4EC203BF2A6BCF549759C92875BE633E20801C130874CAAEB8FD7276BE101898F12FE7648FB2C76C09D3643386B7F99B26FFA462DF54DBF401C8B73B5101B0A45D83774DC307ABB9F52A54E152E6FE5EBA1F5C91E0CFC2B84791413448023D6BAA4BB3B765DAC3C7D679832378C6737EAAFDB1395DDA29887653C17A4F16330550D9F71C99BD7D81777943CA1437F0C6C9AE89472A5CDC304DA0952AB154FF9F0B73A68FB5B7E661389AF3E407B4456A9FB5EC7FD7F78D0FA97C09DF4BEE8048687BFFA8B6231D9893ACCB75904A16BEB45C9B58DD4C63C9902D0C5015D2325EF7294EFD442CF7C6F837A8D6661DA492459429F9D7CE1251CA3375CFDD74A1C8EF5FFB8370245295E3C09B47D697EA3B5924F7181B1F9810FE479D2B17E63F1886053C32E53B4D6990DFE036BB50CCE32E76EB5BE5F2E0B40EFAB32D19968C0275277A4FFBABD2C5A24C982218DA3B32703381399A653A3CB9FA3009695581F18A7023727D6BE13E5D9829E9788BBC94DCFC493307101A0D7A8A212AA5FFAA45CAC31212202C23DC6ABB39D6192FEFF6FC3C084547036E2DBD6AE3DBB16D0101E10E899B32C70CEF1056DDFAD104E165F7D147C8143C6E6D08C0315D01D41D5FCDF528A8E2AF9B68526949742C691DD8EA545D9F450B54F447E52ACF59172A9BA2284703A6A525BA28664EAADA9CC9A1AA7875CE0273AA591F2AA8C20925EAA62037D109CA0EEA3824C1ABC869978B5B62F853980C,527,0,0:#x9DD4AE7AE3F3E047AB2F86F36EF8F04867BA224C9A45AC9596780B6AAC0290D95539B3274501D2411CDB86482673F35C5E41A4638DF46A28CD54634DEBE64A1E6ACE84760F8CE729DB28303CA956C1AA8026E69908252D172FCFA340B05236EDFBEE17A29F4EA5B964178B1D4EBF8AB92495B4130001D154C0AF742703CE32F707A796896EA61A92758FAAF9E4705B13ADF05F952998701DD5FC07CCC198FE167A50D95812D3CD5B67AC5AFFCD2A28654365B6C44FE8BC83A8C8856E9ED769A317D54C8E4D53FDC310DAF9677CC203032B951E697C5B26903614D1D448C2A7B3E357C60508FE4886ED8E2B9614F7E72AE7AA0410DE8A9EA328B6885D67423030B6A7388DAE88D78B1F8119A3A00C649F13525064648CB92FC88CF3A1038F2EDDA916ED8D06BA84AB4840D869A372C51EA44BC04179C5F7E08164001776D50CB6038FC968F29E30D97BBBF19C865DBAF7C1AD8D133F7EBFA2A80F5394F77DD530B88F0F323B7FC1A7EF8CEDD48FA12CE9EDAE6F1DF95FFDBD62452B214E4B51467A53A5D3C73E5408632E9E4097AA56DFA5B40BEF9DAB8E518867FD3D972BF155EA5C9A5EB17CD58074A742D5AF3DFCBC664CB5CA3FB4B0613485DFF8B5D03507D1513E399332A257DF90E7E9719BDF125430F7339462926404D531677F42E3E937A9CC83613A86394FFDF89A4BECDBA1CEDD9F6D82C4FB406A16F10E70AFF07927FECA0D60E176BEA8C4312E535AEB11B7665E858517F109BA90CEF68464219955024423D88981BC0DAF6E33B1A9283D412F16D06335EF9D86142025C37B54F0CAC5B5AE55097D9F6DACF547ECBBED87ED107543819C8DBD0099B65EBDD07D04DC74A3E2C2496D9E4E0E61D6615A7E08D3FE996DBE187750C4C1FB9B18B928B6550DAEAFAB8D7E4E8662AEBE7F77299D516BBCC8F2242B9FD28EA7460282819525F63E7E5C66E4D0C73181CE8577D3342D76CABD26D8CFB6,98
+
+div:#xC085B984D2F30F92A00B8814DCF9A909C8672B081225AA88A7293D276E46B240B96E345B76A5E2D4E5CE7F4C208616DD38CA2029C37FAA8EF2561C688DA55150788FF7F34E484596590DDE971387608FB88D92A8868F4A2F165D0222A272C5AC300AFEBB152B25D3FAB5999458B1AC27FAE5E2ED6D50E7F8FDE7E22178E4C954,#xD607ECBE4F4608C16A0FBDF0F4DC57278E937A73D2226CE8388C6B9B9DF9C58464CF3E05F29FC58B59CD34CBA4F47810D92B7B4FCAAD1860C76E091D0B1491B9,=1,=2:#xE64618B3D9A1E9B5DC38D204743994492C406B98FFFFFF95EF529A267E06AF4364F984172C9569DC33FCA47F08195C2937CD1CAAE1F954A2A78AA08AF59EC954,#x5BAA50658645474B56E43FF89470E38A4D69E90D43F0E1D5DBCAFDE8853AEDBAC364F06CE50511DEA15DE5D944D9ABBEA76EF56D1E1E2732AC396304B28CB7A0
+div:#xFFFFFFFF400000002,#x7FFFFFFFE,0,0:#x1FFFFFFFF,0
+
+# Spin in s_udiv(), rep. by Andres Navarro.
+div:45035996273705000,45035996273704960,0,0:1,40
+
+# ISL test failures
+div:-110768999339257979395151936604514986,384307167128540502,0,0:-288230376151711743,0
+div:-1141798147252782335678234520236874583761538254,990352029122461210705605426,0,0:-1152921500311879679,0
+div:-11417981504429295188771564013355757832033533955,2475880072806153026764013565,0,0:-4611686014132420607,0
+div:-19746054620424668475119147467564405598366894325375493578959094720,137015778031883893401769208133025156060969696320,0,0:-144115188075855871,0
+div:81791893044,81791893043,0,0:1,1
+div:66461399479760783996783510248698675,57646074961906893,0,0:1152921504606846975,0
+div:4951760153682756594365890554,8589934586,0,0:576460752303423489,0
+div:9903520270472025075672416238,25769803758,0,0:384307166770626561,0
+div:11141460315522012760862883825,48318382095,0,0:230584300062375935,0
diff --git a/polly/lib/External/isl/imath/tests/egcd.t b/polly/lib/External/isl/imath/tests/egcd.t
new file mode 100644
index 00000000000..8d9b695c54e
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/egcd.t
@@ -0,0 +1,752 @@
+# Extended greatest common divisor tests, including
+# constants satisfying Bezout's identity
+
+egcd:7,328492456,0,0,0:1,-140782481,3
+egcd:930165221,819,0,=1,0:1,-85,96537294
+egcd:166,0,0,0,=1:166,1,0
+egcd:44,37824113,0,0,0:1,12894584,-15
+egcd:0,93,0,0,0:93,0,1
+egcd:70522336,34102,0,0,0:2,-2785,5759331
+egcd:863,25,0,0,0:1,2,-69
+egcd:66399,5533253,0,0,0:1,1229584,-14755
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+egcd:7199869643335698558244236956584912995424331052926,1527691037819148269409741859243943945701460882281456,0,0,0:2,57636477641770614383597878932375099638471007783959,-271635504463112985014746084869260309486608865147
+egcd:455180951765525170462545338088508018606122351276980553,556221150703226579334767784298557173503136268977244374088,0,0,0:9,-26099094677797268547154896038734560717230667555583420223,21358070869183507434245801704745654317447493932633856
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+egcd:3810898561978236785388520202269462480808294248695893,1062108199707865331145512609955734128758543891943,0,0,0:1,339617394241108857543260840698815161575640725383,-1218564586632720303123702323672631316897985132530526
diff --git a/polly/lib/External/isl/imath/tests/emod.t b/polly/lib/External/isl/imath/tests/emod.t
new file mode 100644
index 00000000000..cb2340832d6
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/emod.t
@@ -0,0 +1,756 @@
+# Modular exponentiation tests
+
+emod:26,73,10858,0:4748
+emod:-79268,611,80,0:48
+emod:46600826,0,8487413,0:1
+emod:1465748,364434193,5,0:3
+emod:873883,48223971,348,0:127
+emod:-52,183,125544,0:49784
+emod:-846,394,969114928,0:877192256
+emod:7054382,917,9,0:5
+emod:3768,797,42,0:18
+emod:357352724,2093262,84225,0:72601
+emod:109912993,364,8323,0:5279
+emod:-3224,270672,198893010,0:47016406
+emod:55,95723840,523703559,0:372050017
+emod:791480518,904571,581451304,0:444873224
+emod:7278,922,2398,0:2040
+emod:-5,925972,33496,0:15809
+emod:0,649246,262,0:0
+emod:52004,997,2,0:0
+emod:-4115715,8,234398,0:3299
+emod:-13612,2576,57,0:7
+emod:9843599271,9688373,8509,0:5780
+emod:3415850521,762,3050240,0:760241
+emod:38873,18109,6791035,0:4721568
+emod:504242,66851907,859955181,0:352955933
+emod:67731070,2628566175,7885095,0:6298660
+emod:575051,8,72796410,0:35016721
+emod:3,0,6595628337,0:1
+emod:80479332,13,9,0:0
+emod:35578433693,9608165878,32791508266,0:32575013339
+emod:9998073,96042198253,2871019117,0:94477702
+emod:8502,88462049796,288668,0:234260
+emod:966350153,6125446381,940,0:33
+emod:51249,46773,83983261446,0:2570243355
+emod:2571204,205,4404831987,0:2226688593
+emod:19,439650,888059767,0:247638574
+emod:9485028101,55407,3066739,0:2994746
+emod:868043,56705619,183207,0:162593
+emod:7072012,750002,2399910888,0:1322891440
+emod:55779049,27843295,96,0:25
+emod:57835434,31279309,90443,0:13168
+emod:-6511,7486344275,346901687,0:325667244
+emod:635099,3669199,81611251,0:59252827
+emod:8010935551,34,9927,0:1405
+emod:24885132,4526383,58404,0:15444
+emod:613003,6878758,49607641,0:43381288
+emod:132639004426,81043,587317316867,0:532884551555
+emod:4439384,43262747,10119,0:7922
+emod:689907710540,460484866,82087637,0:30398950
+emod:84094,867130,62309,0:9025
+emod:-33557432,60878,580294942,0:194480658
+emod:3869257,4125450573,456883055,0:129529247
+emod:9878539157,15920872437,4738245,0:2745542
+emod:8582013178,46758,28059,0:26806
+emod:48259672,37258629489,7377276929,0:1692584431
+emod:920350,289730727,1122526,0:471310
+emod:13029603,4503668042,8665149,0:1456593
+emod:6938077836,651113,5180493,0:2235984
+emod:-352466,9169937470,70675259176,0:18255076720
+emod:612474,308813590252,510155,0:383521
+emod:180356946205,710633896291,575721,0:242878
+emod:83046,3518,9768613,0:8644737
+emod:8670878985584,7549509958953,968064799634,0:673069231488
+emod:-37831593555,141781,3434657,0:2539127
+emod:94250019226,452692809515,99738576,0:82526128
+emod:-1746555354919,310727,6925934085655,0:2276231760446
+emod:777851791,126226427233,84192406004,0:11526643943
+emod:3511,42864,89474,0:53201
+emod:611596753206,17005495050,3708625387380,0:2318918831736
+emod:482182433948,97822330605,90338471939,0:27830209373
+emod:8971927,5117957930887,1928567236065,0:1437815380378
+emod:67445777442,3930771,4541604,0:773076
+emod:-52133,16783,3337,0:541
+emod:22,7787554274,823549,0:503210
+emod:46816685133,272923,22855404855,0:4704305202
+emod:181225207536,22085689725,67955765111,0:5581219593
+emod:207971493064,41707850,9115746425,0:6454330776
+emod:42840650,206759347,852090956,0:15995168
+emod:27703329637330,20686092,8407250,0:6767250
+emod:-4953,669615,741766623319,0:660186892888
+emod:11957529241,380932179443,973373783,0:970856001
+emod:78726001052485,62593863,24053,0:22766
+emod:340516095,798875,2556443616,0:348235359
+emod:954899979887,472716,979176,0:21817
+emod:1669727071,1008664017563,37837,0:17717
+emod:592681,30974401363809,65653048867352,0:2039369146553
+emod:-47884,525888510,60238953788774,0:8466915489354
+emod:5643430747,31164,8629966388917,0:7441630635230
+emod:-9134817647675,7013427758650,14493293680,0:10069704265
+emod:16875464,4824632,157384425878173,0:110131510850102
+emod:-1891400577,334033142140099,1461471589,0:500051598
+emod:337804687158388,2747595,7100373337,0:3797937129
+emod:597469670,80669751974,564129148100,0:462736594600
+emod:-373150,494511905,69961532,0:44955656
+emod:541002,396515,663670,0:448308
+emod:7443329,192200716115,819064750,0:75062949
+emod:352331718576096,937115695,121165545639266,0:64461611886972
+emod:94655436207250,2442730,199021327246786,0:98351560387664
+emod:450100,441162616,8233398,0:5554612
+emod:-562161185511147,564556372236072,6280258,0:3623203
+emod:-87181577138119,6636401624,33493475420,0:22078084821
+emod:63757825,675193212577,791180,0:551825
+emod:75719430,7834623506500,81861106,0:11697720
+emod:7636428268,342917,424075566,0:382499056
+emod:2847036414481441,3486471,96180960030,0:59848561291
+emod:7605693821889,6857556039961694,9340113967299,0:2791642232520
+emod:8460275679756,1249967640481565,3530286614,0:1855000398
+emod:-21034390,9455714874055,499517062,0:436407454
+emod:-67878984199,6771837956,26518969637,0:25569641018
+emod:24845854,38512788750888,97880493,0:44709115
+emod:913222119920236,89756684,24503538,0:2358514
+emod:282825837874763,49707087,4419250635,0:2888992652
+emod:63007338,778023537730086,51807416,0:42047040
+emod:86515727,37854772745819,932061443536802,0:913958455659941
+emod:16752138958,84132272429,883645897,0:66146743
+emod:76020717,31715587905196,205759754,0:84248649
+emod:408229656454238,503345091356,5122409740,0:1428721156
+emod:335638746639,5761369582225304,4092173299,0:3495108299
+emod:4118216,163393347759,33772615,0:18314396
+emod:6140126413594,8416471601616,48576280861410361,0:26480949886245014
+emod:-654797902884,3118772351656763,49622473264915,0:8452790798791
+emod:724850698,6033802538378749,5080656658,0:3131590110
+emod:-67747240120320,5206803331886510,16513427532371,0:9305404239198
+emod:571096000432,140739165,2954342334466067,0:1353110426217941
+emod:-99625731171,97620037702645811,44537854660922914,0:235109017589213
+emod:33191179298252,90227008,7699370104843722,0:3200733061202818
+emod:63893120170170773,976336027,635053357,0:242474238
+emod:42622029452,37698167195197,75773078526,0:28876796648
+emod:83620114446490,8610382003,464773010095268,0:378308871158308
+emod:9574472571,9560271542,228122075654,0:119978765197
+emod:56753833744514594,333744771296,46684214084,0:22141695456
+emod:-72021302,2716854503311092,329338426,0:308255178
+emod:-12837777900800,17117883244,11645469,0:4924588
+emod:517104019809,96545405933271150,48276368795,0:10960026506
+emod:749819481,17040593450696047,13586015736,0:2311264233
+emod:-3050009718172511,728537309,5801457671812805,0:5155502323398924
+emod:583365344369904345,274298196969,206968876678,0:178154366245
+emod:38505842262,85917484751521,136556283742,0:107808326812
+emod:15796803,666915498303283,9286239329074,0:2947240710171
+emod:1386598857453,2520081676103,9050689134,0:1960415451
+emod:837113484763,3657616517904,83176471315,0:52463786271
+emod:4678899921,2786079987037,15314196040,0:13824191401
+emod:326751076,992797863718,911067433083695337,0:303879844417619866
+emod:756697895735,14491908578947766,2765309614060,0:864557526445
+emod:11248294,4566321352111312,1353192745166,0:172343816040
+emod:7099878002228869,59309793877,927000340655945159,0:5356010609771131
+emod:-52873339014,197683370410545570,9868414456,0:6992876320
+emod:413327450631,3360204571,9303944255,0:1706685666
+emod:-50582368810,137417275741,67023586922155120,0:46974101556682880
+emod:557831168094776659,36136648411374,6960687433,0:6184527385
+emod:7693794217685,34977654183341149,7387705549462024391,0:5937660734860285181
+emod:1230160392609995,985408024602754529,44727668402979310,0:16636048945342015
+emod:20103166039699,5221451032884,543018728744286356,0:478895368134894537
+emod:87410236728907840,255693035925652,37882890299906,0:34530311513908
+emod:-302501648157631,240206529977,230243222283595,0:46069838353259
+emod:71032979947810,4416429704,2419723038477582,0:15218189462248
+emod:355188737187056,6605255426444793,75874689681,0:60034193666
+emod:6806384607660,60639456674,98209051459683,0:1432252996158
+emod:65303175086889,48887808654898,73427910350,0:4396384481
+emod:7752516269162058,62869883943828471,54774176016,0:2099905536
+emod:4326860229332706,2310571812316625763,54001071080,0:33511865496
+emod:6209542490007593568,8412972472856318,788629588235,0:362055759304
+emod:49913992353929542,2486110799,4585324963758433261,0:995513978680870140
+emod:349243489434121,6540085499657705,1389913007264228051,0:177284383643046305
+emod:375679038462457,2350871880932563,6382236922719413,0:6083733543560031
+emod:-30721033347396,77626742323874,523758275336801,0:84240020331255
+emod:5192099267362805,550370357942059,3061410194809,0:1707076237630
+emod:3546331202177649495,63420031374822,91402241031313,0:53901671764582
+emod:-79044395581,9116190241482576941,239200013057,0:188422584849
+emod:286354276465,8227634038665534,8044901060955,0:5958399217675
+emod:826012386295,49355018525,20566214354390,0:4083971754695
+emod:557522956445489849,32695219802913423753,6644227791725857,0:4886117301253388
+emod:-970081051547602,98172198401841564685,928763975802,0:22343542292
+emod:321415667027595,92728916953004350117,4993968573660723,0:2876297201989143
+emod:-8699136317926938091,34913749137460,747114775995,0:558798686791
+emod:-750347276056433413,65602570699247,72710009984856228277,0:105504442183260111
+emod:796490992759,979812498143888903,84011139142607244,0:27172815871253503
+emod:11599197279102044,36330854506634,5869662463842692,0:4784840887360516
+emod:-34727277565261165051,53891223619,2799195351079917048,0:2607666110843551325
+emod:36836705996677,91726402457474120970,787136088375328120,0:479138071357571209
+emod:786215427840767085,6538388351143408779,402285004679,0:145021397316
+emod:970294384045960,339246418487,392505811982975275,0:228766017656314150
+emod:346820606266075,751258033318226,999763369665239019738,0:843347570419357235431
+emod:5301442588198654136,6353479688437885432,36561155260520,0:30332565944536
+emod:-983848133889575,94515815574826812,64461274919837567,0:37200063729864517
+emod:1440437495907,707949003595768210252,773151956245425964025,0:534921841457783741126
+emod:9193524500595,90938893533,22068045096454773348,0:2946366481749712299
+emod:380288714663108486927,73556495897685,78844051786179,0:11766474674606
+emod:-26813489896157,716398856020,4739728890232268011,0:1219466999743008960
+emod:70217805755997,315269791230838613,7083375858304586,0:3059227926161215
+emod:-504603375916094,8554599200483,203113429839514225,0:156384813943715991
+emod:165732904527293745,925640726772320963,700604179708906,0:676451920881317
+emod:8166221848180,76217001334707,46983566031377519480,0:36926290142288602000
+emod:-709635175190425441750,1295562689422083413,78349035380937817045,0:39271509694798926445
+emod:-2183184852451752183835,379414486190148900,724528035479902191477,0:637578758017403897095
+emod:4238159883750584463888,7386606561456,5728437931813448,0:4306066978435024
+emod:1805409007202,743465586820332156,626695586336006980,0:577226884938331876
+emod:941901705860483,972657766381340158572,16833834708698,0:382737644745
+emod:36719313097210,86800505504547056103,565303478138221,0:473055435424504
+emod:-2666022326987011,3591998328200,40859674965133754,0:23789375408884171
+emod:7645153844358510457,209771482022016006,905933746892551078,0:296096846282298881
+emod:7851918408989360228,84734870178640245420,17989167921034697586,0:903417754686768694
+emod:46165799974266,2935049136348,8423621229264979968786,0:6706766736891832422390
+emod:546378248247723,8703795888148,20029030818332,0:13614240516425
+emod:811349021801350852565,48747346545449,8830100158031,0:5382741806069
+emod:885481101754677356504,7096462381701,455223655560834,0:43346926638854
+emod:-80474402802255,3514824850066,78159132039306813,0:54916880583163335
+emod:7131297407676,3315297183633054184085,934198951388076635723,0:651097991912957628236
+emod:12957689764990,546447634520927,3127766567382808,0:233348459584880
+emod:52914542138895536,376010137099253465,490953375293336830,0:36400103844339446
+emod:45685344805562074769093,61394436169997,80543985329602650474818,0:71110063880403297977149
+emod:-3761985423518632726760,956482080293262442007,25753912455108628113,0:8836130744460244129
+emod:80642483462801,144058405752215292120,106396321855736640702,0:8066898094483615543
+emod:529953061572415,79301418791778407635,92480175275263158380,0:39970833689504413995
+emod:74684906365344856040383,442523186603390862631,9182842638237756,0:8987946408749887
+emod:-71808994356961094403637,86526866467357,8474332396093731,0:2594467781892122
+emod:407234425686004311062,856447201531274,99637771372094,0:11888452937044
+emod:93295648386572486115412,25979074094596934000,83211005582689,0:77383113960508
+emod:9427365949717166,750902522043226813,9221846367078539073,0:7648128015969044021
+emod:92605054801073,5909052810075662,88928490696637500993,0:37110323432146043137
+emod:-82692010434673,814335630290109048818,53615009335053,0:46299654891349
+emod:45016205353589874668,95117600743459,2986728093611093895058,0:319442824024520549342
+emod:-47452361244427905002,7926505809295224807,3915672967969228,0:229192748539424
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+emod:936948855951886126122485690093336688820905438,952037324756179560869502130218077067074192084922,33269225773280332072695417422958575761899978253499,0:31343729843026164021419525495602740164653732233547
+emod:1825784156631469036083553494921853600892959329084,1792152855908535702866815908479066064573203506328,7584434492468684030603996404862850988077230764966,0:5668645854080378677534840608739670840420553612804
+emod:4771542980687362499528587321796090255401127670935,49673918430106064176014347389103358990211599558,566808710453386184370243734899518529679284370157755,0:122614051205229170899163057352055351000892027836640
+emod:1823975371238131949336117473697450176001493124551422412,105862300137903514746181350899950234467623927399,872150475093096064129104506408316916638215646399,0:41655575950454700889468004661814966001328043635
+emod:-7403599133408228298723066626194976743023770563,8091134022905952963104699681073539225469153697858,5425355350845048150843160638835947264066541393,0:2741144821308534612335116015287775234294403710
+emod:890646208254935784836741265729061048252488750371261965,3014682556575993233966766524815533447095651265,546627202244553599303777629347742865125211029412308759,0:129189292373035871943643466963326800150883513587943803
+emod:-478301676028477872037215528133488673909770705969021,4776644958028138697511933491453433971900379293,9804006501217419003156374248597140990532373604359741200,0:4256526039075877642788870040517474793472829088151023139
+emod:-2685385133378038265400322588836701680595103090,6682051156600485359090971175835317671334686963274670,272381294217617274132435187599076952898337370816,0:160448248258559406045792992578310861095674214848
+emod:5610352369978196066557531911583288845257728872885,29105174820600183606830671874489670474983812495,10153364462051971143078318734983556641603678006,0:5654495476089063930689151404880397736789807021
+emod:226106902099977963681448571570521911815066120995317231,1707966440323110108128795246285372866871484571190343733,3022313205119289698061840897980713840817653246137,0:1577789001757918961027109729382551177312942381192
+emod:56994398320188483459448108143592444430607410910,776571171150667113541269029583357392475898590899457,2952252481199115860737553986433506978840458215941629739,0:2115048663787786340261444189400738968421587522302410834
+emod:3490308621678382172799094687550532898406999135779394,92096452137945096831873480789275081374251295752,41859985473397465016337344479699193310470254681186836,0:13883201384282345984019723588819996850955384028405568
+emod:78199345624452323665538899878571977010103711120353586,426889339090868016603770192089391858795483010444,614591398724869622540933622810612317245462099,0:522221021562810412369119229465715299116874680
+emod:24494234546606769948911248691757492347200292227140712,737664088655731527937609365429682453415687199299030,41867909567513789892657868211831771169136305959589936,0:29639257666757184257116019424419930205524205997421488
+emod:56340168172071966259164709446976804762407806722929085,9330607041404161548877129661058575432184418559,9073094112420733297526890865499708444478527878927152123,0:154033683820095649526112863414229406070106242714564017
+emod:-98557586380727162991959870302640551435620563142855,352179144624443366108174660913068150097236769961283,82957931261705630469780521065649609232909069967906297,0:4611469772581670990273871321012941407812443591119341
+emod:-327557046252127633720556940076118229316928253056031837,87561673782562800570041955183322599955629735731,999914991747113895498838412296608587151797268932915993,0:394844471425716300825167828282861382716619299699558452
+emod:62239294453310399816462667483598707872602240932142297,4658914995757017485299221864332093690860930515,2486140058721179216992139245736934931657664228105707838,0:1785631544920858868455392242463320012563257587191303743
+emod:98844248773908427486663349850376500066907267527022459,30192206183400841123051111248262548983869820657370418,68953646632746416359816661196045006942249287619,0:46348894487037788897865804485372390262476609587
+emod:157314584387460613339515315779646913728585159983563958,7129487243609251992087268468464230874147780513130629,7635331003686312032946464576006029771208342239,0:4241581628875360523816771058381935521234659115
+emod:-20413864406387948000986404651284225843901612271532247,60570204203293653144874911182227469571265725493933396798,74868812887175685382885501310045723065586089753,0:19048798896908808195904371996264452593501059126
+emod:-9994818638674529038541342163911785506433155975282,2441529131191726286407021656054505236538534218505,868763054397265950319354411340564521520500110285341,0:713646341945805538982296660502179127412919111592110
+emod:89928590797284478729252348706855088798121290869809060,59274533239795541803239071337695221071797917061551,458720998639333601494661224903885165927950415283068,0:238541586391208422370502281952541020478014766570928
+emod:95798368921402497294705740910984389802169456614551866,5780238728300624701546948669191486225967922222,17602545836464476089657116165016615105637757631075883410,0:78935491841772453025866169047063528694493328256027716
+emod:-6285988010245529074539471748460012192454923426879038527,4464035763112866760911458677964807167647554031374417,557426143112327094606990154681823173588434990737834515,0:492308894761273719919857937651384253941194670007301468
+emod:9216609859426876786581364775917107940830826147193,2584656223509851604195298909793029088361431700609487947,97854661860659618908160182622420008085466240143877617,0:85437882709473882823892529660893798801236147303713473
+emod:9369310120459129777778887291797647697234777683432051,29114272244461903760267994837317966967186291374,9778907141259037429006274419046033685045015909549812,0:6158258816758612931552439105354775382106793792840429
+emod:121095461268256661739120249224368430821121813080,24216928007688542369493467184086532629768906571974067315,1289101505296127733900239534120001726302383453437972,0:559264053276767206654909871928116713399654543670428
+emod:182013722639261555865084991953687573500429962535766,5312171499532129540711226268300910853763234503893756460,373925090068292154121977949608424249047611823254,0:314492433924568960493434943112384754084000516952
+emod:93122409584336684208634425627697110814180323391,14785829877381197130422634393843365096818009069934141,71932199753627287791733451362196318752438103297,0:8651254301747600278741940327234016885952587642
+emod:47314169772357749238033685118725095245857710174326752251,7737526111797627477363360414613284237413174897695810,22636439121232576533369780871442014748582558494258753099,0:13481881160769756096301137118624702877253559402875253449
+emod:40437135697420911841447145598134529938086874417869482,34142834203311856114775566332299272975314691621831451157,61483400000688184187598139984127464523117551228,0:24693895005298972635867796289890449288509340480
+emod:602105062592660782251188458378445123063805101057628877,8347043677742730027972079328186098141974290663352,26831519142227837662521069711039879610686983918942079815,0:19346421778661788162968173970766981230660937987122859431
+emod:-631401827295443763663013499197680737231639584989,4430590554183773954781048837898216988481838421600006676,2298393896534361874746279427962482679705626081354993,0:902945851002567106949268418213474969113107638762554
+emod:27786418359632188385534950865019584947593363685667,98296086104810263777729111156693306399745439517905843246,885588959591327389647962772187472088293854582596501585,0:219691806105674835153486211154061118027834335467768059
+emod:544755872510174513208896610649845089424204895203973,87613019793007744065404211674234696040702876770839324,4802706142382015625537025588688720114067633075179938214,0:4385825975408070392248170302791202812902625274088936755
+emod:266945202318091617395513307631280644325314217797596447,327566429748960361523726509719626329764280050772,11519336904029441280173947176416866297494364467208,0:9214541334269337101551342582484701318239260425185
+emod:462274575377872280232466472100106379919338255223,55208106570738920386188352962242729542845793547908872955,394947463398055276743090558014108833519958843326036,0:14090943624401742126636618168767136948067183397671
+emod:498588690178207860433541173652626793819053542771822736281,7320383073654496423015034860141911017275032239632,864143543362782361157239005190574446540608427530,0:405837700105174778010475141584205874050426805841
+emod:9889983273673267055836467894511050393862566401427167421,9249465646754817541871880579390030265752574382697001,25959701663622988366438447815418575609294742760915,0:14146692686195759173905915586727366427907029148496
+emod:8977490701791885156174343111351369788045969359197758,310958534312954656622864682281335777958703646747002872696,174885990030766375557335981487351371779234545791,0:172268700100468974308770735824988553302287984097
+emod:75891699691571906472460353214537778119193566652164258472,75642710398299172975011498854625838470353049691991252,955726338299637204220414390179262210175010011084260057,0:871551220400711242628454200406202564111664531240532043
+emod:817687870329204460156610156013195890028396416937162234,71199838789107588122529185516473983414598608665610,9797217400749418393402581106919553154845468700907872,0:9732835751266971971517518815617086242024182057719936
+emod:656535389260219421975691249279051158130957449883966,9965820125071850776397934471591602941345422709213,7172697024967323900460450488928712811727935935053426,0:952029660753929926293707648671191250800460252928490
+emod:66765940995445083043566397231259337861214788967954,698949221229345492777837382914330269944860925101870058401,370977162689246158196783733149906357225503977003,0:56780810013712823080333031377883978714594923759
+emod:-24860118467430048016063231699711679455683726879672563278,546329058333516554426292196577372643895447816095329334984,102083942512514956911062347152596366057306100868,0:89329427941719016468252169448693561927858667460
+emod:518281434431704435940150792966272899127164218411,19290830216216089485046976601623655817078004660590796816,216069676358948655463888215158143195786697578360538582,0:86506166688482117996142986160440612354876384284094157
+emod:517542901438648396791987658999334820221431246981278,4954735021213009892606598900462212813613793939697004110,5546708620630276172568015746256008412046274679011721163,0:114558840275328029858937724196533085168759209383758948
+emod:97796431105417086121597265848245224080646880891314620699,3566309852926516051758561425477409449674731444065,4237358411463841472327670289343773756088420869185382928335,0:1728379035119996080675313770460691892171354788520339293899
+emod:8797217794906750966627546899905389746578400755524,87461473070045326370501974269638901791524473673607812671,63579576817694957532492907897275166360331019259167287988,0:10068755237067361214693799084814132014570304060924282180
+emod:3559874918456198560844107856956492116344121248206,45190612746644425408414684724540657030439304479324039,684772758312255825130987448842421703263477856778082,0:500105229544177258096786497048655528691069662344352
+emod:244693542815021761876325478432698673186866834942911366704,569049151750280294347914587638406693043104703456596078,40418336644392592645116495825339904699620388961770,0:29946736018608816763137681085572521922498414649586
+emod:9552030352913538449959248911819718532447389423411634,15303289522386175735929093158193781525599040386989921,17905297851878021472325954407339512702437837940804355,0:7227207308763570886905006773221274115571235787935329
+emod:2427953611523095508885901603899411553216124564768970531,84219397598182701873267938504642229898519369455120690652,5124565221346746692094768907421280792680721722663049,0:2356962294194965054880021165404451023189256523983359
+emod:4450272023458109693579519823508011943468990250887770429477,9852180148050086854797451708622242708750405652182828950510,53218591961090046740426308696077544888041718958796756705,0:20722361228858746975332642022116538507546307084688876639
+emod:-3004436507840212393446088399233355090512633931853,49762352101558708510764823100128412949214842904414503,62339832791097134323794421391092834208651832053582621969,0:18644587060060960018044094951299852000053162441627275349
+emod:88466522329785151709268834138668446416231472463853002,84423940822580221018912007091908955924620089965579353300,4272746558814012659752446371362908792507498696285890777,0:2076687497522815194513922920413442770240123518700437310
+emod:-452137704321173773725358540484655780134513962685597,108536825158896375705138771152627679486775845056880757255,7443505375179903958445129175383134980666720252160978,0:2626655033625685458909447520232601675412626580083149
+emod:3646226144731352080332815382512112527325988973642,736730185733372156433101295886081895604485817722725,187559548105706428460460100309284194685793795827962,0:153703862976742792640135721867824635074042235940792
+emod:7128578973288696116455318246508827884395148700024035843,120044617176238734085406385417402387651027645325661,437777976047468143063879147080739099513994845762621846093,0:383512808517870785600076597930828756369001886406596109329
+emod:92651404329219142594991415600374429189694983250618442,1831956594370082650320589803647600704949512437508530740,989079198090857685320779737579388665165981389775724354,0:429473741854566274661236706264645428539362137664132958
+emod:783048980600864030214223520282343154949582263988130988029,5640798680918146067495300825091961866365823288705,712576055151408429943237184396668347415006718010549447,0:132303576760619346367560792334725765196706066347238184
+emod:54419766013288244584593400647946050509291116271357,37442044497849759156397618484039469808263267657304,55409955411797081810766022756739065410388681698701498,0:625989815794948434601895148375843191288774735399931
+
+# Regression for overflow bug
+emod:#x02,#x6FDD97BC0A760282D5A180D2308BE3B075BB6456F26B1DD0BEB2C1B0F880CF2C8EB724F2AD0567F7899D9A980C7DB93E418ED09AA3E35357F4F2E05963CB636A0C91B1BC7A000ADAC40F332D6ACCF4A3285112EDBAA95BB27904794ACC8D6FF188E2B04888ABFB11665BD1D915DEC8C6575706067AF3F7179BA57D1465D78129,#xFFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD129024E088A67CC74020BBEA63B139B22514A08798E3404DDEF9519B3CD3A431B302B0A6DF25F14374FE1356D6D51C245E485B576625E7EC6F44C42E9A637ED6B0BFF5CB6F406B7EDEE386BFB5A899FA5AE9F24117C4B1FE649286651ECE65381FFFFFFFFFFFFFFFF,0:#xD2C80DADBC0C72DBAC45F380064FDC5FECEEED268C7E86368DFC436908B2D33426C67DFAFAA56FAE89722EAE3644270B0512BE69DE61FDE07585001B5EA4FFDA1B8B98393D657E2EC4073CE4A6953093EF7C3B130CE1BB316C97CAAA88F83C32575D6E1128E107BF66910AAD2B84C44128370DEACFE7EC8A98B2CEA4DB7B09F1
+
+# Jacob Refstrup reported this failed in 1.11.
+emod:2,#x10001,#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,0:#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
diff --git a/polly/lib/External/isl/imath/tests/emodv.t b/polly/lib/External/isl/imath/tests/emodv.t
new file mode 100644
index 00000000000..4600a31feef
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/emodv.t
@@ -0,0 +1,305 @@
+# Modular exponentiation tests with small exponents
+
+emodev:19,2,84,0:25
+emodev:19,2,84,=1:25
+emodev:19,2,84,=3:25
+emodev:92,693930,16,0:0
+emodev:92,693930,16,=1:0
+emodev:92,693930,16,=3:0
+emodev:522181,0,166284,0:1
+emodev:522181,0,166284,=1:1
+emodev:522181,0,166284,=3:1
+emodev:437824,312,93008,0:26544
+emodev:437824,312,93008,=1:26544
+emodev:437824,312,93008,=3:26544
+emodev:5705223363,2,3286325576,0:818318913
+emodev:5705223363,2,3286325576,=1:818318913
+emodev:5705223363,2,3286325576,=3:818318913
+emodev:6399553325,6,4107014863,0:3307178134
+emodev:6399553325,6,4107014863,=1:3307178134
+emodev:6399553325,6,4107014863,=3:3307178134
+emodev:29074823374249,77004,71683909277697,0:8940258342532
+emodev:29074823374249,77004,71683909277697,=1:8940258342532
+emodev:29074823374249,77004,71683909277697,=3:8940258342532
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+emodev:50271884481278417471936841270780965441117412446282412169100969024758168373693942106624,3929,81963066766847673205247669622355518222939970750639201358276824482440704767576564991009,0:62281703540827707015524754807406004815705275005421409683659540143504669055166031995074
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+emodev:50271884481278417471936841270780965441117412446282412169100969024758168373693942106624,3929,81963066766847673205247669622355518222939970750639201358276824482440704767576564991009,=3:62281703540827707015524754807406004815705275005421409683659540143504669055166031995074
+emodev:-579340159717388367484103071217417305401854676291827723561829153224200380081237995138900716,651,307750673565730651787738866338890216282311653914599395911124646576595981539992209095900511,0:151509745004628309027577206278564432986410874344501872366280516978352818925422691489195213
+emodev:-579340159717388367484103071217417305401854676291827723561829153224200380081237995138900716,651,307750673565730651787738866338890216282311653914599395911124646576595981539992209095900511,=1:151509745004628309027577206278564432986410874344501872366280516978352818925422691489195213
+emodev:-579340159717388367484103071217417305401854676291827723561829153224200380081237995138900716,651,307750673565730651787738866338890216282311653914599395911124646576595981539992209095900511,=3:151509745004628309027577206278564432986410874344501872366280516978352818925422691489195213
+emodev:-97916243956905636477652083813766624733750918729402656599354295427509394488660812811650986,0,504214226951518003387724255017508609607833311675048629650486096975331876439736067207074328,0:1
+emodev:-97916243956905636477652083813766624733750918729402656599354295427509394488660812811650986,0,504214226951518003387724255017508609607833311675048629650486096975331876439736067207074328,=1:1
+emodev:-97916243956905636477652083813766624733750918729402656599354295427509394488660812811650986,0,504214226951518003387724255017508609607833311675048629650486096975331876439736067207074328,=3:1
+emodev:3292590583724934273152632680100030114145522324231651230586226973102509131367913652288365962855,0,3489305948459500758622220373494821215068462533405866936756424017653969429333910353069394383604,0:1
+emodev:3292590583724934273152632680100030114145522324231651230586226973102509131367913652288365962855,0,3489305948459500758622220373494821215068462533405866936756424017653969429333910353069394383604,=1:1
+emodev:3292590583724934273152632680100030114145522324231651230586226973102509131367913652288365962855,0,3489305948459500758622220373494821215068462533405866936756424017653969429333910353069394383604,=3:1
+emodev:-7978102455985676350366679275953460048435607670228140092963222007703573212864449715877925597841,29856,6437496673370692769029520191499051311630305921790132367949451008467412252240244679740653218546,0:5317938765885037665302910560037583058809520625824168737742040261509540549928785850499941206729
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+emodev:-7978102455985676350366679275953460048435607670228140092963222007703573212864449715877925597841,29856,6437496673370692769029520191499051311630305921790132367949451008467412252240244679740653218546,=3:5317938765885037665302910560037583058809520625824168737742040261509540549928785850499941206729
+emodev:67756737479519149309169118189597305480464085950144437770651525337717619238974063965848555442893510,0,61721185386054729344028716311850202684952560017220443047936445752013986984883181511511066630902828,0:1
+emodev:67756737479519149309169118189597305480464085950144437770651525337717619238974063965848555442893510,0,61721185386054729344028716311850202684952560017220443047936445752013986984883181511511066630902828,=1:1
+emodev:67756737479519149309169118189597305480464085950144437770651525337717619238974063965848555442893510,0,61721185386054729344028716311850202684952560017220443047936445752013986984883181511511066630902828,=3:1
+emodev:13622579220271678876766048827129463577857747656710532915693612564806742487161109510958801491048115,28000,84682569215903086997370780472826384755153630829606521208326331712119423387083844442113181771232731,0:61921681237911684957734827136839012206257886339643336895704703733211281700691618371720081605408837
+emodev:13622579220271678876766048827129463577857747656710532915693612564806742487161109510958801491048115,28000,84682569215903086997370780472826384755153630829606521208326331712119423387083844442113181771232731,=1:61921681237911684957734827136839012206257886339643336895704703733211281700691618371720081605408837
+emodev:13622579220271678876766048827129463577857747656710532915693612564806742487161109510958801491048115,28000,84682569215903086997370780472826384755153630829606521208326331712119423387083844442113181771232731,=3:61921681237911684957734827136839012206257886339643336895704703733211281700691618371720081605408837
+
+# Modular exponentation with small bases
+
+emodbv:9,32,84,0:81
+emodbv:9,32,84,=2:81
+emodbv:9,32,84,=3:81
+emodbv:0,45,69,0:0
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+emodbv:0,45,69,=3:0
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+emodbv:-93016,221819,301662,=3:211706
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+emodbv:44378,411312,93008,=2:33472
+emodbv:44378,411312,93008,=3:33472
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+emodbv:70,2233634102,3286325576,=2:2426711328
+emodbv:70,2233634102,3286325576,=3:2426711328
+emodbv:39,5533253464,1070148632,0:416533177
+emodbv:39,5533253464,1070148632,=2:416533177
+emodbv:39,5533253464,1070148632,=3:416533177
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+emodbv:7482,37424954770047,16839092776977,=2:7609954661820
+emodbv:7482,37424954770047,16839092776977,=3:7609954661820
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+emodbv:35080,873804207633443539517026163606065,7700271966945032910098505871172260,=2:5423974510417358397307357591894900
+emodbv:35080,873804207633443539517026163606065,7700271966945032910098505871172260,=3:5423974510417358397307357591894900
+emodbv:6,73591715017125540836707619882023150680,60596719455443436752409094193881404578,0:19278231891294150546581628518996450870
+emodbv:6,73591715017125540836707619882023150680,60596719455443436752409094193881404578,=2:19278231891294150546581628518996450870
+emodbv:6,73591715017125540836707619882023150680,60596719455443436752409094193881404578,=3:19278231891294150546581628518996450870
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+emodbv:0,571664949666413150109110287956385784303120,899228089361406308796439491202242113692801,=2:0
+emodbv:0,571664949666413150109110287956385784303120,899228089361406308796439491202242113692801,=3:0
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+emodbv:-559530,4636346354116451435163995696044394621599926066887606404471,2007044169839741253208493501852117173644320085073436239558,0:980083462330793649852651395567038357390956681750027115900
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diff --git a/polly/lib/External/isl/imath/tests/expt.t b/polly/lib/External/isl/imath/tests/expt.t
new file mode 100644
index 00000000000..c778929860e
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/expt.t
@@ -0,0 +1,154 @@
+# Exponentiation tests
+
+exptv:25,37,0:5293955920339377119177015629247762262821197509765625
+exptv:0,5,0:0
+exptv:10,5,0:100000
+exptv:1,-1,0:$MP_RANGE
+
+expt:-1,0,0:1
+expt:-1,1,0:-1
+expt:1,1000,0:1
+expt:10,5,0:100000
+expt:10,5,=1:100000
+expt:16,2,0:256
+expt:16,-2,0:$MP_RANGE
+expt:9,114,0:6076396096647706909168138770838836135530328017648434830996201971201776350890241322455818405320466786549738961
+expt:-86,476,0:662617332696362860524067230796827820618815284881665908870595313490845384402730799402221740741547892729430741053639014274968962128613995528237195069353968871860065099842959580048834695510748974202424908905352730826874949676923389706328585129209694843598031316497606363273659659518137129001683914681077900538211178870670637655474948719356656149928857724507047191532188701873201466847680543186709462811132458087950485580962981785390351911145306017418845982546035812744084364382432586452339098567898296989357418159116783507112574874002896769395516552960579886303506398054290885811079444534809660731243125017742263312078176886504861185566719851630353701862941682854137129215263285153620969438652988427729550528379121310832057528645876351114476064584186293636620546183047738657594287749786268444920697293633564476272776432769083439149706832453778408583421346390651383448852253784001810843070684412323782386158601772005993742336
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diff --git a/polly/lib/External/isl/imath/tests/gcd.t b/polly/lib/External/isl/imath/tests/gcd.t
new file mode 100644
index 00000000000..df7537e650a
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/gcd.t
@@ -0,0 +1,761 @@
+# Greatest common divisor tests
+
+gcd:0,0,0:$MP_UNDEF
+gcd:0,0,=1:$MP_UNDEF
+gcd:0,0,=2:$MP_UNDEF
+gcd:0,=1,0:$MP_UNDEF
+gcd:0,=1,99999:$MP_UNDEF
+gcd:-7,328492456,0:1
+gcd:16522181,930,0:1
+gcd:844378,24,0:2
+gcd:209,3,0:1
+gcd:70522336,34102,0:2
+gcd:32557663,995533,0:1
+gcd:641,701,0:1
+gcd:290748,-233,0:1
+gcd:-95,4770,0:5
+gcd:6839092,7,0:1
+gcd:787590783,-6789049,0:1
+gcd:721,15845,0:1
+gcd:-37028,-949534667,0:1
+gcd:618,-3,0:3
+gcd:16657,-544851,0:1
+gcd:28271,6,0:1
+gcd:-35357673,5207718,0:3
+gcd:99,-565,0:1
+gcd:9311525,-5,0:5
+gcd:6221599,7764515429,0:1
+gcd:21590339,486739392,0:1
+gcd:335,3699183948,0:1
+gcd:14,8755508782,0:2
+gcd:-5302,9,0:1
+gcd:8236,9462435,0:1
+gcd:43040,2766705,0:5
+gcd:863478587,63,0:1
+gcd:8874055124,10,0:2
+gcd:8204,-89955,0:1
+gcd:3303887262,3,0:3
+gcd:75504,-62,0:2
+gcd:436,2123372,0:4
+gcd:3361227,943089540,0:3
+gcd:37340421,223177728,0:3
+gcd:-35694701,238293550,0:1
+gcd:4263,15500547338,0:1
+gcd:1493,38730799,0:1
+gcd:-635007,298501,0:1
+gcd:4968,-98492710997,0:1
+gcd:3367542606,17576297,0:1
+gcd:232155550,3921719,0:1
+gcd:4860966,700,0:2
+gcd:2323372237,867,0:1
+gcd:4268,-29035,0:1
+gcd:8,-73804207633,0:1
+gcd:395170,26163606,0:2
+gcd:-70027196,6945032910,0:2
+gcd:5871172260,26773591,0:1
+gcd:17125540,836,0:4
+gcd:1988202315,-68060596,0:1
+gcd:554434367524,-909419,0:1
+gcd:-40457802353,3398,0:1
+gcd:410791452776,4492144466,0:2
+gcd:135751970,7417878752,0:2
+gcd:9593,31047471995,0:1
+gcd:-18387143,50226,0:1
+gcd:413450673851,-124,0:1
+gcd:-618513698871,-318914624,0:1
+gcd:37416674596,-8817,0:1
+gcd:6242382,-943205,0:1
+gcd:-4949666413,1501091102,0:1
+gcd:-6385784303120,899228089,0:1
+gcd:6308,-79643949,0:1
+gcd:2421,136928,0:1
+gcd:333180111,442438,0:1
+gcd:723761305,611024432789,0:1
+gcd:-91289,-9223787613074,0:1
+gcd:10944,57098,0:2
+gcd:57145099,-94544476,0:1
+gcd:9327105307,88173695233,0:13
+gcd:73230770347,67182928,0:1
+gcd:4810254287,-4310,0:1
+gcd:7200253992,9893,0:1
+gcd:-92709373682,-664752121448,0:2
+gcd:-2236424417,-69570761,0:1
+gcd:781963714,61084006246,0:2
+gcd:78553,-3773682081509,0:1
+gcd:24811454802,-26314981001549,0:1
+gcd:805866671,8061826,0:1
+gcd:4281823377004,2916710,0:2
+gcd:15229,320560332911,0:1
+gcd:6093153060,357460,0:20
+gcd:624794507388,3264784,0:4
+gcd:2632503413711,-514594728450,0:1
+gcd:7216089,86034,0:3
+gcd:112578025,4532680608704,0:1
+gcd:-972617496,-1100642928524,0:4
+gcd:86658,-56860,0:2
+gcd:2292080,40552081391276,0:28
+gcd:57144144326678,3579498119,0:1
+gcd:77390585076,614924,0:4
+gcd:685579,959286825595306,0:1
+gcd:346354116,451435163995,0:1
+gcd:443946215999,260668,0:1
+gcd:640447120070,-441698,0:2
+gcd:1253208493501,-8521171736,0:1
+gcd:8507343,62395589,0:1
+gcd:81126006,461820852819773,0:1
+gcd:87891707090,40954510522233,0:11
+gcd:-767589,-991593,0:3
+gcd:86229155,43027930,0:5
+gcd:-431963821,48986055721,0:1
+gcd:101557073166691,-74080192,0:1
+gcd:-34269048183,250074,0:3
+gcd:7650667716360,-959482143,0:3
+gcd:-176432105527291,712269105001105,0:1
+gcd:91973525400419,-52180215632622,0:1
+gcd:9054526,-21388581343089,0:19
+gcd:19326311972,3180878,0:2
+gcd:501569850634,39035351302,0:2
+gcd:-3470290487760846,622612793613,0:3
+gcd:827793089252612,77591849874263,0:1
+gcd:19866668332,4414961,0:1
+gcd:136274450771,8659380001,0:1
+gcd:952660786088689,171372582380,0:1
+gcd:429415374,9140880954,0:6
+gcd:-368107346,2945634,0:2
+gcd:2147845929,79983807840463,0:1
+gcd:77885771608,-51236668812562,0:2
+gcd:62272695670,208399244,0:2
+gcd:61973019,-1405255816878779,0:1
+gcd:286206733986,212837117931767,0:1
+gcd:-131938894,1048695278656631,0:1
+gcd:27109492,306933238851468,0:4
+gcd:-314477095129839,3837918787896,0:3
+gcd:4116345,9101435151,0:3
+gcd:63853435249810,902940229903,0:1
+gcd:80076033,14791385,0:1
+gcd:-31038678,1454704764,0:6
+gcd:20638192722,-14267370499730,0:2
+gcd:8065790684,-1977228453431,0:7
+gcd:362687955673,99742877979696,0:1
+gcd:1235326072088,201675880263044,0:4
+gcd:568009735,49546482048450,0:5
+gcd:849579042,-3197920541,0:1
+gcd:1808291492597692,96814519920612,0:4
+gcd:6275712123233717,739880208952795162,0:1
+gcd:774075573665981,7523229210171,0:1
+gcd:2944466357361,497012903828640,0:3
+gcd:282862415058866,-6240991940068,0:2
+gcd:571225883598025,189932649930728,0:1
+gcd:815309478,-371377784,0:2
+gcd:320172590239,777220834144,0:1
+gcd:654332382,7919190746613,0:9
+gcd:-5094673003562,268414312780910,0:2
+gcd:-77841790209187781,35267718271550,0:1
+gcd:735762990564598234,-32739891496,0:2
+gcd:-5999905013129575,144027889596460,0:5
+gcd:6861193105404525,-4725324526382425,0:25
+gcd:859748569384694164,98523184164,0:4
+gcd:570460270963,-54720108308,0:1
+gcd:61750383079701,3452251304617,0:1
+gcd:-1738716695,8560399484936237,0:1
+gcd:9953417006786660,90085339160,0:20
+gcd:188523146978,954842405502,0:2
+gcd:198834734074395695,171072375577893,0:1
+gcd:-102893613303732727,155104650554,0:7
+gcd:90836124710855,-98332551310,0:5
+gcd:10512242068892,84377075433,0:1
+gcd:-726835478343584190,29339956106373,0:3
+gcd:5033847480079150451,-69469662902517,0:1
+gcd:3869558083241,-8770208065,0:1
+gcd:46079368203995132,23609959311256,0:4
+gcd:979137194422864421,-1378816153373947,0:1
+gcd:-72618673468,-244198092166982349,0:11
+gcd:529564016606,376298866055896536,0:2
+gcd:554186920892,-629719688885314,0:2
+gcd:3455963977845612750,271884481278417471,0:3
+gcd:-41270780965441117,-4124462824121691009,0:1
+gcd:47581683736,9394210662450,0:2
+gcd:8196306676684,76732052476696223555,0:1
+gcd:2939970750639,-2013582768244,0:1
+gcd:70476757656499,-100957934015971,0:1
+gcd:3674841030712174173,540185467629182772,0:3
+gcd:82915322420038008,123799513890,0:6
+gcd:466513077506,73565730651787738,0:14
+gcd:-38890216282311653,91459939591112,0:1
+gcd:576595981539992,20909590051109791,0:1
+gcd:956905636477652,8381376662473,0:1
+gcd:9187294026565993,54295427509,0:1
+gcd:8866081281165098,6545042142269515,0:1
+gcd:338772425501,-750860960783,0:1
+gcd:6750486296504,8609697533187,0:1
+gcd:736067207074328,329259058372493427315,0:1
+gcd:680100030114145,52232423165123,0:1
+gcd:22697310250913136791,365228836596285515,0:1
+gcd:30594845950075862222,37349482121506846253,0:1
+gcd:866936756424,-1765396942933391,0:1
+gcd:6939438360479781,24559856763503,0:1
+gcd:927595346004843560,7670228140092963222,0:2
+gcd:-357321286444971587,7925597841132985664,0:1
+gcd:6673370692769029,520191499051311630305,0:1
+gcd:9013236794945,1008467412252240244,0:1
+gcd:406532185466775673747,9519149309169118189,0:1
+gcd:548046408595014443,777065152533771,0:1
+gcd:23897406396584,-8555442893510156172118,0:2
+gcd:54729344028716311850,2026849525600172204,0:2
+gcd:-7936445752013,-98698488318151151,0:1
+gcd:6309028281362257922,271678876766048827,0:1
+gcd:6357785774765671053291,56936125648067424,0:33
+gcd:11095109588014,9104811531028000846,0:2
+gcd:-921590308699737078,472826384755153630,0:2
+gcd:652120832633171211942,-3387083844442113181,0:1
+gcd:32731949302549,335896184427699,0:7
+gcd:3635516318567409721998,16713986701316425,0:1
+gcd:6746198240933100303896,2007405485449,0:1
+gcd:-19072614334803913032,-240759760944833349,0:3
+gcd:828472954262165857,127978158379025077,0:1
+gcd:8306750095431507743,6378030328941226919,0:1
+gcd:7938532173109072,7646615157112938170608,0:16
+gcd:25274397567311,3125247753747112,0:1
+gcd:0,-6378030328941226919,0:6378030328941226919
+gcd:-6378030328941226919,0,0:6378030328941226919
+gcd:-6378030328941226919,0,=1:6378030328941226919
+gcd:-6378030328941226919,0,=2:6378030328941226919
+gcd:400327281633151431,37147489147345586,0:1
+gcd:28164658061513141691,818613450005821036296,0:3
+gcd:476291935913056166817,170589720180888384,0:3
+gcd:84130837655605609,16051948347568699941145,0:1
+gcd:438146043336536562159,-2206822496905640873917,0:1
+gcd:8120293725815373571,24028314574527126,0:1
+gcd:-667658265953408,8440015504605457600699,0:1
+gcd:53314441150262832895,283247085055963859,0:1
+gcd:6658153298145442,2140281300389057549,0:1
+gcd:213046498706732967,784948480361985741,0:3
+gcd:-3857522843819,419578526133778265155,0:1
+gcd:-4948941087660882967,-779517282262677550602,0:1
+gcd:4277076043224464,448010594023993,0:1
+gcd:-644113523059324254840,-71225587742718143,0:1
+gcd:264902435468141506,-823887292259809475,0:1
+gcd:1538706915662950684,448326823774496344436,0:4
+gcd:2036808313025110180,30422160628736748027,0:1
+gcd:7385426850078311,9871854919284839,0:1
+gcd:804644713550364054117828,8379163811459445,0:3
+gcd:981008037679996757191201,-641409668917881464,0:1
+gcd:64201713053871850,879119102660231895102,0:2
+gcd:1033957429367195,1113485105415049436,0:1
+gcd:14455075580451942850,-1885962875546302716,0:2
+gcd:2615475057635555327591,71606675442458157933637,0:1
+gcd:596676024530775,2621192750454214313695,0:5
+gcd:71665682789483848832034,351563529541544408090967,0:3
+gcd:5092055214749737510271,-1833800774226189505,0:1
+gcd:600508413963341945,5057063373752305154,0:1
+gcd:-92318043858914149,958444935645415778103,0:1
+gcd:385447683586530283,1775981280157037775814,0:1
+gcd:80717517906672139,362327909576030289130356,0:1
+gcd:31249856290116188,7875872084416640,0:4
+gcd:5339381130045070,-287550419156538713298378,0:2
+gcd:-28993892995370333046,698005135756870427,0:1
+gcd:9179674525723481348,81469928617693608886,0:2
+gcd:7522547789649675160016,285918417200326088447794,0:2
+gcd:-901774653467788803,236000195734098992,0:1
+gcd:142671415131788790,-2583777714764598301032762,0:6
+gcd:951078283105727024828,1487627587101952300275691,0:1
+gcd:19185476025869672486,2445736876347152429980675,0:11
+gcd:6741226163186521681303,2568160083399218078514720,0:1
+gcd:3987919571578399660412212,9884144363611069680876,0:4
+gcd:99431631881940059268,-291965406586754754564989,0:1
+gcd:36727416735327035,-1612205593089728906670,0:5
+gcd:19590774006340220367562,984968880458891391,0:1
+gcd:70695613219926859099201753,62591913993270611799645,0:1
+gcd:77425442645246583405,-296832015660977965,0:35
+gcd:807971872592227526408,40230405885131399744453603,0:1
+gcd:965370212546292322,92058451234918953535616,0:2
+gcd:418561885926860582,239657399821326936,0:2
+gcd:4657697584188823828175,78713797195009784682708,0:1
+gcd:-38091455008728662,5763777114003492300442812,0:2
+gcd:54943065861736600767,6268390929335695219433,0:1
+gcd:-697639190379957186138,2936028900796584636,0:6
+gcd:96018089280939789090187,29566309848085271814423786,0:1
+gcd:-1112368751522257152114,1995174854903534746993983,0:9
+gcd:-84342841213955171500540,5699084396809137598518,0:2
+gcd:225675029492277024520800,92998377812004114915728728,0:8
+gcd:7655269162163547244,594548791001649285,0:1
+gcd:17608380663675562604,264448308193844308642824889,0:1
+gcd:223093179893694063251643,17908241494426478956477995,0:3
+gcd:-9501451313199296478183,164266650410143061944307590,0:1
+gcd:4558730178677382262,3122005086984557462,0:2
+gcd:54440882483631228680923933,4478571958703194374379,0:7
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diff --git a/polly/lib/External/isl/imath/tests/gmp-compat-test/.gitignore b/polly/lib/External/isl/imath/tests/gmp-compat-test/.gitignore
new file mode 100644
index 00000000000..64a46a937a9
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/gmp-compat-test/.gitignore
@@ -0,0 +1,10 @@
+*.o
+*.so
+*.pyc
+#for now...
+test.list
+gmp_test.c
+imath_test.c
+wrappers.py
+*.tests
+_T*
diff --git a/polly/lib/External/isl/imath/tests/gmp-compat-test/README b/polly/lib/External/isl/imath/tests/gmp-compat-test/README
new file mode 100644
index 00000000000..b03282517c0
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/gmp-compat-test/README
@@ -0,0 +1,106 @@
+Overview
+==================================================
+This directory contains a random test generator for the gmp
+compatibility layer of imath. The tests cases are randomly generated
+and run with both gmp and imath. The results are compared and any
+mismatching results are flagged as failures.
+
+You should not see any failures when running these tests.
+
+Requirements
+==================================================
+These tests use the python ffi to run the imath and gmp functions. To
+run these tests you will need the following items
+
+  * libimath.so
+  * python 3
+  * gmp library and header files
+
+
+Running
+==================================================
+All the tests cases will be generated and run automatically by the
+makefile. First, make sure you have built libimath.so in the top level
+imath directory.
+
+Use the following command to generate and run the tests
+
+    $ make TESTS=random.tests
+
+This should generate all the needed wrapper files along with the
+randomized unit tests. By default the unit tests are output to the
+random.tests file. The tests can be run by hand using the following
+command line
+
+    $ ./runtest random.tests
+
+You can also write your own unit tests and run them by passing the
+file name to the runtest script.
+
+
+Testing Methodology
+==================================================
+The goal of our testing is to ensure that we are compatible with the
+gmp api. To test our compatibility layer we generate inputs for each
+gmp api that we wrap and then call the gmp version of the api and
+compare the results to the imath version. The results should be
+identical. Any output mismatch is considered an error.
+
+
+Testcase Generation
+--------------------
+The test data generation is inspired by the QuickCheck testing
+methodolgy. We want to test both random data and important values such
+as 0,1,-1, etc. We generate input data based on the type of the input
+parameter. Most apis take either an mpz_t or a signed/unsigned long.
+
+For each parameter of the given type we generate the following values:
+
+  mpz_t:  0,1,-1, all min/max values, all min/max values +/- 1
+          small  numbers: 1-4    digits
+          medium numbers: 5-20   digits
+          large  numbers: 20-100 digits
+  long :  0,1,-1 short,int,long min/max values
+          random long values
+  unsigned long: 0,1,-1, unsigned short, unsigned int, unsigned long min/max values
+                 random unsigned long values
+
+The generated data for each paramater is combined to produce a series
+of inputs to the function. The input data format looks like:
+
+mpz_add|0,1,2
+
+Which represents the call mpz_add(0, 1, 2). Additional test cases can be written by hand.
+
+Test Structure
+--------------------
+The tests are run using the python ffi (the ctypes module). We have a
+single description of each api that is used to generate the following:
+
+  * intput data
+  * c function wrapper to call the libgmp or libimath function
+  * python wrapper to call both libgmp and libimath wrappers and compare results
+
+The test_gmp.so and test_imath.so libraries are loaded at runtime by
+the python runner scrip and used to run each test input and compare
+the results.
+
+The code generation pattern looks something like this:
+
+genctest.py  ~~generates~~> gmp_test.c   <--includes--  gmp_custom_test.c
+gmp_test.c   ~~generates~~> gmp_tests.so <--links--     libgmp.so
+genpytest.py ~~generates~~> wrappers.py  <--calls--     gmp_test.so
+gmp_test.so      --loads--> runtest.py   <--includes--  wrappers.py
+                               ^
+                               |
+                             reads
+                               |
+gendata.py  ~~generates~~> random.tests
+
+Adding tests for new APIs
+==================================================
+New apis can be tested by modifying the gmpapy.py file. This file
+contains a list of all apis that will be tested. Adding an api to the
+apis list will cause code and input data to be generated for that api.
+Rerunning make will generate the test data and code to run the test.
+
diff --git a/polly/lib/External/isl/imath/tests/gmp-compat-test/genctest.py b/polly/lib/External/isl/imath/tests/gmp-compat-test/genctest.py
new file mode 100755
index 00000000000..67c140e2a21
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/gmp-compat-test/genctest.py
@@ -0,0 +1,266 @@
+#!/usr/bin/env python3
+import sys
+import gmpapi
+from gmpapi import void
+from gmpapi import ilong
+from gmpapi import iint
+from gmpapi import ulong
+from gmpapi import mpz_t
+from gmpapi import size_t
+from gmpapi import charp
+from gmpapi import mpq_t
+
+class APITest:
+  def __init__(self, gmpapi):
+    self.api = gmpapi
+
+  def test_prefix(self):
+    return "test"
+
+  def test_param_name(self, ty, i):
+    if ty == mpz_t:
+      pname = "p_zs"
+    elif ty == ilong:
+      pname = "p_si"
+    elif ty == ulong:
+      pname = "p_ui"
+    elif ty == iint:
+      pname = "p_i"
+    elif ty == charp:
+      pname = "p_cs"
+    elif ty == mpq_t:
+      pname = "p_qs"
+    else:
+      raise RuntimeError("Unknown param type: "+str(ty))
+    return pname + str(i)
+
+  def test_param_type(self, ty):
+    if ty == mpz_t or ty == mpq_t:
+      pty_name = "char *"
+    else:
+      pty_name = str(ty)
+    return pty_name
+
+  def test_var_name(self, ty, i):
+    if ty == mpz_t:
+      vname = "v_z"
+    elif ty == ilong:
+      vname = "v_si"
+    elif ty == ulong:
+      vname = "v_ui"
+    elif ty == iint:
+      vname = "v_i"
+    elif ty == size_t:
+      vname = "v_st"
+    elif ty == charp:
+      vname = "v_cs"
+    elif ty == mpq_t:
+      vname = "v_q"
+    else:
+      raise RuntimeError("Unknown param type: "+str(ty))
+    return vname + str(i)
+
+  def test_var_type(self, ty):
+    if ty == mpz_t:
+      return self.mpz_type()
+    elif ty == mpq_t:
+      return self.mpq_type()
+    else:
+      return str(ty)
+
+  def init_var_from_param(self, ty, var, param):
+    code = "\t"
+    if ty == mpz_t or ty == mpq_t:
+      code += self.api_call_prefix(ty)+"init("+var+");\n\t"
+      code += self.api_call_prefix(ty)+"set_str("+",".join([var,param,"10"])+")"
+      if ty == mpq_t:
+        code += ";\n\t"
+        code += self.api_call_prefix(ty)+"canonicalize("+var+")"
+    else:
+      code += var + "=" + param
+    return code
+
+  def init_vars_from_params(self):
+    code = ""
+    for (i,p) in enumerate(self.api.params):
+      param = self.test_param_name(p, i)
+      code += "\t"
+      code += self.test_var_type(p) + " "
+      var   = self.test_var_name(p, i)
+      code += var + ";\n"
+      code += self.init_var_from_param(p, var, param) + ";\n\n"
+    return code
+
+  def make_api_call(self):
+    bare_name = self.api.name.replace("mpz_", "", 1).replace("mpq_", "", 1)
+    call_params = [self.test_var_name(p,i) for (i,p) in enumerate(self.api.params)]
+    ret = "\t"
+    ret_ty = self.api.ret_ty
+    if ret_ty != void:
+      ret += self.test_var_type(ret_ty)+" "+self.test_var_name(ret_ty, "_ret") + " = "
+    # call mpq or mpz function
+    if self.api.name.startswith("mpz_"):
+      prefix = self.api_call_prefix(mpz_t)
+    else:
+      prefix = self.api_call_prefix(mpq_t)
+    return ret + prefix + bare_name + "(" + ",".join(call_params) + ");\n"
+
+  def normalize_cmp(self, ty):
+    cmpval = self.test_var_name(ty, "_ret")
+    code = ""
+    code += """
+	if ({var} > 0)
+	  {var} = 1;
+	else if ({var} < 0)
+	  {var} = -1;\n\t
+""".format(var=cmpval)
+    return code
+
+  def extract_result(self, ty, pos):
+    code = ""
+    if ty == mpz_t or ty == mpq_t:
+      var = self.test_var_name(ty, pos)
+      code += self.api_call_prefix(ty) + "get_str(out+offset, 10,"+var+");\n"
+      code += "\toffset = offset + strlen(out); "
+      code += "out[offset] = ' '; out[offset+1] = 0; offset += 1;"
+    else:
+      assert pos == -1, "expected a return value, not a param value"
+      if ty == ilong:
+        var = self.test_var_name(ty, "_ret")
+        code += 'offset = sprintf(out+offset, " %ld ", '+var+');'
+      elif ty == ulong:
+        var = self.test_var_name(ty, "_ret")
+        code += 'offset = sprintf(out+offset, " %lu ", '+var+');'
+      elif ty == iint:
+        var = self.test_var_name(ty, "_ret")
+        code += 'offset = sprintf(out+offset, " %d ", '+var+');'
+      elif ty == size_t:
+        var = self.test_var_name(ty, "_ret")
+        code += 'offset = sprintf(out+offset, " %zu ", '+var+');'
+      elif ty == charp:
+        var = self.test_var_name(ty, "_ret")
+        code += 'offset = sprintf(out+offset, " %s ", '+var+');'
+      else:
+        raise RuntimeError("Unknown param type: "+str(ty))
+    return code
+
+  def extract_results(self):
+    ret_ty = self.api.ret_ty
+    code = "\tint offset = 0;\n\t"
+
+    # normalize cmp return values
+    if ret_ty == iint and "cmp" in self.api.name:
+      code += self.normalize_cmp(ret_ty)
+
+    # call canonicalize for mpq_set_ui
+    if self.api.name == "mpq_set_ui":
+      code += self.api_call_prefix(mpq_t) + "canonicalize("+self.test_var_name(mpq_t, 0)+");\n\t"
+
+    # get return value
+    if ret_ty != void:
+      code += self.extract_result(ret_ty, -1) + "\n"
+
+    # get out param values
+    for pos in self.api.out_params:
+      code += "\t"
+      code += self.extract_result(self.api.params[pos], pos) + "\n"
+
+    return code + "\n"
+
+  def clear_local_vars(self):
+    code = ""
+    for (i,p) in enumerate(self.api.params):
+      if p == mpz_t or p == mpq_t:
+        var   = self.test_var_name(p, i)
+        code += "\t"+self.api_call_prefix(p)+"clear("+var+");\n"
+    return code
+
+  def print_test_code(self, outf):
+    api = self.api
+    params = [self.test_param_type(p)+" "+self.test_param_name(p, i) for (i, p) in enumerate(api.params)]
+    code = "void {}_{}(char *out, {})".format(self.test_prefix(), api.name, ", ".join(params))
+    code += "{\n"
+    code += self.init_vars_from_params()
+    code += self.make_api_call()
+    code += self.extract_results()
+    code += self.clear_local_vars()
+    code += "}\n"
+    outf.write(code)
+    outf.write("\n")
+
+class GMPTest(APITest):
+  def __init__(self, gmpapi):
+    super().__init__(gmpapi)
+
+  def api_call_prefix(self, kind):
+    if kind == mpz_t:
+      return "mpz_"
+    elif kind == mpq_t:
+      return "mpq_"
+    else:
+      raise RuntimeError("Unknown call kind: "+str(kind))
+
+  def mpz_type(self):
+    return "mpz_t"
+
+  def mpq_type(self):
+    return "mpq_t"
+
+class ImathTest(APITest):
+  def __init__(self, gmpapi):
+    super().__init__(gmpapi)
+
+  def api_call_prefix(self, kind):
+    if kind == mpz_t:
+      return "impz_"
+    elif kind == mpq_t:
+      return "impq_"
+    else:
+      raise RuntimeError("Unknown call kind: "+str(kind))
+
+  def mpz_type(self):
+    return "impz_t"
+
+  def mpq_type(self):
+    return "impq_t"
+
+def print_gmp_header(outf):
+  code  = ""
+  code += "#include <gmp.h>\n"
+  code += "#include <stdio.h>\n"
+  code += "#include <string.h>\n"
+  code += '#include "gmp_custom_test.c"\n'
+  outf.write(code)
+
+def print_imath_header(outf):
+  code  = ""
+  code += "#include <gmp_compat.h>\n"
+  code += "#include <stdio.h>\n"
+  code += "#include <string.h>\n"
+  code += "typedef mpz_t impz_t[1];\n"
+  code += "typedef mpq_t impq_t[1];\n"
+  code += '#include "imath_custom_test.c"\n'
+  outf.write(code)
+
+def print_gmp_tests(outf):
+    print_gmp_header(outf)
+    for api in gmpapi.apis:
+      if not api.custom_test:
+        GMPTest(api).print_test_code(outf)
+
+def print_imath_tests(outf):
+    print_imath_header(outf)
+    for api in gmpapi.apis:
+      if not api.custom_test:
+        ImathTest(api).print_test_code(outf)
+
+def main():
+  test = sys.argv[1]
+
+  if test == "gmp":
+    print_gmp_tests(sys.stdout)
+  elif test == "imath":
+    print_imath_tests(sys.stdout)
+
+if __name__ == "__main__":
+    main()
diff --git a/polly/lib/External/isl/imath/tests/gmp-compat-test/gendata.py b/polly/lib/External/isl/imath/tests/gmp-compat-test/gendata.py
new file mode 100755
index 00000000000..225f71b7aa0
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/gmp-compat-test/gendata.py
@@ -0,0 +1,338 @@
+#!/usr/bin/env python3
+import random
+import gmpapi
+
+MAX_SLONG="9223372036854775807"
+MIN_SLONG="-9223372036854775808"
+MAX_ULONG="18446744073709551615"
+MAX_SINT="2147483647"
+MIN_SINT="-2147483648"
+MAX_UINT="4294967295"
+MAX_SSHORT="32767"
+MIN_SSHORT="-32768"
+MAX_USHORT="65535"
+
+def plus1(x):  return str(int(x)+1)
+def minus1(x): return str(int(x)-1)
+def apply(fun, lst):
+  return list(map(str, map(fun, lst)))
+
+mzero_one = ["-0", "-1"]
+zero_one  = ["0", "1"]
+mm_slong   = [MAX_SLONG, MIN_SLONG]
+mm_slong1  = [minus1(MAX_SLONG), plus1(MIN_SLONG)]
+mm_ulong  = [MAX_ULONG]
+mm_ulong1 = [minus1(MAX_ULONG)]
+mm_sint   = [MAX_SINT, MIN_SINT]
+mm_sint1  = [minus1(MAX_SINT), plus1(MIN_SINT)]
+mm_uint   = [MAX_UINT]
+mm_uint1  = [minus1(MAX_UINT)]
+mm_sshort = [MAX_SSHORT, MIN_SSHORT]
+mm_sshort1= [minus1(MAX_SSHORT), plus1(MIN_SSHORT)]
+mm_ushort = [MAX_USHORT]
+mm_ushort1= [minus1(MAX_USHORT)]
+mm_all    = mm_slong + mm_ulong + mm_sint + mm_uint + mm_sshort + mm_ushort
+zero_one_all = mzero_one + zero_one
+
+
+mpz_std_list = zero_one_all + mm_all + apply(plus1, mm_all) + apply(minus1, mm_all)
+si_std_list  = zero_one     + mm_slong + mm_sint + mm_sshort + mm_slong1 + mm_sint1 + mm_sshort1
+ui_std_list  = zero_one     + mm_ulong + mm_uint + mm_ushort + mm_ulong1 + mm_uint1 + mm_ushort1
+
+def gen_random_mpz(mindigits=1, maxdigits=100, allowneg=True):
+  sign = random.choice(["", "-"])
+  if not allowneg:
+    sign = ""
+  return sign + gen_digits(random.randint(mindigits, maxdigits))
+
+def gen_random_si():
+  si = gen_random_mpz(mindigits=1,maxdigits=19)
+  while int(si) > int(MAX_SLONG) or int(si) < int(MIN_SLONG):
+    si = gen_random_mpz(mindigits=1,maxdigits=19)
+  return si
+
+def gen_random_ui():
+  ui = gen_random_mpz(mindigits=1,maxdigits=20, allowneg=False)
+  while int(ui) > int(MAX_ULONG):
+    ui = gen_random_mpz(mindigits=1,maxdigits=20, allowneg=False)
+  return ui
+
+def gen_digits(length):
+  if length == 1:
+    i = random.randint(1, 9)
+  else:
+    digits = [random.randint(1,9)] + [random.randint(0,9) for x in range(length-1)]
+    digits = map(str, digits)
+    i = "".join(digits)
+  return str(i)
+
+def gen_mpzs(mindigits=1, maxdigits=100, count=10):
+  return [gen_random_mpz(mindigits=mindigits, maxdigits=maxdigits) for x in range(count)]
+
+default_count = 10
+def gen_sis(count=default_count):
+  return [gen_random_si() for x in range(count)]
+
+def gen_uis(count=default_count):
+  return [gen_random_ui() for x in range(count)]
+
+def gen_small_mpzs(count=default_count):
+  return gen_mpzs(mindigits=1,maxdigits=4,count=count)
+
+def is_small_mpz(s):
+  return len(s) >= 1 and len(s) <= 4
+
+def gen_medium_mpzs(count=default_count):
+  return gen_mpzs(mindigits=5,maxdigits=20,count=count)
+
+def is_medium_mpz(s):
+  return len(s) >= 5 and len(s) <= 20
+
+def gen_large_mpzs(count=default_count):
+  return gen_mpzs(mindigits=21,maxdigits=100,count=count)
+
+def is_large_mpz(s):
+  return len(s) >= 21
+
+def gen_mpz_spread(count=default_count):
+  return gen_small_mpzs(count) + gen_medium_mpzs(count) + gen_large_mpzs(count)
+
+def gen_mpz_args(count=default_count):
+    return mpz_std_list + gen_mpz_spread(count)
+
+def gen_mpq_args(count=4):
+  nums = zero_one + gen_mpz_spread(count)
+  dens = ["1"]    + gen_mpz_spread(count)
+  return [n+"/"+d for n in nums for d in dens if int(d) != 0]
+
+def gen_si_args():
+  return si_std_list + gen_sis()
+
+def gen_ui_args():
+  return ui_std_list + gen_uis()
+
+def gen_list_for_type(t, is_write_only):
+  if (t == gmpapi.mpz_t or t == gmpapi.mpq_t) and is_write_only:
+    return ["0"]
+  elif t == gmpapi.mpz_t:
+      return gen_mpz_args()
+  elif t == gmpapi.ilong:
+    return gen_si_args()
+  elif t == gmpapi.ulong:
+    return gen_ui_args()
+  elif t == gmpapi.mpq_t:
+    return gen_mpq_args()
+  else:
+    raise RuntimeError("Unknown type: {}".format(t))
+
+def gen_args(api):
+  if api.custom_test or api.name in custom:
+    return custom[api.name](api)
+  types = api.params
+  if len(types) == 1:
+    return [[a] for a in gen_list_for_type(types[0], api.is_write_only(0))]
+  elif len(types) == 2:
+    t1 = gen_list_for_type(types[0], api.is_write_only(0))
+    t2 = gen_list_for_type(types[1], api.is_write_only(1))
+    return [(a,b) for a in t1 for b in t2]
+  elif len(types) == 3:
+    t1 = gen_list_for_type(types[0], api.is_write_only(0))
+    t2 = gen_list_for_type(types[1], api.is_write_only(1))
+    t3 = gen_list_for_type(types[2], api.is_write_only(2))
+    return [(a,b,c) for a in t1 for b in t2 for c in t3]
+  elif len(types) == 4:
+    t1 = gen_list_for_type(types[0], api.is_write_only(0))
+    t2 = gen_list_for_type(types[1], api.is_write_only(1))
+    t3 = gen_list_for_type(types[2], api.is_write_only(2))
+    t4 = gen_list_for_type(types[3], api.is_write_only(3))
+    return [(a,b,c,d) for a in t1 for b in t2 for c in t3 for d in t4]
+  else:
+    raise RuntimeError("Too many args: {}".format(len(types)))
+
+###################################################################
+#
+# Fixup and massage random data for better test coverage
+#
+###################################################################
+def mul_mpzs(a, b):
+  return str(int(a) * int(b))
+
+def mpz_divexact_data(args):
+  # set n = n * d
+  divisible = mul_mpzs(args[1],(args[2]))
+  return [(args[0], divisible, args[2])]
+
+def mpz_divisible_p_data(args):
+  (n,d) = get_div_data(args[0], args[1],rate=1.0)
+  return [(n,d), (args[0], args[1])]
+
+def mpz_div3_data(args):
+  q     = args[0]
+  (n,d) = get_div_data(args[1], args[2],rate=1.0)
+  return [(q,n,d), (q,args[1], args[2])]
+
+def mpz_pow_data(args, alwaysallowbase1=True):
+  base = int(args[1])
+  exp  = int(args[2])
+  # allow special numbers
+  if base == 0 or exp == 0 or exp == 1:
+    return [args]
+  if base == 1 and alwaysallowbase1:
+    return [args]
+
+  # disallow too big numbers
+  if base > 1000 or base < -1000:
+    base = gen_random_mpz(maxdigits=3)
+  if exp > 1000:
+    exp = gen_random_mpz(maxdigits=3, allowneg=False)
+
+  return [(args[0], str(base), str(exp))]
+
+def mpz_mul_2exp_data(args):
+  return mpz_pow_data(args, alwaysallowbase1=False)
+
+def mpz_gcd_data(args):
+  r = args[0]
+  a = args[1]
+  b = args[2]
+  s_ = gen_small_mpzs(1)[0]
+  m_ = gen_medium_mpzs(1)[0]
+  l_ = gen_large_mpzs(1)[0]
+
+  return [ (r, a, b),
+           (r, mul_mpzs(a, b),  b),
+           (r, mul_mpzs(a, s_), mul_mpzs(b, s_)),
+           (r, mul_mpzs(a, m_), mul_mpzs(b, m_)),
+           (r, mul_mpzs(a, l_), mul_mpzs(b, l_)),]
+
+def mpz_export_data(api):
+  rop    = ["0"]
+  countp = ["0"]
+  order  = ["-1","1"]
+  size   = ["1","2","4","8"]
+  endian = ["0"]
+  nails  = ["0"]
+  ops    = gen_mpz_args(1000) + gen_mpzs(count=100,mindigits=100,maxdigits=1000)
+
+  args = []
+  for r in rop:
+    for c in countp:
+      for o in order:
+        for s in size:
+          for e in endian:
+            for n in nails:
+              for op in ops:
+                args.append((r,c,o,s,e,n,op))
+  return args
+
+def mpz_sizeinbase_data(api):
+  bases = list(map(str, range(2, 37)))
+  ops    = gen_mpz_args(1000) + gen_mpzs(count=1000,mindigits=100,maxdigits=2000)
+  return [(op, b) for op in ops for b in bases]
+
+def get_str_data(ty):
+  bases = list(range(2,37)) + list(range(-2, -37, -1))
+  bases = list(map(str, bases))
+  if ty == gmpapi.mpz_t:
+    ops = gen_mpz_args(1000)
+  elif ty == gmpapi.mpq_t:
+    ops = gen_mpq_args(20)
+  else:
+    raise RuntimeError("Unsupported get_str type: "+str(ty))
+  return [("NULL", b, op) for b in bases for op in ops]
+
+def mpz_get_str_data(api):
+  return get_str_data(gmpapi.mpz_t)
+
+def mpq_get_str_data(api):
+  return get_str_data(gmpapi.mpq_t)
+
+def mpq_set_str_data(api):
+  args = gen_mpq_args(20) + gen_mpz_args()
+  # zero does not match results exactly because the
+  # results are not canonicalized first. We choose to
+  # exclude zero from test results. The other option is
+  # to canonicalize the results after parsing the strings.
+  # Instead we exclude zero so that we can independently
+  # test correctness of set_str and canonicalization
+  nonzero = []
+  for arg in args:
+    if "/" in arg:
+      pos = arg.find("/")
+      if int(arg[:pos]) != 0:
+        nonzero.append(arg)
+    elif int(arg) != 0:
+        nonzero.append(arg)
+
+  return [("0", q, "10") for q in nonzero]
+
+def get_div_data(n, d, rate=0.2):
+  """Generate some inputs that are perfectly divisible"""
+  if random.random() < rate:
+    n = mul_mpzs(n, d)
+  return (n,d)
+
+def allow(name, args):
+  if name not in blacklists:
+    return True
+  filters = blacklists[name]
+  for (pos, disallow) in filters:
+    if args[pos] in disallow:
+      return False
+  return True
+
+def fixup_args(name, args):
+  if name not in fixups:
+    return [args]
+  return fixups[name](args)
+
+
+# list of values to be excluded for various api calls
+# list format is (pos, [list of values to exclude])
+blacklists = {
+  "mpz_cdiv_q"      : [(2, ["0", "-0"])],
+  "mpz_fdiv_q"      : [(2, ["0", "-0"])],
+  "mpz_fdiv_r"      : [(2, ["0", "-0"])],
+  "mpz_tdiv_q"      : [(2, ["0", "-0"])],
+  "mpz_fdiv_q_ui"   : [(2, ["0", "-0"])],
+  "mpz_divexact"    : [(2, ["0", "-0"])],
+  "mpz_divisible_p" : [(1, ["0", "-0"])],
+  "mpz_divexact_ui" : [(2, ["0", "-0"])],
+  "mpq_set_ui"      : [(2, ["0", "-0"])],
+}
+
+fixups = {
+  "mpz_divexact"    : mpz_divexact_data,
+  "mpz_divisible_p" : mpz_divisible_p_data,
+  "mpz_cdiv_q"      : mpz_div3_data,
+  "mpz_fdiv_q"      : mpz_div3_data,
+  "mpz_fdiv_r"      : mpz_div3_data,
+  "mpz_tdiv_q"      : mpz_div3_data,
+  "mpz_fdiv_q_ui"   : mpz_div3_data,
+  "mpz_divexact_ui" : mpz_divexact_data,
+  "mpz_pow_ui"      : mpz_pow_data,
+  "mpz_gcd"         : mpz_gcd_data,
+  "mpz_lcm"         : mpz_gcd_data,
+  "mpz_mul_2exp"    : mpz_mul_2exp_data,
+}
+
+custom = {
+  "mpz_export" : mpz_export_data,
+  "mpz_import" : mpz_export_data,
+  "mpz_sizeinbase" : mpz_sizeinbase_data,
+  "mpz_get_str" : mpz_get_str_data,
+  "mpq_set_str" : mpq_set_str_data,
+  "mpq_get_str" : mpq_get_str_data,
+}
+
+if __name__ == "__main__":
+  #apis = [gmpapi.get_api("mpq_set_str"),]
+  apis = gmpapi.apis
+  for api in apis:
+    tests = gen_args(api)
+    for args in tests:
+      expanded_args = fixup_args(api.name, args)
+      for args in expanded_args:
+        if allow(api.name, args):
+          print("{}|{}".format(api.name, ",".join(args)))
+
diff --git a/polly/lib/External/isl/imath/tests/gmp-compat-test/genpytest.py b/polly/lib/External/isl/imath/tests/gmp-compat-test/genpytest.py
new file mode 100755
index 00000000000..16363980c1e
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/gmp-compat-test/genpytest.py
@@ -0,0 +1,134 @@
+#!/usr/bin/env python3
+
+import gmpapi
+import sys
+
+def print_header(outf):
+  outf.write("""
+#AUTOGENERATED FILE
+import ctypes
+import os
+import sys
+import binascii
+
+verbose = False
+fork    = False
+  """)
+
+def print_cmp(outf):
+  outf.write("""
+def passt(line, name, a_s, b_s):
+    if verbose:
+      print("PASS: {}@{} {} == {}".format(line, name, a_s, b_s))
+    return True
+
+def fail(line, name, a_s, b_s):
+    print("FAIL: {}@{} {} != {}".format(line, name, a_s,b_s))
+    return False
+
+def cstr_eq(line, name, a, b):
+  a_s = a.value.decode("utf-8")
+  b_s = b.value.decode("utf-8")
+
+  if a_s == b_s:
+    return passt(line, name, a_s, b_s)
+  else:
+    return fail(line, name, a_s, b_s)
+
+def bytes_eq(line, name, a, b):
+  if a == b:
+    return passt(line, name, a, b)
+  else:
+    return fail(line, name, a, b)
+
+def run_test(test, line, name, gmp_test_so, imath_test_so, *args):
+  if fork:
+    childpid = os.fork()
+  else:
+    childpid = 0
+  if childpid == 0:
+    eq = test(line, name, gmp_test_so, imath_test_so, *args)
+    if fork:
+      sys.exit(eq != True)
+    else:
+      return eq
+  else:
+    (pid, status) = os.waitpid(childpid, 0)
+    return status == 0
+
+# custom tests
+def test_mpz_export(line, name, gmp_test_so, imath_test_so, *args):
+  # do not use first two args from the test. need to create our own pointers
+  used_args = args[2:]
+  gbuf  = ctypes.create_string_buffer(b'0xdeadbeef'*1024);
+  gsize = ctypes.c_size_t()
+  gout  = ctypes.c_void_p()
+  ibuf  = ctypes.create_string_buffer(b'0xdeadbeef'*1024);
+  isize = ctypes.c_size_t()
+  iout  = ctypes.c_void_p()
+  word_size = args[3].value
+
+  #Test with a NULL pointer
+  gmp_test_so.test_mpz_export(ctypes.byref(gout), None, ctypes.byref(gsize), *used_args)
+  imath_test_so.test_mpz_export(ctypes.byref(iout), None, ctypes.byref(isize), *used_args)
+  gb = ctypes.string_at(gout.value, gsize.value * word_size)
+  ib = ctypes.string_at(iout.value, isize.value * word_size)
+  if not bytes_eq(line, name, gb, ib):
+    return False
+
+  #Test with a provided buffer
+  gmp_test_so.test_mpz_export(ctypes.byref(gout), gbuf, ctypes.byref(gsize), *used_args)
+  imath_test_so.test_mpz_export(ctypes.byref(iout), ibuf, ctypes.byref(isize), *used_args)
+  #print("G:gbuf", gbuf.raw[:gsize.value * word_size])
+  #print("I:ibuf", ibuf.raw[:isize.value * word_size])
+  gb = ctypes.string_at(gout.value, gsize.value * word_size)
+  ib = ctypes.string_at(iout.value, isize.value * word_size)
+  if not bytes_eq(line, name, gb, ib):
+    return False
+
+  return True
+
+def test_mpz_import(line, name, gmp_test_so, imath_test_so, *args):
+  # do not use first two args from the test. need to create our own pointers
+  gout  = ctypes.create_string_buffer(b'0xdeadbeef'*1024);
+  iout  = ctypes.create_string_buffer(b'0xdeadbeef'*1024);
+
+  gmp_test_so.test_mpz_import(gout, *args)
+  imath_test_so.test_mpz_import(iout, *args)
+  #print(gout.raw[:70])
+  #print(iout.raw[:70])
+  return cstr_eq(line, name, gout, iout)
+
+""")
+
+def print_api(name, outf):
+  outf.write("""
+def test_{0}(line, name, gmp_test_so, imath_test_so, *args):
+  gout = ctypes.create_string_buffer(1024*4);
+  iout = ctypes.create_string_buffer(1024*4);
+  gmp_test_so.test_{0}(gout, *args)
+  imath_test_so.test_{0}(iout, *args)
+  eq = cstr_eq(line, name, gout, iout)
+  return eq
+""".format(name))
+
+def print_api_map(outf):
+  outf.write("""
+def get_wrapper(name):
+  test_wrappers = {
+""")
+  for api in gmpapi.apis:
+    outf.write('    "{}" : {},\n'.format(api.name, "test_"+api.name))
+  outf.write("  }\n")
+  outf.write("  return test_wrappers[name]\n")
+
+
+if __name__ == "__main__":
+  outf = sys.stdout
+  print_header(outf)
+  print_cmp(outf)
+  for api in gmpapi.apis:
+    if not api.custom_test:
+      print_api(api.name, outf)
+  print_api_map(outf)
+
diff --git a/polly/lib/External/isl/imath/tests/gmp-compat-test/gmp_custom_test.c b/polly/lib/External/isl/imath/tests/gmp-compat-test/gmp_custom_test.c
new file mode 100644
index 00000000000..1a0739bdfd8
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/gmp-compat-test/gmp_custom_test.c
@@ -0,0 +1,24 @@
+
+void test_mpz_export(char **out, char *rop, size_t *countp, int order, size_t size, int endian, size_t nails, char* mpzstr) {
+  mpz_t op;
+  mpz_init(op);
+  mpz_set_str(op, mpzstr, 10);
+  //printf("%p,%p,%d,%zi,%d,%zi,%s\n", rop, countp, order, size, endian, nails, mpzstr);
+  *out = mpz_export(rop, countp, order, size, endian, nails, op);
+}
+
+void test_mpz_import(char *out, void *unused, size_t count, int order, size_t size, int endian, size_t nails, char* mpzstr) {
+  mpz_t op;
+  mpz_t rop;
+  mpz_init(op);
+  mpz_init(rop);
+  mpz_set_str(op, mpzstr, 10);
+  char *data;
+
+  //printf("%p,%p,%d,%zi,%d,%zi,%s\n", rop, countp, order, size, endian, nails, mpzstr);
+  data = mpz_export(NULL, &count, order, size, endian, nails, op);
+  mpz_import(rop, count, order, size, endian, nails, data);
+  int eq = mpz_cmpabs(op, rop);
+  sprintf(out, "%2d:", eq);
+  mpz_get_str(out+3, 10, rop);
+}
diff --git a/polly/lib/External/isl/imath/tests/gmp-compat-test/gmpapi.py b/polly/lib/External/isl/imath/tests/gmp-compat-test/gmpapi.py
new file mode 100644
index 00000000000..27f94fcced9
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/gmp-compat-test/gmpapi.py
@@ -0,0 +1,103 @@
+#!/usr/bin/env python3
+
+import sys
+
+class CType:
+  def __init__(self, name):
+    self.name = name
+
+  def __str__(self):
+    return self.name
+
+class GMPAPI:
+  def __init__(self, ret_ty, name, *params, out=[0], inout=[], mixed=False,custom=False):
+    self.name = name
+    self.ret_ty = ret_ty
+    self.params = params
+    self.inout_params = inout
+    self.custom_test = custom
+    # most functions with return results dont need extra out params
+    # set mixed to true to check both the return value and an out param
+    if self.ret_ty != void and not mixed:
+      self.out_params = []
+    else:
+      self.out_params = out #param location of the output result
+
+  def is_write_only(self, pos):
+    if pos in self.out_params and pos not in self.inout_params:
+      return True
+    return False
+
+  def __str__(self):
+    return("{} {}({})".format(self.ret_ty, self.name, ",".join(map(str,self.params))))
+
+  def __repr__(self):
+    return str(self)
+
+void  = CType("void")
+voidp = CType("void *")
+charp = CType("char *")
+iint  = CType("int")
+size_t= CType("size_t")
+size_tp= CType("size_t*")
+ilong = CType("long")
+ulong = CType("unsigned long")
+mpz_t = CType("mpz_t")
+mpq_t = CType("mpq_t")
+
+apis = [
+          GMPAPI(void,  "mpz_abs", mpz_t, mpz_t),
+          GMPAPI(void,  "mpz_add", mpz_t, mpz_t, mpz_t),
+          GMPAPI(iint,  "mpz_cmp_si", mpz_t, ilong),
+          GMPAPI(iint,  "mpz_cmpabs", mpz_t, mpz_t),
+          GMPAPI(iint,  "mpz_cmp", mpz_t, mpz_t),
+          GMPAPI(void,  "mpz_mul", mpz_t, mpz_t, mpz_t),
+          GMPAPI(void,  "mpz_neg", mpz_t, mpz_t),
+          GMPAPI(void,  "mpz_set_si", mpz_t, ilong),
+          GMPAPI(void,  "mpz_set", mpz_t, mpz_t),
+          GMPAPI(void,  "mpz_sub", mpz_t, mpz_t, mpz_t),
+          GMPAPI(void,  "mpz_swap", mpz_t, mpz_t, out=[0,1], inout=[0,1]),
+          GMPAPI(iint,  "mpz_sgn", mpz_t),
+          GMPAPI(void,  "mpz_addmul", mpz_t, mpz_t, mpz_t, inout=[0]),
+          GMPAPI(void,  "mpz_divexact", mpz_t, mpz_t, mpz_t),
+          GMPAPI(iint,  "mpz_divisible_p", mpz_t, mpz_t),
+          GMPAPI(void,  "mpz_submul", mpz_t, mpz_t, mpz_t, inout=[0]),
+          GMPAPI(void,  "mpz_set_ui", mpz_t, ulong),
+          GMPAPI(void,  "mpz_add_ui", mpz_t, mpz_t, ulong),
+          GMPAPI(void,  "mpz_divexact_ui", mpz_t, mpz_t, ulong),
+          GMPAPI(void,  "mpz_mul_ui", mpz_t, mpz_t, ulong),
+          GMPAPI(void,  "mpz_pow_ui", mpz_t, mpz_t, ulong),
+          GMPAPI(void,  "mpz_sub_ui", mpz_t, mpz_t, ulong),
+          GMPAPI(void,  "mpz_cdiv_q", mpz_t, mpz_t, mpz_t),
+          GMPAPI(void,  "mpz_fdiv_q", mpz_t, mpz_t, mpz_t),
+          GMPAPI(void,  "mpz_fdiv_r", mpz_t, mpz_t, mpz_t),
+          GMPAPI(void,  "mpz_tdiv_q", mpz_t, mpz_t, mpz_t),
+          GMPAPI(ulong, "mpz_fdiv_q_ui", mpz_t, mpz_t, ulong, out=[0], mixed=True),
+          GMPAPI(ilong, "mpz_get_si", mpz_t),
+          GMPAPI(ulong, "mpz_get_ui", mpz_t),
+          GMPAPI(void,  "mpz_gcd", mpz_t, mpz_t, mpz_t),
+          GMPAPI(void,  "mpz_lcm", mpz_t, mpz_t, mpz_t),
+          GMPAPI(void,  "mpz_mul_2exp", mpz_t, mpz_t, ulong),
+          GMPAPI(void,  "mpz_export", voidp, size_tp, iint, size_t, iint, size_t, mpz_t, custom=True),
+          # The mpz_import signature is a bit of a lie, but it is ok because it is custom
+          GMPAPI(void,  "mpz_import", voidp, size_t,  iint, size_t, iint, size_t, mpz_t, custom=True),
+          GMPAPI(size_t,"mpz_sizeinbase", mpz_t, iint),
+          GMPAPI(charp, "mpz_get_str", charp, iint, mpz_t),
+
+          # mpq functions
+          GMPAPI(iint, "mpq_set_str", mpq_t, charp, iint, out=[0], mixed=True),
+          GMPAPI(void, "mpq_canonicalize", mpq_t, inout=[0]),
+          GMPAPI(iint, "mpq_cmp", mpq_t, mpq_t),
+          GMPAPI(void, "mpq_mul", mpq_t, mpq_t, mpq_t),
+          GMPAPI(void, "mpq_set", mpq_t, mpq_t),
+          GMPAPI(void, "mpq_set_ui", mpq_t, ulong, ulong),
+          GMPAPI(iint, "mpq_sgn", mpq_t),
+          GMPAPI(charp,"mpq_get_str", charp, iint, mpq_t),
+       ]
+
+def get_api(name):
+  for a in apis:
+    if a.name == name:
+      return a
+  raise RuntimeError("Unknown api: {}".format(name))
+
diff --git a/polly/lib/External/isl/imath/tests/gmp-compat-test/imath_custom_test.c b/polly/lib/External/isl/imath/tests/gmp-compat-test/imath_custom_test.c
new file mode 100644
index 00000000000..8772ae8c6a2
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/gmp-compat-test/imath_custom_test.c
@@ -0,0 +1,24 @@
+
+void test_mpz_export(char **out, char *rop, size_t *countp, int order, size_t size, int endian, size_t nails, char* mpzstr) {
+  impz_t op;
+  impz_init(op);
+  impz_set_str(op, mpzstr, 10);
+  //printf("%p,%p,%d,%zi,%d,%zi,%s\n", rop, countp, order, size, endian, nails, mpzstr);
+  *out = impz_export(rop, countp, order, size, endian, nails, op);
+}
+
+void test_mpz_import(char *out, void *unused, size_t count, int order, size_t size, int endian, size_t nails, char* mpzstr) {
+  impz_t op;
+  impz_t rop;
+  impz_init(op);
+  impz_init(rop);
+  impz_set_str(op, mpzstr, 10);
+  char *data;
+
+  //printf("%p,%p,%d,%zi,%d,%zi,%s\n", rop, countp, order, size, endian, nails, mpzstr);
+  data = impz_export(NULL, &count, order, size, endian, nails, op);
+  impz_import(rop, count, order, size, endian, nails, data);
+  int eq = impz_cmpabs(op, rop);
+  sprintf(out, "%2d:", eq);
+  impz_get_str(out+3, 10, rop);
+}
diff --git a/polly/lib/External/isl/imath/tests/gmp-compat-test/runtest b/polly/lib/External/isl/imath/tests/gmp-compat-test/runtest
new file mode 100755
index 00000000000..4a9f5854a5a
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/gmp-compat-test/runtest
@@ -0,0 +1,3 @@
+#!/bin/bash
+export LD_LIBRARY_PATH=.:../..
+python3 runtest.py $@
diff --git a/polly/lib/External/isl/imath/tests/gmp-compat-test/runtest.py b/polly/lib/External/isl/imath/tests/gmp-compat-test/runtest.py
new file mode 100755
index 00000000000..2c1a35f0f2e
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/gmp-compat-test/runtest.py
@@ -0,0 +1,117 @@
+#!/usr/bin/env python3
+
+import ctypes
+import random
+import gmpapi
+import wrappers
+import glob
+import sys
+import os
+from optparse import OptionParser
+from gmpapi import void
+from gmpapi import ilong
+from gmpapi import ulong
+from gmpapi import mpz_t
+from gmpapi import voidp
+from gmpapi import size_t
+from gmpapi import size_tp
+from gmpapi import iint
+from gmpapi import charp
+from gmpapi import mpq_t
+
+def print_failure(line, test):
+  print("FAIL: {}@{}".format(line,test))
+
+
+def run_tests(test_file, options):
+  passes = 0
+  failures = 0
+  fail_lines = []
+  for (line, test) in enumerate(open(test_file), start=1):
+    if test.startswith("#"):
+      continue
+    if options.skip > 0 and line < options.skip:
+      continue
+    name,args = test.split("|")
+    if options.verbose or (options.progress > 0 and line % options.progress == 0):
+      print("TEST: {}@{}".format(line, test), end="")
+    api = gmpapi.get_api(name)
+    wrapper = wrappers.get_wrapper(name)
+    input_args = args.split(",")
+    if len(api.params) != len(input_args):
+      raise RuntimeError("Mismatch in args length: {} != {}".format(len(api.params), len(input_args)))
+
+    call_args = []
+    for i in range(len(api.params)):
+      param = api.params[i]
+      if param == mpz_t:
+        call_args.append(bytes(input_args[i],"utf-8"))
+      elif param == mpq_t:
+        call_args.append(bytes(input_args[i],"utf-8"))
+      elif param == ulong:
+        call_args.append(ctypes.c_ulong(int(input_args[i])))
+      elif param == ilong:
+        call_args.append(ctypes.c_long(int(input_args[i])))
+      elif param == voidp or param == size_tp:
+        call_args.append(ctypes.c_void_p(None))
+      elif param == size_t:
+        call_args.append(ctypes.c_size_t(int(input_args[i])))
+      elif param == iint:
+        call_args.append(ctypes.c_int(int(input_args[i])))
+      # pass null for charp
+      elif param == charp:
+        if input_args[i] == "NULL":
+          call_args.append(ctypes.c_void_p(None))
+        else:
+          call_args.append(bytes(input_args[i],"utf-8"))
+      else:
+        raise RuntimeError("Unknown param type: {}".format(param))
+
+    res = wrappers.run_test(wrapper, line, name, gmp_test_so, imath_test_so, *call_args)
+    if not res:
+      failures += 1
+      print_failure(line,test)
+      fail_lines.append((line,test))
+    else:
+      passes += 1
+  return (passes,failures, fail_lines)
+
+def parse_args():
+  parser = OptionParser()
+  parser.add_option("-f", "--fork", help="fork() before each operation",
+                    action="store_true", default=False)
+  parser.add_option("-v", "--verbose", help="print PASS and FAIL tests",
+                    action="store_true", default=False)
+  parser.add_option("-p", "--progress", help="print progress every N tests ",
+                    metavar="N", type="int", default=0)
+  parser.add_option("-s", "--skip", help="skip to test N",
+                    metavar="N", type="int", default=0)
+  return parser.parse_args()
+
+if __name__ == "__main__":
+  (options, tests) = parse_args()
+  gmp_test_so = ctypes.cdll.LoadLibrary("gmp_test.so")
+  imath_test_so = ctypes.cdll.LoadLibrary("imath_test.so")
+
+  wrappers.verbose = options.verbose
+  wrappers.fork    = options.fork
+
+  total_pass = 0
+  total_fail = 0
+  all_fail_lines = []
+  for test_file in tests:
+    print("Running tests in {}".format(test_file))
+    (passes,failures,fail_lines) = run_tests(test_file, options)
+    print("  Tests: {}. Passes: {}. Failures: {}.".format(passes+failures, passes, failures))
+    total_pass += passes
+    total_fail += failures
+    all_fail_lines += fail_lines
+
+  print("="*70)
+  print("Total")
+  print("  Tests: {}. Passes: {}. Failures: {}.".format(total_pass+total_fail, total_pass, total_fail))
+  if len(all_fail_lines) > 0:
+    print("Failing Tests:")
+    for (line, test) in all_fail_lines:
+      print(test.rstrip())
+
diff --git a/polly/lib/External/isl/imath/tests/init.t b/polly/lib/External/isl/imath/tests/init.t
new file mode 100644
index 00000000000..074f5d61f07
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/init.t
@@ -0,0 +1,12 @@
+#Unsigned integer initialization. Assumes sizeof(long) == 8.
+initu:0,0:0
+initu:0,18446744073709551615:18446744073709551615
+initu:0,567:567
+
+#Signed integer initialization Assumes sizeof(long) == 8.
+initv:0,0:0
+initv:0,-1:-1
+initv:0,9223372036854775807:9223372036854775807
+initv:0,-9223372036854775808:-9223372036854775808
+initv:0,4324:4324
+
diff --git a/polly/lib/External/isl/imath/tests/invmod.t b/polly/lib/External/isl/imath/tests/invmod.t
new file mode 100644
index 00000000000..f969b039d2d
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/invmod.t
@@ -0,0 +1,452 @@
+# Modular inverse tests (no undefined cases)
+
+invmod:-25,37,=1:34
+invmod:-25,37,=2:34
+invmod:-5,11,0:2
+invmod:22153,514,0:383
+invmod:6521,442674856,0:28104185
+invmod:83,8990271,0:2924546
+invmod:643111,73,0:23
+invmod:6756751,5,0:1
+invmod:4310757,655173482,0:578093623
+invmod:67658,37989,0:29720
+invmod:140,456739,0:394753
+invmod:8363,2152,0:1827
+invmod:6067,5554,0:1061
+invmod:134706247,840181,0:713245
+invmod:581,8525,0:3096
+invmod:991357,2885,0:2243
+invmod:9820551,53210,0:5931
+invmod:366,3905684875,0:1846129736
+invmod:4582993,6971170,0:6590887
+invmod:4261,5,0:1
+invmod:2089582769,72370,0:34059
+invmod:1385486874,274843,0:100773
+invmod:2496082853,33877892272,0:25370727693
+invmod:29,77842490059,0:18789566566
+invmod:160735723,837433,0:536522
+invmod:33,92874448,0:39401281
+invmod:17,801910,0:518883
+invmod:1243,47177637,0:28769629
+invmod:5553626,17225545633,0:4551550593
+invmod:493375452,7744903,=1:5021402
+invmod:505,75126821802,0:15917960263
+invmod:8082104357,6160949,0:64657
+invmod:4793,63911,0:44243
+invmod:846,4453189,0:689560
+invmod:5475,596527544653,0:311392826049
+invmod:239,28531,0:9192
+invmod:71044,91829,0:4016
+invmod:2923052473,673,0:362
+invmod:40113371,396,0:23
+invmod:799762079,671587,0:280457
+invmod:6137106,72083,0:61684
+invmod:2663959394312,7297021,0:6396652
+invmod:16314,437519,0:121837
+invmod:5409920801970,224827,0:146917
+invmod:1701531745,565383113204,0:392963934477
+invmod:6702285244447,4818940,0:3886523
+invmod:3853762105973,7986397045899,0:4649165556065
+invmod:83203877,17340640,=2:1474413
+invmod:349,3756,0:1765
+invmod:513237329,6426867498383,0:5161899757325
+invmod:218717486055,4764721,0:3629102
+invmod:5697807566182,23837,0:17052
+invmod:640499,3706953268445,0:862821487954
+invmod:74365895080186,37008968409199,0:35062440514236
+invmod:5447180,17521,0:6484
+invmod:87747,309211711,0:64843296
+invmod:65052052,72604563,0:26029330
+invmod:1200121,9944190318201,=2:5066700447199
+invmod:487321393172,41829673981,=1:18173847806
+invmod:3762709971997,877939264220,=1:575680399733
+invmod:78323027,3830390625030,0:1772587463183
+invmod:1324701179,57116535299,0:4144192403
+invmod:663267333048769,9458708,0:1445501
+invmod:925655,3262585612,0:2494330419
+invmod:809760047,253746243247,0:109619303279
+invmod:6962181619,44237293531835,0:34688460868884
+invmod:32137986591347,450053710,0:207388563
+invmod:171464220392,1821508963,0:85136287
+invmod:4866698,1559814931871,0:875727775988
+invmod:83084876,852948279591,=2:67614759050
+invmod:3087717,3533808897939368,0:3296313572604861
+invmod:685072,47729803,0:7913327
+invmod:3403087579775453,307599045,0:85279967
+invmod:618837304,2026545,0:580654
+invmod:1617142737,5396772102241213,0:309489822606449
+invmod:398279389,695426305815576,0:548019512309869
+invmod:807807859,101277481,0:1799573
+invmod:4399921090,93999639721573,0:23138878622757
+invmod:6198378859,5720311022,0:2650981593
+invmod:86572714,9776668159093959,0:99027417792355
+invmod:1106685402600,781883021,0:593993010
+invmod:788429239716,54846845,0:23729486
+invmod:26682164884969,84892199507532,0:38639429314105
+invmod:490223409,87336215488172224,0:60928799099783953
+invmod:11687153,155174158,0:112979329
+invmod:81995673521,1680666567,0:4778198
+invmod:36278417230108,2563749243683,0:97972077091
+invmod:81100161947491318,626490251,0:10036586
+invmod:84353708437,789691774074278,0:384046667492909
+invmod:4784222527,220196213,0:137693182
+invmod:656338347,17114903662,0:2025727201
+invmod:205944554689,55544669634,0:45790392391
+invmod:58311406949710,684448841,0:398362627
+invmod:2984553893,39836508385650164,0:23063415225053961
+invmod:1226833387651,868339671575144482,0:861301708657625181
+invmod:7850491405,3995878481964,0:222583321669
+invmod:60187945992,93467753554255,0:39782639687763
+invmod:26790048589,54188485741,0:37088225013
+invmod:2047388699,2277281699,0:1326800159
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diff --git a/polly/lib/External/isl/imath/tests/lcm.t b/polly/lib/External/isl/imath/tests/lcm.t
new file mode 100644
index 00000000000..14eb20b514a
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/lcm.t
@@ -0,0 +1,780 @@
+# Least common multiple tests
+
+# Basic tests contributed by Hal Finkel 
+lcm:2,3,0:6
+lcm:2,3,=1:6
+lcm:2,=1,0:2
+lcm:123456,34,0:2098752
+lcm:123456,34,=1:2098752
+lcm:123456,34,=2:2098752
+lcm:-568,34,0:-9656
+lcm:25,=1,=1:25
+
+# More involved tests extracted from gcd.t
+lcm:0,0,0:$MP_UNDEF
+lcm:0,0,=1:$MP_UNDEF
+lcm:0,0,=2:$MP_UNDEF
+lcm:0,=1,0:$MP_UNDEF
+lcm:0,=1,99999:$MP_UNDEF
+
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+lcm:32557663,995533,0:32412227919379
+lcm:641,701,0:449341
+lcm:290748,-233,0:-67744284
+lcm:290748,-233,=1:-67744284
+lcm:290748,-233,=2:-67744284
+lcm:-95,4770,0:-90630
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diff --git a/polly/lib/External/isl/imath/tests/mod.t b/polly/lib/External/isl/imath/tests/mod.t
new file mode 100644
index 00000000000..7ca2635d1e9
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/mod.t
@@ -0,0 +1,754 @@
+# Modulo (remainder) tests
+
+mod:-52,67,0:15
+mod:-54090198,1,=1:0
+mod:-76,427,0:351
+mod:-84,576007732,=2:576007648
+mod:0,9268611,=2:0
+mod:0,807,0:0
+mod:8268,487413,0:8268
+mod:1146574,836443419,=1:1146574
+mod:68387,38,0:25
+mod:822,3971,0:822
+mod:6236,52183125,0:6236
+mod:3339,8463,0:3339
+mod:-6911,492807317,0:492800406
+mod:38291,7924,0:6595
+mod:768,7974,0:768
+mod:-753573,5272420932,0:5271667359
+mod:422,509441099,0:422
+mod:930,3648323446,0:930
+mod:2427,67,0:15
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diff --git a/polly/lib/External/isl/imath/tests/mul.t b/polly/lib/External/isl/imath/tests/mul.t
new file mode 100644
index 00000000000..b113018c569
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/mul.t
@@ -0,0 +1,864 @@
+## Multiplication tests
+## Generated and verified with GNU bc 1.05
+
+# Regression for zero-sign bug
+mul:0,5000,0:0
+mul:0,10235,=1:0
+mul:0,-58382939939929385885,=2:0
+
+# Basic functionality tests
+mul:0,0,0:0
+mul:100000000000000000,0,=1:0
+mul:102328632557663,995533253464107,0:101871556492663391138178301941
+mul:1099511627775,-65839281,=2:-72391055023845629775
+mul:424417695707616156,78196371461084006246,0:33187923788210071918733291062874510376
+mul:8785533773682081,509516924811454802,0:4476378151193669633826349848802962
+mul:3149810015499242805866,671806182669824281823,=1:2116061842647726364810477474851492861573718
+mul:7004291671016071522932056033,291108610609315306035746008,=2:2039009616651887991790118373228449729743842556412066264
+mul:62479450738832647,8434970263250341371151,0:527012309046267948145296885851102766697
+mul:9472845009207216089,8603477481125780254532680,=1:81499408718909018140819102866811806472288520
+mul:8704164997261749,601100642928524,=2:5232079176009991665860326228476
+mul:-18665805686013,-2922920804055208139127600955,0:54558671764099393911694170691186688942415
+mul:44144326678357949,-81195051773905850760614924,0:-3584300890173486959579517806592053823430876
+mul:9685579959286825,5953064636346354,0:57658883538135357049439528986050
+mul:-7,328492456,0:-2299447192
+mul:16522181,930,0:15365628330
+mul:844378,24,0:20265072
+mul:209,3,0:627
+mul:70522336,34102,0:2404952702272
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+mul:641,701,0:449341
+mul:290748,-233,0:-67744284
+mul:-95,4770,0:-453150
+mul:6839092,7,0:47873644
+mul:787590783,-6789049,0:-5346992417735367
+mul:721,15845,0:11424245
+mul:-37028,-949534667,0:35159369649676
+mul:618,-3,0:-1854
+mul:16657,-544851,0:-9075583107
+mul:28271,6,0:169626
+mul:-35357673,5207718,0:-184132790120214
+mul:99,-565,0:-55935
+mul:9311525,-5,0:-46557625
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+mul:-5302,9,0:-47718
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+mul:43040,2766705,0:119078983200
+mul:863478587,63,0:54399150981
+mul:8874055124,10,0:88740551240
+mul:8204,-89955,0:-737990820
+mul:3303887262,3,0:9911661786
+mul:75504,-62,0:-4681248
+mul:436,2123372,0:925790192
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+mul:-767589,-991593,0:761135879277
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+mul:987205647053507741802554416690401353287928887038,99876648283232575129630115679242229378848334523759990898,0:98598791193984227490766355696552554241176001533681327979309043773446027278571920948873010940294150180124
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+mul:8086945629611784458016532027184114429962417844094,915144274243273504802165174453120299631299876477769004475,0:7400721989055888996136233798580013159529233408791044035035967083245349723159601722480356017380766138320650
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+mul:45518095176552517046254533808850801860612235127698055,35562211507032265793347677842985571735031362689772443740,0:1618724128065785795744673208584473680751616875603258142963030074995341676160450789993590567976668358194925700
+mul:9347651927739244225461883666785850681796998039204,-49878021756813856103391637115956958358002286487531,0:-466242386226901006946356389028134676636414341103795172547644257145552025595817036151721871395165324
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+mul:1070320750308710183653165839089007570013512879851785,1457798154372592051774290334996812066679847800239,0:1560311614386725640264706359282306460222595994684719021358994107802507336028163090509939936607576615
+mul:4878833860261622243309660185570248437064542060170266009303,2713399212910344466363698742990679778889654409261764714,0:13238223956354223315835115083131058404743853065832322244195207641759173433600815039556809272534689440156121134342
+mul:-153727669883775746511963322850970227020812560900211022,967696702028565741720227898697068313091723784217356354,0:-148761759157065858120367305796775921605388922829212089475409575708151324377711363197839787691020073572533788
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+mul:6147134348286513705872901614503400707429078163757093529,477008107151085089956583728568535862114663515203848020888,0:2932232919879568942454061197378668098753424444460385986524578717121657294266275306745807650523035121257761633752
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+mul:1973284924569393016522181930166284437824113120930085705223363410232863255766399553325346410701486329074823374249547700471683909277697787590783678904983357211584574690370289495346677331618312561665754485123402827162476353576735207718451299565526093115255556962215997764515429748902159033904867393928929335369918394864191487555087825530530292436823694624357846430402766705368186347858763649288740551240105434820489955559033038872623623075504620415436212337263593361227009430895404467373404212231777284568356947010238293550872942631550054733886261493387307990754063500729850132294968984927109971586336754260617576297578,5023215555039217191451486096670046812323372237867852342682903508090873804207633443539517026163606065770027196694503291009850587117226026773591715017125540836707619882023150680605967194554434367524090941938814045780235333982097410791452776449214446635671357519707417878752761895933104747199582531838714305022614904134506738511247696618513698871318914624938137416674596881724436242382943205716649496664131501091102879563857843031208992280893614063087964394912022421136928015233318011144243860597237613050611024432789921991289922378761307428111094457098264457145099945444765267932710530788173695233378407323077034767182,0:9912235527621363370177765461045903296757346464251250997934314384765381968455767811674327750035883028560327843635490083086230089733649800500491928845965808260177110097000550870708488183060088779215210481900113545281854124601490142211180660492441361405426298053896802091578235012988730731854809702026093664780523793649405978324745399501432045298507869343282835244908856693641856464415460033802080004246708824900236489959461486545994943320360059459917374730418136788819890973086342429995201653932338829740545718352667527142958499428354077966431655595175455785104618218647409782931121099857213491801531717606789462812743975739913830932552436532349361163108142395102143904119547756730321871557900417573185916761969667939077354954258105548072267528998411702020631497735552158986372677452750936969993423245268272976429454770189221077520557005286622417163333812030510164503314440581497599501417407932697447376958591635071781976491704916693799964449632893911830806101578411062044010045678256608720149155356874608716245388282176400921764516344436742619757576145128777884235797616593795005421758664915151768829194432974835591116318659671888399376705850603804666493415116788205809451378581533592708248188469927900316495904963905315042780485196
+mul:5693930165221819,3016628443782411,0:17176471693318822194099185625609
+mul:19732849245693930165,22181930166284437824,0:437712683949801303830876080256140560960
+
+# Power of two multiplication
+mulp2:7328492456,15,0:240140040798208
+mulp2:9301,45,0:327249844798226432
+mulp2:221819,104,0:4499023815872409881769555333012783104
+mulp2:1,3,0:8
+mulp2:28443782,108,0:9230534995215592279593651121258540040192
+mulp2:131,76,0:9898080148094776367906816
+mulp2:93,36,0:6390911336448
+mulp2:85,4,0:1360
+mulp2:522,113,0:5420757920310360044185838167705780224
+mulp2:634102,29,0:340430919041024
+mulp2:86325,31,0:185381525913600
+mulp2:6639955332,101,0:16834286724196864462774406457915724529664
+mulp2:4641070,109,0:3012226647655029057743003824831688867840
+mulp2:86329074,43,0:759358565425078075392
+mulp2:337424,112,0:1752005575194255677729274193390761672704
+mulp2:477004716,31,0:1024359827628883968
+mulp2:90927769,62,0:419330320984095270608306176
+mulp2:875907836789,55,0:31557905658990701020815818752
+mulp2:983357211,56,0:70858354704507043363946496
+mulp2:4574690370289,41,0:10059850511215301155094528
+mulp2:534667733161831,94,0:10590185513519716577933182844778629828706304
+mulp2:61665754485,104,0:1250730090982990067823267062394063169781760
+mulp2:34028271,97,0:5391994769734856490537202784887898112
+mulp2:47635357,28,0:12787018778017792
+mulp2:35207718451299,38,0:9677823956166718317920256
+mulp2:5526093115255,15,0:181079019200675840
+mulp2:696221599776,31,0:1495124500903360462848
+mulp2:154297489021,86,0:11938189976268266978253598864697196544
+mulp2:3390486739392892,71,0:8005560469361491387429364912321724416
+mulp2:35369918394,31,0:75956321382209421312
+mulp2:41914875550878,28,0:11251438731683187130368
+mulp2:5305302924368,42,0:23332969016866530784182272
+mulp2:694624357846,102,0:3522163936626521236224897501281415419920384
+mulp2:40276670536,50,0:45347499604413458849726464
+mulp2:8634785876,54,0:155550473814310487519657984
+mulp2:492887405512401,103,0:4998472123541838661247665060074167267342942208
+
+
+
+# Multiplication by small integers
+
+mulv:39,34165,0:1332435
+mulv:1331169483,45488,0:60552237442704
+mulv:967,3873,0:3745191
+mulv:4609944,12208,0:56278196352
+mulv:-9187510646,-19674,0:180755084449404
+mulv:-9187510646,-19674,=1:180755084449404
+mulv:-705904,-32646,0:23044941984
+mulv:-494,61105,0:-30185870
+mulv:31884502736,41742,0:1330922913206112
+mulv:124758920,14754,0:1840693105680
+mulv:1578359,-34693,0:-54758008787
+mulv:66733360927656,-31832,0:-2124256345049145792
+mulv:25747369,-31317,0:-806330354973
+mulv:241723337947,63112,0:15255643304511064
+mulv:7902551,11307,0:89354144157
+mulv:-9354271604,39887,0:-373113831468748
+mulv:952191019,55813,0:53144637343447
+mulv:809248646915194,9772,0:7907977777655275768
+mulv:-2633237043,16406,0:-43200886927458
+mulv:764186567812009651,1311,0:1001848590401544652461
+mulv:866541923,29701,0:25737161655023
+mulv:-834567728673833890,4681,0:-3906611537922216439090
+mulv:332307463075,323,0:107335310573225
+mulv:27691642758,54832,0:1518388155706656
+mulv:2202473525085705,16772,0:36939885962737444260
+mulv:2202473525085705,16772,=1:36939885962737444260
+mulv:-16583738845087,37131,0:-615770807056925397
+mulv:9196926997210,10181,0:93633913758595010
+mulv:514078180671038315,62002,0:31873875357965717606630
+mulv:514078180671038315,62002,=1:31873875357965717606630
+mulv:-3412120508850357,36908,0:-125934543740648976156
+mulv:549214983476935,10691,0:5871657388351912085
+mulv:651367312269732,44154,0:28760472305957746728
+mulv:-9584205879196113,22814,0:-218654072927980121982
+mulv:987433915942054416876759,34616,0:34181012434250155694605889544
+mulv:98036280382191791,23951,0:2348066951433875586241
+mulv:80662726244584753,2712,0:218757313575313850136
+mulv:47866183268822904587301,56393,0:2699317673078730058391665293
+mulv:-6912574313961965048871386,30037,0:-207632994668475544172949821282
+mulv:-6912574313961965048871386,30037,=1:-207632994668475544172949821282
+mulv:727313511777411140493,-48518,0:-35287796964416433714439374
+mulv:465780278683330104360026,58793,0:27384619924629026825639008618
+mulv:9089195999415213350558115,62437,0:567502130615487675968797026255
+mulv:9089195999415213350558115,62437,=1:567502130615487675968797026255
+mulv:1956713589673152168482712,-63098,0:-123464714081196555526922161776
diff --git a/polly/lib/External/isl/imath/tests/neg.t b/polly/lib/External/isl/imath/tests/neg.t
new file mode 100644
index 00000000000..6feabb3d71c
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/neg.t
@@ -0,0 +1,9 @@
+#
+# Negation tests
+#
+
+neg:0,0:0
+neg:0,=1:0
+neg:-0,-0:-0
+neg:-0,=1:0
+neg:-1050000000000000000,58392:1050000000000000000
diff --git a/polly/lib/External/isl/imath/tests/pi1024.txt b/polly/lib/External/isl/imath/tests/pi1024.txt
new file mode 100644
index 00000000000..204ad8b5902
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/pi1024.txt
@@ -0,0 +1 @@
+3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788
\ No newline at end of file
diff --git a/polly/lib/External/isl/imath/tests/pi1500-10.txt b/polly/lib/External/isl/imath/tests/pi1500-10.txt
new file mode 100644
index 00000000000..aa774a008fb
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/pi1500-10.txt
@@ -0,0 +1 @@
+3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955
\ No newline at end of file
diff --git a/polly/lib/External/isl/imath/tests/pi1698-16.txt b/polly/lib/External/isl/imath/tests/pi1698-16.txt
new file mode 100644
index 00000000000..7768ad3fb95
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/pi1698-16.txt
@@ -0,0 +1 @@
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
\ No newline at end of file
diff --git a/polly/lib/External/isl/imath/tests/qadd.t b/polly/lib/External/isl/imath/tests/qadd.t
new file mode 100644
index 00000000000..5a98e52d89f
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/qadd.t
@@ -0,0 +1,803 @@
+## Rational addition tests
+## Generated and verified in PLT Scheme (see imath-test.scm)
+
+qadd:3/13356872859954085399,9/62913230326443414581,6/70815656362754309351:308951546718917012334/840324018679312348674627422530435802819
+qadd:-1/10707768797352318399,-7/98320256665411314243,1/73359800109931088847:-57758212748959181012/350930192156520858413194622860526552319
+qadd:3/8871935578221811012,=1,-3/4866674049828142891:3/4435967789110905506
+qadd:9/4322444789227983625,0,-1/968244430692371869:9/4322444789227983625
+qadd:2/225379267690226671,1/650264369025696376,1/34848592015687152:217986857963088489/20936586756579823441422616437606328
+qadd:-3/68292234169742612,0,=2:-3/68292234169742612
+qadd:-6/86190045205685135,4/67524620320880421,7/25266746450126878:-60387541102541986/5819950077953408572132977712241835
+qadd:3/32384565517809325,1/12077081878824172,-5/29092772826669868:68615811154281841/391111049368729137348856297003900
+qadd:7/8696576917665055,-1/7581339998976754,3/1629599527000648:44372803075172223/65931706440072050529006903131470
+qadd:2/2271361357023875,3/7926992193500824,=2:22668068458073273/18005063745747695119878000173000
+qadd:1/149242689846961,=1,-8/422836348382141:2/149242689846961
+qadd:1/8658078704679,-3/951317712555992,2/558713214761595:925343476441955/8236583628464972469619886568
+qadd:1/16676404623347,-5/33326850326497,9/43833507346994:-7150738970034/79396005838055435322703637
+qadd:9/79221016041680,=1,=1:9/39610508020840
+qadd:-2/2334176395033,=1,3/4196164230632:-4/2334176395033
+qadd:6/8258476677649,3/2022168438715,7/2650901397286:36908440665237/16700030889405718666781035
+qadd:2/89237876621,3/593210741974,0:1454135113811/52936867002527677989854
+qadd:-1/86244958313,4/307620567537,7/284049788408:37359265715/26530723023449966085081
+qadd:7/30268895338,4/83657377701,8/72832717953:706677225259/2532216409883104057938
+qadd:-2/16397609979,9/46846025410,=1:53886438991/768162853739503566390
+qadd:0,1/192040267,0:1/192040267
+qadd:9/9089766539,=1,0:18/9089766539
+qadd:2/331584881,4/272560309,=2:1871460142/90376877625088229
+qadd:5/832601742,-3/639383129,=2:99872917/76050215287258674
+qadd:9/36166996,3/22041145,1/5580628:306871293/797162003050420
+qadd:8/17901539,=1,3/89155358:16/17901539
+qadd:5/5040389,4/3388433,-1/7168517:37103721/17079020420437
+qadd:-3/1713461,=1,1/4488431:-6/1713461
+qadd:5/516179,-1/59578,=1:-218289/30752912462
+qadd:-1/244201,2/201199,9/174458:287203/49132996999
+qadd:-1/41212,3/27175,=2:96461/1119936100
+qadd:-3/13547,4/29889,0:-35479/404906283
+qadd:2/1907,0,-1/9340:2/1907
+qadd:4/1341,4/9919,=2:45040/13301379
+qadd:3/160,-1/799,1/705:2237/127840
+qadd:4/85,2/265,1/5:246/4505
+qadd:-2/59,3/89,=2:-1/5251
+qadd:-7/55,9/74,-4/33:-23/4070
+qadd:1,1/2,1:3/2
+qadd:2,4/5,-9/7:14/5
+qadd:53/70893980658822810696,=1,9/9193530480905424214:53/35446990329411405348
+qadd:86/71933586324624884067,0,7/18408383381035284932:86/71933586324624884067
+qadd:35/378921420794267711,24/6551623471410322667,=2:238400935598423718409/2482550474295871845608778066489505237
+qadd:36/3722553911462108789,7/3244440092917636530,65/8189167080175627777:142857720725269676603/12077603158395035547949185323620462170
+qadd:74/204211327800791583,=1,-4/261650327849481535:148/204211327800791583
+qadd:14/190596241295804639,=1,5/7319468293717642:28/190596241295804639
+qadd:11/20832946214725250,=1,34/81808092645053823:11/10416473107362625
+qadd:-2/36273895397703103,21/88730391521225765,-11/10905700152968561:584291020309313633/3218596940638185709301052104048795
+qadd:39/8019234291976340,-41/686183563094027,47/1842385817838468:-302027447010362887/5502666759754131835671079321180
+qadd:27/3512663969871511,=1,-21/6043832798535571:54/3512663969871511
+qadd:-5/2985747324014,45/607698145828948,29/712218076096765:-1452066049782055/907216556358525513388378636
+qadd:19/46107312122174,92/20099305290153,=2:4623759515752915/926724942451947423255152622
+qadd:17/10280669487307,=1,29/71287679743652:34/10280669487307
+qadd:7/48006164806589,=1,=2:14/48006164806589
+qadd:-21/5174158635853,55/5726484671487,45/4364076509257:164322546870688/29629740116054290765023411
+qadd:2/515061581267,81/3885600123095,62/1777317903213:49491188328817/2001323343572560546061365
+qadd:59/587566787929,17/530520857961,37/408187963651:41289366014492/311716436441482018352769
+qadd:87/16848388346,=1,-29/793777748404:87/8424194173
+qadd:13/40356316605,-16/35824973661,12/36053750683:-59992136029/481921326476367313635
+qadd:62/58964244603,=1,-41/7476886656:124/58964244603
+qadd:49/2379119051,-9/250852979,16/749619475:-9120275488/596809101339002929
+qadd:35/1381737807,13/332249103,=1:365324816/5667668480949841
+qadd:-29/262819147,-44/212418643,12/429833273:-17724183115/55827686560157521
+qadd:14/189466127,31/78740185,-7/602930208:6975812527/14918597891213495
+qadd:19/15866903,3/8919802,67/8901141:217076947/141529633113206
+qadd:8/44756143,33/6935255,44/26346673:1532434759/310395264521465
+qadd:9/9886228,0,=2:9/9886228
+qadd:-31/5061520,-19/3104202,43/3631883:-96199571/7855990253520
+qadd:-31/302786,=1,31/132480:-31/151393
+qadd:10/50561,56/854269,-17/764873:11374106/43192694909
+qadd:-12/40781,=1,83/13428:-24/40781
+qadd:-7/5458,17/6090,1/14614:12539/8309805
+qadd:85/8919,25/1261,=2:330160/11246859
+qadd:25/2609,3/832,-17/7121:28627/2170688
+qadd:29/910,=1,-19/91:29/455
+qadd:41/715,2/15,64/97:409/2145
+qadd:22/3,-3/31,22:673/93
+qadd:35/29,=1,4:70/29
+qadd:-14,9/2,-20:-19/2
+qadd:-60,=1,43/2:-120
+qadd:375/18696391582109365451,-131/32949770573031503434,166/56750232802998421883:9906936667630486913669/616041813174061083543307811398267458734
+qadd:615/80348516296708248277,=1,=2:1230/80348516296708248277
+qadd:-150/1530888833706527119,136/9537020524274172927,=1:-1222352197257038250866/14600118227441300399114525155221107313
+qadd:89/1394353050192727085,=1,819/2322159924711043043:178/1394353050192727085
+qadd:62/46458468892037221,813/119745041246081149,-121/356270572095961665:45194927766483291911/5563171273706775007904337294446929
+qadd:809/639945078105944091,400/118384885068174771,-713/209608760084814514:117250467754177008713/25253274840505439393971462380909387
+qadd:1/576267190309129,=1,207/61006148560056413:2/576267190309129
+qadd:-515/3615716945775737,83/23367471127892459,454/27760785001119423:-11734143124365230214/84490161337046038094355667467283
+qadd:73/835205561268243,353/4576154733864164,=1:5197412055370031/31587023826011126162743214412
+qadd:159/1356415561313968,=1,610/1188572111871663:159/678207780656984
+qadd:17/27804219463543,=1,480/45536329454957:34/27804219463543
+qadd:-313/103858729128189,97/192554775266034,53/819834763511200:-16731782644278103/6666164748898110732454544142
+qadd:-497/99654168988628,739/9692326546155,144/61549637278907:68827344589157057/965880747523495532712125340
+qadd:-215/35851399647701,309/1276360474687,-79/2088545670863:10803664989081904/45759309472532992928244587
+qadd:-121/9491608537990,-109/8274027606869,1/42153880273:-2035742671072059/78533831076922767571453310
+qadd:513/9899067246839,413/3814421378956,43/320124503991:6045112940348935/37759213738065792812120084
+qadd:371/929964631846,19/164386836347,-347/394487184823:78656844289811/152873943743766506506562
+qadd:149/158337939815,305/905893441427,-73/931434015171:183271194416198/143437301207471553716005
+qadd:601/39399444858,=1,100/29792912589:601/19699722429
+qadd:7/24017912572,-55/47728203251,=2:-986887768703/1146331812901164171572
+qadd:497/5699936840,-145/2007901849,=1:171436377153/11444913720219217160
+qadd:-307/2244804442,112/2757559041,399/2539679446:-595152528083/6190180784314060122
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diff --git a/polly/lib/External/isl/imath/tests/qaddz.t b/polly/lib/External/isl/imath/tests/qaddz.t
new file mode 100644
index 00000000000..584ffcc5ebf
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/qaddz.t
@@ -0,0 +1,800 @@
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+qaddz:-93473235956409121974/3521849589372767293,14176089406854262878,86358338521892655661/8513932339338708236:49926054656441321879828060404133327280/3521849589372767293
+qaddz:-75776234900005146680/1934924867434433667,58779385165176736174,=1:113733694048607113038507934157157223378/1934924867434433667
+qaddz:10754058131593805677/10119651272346505,54504300977159159643,95585489620287220450/323814423064533371:551564518731865556391884682101903392/10119651272346505
+qaddz:-26320122648161591867/118955488311945989,36439044282055136810,19850642639958356517/200120485264504867:4334624306192492122600675236564163223/118955488311945989
+qaddz:21672449029428526941/15150265559827046,99277026381166669676,18730584597709492865/69655516304512819:1504073313664630491063919887801384037/15150265559827046
+qaddz:85792494086863781236/50064798708777621,-76374367048497097864,-76196863606569205097/60267456232905085:-3823667312793305504369462034586320308/50064798708777621
+qaddz:82198870205394916580/3448850359841959,92122854365946272671,-9499119469817233118/381994720733569:317717939429662268414058202075719069/3448850359841959
+qaddz:6940206776533138781/442725248202698,52972508975617224025,5617429131950608687/3366289783208936:23452267184149790024086119808558231/442725248202698
+qaddz:55599986966946799564/351405696052723,81353132501416252986,2145222144601366685/198829739629804:28587954152729636171116544308980442/351405696052723
+qaddz:94458271410973778847/361353862043500,54761990849557223858,-14653194369734462194/40666523642913:19788456886678404891313989807601847/361353862043500
+qaddz:-84180062330049702321/10106997571532,34216878817519062264,43542034768301689369/65407469845390:345829911113985714017035852166127/10106997571532
+qaddz:-24742695506396929785/17404029796744,41450961372253719353,=1:721413766826363551856884212256847/17404029796744
+qaddz:34674970696505269672/434120698329,90706045368333568168,4047354904123103766/5628094003009:39377371757997599606800990460944/434120698329
+qaddz:16761364796027525373/5854257789796,65993384007951488766,2006868112336026267/163736160281:386342282403565536077626910957109/5854257789796
+qaddz:13213233170358417707/28953389411,-84133659565012740924,38441120866807154861/611830462605:-2435954607945105525978669538057/28953389411
+qaddz:-64998396976150810447/657173799966,68741723486105419807,29966320909708913126/113185344833:45175259639510928939657721516115/657173799966
+qaddz:-75993621747015299032/49530111977,64950591439466856818,=1:3217010066893177413628750610154/49530111977
+qaddz:-20552895783534231165/77134912822,22987772657101552827,59095027841080059905/57879333453:1773159839856930680727938416629/77134912822
+qaddz:23575585295873866365/354640952,10572907047388947403,63974647548865514804/2902818571:3749585844269110717151714021/354640952
+qaddz:-3423087053862335915/3563944811,14776086984480635327,87473419453510292251/150143797:52661158531801310749782602282/3563944811
+qaddz:20175953575026075815/258856713,-707977887059042238,-88657359408605704674/445245889:-183264808544835335630767879/258856713
+qaddz:5641506401094150499/81605428,27796153257209226730,-31152461920419370958/183221007:2268316988949659433944840939/81605428
+qaddz:64291966735347840169/1472255,-37813006182818155116,58751143690989021203/3131508:-55670323125718207612466411/1472255
+qaddz:-40829665387689228845/40252968,65646528058726294873,48548143291840415697/19567973:2642467552429346280592204219/40252968
+qaddz:84643493739932974309/2975141,-85925500041436529739,29931128280261616739/3657301:-255640393475285778591243890/2975141
+qaddz:49533459783522540811/7076331,28143218875743097472,=1:199150781703665812199676043/7076331
+qaddz:45584750001594149227/984336,41115590339049176247,15975001635513275306/154305:40471601316728311544416219/984336
+qaddz:6418669402794912089/164220,73293587179171456413,-2508617797150506151/19201:12036279305232939367054949/164220
+qaddz:9535512106815022878/7681,28863162607332458363,16912585483283156773/19761:221707487499027427709081/7681
+qaddz:31040563813369589503/9684,-10330625966877227525,58520741913037533005/66343:-100010741299425701762597/9684
+qaddz:25406696906821168845/637,-96989536360380372540,89275628731812608081/8978:-61756927964655476139135/637
+qaddz:12530150187862657939/1219,40341242697614742420,=1:49188504998580233667919/1219
+qaddz:80569150582987599935/2,-15179790697208915754,82993038152425243161/355:50209569188569768427/2
+qaddz:36661516357580891535/19,-66347692158873159944,2979405734116612133/33:-1223944634661009147401/19
+qaddz:-77628572518590925665/92,9553267737685350880,69073840554559304981/47:801272059348461355295/92
+qaddz:94231430591775515069/83,1462408193524266778,-64327046129355071262/11:215611310654289657643/83
+qaddz:-6524291256793749081/5,46538712122784125243,=1:226169269357126877134/5
+qaddz:1240850785353652095/2,42205460810156144332,=1:85651772405665940759/2
diff --git a/polly/lib/External/isl/imath/tests/qdiv.t b/polly/lib/External/isl/imath/tests/qdiv.t
new file mode 100644
index 00000000000..f0dd12f9996
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/qdiv.t
@@ -0,0 +1,803 @@
+## Rational division tests
+## Generated and verified in PLT Scheme (see imath-test.scm)
+
+qdiv:3/8827981081671378268,0,3/3926508931624275206:$MP_UNDEF
+qdiv:-3/474373049650626538,-3/30777320180374251050,9/1226655408788630648:15388660090187125525/237186524825313269
+qdiv:1/1306590006879123716,=1,0:1
+qdiv:4/687201385528545547,8/9869825196514386591,-3/5975430289097816269:9869825196514386591/1374402771057091094
+qdiv:1/61423611473194332,1/15455670029280858,9/557484359857054244:2575945004880143/10237268578865722
+qdiv:-5/979333377476953559,1/99869524971840393,=1:-499347624859201965/979333377476953559
+qdiv:-7/30691138649930912,1/11634392715914207,3/17315717048169457:-81440749011399449/30691138649930912
+qdiv:3/20309884568576860,5/77811383120554786,-1/10721697997657271:116717074680832179/50774711421442150
+qdiv:9/5449666303362005,-5/2861847982830984,1/1826459823357042:-25756631845478856/27248331516810025
+qdiv:7/8847841671787170,2/1586192862907947,=1:3701116680118543/5898561114524780
+qdiv:8/289082769132563,=1,0:1
+qdiv:4/404037054778449,-3/231560563043393,7/849880896656765:-926242252173572/1212111164335347
+qdiv:1/48469516794680,-1/28088185111303,2/21754840126411:-28088185111303/48469516794680
+qdiv:8/50407054689999,=1,=1:1
+qdiv:-5/3164747219031,=1,1/3512888487599:1
+qdiv:5/5486024696808,3/3947571370957,4/5612399333251:19737856854785/16458074090424
+qdiv:0,1/41619742838,3/808926099199:0
+qdiv:-7/908640056265,1/159999865095,4/624715801203:-74666603711/60576003751
+qdiv:3/31854541511,2/14382200067,7/81824133981:43146600201/63709083022
+qdiv:-2/53726093607,2/89209379771,=2:-2175838531/1310392527
+qdiv:-1/6605722316,1/2540812616,9/2982865172:-635203154/1651430579
+qdiv:9/9991323937,=1,0:1
+qdiv:3/179688887,9/702152629,=1:702152629/539066661
+qdiv:3/404963252,-5/696394373,=2:-2089183119/2024816260
+qdiv:1/6333273,1/37207089,2/31767299:4134121/703697
+qdiv:3/90875674,=1,3/86752486:1
+qdiv:5/7636188,2/3416475,-8/1655405:5694125/5090792
+qdiv:-2/244339,=1,9/4733791:1
+qdiv:1/107213,-1/229854,=1:-7926/3697
+qdiv:-1/191856,2/941299,4/169459:-941299/383712
+qdiv:-1/11772,1/30223,=2:-30223/11772
+qdiv:-7/11960,6/95509,8/90341:-668563/71760
+qdiv:7/6422,4/4147,0:2233/1976
+qdiv:8/9005,1/5326,=2:42608/9005
+qdiv:1/665,-4/217,3/278:-31/380
+qdiv:3/140,1/227,1/100:681/140
+qdiv:-5/27,2/11,=2:-55/54
+qdiv:-7/26,1/5,-7/78:-35/26
+qdiv:1,7/2,8/9:2/7
+qdiv:1,1/4,-5/2:4
+qdiv:55/20700362000632320869,=1,69/80225473900126748965:1
+qdiv:14/3091014050673014971,-28/81380133061144194395,84/40859471274765910171:-81380133061144194395/6182028101346029942
+qdiv:1/314668662662881104,98/4475605721586782695,=1:4475605721586782695/30837528940962348192
+qdiv:3/1845567143863582553,11/786490044553073944,18/1855454337971843903:2359470133659221832/20301238582499408083
+qdiv:63/385333292496806971,=1,-1/32472264092720881:1
+qdiv:3/149509460292616220,=1,89/655644037836531409:1
+qdiv:59/60850170481238094,=1,31/16962204174702505:1
+qdiv:-21/11323738586240236,89/72739976921345915,-68/47763887738600551:-1527539515348264215/1007812734175381004
+qdiv:42/8561938043647535,-93/6809871846010066,17/9985561445050651:-95338205844140924/265420079353073585
+qdiv:74/824445644271491,=1,-10/462004715375339:1
+qdiv:-91/158705946232261,66/377268980338471,1/180572218716832:-34331477210800861/10474592451329226
+qdiv:13/294374634105374,17/137508734190942,=2:893806772241123/2502184389895679
+qdiv:60/52806122696123,=1,25/35647894318199:1
+qdiv:2/70084255188433,=1,=1:1
+qdiv:-11/522366309775,29/712593081308,89/4408829724580:-7838523894388/15148622983475
+qdiv:84/102632540711,61/53857049409,15/1404297586412:4523992150356/6260584983371
+qdiv:70/902531223717,92/698425518183,18/41071684813:8148297712135/13838812096994
+qdiv:9/100549718122,=1,-29/262025912406:1
+qdiv:5/78979980317,-29/15117078653,65/7124115824:-75585393265/2290419429193
+qdiv:31/30756865282,=1,-11/69831756369:1
+qdiv:43/1638160013,-9/6652991074,36/1999878215:-286078616182/14743440117
+qdiv:4/7372173731,11/274587688,=1:1098350752/81093911041
+qdiv:-75/216042979,-19/338590753,33/292120913:25394306475/4104816601
+qdiv:5/521918956,74/512764585,-35/161566463:2563822925/38622002744
+qdiv:18/42121601,28/26500085,65/84650518:238500765/589702414
+qdiv:41/41853940,11/85956056,53/26801053:881049574/115098335
+qdiv:96/7914185,0,=1:$MP_UNDEF
+qdiv:-18/2408419,-29/2331289,39/1991248:41963202/69844151
+qdiv:-21/209413,=1,38/45091:1
+qdiv:73/941827,31/311506,-28/306503:22739938/29196637
+qdiv:-80/34347,=1,76/60617:1
+qdiv:-27/34016,44/3123,11/36759:-84321/1496704
+qdiv:45/4447,95/7813,=2:70317/84493
+qdiv:31/107,16/1959,-2/1335:60729/1712
+qdiv:7/6,=1,-72/455:1
+qdiv:82/415,5/846,4/63:69372/2075
+qdiv:95/28,-25/38,28/19:-361/70
+qdiv:89/43,=1,79/14:1
+qdiv:-49,8,-71/9:-49/8
+qdiv:-10,=1,1/4:1
+qdiv:808/45524274987585732633,-72/3230580501556764385,69/50585273432898431555:-326288630657233202885/409718474888271593697
+qdiv:319/66978137654707641182,=1,=1:1
+qdiv:-669/4359545352267142613,389/6178257053522267612,=1:-4133253968806397032428/1695863142031918476457
+qdiv:991/3477727818712482910,=1,529/5027018252141619996:1
+qdiv:3/23180871072888724,592/673665479785224825,-596/518946796253532009:2020996439355674475/13723075675150124608
+qdiv:179/793396855429515116,931/387517624716045840,-695/566279872123894778:17341413706043051340/184663118101219643249
+qdiv:221/3427690629405359,=1,598/80947651271959489:1
+qdiv:-724/52750058649448927,153/12336232443953527,67/55314453014305481:-8931432289422353548/8070758973365685831
+qdiv:575/8856014113088424,77/1046092009770736,=1:75187863202271650/85239135838476081
+qdiv:639/7628460189090482,=1,3/187674005095612:1
+qdiv:167/57172907255471,=1,458/490173841920009:1
+qdiv:-64/133181712979849,571/541047169178329,495/108993178847291:-34627018827413056/76046758111493779
+qdiv:-527/33346321735789,13/1536370494056,616/27148895576567:-809667250367512/433502182565257
+qdiv:-600/42483246609437,177/7709155575481,-274/29153462781241:-1541831115096200/2506511549956783
+qdiv:-891/3797928330707,-557/8539817631461,44/93582862751:7608977509631751/2115446080203799
+qdiv:30/101373302507,64/912736019917,127/1421964160402:13691040298755/3243945680224
+qdiv:95/811213545591,967/271585517890,-221/724077624393:25800624199550/784443498586497
+qdiv:482/121409952851,873/176693357788,-173/379909953056:85166198453816/105990888838923
+qdiv:211/83869812582,=1,425/21426470616:1
+qdiv:89/16604301838,-370/14168930881,=1:-1261034848409/6143591680060
+qdiv:357/8146855082,-657/9471809692,=1:-563572676674/892080631479
+qdiv:-21/2562596293,903/2881728821,156/164990119:-2881728821/110191640599
+qdiv:-579/609663644,=1,157/25380668:1
+qdiv:-39/383217583,929/665052497,817/218315394:-25937047383/356009134607
+qdiv:937/74356752,422/14604731,959/48789395:13684632947/31378549344
+qdiv:81/64759993,-169/12649693,-554/43474193:-1024625133/10944438817
+qdiv:-277/389415,=1,=2:1
+qdiv:251/2232087,-151/728262,=1:-60931254/112348379
+qdiv:-61/65749,-207/485432,=1:29611352/13610043
+qdiv:363/228754,227/70325,569/103195:25527975/51927158
+qdiv:159/83267,-157/2354,327/82759:-374286/13072919
+qdiv:41/1099,471/99860,201/1723:4094260/517629
+qdiv:14/23,27/70,=1:980/621
+qdiv:775/8954,916/3399,-956/2393:239475/745624
+qdiv:-16/17,-337/37,753/524:592/5729
+qdiv:680/89,955/296,-767/913:40256/16999
+qdiv:671/29,69/71,=1:47641/2001
+qdiv:45/7,227/47,-588/17:2115/1589
+qdiv:309/4,=1,477/7:1
+qdiv:995/8,-542/3,=2:-2985/4336
+qdiv:709/12248686694307459921,-359/3012348473905143856,3209/3205529400054125663:-305107866856963856272/628182646179482587377
+qdiv:7039/48532814476573486698,1842/19665238977050464607,-686/59623200188183875815:138423617159458220368673/89397444265848362497716
+qdiv:451/329309233590151305,=1,=2:1
+qdiv:809/682008587763566662,8909/1007199215258353342,2888/6648659691864956039:407412082572003926839/3038007254192807695879
+qdiv:7143/477997774542113905,8817/680997252822749635,-1905/134302220820939382:324290891794193376187/280967091875854553359
+qdiv:347/219003978754231945,-8579/862605866526757805,1879/749017786570395462:-59864847136956991667/375767026746511171231
+qdiv:2252/19919238262742963,3008/30104309448888197,8998/18293466522564645:16948726219724054911/14979267173582708176
+qdiv:3132/7632328378838467,-7891/65522084848474051,-6375/16422802718592713:-205215169745420727732/60226703237414343097
+qdiv:3091/6666475760056275,-4948/972687207365139,-1089/869334750550498:-1002192052655214883/10995240686919482900
+qdiv:-1181/1328394658282401,-4449/5267675168926367,2101/2626633098162958:6221124374502039427/5910027834698402049
+qdiv:7323/118964685899438,-259/116805460794557,9535/306412793639648:-855366389398540911/30811853647954442
+qdiv:2433/121526971245398,360/82851643972451,-3062/418315498791959:67192683261657761/14583236549447760
+qdiv:6067/57796241878231,964/16961015496389,-6693/83462416525900:102902481016592063/55715577170614684
+qdiv:4159/18938105171021,-3304/15702563734975,-497/5886368665864:-65306962573761025/62571499485053384
+qdiv:-2339/6560171909556,-3315/879777839282,1570/7171072558101:1028900183040299/10873484940089070
+qdiv:1225/584214253087,-6283/3835506609411,1613/7130518143781:-4698495596528475/3670618152145621
+qdiv:608/483278255319,2413/247017404385,-9588/346021249721:2634852313440/20458779475171
+qdiv:5057/839656743164,-8443/6430957362,3191/797476829669:-16260675689817/3544610941266826
+qdiv:8/2915377743,-2073/42761112191,3028/25982265273:-342088897528/6043578061239
+qdiv:-3080/48285456271,7722/56966382901,4950/16693686061:-7975293606140/16948195151121
+qdiv:2506/7829482533,3738/894286585,1315/9318157842:160077298715/2090471836311
+qdiv:395/539727584,3101/2435581364,8543/4140459726:240513659695/418423809496
+qdiv:-2933/85901348,1075/31492019,5089/657510451:-7105083979/7103380700
+qdiv:-109/7944981,1560/35171599,8205/374231336:-3833704291/12394170360
+qdiv:351/84445888,=1,-972/45463937:1
+qdiv:1896/15952037,1648/24133709,-8841/97126097:5719689033/3286119622
+qdiv:-7849/786920,8468/8925921,-3114/6765635:-70059553929/6663638560
+qdiv:245/566264,=1,8221/9131472:1
+qdiv:79/17927,=1,=2:1
+qdiv:1970/222369,-9680/117859,-333/53716:-3316889/30750456
+qdiv:1255/31098,5871/80299,581/4346:100775245/182576358
+qdiv:2551/88863,310/16021,937/5727:40869571/27547530
+qdiv:8096/4615,=1,359/1549:1
+qdiv:4380/6089,-1293/32,=2:-46720/2624359
+qdiv:4841/137,=1,63/127:1
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diff --git a/polly/lib/External/isl/imath/tests/qdivz.t b/polly/lib/External/isl/imath/tests/qdivz.t
new file mode 100644
index 00000000000..b4f010d437c
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/qdivz.t
@@ -0,0 +1,800 @@
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diff --git a/polly/lib/External/isl/imath/tests/qmisc.t b/polly/lib/External/isl/imath/tests/qmisc.t
new file mode 100644
index 00000000000..2005b0f8519
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/qmisc.t
@@ -0,0 +1,49 @@
+## Miscellaneous rational arithmetic tests
+## Generated and verified by hand
+
+# Basic negation tests
+qneg:0,0:0
+qneg:0,=1:0
+qneg:1/37,15/16:-1/37
+qneg:65/18,=1:-65/18
+qneg:-6/7,=1:6/7
+
+# Basic absolute value tests
+qabs:0,0:0
+qabs:0,=1:0
+qabs:1/37,15/16:1/37
+qabs:-592384293849818/1000001,55:592384293849818/1000001
+
+# Reciprocal tests
+qrecip:0,0:$MP_UNDEF
+qrecip:1/2,5:2
+qrecip:-3/2,100/995:-2/3
+qrecip:655378011918000000,=1:1/655378011918000000
+
+# Reading decimal fractions
+qrdec:0,10:0
+qrdec:1.10000,10:11/10
+qrdec:1.01000,10:101/100
+qrdec:1001.001,10:1001001/1000
+qrdec:1001.001,2:73/8
+qrdec:12.05,10:241/20
+qrdec:-5.00138,10:-250069/50000
+qrdec:-00000.5,10:-1/2
+qrdec:.5,10:1/2
+qrdec:-.005,10:-1/200
+
+# Rational exponentiation
+qexpt:-5/8,0,0:1
+qexpt:-5/8,0,=1:1
+qexpt:13/8,0,0:1
+qexpt:13/8,0,=1:1
+qexpt:509252236125428/47889593208724507642867,0,0:1
+qexpt:509252236125428/47889593208724507642867,0,=1:1
+qexpt:-5/8,1,0:-5/8
+qexpt:-5/8,1,=1:-5/8
+qexpt:13/8,1,0:13/8
+qexpt:13/8,1,=1:13/8
+qexpt:509252236125428/47889593208724507642867,1,0:509252236125428/47889593208724507642867
+qexpt:509252236125428/47889593208724507642867,1,=1:509252236125428/47889593208724507642867
+qexpt:509252236125428/47889593208724507642867,4,15/39:67256115255216568118287656956157205834289880165948820377856/5259743820161714626344840164714583399959974116151536869550562774750827767486998189660536721
+qexpt:509252236125428/47889593208724507642867,4,=1:67256115255216568118287656956157205834289880165948820377856/5259743820161714626344840164714583399959974116151536869550562774750827767486998189660536721
diff --git a/polly/lib/External/isl/imath/tests/qmul.t b/polly/lib/External/isl/imath/tests/qmul.t
new file mode 100644
index 00000000000..db1f3908f49
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/qmul.t
@@ -0,0 +1,803 @@
+## Rational multiplication tests
+## Generated and verified in PLT Scheme (see imath-test.scm)
+
+qmul:-3/26206484091896184128,1/693472590887743985,-4/70335584174882085859:-3/18173478421265693438940076224084470080
+qmul:3/91130421678723294601,2/1511297448571716895,-9/67416878129472613559:6/137725173770319192750843847801453983895
+qmul:3/3660941890584665578,-9/8380105656182991401,5/2848423294177090248:-27/30679079844245810042603985831234694778
+qmul:7/2164077499217694058,0,-1/333401876099655555:0
+qmul:4/904017211204226053,=1,-1/248448496161174245:16/817247118153466254734904646723958809
+qmul:0,-4/100568637457389843,0:0
+qmul:2/73650052097202983,5/4552230994414799,0:10/335272049897152087881722802145417
+qmul:7/999642011115757,=1,7/12191208784714605:49/999284150387555241548053683049
+qmul:4/4748171647596839,7/4194009713619088,1/878273774653032:1/711209928998281855823737102244
+qmul:-1/3272725541181013,3/1299116858075758,9/9062356618866682:-3/4251652922403362359365945182854
+qmul:1/107669261142969,=1,2/427008086776155:1/11592669795072854172258134961
+qmul:9/886384410786908,=1,1/143876368001899:81/785677323686054067227784200464
+qmul:3/35807911942040,=1,-5/90295680789497:9/1282206557648890844319361600
+qmul:2/68986921725007,5/77598287457417,3/43453045339831:10/5353266982819419037696526919
+qmul:-1/1109481330540,-4/9404058215555,=2:1/2608406755367394881137425
+qmul:0,=1,5/4715212234981:0
+qmul:1/68986716212,-3/2931468080,=1:-3/202232356519496512960
+qmul:1/26808054670,=1,=1:1/718671795189708808900
+qmul:7/43068832781,0,7/19211808304:0
+qmul:1/9608906275,4/5041381803,=1:4/48442165241517513825
+qmul:1/1228681579,6/2225293505,2/749381381:6/2734177137461844395
+qmul:1/8574959114,=1,6/1956551185:1/73529923806771664996
+qmul:-5/861495024,=1,-2/67979545:25/742173676376760576
+qmul:-2/140727827,1/692019033,-7/753572629:-2/97386334756731291
+qmul:3/95516372,=1,0:9/9123377320042384
+qmul:3/21198931,1/3432139,1/3530668:3/72757677843409
+qmul:7/6222866,4/1833695,0:14/5705419134935
+qmul:6/615053,=1,0:36/378290192809
+qmul:5/568461,4/392783,-2/643261:20/223281816963
+qmul:-5/510613,=1,0:25/260725635769
+qmul:2/14663,-8/93277,=1:-16/1367720651
+qmul:9/49612,8/73143,1/20024:2/100799181
+qmul:3/697,0,=1:0
+qmul:-4/2403,0,1/2247:0
+qmul:3/196,9/422,3/424:27/82712
+qmul:-1/309,2/257,4/233:-2/79413
+qmul:5/67,9/11,7/8:45/737
+qmul:1,-1/22,=2:-1/22
+qmul:0,=1,3/8:0
+qmul:0,-4/7,=1:0
+qmul:-85/86153854226096021421,=1,-17/21068656717268700162:7225/7422486598011403290090823970405290859241
+qmul:-5/43607938850040716426,27/16056333585184134128,32/52338799857236448113:-135/700183613138564743562274537293596786528
+qmul:17/1822698577891414141,=1,-7/8417742332722538862:289/3322230105847383502431373884774767881
+qmul:16/4598825929956443605,49/4847943197216974161,27/3451375546922816345:112/3184978126045337954130178471819812915
+qmul:-13/308229631559916894,7/23027204844472228,=2:-91/7097666865066408240206510971019832
+qmul:9/407734736332167529,=1,=1:81/166247615211862175599509955321965841
+qmul:85/4942792842537911,76/69829540258955019,-91/17762034363033874:6460/345152951789675772143408751225309
+qmul:54/48159800199427181,=1,5/15794694767846184:2916/2319366355248746342818800521606761
+qmul:-1/11155623059888,5/2213190858263331,74/4478858739544117:-5/24689522974375729479887366928
+qmul:28/7444112023597965,94/2835981424363017,44/965216022716111:376/3015909059971602514096088922915
+qmul:61/275448071365986,=1,39/801040431549668:3721/75871640019241316559957752196
+qmul:-16/122979260248953,91/861294833601044,69/330950567465674:-364/26480350373125364764230176733
+qmul:-5/15361734038378,-4/21146843499433,-42/41820111648961:10/162426092794746228321619837
+qmul:31/27515949153680,42/8919478678031,-31/98260871319338:651/122713960891016949829402040
+qmul:-96/7797764223407,-71/8834811164017,1/1862795966742:6816/68891774415328515707545919
+qmul:15/5131534941574,57/7059500713061,=1:855/36226074579139089973698014
+qmul:99/412586992259,-70/740258455507,2/72307587241:-6930/305421009651925904920313
+qmul:79/741601059426,-1/218034913785,-91/880841444516:-79/161694923054812571587410
+qmul:-31/7229094496,1/4120337146,=1:-31/29786306583812948416
+qmul:55/83518896547,9/29598222046,-49/35480447526:495/2472010845035008675162
+qmul:-70/3176137243,=1,1/170734393:4900/10087847786371641049
+qmul:19/8816510816,16/688207893,1/526411545:19/379224520768191918
+qmul:-3/77923064,=1,29/420820637:9/6072003903148096
+qmul:7/886358404,71/63529999,3/267607006:497/56310348519761596
+qmul:1/2981721,46/37858403,83/6367099:46/112883195251563
+qmul:-31/938629,-15/8290882,1/1023462:465/7782062280778
+qmul:37/410302,4/1757463,=1:2/9744467349
+qmul:28/8167855,=1,81/7643948:784/66713855301025
+qmul:-97/992628,13/93468,-47/39726:-97/7136842608
+qmul:-23/674165,7/523911,47/942664:-161/353202459315
+qmul:-7/19181,54/85811,=1:-378/1645940791
+qmul:5/7653,=1,-76/63829:25/58568409
+qmul:11/4265,=1,57/520:121/18190225
+qmul:65/6957,41/389,96/2993:2665/2706273
+qmul:-23/297,-9/83,14/891:23/2739
+qmul:2/31,86/985,-4/905:172/30535
+qmul:-12/7,-10/3,62/67:40/7
+qmul:-28/23,16/87,4/15:-448/2001
+qmul:37/2,=1,-16:1369/4
+qmul:-20,=1,35/4:400
+qmul:437/75577860178450821941,=1,572/74570847326022239185:190969/5712012949153462506928348371418487007481
+qmul:-189/57565050795595180562,-59/2655741179562929534,963/98951958258639716842:11151/152877875901493900051494486844412518108
+qmul:392/9223441830913061231,10/126887593668291297,102/1854286196997300457:3920/1170340339264017235848151367305406607
+qmul:255/3855767718884073169,129/2905792024958436842,-235/3219625969384677757:32895/11204059087625523831085398908193292298
+qmul:-47/35122532617610421,-117/122052767037802280,319/272062018205952087:1833/1428934097118272207801649688519960
+qmul:979/892962159830670377,-375/273191235681889997,70/47294962936566903:-367125/243949435861310195658594938880518869
+qmul:117/20991277911775766,=1,-92/47578690454158995:13689/440633748369405163321411464886756
+qmul:-243/11407352500485733,587/76385498874294444,-614/81454021700493267:-47547/290452103861510956341405887722484
+qmul:174/900283464250993,229/9496538222605715,-274/3248450234623149:39846/8549576329439440824503236224995
+qmul:-690/8145214566421669,141/5752215623742809,71/3004747651790415:-97290/46853030487708234314403900528221
+qmul:226/102973529231733,30/68951726615689,=2:2260/2366734212079704520104819679
+qmul:-815/885459835828344,37/163413072330166,-43/147651688219350:-30155/144695712197674089868869025104
+qmul:75/6638760558632,-138/35179390065353,413/82329576156442:-5175/116773773621297956634138548
+qmul:147/95595835551176,6/10399082254439,189/19869218037638:441/497054478539251611118835132
+qmul:-610/7116766665669,370/6178271155283,-170/1991438380331:-225700/43969314209382356444079327
+qmul:-22/3906839603439,=1,495/2629172214638:484/15263395686999402780626721
+qmul:145/9760768489,529/803913134519,3/3085831861:76705/7846809991306273371791
+qmul:30/130533848429,-291/232962676405,413/647890196158:-1746/6081902938292888923549
+qmul:218/45496969487,53/7721267751,312/52099708697:11554/351294283268204113737
+qmul:277/88084217182,-643/95570881719,=2:-178111/8418286301611629495858
+qmul:97/9657661098,472/161930139,400/6505391099:22892/781933202007016311
+qmul:137/1394596885,923/2175332507,-63/538078336:126451/3033711938101440695
+qmul:-99/123912484,299/998630381,359/807136797:-29601/123742771107576404
+qmul:281/377026117,-970/10165277,80/672610947:-272570/3832574915539409
+qmul:641/45327247,95/3702049,-31/16613945:60895/167803689429103
+qmul:-737/25452329,363/67508236,825/73130323:-267531/1718241832881644
+qmul:51/7250464,112/8927185,903/6361594:357/4045389591490
+qmul:199/4639977,588/7971809,322/8874233:39004/12329670136131
+qmul:665/703012,329/294996,787/209117:218785/207385727952
+qmul:133/29650,307/137037,809/690522:40831/4063147050
+qmul:553/42863,161/16139,552/86389:89033/691765957
+qmul:-568/72299,-953/23177,798/52267:541304/1675673923
+qmul:-907/7588,47/7619,=2:-42629/57812972
+qmul:37/1428,-161/1853,-140/3279:-851/378012
+qmul:69/659,194/7,-183/238:13386/4613
+qmul:-365/824,264/503,-421/698:-12045/51809
+qmul:747/74,-15/94,12/19:-11205/6956
+qmul:12/5,113/37,=2:1356/185
+qmul:-807/7,943/8,199/3:-761001/56
+qmul:591/7,85,-156:50235/7
+qmul:853/18759654468263598130,-2841/40750915886836631585,=1:-2423373/764473101302328863323172599481304936050
+qmul:-4752/16314480916934323241,1675/15223840982975523171,=2:-2653200/82789687733065614038692540422933105737
+qmul:-877/6101604969629738233,=1,3917/3506403215407013175:769129/37229583205410318824680729796101962289
+qmul:3511/9447760616772726092,2633/1439700094583390479,-1093/3530171944589695778:9244463/13601941853568925322985028756947678068
+qmul:1947/430992381411198689,=1,6905/150990131770718358:3790809/185754432834496165241461379835318721
+qmul:8719/646714849991103264,=1,5786/734533820409647257:76020961/418240097199015197424619951911453696
+qmul:-901/11042787574357099,-1492/25943871895296763,2003/71983717831437714:1344292/286492666196095454208761540770537
+qmul:248/45424964911300681,-2828/10155871660092285,1922/24482631313347631:-100192/65904301971909291827634263335155
+qmul:2932/167088922177995,4285/8315956341564334,=1:1256362/138950418199124701072469163033
+qmul:63/96864941445455,1217/536600078187501,-1363/813397840648021:25557/17325911717752953043818085985
+qmul:2028/952222976621975,=1,-4447/132247057152414:4112784/906728597206814347382052900625
+qmul:-2672/564653311125411,=1,593/245849676591157:7139584/318833361764890193787369918921
+qmul:8811/12688709770015,-6995/78357482492503,2405/8730288903721:-12326589/198851070731280426018339509
+qmul:-6216/89020593811217,2073/8676827195515,-43/383215257478:-12885768/772416309342061967459091755
+qmul:649/679490122076,-1037/2294439110154,6125/672996983517:-673013/1559048711054490271159704
+qmul:4001/986338046878,5/3558981731,-2511/3763709597086:20005/3510359089429023585818
+qmul:-1523/69241899473,=1,7123/515464482213:2319529/4794440642629037677729
+qmul:-6869/800191115742,938/461384295473,-3029/66737130644:-3221561/184597807090188234817983
+qmul:3554/16260606585,-2909/10951382696,=1:-5169293/89038062790716326580
+qmul:-8706/24347671943,-2333/75528831647,6568/10275254341:20311098/1838951215179232380121
+qmul:8843/2387867375,3806/2482665293,-478/3437714583:33656458/5928275456199515875
+qmul:676/469237349,=1,=2:456976/220183689696547801
+qmul:4381/415783140,=1,-9189/160372247:19193161/172875619508259600
+qmul:5209/647900590,31/76532430,5022/774082651:161479/49585406551133700
+qmul:6785/98653728,1643/10927961,-3657/22982951:11147755/1078084092088608
+qmul:-131/3707540,-3070/95931603,8132/10434367:40217/35567025538662
+qmul:3495/4205702,1768/4495963,642/2009179:3089580/9454340290513
+qmul:-3081/1925708,-557/52210,=2:1716117/100541214680
+qmul:1824/515329,4783/215399,-80/428083:8724192/111001351271
+qmul:-1259/116291,686/450649,=2:-123382/7486631837
+qmul:2158/8127,4919/28127,=2:10615202/228588129
+qmul:-8479/13630,228/39107,-6607/63539:-966606/266514205
+qmul:-3929/4639,1292/907,4288/3241:-5076268/4207573
+qmul:-2277/9067,379/5790,-503/3539:-287661/17499310
+qmul:7172/525,4813/272,4472/631:8629709/35700
+qmul:759/64,=1,1976/105:576081/4096
+qmul:3255/4,-1567/13,4881/26:-5100585/52
+qmul:3089/89,=1,-8984/35:9541921/7921
+qmul:5031,-5481,=2:-27574911
+qmul:1479,6716/7,669/5:9932964/7
+qmul:-79907/42073633953804470386,9649/39994826473890828375,-32061/69201000109873943624:-771022643/1682727689108411078225149789999696002750
+qmul:7869/4137317017220625920,-46300/94718332275370703049,-79642/60022119059401889179:-6072245/6531329466094147648450013992237207168
+qmul:-29138/1677259660137772757,55969/6796400935123913154,-10129/967130263887253020:-815412361/5699664561302987764528483645927572789
+qmul:63296/9530224706297545807,2348/322614345457502213,958/9427355800897598701:148619008/3074587205685099009151203658171370891
+qmul:8775/40604046784024133,-33221/144310822954936081,=2:-291514275/5859603406703248409743984976442773
+qmul:21115/310091896403808544,11297/311633467720059450,=2:47707231/19327002597641657788985081219588160
+qmul:-7541/568197224156233,78686/80834506129161047,=2:-593371126/45929941998629309750410745855951
+qmul:-69371/45833711711077260,45306/56585412779622335,7626/54270702558063797:-174606807/144084416466306503713223267033450
+qmul:-47821/557744909539529,-35381/1596080991630467,27735/4832267108379043:1691954801/890206048294696559875897230043
+qmul:39463/2836203973039415,5893/5755385978253006,39369/9842567552327876:232555459/16323448577896515754725680231490
+qmul:59563/93078116326943,=1,39162/530318464795213:3547750969/8663535738971933065667725249
+qmul:-6422/219350698334991,12009/451668291281128,38643/184068500609632:-12853633/16512292518047925598650058308
+qmul:6353/29487362288887,18649/10109618056586,-739/5084432721670:118477097/298105970236825097644959782
+qmul:82991/13932554844361,44258/40391647302859,-4103/3232141273561:3673015678/562758841301169080175328099
+qmul:68609/2797606749368,71293/8880170450370,=2:4891341437/24843224787493384272866160
+qmul:69352/1282840784633,51360/5155457119681,=2:3561918720/6613630656553360226662073
+qmul:19996/284659901481,91975/649935845164,992/22386886571:459783025/46252668413338677571971
+qmul:95164/291066123381,=1,86173/926768430166:9056186896/84719488180043514871161
+qmul:58599/35278850458,=1,=2:3433842801/1244597289637926809764
+qmul:16793/50880968745,32965/31422146373,5133/4972604243:110716249/319757849501085622377
+qmul:10537/861910652,22310/9378414407,=1:117540235/4041677638131781682
+qmul:79317/9648480088,22265/3941357964,=2:588664335/12676037945111406944
+qmul:11221/182565205,56092/543156657,49073/666425232:629408332/99161506432319685
+qmul:85751/945688766,15325/126345977,14001/74129827:1314134075/119483971078194382
+qmul:-70533/6456184,-11237/1159737,36975/90043007:264193107/2495825154536
+qmul:56162/84955139,=1,=2:3154170244/7217375642509321
+qmul:559/4445450,87073/9891852,=1:48673807/43973733473400
+qmul:2247/126982,=1,=2:5049009/16124428324
+qmul:87205/903493,=1,4007/74486:7604712025/816299601049
+qmul:20400/957959,=1,-9955/111633:416160000/917685445681
+qmul:31836/73595,11609/39450,46021/33218:61597354/483887125
+qmul:84969/75326,-86068/92037,9321/65618:-1218851982/1155463177
+qmul:49337/4359,71033/6220,-25745/2781:3504555121/27112980
+qmul:15077/1290,20527/9232,-259/71:309485579/11909280
+qmul:52867/502,-63033/451,83228/635:-3332365611/226402
+qmul:1852/17,45130/359,67421/50:83580760/6103
+qmul:57233/78,37817/35,-23454:166490797/210
+qmul:9693/26,82259/68,46811/66:797336487/1768
+qmul:-53957/7,=1,57497/7:2911357849/49
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+qmul:43291079967737078989/47,-41261412311052788301/4,81583584176718755579/67:-1786251099939557456367856008793532107689/188
+qmul:-63431471455230924013/94,=1,69816003657973600474/97:4023551570975775516902820423729780024169/8836
+qmul:10484919345725066249/3,=1,-37631393326791641801/9:109933533686359751306619094875438930001/9
+qmul:76285024187842291525,15471681970508023553/4,14287488305535091904/5:1180257633346808064019424884594192288325/4
diff --git a/polly/lib/External/isl/imath/tests/qmulz.t b/polly/lib/External/isl/imath/tests/qmulz.t
new file mode 100644
index 00000000000..233b1258c39
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/qmulz.t
@@ -0,0 +1,800 @@
+qmulz:4/21116887005095188595,0,1/17609283250378275381:0
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+qmulz:0,0,-1/273384650324881125:0
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+qmulz:0,4,=1:0
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+qmulz:3/5927038874,3,-8/86686893619:9/5927038874
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+qmulz:1/7271098513,6,0:6/7271098513
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+qmulz:-4/90120677,4,=1:-16/90120677
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+qmulz:-1/58093,9,7/9595979:-9/58093
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+qmulz:-7/355662,7,-1/161621:-49/355662
+qmulz:-1/55532,-8,3/126767:2/13883
+qmulz:7/6254,8,=1:28/3127
+qmulz:7/22045,2,=1:14/22045
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+qmulz:-63773122944071670311,68924540068336860054,-11050684739856731468:-4395533167641640581052894142988833656794
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diff --git a/polly/lib/External/isl/imath/tests/qsub.t b/polly/lib/External/isl/imath/tests/qsub.t
new file mode 100644
index 00000000000..10d94ea4038
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/qsub.t
@@ -0,0 +1,803 @@
+## Rational subtraction tests
+## Generated and verified in PLT Scheme (see imath-test.scm)
+
+qsub:1/6908363373906559980,-3/3282386453454754261,=2:24007476575174434201/22675918354053873847004295306757074780
+qsub:1/38717857456042735841,4/21180041959354743117,=1:-133691387864816200247/820045845495301036647051443116443956397
+qsub:5/43729125372682174,-9/5619593033754393704,0:14245763648563054043/122869944158248625198135913829316248
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diff --git a/polly/lib/External/isl/imath/tests/qsubz.t b/polly/lib/External/isl/imath/tests/qsubz.t
new file mode 100644
index 00000000000..7af689db540
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/qsubz.t
@@ -0,0 +1,800 @@
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diff --git a/polly/lib/External/isl/imath/tests/qtodec.t b/polly/lib/External/isl/imath/tests/qtodec.t
new file mode 100644
index 00000000000..e037a8dfa65
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/qtodec.t
@@ -0,0 +1,1303 @@
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+qtodec:5738645734057/90951057139915,10,6,0:0.063095
+qtodec:2150180745771/75085058005357,10,10,0:0.0286365996
+qtodec:9211573654621/31326323574601,10,1,0:0.2
+qtodec:-2622066957883/1546353871091,10,6,0:-1.695644
+qtodec:224011912746/207690732319,10,17,0:1.07858405738553459
+qtodec:9555205925617/1205795142091,2,20,0:111.11101100101001011010
+qtodec:-8730617437121/6055175866493,10,24,0:-1.441843743207007133126749
+qtodec:-4064904713342/494211830517,10,24,0:-8.225025105306892432251847
+qtodec:7676618287424/780404569679,35,3,0:9.T9Y
+qtodec:-4358217891649/583281222878,17,13,0:-7.80676F3C2CC10
+qtodec:4328191663159/246276445006,26,9,0:H.EO9M9D2J7
+qtodec:737625000181/4792760124,17,17,0:90.F646D65A829FCC044
+qtodec:-71443761268/988984303,10,8,0:-72.23953004
+qtodec:381451888319/69461063050,16,1,0:5.7
+qtodec:201128820701/3339261143,16,3,0:3C.3B4
+qtodec:4042648911199/1793950960,10,7,0:2253.4890871
+qtodec:5673741769200/852342691,10,0,0:6656
+qtodec:-8389684341647/9150651656,10,15,0:-916.840095879505961
+qtodec:1372726881895/1560653491,16,19,0:36F.95ACC431C89A521C30D
+qtodec:328988903869/101330533,4,1,0:302232.2
+qtodec:6568406520821/800160752,8,17,0:20020.66750526036730745
+qtodec:501238490585/5971867,8,0,0:243735
+qtodec:6266930625937/691212664,8,23,0:21552.44562064316534547754507
+qtodec:4565491072367/11630659,16,17,0:5FD5B.541017CCD8963842F
+qtodec:-1162420020152/24630125,4,22,0:-23201123.0031031110222320020123
+qtodec:5113275296450/89201439,15,21,0:11EB7.BC9038ED2071A71B54669
+qtodec:3315661534768/61059429,10,14,0:54302.20342820434825
+qtodec:1733402780098/4797927,4,12,0:1120031001.212321010002
+qtodec:3283980196433/2438910,18,5,0:CEFF5.0A1C8
+qtodec:8489184557941/8439595,35,7,0:NG4A.SQX37QV
+qtodec:3788804069259/3461851,10,3,0:1094444.581
+qtodec:-3096145670549/510543,7,24,0:-102355342.020600564131455624420322
+qtodec:-3078702213873/276616,10,22,0:-11129877.5698911125892934609711
+qtodec:1748608098871/9819,8,9,0:1247254444.540514032
+qtodec:8487930264661/21627,15,11,0:246C724D.08EEC2CE9E2
+qtodec:740911632886/11719,16,19,0:3C4B546.94752D05B3962EF76A3
+qtodec:563687406299/41136,8,8,0:64213553.32002623
+qtodec:1780375881783/26804,33,7,0:1N09JI.51C9U8J
+qtodec:-3247679347711/26264,2,20,0:-111010111101101010000000000.10010101101101001111
+qtodec:21406766924698/763359226461945,8,7,0:0.0162672
+qtodec:24260209610958/200034339782053,10,24,0:0.121280224372428559322852
+qtodec:1814902384527/227339212768685,4,12,0:0.000200230300
+qtodec:31173219462521/203933404240669,8,6,0:0.116207
+qtodec:18030863259117/23714043196153,16,11,0:0.C2A5FEBA219
+qtodec:12974701069361/70664067852975,10,2,0:0.18
+qtodec:25696621276083/44253078013283,8,4,0:0.4512
+qtodec:9021988651073/35238715595793,10,6,0:0.256024
+qtodec:8048554533578/701038746989,16,11,0:B.7B1C260E450
+qtodec:-23885228112881/6921647732686,10,20,0:-3.45080088373872630271
+qtodec:33478283623865/8474518157089,8,1,0:3.7
+qtodec:-25807603431943/2948804373766,8,19,0:-10.6007565676256043474
+qtodec:89076698620825/964761424737,10,3,0:92.330
+qtodec:76836733524587/591287975017,16,14,0:81.F2B4FA0C640A59
+qtodec:2301284879441/92761811708,8,7,0:30.6357611
+qtodec:-22162325445043/7119385698,10,7,0:-3112.9547386
+qtodec:-88830049403167/77327410704,10,12,0:-1148.752409972677
+qtodec:40330389045243/30915263495,8,10,0:2430.4275130461
+qtodec:-97356183980680/88686324821,8,0,0:-2111
+qtodec:-61508444809/21872771653,4,5,0:-2.30333
+qtodec:-76008039360448/2365100065,2,20,0:-111110110001001.01011000100110100100
+qtodec:4553948402077/3181382511,10,21,0:1431.436926031746831338509
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+qtodec:29508211302267/62743787,22,5,0:203F2.KH16G
+qtodec:20444506756589/73095694,10,8,0:279695.09061079
+qtodec:60358251477113/17276777,2,9,0:1101010100111011100101.111110110
+qtodec:7675515321823/53051166,16,2,0:23529.5E
+qtodec:68145812098865/9201386,6,15,0:422423113.430414512210252
+qtodec:9010952293583/1041530,10,5,0:8651649.29822
+qtodec:64799577298519/4451432,10,22,0:14557018.3479201749010206153884
+qtodec:4766086981551/1109401,4,20,0:100120312122.12111220122321000303
+qtodec:-14985202309069/259277,7,5,0:-1301154532.10024
+qtodec:77527740569441/65903,10,5,0:1176391675.18081
+qtodec:-34544219638590/487433,16,10,0:-43962AD.C208E5DFB8
+qtodec:89606198611077/718691,4,14,0:13123213120300.10302120010032
+qtodec:-20900481598325/19479,8,1,0:-7775046350.6
+qtodec:8694579108687/6527,2,14,0:1001111011001100010010100100000.10100101000000
+qtodec:11008512080877/36688,10,4,0:300057568.7112
+qtodec:-26629103193937/3238,16,24,0:-1EA2F4C19.E0C5562D391507BF7C71372F
+qtodec:320315168187847/67090457824132,16,10,0:4.C63D998DFE
+qtodec:37992496059712/933307870572061,2,15,0:0.000010100110101
+qtodec:-642835243415327/563880960452415,8,2,0:-1.10
+qtodec:-996741207812269/744724604059786,10,19,0:-1.3384024139643589286
+qtodec:865729136060421/18601273799867,4,24,0:232.202221202012312012111231
+qtodec:-190792427336021/12399475938971,8,11,0:-17.30615536546
+qtodec:353117587688196/41602134470867,4,12,0:20.133032231321
+qtodec:223068576005855/34076492509692,15,12,0:6.82D1EA2D84C4
+qtodec:219205101136492/2395575765723,4,5,0:1123.20010
+qtodec:456340378396493/4330214900946,16,21,0:69.6299AAE57B6C08183D61A
+qtodec:3571581360820/3092454154269,2,22,0:1.0010011110101001110001
+qtodec:847831462081399/7150211415400,10,5,0:118.57432
+qtodec:-898578802522565/555835047564,16,16,0:-650.A0E8207F331CE070
+qtodec:663661151253796/453138066205,10,19,0:1464.5892736664396649880
+qtodec:271225450200625/257749526519,34,21,0:UW.9L5RX9FJI38PG83BX6NV9
+qtodec:-165514406278015/4322124123,22,8,0:-3D2E.F462JIGL
+qtodec:969134788868309/87912888352,10,13,0:11023.8078515624311
+qtodec:349857780160166/918601801,10,19,0:380859.0183247049828067994
+qtodec:583417315283519/55625636618,23,7,0:JJ0.6AGDG5C
+qtodec:5198673145663/2582846448,2,16,0:11111011100.1100010011011010
+qtodec:25224504315989/2747113245,10,16,0:9182.1858315815444295
+qtodec:-354734288655699/8606733226,10,5,0:-41215.90379
+qtodec:294622457238131/4699248277,10,15,0:62695.656809649983088
+qtodec:84851645314450/3149545649,8,5,0:64474.72505
+qtodec:162782722239884/940443795,10,5,0:173091.38845
+qtodec:-503919081966818/49953093,10,12,0:-10087845.450667449160
+qtodec:696741265176479/547456417,10,10,0:1272688.0963320208
+qtodec:123373090223114/171645417,10,19,0:718767.1676844945997014298
+qtodec:530906534529992/21770787,10,10,0:24386189.3706457189
+qtodec:25200685975741/10321244,4,3,0:21110012200.213
+qtodec:-58528486417214/9145779,16,23,0:-61A614.0F20EFF0ECB8777D9D089D6
+qtodec:-202870643410118/39728965,4,0,0:-103132222332
+qtodec:226786253503/4854200,10,10,0:46719.5940634914
+qtodec:947663833947325/1199459,8,24,0:5705715226.174676743125024435532213
+qtodec:58508575771048/1950595,10,12,0:29995245.435904429161
+qtodec:-162029920997164/5378993,10,23,0:-30122723.89965259296675046797792
+qtodec:-638908439370581/780673,2,13,0:-110000110001111110011101001101.0011111111011
+qtodec:54048561476435/260446,16,13,0:C5E8D1C.AADE1542405BB
+qtodec:311253898901698/112463,16,23,0:A4F666A8.42F6DD7D2BF402286E84E52
+qtodec:-364014082098529/378836,16,13,0:-3945CA63.2A9FE228A057C
+qtodec:662955978899037/29150,8,10,0:251345254234.0616507470
+qtodec:380819161094198/155,16,19,0:23C0A857C06.F4704F4704F4704F470
+qtodec:19950100398461/41364,3,22,0:1020121112200021201.1102012021021112012001
+qtodec:843068174103353/67398,8,22,0:135145224047.7022451545330470767051
+
+# Writing decimal fractions, with rounding
+qtodec:@0.001,10,1,0:0.0
+qtodec:@-3.5,10,0,0:-3
+qtodec:@12.001,10,2,0:12.00
+qtodec:@0.001,10,1,2:0.1
+qtodec:@-3.2,10,0,2:-4
+qtodec:@12.001,10,2,2:12.01
+qtodec:@0.001,10,1,3:0.0
+qtodec:@0.051,10,1,3:0.1
+qtodec:@0.050,10,1,3:0.0
+qtodec:@-3.5,10,0,3:-3
+qtodec:@12.001,10,2,3:12.00
+qtodec:@0.001,10,1,1:0.0
+qtodec:@0.051,10,1,1:0.1
+qtodec:@0.050,10,1,1:0.1
+qtodec:@-3.5,10,0,1:-4
+qtodec:@12.002,10,2,1:12.00
+qtodec:@12.005,10,2,1:12.01
diff --git a/polly/lib/External/isl/imath/tests/root.t b/polly/lib/External/isl/imath/tests/root.t
new file mode 100644
index 00000000000..b7c5e8b4107
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/root.t
@@ -0,0 +1,44 @@
+# Root-finding tests
+
+# Square root tests
+sqrt:0,0:0
+sqrt:0,=1:0
+sqrt:1,0:1
+sqrt:1,=1:1
+sqrt:1,5:1
+sqrt:256,=1:16
+sqrt:257,=1:16
+sqrt:-1,0:$MP_UNDEF
+sqrt:-1,=1:$MP_UNDEF
+sqrt:-1029384298,66883:$MP_UNDEF
+sqrt:394820398429038402938402,=1:628347354915
+sqrt:394820398429038402938402,-5839823984:628347354915
+sqrt:17179869183,0:131071
+sqrt:4295098369,=1:65537
+sqrt:26965706504352387672564992041751053514175649974280741932736827760729,=1:5192851481060515336695158019653723
+
+# More general roots
+root:0,1,0:0
+root:0,1,=1:0
+root:0,5,0:0
+root:0,5,=1:0
+root:-1,5,0:-1
+root:-1,4,0:$MP_UNDEF
+root:-1,3,=1:-1
+root:-8,3,0:-2
+root:-8,3,=1:-2
+root:17,2,0:4
+root:17,3,0:2
+root:29,2,0:5
+root:29,3,0:3
+root:50,5,=1:2
+root:4295098369,2,0:65537
+root:4295098369,2,43:65537
+root:4295098369,2,=1:65537
+root:-4295098369,2,=1:$MP_UNDEF
+root:-4295098369,4,65537:$MP_UNDEF
+root:-281487861809153,3,0:-65537
+root:-281487861809153,3,=1:-65537
+root:-281487861809153,3,8349283:-65537
+root:-19991036671350700657170189502736817905009200157042372341654483623953031910930270960998207372480741818129399253027601458115528449927512127290460743651463958852225556661727551760552841554362323271980596910944452290359774556447658518439936535499843549082781717193598437061923,7,=1:-571823748273487384719287348728374273472
+root:26965706504352387672564992041751053514175649974280741932736827760729,2,=1:5192851481060515336695158019653723
diff --git a/polly/lib/External/isl/imath/tests/set.t b/polly/lib/External/isl/imath/tests/set.t
new file mode 100644
index 00000000000..0b870a135d0
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/set.t
@@ -0,0 +1,12 @@
+#Unsigned integer assignment. Assumes sizeof(long) == 8.
+setu:0,0:0
+setu:0,18446744073709551615:18446744073709551615
+setu:0,567:567
+
+#Signed integer assignment. Assumes sizeof(long) == 8.
+setv:0,0:0
+setv:0,-1:-1
+setv:0,9223372036854775807:9223372036854775807
+setv:0,-9223372036854775808:-9223372036854775808
+setv:0,4324:4324
+
diff --git a/polly/lib/External/isl/imath/tests/sqr.t b/polly/lib/External/isl/imath/tests/sqr.t
new file mode 100644
index 00000000000..84ef2d2d621
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/sqr.t
@@ -0,0 +1,755 @@
+# Squaring tests
+
+sqr:0,0:0
+sqr:495,0:245025
+sqr:702,0:492804
+sqr:23080,0:532686400
+sqr:-485,0:235225
+sqr:31727877,0:1006658178927129
+sqr:-8234442,0:67806035051364
+sqr:80,0:6400
+sqr:75029676,0:5629452280664976
+sqr:-8153,0:66471409
+sqr:5729957,0:32832407221849
+sqr:641,0:410881
+sqr:-82,0:6724
+sqr:7356,0:54110736
+sqr:0,=1:0
+sqr:7761224,0:60236597978176
+sqr:928319,0:861776165761
+sqr:912,0:831744
+sqr:334190449,0:111683256202821601
+sqr:#xAC46D3FD,0:#x73EF407495E70809
+sqr:-180,0:32400
+sqr:8965915,0:80387631787225
+sqr:1771,0:3136441
+sqr:131,0:17161
+sqr:2518088106,0:6340767709578667236
+sqr:31019,0:962178361
+sqr:460877681,0:212408236843937761
+sqr:2,0:4
+sqr:2,=1:4
+sqr:-43189,0:1865289721
+sqr:400,0:160000
+sqr:400,=1:160000
+sqr:16978,0:288252484
+sqr:32,0:1024
+sqr:4185,0:17514225
+sqr:246082426,0:60556560386045476
+sqr:5024,0:25240576
+sqr:4283818927,0:18351104599323431329
+sqr:79803747,0:6368638035240009
+sqr:27,0:729
+sqr:-294872579,0:86949837846111241
+sqr:5431,0:29495761
+sqr:-64,0:4096
+sqr:-27347593472,0:747890868709777014784
+sqr:-93,0:8649
+sqr:5941698,0:35303775123204
+sqr:68188,0:4649603344
+sqr:-10073,0:101465329
+sqr:712007,0:506953968049
+sqr:74981,0:5622150361
+sqr:590060947429,0:348171921680809101710041
+sqr:3375362,0:11393068631044
+sqr:6288,0:39538944
+sqr:-14938,0:223143844
+sqr:818,0:669124
+sqr:-61517636,0:3784419539028496
+sqr:-485757743,0:235960584884454049
+sqr:52047149,0:2708905719028201
+sqr:880295,0:774919287025
+sqr:23268913856,0:541442352037948788736
+sqr:8611582804,0:74159358390148502416
+sqr:-35440071,0:1255998632485041
+sqr:-123,0:15129
+sqr:-198239315,0:39298826011669225
+sqr:-18792,0:353139264
+sqr:42334,0:1792167556
+sqr:8871716,0:78707344784656
+sqr:-742998638,0:552046976069855044
+sqr:599309793653,0:359172228768401439084409
+sqr:92330620850,0:8524943546546454722500
+sqr:-21253,0:451690009
+sqr:-8591240,0:73809404737600
+sqr:-4406721416023,0:19419193638435754241136529
+sqr:7093605833487,0:50319243720880795970579169
+sqr:540053945630,0:291658264190530996096900
+sqr:6679643648234,0:44617639267392821131318756
+sqr:521589,0:272055084921
+sqr:13144286787,0:172772275138902783369
+sqr:-47469,0:2253305961
+sqr:480758624,0:231128854550373376
+sqr:-8238434,0:67871794772356
+sqr:-867204631,0:752043872027846161
+sqr:84079798766,0:7069412560531055122756
+sqr:14048677,0:197365325450329
+sqr:647598376181,0:419383656832267988144761
+sqr:89538606,0:8017161964423236
+sqr:8775965118,0:77017563752352753924
+sqr:135957585,0:18484464919032225
+sqr:76500157735,0:5852274133479880330225
+sqr:821777330800,0:675317981416772628640000
+sqr:38741,0:1500865081
+sqr:262422499893,0:68865568450091585011449
+sqr:-114911750278913,0:13204710352163261933292461569
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+sqr:5925798128737745925759715663331423546607679889144875112930,0:35115083462551771236157369162138315687265847656064728641830503311468988183199467328159949805669318527886480253184900
+sqr:33488647002377175259885536134537076443400396426003731,0:1121489478049825766276530922246725189111830952465458275864281151986139799264328218398374227234130825920361
+sqr:2123735232819297700815836330311374814786388785356210060,0:4510251339118036610109622239017828107705345092062097267522409602229318236281318584088873097703239806845203600
+sqr:29872141162702890384989735017623770258183788742075,0:892344817644448391848145835628174663828417858780217507228848289828118687946707014712413510875305625
+sqr:7532139657883258576470960593627865781553593844952,0:56733127825857731552224724505206926449807777229886757388281330093349393178013626920192739015882304
+sqr:845457525713874120263223171268947077567681659698840902652,0:714798427786226119268210281388981030688454506537030966614579539102544522728409887855024503885468326555462140633104
+sqr:521768158545019160169259844676904044401662888508547040917,0:272242011271460250849489239712473973544134037328971616162419516920558472338192391022931027231772869417436872200889
+sqr:8452712498170643941637436558664265704301557216577944354047371344426782440907597751590676094202515006314790319892114058862117560952042968596008623655407033230534186943984081346699704282822823056848387726531379014466368452684024987821414350380272583623832617294363807973375,=1:71448348576730208360402604523024658663907311448489024669693316988935593287322878666163481950176220037593478347105937422686501991894419788796088422137966026262523598150372719976137911322484446114613284904383977643176193557817897027023063420124852033989626806764509137929897881780377071828193979369536094745977623512725964279515528917451111552087757216338484610394583686853163069435510427134801022221259480105632478261997058761512146362996413333996057383868504433356414886358698618132135318758644287722813462338791398132036589335991224708890625
diff --git a/polly/lib/External/isl/imath/tests/sub.t b/polly/lib/External/isl/imath/tests/sub.t
new file mode 100644
index 00000000000..ef5f97a061b
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/sub.t
@@ -0,0 +1,849 @@
+# Subtraction tests
+
+sub:0,0,0:0
+sub:0,0,=1:0
+sub:1,=1,=1:0
+sub:1,-1,0:2
+sub:1,-1,=2:2
+sub:1,-1,=1:2
+sub:-1,1,0:-2
+sub:-1,1,=1:-2
+
+sub:103427990038,909510006269847,0:-909406578279809
+sub:128593002,-9007199254740992,0:9007199383333994
+sub:-65537,70000,0:-135537
+sub:-1000000,-6543210,0:5543210
+sub:#x29C932F25E6CEF8046FA,#x29C932F25E6CEF8046FA,0:0
+sub:#x800000001,2,0:#x7FFFFFFFF
+sub:95031018,=1,=1:0
+sub:#x-176E7F8018D6BA4B550AB36AAAFDC78A9,#x186DAD2485A41B18AAF5EF17922D7E5F1F,=1:#x-19E4951C873186BD60469A4E3CDD5AD7C8
+sub:-628443782411312,-301652218193016,=2:-326791564218296
+sub:950147,37133897,0:-36183750
+sub:649972385,297,0:649972088
+sub:505,9,0:496
+sub:33,-2638,0:2671
+sub:-5,71542573,0:-71542578
+sub:4,475357,0:-475353
+sub:5,775166,0:-775161
+sub:0,20805741,0:-20805741
+sub:1,39,0:-38
+sub:76825770,751287444,0:-674461674
+sub:-5846049,574445,0:-6420494
+sub:501105944,54475,0:501051469
+sub:84350933,594,0:84350339
+sub:1367,-96716998,0:96718365
+sub:49879,-426695,0:476574
+sub:4,2,0:2
+sub:710883222,2440893,0:708442329
+sub:-68,2096377330,0:-2096377398
+sub:-87838,-485655,0:397817
+sub:4683739592,822,0:4683738770
+sub:2849496,766332,0:2083164
+sub:60422837,52,0:60422785
+sub:-13318,250,0:-13568
+sub:1,2001827,0:-2001826
+sub:97481748,62763650,0:34718098
+sub:1643,636255899,0:-636254256
+sub:-400522,-40393280,0:39992758
+sub:45568099,22,0:45568077
+sub:645024401,860182184,0:-215157783
+sub:37244191,-976205,0:38220396
+sub:-27,359887,0:-359914
+sub:27528,8076231,0:-8048703
+sub:7298,58548352177,0:-58548344879
+sub:71847293055,573,0:71847292482
+sub:673824473,27805877,0:646018596
+sub:449,53709067,0:-53708618
+sub:40569,6538078687,0:-6538038118
+sub:-642338,9669221067,0:-9669863405
+sub:209745,-28,0:209773
+sub:2488518646,-1846396,0:2490365042
+sub:581,539581676,0:-539581095
+sub:-86,16372077246,0:-16372077332
+sub:82972869,96,0:82972773
+sub:81666,-224,0:81890
+sub:3353174666,6467525,0:3346707141
+sub:178,76264272942,0:-76264272764
+sub:67786255,18533457,0:49252798
+sub:-59339460,8103,0:-59347563
+sub:427117459,-6532,0:427123991
+sub:-514830072898,10894,0:-514830083792
+sub:4160587,946181096,0:-942020509
+sub:76936,1066849,0:-989913
+sub:-776709551,28397881,0:-805107432
+sub:917,6034390,0:-6033473
+sub:5528,-1297,0:6825
+sub:303,37728739196,0:-37728738893
+sub:920355,13059,0:907296
+sub:55576,-56300,0:111876
+sub:-85278759,67364939177,0:-67450217936
+sub:50472201146,-89468966856,0:139941168002
+sub:7257002442515,-4541182637,0:7261543625152
+sub:56005,-57203,0:113208
+sub:1508,8778323669,0:-8778322161
+sub:7477,1777344,0:-1769867
+sub:619620595216,-271826,0:619620867042
+sub:503021,559324,0:-56303
+sub:46990,601026675862,0:-601026628872
+sub:630276867,92393884,0:537882983
+sub:64847122,-1741761,0:66588883
+sub:-901885041650,6063747686,0:-907948789336
+sub:9090817588683,-74217059906,0:9165034648589
+sub:4643353802,39676050,0:4603677752
+sub:61222313,-991211846100,0:991273068413
+sub:8145673172195,79545920286,0:8066127251909
+sub:4326733518687,315334416,0:4326418184271
+sub:65122622889524,686861271,0:65121936028253
+sub:91927935495,815931,0:91927119564
+sub:360036184139,-929647,0:360037113786
+sub:851769193,5104213614039,0:-5103361844846
+sub:49764,-1370901,0:1420665
+sub:543849288,7706203685,0:-7162354397
+sub:62844,8576091,0:-8513247
+sub:-809058073978,541423856,0:-809599497834
+sub:-2639589,58729,0:-2698318
+sub:-79585473,1798247,0:-81383720
+sub:316852976150,-860129592,0:317713105742
+sub:1273939285186,11601,0:1273939273585
+sub:80537932950,292617381640,0:-212079448690
+sub:5467548,5430035835,0:-5424568287
+sub:-5937877,24003123599412,0:-24003129537289
+sub:684651821908033,41369461672136,0:643282360235897
+sub:1137993337,2322246811872,0:-2321108818535
+sub:7374342836,64650413647,0:-57276070811
+sub:432132,-42261483441184,0:42261483873316
+sub:83231980056423,2783677270377,0:80448302786046
+sub:2587541968775,479202225,0:2587062766550
+sub:-928954175477668,-1742785659277,0:-927211389818391
+sub:6404997096,7343350521970,0:-7336945524874
+sub:727251429810,3193503,0:727248236307
+sub:77165840,-457577668996931,0:457577746162771
+sub:33934095603840,74283623494449,0:-40349527890609
+sub:3846088301735,-44249412464855,0:48095500766590
+sub:24497022452,4724262825964,0:-4699765803512
+sub:463651487662200,-753243402,0:463652240905602
+sub:71713522960,15380043873197,0:-15308330350237
+sub:8294630828927135,31183461626,0:8294599645465509
+sub:6703541383230,44950919818203,0:-38247378434973
+sub:868891910767,8481827,0:868883428940
+sub:6321902376793877,-477388684714663,0:6799291061508540
+sub:-32763637641,66549351522069,0:-66582115159710
+sub:288533359019263,58471469888811,0:230061889130452
+sub:-751498035858161,527076997758,0:-752025112855919
+sub:1940356995569832,103761002,0:1940356891808830
+sub:11785051,32863482880064,0:-32863471095013
+sub:6832388,998030894437455,0:-998030887605067
+sub:-29992855,4391685016,0:-4421677871
+sub:-60108503487,-15310122633,0:-44798380854
+sub:-6477442216968621,60885516917,0:-6477503102485538
+sub:31744989963367,971914401,0:31744018048966
+sub:4773395,-699494329,0:704267724
+sub:158083905091,153315973803,0:4767931288
+sub:9257392592301,778762452,0:9256613829849
+sub:-869525659304,751325704189605,0:-752195229848909
+sub:-49703875573842714,4505925018715,0:-49708381498861429
+sub:-91664099,25339513330017666,0:-25339513421681765
+sub:469453608,-325915913621728,0:325916383075336
+sub:-6329249891,453489665443616,0:-453495994693507
+sub:1555896943,216164615094,0:-214608718151
+sub:7867736225,6515990968510638,0:-6515983100774413
+sub:47191079551,22909405,0:47168170146
+sub:480598620144,-630471272,0:481229091416
+sub:-44964236,4129791840132424,0:-4129791885096660
+sub:2078461471836,9510166882939211,0:-9508088421467375
+sub:3399372328,4480430653,0:-1081058325
+sub:37397829728829741,55828595309,0:37397773900234432
+sub:932684640397258009,-748484093750,0:932685388881351759
+sub:840884486710412,70796060480,0:840813690649932
+sub:2539205,16161203759,0:-16158664554
+sub:713637461058,6441899124,0:707195561934
+sub:-8173074559441,-280676331618304941,0:280668158543745500
+sub:430261608,4233937508,0:-3803675900
+sub:-131645380114748755,77441408953418,0:-131722821523702173
+sub:-31661643391,786373077474234,0:-786404739117625
+sub:278686734235471032,120592814722,0:278686613642656310
+sub:-2931231599090,18746712502644976,0:-18749643734244066
+sub:61029444723255769,-2293742672243933,0:63323187395499702
+sub:816220890520,640524790657208413,0:-640523974436317893
+sub:16665591262,113580566277835849,0:-113580549612244587
+sub:-868305701832,6230551669213729,0:-6231419974915561
+sub:11999732039934306,23827304849399,0:11975904735084907
+sub:91534672684829653,795520608093438312,0:-703985935408608659
+sub:-8985197818103126,141826190572801,0:-9127024008675927
+sub:390870406977082056,3249082110578,0:390867157894971478
+sub:88089340380,339939269745765,0:-339851180405385
+sub:-9559600885723,1043929655718,0:-10603530541441
+sub:7521666468386587380,-6553266884,0:7521666474939854264
+sub:8481985305837,8883976116,0:8473101329721
+sub:8322745787655471,-2367148822664118142,0:2375471568451773613
+sub:-1536130411,5098377844876093,0:-5098379381006504
+sub:-80409580247,-379667394325917,0:379586984745670
+sub:2780978527253898223,-793585194380805771,0:3574563721634703994
+sub:983228849414820623,411233979567,0:983228438180841056
+sub:4312344973,8504030055,0:-4191685082
+sub:6929160648081603523,-9139369237550637,0:6938300017319154160
+sub:40543447104286,922867012966969520,0:-922826469519865234
+sub:-32492268811,-52539511415,0:20047242604
+sub:-837736542959704435,-11698939306808,0:-837724844020397627
+sub:30695570430,6965534034801150,0:-6965503339230720
+sub:743449156492,-892800013061,0:1636249169553
+sub:4650491989907718938,9119100370137466,0:4641372889537581472
+sub:475327493368,391070618301538,0:-390595290808170
+sub:9789197909023,-2250068166694,0:12039266075717
+sub:3951486522961630212,41419130651165087030,0:-37467644128203456818
+sub:36289729721981,609158820292,0:35680570901689
+sub:5238096716643,857369737080,0:4380726979563
+sub:20927421747352,7859500562781,0:13067921184571
+sub:853523362775,-58650620319893709899,0:58650621173417072674
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+sub:39332562720628190237986833106169417514936299004,3977882056567658721057988043991788532782214317,0:35354680664060531516928845062177628982154084687
+sub:4875943313527670330837641575224800871607771346650,3006293236431797162023390328257380215745687023197,0:1869650077095873168814251246967420655862084323453
+sub:1046076579612442168236157596521887526129291572,56509310551736435818443805857748077305452153240874241,0:-56509309505659856206001637621590480783564627111582669
+sub:68801970722430443700036017990845232711747541145506,3921528808796839431326261733514349542141553429,0:68798049193621646860604691729111718362205399592077
+sub:7996124256938877464175310613395477260421185912077919187,43920453269964575011284701085185099530091428989544,0:7996080336485607499600299328694392075321655820648929643
+sub:437847871205482940383237227334352005149025155566700,8912640649325787853768245140880891930970131371688993,0:-8474792778120304913385007913546539925821106216122293
+sub:-189562142066164910097303467426752455631983999971824429,-3126396058243394757216280555044878821697145366839147,0:-186435746007921515340087186871707576810286854604985282
+sub:3738151126960474749651813623570168405472279315305,16849954488569956342093577361010810357803346759,0:3721301172471904793309720046209157595114475968546
+sub:326436788465045244627154191800265908625864556666530,8436920985951650690723103995113757167828613977782053802,0:-8436594549163185645478476840921956901919988113225387272
+sub:73862903715165018158730950116899815638213415384,3605176405676828750083989599730691253025429971416,0:-3531313501961663731925258649613791437387216556032
+sub:-98508500558610553174664717035831451337440647751,-85349349113075502208578338510421486004026291637018909,0:85349250604574943598025163845704450172574954196371158
+sub:-3958479771038348317047929281014493361412362156,74460379620572355406333334733218930851102604423,0:-78418859391610703723381264014233424212514966579
+sub:-2678832891454610091209266352796918435982227896,844180754012069973260601080737112141842619632733391282,0:-844180756690902864715211171946378494639538068715619178
+sub:62145977984087344899814827854294051292745009920192834974,7334739588662979743633482328597512684449095879737,0:62145970649347756236835084220811722695232325471096955237
+sub:829628512088735025241917875963770566118880397874,2778273578473650916716745944557448674527619910366636167,0:-2778272748845138827981720702639572710757053791486238293
+sub:5475358208178590127449325322332321821544579907242021829,5101549647265375935536396006666544834314249747938912445,0:373808560913214191912929315665776987230330159303109384
+sub:-327933746445993810109526304208281604193537699541388,-17513330151263794083025328269239503268164355683299,0:-310420416294730016026500975939042100925373343858089
+sub:-97299271106272768099051589856225693996221199008328,-683640181471185906440394924792222282513250833654408,0:586340910364913138341343334935996588517029634646080
+sub:81950415706753285772469335186046579994901810195099988,-695576496751235568817818414861599429774724685935,0:81951111283250037008038153004461441594331584919785923
+sub:11733022337838616680086438101577803026869925229236251633,4209981603742836477443590186310139108935024708250714,0:11728812356234873843608994511391492887760990204528000919
+sub:2993252088056118669505330045592351552378298855615136540,12751727236236955721686048352301817814151141431,0:2993252075304391433268374323906303200076481041463995109
+sub:39376759077330221210471049201931709948853198615,-9493608874721279041170835115951049053361083855467392,0:9493648251480356371392045587000250985071032708666007
+sub:-5280130155085387575883303159292113846385931782810769147,-1019810672320943407741389860781837676645351883469636964,0:-4260319482764444168141913298510276169740579899341132183
+sub:8457922806422768126334853404146638292374687915778812,995127687303275765352581834049367391219012453856914,0:7462795119119492360982271570097270901155675461921898
+sub:34992662458077134231839550916665303135384048742,-8152964279502937641561776276851678598174407814,0:43145626737580071873401327193516981733558456556
+sub:8237492058118191763481523977841344753346311488313,1647118892275027900093320780430922273315158677508003654,0:-1647110654782969781901557298906944431970405331196515341
+sub:-440042339580320057764506833117106015591230779324161,513962930617733479729549861446740257284856596728706,0:-954005270198053537494056694563846272876087376052867
+sub:34635151931992497392391616758693644900635250427499823,287128980133325949899403223688828310849990578079457307,0:-252493828201333452507011606930134665949355327651957484
+sub:48138028043471938379654489799905392276347294678190375860,8955461479465751553200394404645878189278984773776867,0:48129072581992472628101289405500746398158015693416598993
+sub:7624317887859192567599023956675797189693348039462742,-973928186389609942202650295012119719585450666839535260,0:981552504277469134770249318968795516775144014878998002
+sub:775402642132715473344595954662582545794117369927872,63796684229986115177275440213147384846517231365197485,0:-63021281587853399703930844258484802300723113995269613
+sub:37362432357418646911115330814724472853415155235293,87036796847105652716376950065916449694424391617602112013,0:-87036759484673295297730038950585634969951538202446876720
+sub:-300945445086734604702237356948834421909997080585835251,86543393214880176913230280671573673375238923042066,0:-301031988479949484879150587229505995583372319508877317
+sub:-654047435982290398129267970179757606710004276417490112,86422930881746427524648992269920847889306495489,0:-654047522405221279875695494828749876630852165723985601
+sub:794552108714326403336258405676646043844624401498072689,544976653015779028031808690831373018858631899320,0:794551563737673387557230373867955212471605542866173369
+sub:-195959798494693192093817249845807727728979751653,18593677965094459754532848130637776457537309190937253673,0:-18593678161054258249226040224455026303345036919917005326
+sub:57601304615929231199752161276702979414912411756145,21951958716061857914468826747760163180070915240014675524,0:-21951901114757241985237626995598886477091500327602919379
+sub:311215074108149476976205428525338649682414980832,6243249733271681693520367645680387625166806885842687456,0:-6243249422056607585370890669474959099828157203427706624
+sub:-60089258178336658544420554675373945148810656687928409220,-940949437845306471952666460412774999437620754784474,0:-60088317228898813237948602008913532373811219067173624746
+sub:3844895658482592533059556269556968661475827694950384997,527322363491124024058755005552497077118499938802,0:3844895131160229041935532210801963108978750576450446195
+sub:-6028944995177498518136854918757906591099625985414,15888968609159402431030920451645796896759203224067431,0:-15894997554154579929549057306564554803350302850052845
+sub:996013497441974701923950607149438472739078933691,112407411020615162605964831561295393767448218127,0:883606086421359539317985775588143078971630715564
+sub:7719318511816366427665580178076378029426275595533,163667549683108101581704958685911902179912256741218,0:-155948231171291735154039378507835524150485981145685
+sub:123382564431290774896664572552493052435404643704458759934,12531798191484951858836698658135874099473672299388553,0:123370032633099289944805735853834916561305170032159371381
+sub:942048756549299165967164928739420673047263758767141,3639275805983487223033248868875293658626128636190227,0:-2697227049434188057066083940135872985578864877423086
+sub:240418823794626925772533604032962688702579243574687,4907674204394917198714890794857507854118108024516422187,0:-4907433785571122571789118261253474891429405445272847500
+sub:3950213238876863129818663666502819327768719939127457701,29991402193339371342321385670768500228753395231039352972,0:-26041188954462508212502722004265680900984675291911895271
+sub:1920667868219534801099538208015399489683153418341349572065,-233412004414149707586851107620826952570732551540827551954,0:2154079872633684508686389315636226442253885969882177124019
+sub:7784321585128756839986661147453831080387260908002873,586237266875238457979621090825032494806800315405801,0:7198084318253518382007040056628798585580460592597072
+sub:-34225848774985487393407814198681047162140030645991253741,-789489361585689258214473176784928536166156096643657,0:-34225059285623901704149599725504262233603864489894610084
+sub:-944338971240124849797024756328867740674305651345965,649970029057878269623349411694378680589331156335769959,0:-650914368029118394473146436450707548330005461987115924
+sub:25816073494421415375915233416338690441492640884943818729,764159631162282034769988362928740167840568754650018844,0:25051913863259133341145245053409950273652072130293799885
+sub:7111381553325219432932289055091214481651627513743376,7385563406175559235314072692841151936136627185182905,0:-274181852850339802381783637749937454484999671439529
+sub:-438137611853268269308435226684643695607887735487389,5824616025576441892684517623276732904299681537303913,0:-6262753637429710161992952849961376599907569272791302
+sub:-9680790659167697968786181797074343179561171865505840566,-932899327759500157755319258322165056527752927868968726,0:-8747891331408197811030862538752178123033418937636871840
+sub:567793364811051880825781313748966696400236576537363041402,-976578291468928773578542746545826341079985184892523837,0:568769943102520809599359856495512522741316561722255565239
+sub:38007779187090670273244552314378701844162920433585869254,7187767815567041059760334263355184283280579165883650238,0:30820011371523629213484218051023517560882341267702219016
+sub:9014053603553462724377168305482782158101121369010869473906,755601637469656651454003843855153366578358802242924461,0:9013298001915993067725714301638927004734543010208626549445
+
+# Small integer subtraction tests
+
+subv:3358604991,36411,0:3358568580
+subv:674222429,18986,0:674203443
+subv:-3,-29805,=1:29802
+subv:3,-61063,=1:61066
+subv:2717723,-25258,=1:2742981
+subv:394,-40508,=2:40902
+subv:4744781967,21435,0:4744760532
+subv:-14,50287,0:-50301
+subv:2197,3582,=2:-1385
+subv:-5296,38162,=2:-43458
+subv:-258774,57544,=2:-316318
+subv:7417476246,-58958,=2:7417535204
+subv:6714936,-40949,0:6755885
+subv:15862,-58672,0:74534
+subv:958081,21046,0:937035
+subv:82515,-11287,0:93802
+subv:72715,-25449,0:98164
+subv:-8346662975566,52150,0:-8346663027716
+subv:96216,-33648,=2:129864
+subv:71597071124556,40786,0:71597071083770
+subv:2795133710510,-24996,=1:2795133735506
+subv:8501300888546,1685,0:8501300886861
+subv:3877092429,45143,0:3877047286
+subv:-9123902,27698,0:-9151600
+subv:-71914098152,-49684,0:-71914048468
+subv:-5633175963555389,-13681,0:-5633175963541708
+subv:-68774492,24946,0:-68799438
+subv:2837978518,2447,=1:2837976071
+subv:9527903026793981,34045,=2:9527903026759936
+subv:-719766995836501,-14683,=1:-719766995821818
+subv:6826892934542,-9599,0:6826892944141
+subv:474765520,-36545,=2:474802065
+subv:137980222,12896,0:137967326
+subv:4124108674,61743,0:4124046931
+subv:-4977334202151,16161,0:-4977334218312
+subv:-6764356933,33130,=1:-6764390063
+subv:6639990082377605303,63220,0:6639990082377542083
+subv:68384248282,-28652,0:68384276934
+subv:-64782488650,-31866,=2:-64782456784
+subv:84126983615,63328,0:84126920287
+subv:280281352217,19756,=1:280281332461
+subv:-337730569530652702,55596,0:-337730569530708298
+subv:750462053456012,-18056,=1:750462053474068
+subv:-859984251764611,60341,0:-859984251824952
+subv:781147832162694072159,10186,0:781147832162694061973
+subv:414707731902540,13874,0:414707731888666
+subv:4749293013793395,-27594,0:4749293013820989
+subv:-476780935041195291,-51859,0:-476780935041143432
+subv:-3619074244886709,38416,=1:-3619074244925125
+subv:97855199057493,40752,0:97855199016741
+subv:777675086535345464853,9892,=2:777675086535345454961
+subv:3103441257588144,4992,0:3103441257583152
+subv:136564148041720,53068,=2:136564147988652
+subv:-4664978338218772692212,46644,0:-4664978338218772738856
+subv:235299291342838779459,27155,=1:235299291342838752304
+subv:45286262426870,28782,0:45286262398088
+subv:422289955984703063,27236,=2:422289955984675827
+subv:-6379926829937406097,-63773,0:-6379926829937342324
+subv:830836702855178852980658,-61406,=1:830836702855178853042064
+subv:4000074483936047411,59077,=2:4000074483935988334
+subv:1622930723651719839167208,63514,0:1622930723651719839103694
+subv:532951482400205192926144,57956,0:532951482400205192868188
+subv:148980429628794512840,63138,=2:148980429628794449702
+subv:-9890648040727702186064,-46766,=1:-9890648040727702139298
+subv:-10934480308082214085686,63224,=2:-10934480308082214148910
+subv:355809482274653919711872,32446,0:355809482274653919679426
+subv:-33926193961684655624,11389,=2:-33926193961684667013
+subv:-7186493575578045620,15528,=1:-7186493575578061148
+subv:34706765155128499193360248,83,=1:34706765155128499193360165
+subv:4328479538590740673745630,14623,0:4328479538590740673731007
+subv:-963713498430054741223932789,30239,0:-963713498430054741223963028
+subv:-831264221344401371,39516,0:-831264221344440887
+subv:318721481318817753735405,52833,0:318721481318817753682572
+subv:-62693787119478234447,27553,=1:-62693787119478262000
+subv:-7016655669525046977522194886,32638,=2:-7016655669525046977522227524
+subv:608134301020199755515,29066,0:608134301020199726449
diff --git a/polly/lib/External/isl/imath/tests/test.bc b/polly/lib/External/isl/imath/tests/test.bc
new file mode 100644
index 00000000000..f067be8c483
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/test.bc
@@ -0,0 +1,155 @@
+/* e(a, b, m) := a^b (mod m) */
+define e(a, b, m) {
+  auto s, sb;
+
+  sb = obase; obase = 16;
+  s = 1;
+  while(b != 0) {
+    if((b % 2) == 1) {
+      s = (s * a) % m;
+    }
+    b /= 2;
+    a = (a * a) % m;
+  }
+  obase = sb;
+  return (s);
+}
+
+/* g(a, b) := (a, b) */
+define g(a, b) {
+  auto r;
+
+  while(b != 0) {
+    r = a % b;
+    a = b;
+    b = r;
+  }
+  if(a < 0) {
+    return(-a);
+  } else {
+    return(a);
+  }
+}
+
+define a(x) {
+  if(x < 0) {
+    return (-x);
+  } else {
+    return (x);
+  }
+}
+
+xgu = 0; xgv = 0;
+define x(a, b) {
+  auto u, v, x, y, t, r;
+
+  if(a(b) > a(a)) {
+    t = b; b = a; a = t; r = 1;
+  } else {
+    r = 0;
+  }
+
+  u = 1; v = 0;
+  x = 0; y = 1;
+
+  while(b != 0) {
+    t = a / b;
+
+    a = a - (t * b);
+    v = v - (t * y);
+    u = u - (t * x);
+
+    if(a(a) < a(b)) {
+      t = a; a = b; b = t;
+      t = v; v = y; y = t;
+      t = u; u = x; x = t;
+    }
+  }
+
+  if(r) {
+    xgu = v; xgv = u;
+  } else {
+    xgu = u; xgv = v;
+  }
+
+  if(a < 0) {
+    return (-a);
+  } else {
+    return (a);
+  }
+}
+
+define i(a, m) {
+  auto c;
+
+  c = x(a, m);
+  if(c != 1)
+    return(0);
+
+  return ((xgu + m) % m);
+}
+
+scale = 0
+rand_modulus = 2860385147
+rand_base = 129
+rand_seed = 10342799
+
+define srand(s) {
+  rand_seed = s;
+}
+
+define rand(n) {
+  auto r;
+
+  r = (rand_seed * rand_base) % rand_modulus;
+  rand_seed = r;
+  return(r % n);
+}
+
+define rand_digits(k) {
+  auto s;
+
+  s = 0;
+  while(k > 0) {
+    s = (s * 10) + rand(10);
+    k = k - 1;
+  }
+
+  return(s);
+}
+
+define rval(ndigits, probneg) {
+  auto x;
+  x = rand_digits(ndigits);
+  if(probneg > 0 && rand(100) < probneg)
+    x = x * -1;
+  return(x);
+}
+
+for(i = 2; i < 100; i += 4) {
+  value = rand_digits(i);
+  obase = 256;
+  print "readuns:", value;
+  obase = 10;
+  print ":", value, "\n"  
+}
+
+/*
+for(i = 2; i < 100; i += 4) {
+  for(j = 0; j < 2; ++j) {
+    ndig = rand(7);
+    base = rval(ndig, 20);
+    expt = rand_digits(i);
+    mod = rand_digits(i);
+
+    result = e(base, expt, mod);
+    if(result < 0) {
+      result = result + mod;
+    }
+    print "emodbv:", base, ",", expt, ",", mod, ",0:", result, "\n";
+    print "emodbv:", base, ",", expt, ",", mod, ",=2:", result, "\n";
+    print "emodbv:", base, ",", expt, ",", mod, ",=3:", result, "\n";
+  }
+}
+*/
+halt
diff --git a/polly/lib/External/isl/imath/tests/test.sh b/polly/lib/External/isl/imath/tests/test.sh
new file mode 100755
index 00000000000..d1ced881e34
--- /dev/null
+++ b/polly/lib/External/isl/imath/tests/test.sh
@@ -0,0 +1,77 @@
+#!/bin/sh
+## 
+## Name:     test.sh
+## Purpose:  Run test suites for IMath library.
+## Author:   M. J. Fromberger <http://spinning-yarns.org/michael/>
+##
+
+if [ ! -f ../imtest ] ; then
+  echo "I can't find the imath test driver 'imtest', did you build it?"
+  echo "I can't proceed with the unit tests until you do so, sorry."
+  exit 1
+fi
+
+echo "-- Running all available unit tests"
+../imtest *.t | grep -v OK
+
+echo ""
+echo "-- Running test to compute 1024 decimal digits of pi"
+if [ ! -f ../pi ] ; then
+  echo "I can't find the pi computing program, did you build it?"
+  echo "I can't proceed with the pi test until you do so, sorry."
+  exit 1
+fi
+
+tempfile="/tmp/pi.1024.$$"
+
+../pi 1024 | tr -d '\r\n' > ${tempfile}
+if cmp -s ${tempfile} ./pi1024.txt ; then
+  echo "  PASSED 1024 digits"
+else
+  echo "  FAILED"
+  echo "Obtained:"
+  cat ${tempfile}
+  echo "Expected:"
+  cat ./pi1024.txt
+fi
+rm -f ${tempfile}
+
+tempfile="/tmp/pi.1698.$$"
+
+echo "-- Running test to compute 1698 hexadecimal digits of pi"
+
+../pi 1698 16 | tr -d '\r\n' > ${tempfile}
+if cmp -s ${tempfile} ./pi1698-16.txt ; then
+  echo "  PASSED 1698 digits"
+else
+  echo "  FAILED"
+  echo "Obtained:"
+  cat ${tempfile}
+  echo "Expected:"
+  cat ./pi1698-16.txt
+fi
+rm -f ${tempfile}
+
+tempfile="/tmp/pi.1500.$$"
+
+echo "-- Running test to compute 1500 decimal digits of pi"
+
+../pi 1500 10 | tr -d '\r\n' > ${tempfile}
+if cmp -s ${tempfile} ./pi1500-10.txt ; then
+  echo "  PASSED 1500 digits"
+else
+  echo "  FAILED"
+  echo "Obtained:"
+  cat ${tempfile}
+  echo "Expected:"
+  cat ./pi1500-10.txt
+fi
+rm -f ${tempfile}
+
+echo "-- Running regression tests"
+
+for bug in bug-swap ; do
+    ../${bug}
+done
+
+exit 0
diff --git a/polly/lib/External/isl/imath_wrap/gmp_compat.c b/polly/lib/External/isl/imath_wrap/gmp_compat.c
new file mode 100644
index 00000000000..a116913a44c
--- /dev/null
+++ b/polly/lib/External/isl/imath_wrap/gmp_compat.c
@@ -0,0 +1,2 @@
+#include "wrap.h"
+#include "../imath/gmp_compat.c"
diff --git a/polly/lib/External/isl/imath_wrap/gmp_compat.h b/polly/lib/External/isl/imath_wrap/gmp_compat.h
new file mode 100644
index 00000000000..11a332daf54
--- /dev/null
+++ b/polly/lib/External/isl/imath_wrap/gmp_compat.h
@@ -0,0 +1,2 @@
+#include "wrap.h"
+#include "../imath/gmp_compat.h"
diff --git a/polly/lib/External/isl/imath_wrap/imath.c b/polly/lib/External/isl/imath_wrap/imath.c
new file mode 100644
index 00000000000..c4e35c1ac8b
--- /dev/null
+++ b/polly/lib/External/isl/imath_wrap/imath.c
@@ -0,0 +1,2 @@
+#include "wrap.h"
+#include "../imath/imath.c"
diff --git a/polly/lib/External/isl/imath_wrap/imath.h b/polly/lib/External/isl/imath_wrap/imath.h
new file mode 100644
index 00000000000..12029ce5b67
--- /dev/null
+++ b/polly/lib/External/isl/imath_wrap/imath.h
@@ -0,0 +1,2 @@
+#include "wrap.h"
+#include "../imath/imath.h"
diff --git a/polly/lib/External/isl/imath_wrap/imrat.c b/polly/lib/External/isl/imath_wrap/imrat.c
new file mode 100644
index 00000000000..0c7feaaeadb
--- /dev/null
+++ b/polly/lib/External/isl/imath_wrap/imrat.c
@@ -0,0 +1,2 @@
+#include "wrap.h"
+#include "../imath/imrat.c"
diff --git a/polly/lib/External/isl/imath_wrap/imrat.h b/polly/lib/External/isl/imath_wrap/imrat.h
new file mode 100644
index 00000000000..c9983900a45
--- /dev/null
+++ b/polly/lib/External/isl/imath_wrap/imrat.h
@@ -0,0 +1,2 @@
+#include "wrap.h"
+#include "../imath/imrat.h"
diff --git a/polly/lib/External/isl/imath_wrap/wrap.h b/polly/lib/External/isl/imath_wrap/wrap.h
new file mode 100644
index 00000000000..f68e50d20f4
--- /dev/null
+++ b/polly/lib/External/isl/imath_wrap/wrap.h
@@ -0,0 +1,172 @@
+#ifndef ISL_IMATH_WRAP
+#define ISL_IMATH_WRAP
+
+#define MP_BADARG	ISL_MP_BADARG
+#define MP_FALSE	ISL_MP_FALSE
+#define MP_MEMORY	ISL_MP_MEMORY
+#define MP_MINERR	ISL_MP_MINERR
+#define MP_NEG	ISL_MP_NEG
+#define MP_OK	ISL_MP_OK
+#define MP_RANGE	ISL_MP_RANGE
+#define MP_TRUE	ISL_MP_TRUE
+#define MP_TRUNC	ISL_MP_TRUNC
+#define MP_UNDEF	ISL_MP_UNDEF
+#define MP_ZPOS	ISL_MP_ZPOS
+
+#define impq_canonicalize	isl_impq_canonicalize
+#define impq_clear	isl_impq_clear
+#define impq_cmp	isl_impq_cmp
+#define impq_denref	isl_impq_denref
+#define impq_get_str	isl_impq_get_str
+#define impq_init	isl_impq_init
+#define impq_mul	isl_impq_mul
+#define impq_numref	isl_impq_numref
+#define impq_set	isl_impq_set
+#define impq_set_str	isl_impq_set_str
+#define impq_set_ui	isl_impq_set_ui
+#define impq_sgn	isl_impq_sgn
+#define impz_abs	isl_impz_abs
+#define impz_add	isl_impz_add
+#define impz_addmul	isl_impz_addmul
+#define impz_add_ui	isl_impz_add_ui
+#define impz_cdiv_q	isl_impz_cdiv_q
+#define impz_clear	isl_impz_clear
+#define impz_cmp	isl_impz_cmp
+#define impz_cmpabs	isl_impz_cmpabs
+#define impz_cmp_si	isl_impz_cmp_si
+#define impz_divexact	isl_impz_divexact
+#define impz_divexact_ui	isl_impz_divexact_ui
+#define impz_divisible_p	isl_impz_divisible_p
+#define impz_export	isl_impz_export
+#define impz_fdiv_q	isl_impz_fdiv_q
+#define impz_fdiv_q_ui	isl_impz_fdiv_q_ui
+#define impz_fdiv_r	isl_impz_fdiv_r
+#define impz_gcd	isl_impz_gcd
+#define impz_get_si	isl_impz_get_si
+#define impz_get_str	isl_impz_get_str
+#define impz_get_ui	isl_impz_get_ui
+#define impz_import	isl_impz_import
+#define impz_init	isl_impz_init
+#define impz_lcm	isl_impz_lcm
+#define impz_mul	isl_impz_mul
+#define impz_mul_2exp	isl_impz_mul_2exp
+#define impz_mul_ui	isl_impz_mul_ui
+#define impz_neg	isl_impz_neg
+#define impz_pow_ui	isl_impz_pow_ui
+#define impz_set	isl_impz_set
+#define impz_set_si	isl_impz_set_si
+#define impz_set_str	isl_impz_set_str
+#define impz_set_ui	isl_impz_set_ui
+#define impz_sgn	isl_impz_sgn
+#define impz_sizeinbase	isl_impz_sizeinbase
+#define impz_sub	isl_impz_sub
+#define impz_submul	isl_impz_submul
+#define impz_sub_ui	isl_impz_sub_ui
+#define impz_swap	isl_impz_swap
+#define impz_tdiv_q	isl_impz_tdiv_q
+#define mp_error_string	isl_mp_error_string
+#define mp_int_abs	isl_mp_int_abs
+#define mp_int_add	isl_mp_int_add
+#define mp_int_add_value	isl_mp_int_add_value
+#define mp_int_alloc	isl_mp_int_alloc
+#define mp_int_binary_len	isl_mp_int_binary_len
+#define mp_int_clear	isl_mp_int_clear
+#define mp_int_compare	isl_mp_int_compare
+#define mp_int_compare_unsigned	isl_mp_int_compare_unsigned
+#define mp_int_compare_uvalue	isl_mp_int_compare_uvalue
+#define mp_int_compare_value	isl_mp_int_compare_value
+#define mp_int_compare_zero	isl_mp_int_compare_zero
+#define mp_int_copy	isl_mp_int_copy
+#define mp_int_count_bits	isl_mp_int_count_bits
+#define mp_int_div	isl_mp_int_div
+#define mp_int_divisible_value	isl_mp_int_divisible_value
+#define mp_int_div_pow2	isl_mp_int_div_pow2
+#define mp_int_div_value	isl_mp_int_div_value
+#define mp_int_egcd	isl_mp_int_egcd
+#define mp_int_expt	isl_mp_int_expt
+#define mp_int_expt_full	isl_mp_int_expt_full
+#define mp_int_exptmod	isl_mp_int_exptmod
+#define mp_int_exptmod_bvalue	isl_mp_int_exptmod_bvalue
+#define mp_int_exptmod_evalue	isl_mp_int_exptmod_evalue
+#define mp_int_exptmod_known	isl_mp_int_exptmod_known
+#define mp_int_expt_value	isl_mp_int_expt_value
+#define mp_int_free	isl_mp_int_free
+#define mp_int_gcd	isl_mp_int_gcd
+#define mp_int_init	isl_mp_int_init
+#define mp_int_init_copy	isl_mp_int_init_copy
+#define mp_int_init_size	isl_mp_int_init_size
+#define mp_int_init_uvalue	isl_mp_int_init_uvalue
+#define mp_int_init_value	isl_mp_int_init_value
+#define mp_int_invmod	isl_mp_int_invmod
+#define mp_int_is_pow2	isl_mp_int_is_pow2
+#define mp_int_lcm	isl_mp_int_lcm
+#define mp_int_mod	isl_mp_int_mod
+#define mp_int_mul	isl_mp_int_mul
+#define mp_int_mul_pow2	isl_mp_int_mul_pow2
+#define mp_int_mul_value	isl_mp_int_mul_value
+#define mp_int_neg	isl_mp_int_neg
+#define mp_int_read_binary	isl_mp_int_read_binary
+#define mp_int_read_cstring	isl_mp_int_read_cstring
+#define mp_int_read_string	isl_mp_int_read_string
+#define mp_int_read_unsigned	isl_mp_int_read_unsigned
+#define mp_int_redux_const	isl_mp_int_redux_const
+#define mp_int_root	isl_mp_int_root
+#define mp_int_set_uvalue	isl_mp_int_set_uvalue
+#define mp_int_set_value	isl_mp_int_set_value
+#define mp_int_sqr	isl_mp_int_sqr
+#define mp_int_string_len	isl_mp_int_string_len
+#define mp_int_sub	isl_mp_int_sub
+#define mp_int_sub_value	isl_mp_int_sub_value
+#define mp_int_swap	isl_mp_int_swap
+#define mp_int_to_binary	isl_mp_int_to_binary
+#define mp_int_to_int	isl_mp_int_to_int
+#define mp_int_to_string	isl_mp_int_to_string
+#define mp_int_to_uint	isl_mp_int_to_uint
+#define mp_int_to_unsigned	isl_mp_int_to_unsigned
+#define mp_int_unsigned_len	isl_mp_int_unsigned_len
+#define mp_int_zero	isl_mp_int_zero
+#define mp_rat_abs	isl_mp_rat_abs
+#define mp_rat_add	isl_mp_rat_add
+#define mp_rat_add_int	isl_mp_rat_add_int
+#define mp_rat_alloc	isl_mp_rat_alloc
+#define mp_rat_clear	isl_mp_rat_clear
+#define mp_rat_compare	isl_mp_rat_compare
+#define mp_rat_compare_unsigned	isl_mp_rat_compare_unsigned
+#define mp_rat_compare_value	isl_mp_rat_compare_value
+#define mp_rat_compare_zero	isl_mp_rat_compare_zero
+#define mp_rat_copy	isl_mp_rat_copy
+#define mp_rat_decimal_len	isl_mp_rat_decimal_len
+#define mp_rat_denom	isl_mp_rat_denom
+#define mp_rat_denom_ref	isl_mp_rat_denom_ref
+#define mp_rat_div	isl_mp_rat_div
+#define mp_rat_div_int	isl_mp_rat_div_int
+#define mp_rat_expt	isl_mp_rat_expt
+#define mp_rat_free	isl_mp_rat_free
+#define mp_rat_init	isl_mp_rat_init
+#define mp_rat_init_copy	isl_mp_rat_init_copy
+#define mp_rat_init_size	isl_mp_rat_init_size
+#define mp_rat_is_integer	isl_mp_rat_is_integer
+#define mp_rat_mul	isl_mp_rat_mul
+#define mp_rat_mul_int	isl_mp_rat_mul_int
+#define mp_rat_neg	isl_mp_rat_neg
+#define mp_rat_numer	isl_mp_rat_numer
+#define mp_rat_numer_ref	isl_mp_rat_numer_ref
+#define mp_rat_read_cdecimal	isl_mp_rat_read_cdecimal
+#define mp_rat_read_cstring	isl_mp_rat_read_cstring
+#define mp_rat_read_decimal	isl_mp_rat_read_decimal
+#define mp_rat_read_string	isl_mp_rat_read_string
+#define mp_rat_read_ustring	isl_mp_rat_read_ustring
+#define mp_rat_recip	isl_mp_rat_recip
+#define mp_rat_reduce	isl_mp_rat_reduce
+#define mp_rat_set_uvalue	isl_mp_rat_set_uvalue
+#define mp_rat_set_value	isl_mp_rat_set_value
+#define mp_rat_sign	isl_mp_rat_sign
+#define mp_rat_string_len	isl_mp_rat_string_len
+#define mp_rat_sub	isl_mp_rat_sub
+#define mp_rat_sub_int	isl_mp_rat_sub_int
+#define mp_rat_to_decimal	isl_mp_rat_to_decimal
+#define mp_rat_to_ints	isl_mp_rat_to_ints
+#define mp_rat_to_string	isl_mp_rat_to_string
+#define mp_rat_zero	isl_mp_rat_zero
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/aff.h b/polly/lib/External/isl/include/isl/aff.h
new file mode 100644
index 00000000000..4ed02854d1c
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/aff.h
@@ -0,0 +1,663 @@
+#ifndef ISL_AFF_H
+#define ISL_AFF_H
+
+#include <isl/local_space.h>
+#include <isl/printer.h>
+#include <isl/set_type.h>
+#include <isl/aff_type.h>
+#include <isl/list.h>
+#include <isl/multi.h>
+#include <isl/union_set_type.h>
+#include <isl/val.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+__isl_give isl_aff *isl_aff_zero_on_domain(__isl_take isl_local_space *ls);
+__isl_give isl_aff *isl_aff_val_on_domain(__isl_take isl_local_space *ls,
+	__isl_take isl_val *val);
+__isl_give isl_aff *isl_aff_var_on_domain(__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_aff *isl_aff_nan_on_domain(__isl_take isl_local_space *ls);
+
+__isl_give isl_aff *isl_aff_copy(__isl_keep isl_aff *aff);
+__isl_null isl_aff *isl_aff_free(__isl_take isl_aff *aff);
+
+isl_ctx *isl_aff_get_ctx(__isl_keep isl_aff *aff);
+
+int isl_aff_dim(__isl_keep isl_aff *aff, enum isl_dim_type type);
+int isl_aff_involves_dims(__isl_keep isl_aff *aff,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+__isl_give isl_space *isl_aff_get_domain_space(__isl_keep isl_aff *aff);
+__isl_give isl_space *isl_aff_get_space(__isl_keep isl_aff *aff);
+__isl_give isl_local_space *isl_aff_get_domain_local_space(
+	__isl_keep isl_aff *aff);
+__isl_give isl_local_space *isl_aff_get_local_space(__isl_keep isl_aff *aff);
+
+const char *isl_aff_get_dim_name(__isl_keep isl_aff *aff,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_val *isl_aff_get_constant_val(__isl_keep isl_aff *aff);
+__isl_give isl_val *isl_aff_get_coefficient_val(__isl_keep isl_aff *aff,
+	enum isl_dim_type type, int pos);
+int isl_aff_coefficient_sgn(__isl_keep isl_aff *aff,
+	enum isl_dim_type type, int pos);
+__isl_give isl_val *isl_aff_get_denominator_val(__isl_keep isl_aff *aff);
+__isl_give isl_aff *isl_aff_set_constant_si(__isl_take isl_aff *aff, int v);
+__isl_give isl_aff *isl_aff_set_constant_val(__isl_take isl_aff *aff,
+	__isl_take isl_val *v);
+__isl_give isl_aff *isl_aff_set_coefficient_si(__isl_take isl_aff *aff,
+	enum isl_dim_type type, int pos, int v);
+__isl_give isl_aff *isl_aff_set_coefficient_val(__isl_take isl_aff *aff,
+	enum isl_dim_type type, int pos, __isl_take isl_val *v);
+__isl_give isl_aff *isl_aff_add_constant_si(__isl_take isl_aff *aff, int v);
+__isl_give isl_aff *isl_aff_add_constant_val(__isl_take isl_aff *aff,
+	__isl_take isl_val *v);
+__isl_give isl_aff *isl_aff_add_constant_num_si(__isl_take isl_aff *aff, int v);
+__isl_give isl_aff *isl_aff_add_coefficient_si(__isl_take isl_aff *aff,
+	enum isl_dim_type type, int pos, int v);
+__isl_give isl_aff *isl_aff_add_coefficient_val(__isl_take isl_aff *aff,
+	enum isl_dim_type type, int pos, __isl_take isl_val *v);
+
+int isl_aff_is_cst(__isl_keep isl_aff *aff);
+
+__isl_give isl_aff *isl_aff_set_tuple_id(__isl_take isl_aff *aff,
+	enum isl_dim_type type, __isl_take isl_id *id);
+__isl_give isl_aff *isl_aff_set_dim_name(__isl_take isl_aff *aff,
+	enum isl_dim_type type, unsigned pos, const char *s);
+__isl_give isl_aff *isl_aff_set_dim_id(__isl_take isl_aff *aff,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id);
+
+int isl_aff_find_dim_by_name(__isl_keep isl_aff *aff, enum isl_dim_type type,
+	const char *name);
+
+int isl_aff_plain_is_equal(__isl_keep isl_aff *aff1, __isl_keep isl_aff *aff2);
+int isl_aff_plain_is_zero(__isl_keep isl_aff *aff);
+int isl_aff_is_nan(__isl_keep isl_aff *aff);
+
+__isl_give isl_aff *isl_aff_get_div(__isl_keep isl_aff *aff, int pos);
+
+__isl_give isl_aff *isl_aff_neg(__isl_take isl_aff *aff);
+__isl_give isl_aff *isl_aff_ceil(__isl_take isl_aff *aff);
+__isl_give isl_aff *isl_aff_floor(__isl_take isl_aff *aff);
+__isl_give isl_aff *isl_aff_mod_val(__isl_take isl_aff *aff,
+	__isl_take isl_val *mod);
+
+__isl_give isl_aff *isl_aff_mul(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2);
+__isl_give isl_aff *isl_aff_div(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2);
+__isl_give isl_aff *isl_aff_add(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2);
+__isl_give isl_aff *isl_aff_sub(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2);
+
+__isl_give isl_aff *isl_aff_scale_val(__isl_take isl_aff *aff,
+	__isl_take isl_val *v);
+__isl_give isl_aff *isl_aff_scale_down_ui(__isl_take isl_aff *aff, unsigned f);
+__isl_give isl_aff *isl_aff_scale_down_val(__isl_take isl_aff *aff,
+	__isl_take isl_val *v);
+
+__isl_give isl_aff *isl_aff_insert_dims(__isl_take isl_aff *aff,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_aff *isl_aff_add_dims(__isl_take isl_aff *aff,
+	enum isl_dim_type type, unsigned n);
+__isl_give isl_aff *isl_aff_move_dims(__isl_take isl_aff *aff,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n);
+__isl_give isl_aff *isl_aff_drop_dims(__isl_take isl_aff *aff,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_aff *isl_aff_project_domain_on_params(__isl_take isl_aff *aff);
+
+__isl_give isl_aff *isl_aff_align_params(__isl_take isl_aff *aff,
+	__isl_take isl_space *model);
+
+__isl_give isl_aff *isl_aff_gist(__isl_take isl_aff *aff,
+	__isl_take isl_set *context);
+__isl_give isl_aff *isl_aff_gist_params(__isl_take isl_aff *aff,
+	__isl_take isl_set *context);
+
+__isl_give isl_aff *isl_aff_pullback_aff(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2);
+__isl_give isl_aff *isl_aff_pullback_multi_aff(__isl_take isl_aff *aff,
+	__isl_take isl_multi_aff *ma);
+
+__isl_give isl_basic_set *isl_aff_zero_basic_set(__isl_take isl_aff *aff);
+__isl_give isl_basic_set *isl_aff_neg_basic_set(__isl_take isl_aff *aff);
+
+__isl_give isl_basic_set *isl_aff_le_basic_set(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2);
+__isl_give isl_basic_set *isl_aff_ge_basic_set(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2);
+
+__isl_give isl_aff *isl_aff_read_from_str(isl_ctx *ctx, const char *str);
+__isl_give isl_printer *isl_printer_print_aff(__isl_take isl_printer *p,
+	__isl_keep isl_aff *aff);
+void isl_aff_dump(__isl_keep isl_aff *aff);
+
+isl_ctx *isl_pw_aff_get_ctx(__isl_keep isl_pw_aff *pwaff);
+__isl_give isl_space *isl_pw_aff_get_domain_space(__isl_keep isl_pw_aff *pwaff);
+__isl_give isl_space *isl_pw_aff_get_space(__isl_keep isl_pw_aff *pwaff);
+
+__isl_give isl_pw_aff *isl_pw_aff_from_aff(__isl_take isl_aff *aff);
+__isl_give isl_pw_aff *isl_pw_aff_empty(__isl_take isl_space *dim);
+__isl_give isl_pw_aff *isl_pw_aff_alloc(__isl_take isl_set *set,
+	__isl_take isl_aff *aff);
+__isl_give isl_pw_aff *isl_pw_aff_zero_on_domain(
+	__isl_take isl_local_space *ls);
+__isl_give isl_pw_aff *isl_pw_aff_var_on_domain(__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_pw_aff *isl_pw_aff_nan_on_domain(__isl_take isl_local_space *ls);
+__isl_give isl_pw_aff *isl_pw_aff_val_on_domain(__isl_take isl_set *domain,
+	__isl_take isl_val *v);
+
+__isl_give isl_pw_aff *isl_set_indicator_function(__isl_take isl_set *set);
+
+const char *isl_pw_aff_get_dim_name(__isl_keep isl_pw_aff *pa,
+	enum isl_dim_type type, unsigned pos);
+int isl_pw_aff_has_dim_id(__isl_keep isl_pw_aff *pa,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_id *isl_pw_aff_get_dim_id(__isl_keep isl_pw_aff *pa,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_pw_aff *isl_pw_aff_set_dim_id(__isl_take isl_pw_aff *pma,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id);
+
+int isl_pw_aff_find_dim_by_name(__isl_keep isl_pw_aff *pa,
+	enum isl_dim_type type, const char *name);
+
+int isl_pw_aff_is_empty(__isl_keep isl_pw_aff *pwaff);
+int isl_pw_aff_involves_nan(__isl_keep isl_pw_aff *pa);
+int isl_pw_aff_plain_cmp(__isl_keep isl_pw_aff *pa1,
+	__isl_keep isl_pw_aff *pa2);
+int isl_pw_aff_plain_is_equal(__isl_keep isl_pw_aff *pwaff1,
+	__isl_keep isl_pw_aff *pwaff2);
+int isl_pw_aff_is_equal(__isl_keep isl_pw_aff *pa1, __isl_keep isl_pw_aff *pa2);
+
+__isl_give isl_pw_aff *isl_pw_aff_union_min(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2);
+__isl_give isl_pw_aff *isl_pw_aff_union_max(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2);
+__isl_give isl_pw_aff *isl_pw_aff_union_add(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2);
+
+__isl_give isl_pw_aff *isl_pw_aff_copy(__isl_keep isl_pw_aff *pwaff);
+__isl_null isl_pw_aff *isl_pw_aff_free(__isl_take isl_pw_aff *pwaff);
+
+unsigned isl_pw_aff_dim(__isl_keep isl_pw_aff *pwaff, enum isl_dim_type type);
+int isl_pw_aff_involves_dims(__isl_keep isl_pw_aff *pwaff,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+int isl_pw_aff_is_cst(__isl_keep isl_pw_aff *pwaff);
+
+__isl_give isl_pw_aff *isl_pw_aff_align_params(__isl_take isl_pw_aff *pwaff,
+	__isl_take isl_space *model);
+
+int isl_pw_aff_has_tuple_id(__isl_keep isl_pw_aff *pa, enum isl_dim_type type);
+__isl_give isl_id *isl_pw_aff_get_tuple_id(__isl_keep isl_pw_aff *pa,
+	enum isl_dim_type type);
+__isl_give isl_pw_aff *isl_pw_aff_set_tuple_id(__isl_take isl_pw_aff *pwaff,
+	enum isl_dim_type type, __isl_take isl_id *id);
+__isl_give isl_pw_aff *isl_pw_aff_reset_tuple_id(__isl_take isl_pw_aff *pa,
+	enum isl_dim_type type);
+
+__isl_give isl_set *isl_pw_aff_params(__isl_take isl_pw_aff *pwa);
+__isl_give isl_set *isl_pw_aff_domain(__isl_take isl_pw_aff *pwaff);
+__isl_give isl_pw_aff *isl_pw_aff_from_range(__isl_take isl_pw_aff *pwa);
+
+__isl_give isl_pw_aff *isl_pw_aff_min(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2);
+__isl_give isl_pw_aff *isl_pw_aff_max(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2);
+__isl_give isl_pw_aff *isl_pw_aff_mul(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2);
+__isl_give isl_pw_aff *isl_pw_aff_div(__isl_take isl_pw_aff *pa1,
+	__isl_take isl_pw_aff *pa2);
+__isl_give isl_pw_aff *isl_pw_aff_add(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2);
+__isl_give isl_pw_aff *isl_pw_aff_sub(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2);
+__isl_give isl_pw_aff *isl_pw_aff_neg(__isl_take isl_pw_aff *pwaff);
+__isl_give isl_pw_aff *isl_pw_aff_ceil(__isl_take isl_pw_aff *pwaff);
+__isl_give isl_pw_aff *isl_pw_aff_floor(__isl_take isl_pw_aff *pwaff);
+__isl_give isl_pw_aff *isl_pw_aff_mod_val(__isl_take isl_pw_aff *pa,
+	__isl_take isl_val *mod);
+__isl_give isl_pw_aff *isl_pw_aff_tdiv_q(__isl_take isl_pw_aff *pa1,
+	__isl_take isl_pw_aff *pa2);
+__isl_give isl_pw_aff *isl_pw_aff_tdiv_r(__isl_take isl_pw_aff *pa1,
+	__isl_take isl_pw_aff *pa2);
+
+__isl_give isl_pw_aff *isl_pw_aff_intersect_params(__isl_take isl_pw_aff *pa,
+	__isl_take isl_set *set);
+__isl_give isl_pw_aff *isl_pw_aff_intersect_domain(__isl_take isl_pw_aff *pa,
+	__isl_take isl_set *set);
+
+__isl_give isl_pw_aff *isl_pw_aff_cond(__isl_take isl_pw_aff *cond,
+	__isl_take isl_pw_aff *pwaff_true, __isl_take isl_pw_aff *pwaff_false);
+
+__isl_give isl_pw_aff *isl_pw_aff_scale_val(__isl_take isl_pw_aff *pa,
+	__isl_take isl_val *v);
+__isl_give isl_pw_aff *isl_pw_aff_scale_down_val(__isl_take isl_pw_aff *pa,
+	__isl_take isl_val *f);
+
+__isl_give isl_pw_aff *isl_pw_aff_insert_dims(__isl_take isl_pw_aff *pwaff,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_pw_aff *isl_pw_aff_add_dims(__isl_take isl_pw_aff *pwaff,
+	enum isl_dim_type type, unsigned n);
+__isl_give isl_pw_aff *isl_pw_aff_move_dims(__isl_take isl_pw_aff *pa,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n);
+__isl_give isl_pw_aff *isl_pw_aff_drop_dims(__isl_take isl_pw_aff *pwaff,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+__isl_give isl_pw_aff *isl_pw_aff_coalesce(__isl_take isl_pw_aff *pwqp);
+__isl_give isl_pw_aff *isl_pw_aff_gist(__isl_take isl_pw_aff *pwaff,
+	__isl_take isl_set *context);
+__isl_give isl_pw_aff *isl_pw_aff_gist_params(__isl_take isl_pw_aff *pwaff,
+	__isl_take isl_set *context);
+
+__isl_give isl_pw_aff *isl_pw_aff_pullback_multi_aff(
+	__isl_take isl_pw_aff *pa, __isl_take isl_multi_aff *ma);
+__isl_give isl_pw_aff *isl_pw_aff_pullback_pw_multi_aff(
+	__isl_take isl_pw_aff *pa, __isl_take isl_pw_multi_aff *pma);
+__isl_give isl_pw_aff *isl_pw_aff_pullback_multi_pw_aff(
+	__isl_take isl_pw_aff *pa, __isl_take isl_multi_pw_aff *mpa);
+
+int isl_pw_aff_n_piece(__isl_keep isl_pw_aff *pwaff);
+int isl_pw_aff_foreach_piece(__isl_keep isl_pw_aff *pwaff,
+	int (*fn)(__isl_take isl_set *set, __isl_take isl_aff *aff,
+		    void *user), void *user);
+
+__isl_give isl_set *isl_set_from_pw_aff(__isl_take isl_pw_aff *pwaff);
+__isl_give isl_map *isl_map_from_pw_aff(__isl_take isl_pw_aff *pwaff);
+
+__isl_give isl_set *isl_pw_aff_nonneg_set(__isl_take isl_pw_aff *pwaff);
+__isl_give isl_set *isl_pw_aff_zero_set(__isl_take isl_pw_aff *pwaff);
+__isl_give isl_set *isl_pw_aff_non_zero_set(__isl_take isl_pw_aff *pwaff);
+
+__isl_give isl_set *isl_pw_aff_eq_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2);
+__isl_give isl_set *isl_pw_aff_ne_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2);
+__isl_give isl_set *isl_pw_aff_le_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2);
+__isl_give isl_set *isl_pw_aff_lt_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2);
+__isl_give isl_set *isl_pw_aff_ge_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2);
+__isl_give isl_set *isl_pw_aff_gt_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2);
+
+__isl_give isl_pw_aff *isl_pw_aff_read_from_str(isl_ctx *ctx, const char *str);
+__isl_give isl_printer *isl_printer_print_pw_aff(__isl_take isl_printer *p,
+	__isl_keep isl_pw_aff *pwaff);
+void isl_pw_aff_dump(__isl_keep isl_pw_aff *pwaff);
+
+__isl_give isl_pw_aff *isl_pw_aff_list_min(__isl_take isl_pw_aff_list *list);
+__isl_give isl_pw_aff *isl_pw_aff_list_max(__isl_take isl_pw_aff_list *list);
+
+__isl_give isl_set *isl_pw_aff_list_eq_set(__isl_take isl_pw_aff_list *list1,
+	__isl_take isl_pw_aff_list *list2);
+__isl_give isl_set *isl_pw_aff_list_ne_set(__isl_take isl_pw_aff_list *list1,
+	__isl_take isl_pw_aff_list *list2);
+__isl_give isl_set *isl_pw_aff_list_le_set(__isl_take isl_pw_aff_list *list1,
+	__isl_take isl_pw_aff_list *list2);
+__isl_give isl_set *isl_pw_aff_list_lt_set(__isl_take isl_pw_aff_list *list1,
+	__isl_take isl_pw_aff_list *list2);
+__isl_give isl_set *isl_pw_aff_list_ge_set(__isl_take isl_pw_aff_list *list1,
+	__isl_take isl_pw_aff_list *list2);
+__isl_give isl_set *isl_pw_aff_list_gt_set(__isl_take isl_pw_aff_list *list1,
+	__isl_take isl_pw_aff_list *list2);
+
+ISL_DECLARE_MULTI(aff)
+
+__isl_give isl_multi_aff *isl_multi_aff_from_aff(__isl_take isl_aff *aff);
+__isl_give isl_multi_aff *isl_multi_aff_identity(__isl_take isl_space *space);
+__isl_give isl_multi_aff *isl_multi_aff_domain_map(__isl_take isl_space *space);
+__isl_give isl_multi_aff *isl_multi_aff_range_map(__isl_take isl_space *space);
+__isl_give isl_multi_aff *isl_multi_aff_project_out_map(
+	__isl_take isl_space *space, enum isl_dim_type type,
+	unsigned first, unsigned n);
+
+__isl_give isl_multi_aff *isl_multi_aff_multi_val_on_space(
+	__isl_take isl_space *space, __isl_take isl_multi_val *mv);
+
+__isl_give isl_multi_aff *isl_multi_aff_floor(__isl_take isl_multi_aff *ma);
+
+__isl_give isl_multi_aff *isl_multi_aff_add(__isl_take isl_multi_aff *maff1,
+	__isl_take isl_multi_aff *maff2);
+__isl_give isl_multi_aff *isl_multi_aff_sub(__isl_take isl_multi_aff *ma1,
+	__isl_take isl_multi_aff *ma2);
+
+__isl_give isl_multi_aff *isl_multi_aff_product(
+	__isl_take isl_multi_aff *ma1, __isl_take isl_multi_aff *ma2);
+
+__isl_give isl_multi_aff *isl_multi_aff_gist_params(
+	__isl_take isl_multi_aff *maff, __isl_take isl_set *context);
+__isl_give isl_multi_aff *isl_multi_aff_gist(__isl_take isl_multi_aff *maff,
+	__isl_take isl_set *context);
+
+__isl_give isl_multi_aff *isl_multi_aff_lift(__isl_take isl_multi_aff *maff,
+	__isl_give isl_local_space **ls);
+
+__isl_give isl_multi_aff *isl_multi_aff_pullback_multi_aff(
+	__isl_take isl_multi_aff *ma1, __isl_take isl_multi_aff *ma2);
+
+__isl_give isl_multi_aff *isl_multi_aff_move_dims(__isl_take isl_multi_aff *ma,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n);
+
+__isl_give isl_set *isl_multi_aff_lex_le_set(__isl_take isl_multi_aff *ma1,
+	__isl_take isl_multi_aff *ma2);
+__isl_give isl_set *isl_multi_aff_lex_ge_set(__isl_take isl_multi_aff *ma1,
+	__isl_take isl_multi_aff *ma2);
+
+__isl_give char *isl_multi_aff_to_str(__isl_keep isl_multi_aff *aff);
+__isl_give isl_printer *isl_printer_print_multi_aff(__isl_take isl_printer *p,
+	__isl_keep isl_multi_aff *maff);
+
+__isl_give isl_multi_aff *isl_multi_aff_read_from_str(isl_ctx *ctx,
+		const char *str);
+void isl_multi_aff_dump(__isl_keep isl_multi_aff *maff);
+
+ISL_DECLARE_MULTI(pw_aff)
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_zero(__isl_take isl_space *space);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_identity(
+	__isl_take isl_space *space);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_project_out_map(
+	__isl_take isl_space *space, enum isl_dim_type type,
+	unsigned first, unsigned n);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_multi_aff(
+	__isl_take isl_multi_aff *ma);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_pw_aff(
+	__isl_take isl_pw_aff *pa);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_alloc(__isl_take isl_set *set,
+	__isl_take isl_multi_aff *maff);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_copy(
+	__isl_keep isl_pw_multi_aff *pma);
+__isl_null isl_pw_multi_aff *isl_pw_multi_aff_free(
+	__isl_take isl_pw_multi_aff *pma);
+
+unsigned isl_pw_multi_aff_dim(__isl_keep isl_pw_multi_aff *pma,
+	enum isl_dim_type type);
+__isl_give isl_pw_aff *isl_pw_multi_aff_get_pw_aff(
+	__isl_keep isl_pw_multi_aff *pma, int pos);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_set_pw_aff(
+	__isl_take isl_pw_multi_aff *pma, unsigned pos,
+	__isl_take isl_pw_aff *pa);
+
+isl_ctx *isl_pw_multi_aff_get_ctx(__isl_keep isl_pw_multi_aff *pma);
+__isl_give isl_space *isl_pw_multi_aff_get_domain_space(
+	__isl_keep isl_pw_multi_aff *pma);
+__isl_give isl_space *isl_pw_multi_aff_get_space(
+	__isl_keep isl_pw_multi_aff *pma);
+int isl_pw_multi_aff_has_tuple_name(__isl_keep isl_pw_multi_aff *pma,
+	enum isl_dim_type type);
+const char *isl_pw_multi_aff_get_tuple_name(__isl_keep isl_pw_multi_aff *pma,
+	enum isl_dim_type type);
+__isl_give isl_id *isl_pw_multi_aff_get_tuple_id(
+	__isl_keep isl_pw_multi_aff *pma, enum isl_dim_type type);
+int isl_pw_multi_aff_has_tuple_id(__isl_keep isl_pw_multi_aff *pma,
+	enum isl_dim_type type);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_set_tuple_id(
+	__isl_take isl_pw_multi_aff *pma,
+	enum isl_dim_type type, __isl_take isl_id *id);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_reset_tuple_id(
+	__isl_take isl_pw_multi_aff *pma, enum isl_dim_type type);
+
+int isl_pw_multi_aff_find_dim_by_name(__isl_keep isl_pw_multi_aff *pma,
+	enum isl_dim_type type, const char *name);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_drop_dims(
+	__isl_take isl_pw_multi_aff *pma,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+__isl_give isl_set *isl_pw_multi_aff_domain(__isl_take isl_pw_multi_aff *pma);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_empty(__isl_take isl_space *space);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_domain(
+	__isl_take isl_set *set);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_multi_val_on_domain(
+	__isl_take isl_set *domain, __isl_take isl_multi_val *mv);
+
+const char *isl_pw_multi_aff_get_dim_name(__isl_keep isl_pw_multi_aff *pma,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_id *isl_pw_multi_aff_get_dim_id(
+	__isl_keep isl_pw_multi_aff *pma, enum isl_dim_type type,
+	unsigned pos);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_set_dim_id(
+	__isl_take isl_pw_multi_aff *pma,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id);
+
+int isl_pw_multi_aff_plain_is_equal(__isl_keep isl_pw_multi_aff *pma1,
+	__isl_keep isl_pw_multi_aff *pma2);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_fix_si(
+	__isl_take isl_pw_multi_aff *pma, enum isl_dim_type type,
+	unsigned pos, int value);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_add(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_add(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_sub(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_scale_val(
+	__isl_take isl_pw_multi_aff *pma, __isl_take isl_val *v);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_scale_down_val(
+	__isl_take isl_pw_multi_aff *pma, __isl_take isl_val *v);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_scale_multi_val(
+	__isl_take isl_pw_multi_aff *pma, __isl_take isl_multi_val *mv);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_lexmin(
+	__isl_take isl_pw_multi_aff *pma1,
+	__isl_take isl_pw_multi_aff *pma2);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_lexmax(
+	__isl_take isl_pw_multi_aff *pma1,
+	__isl_take isl_pw_multi_aff *pma2);
+
+__isl_give isl_multi_aff *isl_multi_aff_flatten_domain(
+	__isl_take isl_multi_aff *ma);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_range_product(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_flat_range_product(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_product(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_intersect_params(
+	__isl_take isl_pw_multi_aff *pma, __isl_take isl_set *set);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_intersect_domain(
+	__isl_take isl_pw_multi_aff *pma, __isl_take isl_set *set);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_project_domain_on_params(
+	__isl_take isl_pw_multi_aff *pma);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_align_params(
+	__isl_take isl_pw_multi_aff *pma, __isl_take isl_space *model);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_coalesce(
+	__isl_take isl_pw_multi_aff *pma);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_gist_params(
+	__isl_take isl_pw_multi_aff *pma, __isl_take isl_set *set);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_gist(
+	__isl_take isl_pw_multi_aff *pma, __isl_take isl_set *set);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_pullback_multi_aff(
+	__isl_take isl_pw_multi_aff *pma, __isl_take isl_multi_aff *ma);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_pullback_pw_multi_aff(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2);
+
+int isl_pw_multi_aff_foreach_piece(__isl_keep isl_pw_multi_aff *pma,
+	int (*fn)(__isl_take isl_set *set, __isl_take isl_multi_aff *maff,
+		    void *user), void *user);
+
+__isl_give isl_map *isl_map_from_pw_multi_aff(__isl_take isl_pw_multi_aff *pma);
+__isl_give isl_set *isl_set_from_pw_multi_aff(__isl_take isl_pw_multi_aff *pma);
+
+__isl_give isl_printer *isl_printer_print_pw_multi_aff(__isl_take isl_printer *p,
+	__isl_keep isl_pw_multi_aff *pma);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_set(__isl_take isl_set *set);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_map(__isl_take isl_map *map);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_read_from_str(isl_ctx *ctx,
+	const char *str);
+void isl_pw_multi_aff_dump(__isl_keep isl_pw_multi_aff *pma);
+
+
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_empty(
+	__isl_take isl_space *space);
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_from_pw_multi_aff(
+	__isl_take isl_pw_multi_aff *pma);
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_from_domain(
+	__isl_take isl_union_set *uset);
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_multi_val_on_domain(
+	__isl_take isl_union_set *domain, __isl_take isl_multi_val *mv);
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_copy(
+	__isl_keep isl_union_pw_multi_aff *upma);
+__isl_null isl_union_pw_multi_aff *isl_union_pw_multi_aff_free(
+	__isl_take isl_union_pw_multi_aff *upma);
+
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_add_pw_multi_aff(
+	__isl_take isl_union_pw_multi_aff *upma,
+	__isl_take isl_pw_multi_aff *pma);
+
+isl_ctx *isl_union_pw_multi_aff_get_ctx(
+	__isl_keep isl_union_pw_multi_aff *upma);
+__isl_give isl_space *isl_union_pw_multi_aff_get_space(
+	__isl_keep isl_union_pw_multi_aff *upma);
+
+unsigned isl_union_pw_multi_aff_dim(__isl_keep isl_union_pw_multi_aff *upma,
+	enum isl_dim_type type);
+int isl_union_pw_multi_aff_find_dim_by_name(
+	__isl_keep isl_union_pw_multi_aff *upma, enum isl_dim_type type,
+	const char *name);
+
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_coalesce(
+	__isl_take isl_union_pw_multi_aff *upma);
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_gist_params(
+	__isl_take isl_union_pw_multi_aff *upma, __isl_take isl_set *context);
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_gist(
+	__isl_take isl_union_pw_multi_aff *upma,
+	__isl_take isl_union_set *context);
+
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_align_params(
+	__isl_take isl_union_pw_multi_aff *upma, __isl_take isl_space *model);
+
+int isl_union_pw_multi_aff_foreach_pw_multi_aff(
+	__isl_keep isl_union_pw_multi_aff *upma,
+	int (*fn)(__isl_take isl_pw_multi_aff *pma, void *user), void *user);
+__isl_give isl_pw_multi_aff *isl_union_pw_multi_aff_extract_pw_multi_aff(
+	__isl_keep isl_union_pw_multi_aff *upma, __isl_take isl_space *space);
+
+int isl_union_pw_multi_aff_plain_is_equal(
+	__isl_keep isl_union_pw_multi_aff *upma1,
+	__isl_keep isl_union_pw_multi_aff *upma2);
+
+__isl_give isl_union_set *isl_union_pw_multi_aff_domain(
+	__isl_take isl_union_pw_multi_aff *upma);
+
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_add(
+	__isl_take isl_union_pw_multi_aff *upma1,
+	__isl_take isl_union_pw_multi_aff *upma2);
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_union_add(
+	__isl_take isl_union_pw_multi_aff *upma1,
+	__isl_take isl_union_pw_multi_aff *upma2);
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_sub(
+	__isl_take isl_union_pw_multi_aff *upma1,
+	__isl_take isl_union_pw_multi_aff *upma2);
+
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_scale_val(
+	__isl_take isl_union_pw_multi_aff *upma, __isl_take isl_val *val);
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_scale_down_val(
+	__isl_take isl_union_pw_multi_aff *upma, __isl_take isl_val *val);
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_scale_multi_val(
+	__isl_take isl_union_pw_multi_aff *upma, __isl_take isl_multi_val *mv);
+
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_flat_range_product(
+	__isl_take isl_union_pw_multi_aff *upma1,
+	__isl_take isl_union_pw_multi_aff *upma2);
+
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_intersect_params(
+	__isl_take isl_union_pw_multi_aff *upma, __isl_take isl_set *set);
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_intersect_domain(
+	__isl_take isl_union_pw_multi_aff *upma,
+	__isl_take isl_union_set *uset);
+
+__isl_give isl_union_map *isl_union_map_from_union_pw_multi_aff(
+	__isl_take isl_union_pw_multi_aff *upma);
+
+__isl_give isl_printer *isl_printer_print_union_pw_multi_aff(
+	__isl_take isl_printer *p, __isl_keep isl_union_pw_multi_aff *upma);
+
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_from_union_set(
+	__isl_take isl_union_set *uset);
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_from_union_map(
+	__isl_take isl_union_map *umap);
+
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_read_from_str(
+	isl_ctx *ctx, const char *str);
+void isl_union_pw_multi_aff_dump(__isl_keep isl_union_pw_multi_aff *upma);
+__isl_give char *isl_union_pw_multi_aff_to_str(
+	__isl_keep isl_union_pw_multi_aff *upma);
+
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_identity(
+	__isl_take isl_space *space);
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_from_multi_aff(
+	__isl_take isl_multi_aff *ma);
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_from_pw_aff(
+	__isl_take isl_pw_aff *pa);
+__isl_give isl_set *isl_multi_pw_aff_domain(__isl_take isl_multi_pw_aff *mpa);
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_intersect_params(
+	__isl_take isl_multi_pw_aff *mpa, __isl_take isl_set *set);
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_intersect_domain(
+	__isl_take isl_multi_pw_aff *mpa, __isl_take isl_set *domain);
+
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_coalesce(
+	__isl_take isl_multi_pw_aff *mpa);
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_gist(
+	__isl_take isl_multi_pw_aff *mpa, __isl_take isl_set *set);
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_gist_params(
+	__isl_take isl_multi_pw_aff *mpa, __isl_take isl_set *set);
+
+int isl_multi_pw_aff_is_equal(__isl_keep isl_multi_pw_aff *mpa1,
+	__isl_keep isl_multi_pw_aff *mpa2);
+
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_pullback_multi_aff(
+	__isl_take isl_multi_pw_aff *mpa, __isl_take isl_multi_aff *ma);
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_pullback_pw_multi_aff(
+	__isl_take isl_multi_pw_aff *mpa, __isl_take isl_pw_multi_aff *pma);
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_pullback_multi_pw_aff(
+	__isl_take isl_multi_pw_aff *mpa1, __isl_take isl_multi_pw_aff *mpa2);
+
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_move_dims(
+	__isl_take isl_multi_pw_aff *pma,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n);
+
+__isl_give isl_set *isl_set_from_multi_pw_aff(__isl_take isl_multi_pw_aff *mpa);
+__isl_give isl_map *isl_map_from_multi_pw_aff(__isl_take isl_multi_pw_aff *mpa);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_multi_pw_aff(
+	__isl_take isl_multi_pw_aff *mpa);
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_from_pw_multi_aff(
+	__isl_take isl_pw_multi_aff *pma);
+
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_read_from_str(isl_ctx *ctx,
+	const char *str);
+__isl_give isl_printer *isl_printer_print_multi_pw_aff(
+	__isl_take isl_printer *p, __isl_keep isl_multi_pw_aff *mpa);
+void isl_multi_pw_aff_dump(__isl_keep isl_multi_pw_aff *mpa);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#include <isl/dim.h>
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/aff_type.h b/polly/lib/External/isl/include/isl/aff_type.h
new file mode 100644
index 00000000000..ce19f1dfa18
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/aff_type.h
@@ -0,0 +1,36 @@
+#ifndef ISL_AFF_TYPE_H
+#define ISL_AFF_TYPE_H
+
+#include <isl/list.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_aff;
+typedef struct isl_aff isl_aff;
+
+ISL_DECLARE_LIST(aff)
+
+struct isl_pw_aff;
+typedef struct isl_pw_aff isl_pw_aff;
+
+ISL_DECLARE_LIST(pw_aff)
+
+struct isl_multi_aff;
+typedef struct isl_multi_aff isl_multi_aff;
+
+struct isl_pw_multi_aff;
+typedef struct isl_pw_multi_aff isl_pw_multi_aff;
+
+struct isl_union_pw_multi_aff;
+typedef struct isl_union_pw_multi_aff isl_union_pw_multi_aff;
+
+struct isl_multi_pw_aff;
+typedef struct isl_multi_pw_aff isl_multi_pw_aff;
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/arg.h b/polly/lib/External/isl/include/isl/arg.h
new file mode 100644
index 00000000000..896293a9fc5
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/arg.h
@@ -0,0 +1,314 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_ARG_H
+#define ISL_ARG_H
+
+#include <stddef.h>
+#include <stdlib.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_arg_choice {
+	const char	*name;
+	unsigned	 value;
+};
+
+struct isl_arg_flags {
+	const char	*name;
+	unsigned	 mask;
+	unsigned	 value;
+};
+
+enum isl_arg_type {
+	isl_arg_end,
+	isl_arg_alias,
+	isl_arg_arg,
+	isl_arg_bool,
+	isl_arg_child,
+	isl_arg_choice,
+	isl_arg_flags,
+	isl_arg_footer,
+	isl_arg_int,
+	isl_arg_user,
+	isl_arg_long,
+	isl_arg_ulong,
+	isl_arg_str,
+	isl_arg_str_list,
+	isl_arg_version
+};
+
+struct isl_args;
+
+struct isl_arg {
+	enum isl_arg_type	 type;
+	char			 short_name;
+	const char		*long_name;
+	const char		*argument_name;
+	size_t			 offset;
+	const char		*help_msg;
+#define ISL_ARG_SINGLE_DASH	(1 << 0)
+#define ISL_ARG_BOOL_ARG	(1 << 1)
+#define ISL_ARG_HIDDEN		(1 << 2)
+	unsigned		 flags;
+	union {
+	struct {
+		struct isl_arg_choice	*choice;
+		unsigned	 	 default_value;
+		unsigned	 	 default_selected;
+		int (*set)(void *opt, unsigned val);
+	} choice;
+	struct {
+		struct isl_arg_flags	*flags;
+		unsigned	 	 default_value;
+	} flags;
+	struct {
+		unsigned		 default_value;
+		int (*set)(void *opt, unsigned val);
+	} b;
+	struct {
+		int			default_value;
+	} i;
+	struct {
+		long		 	default_value;
+		long		 	default_selected;
+		int (*set)(void *opt, long val);
+	} l;
+	struct {
+		unsigned long		default_value;
+	} ul;
+	struct {
+		const char		*default_value;
+	} str;
+	struct {
+		size_t			 offset_n;
+	} str_list;
+	struct {
+		struct isl_args		*child;
+	} child;
+	struct {
+		void (*print_version)(void);
+	} version;
+	struct {
+		int (*init)(void*);
+		void (*clear)(void*);
+	} user;
+	} u;
+};
+
+struct isl_args {
+	size_t			 options_size;
+	struct isl_arg		*args;
+};
+
+#define ISL_ARGS_START(s,name)						\
+	struct isl_arg name ## LIST[];					\
+	struct isl_args name = { sizeof(s), name ## LIST };		\
+	struct isl_arg name ## LIST[] = {
+#define ISL_ARGS_END							\
+	{ isl_arg_end } };
+
+#define ISL_ARG_ALIAS(l)	{					\
+	.type = isl_arg_alias,						\
+	.long_name = l,							\
+},
+#define ISL_ARG_ARG(st,f,a,d)	{					\
+	.type = isl_arg_arg,						\
+	.argument_name = a,						\
+	.offset = offsetof(st, f),					\
+	.u = { .str = { .default_value = d } }				\
+},
+#define ISL_ARG_FOOTER(h)	{					\
+	.type = isl_arg_footer,						\
+	.help_msg = h,							\
+},
+#define ISL_ARG_CHOICE(st,f,s,l,c,d,h)	{				\
+	.type = isl_arg_choice,						\
+	.short_name = s,						\
+	.long_name = l,							\
+	.offset = offsetof(st, f),					\
+	.help_msg = h,							\
+	.u = { .choice = { .choice = c, .default_value = d,		\
+			    .default_selected = d, .set = NULL } }	\
+},
+#define ISL_ARG_OPT_CHOICE(st,f,s,l,c,d,ds,h)	{			\
+	.type = isl_arg_choice,						\
+	.short_name = s,						\
+	.long_name = l,							\
+	.offset = offsetof(st, f),					\
+	.help_msg = h,							\
+	.u = { .choice = { .choice = c, .default_value = d,		\
+			    .default_selected = ds, .set = NULL } }	\
+},
+#define ISL_ARG_USER_OPT_CHOICE(st,f,s,l,c,setter,d,ds,h)	{	\
+	.type = isl_arg_choice,						\
+	.short_name = s,						\
+	.long_name = l,							\
+	.offset = offsetof(st, f),					\
+	.help_msg = h,							\
+	.u = { .choice = { .choice = c, .default_value = d,		\
+			    .default_selected = ds, .set = setter } }	\
+},
+#define _ISL_ARG_BOOL_F(o,s,l,setter,d,h,fl)	{			\
+	.type = isl_arg_bool,						\
+	.short_name = s,						\
+	.long_name = l,							\
+	.offset = o,							\
+	.help_msg = h,							\
+	.flags = fl,							\
+	.u = { .b = { .default_value = d, .set = setter } }		\
+},
+#define ISL_ARG_BOOL_F(st,f,s,l,d,h,fl)					\
+	_ISL_ARG_BOOL_F(offsetof(st, f),s,l,NULL,d,h,fl)
+#define ISL_ARG_BOOL(st,f,s,l,d,h)					\
+	ISL_ARG_BOOL_F(st,f,s,l,d,h,0)
+#define ISL_ARG_PHANTOM_BOOL_F(s,l,setter,h,fl)				\
+	_ISL_ARG_BOOL_F(-1,s,l,setter,0,h,fl)
+#define ISL_ARG_PHANTOM_BOOL(s,l,setter,h)				\
+	ISL_ARG_PHANTOM_BOOL_F(s,l,setter,h,0)
+#define ISL_ARG_INT_F(st,f,s,l,a,d,h,fl)	{			\
+	.type = isl_arg_int,						\
+	.short_name = s,						\
+	.long_name = l,							\
+	.argument_name = a,						\
+	.offset = offsetof(st, f),					\
+	.help_msg = h,							\
+	.flags = fl,							\
+	.u = { .ul = { .default_value = d } }				\
+},
+#define ISL_ARG_INT(st,f,s,l,a,d,h)					\
+	ISL_ARG_INT_F(st,f,s,l,a,d,h,0)
+#define ISL_ARG_LONG(st,f,s,lo,d,h)	{				\
+	.type = isl_arg_long,						\
+	.short_name = s,						\
+	.long_name = lo,						\
+	.offset = offsetof(st, f),					\
+	.help_msg = h,							\
+	.u = { .l = { .default_value = d, .default_selected = d,	\
+		      .set = NULL } }					\
+},
+#define ISL_ARG_USER_LONG(st,f,s,lo,setter,d,h)	{			\
+	.type = isl_arg_long,						\
+	.short_name = s,						\
+	.long_name = lo,						\
+	.offset = offsetof(st, f),					\
+	.help_msg = h,							\
+	.u = { .l = { .default_value = d, .default_selected = d,	\
+		      .set = setter } }					\
+},
+#define ISL_ARG_OPT_LONG(st,f,s,lo,d,ds,h)	{			\
+	.type = isl_arg_long,						\
+	.short_name = s,						\
+	.long_name = lo,						\
+	.offset = offsetof(st, f),					\
+	.help_msg = h,							\
+	.u = { .l = { .default_value = d, .default_selected = ds,	\
+		      .set = NULL } }					\
+},
+#define ISL_ARG_ULONG(st,f,s,l,d,h)	{				\
+	.type = isl_arg_ulong,						\
+	.short_name = s,						\
+	.long_name = l,							\
+	.offset = offsetof(st, f),					\
+	.help_msg = h,							\
+	.u = { .ul = { .default_value = d } }				\
+},
+#define ISL_ARG_STR_F(st,f,s,l,a,d,h,fl)	{			\
+	.type = isl_arg_str,						\
+	.short_name = s,						\
+	.long_name = l,							\
+	.argument_name = a,						\
+	.offset = offsetof(st, f),					\
+	.help_msg = h,							\
+	.flags = fl,							\
+	.u = { .str = { .default_value = d } }				\
+},
+#define ISL_ARG_STR(st,f,s,l,a,d,h)					\
+	ISL_ARG_STR_F(st,f,s,l,a,d,h,0)
+#define ISL_ARG_STR_LIST(st,f_n,f_l,s,l,a,h)	{			\
+	.type = isl_arg_str_list,					\
+	.short_name = s,						\
+	.long_name = l,							\
+	.argument_name = a,						\
+	.offset = offsetof(st, f_l),					\
+	.help_msg = h,							\
+	.u = { .str_list = { .offset_n = offsetof(st, f_n) } }		\
+},
+#define _ISL_ARG_CHILD(o,l,c,h,fl)	{				\
+	.type = isl_arg_child,						\
+	.long_name = l,							\
+	.offset = o,							\
+	.help_msg = h,							\
+	.flags = fl,							\
+	.u = { .child = { .child = c } }				\
+},
+#define ISL_ARG_CHILD(st,f,l,c,h)					\
+	_ISL_ARG_CHILD(offsetof(st, f),l,c,h,0)
+#define ISL_ARG_GROUP_F(l,c,h,fl)					\
+	_ISL_ARG_CHILD(-1,l,c,h,fl)
+#define ISL_ARG_GROUP(l,c,h)						\
+	ISL_ARG_GROUP_F(l,c,h,0)
+#define ISL_ARG_FLAGS(st,f,s,l,c,d,h)	{				\
+	.type = isl_arg_flags,						\
+	.short_name = s,						\
+	.long_name = l,							\
+	.offset = offsetof(st, f),					\
+	.help_msg = h,							\
+	.u = { .flags = { .flags = c, .default_value = d } }		\
+},
+#define ISL_ARG_USER(st,f,i,c) {					\
+	.type = isl_arg_user,						\
+	.offset = offsetof(st, f),					\
+	.u = { .user = { .init = i, .clear = c} }			\
+},
+#define ISL_ARG_VERSION(print) {					\
+	.type = isl_arg_version,					\
+	.u = { .version = { .print_version = print } }			\
+},
+
+#define ISL_ARG_ALL		(1 << 0)
+#define ISL_ARG_SKIP_HELP	(1 << 1)
+
+void isl_args_set_defaults(struct isl_args *args, void *opt);
+void isl_args_free(struct isl_args *args, void *opt);
+int isl_args_parse(struct isl_args *args, int argc, char **argv, void *opt,
+	unsigned flags);
+
+#define ISL_ARG_DECL(prefix,st,args)					\
+extern struct isl_args args;						\
+st *prefix ## _new_with_defaults(void);					\
+void prefix ## _free(st *opt);						\
+int prefix ## _parse(st *opt, int argc, char **argv, unsigned flags);
+
+#define ISL_ARG_DEF(prefix,st,args)					\
+st *prefix ## _new_with_defaults()					\
+{									\
+	st *opt = (st *)calloc(1, sizeof(st));				\
+	if (opt)							\
+		isl_args_set_defaults(&(args), opt);			\
+	return opt;							\
+}									\
+									\
+void prefix ## _free(st *opt)						\
+{									\
+	isl_args_free(&(args), opt);					\
+}									\
+									\
+int prefix ## _parse(st *opt, int argc, char **argv, unsigned flags)	\
+{									\
+	return isl_args_parse(&(args), argc, argv, opt, flags);		\
+}
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/ast.h b/polly/lib/External/isl/include/isl/ast.h
new file mode 100644
index 00000000000..cc3008bbfda
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/ast.h
@@ -0,0 +1,159 @@
+#ifndef ISL_AST_H
+#define ISL_AST_H
+
+#include <isl/ctx.h>
+#include <isl/ast_type.h>
+#include <isl/id.h>
+#include <isl/id_to_ast_expr.h>
+#include <isl/val.h>
+#include <isl/list.h>
+#include <isl/printer.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+int isl_options_set_ast_iterator_type(isl_ctx *ctx, const char *val);
+const char *isl_options_get_ast_iterator_type(isl_ctx *ctx);
+
+int isl_options_set_ast_always_print_block(isl_ctx *ctx, int val);
+int isl_options_get_ast_always_print_block(isl_ctx *ctx);
+
+__isl_give isl_ast_expr *isl_ast_expr_from_val(__isl_take isl_val *v);
+__isl_give isl_ast_expr *isl_ast_expr_from_id(__isl_take isl_id *id);
+__isl_give isl_ast_expr *isl_ast_expr_neg(__isl_take isl_ast_expr *expr);
+__isl_give isl_ast_expr *isl_ast_expr_add(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2);
+__isl_give isl_ast_expr *isl_ast_expr_sub(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2);
+__isl_give isl_ast_expr *isl_ast_expr_mul(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2);
+__isl_give isl_ast_expr *isl_ast_expr_div(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2);
+__isl_give isl_ast_expr *isl_ast_expr_and(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2);
+__isl_give isl_ast_expr *isl_ast_expr_or(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2);
+__isl_give isl_ast_expr *isl_ast_expr_le(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2);
+__isl_give isl_ast_expr *isl_ast_expr_lt(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2);
+__isl_give isl_ast_expr *isl_ast_expr_ge(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2);
+__isl_give isl_ast_expr *isl_ast_expr_gt(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2);
+__isl_give isl_ast_expr *isl_ast_expr_eq(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2);
+__isl_give isl_ast_expr *isl_ast_expr_access(__isl_take isl_ast_expr *array,
+	__isl_take isl_ast_expr_list *indices);
+__isl_give isl_ast_expr *isl_ast_expr_address_of(__isl_take isl_ast_expr *expr);
+
+__isl_give isl_ast_expr *isl_ast_expr_copy(__isl_keep isl_ast_expr *expr);
+__isl_null isl_ast_expr *isl_ast_expr_free(__isl_take isl_ast_expr *expr);
+
+isl_ctx *isl_ast_expr_get_ctx(__isl_keep isl_ast_expr *expr);
+enum isl_ast_expr_type isl_ast_expr_get_type(__isl_keep isl_ast_expr *expr);
+__isl_give isl_val *isl_ast_expr_get_val(__isl_keep isl_ast_expr *expr);
+__isl_give isl_id *isl_ast_expr_get_id(__isl_keep isl_ast_expr *expr);
+
+enum isl_ast_op_type isl_ast_expr_get_op_type(__isl_keep isl_ast_expr *expr);
+int isl_ast_expr_get_op_n_arg(__isl_keep isl_ast_expr *expr);
+__isl_give isl_ast_expr *isl_ast_expr_get_op_arg(__isl_keep isl_ast_expr *expr,
+	int pos);
+__isl_give isl_ast_expr *isl_ast_expr_set_op_arg(__isl_take isl_ast_expr *expr,
+	int pos, __isl_take isl_ast_expr *arg);
+
+int isl_ast_expr_is_equal(__isl_keep isl_ast_expr *expr1,
+	__isl_keep isl_ast_expr *expr2);
+
+__isl_give isl_ast_expr *isl_ast_expr_substitute_ids(
+	__isl_take isl_ast_expr *expr, __isl_take isl_id_to_ast_expr *id2expr);
+
+__isl_give isl_printer *isl_printer_print_ast_expr(__isl_take isl_printer *p,
+	__isl_keep isl_ast_expr *expr);
+void isl_ast_expr_dump(__isl_keep isl_ast_expr *expr);
+__isl_give char *isl_ast_expr_to_str(__isl_keep isl_ast_expr *expr);
+
+__isl_give isl_ast_node *isl_ast_node_alloc_user(__isl_take isl_ast_expr *expr);
+__isl_give isl_ast_node *isl_ast_node_copy(__isl_keep isl_ast_node *node);
+__isl_null isl_ast_node *isl_ast_node_free(__isl_take isl_ast_node *node);
+
+isl_ctx *isl_ast_node_get_ctx(__isl_keep isl_ast_node *node);
+enum isl_ast_node_type isl_ast_node_get_type(__isl_keep isl_ast_node *node);
+
+__isl_give isl_ast_node *isl_ast_node_set_annotation(
+	__isl_take isl_ast_node *node, __isl_take isl_id *annotation);
+__isl_give isl_id *isl_ast_node_get_annotation(__isl_keep isl_ast_node *node);
+
+__isl_give isl_ast_expr *isl_ast_node_for_get_iterator(
+	__isl_keep isl_ast_node *node);
+__isl_give isl_ast_expr *isl_ast_node_for_get_init(
+	__isl_keep isl_ast_node *node);
+__isl_give isl_ast_expr *isl_ast_node_for_get_cond(
+	__isl_keep isl_ast_node *node);
+__isl_give isl_ast_expr *isl_ast_node_for_get_inc(
+	__isl_keep isl_ast_node *node);
+__isl_give isl_ast_node *isl_ast_node_for_get_body(
+	__isl_keep isl_ast_node *node);
+int isl_ast_node_for_is_degenerate(__isl_keep isl_ast_node *node);
+
+__isl_give isl_ast_expr *isl_ast_node_if_get_cond(
+	__isl_keep isl_ast_node *node);
+__isl_give isl_ast_node *isl_ast_node_if_get_then(
+	__isl_keep isl_ast_node *node);
+int isl_ast_node_if_has_else(__isl_keep isl_ast_node *node);
+__isl_give isl_ast_node *isl_ast_node_if_get_else(
+	__isl_keep isl_ast_node *node);
+
+__isl_give isl_ast_node_list *isl_ast_node_block_get_children(
+	__isl_keep isl_ast_node *node);
+
+__isl_give isl_ast_expr *isl_ast_node_user_get_expr(
+	__isl_keep isl_ast_node *node);
+
+__isl_give isl_printer *isl_printer_print_ast_node(__isl_take isl_printer *p,
+	__isl_keep isl_ast_node *node);
+void isl_ast_node_dump(__isl_keep isl_ast_node *node);
+
+__isl_give isl_ast_print_options *isl_ast_print_options_alloc(isl_ctx *ctx);
+__isl_give isl_ast_print_options *isl_ast_print_options_copy(
+	__isl_keep isl_ast_print_options *options);
+__isl_null isl_ast_print_options *isl_ast_print_options_free(
+	__isl_take isl_ast_print_options *options);
+isl_ctx *isl_ast_print_options_get_ctx(
+	__isl_keep isl_ast_print_options *options);
+
+__isl_give isl_ast_print_options *isl_ast_print_options_set_print_user(
+	__isl_take isl_ast_print_options *options,
+	__isl_give isl_printer *(*print_user)(__isl_take isl_printer *p,
+		__isl_take isl_ast_print_options *options,
+		__isl_keep isl_ast_node *node, void *user),
+	void *user);
+__isl_give isl_ast_print_options *isl_ast_print_options_set_print_for(
+	__isl_take isl_ast_print_options *options,
+	__isl_give isl_printer *(*print_for)(__isl_take isl_printer *p,
+		__isl_take isl_ast_print_options *options,
+		__isl_keep isl_ast_node *node, void *user),
+	void *user);
+
+int isl_ast_node_foreach_ast_op_type(__isl_keep isl_ast_node *node,
+	int (*fn)(enum isl_ast_op_type type, void *user), void *user);
+__isl_give isl_printer *isl_ast_op_type_print_macro(
+	enum isl_ast_op_type type, __isl_take isl_printer *p);
+__isl_give isl_printer *isl_ast_node_print_macros(
+	__isl_keep isl_ast_node *node, __isl_take isl_printer *p);
+__isl_give isl_printer *isl_ast_node_print(__isl_keep isl_ast_node *node,
+	__isl_take isl_printer *p,
+	__isl_take isl_ast_print_options *options);
+__isl_give isl_printer *isl_ast_node_for_print(__isl_keep isl_ast_node *node,
+	__isl_take isl_printer *p,
+	__isl_take isl_ast_print_options *options);
+__isl_give isl_printer *isl_ast_node_if_print(__isl_keep isl_ast_node *node,
+	__isl_take isl_printer *p,
+	__isl_take isl_ast_print_options *options);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/ast_build.h b/polly/lib/External/isl/include/isl/ast_build.h
new file mode 100644
index 00000000000..4fa7b454909
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/ast_build.h
@@ -0,0 +1,100 @@
+#ifndef ISL_AST_CONTEXT_H
+#define ISL_AST_CONTEXT_H
+
+#include <isl/ctx.h>
+#include <isl/set.h>
+#include <isl/ast.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_ast_build;
+typedef struct isl_ast_build isl_ast_build;
+
+
+int isl_options_set_ast_build_atomic_upper_bound(isl_ctx *ctx, int val);
+int isl_options_get_ast_build_atomic_upper_bound(isl_ctx *ctx);
+
+int isl_options_set_ast_build_prefer_pdiv(isl_ctx *ctx, int val);
+int isl_options_get_ast_build_prefer_pdiv(isl_ctx *ctx);
+
+int isl_options_set_ast_build_exploit_nested_bounds(isl_ctx *ctx, int val);
+int isl_options_get_ast_build_exploit_nested_bounds(isl_ctx *ctx);
+
+int isl_options_set_ast_build_group_coscheduled(isl_ctx *ctx, int val);
+int isl_options_get_ast_build_group_coscheduled(isl_ctx *ctx);
+
+#define ISL_AST_BUILD_SEPARATION_BOUNDS_EXPLICIT		0
+#define ISL_AST_BUILD_SEPARATION_BOUNDS_IMPLICIT		1
+int isl_options_set_ast_build_separation_bounds(isl_ctx *ctx, int val);
+int isl_options_get_ast_build_separation_bounds(isl_ctx *ctx);
+
+int isl_options_set_ast_build_scale_strides(isl_ctx *ctx, int val);
+int isl_options_get_ast_build_scale_strides(isl_ctx *ctx);
+
+int isl_options_set_ast_build_allow_else(isl_ctx *ctx, int val);
+int isl_options_get_ast_build_allow_else(isl_ctx *ctx);
+
+int isl_options_set_ast_build_allow_or(isl_ctx *ctx, int val);
+int isl_options_get_ast_build_allow_or(isl_ctx *ctx);
+
+isl_ctx *isl_ast_build_get_ctx(__isl_keep isl_ast_build *build);
+
+__isl_give isl_ast_build *isl_ast_build_from_context(__isl_take isl_set *set);
+
+__isl_give isl_space *isl_ast_build_get_schedule_space(
+	__isl_keep isl_ast_build *build);
+__isl_give isl_union_map *isl_ast_build_get_schedule(
+	__isl_keep isl_ast_build *build);
+
+__isl_give isl_ast_build *isl_ast_build_restrict(
+	__isl_take isl_ast_build *build, __isl_take isl_set *set);
+
+__isl_give isl_ast_build *isl_ast_build_copy(
+	__isl_keep isl_ast_build *build);
+__isl_null isl_ast_build *isl_ast_build_free(
+	__isl_take isl_ast_build *build);
+
+__isl_give isl_ast_build *isl_ast_build_set_options(
+	__isl_take isl_ast_build *build,
+	__isl_take isl_union_map *options);
+__isl_give isl_ast_build *isl_ast_build_set_iterators(
+	__isl_take isl_ast_build *build,
+	__isl_take isl_id_list *iterators);
+__isl_give isl_ast_build *isl_ast_build_set_at_each_domain(
+	__isl_take isl_ast_build *build,
+	__isl_give isl_ast_node *(*fn)(__isl_take isl_ast_node *node,
+		__isl_keep isl_ast_build *build, void *user), void *user);
+__isl_give isl_ast_build *isl_ast_build_set_before_each_for(
+	__isl_take isl_ast_build *build,
+	__isl_give isl_id *(*fn)(__isl_keep isl_ast_build *build,
+		void *user), void *user);
+__isl_give isl_ast_build *isl_ast_build_set_after_each_for(
+	__isl_take isl_ast_build *build,
+	__isl_give isl_ast_node *(*fn)(__isl_take isl_ast_node *node,
+		__isl_keep isl_ast_build *build, void *user), void *user);
+__isl_give isl_ast_build *isl_ast_build_set_create_leaf(
+	__isl_take isl_ast_build *build,
+	__isl_give isl_ast_node *(*fn)(__isl_take isl_ast_build *build,
+		void *user), void *user);
+
+__isl_give isl_ast_expr *isl_ast_build_expr_from_pw_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_pw_aff *pa);
+__isl_give isl_ast_expr *isl_ast_build_access_from_pw_multi_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_pw_multi_aff *pma);
+__isl_give isl_ast_expr *isl_ast_build_access_from_multi_pw_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_multi_pw_aff *mpa);
+__isl_give isl_ast_expr *isl_ast_build_call_from_pw_multi_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_pw_multi_aff *pma);
+__isl_give isl_ast_expr *isl_ast_build_call_from_multi_pw_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_multi_pw_aff *mpa);
+
+__isl_give isl_ast_node *isl_ast_build_ast_from_schedule(
+	__isl_keep isl_ast_build *build, __isl_take isl_union_map *schedule);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/ast_type.h b/polly/lib/External/isl/include/isl/ast_type.h
new file mode 100644
index 00000000000..19b14b3b7dc
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/ast_type.h
@@ -0,0 +1,71 @@
+#ifndef ISL_AST_TYPE_H
+#define ISL_AST_TYPE_H
+
+#include <isl/list.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_ast_expr;
+typedef struct isl_ast_expr isl_ast_expr;
+
+struct isl_ast_node;
+typedef struct isl_ast_node isl_ast_node;
+
+enum isl_ast_op_type {
+	isl_ast_op_error = -1,
+	isl_ast_op_and,
+	isl_ast_op_and_then,
+	isl_ast_op_or,
+	isl_ast_op_or_else,
+	isl_ast_op_max,
+	isl_ast_op_min,
+	isl_ast_op_minus,
+	isl_ast_op_add,
+	isl_ast_op_sub,
+	isl_ast_op_mul,
+	isl_ast_op_div,
+	isl_ast_op_fdiv_q,	/* Round towards -infty */
+	isl_ast_op_pdiv_q,	/* Dividend is non-negative */
+	isl_ast_op_pdiv_r,	/* Dividend is non-negative */
+	isl_ast_op_zdiv_r,	/* Result only compared against zero */
+	isl_ast_op_cond,
+	isl_ast_op_select,
+	isl_ast_op_eq,
+	isl_ast_op_le,
+	isl_ast_op_lt,
+	isl_ast_op_ge,
+	isl_ast_op_gt,
+	isl_ast_op_call,
+	isl_ast_op_access,
+	isl_ast_op_member,
+	isl_ast_op_address_of
+};
+
+enum isl_ast_expr_type {
+	isl_ast_expr_error = -1,
+	isl_ast_expr_op,
+	isl_ast_expr_id,
+	isl_ast_expr_int
+};
+
+enum isl_ast_node_type {
+	isl_ast_node_error = -1,
+	isl_ast_node_for = 1,
+	isl_ast_node_if,
+	isl_ast_node_block,
+	isl_ast_node_user
+};
+
+struct isl_ast_print_options;
+typedef struct isl_ast_print_options isl_ast_print_options;
+
+ISL_DECLARE_LIST(ast_expr)
+ISL_DECLARE_LIST(ast_node)
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/band.h b/polly/lib/External/isl/include/isl/band.h
new file mode 100644
index 00000000000..cb2bfeebfc1
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/band.h
@@ -0,0 +1,56 @@
+#ifndef ISL_BAND_H
+#define ISL_BAND_H
+
+#include <isl/printer.h>
+#include <isl/list.h>
+#include <isl/union_map_type.h>
+#include <isl/vec.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_band;
+typedef struct isl_band isl_band;
+
+ISL_DECLARE_LIST(band)
+
+__isl_give isl_band *isl_band_copy(__isl_keep isl_band *band);
+__isl_null isl_band *isl_band_free(__isl_take isl_band *band);
+
+isl_ctx *isl_band_get_ctx(__isl_keep isl_band *band);
+
+int isl_band_has_children(__isl_keep isl_band *band);
+__isl_give isl_band_list *isl_band_get_children(
+	__isl_keep isl_band *band);
+
+__isl_give isl_union_map *isl_band_get_prefix_schedule(
+	__isl_keep isl_band *band);
+__isl_give isl_union_map *isl_band_get_partial_schedule(
+	__isl_keep isl_band *band);
+__isl_give isl_union_map *isl_band_get_suffix_schedule(
+	__isl_keep isl_band *band);
+
+int isl_options_set_tile_scale_tile_loops(isl_ctx *ctx, int val);
+int isl_options_get_tile_scale_tile_loops(isl_ctx *ctx);
+int isl_options_set_tile_shift_point_loops(isl_ctx *ctx, int val);
+int isl_options_get_tile_shift_point_loops(isl_ctx *ctx);
+
+int isl_band_tile(__isl_keep isl_band *band, __isl_take isl_vec *sizes);
+int isl_band_split(__isl_keep isl_band *band, int pos);
+
+int isl_band_n_member(__isl_keep isl_band *band);
+int isl_band_member_is_coincident(__isl_keep isl_band *band, int pos);
+
+int isl_band_list_foreach_band(__isl_keep isl_band_list *list,
+	int (*fn)(__isl_keep isl_band *band, void *user), void *user);
+
+__isl_give isl_printer *isl_printer_print_band(__isl_take isl_printer *p,
+	__isl_keep isl_band *band);
+void isl_band_dump(__isl_keep isl_band *band);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/constraint.h b/polly/lib/External/isl/include/isl/constraint.h
new file mode 100644
index 00000000000..70de5b5a823
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/constraint.h
@@ -0,0 +1,148 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_CONSTRAINT_H
+#define ISL_CONSTRAINT_H
+
+#include <isl/local_space.h>
+#include <isl/space.h>
+#include <isl/aff_type.h>
+#include <isl/set_type.h>
+#include <isl/list.h>
+#include <isl/val.h>
+#include <isl/printer.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_constraint;
+typedef struct isl_constraint isl_constraint;
+
+ISL_DECLARE_LIST(constraint)
+
+isl_ctx *isl_constraint_get_ctx(__isl_keep isl_constraint *c);
+
+__isl_give isl_constraint *isl_equality_alloc(__isl_take isl_local_space *ls);
+__isl_give isl_constraint *isl_inequality_alloc(__isl_take isl_local_space *ls);
+
+struct isl_constraint *isl_constraint_cow(struct isl_constraint *c);
+struct isl_constraint *isl_constraint_copy(struct isl_constraint *c);
+__isl_null isl_constraint *isl_constraint_free(__isl_take isl_constraint *c);
+
+int isl_basic_map_n_constraint(__isl_keep isl_basic_map *bmap);
+int isl_basic_set_n_constraint(__isl_keep isl_basic_set *bset);
+int isl_basic_map_foreach_constraint(__isl_keep isl_basic_map *bmap,
+	int (*fn)(__isl_take isl_constraint *c, void *user), void *user);
+int isl_basic_set_foreach_constraint(__isl_keep isl_basic_set *bset,
+	int (*fn)(__isl_take isl_constraint *c, void *user), void *user);
+__isl_give isl_constraint_list *isl_basic_map_get_constraint_list(
+	__isl_keep isl_basic_map *bmap);
+__isl_give isl_constraint_list *isl_basic_set_get_constraint_list(
+	__isl_keep isl_basic_set *bset);
+int isl_constraint_is_equal(struct isl_constraint *constraint1,
+			    struct isl_constraint *constraint2);
+
+int isl_basic_set_foreach_bound_pair(__isl_keep isl_basic_set *bset,
+	enum isl_dim_type type, unsigned pos,
+	int (*fn)(__isl_take isl_constraint *lower,
+		  __isl_take isl_constraint *upper,
+		  __isl_take isl_basic_set *bset, void *user), void *user);
+
+__isl_give isl_basic_map *isl_basic_map_add_constraint(
+	__isl_take isl_basic_map *bmap, __isl_take isl_constraint *constraint);
+__isl_give isl_basic_set *isl_basic_set_add_constraint(
+	__isl_take isl_basic_set *bset, __isl_take isl_constraint *constraint);
+__isl_give isl_map *isl_map_add_constraint(__isl_take isl_map *map,
+	__isl_take isl_constraint *constraint);
+__isl_give isl_set *isl_set_add_constraint(__isl_take isl_set *set,
+	__isl_take isl_constraint *constraint);
+
+int isl_basic_map_has_defining_equality(
+	__isl_keep isl_basic_map *bmap, enum isl_dim_type type, int pos,
+	__isl_give isl_constraint **c);
+int isl_basic_set_has_defining_equality(
+	struct isl_basic_set *bset, enum isl_dim_type type, int pos,
+	struct isl_constraint **constraint);
+int isl_basic_set_has_defining_inequalities(
+	struct isl_basic_set *bset, enum isl_dim_type type, int pos,
+	struct isl_constraint **lower,
+	struct isl_constraint **upper);
+
+__isl_give isl_space *isl_constraint_get_space(
+	__isl_keep isl_constraint *constraint);
+__isl_give isl_local_space *isl_constraint_get_local_space(
+	__isl_keep isl_constraint *constraint);
+int isl_constraint_dim(struct isl_constraint *constraint,
+	enum isl_dim_type type);
+
+int isl_constraint_involves_dims(__isl_keep isl_constraint *constraint,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+const char *isl_constraint_get_dim_name(__isl_keep isl_constraint *constraint,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_val *isl_constraint_get_constant_val(
+	__isl_keep isl_constraint *constraint);
+__isl_give isl_val *isl_constraint_get_coefficient_val(
+	__isl_keep isl_constraint *constraint, enum isl_dim_type type, int pos);
+__isl_give isl_constraint *isl_constraint_set_constant_si(
+	__isl_take isl_constraint *constraint, int v);
+__isl_give isl_constraint *isl_constraint_set_constant_val(
+	__isl_take isl_constraint *constraint, __isl_take isl_val *v);
+__isl_give isl_constraint *isl_constraint_set_coefficient_si(
+	__isl_take isl_constraint *constraint,
+	enum isl_dim_type type, int pos, int v);
+__isl_give isl_constraint *isl_constraint_set_coefficient_val(
+	__isl_take isl_constraint *constraint,
+	enum isl_dim_type type, int pos, __isl_take isl_val *v);
+
+__isl_give isl_aff *isl_constraint_get_div(__isl_keep isl_constraint *constraint,
+	int pos);
+
+struct isl_constraint *isl_constraint_negate(struct isl_constraint *constraint);
+
+int isl_constraint_is_equality(__isl_keep isl_constraint *constraint);
+int isl_constraint_is_div_constraint(__isl_keep isl_constraint *constraint);
+
+int isl_constraint_is_lower_bound(__isl_keep isl_constraint *constraint,
+	enum isl_dim_type type, unsigned pos);
+int isl_constraint_is_upper_bound(__isl_keep isl_constraint *constraint,
+	enum isl_dim_type type, unsigned pos);
+
+__isl_give isl_basic_map *isl_basic_map_from_constraint(
+	__isl_take isl_constraint *constraint);
+struct isl_basic_set *isl_basic_set_from_constraint(
+	struct isl_constraint *constraint);
+
+__isl_give isl_aff *isl_constraint_get_bound(
+	__isl_keep isl_constraint *constraint, enum isl_dim_type type, int pos);
+__isl_give isl_aff *isl_constraint_get_aff(
+	__isl_keep isl_constraint *constraint);
+__isl_give isl_constraint *isl_equality_from_aff(__isl_take isl_aff *aff);
+__isl_give isl_constraint *isl_inequality_from_aff(__isl_take isl_aff *aff);
+
+__isl_give isl_basic_set *isl_basic_set_drop_constraint(
+	__isl_take isl_basic_set *bset, __isl_take isl_constraint *constraint);
+
+int isl_constraint_plain_cmp(__isl_keep isl_constraint *c1,
+	__isl_keep isl_constraint *c2);
+int isl_constraint_cmp_last_non_zero(__isl_keep isl_constraint *c1,
+	__isl_keep isl_constraint *c2);
+
+__isl_give isl_printer *isl_printer_print_constraint(__isl_take isl_printer *p,
+	__isl_keep isl_constraint *c);
+void isl_constraint_dump(__isl_keep isl_constraint *c);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#include <isl/dim.h>
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/ctx.h b/polly/lib/External/isl/include/isl/ctx.h
new file mode 100644
index 00000000000..44680b10a08
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/ctx.h
@@ -0,0 +1,246 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_CTX_H
+#define ISL_CTX_H
+
+#include <stdio.h>
+#include <stdlib.h>
+
+#include <isl/arg.h>
+#include <isl/hash.h>
+
+#ifndef __isl_give
+#define __isl_give
+#endif
+#ifndef __isl_take
+#define __isl_take
+#endif
+#ifndef __isl_keep
+#define __isl_keep
+#endif
+#ifndef __isl_null
+#define __isl_null
+#endif
+#ifndef __isl_export
+#define __isl_export
+#endif
+#ifndef __isl_constructor
+#define __isl_constructor
+#endif
+#ifndef __isl_subclass
+#define __isl_subclass(super)
+#endif
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+/* Nearly all isa functions require a struct isl_ctx allocated using
+ * isl_ctx_alloc.  This ctx contains (or will contain) options that
+ * control the behavior of the library and some caches.
+ *
+ * An object allocated within a given ctx should never be used inside
+ * another ctx.  Functions for moving objects from one ctx to another
+ * will be added as the need arises.
+ *
+ * A given context should only be used inside a single thread.
+ * A global context for synchronization between different threads
+ * as well as functions for moving a context to a different thread
+ * will be added as the need arises.
+ *
+ * If anything goes wrong (out of memory, failed assertion), then
+ * the library will currently simply abort.  This will be made
+ * configurable in the future.
+ * Users of the library should expect functions that return
+ * a pointer to a structure, to return NULL, indicating failure.
+ * Any function accepting a pointer to a structure will treat
+ * a NULL argument as a failure, resulting in the function freeing
+ * the remaining structures (if any) and returning NULL itself
+ * (in case of pointer return type).
+ * The only exception is the isl_ctx argument, which should never be NULL.
+ */
+struct isl_stats {
+	long	gbr_solved_lps;
+};
+enum isl_error {
+	isl_error_none = 0,
+	isl_error_abort,
+	isl_error_alloc,
+	isl_error_unknown,
+	isl_error_internal,
+	isl_error_invalid,
+	isl_error_quota,
+	isl_error_unsupported
+};
+struct isl_ctx;
+typedef struct isl_ctx isl_ctx;
+
+/* Some helper macros */
+
+#if __GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1)
+#define ISL_DEPRECATED	__attribute__((__deprecated__))
+#else
+#define ISL_DEPRECATED
+#endif
+
+#define ISL_FL_INIT(l, f)   (l) = (f)               /* Specific flags location. */
+#define ISL_FL_SET(l, f)    ((l) |= (f))
+#define ISL_FL_CLR(l, f)    ((l) &= ~(f))
+#define ISL_FL_ISSET(l, f)  (!!((l) & (f)))
+
+#define ISL_F_INIT(p, f)    ISL_FL_INIT((p)->flags, f)  /* Structure element flags. */
+#define ISL_F_SET(p, f)     ISL_FL_SET((p)->flags, f)
+#define ISL_F_CLR(p, f)     ISL_FL_CLR((p)->flags, f)
+#define ISL_F_ISSET(p, f)   ISL_FL_ISSET((p)->flags, f)
+
+void *isl_malloc_or_die(isl_ctx *ctx, size_t size);
+void *isl_calloc_or_die(isl_ctx *ctx, size_t nmemb, size_t size);
+void *isl_realloc_or_die(isl_ctx *ctx, void *ptr, size_t size);
+
+#define isl_alloc(ctx,type,size)	((type *)isl_malloc_or_die(ctx, size))
+#define isl_calloc(ctx,type,size)	((type *)isl_calloc_or_die(ctx,\
+								    1, size))
+#define isl_realloc(ctx,ptr,type,size)	((type *)isl_realloc_or_die(ctx,\
+								    ptr, size))
+#define isl_alloc_type(ctx,type)	isl_alloc(ctx,type,sizeof(type))
+#define isl_calloc_type(ctx,type)	isl_calloc(ctx,type,sizeof(type))
+#define isl_realloc_type(ctx,ptr,type)	isl_realloc(ctx,ptr,type,sizeof(type))
+#define isl_alloc_array(ctx,type,n)	isl_alloc(ctx,type,(n)*sizeof(type))
+#define isl_calloc_array(ctx,type,n)	((type *)isl_calloc_or_die(ctx,\
+							    n, sizeof(type)))
+#define isl_realloc_array(ctx,ptr,type,n) \
+				    isl_realloc(ctx,ptr,type,(n)*sizeof(type))
+
+#define isl_die(ctx,errno,msg,code)					\
+	do {								\
+		isl_handle_error(ctx, errno, msg, __FILE__, __LINE__);	\
+		code;							\
+	} while (0)
+
+void isl_handle_error(isl_ctx *ctx, enum isl_error error, const char *msg,
+	const char *file, int line);
+
+#define isl_assert4(ctx,test,code,errno)				\
+	do {								\
+		if (test)						\
+			break;						\
+		isl_die(ctx, errno, "Assertion \"" #test "\" failed", code);	\
+	} while (0)
+#define isl_assert(ctx,test,code)					\
+	isl_assert4(ctx,test,code,isl_error_unknown)
+
+#define isl_min(a,b)			((a < b) ? (a) : (b))
+
+/* struct isl_ctx functions */
+
+struct isl_options *isl_ctx_options(isl_ctx *ctx);
+
+isl_ctx *isl_ctx_alloc_with_options(struct isl_args *args,
+	__isl_take void *opt);
+isl_ctx *isl_ctx_alloc(void);
+void *isl_ctx_peek_options(isl_ctx *ctx, struct isl_args *args);
+int isl_ctx_parse_options(isl_ctx *ctx, int argc, char **argv, unsigned flags);
+void isl_ctx_ref(struct isl_ctx *ctx);
+void isl_ctx_deref(struct isl_ctx *ctx);
+void isl_ctx_free(isl_ctx *ctx);
+
+void isl_ctx_abort(isl_ctx *ctx);
+void isl_ctx_resume(isl_ctx *ctx);
+int isl_ctx_aborted(isl_ctx *ctx);
+
+void isl_ctx_set_max_operations(isl_ctx *ctx, unsigned long max_operations);
+unsigned long isl_ctx_get_max_operations(isl_ctx *ctx);
+void isl_ctx_reset_operations(isl_ctx *ctx);
+
+#define ISL_ARG_CTX_DECL(prefix,st,args)				\
+st *isl_ctx_peek_ ## prefix(isl_ctx *ctx);
+
+#define ISL_ARG_CTX_DEF(prefix,st,args)					\
+st *isl_ctx_peek_ ## prefix(isl_ctx *ctx)				\
+{									\
+	return (st *)isl_ctx_peek_options(ctx, &(args));		\
+}
+
+#define ISL_CTX_GET_INT_DEF(prefix,st,args,field)			\
+int prefix ## _get_ ## field(isl_ctx *ctx)				\
+{									\
+	st *options;							\
+	options = isl_ctx_peek_ ## prefix(ctx);				\
+	if (!options)							\
+		isl_die(ctx, isl_error_invalid,				\
+			"isl_ctx does not reference " #prefix,		\
+			return -1);					\
+	return options->field;						\
+}
+
+#define ISL_CTX_SET_INT_DEF(prefix,st,args,field)			\
+int prefix ## _set_ ## field(isl_ctx *ctx, int val)			\
+{									\
+	st *options;							\
+	options = isl_ctx_peek_ ## prefix(ctx);				\
+	if (!options)							\
+		isl_die(ctx, isl_error_invalid,				\
+			"isl_ctx does not reference " #prefix,		\
+			return -1);					\
+	options->field = val;						\
+	return 0;							\
+}
+
+#define ISL_CTX_GET_STR_DEF(prefix,st,args,field)			\
+const char *prefix ## _get_ ## field(isl_ctx *ctx)			\
+{									\
+	st *options;							\
+	options = isl_ctx_peek_ ## prefix(ctx);				\
+	if (!options)							\
+		isl_die(ctx, isl_error_invalid,				\
+			"isl_ctx does not reference " #prefix,		\
+			return NULL);					\
+	return options->field;						\
+}
+
+#define ISL_CTX_SET_STR_DEF(prefix,st,args,field)			\
+int prefix ## _set_ ## field(isl_ctx *ctx, const char *val)		\
+{									\
+	st *options;							\
+	options = isl_ctx_peek_ ## prefix(ctx);				\
+	if (!options)							\
+		isl_die(ctx, isl_error_invalid,				\
+			"isl_ctx does not reference " #prefix,		\
+			return -1);					\
+	if (!val)							\
+		return -1;						\
+	free(options->field);						\
+	options->field = strdup(val);					\
+	if (!options->field)						\
+		return -1;						\
+	return 0;							\
+}
+
+#define ISL_CTX_GET_BOOL_DEF(prefix,st,args,field)			\
+	ISL_CTX_GET_INT_DEF(prefix,st,args,field)
+
+#define ISL_CTX_SET_BOOL_DEF(prefix,st,args,field)			\
+	ISL_CTX_SET_INT_DEF(prefix,st,args,field)
+
+#define ISL_CTX_GET_CHOICE_DEF(prefix,st,args,field)			\
+	ISL_CTX_GET_INT_DEF(prefix,st,args,field)
+
+#define ISL_CTX_SET_CHOICE_DEF(prefix,st,args,field)			\
+	ISL_CTX_SET_INT_DEF(prefix,st,args,field)
+
+enum isl_error isl_ctx_last_error(isl_ctx *ctx);
+void isl_ctx_reset_error(isl_ctx *ctx);
+void isl_ctx_set_error(isl_ctx *ctx, enum isl_error error);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/deprecated/aff_int.h b/polly/lib/External/isl/include/isl/deprecated/aff_int.h
new file mode 100644
index 00000000000..cbab1f98717
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/deprecated/aff_int.h
@@ -0,0 +1,45 @@
+#ifndef ISL_DEPRECATED_AFF_INT_H
+#define ISL_DEPRECATED_AFF_INT_H
+
+#include <isl/deprecated/int.h>
+#include <isl/aff_type.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+int isl_aff_get_constant(__isl_keep isl_aff *aff, isl_int *v);
+int isl_aff_get_coefficient(__isl_keep isl_aff *aff,
+	enum isl_dim_type type, int pos, isl_int *v);
+int isl_aff_get_denominator(__isl_keep isl_aff *aff, isl_int *v);
+__isl_give isl_aff *isl_aff_set_constant(__isl_take isl_aff *aff, isl_int v);
+__isl_give isl_aff *isl_aff_set_coefficient(__isl_take isl_aff *aff,
+	enum isl_dim_type type, int pos, isl_int v);
+__isl_give isl_aff *isl_aff_set_denominator(__isl_take isl_aff *aff, isl_int v);
+__isl_give isl_aff *isl_aff_add_constant(__isl_take isl_aff *aff, isl_int v);
+__isl_give isl_aff *isl_aff_add_constant_num(__isl_take isl_aff *aff,
+	isl_int v);
+__isl_give isl_aff *isl_aff_add_coefficient(__isl_take isl_aff *aff,
+	enum isl_dim_type type, int pos, isl_int v);
+
+__isl_give isl_aff *isl_aff_mod(__isl_take isl_aff *aff, isl_int mod);
+
+__isl_give isl_aff *isl_aff_scale(__isl_take isl_aff *aff, isl_int f);
+__isl_give isl_aff *isl_aff_scale_down(__isl_take isl_aff *aff, isl_int f);
+
+__isl_give isl_pw_aff *isl_pw_aff_mod(__isl_take isl_pw_aff *pwaff,
+	isl_int mod);
+
+__isl_give isl_pw_aff *isl_pw_aff_scale(__isl_take isl_pw_aff *pwaff,
+	isl_int f);
+__isl_give isl_pw_aff *isl_pw_aff_scale_down(__isl_take isl_pw_aff *pwaff,
+	isl_int f);
+
+__isl_give isl_multi_aff *isl_multi_aff_scale(__isl_take isl_multi_aff *maff,
+	isl_int f);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/deprecated/ast_int.h b/polly/lib/External/isl/include/isl/deprecated/ast_int.h
new file mode 100644
index 00000000000..7d2039be0a5
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/deprecated/ast_int.h
@@ -0,0 +1,17 @@
+#ifndef ISL_DEPRECATED_AST_INT_H
+#define ISL_DEPRECATED_AST_INT_H
+
+#include <isl/deprecated/int.h>
+#include <isl/ast.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+int isl_ast_expr_get_int(__isl_keep isl_ast_expr *expr, isl_int *v);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/deprecated/constraint_int.h b/polly/lib/External/isl/include/isl/deprecated/constraint_int.h
new file mode 100644
index 00000000000..c5ab0aabc41
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/deprecated/constraint_int.h
@@ -0,0 +1,25 @@
+#ifndef ISL_DEPRECATED_CONSTRAINT_INT_H
+#define ISL_DEPRECATED_CONSTRAINT_INT_H
+
+#include <isl/deprecated/int.h>
+#include <isl/constraint.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+void isl_constraint_get_constant(__isl_keep isl_constraint *constraint,
+	isl_int *v);
+void isl_constraint_get_coefficient(__isl_keep isl_constraint *constraint,
+	enum isl_dim_type type, int pos, isl_int *v);
+__isl_give isl_constraint *isl_constraint_set_constant(
+	__isl_take isl_constraint *constraint, isl_int v);
+__isl_give isl_constraint *isl_constraint_set_coefficient(
+	__isl_take isl_constraint *constraint,
+	enum isl_dim_type type, int pos, isl_int v);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/deprecated/ilp_int.h b/polly/lib/External/isl/include/isl/deprecated/ilp_int.h
new file mode 100644
index 00000000000..b952e1b17ac
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/deprecated/ilp_int.h
@@ -0,0 +1,23 @@
+#ifndef ISL_DEPRECATED_ILP_INT_H
+#define ISL_DEPRECATED_ILP_INT_H
+
+#include <isl/deprecated/int.h>
+#include <isl/lp.h>
+#include <isl/ilp.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+enum isl_lp_result isl_basic_set_max(__isl_keep isl_basic_set *bset,
+	__isl_keep isl_aff *obj, isl_int *opt);
+enum isl_lp_result isl_set_min(__isl_keep isl_set *set,
+	__isl_keep isl_aff *obj, isl_int *opt);
+enum isl_lp_result isl_set_max(__isl_keep isl_set *set,
+	__isl_keep isl_aff *obj, isl_int *opt);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/deprecated/int.h b/polly/lib/External/isl/include/isl/deprecated/int.h
new file mode 100644
index 00000000000..98782346192
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/deprecated/int.h
@@ -0,0 +1,136 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_DEPRECATED_INT_H
+#define ISL_DEPRECATED_INT_H
+
+#include <isl/hash.h>
+#include <string.h>
+#include <gmp.h>
+#if defined(__cplusplus)
+#include <iostream>
+#endif
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+#ifndef mp_get_memory_functions
+void mp_get_memory_functions(
+		void *(**alloc_func_ptr) (size_t),
+		void *(**realloc_func_ptr) (void *, size_t, size_t),
+		void (**free_func_ptr) (void *, size_t));
+#endif
+
+/* isl_int is the basic integer type.  It currently always corresponds
+ * to a gmp mpz_t, but in the future, different types such as long long
+ * or cln::cl_I will be supported.
+ */
+typedef mpz_t	isl_int;
+
+#define isl_int_init(i)		mpz_init(i)
+#define isl_int_clear(i)	mpz_clear(i)
+
+#define isl_int_set(r,i)	mpz_set(r,i)
+#define isl_int_set_gmp(r,i)	mpz_set(r,i)
+#define isl_int_set_si(r,i)	mpz_set_si(r,i)
+#define isl_int_set_ui(r,i)	mpz_set_ui(r,i)
+#define isl_int_get_gmp(i,g)	mpz_set(g,i)
+#define isl_int_get_si(r)	mpz_get_si(r)
+#define isl_int_get_ui(r)	mpz_get_ui(r)
+#define isl_int_get_d(r)	mpz_get_d(r)
+#define isl_int_get_str(r)	mpz_get_str(0, 10, r)
+typedef void (*isl_int_print_gmp_free_t)(void *, size_t);
+#define isl_int_free_str(s)					\
+	do {								\
+		isl_int_print_gmp_free_t gmp_free;			\
+		mp_get_memory_functions(NULL, NULL, &gmp_free);		\
+		(*gmp_free)(s, strlen(s) + 1);				\
+	} while (0)
+#define isl_int_abs(r,i)	mpz_abs(r,i)
+#define isl_int_neg(r,i)	mpz_neg(r,i)
+#define isl_int_swap(i,j)	mpz_swap(i,j)
+#define isl_int_swap_or_set(i,j)	mpz_swap(i,j)
+#define isl_int_add_ui(r,i,j)	mpz_add_ui(r,i,j)
+#define isl_int_sub_ui(r,i,j)	mpz_sub_ui(r,i,j)
+
+#define isl_int_add(r,i,j)	mpz_add(r,i,j)
+#define isl_int_sub(r,i,j)	mpz_sub(r,i,j)
+#define isl_int_mul(r,i,j)	mpz_mul(r,i,j)
+#define isl_int_mul_2exp(r,i,j)	mpz_mul_2exp(r,i,j)
+#define isl_int_mul_ui(r,i,j)	mpz_mul_ui(r,i,j)
+#define isl_int_pow_ui(r,i,j)	mpz_pow_ui(r,i,j)
+#define isl_int_addmul(r,i,j)	mpz_addmul(r,i,j)
+#define isl_int_submul(r,i,j)	mpz_submul(r,i,j)
+
+#define isl_int_gcd(r,i,j)	mpz_gcd(r,i,j)
+#define isl_int_lcm(r,i,j)	mpz_lcm(r,i,j)
+#define isl_int_divexact(r,i,j)	mpz_divexact(r,i,j)
+#define isl_int_divexact_ui(r,i,j)	mpz_divexact_ui(r,i,j)
+#define isl_int_tdiv_q(r,i,j)	mpz_tdiv_q(r,i,j)
+#define isl_int_cdiv_q(r,i,j)	mpz_cdiv_q(r,i,j)
+#define isl_int_fdiv_q(r,i,j)	mpz_fdiv_q(r,i,j)
+#define isl_int_fdiv_r(r,i,j)	mpz_fdiv_r(r,i,j)
+#define isl_int_fdiv_q_ui(r,i,j)	mpz_fdiv_q_ui(r,i,j)
+
+#define isl_int_read(r,s)	mpz_set_str(r,s,10)
+#define isl_int_print(out,i,width)					\
+	do {								\
+		char *s;						\
+		s = mpz_get_str(0, 10, i);				\
+		fprintf(out, "%*s", width, s);				\
+		isl_int_free_str(s);                                        \
+	} while (0)
+
+#define isl_int_sgn(i)		mpz_sgn(i)
+#define isl_int_cmp(i,j)	mpz_cmp(i,j)
+#define isl_int_cmp_si(i,si)	mpz_cmp_si(i,si)
+#define isl_int_eq(i,j)		(mpz_cmp(i,j) == 0)
+#define isl_int_ne(i,j)		(mpz_cmp(i,j) != 0)
+#define isl_int_lt(i,j)		(mpz_cmp(i,j) < 0)
+#define isl_int_le(i,j)		(mpz_cmp(i,j) <= 0)
+#define isl_int_gt(i,j)		(mpz_cmp(i,j) > 0)
+#define isl_int_ge(i,j)		(mpz_cmp(i,j) >= 0)
+#define isl_int_abs_eq(i,j)	(mpz_cmpabs(i,j) == 0)
+#define isl_int_abs_ne(i,j)	(mpz_cmpabs(i,j) != 0)
+#define isl_int_abs_lt(i,j)	(mpz_cmpabs(i,j) < 0)
+#define isl_int_abs_gt(i,j)	(mpz_cmpabs(i,j) > 0)
+#define isl_int_abs_ge(i,j)	(mpz_cmpabs(i,j) >= 0)
+
+
+#define isl_int_is_zero(i)	(isl_int_sgn(i) == 0)
+#define isl_int_is_one(i)	(isl_int_cmp_si(i,1) == 0)
+#define isl_int_is_negone(i)	(isl_int_cmp_si(i,-1) == 0)
+#define isl_int_is_pos(i)	(isl_int_sgn(i) > 0)
+#define isl_int_is_neg(i)	(isl_int_sgn(i) < 0)
+#define isl_int_is_nonpos(i)	(isl_int_sgn(i) <= 0)
+#define isl_int_is_nonneg(i)	(isl_int_sgn(i) >= 0)
+#define isl_int_is_divisible_by(i,j)	mpz_divisible_p(i,j)
+
+uint32_t isl_gmp_hash(mpz_t v, uint32_t hash);
+#define isl_int_hash(v,h)	isl_gmp_hash(v,h)
+
+#if defined(__cplusplus)
+}
+#endif
+
+#if defined(__cplusplus)
+extern "C" { typedef void (*isl_gmp_free_t)(void *, size_t); }
+
+static inline std::ostream &operator<<(std::ostream &os, isl_int i)
+{
+	char *s;
+	s = mpz_get_str(0, 10, i);
+	os << s;
+	isl_int_free_str(s);
+	return os;
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/deprecated/map_int.h b/polly/lib/External/isl/include/isl/deprecated/map_int.h
new file mode 100644
index 00000000000..915afbf8661
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/deprecated/map_int.h
@@ -0,0 +1,27 @@
+#ifndef ISL_DEPRECATED_MAP_INT_H
+#define ISL_DEPRECATED_MAP_INT_H
+
+#include <isl/deprecated/int.h>
+#include <isl/map_type.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+int isl_basic_map_plain_is_fixed(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos, isl_int *val);
+
+__isl_give isl_map *isl_map_fix(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned pos, isl_int value);
+int isl_map_plain_is_fixed(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos, isl_int *val);
+int isl_map_fast_is_fixed(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos, isl_int *val);
+
+__isl_give isl_map *isl_map_fixed_power(__isl_take isl_map *map, isl_int exp);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/deprecated/mat_int.h b/polly/lib/External/isl/include/isl/deprecated/mat_int.h
new file mode 100644
index 00000000000..efacd4e5882
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/deprecated/mat_int.h
@@ -0,0 +1,19 @@
+#ifndef ISL_DEPRECATED_MAT_INT_H
+#define ISL_DEPRECATED_MAT_INT_H
+
+#include <isl/deprecated/int.h>
+#include <isl/mat.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+int isl_mat_get_element(__isl_keep isl_mat *mat, int row, int col, isl_int *v);
+__isl_give isl_mat *isl_mat_set_element(__isl_take isl_mat *mat,
+	int row, int col, isl_int v);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/deprecated/point_int.h b/polly/lib/External/isl/include/isl/deprecated/point_int.h
new file mode 100644
index 00000000000..2e1a98b6d6f
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/deprecated/point_int.h
@@ -0,0 +1,20 @@
+#ifndef ISL_DEPRECATED_POINT_INT_H
+#define ISL_DEPRECATED_POINT_INT_H
+
+#include <isl/deprecated/int.h>
+#include <isl/point.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+int isl_point_get_coordinate(__isl_keep isl_point *pnt,
+	enum isl_dim_type type, int pos, isl_int *v);
+__isl_give isl_point *isl_point_set_coordinate(__isl_take isl_point *pnt,
+	enum isl_dim_type type, int pos, isl_int v);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/deprecated/polynomial_int.h b/polly/lib/External/isl/include/isl/deprecated/polynomial_int.h
new file mode 100644
index 00000000000..b86fcf29f48
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/deprecated/polynomial_int.h
@@ -0,0 +1,32 @@
+#ifndef ISL_DEPRECATED_POLYNOMIAL_INT_H
+#define ISL_DEPRECATED_POLYNOMIAL_INT_H
+
+#include <isl/deprecated/int.h>
+#include <isl/polynomial.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+__isl_give isl_qpolynomial *isl_qpolynomial_rat_cst_on_domain(
+	__isl_take isl_space *space, const isl_int n, const isl_int d);
+int isl_qpolynomial_is_cst(__isl_keep isl_qpolynomial *qp,
+	isl_int *n, isl_int *d);
+__isl_give isl_qpolynomial *isl_qpolynomial_scale(
+	__isl_take isl_qpolynomial *qp, isl_int v);
+
+void isl_term_get_num(__isl_keep isl_term *term, isl_int *n);
+void isl_term_get_den(__isl_keep isl_term *term, isl_int *d);
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_scale(
+	__isl_take isl_qpolynomial_fold *fold, isl_int v);
+
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_fix_dim(
+	__isl_take isl_pw_qpolynomial_fold *pwf,
+	enum isl_dim_type type, unsigned n, isl_int v);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/deprecated/set_int.h b/polly/lib/External/isl/include/isl/deprecated/set_int.h
new file mode 100644
index 00000000000..84bb88ae3d4
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/deprecated/set_int.h
@@ -0,0 +1,27 @@
+#ifndef ISL_DEPRECATED_SET_INT_H
+#define ISL_DEPRECATED_SET_INT_H
+
+#include <isl/deprecated/int.h>
+#include <isl/set_type.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+__isl_give isl_basic_set *isl_basic_set_fix(__isl_take isl_basic_set *bset,
+		enum isl_dim_type type, unsigned pos, isl_int value);
+__isl_give isl_set *isl_set_lower_bound(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, isl_int value);
+__isl_give isl_set *isl_set_upper_bound(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, isl_int value);
+__isl_give isl_set *isl_set_fix(__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned pos, isl_int value);
+
+int isl_set_plain_is_fixed(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos, isl_int *val);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/deprecated/union_map_int.h b/polly/lib/External/isl/include/isl/deprecated/union_map_int.h
new file mode 100644
index 00000000000..4a57997a91f
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/deprecated/union_map_int.h
@@ -0,0 +1,18 @@
+#ifndef ISL_DEPRECATED_UNION_MAP_INT_H
+#define ISL_DEPRECATED_UNION_MAP_INT_H
+
+#include <isl/deprecated/int.h>
+#include <isl/union_map_type.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+__isl_give isl_union_map *isl_union_map_fixed_power(
+	__isl_take isl_union_map *umap, isl_int exp);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/deprecated/val_int.h b/polly/lib/External/isl/include/isl/deprecated/val_int.h
new file mode 100644
index 00000000000..ef2eba221d2
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/deprecated/val_int.h
@@ -0,0 +1,18 @@
+#ifndef ISL_DEPRECATED_VAL_INT_H
+#define ISL_DEPRECATED_VAL_INT_H
+
+#include <isl/deprecated/int.h>
+#include <isl/val.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+__isl_give isl_val *isl_val_int_from_isl_int(isl_ctx *ctx, isl_int n);
+int isl_val_get_num_isl_int(__isl_keep isl_val *v, isl_int *n);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/deprecated/vec_int.h b/polly/lib/External/isl/include/isl/deprecated/vec_int.h
new file mode 100644
index 00000000000..c072997ac0f
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/deprecated/vec_int.h
@@ -0,0 +1,22 @@
+#ifndef ISL_DEPRECATED_VEC_INT_H
+#define ISL_DEPRECATED_VEC_INT_H
+
+#include <isl/deprecated/int.h>
+#include <isl/vec.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+int isl_vec_get_element(__isl_keep isl_vec *vec, int pos, isl_int *v);
+__isl_give isl_vec *isl_vec_set_element(__isl_take isl_vec *vec,
+	int pos, isl_int v);
+
+__isl_give isl_vec *isl_vec_set(__isl_take isl_vec *vec, isl_int v);
+__isl_give isl_vec *isl_vec_fdiv_r(__isl_take isl_vec *vec, isl_int m);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/dim.h b/polly/lib/External/isl/include/isl/dim.h
new file mode 100644
index 00000000000..6c439f5bbcc
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/dim.h
@@ -0,0 +1,174 @@
+#ifndef ISL_DIM_H
+#define ISL_DIM_H
+
+#include <isl/space.h>
+#include <isl/local_space.h>
+#include <isl/aff_type.h>
+#include <isl/constraint.h>
+#include <isl/map_type.h>
+#include <isl/set_type.h>
+#include <isl/point.h>
+#include <isl/union_map.h>
+#include <isl/union_set.h>
+#include <isl/polynomial_type.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+#define isl_dim isl_space
+
+ISL_DEPRECATED
+isl_ctx *isl_dim_get_ctx(__isl_keep isl_space *dim);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_alloc(isl_ctx *ctx,
+			unsigned nparam, unsigned n_in, unsigned n_out);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_set_alloc(isl_ctx *ctx,
+			unsigned nparam, unsigned dim);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_copy(__isl_keep isl_space *dim);
+ISL_DEPRECATED
+void isl_dim_free(__isl_take isl_space *dim);
+
+ISL_DEPRECATED
+unsigned isl_dim_size(__isl_keep isl_space *dim, enum isl_dim_type type);
+
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_set_dim_id(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id);
+ISL_DEPRECATED
+int isl_dim_has_dim_id(__isl_keep isl_space *dim,
+	enum isl_dim_type type, unsigned pos);
+ISL_DEPRECATED
+__isl_give isl_id *isl_dim_get_dim_id(__isl_keep isl_space *dim,
+	enum isl_dim_type type, unsigned pos);
+
+ISL_DEPRECATED
+int isl_dim_find_dim_by_id(__isl_keep isl_space *dim,
+	enum isl_dim_type type, __isl_keep isl_id *id);
+
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_set_tuple_id(__isl_take isl_space *dim,
+	enum isl_dim_type type, __isl_take isl_id *id);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_reset_tuple_id(__isl_take isl_space *dim,
+	enum isl_dim_type type);
+ISL_DEPRECATED
+int isl_dim_has_tuple_id(__isl_keep isl_space *dim, enum isl_dim_type type);
+ISL_DEPRECATED
+__isl_give isl_id *isl_dim_get_tuple_id(__isl_keep isl_space *dim,
+	enum isl_dim_type type);
+
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_set_name(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned pos, __isl_keep const char *name);
+ISL_DEPRECATED
+__isl_keep const char *isl_dim_get_name(__isl_keep isl_space *dim,
+	enum isl_dim_type type, unsigned pos);
+
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_set_tuple_name(__isl_take isl_space *dim,
+	enum isl_dim_type type, const char *s);
+ISL_DEPRECATED
+const char *isl_dim_get_tuple_name(__isl_keep isl_space *dim,
+				 enum isl_dim_type type);
+
+ISL_DEPRECATED
+int isl_dim_is_wrapping(__isl_keep isl_space *dim);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_wrap(__isl_take isl_space *dim);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_unwrap(__isl_take isl_space *dim);
+
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_domain(__isl_take isl_space *dim);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_from_domain(__isl_take isl_space *dim);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_range(__isl_take isl_space *dim);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_from_range(__isl_take isl_space *dim);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_reverse(__isl_take isl_space *dim);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_join(__isl_take isl_space *left,
+	__isl_take isl_space *right);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_align_params(__isl_take isl_space *dim1,
+	__isl_take isl_space *dim2);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_insert(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned pos, unsigned n);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_add(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned n);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_drop(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned first, unsigned n);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_move(__isl_take isl_space *dim,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_map_from_set(
+	__isl_take isl_space *dim);
+ISL_DEPRECATED
+__isl_give isl_space *isl_dim_zip(__isl_take isl_space *dim);
+
+ISL_DEPRECATED
+__isl_give isl_local_space *isl_local_space_from_dim(
+	__isl_take isl_space *dim);
+ISL_DEPRECATED
+__isl_give isl_space *isl_local_space_get_dim(
+	__isl_keep isl_local_space *ls);
+
+ISL_DEPRECATED
+__isl_give isl_space *isl_aff_get_dim(__isl_keep isl_aff *aff);
+ISL_DEPRECATED
+__isl_give isl_space *isl_pw_aff_get_dim(__isl_keep isl_pw_aff *pwaff);
+
+ISL_DEPRECATED
+__isl_give isl_space *isl_constraint_get_dim(
+	__isl_keep isl_constraint *constraint);
+
+ISL_DEPRECATED
+__isl_give isl_space *isl_basic_map_get_dim(__isl_keep isl_basic_map *bmap);
+ISL_DEPRECATED
+__isl_give isl_space *isl_map_get_dim(__isl_keep isl_map *map);
+ISL_DEPRECATED
+__isl_give isl_space *isl_union_map_get_dim(__isl_keep isl_union_map *umap);
+
+ISL_DEPRECATED
+__isl_give isl_space *isl_basic_set_get_dim(__isl_keep isl_basic_set *bset);
+ISL_DEPRECATED
+__isl_give isl_space *isl_set_get_dim(__isl_keep isl_set *set);
+ISL_DEPRECATED
+__isl_give isl_space *isl_union_set_get_dim(__isl_keep isl_union_set *uset);
+
+ISL_DEPRECATED
+__isl_give isl_space *isl_point_get_dim(__isl_keep isl_point *pnt);
+
+ISL_DEPRECATED
+__isl_give isl_space *isl_qpolynomial_get_dim(__isl_keep isl_qpolynomial *qp);
+ISL_DEPRECATED
+__isl_give isl_space *isl_pw_qpolynomial_get_dim(
+	__isl_keep isl_pw_qpolynomial *pwqp);
+ISL_DEPRECATED
+__isl_give isl_space *isl_qpolynomial_fold_get_dim(
+	__isl_keep isl_qpolynomial_fold *fold);
+ISL_DEPRECATED
+__isl_give isl_space *isl_pw_qpolynomial_fold_get_dim(
+	__isl_keep isl_pw_qpolynomial_fold *pwf);
+ISL_DEPRECATED
+__isl_give isl_space *isl_union_pw_qpolynomial_get_dim(
+	__isl_keep isl_union_pw_qpolynomial *upwqp);
+ISL_DEPRECATED
+__isl_give isl_space *isl_union_pw_qpolynomial_fold_get_dim(
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/flow.h b/polly/lib/External/isl/include/isl/flow.h
new file mode 100644
index 00000000000..2aee1949f04
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/flow.h
@@ -0,0 +1,77 @@
+#ifndef ISL_FLOW_H
+#define ISL_FLOW_H
+
+#include <isl/set_type.h>
+#include <isl/map_type.h>
+#include <isl/union_set_type.h>
+#include <isl/union_map_type.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+/* Let n (>= 0) be the number of iterators shared by first and second.
+ * If first precedes second textually return 2 * n + 1,
+ * otherwise return 2 * n.
+ */
+typedef int (*isl_access_level_before)(void *first, void *second);
+
+struct isl_restriction;
+typedef struct isl_restriction isl_restriction;
+
+__isl_null isl_restriction *isl_restriction_free(
+	__isl_take isl_restriction *restr);
+__isl_give isl_restriction *isl_restriction_empty(
+	__isl_take isl_map *source_map);
+__isl_give isl_restriction *isl_restriction_none(
+	__isl_take isl_map *source_map);
+__isl_give isl_restriction *isl_restriction_input(
+	__isl_take isl_set *source_restr, __isl_take isl_set *sink_restr);
+__isl_give isl_restriction *isl_restriction_output(
+	__isl_take isl_set *source_restr);
+
+isl_ctx *isl_restriction_get_ctx(__isl_keep isl_restriction *restr);
+
+typedef __isl_give isl_restriction *(*isl_access_restrict)(
+	__isl_keep isl_map *source_map, __isl_keep isl_set *sink,
+	void *source_user, void *user);
+
+struct isl_access_info;
+typedef struct isl_access_info isl_access_info;
+struct isl_flow;
+typedef struct isl_flow isl_flow;
+
+__isl_give isl_access_info *isl_access_info_alloc(__isl_take isl_map *sink,
+	void *sink_user, isl_access_level_before fn, int max_source);
+__isl_give isl_access_info *isl_access_info_set_restrict(
+	__isl_take isl_access_info *acc, isl_access_restrict fn, void *user);
+__isl_give isl_access_info *isl_access_info_add_source(
+	__isl_take isl_access_info *acc, __isl_take isl_map *source,
+	int must, void *source_user);
+__isl_null isl_access_info *isl_access_info_free(
+	__isl_take isl_access_info *acc);
+
+isl_ctx *isl_access_info_get_ctx(__isl_keep isl_access_info *acc);
+
+__isl_give isl_flow *isl_access_info_compute_flow(__isl_take isl_access_info *acc);
+int isl_flow_foreach(__isl_keep isl_flow *deps,
+	int (*fn)(__isl_take isl_map *dep, int must, void *dep_user, void *user),
+	void *user);
+__isl_give isl_map *isl_flow_get_no_source(__isl_keep isl_flow *deps, int must);
+void isl_flow_free(__isl_take isl_flow *deps);
+
+isl_ctx *isl_flow_get_ctx(__isl_keep isl_flow *deps);
+
+int isl_union_map_compute_flow(__isl_take isl_union_map *sink,
+	__isl_take isl_union_map *must_source,
+	__isl_take isl_union_map *may_source,
+	__isl_take isl_union_map *schedule,
+	__isl_give isl_union_map **must_dep, __isl_give isl_union_map **may_dep,
+	__isl_give isl_union_map **must_no_source,
+	__isl_give isl_union_map **may_no_source);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/hash.h b/polly/lib/External/isl/include/isl/hash.h
new file mode 100644
index 00000000000..791f549ad3b
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/hash.h
@@ -0,0 +1,85 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_HASH_H
+#define ISL_HASH_H
+
+#include <stdlib.h>
+
+// We do not generate isl/stdint.h in the Polly build. The configure logic has
+// not yet been implemented and it is unclear if it actually needed on the
+// platform LLVM is compiled for.
+
+#include <stdint.h>
+// #include <isl/stdint.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+#define isl_hash_init()		(2166136261u)
+#define isl_hash_byte(h,b)	do {					\
+					h *= 16777619;			\
+					h ^= b;				\
+				} while(0)
+#define isl_hash_hash(h,h2)						\
+	do {								\
+		isl_hash_byte(h, (h2) & 0xFF);				\
+		isl_hash_byte(h, ((h2) >> 8) & 0xFF);			\
+		isl_hash_byte(h, ((h2) >> 16) & 0xFF);			\
+		isl_hash_byte(h, ((h2) >> 24) & 0xFF);			\
+	} while(0)
+#define isl_hash_bits(h,bits)						\
+	((bits) == 32) ? (h) :						\
+	((bits) >= 16) ?						\
+	      ((h) >> (bits)) ^ ((h) & (((uint32_t)1 << (bits)) - 1)) :	\
+	      (((h) >> (bits)) ^ (h)) & (((uint32_t)1 << (bits)) - 1)
+
+uint32_t isl_hash_string(uint32_t hash, const char *s);
+uint32_t isl_hash_mem(uint32_t hash, const void *p, size_t len);
+
+#define isl_hash_builtin(h,l)	isl_hash_mem(h, &l, sizeof(l))
+
+struct isl_hash_table_entry
+{
+	uint32_t  hash;
+	void     *data;
+};
+
+struct isl_hash_table {
+	int    bits;
+	int    n;
+	struct isl_hash_table_entry *entries;
+};
+
+struct isl_ctx;
+
+struct isl_hash_table *isl_hash_table_alloc(struct isl_ctx *ctx, int min_size);
+void isl_hash_table_free(struct isl_ctx *ctx, struct isl_hash_table *table);
+
+int isl_hash_table_init(struct isl_ctx *ctx, struct isl_hash_table *table,
+			int min_size);
+void isl_hash_table_clear(struct isl_hash_table *table);
+struct isl_hash_table_entry *isl_hash_table_find(struct isl_ctx *ctx,
+				struct isl_hash_table *table,
+				uint32_t key_hash,
+				int (*eq)(const void *entry, const void *val),
+				const void *val, int reserve);
+int isl_hash_table_foreach(struct isl_ctx *ctx,
+			    struct isl_hash_table *table,
+			    int (*fn)(void **entry, void *user), void *user);
+void isl_hash_table_remove(struct isl_ctx *ctx,
+				struct isl_hash_table *table,
+				struct isl_hash_table_entry *entry);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/hmap.h b/polly/lib/External/isl/include/isl/hmap.h
new file mode 100644
index 00000000000..41cd06f8700
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/hmap.h
@@ -0,0 +1,56 @@
+#include <isl/ctx.h>
+#include <isl/printer.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+#define ISL_xCAT(A,B) A ## B
+#define ISL_CAT(A,B) ISL_xCAT(A,B)
+#define ISL_KEY ISL_CAT(isl_,ISL_KEY_BASE)
+#define ISL_VAL ISL_CAT(isl_,ISL_VAL_BASE)
+#define ISL_xFN(TYPE,NAME) TYPE ## _ ## NAME
+#define ISL_FN(TYPE,NAME) ISL_xFN(TYPE,NAME)
+#define ISL_xHMAP(KEY,VAL_BASE) KEY ## _to_ ## VAL_BASE
+#define ISL_yHMAP(KEY,VAL_BASE) ISL_xHMAP(KEY,VAL_BASE)
+#define ISL_HMAP ISL_yHMAP(ISL_KEY,ISL_VAL_BASE)
+#define ISL_HMAP_BASE ISL_yHMAP(ISL_KEY_BASE,ISL_VAL_BASE)
+
+struct ISL_HMAP;
+typedef struct ISL_HMAP	ISL_HMAP;
+
+__isl_give ISL_HMAP *ISL_FN(ISL_HMAP,alloc)(isl_ctx *ctx, int min_size);
+__isl_give ISL_HMAP *ISL_FN(ISL_HMAP,copy)(__isl_keep ISL_HMAP *hmap);
+__isl_null ISL_HMAP *ISL_FN(ISL_HMAP,free)(__isl_take ISL_HMAP *hmap);
+
+isl_ctx *ISL_FN(ISL_HMAP,get_ctx)(__isl_keep ISL_HMAP *hmap);
+
+int ISL_FN(ISL_HMAP,has)(__isl_keep ISL_HMAP *hmap, __isl_keep ISL_KEY *key);
+__isl_give ISL_VAL *ISL_FN(ISL_HMAP,get)(__isl_keep ISL_HMAP *hmap,
+	__isl_take ISL_KEY *key);
+__isl_give ISL_HMAP *ISL_FN(ISL_HMAP,set)(__isl_take ISL_HMAP *hmap,
+	__isl_take ISL_KEY *key, __isl_take ISL_VAL *val);
+__isl_give ISL_HMAP *ISL_FN(ISL_HMAP,drop)(__isl_take ISL_HMAP *hmap,
+	__isl_take ISL_KEY *key);
+
+int ISL_FN(ISL_HMAP,foreach)(__isl_keep ISL_HMAP *hmap,
+	int (*fn)(__isl_take ISL_KEY *key, __isl_take ISL_VAL *val, void *user),
+	void *user);
+
+__isl_give isl_printer *ISL_FN(isl_printer_print,ISL_HMAP_BASE)(
+	__isl_take isl_printer *p, __isl_keep ISL_HMAP *hmap);
+void ISL_FN(ISL_HMAP,dump)(__isl_keep ISL_HMAP *hmap);
+
+#undef ISL_xCAT
+#undef ISL_CAT
+#undef ISL_KEY
+#undef ISL_VAL
+#undef ISL_xFN
+#undef ISL_FN
+#undef ISL_xHMAP
+#undef ISL_yHMAP
+#undef ISL_HMAP
+
+#if defined(__cplusplus)
+}
+#endif
diff --git a/polly/lib/External/isl/include/isl/id.h b/polly/lib/External/isl/include/isl/id.h
new file mode 100644
index 00000000000..bd81d66562a
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/id.h
@@ -0,0 +1,39 @@
+#ifndef ISL_ID_H
+#define ISL_ID_H
+
+#include <isl/ctx.h>
+#include <isl/list.h>
+#include <isl/printer.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_id;
+typedef struct isl_id isl_id;
+
+ISL_DECLARE_LIST(id)
+
+isl_ctx *isl_id_get_ctx(__isl_keep isl_id *id);
+uint32_t isl_id_get_hash(__isl_keep isl_id *id);
+
+__isl_give isl_id *isl_id_alloc(isl_ctx *ctx,
+	__isl_keep const char *name, void *user);
+__isl_give isl_id *isl_id_copy(isl_id *id);
+__isl_null isl_id *isl_id_free(__isl_take isl_id *id);
+
+void *isl_id_get_user(__isl_keep isl_id *id);
+__isl_keep const char *isl_id_get_name(__isl_keep isl_id *id);
+
+__isl_give isl_id *isl_id_set_free_user(__isl_take isl_id *id,
+	__isl_give void (*free_user)(void *user));
+
+__isl_give isl_printer *isl_printer_print_id(__isl_take isl_printer *p,
+	__isl_keep isl_id *id);
+void isl_id_dump(__isl_keep isl_id *id);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/id_to_ast_expr.h b/polly/lib/External/isl/include/isl/id_to_ast_expr.h
new file mode 100644
index 00000000000..958360c3b63
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/id_to_ast_expr.h
@@ -0,0 +1,13 @@
+#ifndef ISL_ID_TO_AST_EXPR_H
+#define ISL_ID_TO_AST_EXPR_H
+
+#include <isl/id.h>
+#include <isl/ast_type.h>
+
+#define ISL_KEY_BASE	id
+#define ISL_VAL_BASE	ast_expr
+#include <isl/hmap.h>
+#undef ISL_KEY_BASE
+#undef ISL_VAL_BASE
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/id_to_pw_aff.h b/polly/lib/External/isl/include/isl/id_to_pw_aff.h
new file mode 100644
index 00000000000..e54fefb50b8
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/id_to_pw_aff.h
@@ -0,0 +1,13 @@
+#ifndef ISL_ID_TO_PW_AFF_H
+#define ISL_ID_TO_PW_AFF_H
+
+#include <isl/id.h>
+#include <isl/aff_type.h>
+
+#define ISL_KEY_BASE	id
+#define ISL_VAL_BASE	pw_aff
+#include <isl/hmap.h>
+#undef ISL_KEY_BASE
+#undef ISL_VAL_BASE
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/ilp.h b/polly/lib/External/isl/include/isl/ilp.h
new file mode 100644
index 00000000000..c177014fda5
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/ilp.h
@@ -0,0 +1,33 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_ILP_H
+#define ISL_ILP_H
+
+#include <isl/aff_type.h>
+#include <isl/set_type.h>
+#include <isl/val.h>
+#include <isl/vec.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+__isl_give isl_val *isl_basic_set_max_val(__isl_keep isl_basic_set *bset,
+	__isl_keep isl_aff *obj);
+__isl_give isl_val *isl_set_min_val(__isl_keep isl_set *set,
+	__isl_keep isl_aff *obj);
+__isl_give isl_val *isl_set_max_val(__isl_keep isl_set *set,
+	__isl_keep isl_aff *obj);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/list.h b/polly/lib/External/isl/include/isl/list.h
new file mode 100644
index 00000000000..3dbde2e58e5
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/list.h
@@ -0,0 +1,76 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_LIST_H
+#define ISL_LIST_H
+
+#include <isl/ctx.h>
+#include <isl/printer.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+#define ISL_DECLARE_LIST_TYPE(EL)					\
+struct isl_##EL;							\
+struct isl_##EL##_list;							\
+typedef struct isl_##EL##_list isl_##EL##_list;
+#define ISL_DECLARE_LIST_FN(EL)						\
+isl_ctx *isl_##EL##_list_get_ctx(__isl_keep isl_##EL##_list *list);	\
+__isl_give isl_##EL##_list *isl_##EL##_list_from_##EL(			\
+	__isl_take struct isl_##EL *el);				\
+__isl_give isl_##EL##_list *isl_##EL##_list_alloc(isl_ctx *ctx, int n);	\
+__isl_give isl_##EL##_list *isl_##EL##_list_copy(			\
+	__isl_keep isl_##EL##_list *list);				\
+__isl_null isl_##EL##_list *isl_##EL##_list_free(			\
+	__isl_take isl_##EL##_list *list);				\
+__isl_give isl_##EL##_list *isl_##EL##_list_add(			\
+	__isl_take isl_##EL##_list *list,				\
+	__isl_take struct isl_##EL *el);				\
+__isl_give isl_##EL##_list *isl_##EL##_list_insert(			\
+	__isl_take isl_##EL##_list *list, unsigned pos,			\
+	__isl_take struct isl_##EL *el);				\
+__isl_give isl_##EL##_list *isl_##EL##_list_drop(			\
+	__isl_take isl_##EL##_list *list, unsigned first, unsigned n);	\
+__isl_give isl_##EL##_list *isl_##EL##_list_concat(			\
+	__isl_take isl_##EL##_list *list1,				\
+	__isl_take isl_##EL##_list *list2);				\
+int isl_##EL##_list_n_##EL(__isl_keep isl_##EL##_list *list);		\
+__isl_give struct isl_##EL *isl_##EL##_list_get_##EL(			\
+	__isl_keep isl_##EL##_list *list, int index);			\
+__isl_give struct isl_##EL##_list *isl_##EL##_list_set_##EL(		\
+	__isl_take struct isl_##EL##_list *list, int index,		\
+	__isl_take struct isl_##EL *el);				\
+int isl_##EL##_list_foreach(__isl_keep isl_##EL##_list *list,		\
+	int (*fn)(__isl_take struct isl_##EL *el, void *user),		\
+	void *user);							\
+__isl_give isl_##EL##_list *isl_##EL##_list_sort(			\
+	__isl_take isl_##EL##_list *list,				\
+	int (*cmp)(__isl_keep struct isl_##EL *a,			\
+		__isl_keep struct isl_##EL *b,				\
+		void *user), void *user);				\
+int isl_##EL##_list_foreach_scc(__isl_keep isl_##EL##_list *list,	\
+	int (*follows)(__isl_keep struct isl_##EL *a,			\
+			__isl_keep struct isl_##EL *b, void *user),	\
+	void *follows_user,						\
+	int (*fn)(__isl_take isl_##EL##_list *scc, void *user),		\
+	void *fn_user);							\
+__isl_give isl_printer *isl_printer_print_##EL##_list(			\
+	__isl_take isl_printer *p, __isl_keep isl_##EL##_list *list);	\
+void isl_##EL##_list_dump(__isl_keep isl_##EL##_list *list);
+
+#define ISL_DECLARE_LIST(EL)						\
+	ISL_DECLARE_LIST_TYPE(EL)					\
+	ISL_DECLARE_LIST_FN(EL)
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/local_space.h b/polly/lib/External/isl/include/isl/local_space.h
new file mode 100644
index 00000000000..57ac83f10dd
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/local_space.h
@@ -0,0 +1,92 @@
+#ifndef ISL_LOCAL_SPACE_H
+#define ISL_LOCAL_SPACE_H
+
+#include <isl/aff_type.h>
+#include <isl/space.h>
+#include <isl/printer.h>
+#include <isl/map_type.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_local_space;
+typedef struct isl_local_space isl_local_space;
+
+isl_ctx *isl_local_space_get_ctx(__isl_keep isl_local_space *ls);
+
+__isl_give isl_local_space *isl_local_space_from_space(__isl_take isl_space *dim);
+
+__isl_give isl_local_space *isl_local_space_copy(
+	__isl_keep isl_local_space *ls);
+__isl_null isl_local_space *isl_local_space_free(
+	__isl_take isl_local_space *ls);
+
+int isl_local_space_is_params(__isl_keep isl_local_space *ls);
+int isl_local_space_is_set(__isl_keep isl_local_space *ls);
+
+__isl_give isl_local_space *isl_local_space_set_tuple_id(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type type, __isl_take isl_id *id);
+
+int isl_local_space_dim(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type);
+int isl_local_space_has_dim_name(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos);
+const char *isl_local_space_get_dim_name(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_local_space *isl_local_space_set_dim_name(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos, const char *s);
+int isl_local_space_has_dim_id(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_id *isl_local_space_get_dim_id(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_local_space *isl_local_space_set_dim_id(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id);
+__isl_give isl_space *isl_local_space_get_space(__isl_keep isl_local_space *ls);
+__isl_give isl_aff *isl_local_space_get_div(__isl_keep isl_local_space *ls,
+	int pos);
+
+int isl_local_space_find_dim_by_name(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type, const char *name);
+
+__isl_give isl_local_space *isl_local_space_domain(
+	__isl_take isl_local_space *ls);
+__isl_give isl_local_space *isl_local_space_range(
+	__isl_take isl_local_space *ls);
+__isl_give isl_local_space *isl_local_space_from_domain(
+	__isl_take isl_local_space *ls);
+__isl_give isl_local_space *isl_local_space_add_dims(
+	__isl_take isl_local_space *ls, enum isl_dim_type type, unsigned n);
+__isl_give isl_local_space *isl_local_space_drop_dims(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_local_space *isl_local_space_insert_dims(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+__isl_give isl_local_space *isl_local_space_intersect(
+	__isl_take isl_local_space *ls1, __isl_take isl_local_space *ls2);
+
+int isl_local_space_is_equal(__isl_keep isl_local_space *ls1,
+	__isl_keep isl_local_space *ls2);
+
+__isl_give isl_basic_map *isl_local_space_lifting(
+	__isl_take isl_local_space *ls);
+
+__isl_give isl_local_space *isl_local_space_flatten_domain(
+	__isl_take isl_local_space *ls);
+__isl_give isl_local_space *isl_local_space_flatten_range(
+	__isl_take isl_local_space *ls);
+
+__isl_give isl_printer *isl_printer_print_local_space(__isl_take isl_printer *p,
+	__isl_keep isl_local_space *ls);
+void isl_local_space_dump(__isl_keep isl_local_space *ls);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/lp.h b/polly/lib/External/isl/include/isl/lp.h
new file mode 100644
index 00000000000..461ae0098e0
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/lp.h
@@ -0,0 +1,37 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_LP_H
+#define ISL_LP_H
+
+#include <isl/aff.h>
+#include <isl/val.h>
+#include <isl/set_type.h>
+
+enum isl_lp_result {
+	isl_lp_error = -1,
+	isl_lp_ok = 0,
+	isl_lp_unbounded,
+	isl_lp_empty
+};
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+__isl_give isl_val *isl_basic_set_min_lp_val(__isl_keep isl_basic_set *bset,
+	__isl_keep isl_aff *obj);
+__isl_give isl_val *isl_basic_set_max_lp_val(__isl_keep isl_basic_set *bset,
+	__isl_keep isl_aff *obj);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/map.h b/polly/lib/External/isl/include/isl/map.h
new file mode 100644
index 00000000000..a31ddfc8e16
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/map.h
@@ -0,0 +1,697 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_MAP_H
+#define ISL_MAP_H
+
+#include <stdio.h>
+
+#include <isl/ctx.h>
+#include <isl/space.h>
+#include <isl/vec.h>
+#include <isl/mat.h>
+#include <isl/printer.h>
+#include <isl/local_space.h>
+#include <isl/aff_type.h>
+#include <isl/list.h>
+#include <isl/map_type.h>
+#include <isl/val.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+/* General notes:
+ *
+ * All structures are reference counted to allow reuse without duplication.
+ * A *_copy operation will increase the reference count, while a *_free
+ * operation will decrease the reference count and only actually release
+ * the structures when the reference count drops to zero.
+ *
+ * Functions that return an isa structure will in general _destroy_
+ * all argument isa structures (the obvious execption begin the _copy
+ * functions).  A pointer passed to such a function may therefore
+ * never be used after the function call.  If you want to keep a
+ * reference to the old structure(s), use the appropriate _copy function.
+ */
+
+unsigned isl_basic_map_n_in(const struct isl_basic_map *bmap);
+unsigned isl_basic_map_n_out(const struct isl_basic_map *bmap);
+unsigned isl_basic_map_n_param(const struct isl_basic_map *bmap);
+unsigned isl_basic_map_n_div(const struct isl_basic_map *bmap);
+unsigned isl_basic_map_total_dim(const struct isl_basic_map *bmap);
+unsigned isl_basic_map_dim(__isl_keep isl_basic_map *bmap,
+				enum isl_dim_type type);
+
+unsigned isl_map_n_in(const struct isl_map *map);
+unsigned isl_map_n_out(const struct isl_map *map);
+unsigned isl_map_n_param(const struct isl_map *map);
+unsigned isl_map_dim(__isl_keep isl_map *map, enum isl_dim_type type);
+
+isl_ctx *isl_basic_map_get_ctx(__isl_keep isl_basic_map *bmap);
+isl_ctx *isl_map_get_ctx(__isl_keep isl_map *map);
+__isl_give isl_space *isl_basic_map_get_space(__isl_keep isl_basic_map *bmap);
+__isl_give isl_space *isl_map_get_space(__isl_keep isl_map *map);
+
+__isl_give isl_aff *isl_basic_map_get_div(__isl_keep isl_basic_map *bmap,
+	int pos);
+
+__isl_give isl_local_space *isl_basic_map_get_local_space(
+	__isl_keep isl_basic_map *bmap);
+
+__isl_give isl_basic_map *isl_basic_map_set_tuple_name(
+	__isl_take isl_basic_map *bmap, enum isl_dim_type type, const char *s);
+const char *isl_basic_map_get_tuple_name(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type);
+int isl_map_has_tuple_name(__isl_keep isl_map *map, enum isl_dim_type type);
+const char *isl_map_get_tuple_name(__isl_keep isl_map *map,
+	enum isl_dim_type type);
+__isl_give isl_map *isl_map_set_tuple_name(__isl_take isl_map *map,
+	enum isl_dim_type type, const char *s);
+const char *isl_basic_map_get_dim_name(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos);
+int isl_map_has_dim_name(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos);
+const char *isl_map_get_dim_name(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_basic_map *isl_basic_map_set_dim_name(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos, const char *s);
+__isl_give isl_map *isl_map_set_dim_name(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned pos, const char *s);
+
+__isl_give isl_basic_map *isl_basic_map_set_tuple_id(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, __isl_take isl_id *id);
+__isl_give isl_map *isl_map_set_dim_id(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id);
+int isl_basic_map_has_dim_id(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos);
+int isl_map_has_dim_id(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_id *isl_map_get_dim_id(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_map *isl_map_set_tuple_id(__isl_take isl_map *map,
+	enum isl_dim_type type, __isl_take isl_id *id);
+__isl_give isl_map *isl_map_reset_tuple_id(__isl_take isl_map *map,
+	enum isl_dim_type type);
+int isl_map_has_tuple_id(__isl_keep isl_map *map, enum isl_dim_type type);
+__isl_give isl_id *isl_map_get_tuple_id(__isl_keep isl_map *map,
+	enum isl_dim_type type);
+__isl_give isl_map *isl_map_reset_user(__isl_take isl_map *map);
+
+int isl_basic_map_find_dim_by_name(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, const char *name);
+int isl_map_find_dim_by_id(__isl_keep isl_map *map, enum isl_dim_type type,
+	__isl_keep isl_id *id);
+int isl_map_find_dim_by_name(__isl_keep isl_map *map, enum isl_dim_type type,
+	const char *name);
+
+int isl_basic_map_is_rational(__isl_keep isl_basic_map *bmap);
+
+struct isl_basic_map *isl_basic_map_alloc(struct isl_ctx *ctx,
+		unsigned nparam, unsigned in, unsigned out, unsigned extra,
+		unsigned n_eq, unsigned n_ineq);
+__isl_give isl_basic_map *isl_basic_map_identity(__isl_take isl_space *dim);
+struct isl_basic_map *isl_basic_map_identity_like(struct isl_basic_map *model);
+struct isl_basic_map *isl_basic_map_finalize(struct isl_basic_map *bmap);
+__isl_null isl_basic_map *isl_basic_map_free(__isl_take isl_basic_map *bmap);
+__isl_give isl_basic_map *isl_basic_map_copy(__isl_keep isl_basic_map *bmap);
+struct isl_basic_map *isl_basic_map_extend(struct isl_basic_map *base,
+		unsigned nparam, unsigned n_in, unsigned n_out, unsigned extra,
+		unsigned n_eq, unsigned n_ineq);
+struct isl_basic_map *isl_basic_map_extend_constraints(
+		struct isl_basic_map *base, unsigned n_eq, unsigned n_ineq);
+__isl_give isl_basic_map *isl_basic_map_equal(
+	__isl_take isl_space *dim, unsigned n_equal);
+__isl_give isl_basic_map *isl_basic_map_less_at(__isl_take isl_space *dim,
+	unsigned pos);
+__isl_give isl_basic_map *isl_basic_map_more_at(__isl_take isl_space *dim,
+	unsigned pos);
+__isl_give isl_basic_map *isl_basic_map_empty(__isl_take isl_space *dim);
+struct isl_basic_map *isl_basic_map_empty_like(struct isl_basic_map *model);
+struct isl_basic_map *isl_basic_map_empty_like_map(struct isl_map *model);
+__isl_give isl_basic_map *isl_basic_map_universe(__isl_take isl_space *dim);
+__isl_give isl_basic_map *isl_basic_map_nat_universe(__isl_take isl_space *dim);
+__isl_give isl_basic_map *isl_basic_map_universe_like(
+		__isl_keep isl_basic_map *bmap);
+__isl_give isl_basic_map *isl_basic_map_remove_redundancies(
+	__isl_take isl_basic_map *bmap);
+__isl_give isl_map *isl_map_remove_redundancies(__isl_take isl_map *map);
+__isl_give isl_basic_map *isl_map_simple_hull(__isl_take isl_map *map);
+__isl_give isl_basic_map *isl_map_unshifted_simple_hull(
+	__isl_take isl_map *map);
+
+__isl_export
+__isl_give isl_basic_map *isl_basic_map_intersect_domain(
+		__isl_take isl_basic_map *bmap,
+		__isl_take isl_basic_set *bset);
+__isl_export
+__isl_give isl_basic_map *isl_basic_map_intersect_range(
+		__isl_take isl_basic_map *bmap,
+		__isl_take isl_basic_set *bset);
+__isl_export
+__isl_give isl_basic_map *isl_basic_map_intersect(
+		__isl_take isl_basic_map *bmap1,
+		__isl_take isl_basic_map *bmap2);
+__isl_export
+__isl_give isl_map *isl_basic_map_union(
+		__isl_take isl_basic_map *bmap1,
+		__isl_take isl_basic_map *bmap2);
+__isl_export
+__isl_give isl_basic_map *isl_basic_map_apply_domain(
+		__isl_take isl_basic_map *bmap1,
+		__isl_take isl_basic_map *bmap2);
+__isl_export
+__isl_give isl_basic_map *isl_basic_map_apply_range(
+		__isl_take isl_basic_map *bmap1,
+		__isl_take isl_basic_map *bmap2);
+__isl_export
+__isl_give isl_basic_map *isl_basic_map_affine_hull(
+		__isl_take isl_basic_map *bmap);
+__isl_give isl_basic_map *isl_basic_map_preimage_domain_multi_aff(
+	__isl_take isl_basic_map *bmap, __isl_take isl_multi_aff *ma);
+__isl_give isl_basic_map *isl_basic_map_preimage_range_multi_aff(
+	__isl_take isl_basic_map *bmap, __isl_take isl_multi_aff *ma);
+__isl_export
+__isl_give isl_basic_map *isl_basic_map_reverse(__isl_take isl_basic_map *bmap);
+__isl_give isl_basic_set *isl_basic_map_domain(__isl_take isl_basic_map *bmap);
+__isl_give isl_basic_set *isl_basic_map_range(__isl_take isl_basic_map *bmap);
+__isl_give isl_basic_map *isl_basic_map_domain_map(
+	__isl_take isl_basic_map *bmap);
+__isl_give isl_basic_map *isl_basic_map_range_map(
+	__isl_take isl_basic_map *bmap);
+__isl_give isl_basic_map *isl_basic_map_remove_dims(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_basic_map *isl_basic_map_eliminate(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_basic_map *isl_basic_map_from_basic_set(
+	__isl_take isl_basic_set *bset, __isl_take isl_space *dim);
+struct isl_basic_set *isl_basic_set_from_basic_map(struct isl_basic_map *bmap);
+__isl_export
+__isl_give isl_basic_map *isl_basic_map_sample(__isl_take isl_basic_map *bmap);
+struct isl_basic_map *isl_basic_map_simplify(struct isl_basic_map *bmap);
+__isl_export
+__isl_give isl_basic_map *isl_basic_map_detect_equalities(
+						__isl_take isl_basic_map *bmap);
+__isl_give isl_basic_map *isl_basic_map_read_from_file(isl_ctx *ctx,
+	FILE *input);
+__isl_constructor
+__isl_give isl_basic_map *isl_basic_map_read_from_str(isl_ctx *ctx,
+	const char *str);
+__isl_give isl_map *isl_map_read_from_file(isl_ctx *ctx, FILE *input);
+__isl_constructor
+__isl_give isl_map *isl_map_read_from_str(isl_ctx *ctx, const char *str);
+void isl_basic_map_dump(__isl_keep isl_basic_map *bmap);
+void isl_basic_map_print(__isl_keep isl_basic_map *bmap, FILE *out, int indent,
+	const char *prefix, const char *suffix, unsigned output_format);
+void isl_map_dump(__isl_keep isl_map *map);
+void isl_map_print(__isl_keep isl_map *map, FILE *out, int indent,
+	unsigned output_format);
+__isl_give isl_printer *isl_printer_print_basic_map(
+	__isl_take isl_printer *printer, __isl_keep isl_basic_map *bmap);
+__isl_give char *isl_map_to_str(__isl_keep isl_map *map);
+__isl_give isl_printer *isl_printer_print_map(__isl_take isl_printer *printer,
+	__isl_keep isl_map *map);
+__isl_give isl_basic_map *isl_basic_map_fix_si(__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned pos, int value);
+__isl_give isl_basic_map *isl_basic_map_fix_val(__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_val *v);
+__isl_give isl_basic_map *isl_basic_map_lower_bound_si(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned pos, int value);
+__isl_give isl_basic_map *isl_basic_map_upper_bound_si(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos, int value);
+
+struct isl_basic_map *isl_basic_map_sum(
+		struct isl_basic_map *bmap1, struct isl_basic_map *bmap2);
+struct isl_basic_map *isl_basic_map_neg(struct isl_basic_map *bmap);
+
+struct isl_map *isl_map_sum(struct isl_map *map1, struct isl_map *map2);
+struct isl_map *isl_map_neg(struct isl_map *map);
+__isl_give isl_map *isl_map_floordiv_val(__isl_take isl_map *map,
+	__isl_take isl_val *d);
+
+__isl_export
+int isl_basic_map_is_equal(
+		__isl_keep isl_basic_map *bmap1,
+		__isl_keep isl_basic_map *bmap2);
+int isl_basic_map_is_disjoint(__isl_keep isl_basic_map *bmap1,
+	__isl_keep isl_basic_map *bmap2);
+
+__isl_give isl_map *isl_basic_map_partial_lexmax(
+		__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+		__isl_give isl_set **empty);
+__isl_give isl_map *isl_basic_map_partial_lexmin(
+		__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+		__isl_give isl_set **empty);
+__isl_give isl_map *isl_map_partial_lexmax(
+		__isl_take isl_map *map, __isl_take isl_set *dom,
+		__isl_give isl_set **empty);
+__isl_give isl_map *isl_map_partial_lexmin(
+		__isl_take isl_map *map, __isl_take isl_set *dom,
+		__isl_give isl_set **empty);
+__isl_export
+__isl_give isl_map *isl_basic_map_lexmin(__isl_take isl_basic_map *bmap);
+__isl_export
+__isl_give isl_map *isl_basic_map_lexmax(__isl_take isl_basic_map *bmap);
+__isl_export
+__isl_give isl_map *isl_map_lexmin(__isl_take isl_map *map);
+__isl_export
+__isl_give isl_map *isl_map_lexmax(__isl_take isl_map *map);
+__isl_give isl_pw_multi_aff *isl_basic_map_partial_lexmin_pw_multi_aff(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty);
+__isl_give isl_pw_multi_aff *isl_basic_map_partial_lexmax_pw_multi_aff(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty);
+__isl_give isl_pw_multi_aff *isl_basic_map_lexmin_pw_multi_aff(
+	__isl_take isl_basic_map *bmap);
+__isl_give isl_pw_multi_aff *isl_map_lexmin_pw_multi_aff(
+	__isl_take isl_map *map);
+__isl_give isl_pw_multi_aff *isl_map_lexmax_pw_multi_aff(
+	__isl_take isl_map *map);
+
+void isl_basic_map_print_internal(__isl_keep isl_basic_map *bmap,
+	FILE *out, int indent);
+
+struct isl_basic_map *isl_map_copy_basic_map(struct isl_map *map);
+__isl_give isl_map *isl_map_drop_basic_map(__isl_take isl_map *map,
+						__isl_keep isl_basic_map *bmap);
+
+__isl_give isl_val *isl_basic_map_plain_get_val_if_fixed(
+	__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos);
+
+int isl_basic_map_image_is_bounded(__isl_keep isl_basic_map *bmap);
+int isl_basic_map_is_universe(__isl_keep isl_basic_map *bmap);
+int isl_basic_map_plain_is_empty(__isl_keep isl_basic_map *bmap);
+ISL_DEPRECATED
+int isl_basic_map_fast_is_empty(__isl_keep isl_basic_map *bmap);
+__isl_export
+int isl_basic_map_is_empty(__isl_keep isl_basic_map *bmap);
+__isl_export
+int isl_basic_map_is_subset(__isl_keep isl_basic_map *bmap1,
+		__isl_keep isl_basic_map *bmap2);
+int isl_basic_map_is_strict_subset(__isl_keep isl_basic_map *bmap1,
+		__isl_keep isl_basic_map *bmap2);
+
+struct isl_map *isl_map_alloc(struct isl_ctx *ctx,
+		unsigned nparam, unsigned in, unsigned out, int n,
+		unsigned flags);
+__isl_give isl_map *isl_map_universe(__isl_take isl_space *dim);
+__isl_give isl_map *isl_map_nat_universe(__isl_take isl_space *dim);
+__isl_give isl_map *isl_map_empty(__isl_take isl_space *dim);
+struct isl_map *isl_map_empty_like(struct isl_map *model);
+struct isl_map *isl_map_empty_like_basic_map(struct isl_basic_map *model);
+struct isl_map *isl_map_dup(struct isl_map *map);
+__isl_give isl_map *isl_map_add_basic_map(__isl_take isl_map *map,
+						__isl_take isl_basic_map *bmap);
+__isl_give isl_map *isl_map_identity(__isl_take isl_space *dim);
+struct isl_map *isl_map_identity_like(struct isl_map *model);
+struct isl_map *isl_map_identity_like_basic_map(struct isl_basic_map *model);
+__isl_give isl_map *isl_map_lex_lt_first(__isl_take isl_space *dim, unsigned n);
+__isl_give isl_map *isl_map_lex_le_first(__isl_take isl_space *dim, unsigned n);
+__isl_give isl_map *isl_map_lex_lt(__isl_take isl_space *set_dim);
+__isl_give isl_map *isl_map_lex_le(__isl_take isl_space *set_dim);
+__isl_give isl_map *isl_map_lex_gt_first(__isl_take isl_space *dim, unsigned n);
+__isl_give isl_map *isl_map_lex_ge_first(__isl_take isl_space *dim, unsigned n);
+__isl_give isl_map *isl_map_lex_gt(__isl_take isl_space *set_dim);
+__isl_give isl_map *isl_map_lex_ge(__isl_take isl_space *set_dim);
+struct isl_map *isl_map_finalize(struct isl_map *map);
+__isl_null isl_map *isl_map_free(__isl_take isl_map *map);
+__isl_give isl_map *isl_map_copy(__isl_keep isl_map *map);
+struct isl_map *isl_map_extend(struct isl_map *base,
+		unsigned nparam, unsigned n_in, unsigned n_out);
+__isl_export
+__isl_give isl_map *isl_map_reverse(__isl_take isl_map *map);
+__isl_export
+__isl_give isl_map *isl_map_union(
+		__isl_take isl_map *map1,
+		__isl_take isl_map *map2);
+struct isl_map *isl_map_union_disjoint(
+			struct isl_map *map1, struct isl_map *map2);
+__isl_export
+__isl_give isl_map *isl_map_intersect_domain(
+		__isl_take isl_map *map,
+		__isl_take isl_set *set);
+__isl_export
+__isl_give isl_map *isl_map_intersect_range(
+		__isl_take isl_map *map,
+		__isl_take isl_set *set);
+__isl_export
+__isl_give isl_map *isl_map_apply_domain(
+		__isl_take isl_map *map1,
+		__isl_take isl_map *map2);
+__isl_export
+__isl_give isl_map *isl_map_apply_range(
+		__isl_take isl_map *map1,
+		__isl_take isl_map *map2);
+__isl_give isl_map *isl_map_preimage_domain_multi_aff(__isl_take isl_map *map,
+	__isl_take isl_multi_aff *ma);
+__isl_give isl_map *isl_map_preimage_range_multi_aff(__isl_take isl_map *map,
+	__isl_take isl_multi_aff *ma);
+__isl_give isl_map *isl_map_preimage_domain_pw_multi_aff(
+	__isl_take isl_map *map, __isl_take isl_pw_multi_aff *pma);
+__isl_give isl_map *isl_map_preimage_range_pw_multi_aff(
+	__isl_take isl_map *map, __isl_take isl_pw_multi_aff *pma);
+__isl_give isl_map *isl_map_preimage_domain_multi_pw_aff(
+	__isl_take isl_map *map, __isl_take isl_multi_pw_aff *mpa);
+__isl_give isl_basic_map *isl_basic_map_product(
+	__isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2);
+__isl_give isl_map *isl_map_product(__isl_take isl_map *map1,
+	__isl_take isl_map *map2);
+__isl_give isl_basic_map *isl_basic_map_domain_product(
+	__isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2);
+__isl_give isl_basic_map *isl_basic_map_range_product(
+	__isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2);
+__isl_give isl_map *isl_map_domain_product(__isl_take isl_map *map1,
+	__isl_take isl_map *map2);
+__isl_give isl_map *isl_map_range_product(__isl_take isl_map *map1,
+	__isl_take isl_map *map2);
+__isl_give isl_basic_map *isl_basic_map_flat_product(
+	__isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2);
+__isl_give isl_map *isl_map_flat_product(__isl_take isl_map *map1,
+	__isl_take isl_map *map2);
+__isl_give isl_basic_map *isl_basic_map_flat_range_product(
+	__isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2);
+__isl_give isl_map *isl_map_flat_domain_product(__isl_take isl_map *map1,
+	__isl_take isl_map *map2);
+__isl_give isl_map *isl_map_flat_range_product(__isl_take isl_map *map1,
+	__isl_take isl_map *map2);
+int isl_map_domain_is_wrapping(__isl_keep isl_map *map);
+int isl_map_range_is_wrapping(__isl_keep isl_map *map);
+__isl_give isl_map *isl_map_factor_domain(__isl_take isl_map *map);
+__isl_give isl_map *isl_map_factor_range(__isl_take isl_map *map);
+__isl_give isl_map *isl_map_domain_factor_domain(__isl_take isl_map *map);
+__isl_give isl_map *isl_map_domain_factor_range(__isl_take isl_map *map);
+__isl_give isl_map *isl_map_range_factor_domain(__isl_take isl_map *map);
+__isl_give isl_map *isl_map_range_factor_range(__isl_take isl_map *map);
+__isl_export
+__isl_give isl_map *isl_map_intersect(__isl_take isl_map *map1,
+				      __isl_take isl_map *map2);
+__isl_export
+__isl_give isl_map *isl_map_intersect_params(__isl_take isl_map *map,
+		__isl_take isl_set *params);
+__isl_export
+__isl_give isl_map *isl_map_subtract(
+		__isl_take isl_map *map1,
+		__isl_take isl_map *map2);
+__isl_give isl_map *isl_map_subtract_domain(__isl_take isl_map *map,
+	__isl_take isl_set *dom);
+__isl_give isl_map *isl_map_subtract_range(__isl_take isl_map *map,
+	__isl_take isl_set *dom);
+__isl_export
+__isl_give isl_map *isl_map_complement(__isl_take isl_map *map);
+struct isl_map *isl_map_fix_input_si(struct isl_map *map,
+		unsigned input, int value);
+__isl_give isl_map *isl_map_fix_si(__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned pos, int value);
+__isl_give isl_map *isl_map_fix_val(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_val *v);
+__isl_give isl_map *isl_map_lower_bound_si(__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned pos, int value);
+__isl_give isl_map *isl_map_upper_bound_si(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned pos, int value);
+__isl_export
+__isl_give isl_basic_set *isl_basic_map_deltas(__isl_take isl_basic_map *bmap);
+__isl_export
+__isl_give isl_set *isl_map_deltas(__isl_take isl_map *map);
+__isl_give isl_basic_map *isl_basic_map_deltas_map(
+	__isl_take isl_basic_map *bmap);
+__isl_give isl_map *isl_map_deltas_map(__isl_take isl_map *map);
+__isl_export
+__isl_give isl_map *isl_map_detect_equalities(__isl_take isl_map *map);
+__isl_export
+__isl_give isl_basic_map *isl_map_affine_hull(__isl_take isl_map *map);
+__isl_give isl_basic_map *isl_map_convex_hull(__isl_take isl_map *map);
+__isl_export
+__isl_give isl_basic_map *isl_map_polyhedral_hull(__isl_take isl_map *map);
+__isl_give isl_basic_map *isl_basic_map_add(__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned n);
+__isl_give isl_map *isl_map_add_dims(__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned n);
+__isl_give isl_basic_map *isl_basic_map_insert_dims(
+	__isl_take isl_basic_map *bmap, enum isl_dim_type type,
+	unsigned pos, unsigned n);
+__isl_give isl_map *isl_map_insert_dims(__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned pos, unsigned n);
+__isl_give isl_basic_map *isl_basic_map_move_dims(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n);
+__isl_give isl_map *isl_map_move_dims(__isl_take isl_map *map,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n);
+__isl_give isl_basic_map *isl_basic_map_project_out(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_map *isl_map_project_out(__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_basic_map *isl_basic_map_remove_divs(
+	__isl_take isl_basic_map *bmap);
+__isl_give isl_map *isl_map_remove_unknown_divs(__isl_take isl_map *map);
+__isl_give isl_map *isl_map_remove_divs(__isl_take isl_map *map);
+__isl_give isl_map *isl_map_eliminate(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_map *isl_map_remove_dims(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_basic_map *isl_basic_map_remove_divs_involving_dims(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_map *isl_map_remove_divs_involving_dims(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned first, unsigned n);
+struct isl_map *isl_map_remove_inputs(struct isl_map *map,
+	unsigned first, unsigned n);
+
+__isl_give isl_basic_map *isl_basic_map_equate(__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2);
+__isl_give isl_basic_map *isl_basic_map_order_ge(__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2);
+__isl_give isl_map *isl_map_order_ge(__isl_take isl_map *map,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2);
+__isl_give isl_map *isl_map_order_le(__isl_take isl_map *map,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2);
+__isl_give isl_map *isl_map_equate(__isl_take isl_map *map,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2);
+__isl_give isl_map *isl_map_oppose(__isl_take isl_map *map,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2);
+__isl_give isl_map *isl_map_order_lt(__isl_take isl_map *map,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2);
+__isl_give isl_basic_map *isl_basic_map_order_gt(__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2);
+__isl_give isl_map *isl_map_order_gt(__isl_take isl_map *map,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2);
+
+__isl_export
+__isl_give isl_map *isl_set_identity(__isl_take isl_set *set);
+
+__isl_export
+int isl_basic_set_is_wrapping(__isl_keep isl_basic_set *bset);
+__isl_export
+int isl_set_is_wrapping(__isl_keep isl_set *set);
+__isl_give isl_basic_set *isl_basic_map_wrap(__isl_take isl_basic_map *bmap);
+__isl_give isl_set *isl_map_wrap(__isl_take isl_map *map);
+__isl_give isl_basic_map *isl_basic_set_unwrap(__isl_take isl_basic_set *bset);
+__isl_give isl_map *isl_set_unwrap(__isl_take isl_set *set);
+__isl_export
+__isl_give isl_basic_map *isl_basic_map_flatten(__isl_take isl_basic_map *bmap);
+__isl_export
+__isl_give isl_map *isl_map_flatten(__isl_take isl_map *map);
+__isl_export
+__isl_give isl_basic_map *isl_basic_map_flatten_domain(
+	__isl_take isl_basic_map *bmap);
+__isl_export
+__isl_give isl_basic_map *isl_basic_map_flatten_range(
+	__isl_take isl_basic_map *bmap);
+__isl_export
+__isl_give isl_map *isl_map_flatten_domain(__isl_take isl_map *map);
+__isl_export
+__isl_give isl_map *isl_map_flatten_range(__isl_take isl_map *map);
+__isl_export
+__isl_give isl_basic_set *isl_basic_set_flatten(__isl_take isl_basic_set *bset);
+__isl_export
+__isl_give isl_set *isl_set_flatten(__isl_take isl_set *set);
+__isl_give isl_map *isl_set_flatten_map(__isl_take isl_set *set);
+__isl_give isl_set *isl_map_params(__isl_take isl_map *map);
+__isl_give isl_set *isl_map_domain(__isl_take isl_map *bmap);
+__isl_give isl_set *isl_map_range(__isl_take isl_map *map);
+__isl_give isl_map *isl_map_domain_map(__isl_take isl_map *map);
+__isl_give isl_map *isl_map_range_map(__isl_take isl_map *map);
+__isl_give isl_map *isl_set_wrapped_domain_map(__isl_take isl_set *set);
+__isl_constructor
+__isl_give isl_map *isl_map_from_basic_map(__isl_take isl_basic_map *bmap);
+__isl_give isl_map *isl_map_from_domain(__isl_take isl_set *set);
+__isl_give isl_basic_map *isl_basic_map_from_domain(
+	__isl_take isl_basic_set *bset);
+__isl_give isl_basic_map *isl_basic_map_from_range(
+	__isl_take isl_basic_set *bset);
+__isl_give isl_map *isl_map_from_range(__isl_take isl_set *set);
+__isl_give isl_basic_map *isl_basic_map_from_domain_and_range(
+	__isl_take isl_basic_set *domain, __isl_take isl_basic_set *range);
+__isl_give isl_map *isl_map_from_domain_and_range(__isl_take isl_set *domain,
+	__isl_take isl_set *range);
+__isl_give isl_map *isl_map_from_set(__isl_take isl_set *set,
+	__isl_take isl_space *dim);
+struct isl_set *isl_set_from_map(struct isl_map *map);
+__isl_export
+__isl_give isl_basic_map *isl_map_sample(__isl_take isl_map *map);
+
+int isl_map_plain_is_empty(__isl_keep isl_map *map);
+ISL_DEPRECATED
+int isl_map_fast_is_empty(__isl_keep isl_map *map);
+int isl_map_plain_is_universe(__isl_keep isl_map *map);
+__isl_export
+int isl_map_is_empty(__isl_keep isl_map *map);
+__isl_export
+int isl_map_is_subset(__isl_keep isl_map *map1, __isl_keep isl_map *map2);
+__isl_export
+int isl_map_is_strict_subset(__isl_keep isl_map *map1, __isl_keep isl_map *map2);
+__isl_export
+int isl_map_is_equal(__isl_keep isl_map *map1, __isl_keep isl_map *map2);
+__isl_export
+int isl_map_is_disjoint(__isl_keep isl_map *map1, __isl_keep isl_map *map2);
+int isl_basic_map_is_single_valued(__isl_keep isl_basic_map *bmap);
+int isl_map_plain_is_single_valued(__isl_keep isl_map *map);
+__isl_export
+int isl_map_is_single_valued(__isl_keep isl_map *map);
+int isl_map_plain_is_injective(__isl_keep isl_map *map);
+__isl_export
+int isl_map_is_injective(__isl_keep isl_map *map);
+__isl_export
+int isl_map_is_bijective(__isl_keep isl_map *map);
+int isl_map_is_translation(__isl_keep isl_map *map);
+int isl_map_has_equal_space(__isl_keep isl_map *map1, __isl_keep isl_map *map2);
+
+int isl_basic_map_can_zip(__isl_keep isl_basic_map *bmap);
+int isl_map_can_zip(__isl_keep isl_map *map);
+__isl_give isl_basic_map *isl_basic_map_zip(__isl_take isl_basic_map *bmap);
+__isl_give isl_map *isl_map_zip(__isl_take isl_map *map);
+
+int isl_basic_map_can_curry(__isl_keep isl_basic_map *bmap);
+int isl_map_can_curry(__isl_keep isl_map *map);
+__isl_give isl_basic_map *isl_basic_map_curry(__isl_take isl_basic_map *bmap);
+__isl_give isl_map *isl_map_curry(__isl_take isl_map *map);
+
+int isl_basic_map_can_uncurry(__isl_keep isl_basic_map *bmap);
+int isl_map_can_uncurry(__isl_keep isl_map *map);
+__isl_give isl_basic_map *isl_basic_map_uncurry(__isl_take isl_basic_map *bmap);
+__isl_give isl_map *isl_map_uncurry(__isl_take isl_map *map);
+
+__isl_give isl_map *isl_map_make_disjoint(__isl_take isl_map *map);
+__isl_give isl_map *isl_basic_map_compute_divs(__isl_take isl_basic_map *bmap);
+__isl_give isl_map *isl_map_compute_divs(__isl_take isl_map *map);
+__isl_give isl_map *isl_map_align_divs(__isl_take isl_map *map);
+
+__isl_give isl_basic_map *isl_basic_map_drop_constraints_involving_dims(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_map *isl_map_drop_constraints_involving_dims(
+	__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+int isl_basic_map_involves_dims(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned first, unsigned n);
+int isl_map_involves_dims(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+void isl_map_print_internal(__isl_keep isl_map *map, FILE *out, int indent);
+
+__isl_give isl_val *isl_map_plain_get_val_if_fixed(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos);
+
+__isl_export
+__isl_give isl_basic_map *isl_basic_map_gist(__isl_take isl_basic_map *bmap,
+	__isl_take isl_basic_map *context);
+__isl_export
+__isl_give isl_map *isl_map_gist(__isl_take isl_map *map,
+	__isl_take isl_map *context);
+__isl_export
+__isl_give isl_map *isl_map_gist_domain(__isl_take isl_map *map,
+	__isl_take isl_set *context);
+__isl_give isl_map *isl_map_gist_range(__isl_take isl_map *map,
+	__isl_take isl_set *context);
+__isl_give isl_map *isl_map_gist_params(__isl_take isl_map *map,
+	__isl_take isl_set *context);
+__isl_give isl_map *isl_map_gist_basic_map(__isl_take isl_map *map,
+	__isl_take isl_basic_map *context);
+
+__isl_export
+__isl_give isl_map *isl_map_coalesce(__isl_take isl_map *map);
+
+int isl_map_plain_is_equal(__isl_keep isl_map *map1, __isl_keep isl_map *map2);
+ISL_DEPRECATED
+int isl_map_fast_is_equal(__isl_keep isl_map *map1, __isl_keep isl_map *map2);
+
+uint32_t isl_map_get_hash(__isl_keep isl_map *map);
+
+int isl_map_n_basic_map(__isl_keep isl_map *map);
+__isl_export
+int isl_map_foreach_basic_map(__isl_keep isl_map *map,
+	int (*fn)(__isl_take isl_basic_map *bmap, void *user), void *user);
+
+__isl_give isl_map *isl_set_lifting(__isl_take isl_set *set);
+
+__isl_give isl_map *isl_map_fixed_power_val(__isl_take isl_map *map,
+	__isl_take isl_val *exp);
+__isl_give isl_map *isl_map_power(__isl_take isl_map *map, int *exact);
+__isl_give isl_map *isl_map_reaching_path_lengths(__isl_take isl_map *map,
+	int *exact);
+__isl_give isl_map *isl_map_transitive_closure(__isl_take isl_map *map,
+	int *exact);
+
+__isl_give isl_map *isl_map_lex_le_map(__isl_take isl_map *map1,
+	__isl_take isl_map *map2);
+__isl_give isl_map *isl_map_lex_lt_map(__isl_take isl_map *map1,
+	__isl_take isl_map *map2);
+__isl_give isl_map *isl_map_lex_ge_map(__isl_take isl_map *map1,
+	__isl_take isl_map *map2);
+__isl_give isl_map *isl_map_lex_gt_map(__isl_take isl_map *map1,
+	__isl_take isl_map *map2);
+
+__isl_give isl_basic_map *isl_basic_map_align_params(
+	__isl_take isl_basic_map *bmap, __isl_take isl_space *model);
+__isl_give isl_map *isl_map_align_params(__isl_take isl_map *map,
+	__isl_take isl_space *model);
+
+__isl_give isl_mat *isl_basic_map_equalities_matrix(
+		__isl_keep isl_basic_map *bmap, enum isl_dim_type c1,
+		enum isl_dim_type c2, enum isl_dim_type c3,
+		enum isl_dim_type c4, enum isl_dim_type c5);
+__isl_give isl_mat *isl_basic_map_inequalities_matrix(
+		__isl_keep isl_basic_map *bmap, enum isl_dim_type c1,
+		enum isl_dim_type c2, enum isl_dim_type c3,
+		enum isl_dim_type c4, enum isl_dim_type c5);
+__isl_give isl_basic_map *isl_basic_map_from_constraint_matrices(
+	__isl_take isl_space *dim,
+	__isl_take isl_mat *eq, __isl_take isl_mat *ineq, enum isl_dim_type c1,
+	enum isl_dim_type c2, enum isl_dim_type c3,
+	enum isl_dim_type c4, enum isl_dim_type c5);
+
+__isl_give isl_basic_map *isl_basic_map_from_aff(__isl_take isl_aff *aff);
+__isl_give isl_basic_map *isl_basic_map_from_multi_aff(
+	__isl_take isl_multi_aff *maff);
+__isl_give isl_basic_map *isl_basic_map_from_aff_list(
+	__isl_take isl_space *domain_dim, __isl_take isl_aff_list *list);
+
+__isl_give isl_map *isl_map_from_aff(__isl_take isl_aff *aff);
+__isl_give isl_map *isl_map_from_multi_aff(__isl_take isl_multi_aff *maff);
+
+__isl_give isl_pw_aff *isl_map_dim_max(__isl_take isl_map *map, int pos);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#include <isl/dim.h>
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/map_to_basic_set.h b/polly/lib/External/isl/include/isl/map_to_basic_set.h
new file mode 100644
index 00000000000..7df8ee3e0a9
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/map_to_basic_set.h
@@ -0,0 +1,13 @@
+#ifndef ISL_MAP_TO_BASIC_SET_H
+#define ISL_MAP_TO_BASIC_SET_H
+
+#include <isl/set_type.h>
+#include <isl/map_type.h>
+
+#define ISL_KEY_BASE	map
+#define ISL_VAL_BASE	basic_set
+#include <isl/hmap.h>
+#undef ISL_KEY_BASE
+#undef ISL_VAL_BASE
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/map_type.h b/polly/lib/External/isl/include/isl/map_type.h
new file mode 100644
index 00000000000..148dc0d3c70
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/map_type.h
@@ -0,0 +1,35 @@
+#ifndef ISL_MAP_TYPE_H
+#define ISL_MAP_TYPE_H
+
+#include <isl/ctx.h>
+#include <isl/list.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct __isl_subclass(isl_map) isl_basic_map;
+typedef struct isl_basic_map isl_basic_map;
+struct __isl_subclass(isl_union_map) isl_map;
+typedef struct isl_map isl_map;
+
+#ifndef isl_basic_set
+struct __isl_subclass(isl_set) isl_basic_set;
+typedef struct isl_basic_set isl_basic_set;
+ISL_DECLARE_LIST_TYPE(basic_set)
+#endif
+
+#ifndef isl_set
+struct __isl_subclass(isl_union_set) isl_set;
+typedef struct isl_set isl_set;
+ISL_DECLARE_LIST_TYPE(set)
+#endif
+
+ISL_DECLARE_LIST_FN(basic_set)
+ISL_DECLARE_LIST_FN(set)
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/mat.h b/polly/lib/External/isl/include/isl/mat.h
new file mode 100644
index 00000000000..7a43ccbaf64
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/mat.h
@@ -0,0 +1,111 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_MAT_H
+#define ISL_MAT_H
+
+#include <stdio.h>
+
+#include <isl/ctx.h>
+#include <isl/vec.h>
+#include <isl/val.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_mat;
+typedef struct isl_mat	isl_mat;
+
+isl_ctx *isl_mat_get_ctx(__isl_keep isl_mat *mat);
+
+__isl_give isl_mat *isl_mat_alloc(isl_ctx *ctx,
+	unsigned n_row, unsigned n_col);
+struct isl_mat *isl_mat_dup(struct isl_mat *mat);
+struct isl_mat *isl_mat_extend(struct isl_mat *mat,
+	unsigned n_row, unsigned n_col);
+struct isl_mat *isl_mat_identity(struct isl_ctx *ctx, unsigned n_row);
+__isl_give isl_mat *isl_mat_copy(__isl_keep isl_mat *mat);
+struct isl_mat *isl_mat_cow(struct isl_mat *mat);
+__isl_null isl_mat *isl_mat_free(__isl_take isl_mat *mat);
+
+int isl_mat_rows(__isl_keep isl_mat *mat);
+int isl_mat_cols(__isl_keep isl_mat *mat);
+__isl_give isl_val *isl_mat_get_element_val(__isl_keep isl_mat *mat,
+	int row, int col);
+__isl_give isl_mat *isl_mat_set_element_si(__isl_take isl_mat *mat,
+	int row, int col, int v);
+__isl_give isl_mat *isl_mat_set_element_val(__isl_take isl_mat *mat,
+	int row, int col, __isl_take isl_val *v);
+
+struct isl_mat *isl_mat_swap_cols(struct isl_mat *mat, unsigned i, unsigned j);
+struct isl_mat *isl_mat_swap_rows(struct isl_mat *mat, unsigned i, unsigned j);
+
+struct isl_vec *isl_mat_vec_product(struct isl_mat *mat, struct isl_vec *vec);
+struct isl_vec *isl_vec_mat_product(struct isl_vec *vec, struct isl_mat *mat);
+__isl_give isl_vec *isl_mat_vec_inverse_product(__isl_take isl_mat *mat,
+						__isl_take isl_vec *vec);
+struct isl_mat *isl_mat_aff_direct_sum(struct isl_mat *left,
+					struct isl_mat *right);
+__isl_give isl_mat *isl_mat_diagonal(__isl_take isl_mat *mat1,
+	__isl_take isl_mat *mat2);
+struct isl_mat *isl_mat_left_hermite(struct isl_mat *M,
+	int neg, struct isl_mat **U, struct isl_mat **Q);
+struct isl_mat *isl_mat_lin_to_aff(struct isl_mat *mat);
+struct isl_mat *isl_mat_inverse_product(struct isl_mat *left,
+	struct isl_mat *right);
+__isl_give isl_mat *isl_mat_product(__isl_take isl_mat *left,
+	__isl_take isl_mat *right);
+struct isl_mat *isl_mat_transpose(struct isl_mat *mat);
+__isl_give isl_mat *isl_mat_right_inverse(__isl_take isl_mat *mat);
+__isl_give isl_mat *isl_mat_right_kernel(__isl_take isl_mat *mat);
+
+__isl_give isl_mat *isl_mat_normalize(__isl_take isl_mat *mat);
+__isl_give isl_mat *isl_mat_normalize_row(__isl_take isl_mat *mat, int row);
+
+struct isl_mat *isl_mat_drop_cols(struct isl_mat *mat,
+				unsigned col, unsigned n);
+struct isl_mat *isl_mat_drop_rows(struct isl_mat *mat,
+				unsigned row, unsigned n);
+__isl_give isl_mat *isl_mat_insert_cols(__isl_take isl_mat *mat,
+				unsigned col, unsigned n);
+__isl_give isl_mat *isl_mat_insert_rows(__isl_take isl_mat *mat,
+				unsigned row, unsigned n);
+__isl_give isl_mat *isl_mat_move_cols(__isl_take isl_mat *mat,
+	unsigned dst_col, unsigned src_col, unsigned n);
+__isl_give isl_mat *isl_mat_add_rows(__isl_take isl_mat *mat, unsigned n);
+__isl_give isl_mat *isl_mat_insert_zero_cols(__isl_take isl_mat *mat,
+	unsigned first, unsigned n);
+__isl_give isl_mat *isl_mat_add_zero_cols(__isl_take isl_mat *mat, unsigned n);
+__isl_give isl_mat *isl_mat_insert_zero_rows(__isl_take isl_mat *mat,
+	unsigned row, unsigned n);
+__isl_give isl_mat *isl_mat_add_zero_rows(__isl_take isl_mat *mat, unsigned n);
+
+void isl_mat_col_add(__isl_keep isl_mat *mat, int dst_col, int src_col);
+
+struct isl_mat *isl_mat_unimodular_complete(struct isl_mat *M, int row);
+
+__isl_give isl_mat *isl_mat_from_row_vec(__isl_take isl_vec *vec);
+__isl_give isl_mat *isl_mat_concat(__isl_take isl_mat *top,
+	__isl_take isl_mat *bot);
+__isl_give isl_mat *isl_mat_vec_concat(__isl_take isl_mat *top,
+	__isl_take isl_vec *bot);
+
+int isl_mat_is_equal(__isl_keep isl_mat *mat1, __isl_keep isl_mat *mat2);
+
+int isl_mat_initial_non_zero_cols(__isl_keep isl_mat *mat);
+
+void isl_mat_print_internal(__isl_keep isl_mat *mat, FILE *out, int indent);
+void isl_mat_dump(__isl_keep isl_mat *mat);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/multi.h b/polly/lib/External/isl/include/isl/multi.h
new file mode 100644
index 00000000000..ddcdc20fabd
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/multi.h
@@ -0,0 +1,123 @@
+#ifndef ISL_MULTI_H
+#define ISL_MULTI_H
+
+#include <isl/space.h>
+#include <isl/list.h>
+#include <isl/set_type.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+#define ISL_DECLARE_MULTI(BASE)						\
+unsigned isl_multi_##BASE##_dim(__isl_keep isl_multi_##BASE *multi,	\
+	enum isl_dim_type type);					\
+isl_ctx *isl_multi_##BASE##_get_ctx(					\
+	__isl_keep isl_multi_##BASE *multi);				\
+__isl_give isl_space *isl_multi_##BASE##_get_space(			\
+	__isl_keep isl_multi_##BASE *multi);				\
+__isl_give isl_space *isl_multi_##BASE##_get_domain_space(		\
+	__isl_keep isl_multi_##BASE *multi);				\
+int isl_multi_##BASE##_find_dim_by_name(				\
+	__isl_keep isl_multi_##BASE *multi,				\
+	enum isl_dim_type type, const char *name);			\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_from_##BASE##_list(	\
+	__isl_take isl_space *space, __isl_take isl_##BASE##_list *list); \
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_zero(			\
+	__isl_take isl_space *space);					\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_copy(			\
+	__isl_keep isl_multi_##BASE *multi);				\
+__isl_null isl_multi_##BASE *isl_multi_##BASE##_free(			\
+	__isl_take isl_multi_##BASE *multi);				\
+int isl_multi_##BASE##_plain_is_equal(					\
+	__isl_keep isl_multi_##BASE *multi1,				\
+	__isl_keep isl_multi_##BASE *multi2);				\
+int isl_multi_##BASE##_find_dim_by_id(					\
+	__isl_keep isl_multi_##BASE *multi, enum isl_dim_type type,	\
+	__isl_keep isl_id *id);						\
+__isl_give isl_id *isl_multi_##BASE##_get_dim_id(			\
+	__isl_take isl_multi_##BASE *multi,				\
+	enum isl_dim_type type, unsigned pos);				\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_set_dim_name(		\
+	__isl_take isl_multi_##BASE *multi,				\
+	enum isl_dim_type type, unsigned pos, const char *s);		\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_set_dim_id(		\
+	__isl_take isl_multi_##BASE *multi,				\
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id);	\
+const char *isl_multi_##BASE##_get_tuple_name(				\
+	__isl_keep isl_multi_##BASE *multi, enum isl_dim_type type);	\
+int isl_multi_##BASE##_has_tuple_id(__isl_keep isl_multi_##BASE *multi,	\
+	enum isl_dim_type type);					\
+__isl_give isl_id *isl_multi_##BASE##_get_tuple_id(			\
+	__isl_keep isl_multi_##BASE *multi, enum isl_dim_type type);	\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_set_tuple_name(		\
+	__isl_take isl_multi_##BASE *multi,				\
+	enum isl_dim_type type, const char *s);				\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_set_tuple_id(		\
+	__isl_take isl_multi_##BASE *multi,				\
+	enum isl_dim_type type, __isl_take isl_id *id);			\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_reset_tuple_id(		\
+	__isl_take isl_multi_##BASE *multi, enum isl_dim_type type);	\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_reset_user(		\
+	__isl_take isl_multi_##BASE *multi);				\
+int isl_multi_##BASE##_involves_dims(					\
+	__isl_keep isl_multi_##BASE *multi, enum isl_dim_type type,	\
+	unsigned first, unsigned n);					\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_insert_dims(		\
+	__isl_take isl_multi_##BASE *multi, enum isl_dim_type type,	\
+	unsigned first, unsigned n);					\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_add_dims(		\
+	__isl_take isl_multi_##BASE *multi, enum isl_dim_type type,	\
+	unsigned n);							\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_drop_dims(		\
+	__isl_take isl_multi_##BASE *multi, enum isl_dim_type type,	\
+	unsigned first, unsigned n);					\
+__isl_give isl_##BASE *isl_multi_##BASE##_get_##BASE(			\
+	__isl_keep isl_multi_##BASE *multi, int pos);			\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_set_##BASE(		\
+	__isl_take isl_multi_##BASE *multi, int pos,			\
+	__isl_take isl_##BASE *el);					\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_range_splice(		\
+	__isl_take isl_multi_##BASE *multi1, unsigned pos,		\
+	__isl_take isl_multi_##BASE *multi2);				\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_splice(			\
+	__isl_take isl_multi_##BASE *multi1, unsigned in_pos,		\
+	unsigned out_pos, __isl_take isl_multi_##BASE *multi2);		\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_flatten_range(		\
+	__isl_take isl_multi_##BASE *multi);				\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_flat_range_product(	\
+	__isl_take isl_multi_##BASE *multi1,				\
+	__isl_take isl_multi_##BASE *multi2);				\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_range_product(		\
+	__isl_take isl_multi_##BASE *multi1,				\
+	__isl_take isl_multi_##BASE *multi2);				\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_product(		\
+	__isl_take isl_multi_##BASE *multi1,				\
+	__isl_take isl_multi_##BASE *multi2);				\
+int isl_multi_##BASE##_range_is_wrapping(				\
+	__isl_keep isl_multi_##BASE *multi);				\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_range_factor_domain(	\
+	__isl_take isl_multi_##BASE *multi);				\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_range_factor_range(	\
+	__isl_take isl_multi_##BASE *multi);				\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_scale_val(		\
+	__isl_take isl_multi_##BASE *multi, __isl_take isl_val *v);	\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_scale_down_val(		\
+	__isl_take isl_multi_##BASE *multi, __isl_take isl_val *v);	\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_scale_multi_val(	\
+	__isl_take isl_multi_##BASE *multi,				\
+	__isl_take isl_multi_val *mv);					\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_scale_down_multi_val(	\
+	__isl_take isl_multi_##BASE *multi,				\
+	__isl_take isl_multi_val *mv);					\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_align_params(		\
+	__isl_take isl_multi_##BASE *multi,				\
+	__isl_take isl_space *model);					\
+__isl_give isl_multi_##BASE *isl_multi_##BASE##_from_range(		\
+	__isl_take isl_multi_##BASE *multi);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/obj.h b/polly/lib/External/isl/include/isl/obj.h
new file mode 100644
index 00000000000..7c796f3473e
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/obj.h
@@ -0,0 +1,55 @@
+#ifndef ISL_OBJ_H
+#define ISL_OBJ_H
+
+#include <isl/set_type.h>
+#include <isl/map_type.h>
+#include <isl/union_set_type.h>
+#include <isl/union_map_type.h>
+#include <isl/polynomial_type.h>
+#include <isl/printer.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_obj_vtable {
+	void *(*copy)(void *v1);
+	void *(*add)(void *v1, void *v2);
+	__isl_give isl_printer *(*print)(__isl_take isl_printer *p, void *v);
+	void (*free)(void *v);
+};
+typedef struct isl_obj_vtable *isl_obj_type;
+extern struct isl_obj_vtable isl_obj_none_vtable;
+#define isl_obj_none		(&isl_obj_none_vtable)
+extern struct isl_obj_vtable isl_obj_int_vtable;
+#define isl_obj_int		(&isl_obj_int_vtable)
+extern struct isl_obj_vtable isl_obj_val_vtable;
+#define isl_obj_val		(&isl_obj_val_vtable)
+extern struct isl_obj_vtable isl_obj_set_vtable;
+#define isl_obj_set		(&isl_obj_set_vtable)
+extern struct isl_obj_vtable isl_obj_union_set_vtable;
+#define isl_obj_union_set	(&isl_obj_union_set_vtable)
+extern struct isl_obj_vtable isl_obj_map_vtable;
+#define isl_obj_map		(&isl_obj_map_vtable)
+extern struct isl_obj_vtable isl_obj_union_map_vtable;
+#define isl_obj_union_map	(&isl_obj_union_map_vtable)
+extern struct isl_obj_vtable isl_obj_pw_multi_aff_vtable;
+#define isl_obj_pw_multi_aff	(&isl_obj_pw_multi_aff_vtable)
+extern struct isl_obj_vtable isl_obj_pw_qpolynomial_vtable;
+#define isl_obj_pw_qpolynomial	(&isl_obj_pw_qpolynomial_vtable)
+extern struct isl_obj_vtable isl_obj_union_pw_qpolynomial_vtable;
+#define isl_obj_union_pw_qpolynomial	(&isl_obj_union_pw_qpolynomial_vtable)
+extern struct isl_obj_vtable isl_obj_pw_qpolynomial_fold_vtable;
+#define isl_obj_pw_qpolynomial_fold	(&isl_obj_pw_qpolynomial_fold_vtable)
+extern struct isl_obj_vtable isl_obj_union_pw_qpolynomial_fold_vtable;
+#define isl_obj_union_pw_qpolynomial_fold	(&isl_obj_union_pw_qpolynomial_fold_vtable)
+struct isl_obj {
+	isl_obj_type	type;
+	void		*v;
+};
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/options.h b/polly/lib/External/isl/include/isl/options.h
new file mode 100644
index 00000000000..b0c9a436967
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/options.h
@@ -0,0 +1,50 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_OPTIONS_H
+#define ISL_OPTIONS_H
+
+#include <isl/arg.h>
+#include <isl/ctx.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_options;
+
+ISL_ARG_DECL(isl_options, struct isl_options, isl_options_args)
+
+#define			ISL_BOUND_BERNSTEIN	0
+#define			ISL_BOUND_RANGE		1
+int isl_options_set_bound(isl_ctx *ctx, int val);
+int isl_options_get_bound(isl_ctx *ctx);
+
+#define			ISL_ON_ERROR_WARN	0
+#define			ISL_ON_ERROR_CONTINUE	1
+#define			ISL_ON_ERROR_ABORT	2
+int isl_options_set_on_error(isl_ctx *ctx, int val);
+int isl_options_get_on_error(isl_ctx *ctx);
+
+int isl_options_set_gbr_only_first(isl_ctx *ctx, int val);
+int isl_options_get_gbr_only_first(isl_ctx *ctx);
+
+#define		ISL_SCHEDULE_ALGORITHM_ISL		0
+#define		ISL_SCHEDULE_ALGORITHM_FEAUTRIER	1
+int isl_options_set_schedule_algorithm(isl_ctx *ctx, int val);
+int isl_options_get_schedule_algorithm(isl_ctx *ctx);
+
+int isl_options_set_coalesce_bounded_wrapping(isl_ctx *ctx, int val);
+int isl_options_get_coalesce_bounded_wrapping(isl_ctx *ctx);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/point.h b/polly/lib/External/isl/include/isl/point.h
new file mode 100644
index 00000000000..3247ebb27c5
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/point.h
@@ -0,0 +1,45 @@
+#ifndef ISL_POINT_H
+#define ISL_POINT_H
+
+#include <stdio.h>
+#include <isl/space.h>
+#include <isl/val.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_point;
+typedef struct isl_point isl_point;
+
+isl_ctx *isl_point_get_ctx(__isl_keep isl_point *pnt);
+__isl_give isl_space *isl_point_get_space(__isl_keep isl_point *pnt);
+
+__isl_give isl_point *isl_point_zero(__isl_take isl_space *dim);
+__isl_give isl_point *isl_point_copy(__isl_keep isl_point *pnt);
+void isl_point_free(__isl_take isl_point *pnt);
+
+__isl_give isl_val *isl_point_get_coordinate_val(__isl_keep isl_point *pnt,
+	enum isl_dim_type type, int pos);
+__isl_give isl_point *isl_point_set_coordinate_val(__isl_take isl_point *pnt,
+	enum isl_dim_type type, int pos, __isl_take isl_val *v);
+
+__isl_give isl_point *isl_point_add_ui(__isl_take isl_point *pnt,
+	enum isl_dim_type type, int pos, unsigned val);
+__isl_give isl_point *isl_point_sub_ui(__isl_take isl_point *pnt,
+	enum isl_dim_type type, int pos, unsigned val);
+
+__isl_give isl_point *isl_point_void(__isl_take isl_space *dim);
+int isl_point_is_void(__isl_keep isl_point *pnt);
+
+__isl_give isl_printer *isl_printer_print_point(
+	__isl_take isl_printer *printer, __isl_keep isl_point *pnt);
+void isl_point_dump(__isl_keep isl_point *pnt);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#include <isl/dim.h>
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/polynomial.h b/polly/lib/External/isl/include/isl/polynomial.h
new file mode 100644
index 00000000000..f7b55a55bf7
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/polynomial.h
@@ -0,0 +1,623 @@
+#ifndef ISL_POLYNOMIAL_H
+#define ISL_POLYNOMIAL_H
+
+#include <isl/ctx.h>
+#include <isl/constraint.h>
+#include <isl/space.h>
+#include <isl/set_type.h>
+#include <isl/point.h>
+#include <isl/printer.h>
+#include <isl/union_set_type.h>
+#include <isl/aff_type.h>
+#include <isl/polynomial_type.h>
+#include <isl/val.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+isl_ctx *isl_qpolynomial_get_ctx(__isl_keep isl_qpolynomial *qp);
+__isl_give isl_space *isl_qpolynomial_get_domain_space(
+	__isl_keep isl_qpolynomial *qp);
+__isl_give isl_space *isl_qpolynomial_get_space(__isl_keep isl_qpolynomial *qp);
+unsigned isl_qpolynomial_dim(__isl_keep isl_qpolynomial *qp,
+	enum isl_dim_type type);
+int isl_qpolynomial_involves_dims(__isl_keep isl_qpolynomial *qp,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+__isl_give isl_val *isl_qpolynomial_get_constant_val(
+	__isl_keep isl_qpolynomial *qp);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_set_dim_name(
+	__isl_take isl_qpolynomial *qp,
+	enum isl_dim_type type, unsigned pos, const char *s);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_zero_on_domain(__isl_take isl_space *dim);
+__isl_give isl_qpolynomial *isl_qpolynomial_one_on_domain(__isl_take isl_space *dim);
+__isl_give isl_qpolynomial *isl_qpolynomial_infty_on_domain(__isl_take isl_space *dim);
+__isl_give isl_qpolynomial *isl_qpolynomial_neginfty_on_domain(__isl_take isl_space *dim);
+__isl_give isl_qpolynomial *isl_qpolynomial_nan_on_domain(__isl_take isl_space *dim);
+__isl_give isl_qpolynomial *isl_qpolynomial_val_on_domain(
+	__isl_take isl_space *space, __isl_take isl_val *val);
+__isl_give isl_qpolynomial *isl_qpolynomial_var_on_domain(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_qpolynomial *isl_qpolynomial_copy(__isl_keep isl_qpolynomial *qp);
+__isl_null isl_qpolynomial *isl_qpolynomial_free(
+	__isl_take isl_qpolynomial *qp);
+
+int isl_qpolynomial_plain_is_equal(__isl_keep isl_qpolynomial *qp1,
+	__isl_keep isl_qpolynomial *qp2);
+int isl_qpolynomial_is_zero(__isl_keep isl_qpolynomial *qp);
+int isl_qpolynomial_is_nan(__isl_keep isl_qpolynomial *qp);
+int isl_qpolynomial_is_infty(__isl_keep isl_qpolynomial *qp);
+int isl_qpolynomial_is_neginfty(__isl_keep isl_qpolynomial *qp);
+int isl_qpolynomial_sgn(__isl_keep isl_qpolynomial *qp);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_neg(__isl_take isl_qpolynomial *qp);
+__isl_give isl_qpolynomial *isl_qpolynomial_add(__isl_take isl_qpolynomial *qp1,
+	__isl_take isl_qpolynomial *qp2);
+__isl_give isl_qpolynomial *isl_qpolynomial_sub(__isl_take isl_qpolynomial *qp1,
+	__isl_take isl_qpolynomial *qp2);
+__isl_give isl_qpolynomial *isl_qpolynomial_mul(__isl_take isl_qpolynomial *qp1,
+	__isl_take isl_qpolynomial *qp2);
+__isl_give isl_qpolynomial *isl_qpolynomial_pow(__isl_take isl_qpolynomial *qp,
+	unsigned power);
+__isl_give isl_qpolynomial *isl_qpolynomial_scale_val(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_val *v);
+__isl_give isl_qpolynomial *isl_qpolynomial_scale_down_val(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_val *v);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_insert_dims(
+	__isl_take isl_qpolynomial *qp, enum isl_dim_type type,
+	unsigned first, unsigned n);
+__isl_give isl_qpolynomial *isl_qpolynomial_add_dims(
+	__isl_take isl_qpolynomial *qp, enum isl_dim_type type, unsigned n);
+__isl_give isl_qpolynomial *isl_qpolynomial_move_dims(
+	__isl_take isl_qpolynomial *qp,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n);
+__isl_give isl_qpolynomial *isl_qpolynomial_project_domain_on_params(
+	__isl_take isl_qpolynomial *qp);
+__isl_give isl_qpolynomial *isl_qpolynomial_drop_dims(
+	__isl_take isl_qpolynomial *qp,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_substitute(
+	__isl_take isl_qpolynomial *qp,
+	enum isl_dim_type type, unsigned first, unsigned n,
+	__isl_keep isl_qpolynomial **subs);
+
+int isl_qpolynomial_as_polynomial_on_domain(__isl_keep isl_qpolynomial *qp,
+	__isl_keep isl_basic_set *bset,
+	int (*fn)(__isl_take isl_basic_set *bset,
+		  __isl_take isl_qpolynomial *poly, void *user), void *user);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_homogenize(
+	__isl_take isl_qpolynomial *poly);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_align_params(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_space *model);
+
+isl_ctx *isl_term_get_ctx(__isl_keep isl_term *term);
+
+__isl_give isl_term *isl_term_copy(__isl_keep isl_term *term);
+void isl_term_free(__isl_take isl_term *term);
+
+unsigned isl_term_dim(__isl_keep isl_term *term, enum isl_dim_type type);
+__isl_give isl_val *isl_term_get_coefficient_val(__isl_keep isl_term *term);
+int isl_term_get_exp(__isl_keep isl_term *term,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_aff *isl_term_get_div(__isl_keep isl_term *term, unsigned pos);
+
+int isl_qpolynomial_foreach_term(__isl_keep isl_qpolynomial *qp,
+	int (*fn)(__isl_take isl_term *term, void *user), void *user);
+
+__isl_give isl_val *isl_qpolynomial_eval(__isl_take isl_qpolynomial *qp,
+	__isl_take isl_point *pnt);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_gist_params(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_set *context);
+__isl_give isl_qpolynomial *isl_qpolynomial_gist(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_set *context);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_from_constraint(
+	__isl_take isl_constraint *c, enum isl_dim_type type, unsigned pos);
+__isl_give isl_qpolynomial *isl_qpolynomial_from_term(__isl_take isl_term *term);
+__isl_give isl_qpolynomial *isl_qpolynomial_from_aff(__isl_take isl_aff *aff);
+__isl_give isl_basic_map *isl_basic_map_from_qpolynomial(
+	__isl_take isl_qpolynomial *qp);
+
+__isl_give isl_printer *isl_printer_print_qpolynomial(
+	__isl_take isl_printer *p, __isl_keep isl_qpolynomial *qp);
+void isl_qpolynomial_print(__isl_keep isl_qpolynomial *qp, FILE *out,
+	unsigned output_format);
+void isl_qpolynomial_dump(__isl_keep isl_qpolynomial *qp);
+
+isl_ctx *isl_pw_qpolynomial_get_ctx(__isl_keep isl_pw_qpolynomial *pwqp);
+
+int isl_pw_qpolynomial_plain_is_equal(__isl_keep isl_pw_qpolynomial *pwqp1,
+	__isl_keep isl_pw_qpolynomial *pwqp2);
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_zero(__isl_take isl_space *dim);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_alloc(__isl_take isl_set *set,
+	__isl_take isl_qpolynomial *qp);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_from_qpolynomial(
+	__isl_take isl_qpolynomial *qp);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_copy(
+	__isl_keep isl_pw_qpolynomial *pwqp);
+__isl_null isl_pw_qpolynomial *isl_pw_qpolynomial_free(
+	__isl_take isl_pw_qpolynomial *pwqp);
+
+int isl_pw_qpolynomial_is_zero(__isl_keep isl_pw_qpolynomial *pwqp);
+
+__isl_give isl_space *isl_pw_qpolynomial_get_domain_space(
+	__isl_keep isl_pw_qpolynomial *pwqp);
+__isl_give isl_space *isl_pw_qpolynomial_get_space(
+	__isl_keep isl_pw_qpolynomial *pwqp);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_reset_domain_space(
+	__isl_take isl_pw_qpolynomial *pwqp, __isl_take isl_space *dim);
+unsigned isl_pw_qpolynomial_dim(__isl_keep isl_pw_qpolynomial *pwqp,
+	enum isl_dim_type type);
+int isl_pw_qpolynomial_involves_dims(__isl_keep isl_pw_qpolynomial *pwqp,
+	enum isl_dim_type type, unsigned first, unsigned n);
+int isl_pw_qpolynomial_has_equal_space(__isl_keep isl_pw_qpolynomial *pwqp1,
+	__isl_keep isl_pw_qpolynomial *pwqp2);
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_set_dim_name(
+	__isl_take isl_pw_qpolynomial *pwqp,
+	enum isl_dim_type type, unsigned pos, const char *s);
+
+int isl_pw_qpolynomial_find_dim_by_name(__isl_keep isl_pw_qpolynomial *pwqp,
+	enum isl_dim_type type, const char *name);
+
+__isl_give isl_set *isl_pw_qpolynomial_domain(__isl_take isl_pw_qpolynomial *pwqp);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_intersect_domain(
+	__isl_take isl_pw_qpolynomial *pwpq, __isl_take isl_set *set);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_intersect_params(
+	__isl_take isl_pw_qpolynomial *pwpq, __isl_take isl_set *set);
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_project_domain_on_params(
+	__isl_take isl_pw_qpolynomial *pwqp);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_drop_dims(
+	__isl_take isl_pw_qpolynomial *pwqp,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_split_dims(
+	__isl_take isl_pw_qpolynomial *pwqp,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add(
+	__isl_take isl_pw_qpolynomial *pwqp1,
+	__isl_take isl_pw_qpolynomial *pwqp2);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_sub(
+	__isl_take isl_pw_qpolynomial *pwqp1,
+	__isl_take isl_pw_qpolynomial *pwqp2);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add_disjoint(
+	__isl_take isl_pw_qpolynomial *pwqp1,
+	__isl_take isl_pw_qpolynomial *pwqp2);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_neg(
+	__isl_take isl_pw_qpolynomial *pwqp);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_mul(
+	__isl_take isl_pw_qpolynomial *pwqp1,
+	__isl_take isl_pw_qpolynomial *pwqp2);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_scale_val(
+	__isl_take isl_pw_qpolynomial *pwqp, __isl_take isl_val *v);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_scale_down_val(
+	__isl_take isl_pw_qpolynomial *pwqp, __isl_take isl_val *v);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_pow(
+	__isl_take isl_pw_qpolynomial *pwqp, unsigned exponent);
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_insert_dims(
+	__isl_take isl_pw_qpolynomial *pwqp, enum isl_dim_type type,
+	unsigned first, unsigned n);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add_dims(
+	__isl_take isl_pw_qpolynomial *pwqp,
+	enum isl_dim_type type, unsigned n);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_move_dims(
+	__isl_take isl_pw_qpolynomial *pwqp,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n);
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_fix_val(
+	__isl_take isl_pw_qpolynomial *pwqp,
+	enum isl_dim_type type, unsigned n, __isl_take isl_val *v);
+
+__isl_give isl_val *isl_pw_qpolynomial_eval(
+	__isl_take isl_pw_qpolynomial *pwqp, __isl_take isl_point *pnt);
+
+__isl_give isl_val *isl_pw_qpolynomial_max(__isl_take isl_pw_qpolynomial *pwqp);
+__isl_give isl_val *isl_pw_qpolynomial_min(__isl_take isl_pw_qpolynomial *pwqp);
+
+int isl_pw_qpolynomial_foreach_piece(__isl_keep isl_pw_qpolynomial *pwqp,
+	int (*fn)(__isl_take isl_set *set, __isl_take isl_qpolynomial *qp,
+		    void *user), void *user);
+int isl_pw_qpolynomial_foreach_lifted_piece(__isl_keep isl_pw_qpolynomial *pwqp,
+	int (*fn)(__isl_take isl_set *set, __isl_take isl_qpolynomial *qp,
+		    void *user), void *user);
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_from_pw_aff(
+	__isl_take isl_pw_aff *pwaff);
+
+__isl_constructor
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_read_from_str(isl_ctx *ctx,
+		const char *str);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_read_from_file(isl_ctx *ctx,
+		FILE *input);
+__isl_give isl_printer *isl_printer_print_pw_qpolynomial(
+	__isl_take isl_printer *p, __isl_keep isl_pw_qpolynomial *pwqp);
+void isl_pw_qpolynomial_print(__isl_keep isl_pw_qpolynomial *pwqp, FILE *out,
+	unsigned output_format);
+void isl_pw_qpolynomial_dump(__isl_keep isl_pw_qpolynomial *pwqp);
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_coalesce(
+	__isl_take isl_pw_qpolynomial *pwqp);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_gist(
+	__isl_take isl_pw_qpolynomial *pwqp, __isl_take isl_set *context);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_gist_params(
+	__isl_take isl_pw_qpolynomial *pwqp, __isl_take isl_set *context);
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_split_periods(
+	__isl_take isl_pw_qpolynomial *pwqp, int max_periods);
+
+__isl_give isl_pw_qpolynomial *isl_basic_set_multiplicative_call(
+	__isl_take isl_basic_set *bset,
+	__isl_give isl_pw_qpolynomial *(*fn)(__isl_take isl_basic_set *bset));
+
+isl_ctx *isl_qpolynomial_fold_get_ctx(__isl_keep isl_qpolynomial_fold *fold);
+enum isl_fold isl_qpolynomial_fold_get_type(__isl_keep isl_qpolynomial_fold *fold);
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_empty(enum isl_fold type,
+	__isl_take isl_space *dim);
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_alloc(
+	enum isl_fold type, __isl_take isl_qpolynomial *qp);
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_copy(
+	__isl_keep isl_qpolynomial_fold *fold);
+void isl_qpolynomial_fold_free(__isl_take isl_qpolynomial_fold *fold);
+
+int isl_qpolynomial_fold_is_empty(__isl_keep isl_qpolynomial_fold *fold);
+int isl_qpolynomial_fold_plain_is_equal(__isl_keep isl_qpolynomial_fold *fold1,
+	__isl_keep isl_qpolynomial_fold *fold2);
+
+__isl_give isl_space *isl_qpolynomial_fold_get_space(
+	__isl_keep isl_qpolynomial_fold *fold);
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_fold(
+	__isl_take isl_qpolynomial_fold *fold1,
+	__isl_take isl_qpolynomial_fold *fold2);
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_scale_val(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_val *v);
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_scale_down_val(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_val *v);
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_move_dims(
+	__isl_take isl_qpolynomial_fold *fold,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n);
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_substitute(
+	__isl_take isl_qpolynomial_fold *fold,
+	enum isl_dim_type type, unsigned first, unsigned n,
+	__isl_keep isl_qpolynomial **subs);
+
+__isl_give isl_val *isl_qpolynomial_fold_eval(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_point *pnt);
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_gist_params(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_set *context);
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_gist(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_set *context);
+
+int isl_qpolynomial_fold_foreach_qpolynomial(
+	__isl_keep isl_qpolynomial_fold *fold,
+	int (*fn)(__isl_take isl_qpolynomial *qp, void *user), void *user);
+
+__isl_give isl_printer *isl_printer_print_qpolynomial_fold(
+	__isl_take isl_printer *p, __isl_keep isl_qpolynomial_fold *fold);
+void isl_qpolynomial_fold_print(__isl_keep isl_qpolynomial_fold *fold, FILE *out,
+	unsigned output_format);
+void isl_qpolynomial_fold_dump(__isl_keep isl_qpolynomial_fold *fold);
+
+isl_ctx *isl_pw_qpolynomial_fold_get_ctx(__isl_keep isl_pw_qpolynomial_fold *pwf);
+
+int isl_pw_qpolynomial_fold_plain_is_equal(
+	__isl_keep isl_pw_qpolynomial_fold *pwf1,
+	__isl_keep isl_pw_qpolynomial_fold *pwf2);
+
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_from_pw_qpolynomial(
+	enum isl_fold type, __isl_take isl_pw_qpolynomial *pwqp);
+
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_alloc(
+	enum isl_fold type,
+	__isl_take isl_set *set, __isl_take isl_qpolynomial_fold *fold);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_copy(
+	__isl_keep isl_pw_qpolynomial_fold *pwf);
+__isl_null isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_free(
+	__isl_take isl_pw_qpolynomial_fold *pwf);
+
+int isl_pw_qpolynomial_fold_is_zero(__isl_keep isl_pw_qpolynomial_fold *pwf);
+
+__isl_give isl_space *isl_pw_qpolynomial_fold_get_domain_space(
+	__isl_keep isl_pw_qpolynomial_fold *pwf);
+__isl_give isl_space *isl_pw_qpolynomial_fold_get_space(
+	__isl_keep isl_pw_qpolynomial_fold *pwf);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_reset_space(
+	__isl_take isl_pw_qpolynomial_fold *pwf, __isl_take isl_space *dim);
+unsigned isl_pw_qpolynomial_fold_dim(__isl_keep isl_pw_qpolynomial_fold *pwf,
+	enum isl_dim_type type);
+int isl_pw_qpolynomial_fold_has_equal_space(
+	__isl_keep isl_pw_qpolynomial_fold *pwf1,
+	__isl_keep isl_pw_qpolynomial_fold *pwf2);
+
+size_t isl_pw_qpolynomial_fold_size(__isl_keep isl_pw_qpolynomial_fold *pwf);
+
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_zero(
+	__isl_take isl_space *dim, enum isl_fold type);
+
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_set_dim_name(
+	__isl_take isl_pw_qpolynomial_fold *pwf,
+	enum isl_dim_type type, unsigned pos, const char *s);
+
+int isl_pw_qpolynomial_fold_find_dim_by_name(
+	__isl_keep isl_pw_qpolynomial_fold *pwf,
+	enum isl_dim_type type, const char *name);
+
+__isl_give isl_set *isl_pw_qpolynomial_fold_domain(
+	__isl_take isl_pw_qpolynomial_fold *pwf);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_intersect_domain(
+	__isl_take isl_pw_qpolynomial_fold *pwf, __isl_take isl_set *set);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_intersect_params(
+	__isl_take isl_pw_qpolynomial_fold *pwf, __isl_take isl_set *set);
+
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_add(
+	__isl_take isl_pw_qpolynomial_fold *pwf1,
+	__isl_take isl_pw_qpolynomial_fold *pwf2);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_fold(
+	__isl_take isl_pw_qpolynomial_fold *pwf1,
+	__isl_take isl_pw_qpolynomial_fold *pwf2);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_add_disjoint(
+	__isl_take isl_pw_qpolynomial_fold *pwf1,
+	__isl_take isl_pw_qpolynomial_fold *pwf2);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_scale_val(
+	__isl_take isl_pw_qpolynomial_fold *pwf, __isl_take isl_val *v);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_scale_down_val(
+	__isl_take isl_pw_qpolynomial_fold *pwf, __isl_take isl_val *v);
+
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_project_domain_on_params(
+	__isl_take isl_pw_qpolynomial_fold *pwf);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_drop_dims(
+	__isl_take isl_pw_qpolynomial_fold *pwf,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_move_dims(
+	__isl_take isl_pw_qpolynomial_fold *pwf,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n);
+
+__isl_give isl_val *isl_pw_qpolynomial_fold_eval(
+	__isl_take isl_pw_qpolynomial_fold *pwf, __isl_take isl_point *pnt);
+
+int isl_pw_qpolynomial_fold_foreach_piece(
+	__isl_keep isl_pw_qpolynomial_fold *pwf,
+	int (*fn)(__isl_take isl_set *set, __isl_take isl_qpolynomial_fold *fold,
+		    void *user), void *user);
+int isl_pw_qpolynomial_fold_foreach_lifted_piece(
+	__isl_keep isl_pw_qpolynomial_fold *pwf,
+	int (*fn)(__isl_take isl_set *set, __isl_take isl_qpolynomial_fold *fold,
+		    void *user), void *user);
+
+__isl_give isl_printer *isl_printer_print_pw_qpolynomial_fold(
+	__isl_take isl_printer *p, __isl_keep isl_pw_qpolynomial_fold *pwf);
+void isl_pw_qpolynomial_fold_print(__isl_keep isl_pw_qpolynomial_fold *pwf,
+	FILE *out, unsigned output_format);
+void isl_pw_qpolynomial_fold_dump(__isl_keep isl_pw_qpolynomial_fold *pwf);
+
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_coalesce(
+	__isl_take isl_pw_qpolynomial_fold *pwf);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_gist(
+	__isl_take isl_pw_qpolynomial_fold *pwf, __isl_take isl_set *context);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_gist_params(
+	__isl_take isl_pw_qpolynomial_fold *pwf, __isl_take isl_set *context);
+
+__isl_give isl_val *isl_pw_qpolynomial_fold_max(
+	__isl_take isl_pw_qpolynomial_fold *pwf);
+__isl_give isl_val *isl_pw_qpolynomial_fold_min(
+	__isl_take isl_pw_qpolynomial_fold *pwf);
+
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_bound(
+	__isl_take isl_pw_qpolynomial *pwqp, enum isl_fold type, int *tight);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_bound(
+	__isl_take isl_pw_qpolynomial_fold *pwf, int *tight);
+__isl_give isl_pw_qpolynomial_fold *isl_set_apply_pw_qpolynomial_fold(
+	__isl_take isl_set *set, __isl_take isl_pw_qpolynomial_fold *pwf,
+	int *tight);
+__isl_give isl_pw_qpolynomial_fold *isl_map_apply_pw_qpolynomial_fold(
+	__isl_take isl_map *map, __isl_take isl_pw_qpolynomial_fold *pwf,
+	int *tight);
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_to_polynomial(
+	__isl_take isl_pw_qpolynomial *pwqp, int sign);
+
+isl_ctx *isl_union_pw_qpolynomial_get_ctx(
+	__isl_keep isl_union_pw_qpolynomial *upwqp);
+
+unsigned isl_union_pw_qpolynomial_dim(
+	__isl_keep isl_union_pw_qpolynomial *upwqp, enum isl_dim_type type);
+
+int isl_union_pw_qpolynomial_plain_is_equal(
+	__isl_keep isl_union_pw_qpolynomial *upwqp1,
+	__isl_keep isl_union_pw_qpolynomial *upwqp2);
+
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_from_pw_qpolynomial(__isl_take isl_pw_qpolynomial *pwqp);
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_zero(
+	__isl_take isl_space *dim);
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_add_pw_qpolynomial(
+	__isl_take isl_union_pw_qpolynomial *upwqp,
+	__isl_take isl_pw_qpolynomial *pwqp);
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_copy(
+	__isl_keep isl_union_pw_qpolynomial *upwqp);
+__isl_null isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_free(
+	__isl_take isl_union_pw_qpolynomial *upwqp);
+
+__isl_constructor
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_read_from_str(
+	isl_ctx *ctx, const char *str);
+
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_neg(
+	__isl_take isl_union_pw_qpolynomial *upwqp);
+
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_add(
+	__isl_take isl_union_pw_qpolynomial *upwqp1,
+	__isl_take isl_union_pw_qpolynomial *upwqp2);
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_sub(
+	__isl_take isl_union_pw_qpolynomial *upwqp1,
+	__isl_take isl_union_pw_qpolynomial *upwqp2);
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_mul(
+	__isl_take isl_union_pw_qpolynomial *upwqp1,
+	__isl_take isl_union_pw_qpolynomial *upwqp2);
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_scale_val(
+	__isl_take isl_union_pw_qpolynomial *upwqp, __isl_take isl_val *v);
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_scale_down_val(
+	__isl_take isl_union_pw_qpolynomial *upwqp, __isl_take isl_val *v);
+
+__isl_give isl_union_set *isl_union_pw_qpolynomial_domain(
+	__isl_take isl_union_pw_qpolynomial *upwqp);
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_intersect_domain(
+	__isl_take isl_union_pw_qpolynomial *upwpq,
+	__isl_take isl_union_set *uset);
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_intersect_params(
+	__isl_take isl_union_pw_qpolynomial *upwpq,
+	__isl_take isl_set *set);
+
+__isl_give isl_space *isl_union_pw_qpolynomial_get_space(
+	__isl_keep isl_union_pw_qpolynomial *upwqp);
+
+int isl_union_pw_qpolynomial_find_dim_by_name(
+	__isl_keep isl_union_pw_qpolynomial *upwqp,
+	enum isl_dim_type type, const char *name);
+
+__isl_give isl_val *isl_union_pw_qpolynomial_eval(
+	__isl_take isl_union_pw_qpolynomial *upwqp, __isl_take isl_point *pnt);
+
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_coalesce(
+	__isl_take isl_union_pw_qpolynomial *upwqp);
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_gist(
+	__isl_take isl_union_pw_qpolynomial *upwqp,
+	__isl_take isl_union_set *context);
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_gist_params(
+	__isl_take isl_union_pw_qpolynomial *upwqp,
+	__isl_take isl_set *context);
+
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_align_params(
+	__isl_take isl_union_pw_qpolynomial *upwqp,
+	__isl_take isl_space *model);
+
+int isl_union_pw_qpolynomial_foreach_pw_qpolynomial(
+	__isl_keep isl_union_pw_qpolynomial *upwqp,
+	int (*fn)(__isl_take isl_pw_qpolynomial *pwqp, void *user), void *user);
+__isl_give isl_pw_qpolynomial *isl_union_pw_qpolynomial_extract_pw_qpolynomial(
+	__isl_keep isl_union_pw_qpolynomial *upwqp, __isl_take isl_space *dim);
+
+__isl_give isl_printer *isl_printer_print_union_pw_qpolynomial(
+	__isl_take isl_printer *p, __isl_keep isl_union_pw_qpolynomial *upwqp);
+
+isl_ctx *isl_union_pw_qpolynomial_fold_get_ctx(
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf);
+
+unsigned isl_union_pw_qpolynomial_fold_dim(
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf, enum isl_dim_type type);
+
+int isl_union_pw_qpolynomial_fold_plain_is_equal(
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf1,
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf2);
+
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_from_pw_qpolynomial_fold(__isl_take isl_pw_qpolynomial_fold *pwf);
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_zero(
+	__isl_take isl_space *dim, enum isl_fold type);
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_fold_pw_qpolynomial_fold(
+	__isl_take isl_union_pw_qpolynomial_fold *upwqp,
+	__isl_take isl_pw_qpolynomial_fold *pwqp);
+__isl_null isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_free(
+	__isl_take isl_union_pw_qpolynomial_fold *upwf);
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_copy(
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf);
+
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_fold(
+	__isl_take isl_union_pw_qpolynomial_fold *upwf1,
+	__isl_take isl_union_pw_qpolynomial_fold *upwf2);
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_add_union_pw_qpolynomial(
+	__isl_take isl_union_pw_qpolynomial_fold *upwf,
+	__isl_take isl_union_pw_qpolynomial *upwqp);
+__isl_give isl_union_pw_qpolynomial_fold *
+isl_union_pw_qpolynomial_fold_scale_val(
+	__isl_take isl_union_pw_qpolynomial_fold *upwf, __isl_take isl_val *v);
+__isl_give isl_union_pw_qpolynomial_fold *
+isl_union_pw_qpolynomial_fold_scale_down_val(
+	__isl_take isl_union_pw_qpolynomial_fold *upwf, __isl_take isl_val *v);
+
+__isl_give isl_union_set *isl_union_pw_qpolynomial_fold_domain(
+	__isl_take isl_union_pw_qpolynomial_fold *upwf);
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_intersect_domain(
+	__isl_take isl_union_pw_qpolynomial_fold *upwf,
+	__isl_take isl_union_set *uset);
+__isl_give isl_union_pw_qpolynomial_fold *
+isl_union_pw_qpolynomial_fold_intersect_params(
+	__isl_take isl_union_pw_qpolynomial_fold *upwf,
+	__isl_take isl_set *set);
+
+enum isl_fold isl_union_pw_qpolynomial_fold_get_type(
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf);
+__isl_give isl_space *isl_union_pw_qpolynomial_fold_get_space(
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf);
+
+int isl_union_pw_qpolynomial_fold_find_dim_by_name(
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf,
+	enum isl_dim_type type, const char *name);
+
+__isl_give isl_val *isl_union_pw_qpolynomial_fold_eval(
+	__isl_take isl_union_pw_qpolynomial_fold *upwf,
+	__isl_take isl_point *pnt);
+
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_coalesce(
+	__isl_take isl_union_pw_qpolynomial_fold *upwf);
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_gist(
+	__isl_take isl_union_pw_qpolynomial_fold *upwf,
+	__isl_take isl_union_set *context);
+__isl_give isl_union_pw_qpolynomial_fold *
+isl_union_pw_qpolynomial_fold_gist_params(
+	__isl_take isl_union_pw_qpolynomial_fold *upwf,
+	__isl_take isl_set *context);
+
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_align_params(
+	__isl_take isl_union_pw_qpolynomial_fold *upwf,
+	__isl_take isl_space *model);
+
+int isl_union_pw_qpolynomial_fold_foreach_pw_qpolynomial_fold(
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf,
+	int (*fn)(__isl_take isl_pw_qpolynomial_fold *pwf,
+		    void *user), void *user);
+__isl_give isl_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_extract_pw_qpolynomial_fold(
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf, __isl_take isl_space *dim);
+
+__isl_give isl_printer *isl_printer_print_union_pw_qpolynomial_fold(
+	__isl_take isl_printer *p,
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf);
+
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_bound(
+	__isl_take isl_union_pw_qpolynomial *upwqp,
+	enum isl_fold type, int *tight);
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_set_apply_union_pw_qpolynomial_fold(
+	__isl_take isl_union_set *uset,
+	__isl_take isl_union_pw_qpolynomial_fold *upwf, int *tight);
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_map_apply_union_pw_qpolynomial_fold(
+	__isl_take isl_union_map *umap,
+	__isl_take isl_union_pw_qpolynomial_fold *upwf, int *tight);
+
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_to_polynomial(
+	__isl_take isl_union_pw_qpolynomial *upwqp, int sign);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#include <isl/dim.h>
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/polynomial_type.h b/polly/lib/External/isl/include/isl/polynomial_type.h
new file mode 100644
index 00000000000..df432bda911
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/polynomial_type.h
@@ -0,0 +1,31 @@
+#ifndef ISL_POLYNOMIAL_TYPE_H
+#define ISL_POLYNOMIAL_TYPE_H
+
+struct isl_qpolynomial;
+typedef struct isl_qpolynomial isl_qpolynomial;
+
+struct isl_term;
+typedef struct isl_term isl_term;
+
+struct __isl_export isl_pw_qpolynomial;
+typedef struct isl_pw_qpolynomial isl_pw_qpolynomial;
+
+enum isl_fold {
+	isl_fold_min,
+	isl_fold_max,
+	isl_fold_list
+};
+
+struct isl_qpolynomial_fold;
+typedef struct isl_qpolynomial_fold isl_qpolynomial_fold;
+
+struct isl_pw_qpolynomial_fold;
+typedef struct isl_pw_qpolynomial_fold isl_pw_qpolynomial_fold;
+
+struct __isl_export isl_union_pw_qpolynomial;
+typedef struct isl_union_pw_qpolynomial isl_union_pw_qpolynomial;
+
+struct isl_union_pw_qpolynomial_fold;
+typedef struct isl_union_pw_qpolynomial_fold isl_union_pw_qpolynomial_fold;
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/printer.h b/polly/lib/External/isl/include/isl/printer.h
new file mode 100644
index 00000000000..f43bfbc5f24
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/printer.h
@@ -0,0 +1,62 @@
+#ifndef ISL_PRINTER_H
+#define ISL_PRINTER_H
+
+#include <stdio.h>
+#include <isl/ctx.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_printer;
+typedef struct isl_printer isl_printer;
+
+__isl_give isl_printer *isl_printer_to_file(isl_ctx *ctx, FILE *file);
+__isl_give isl_printer *isl_printer_to_str(isl_ctx *ctx);
+__isl_null isl_printer *isl_printer_free(__isl_take isl_printer *printer);
+
+isl_ctx *isl_printer_get_ctx(__isl_keep isl_printer *printer);
+FILE *isl_printer_get_file(__isl_keep isl_printer *printer);
+
+__isl_give char *isl_printer_get_str(__isl_keep isl_printer *printer);
+
+__isl_give isl_printer *isl_printer_set_indent(__isl_take isl_printer *p,
+	int indent);
+__isl_give isl_printer *isl_printer_indent(__isl_take isl_printer *p,
+	int indent);
+
+#define ISL_FORMAT_ISL			0
+#define ISL_FORMAT_POLYLIB		1
+#define ISL_FORMAT_POLYLIB_CONSTRAINTS	2
+#define ISL_FORMAT_OMEGA		3
+#define ISL_FORMAT_C			4
+#define ISL_FORMAT_LATEX		5
+#define ISL_FORMAT_EXT_POLYLIB		6
+__isl_give isl_printer *isl_printer_set_output_format(__isl_take isl_printer *p,
+	int output_format);
+int isl_printer_get_output_format(__isl_keep isl_printer *p);
+
+__isl_give isl_printer *isl_printer_set_indent_prefix(__isl_take isl_printer *p,
+	const char *prefix);
+__isl_give isl_printer *isl_printer_set_prefix(__isl_take isl_printer *p,
+	const char *prefix);
+__isl_give isl_printer *isl_printer_set_suffix(__isl_take isl_printer *p,
+	const char *suffix);
+__isl_give isl_printer *isl_printer_set_isl_int_width(__isl_take isl_printer *p,
+	int width);
+
+__isl_give isl_printer *isl_printer_start_line(__isl_take isl_printer *p);
+__isl_give isl_printer *isl_printer_end_line(__isl_take isl_printer *p);
+__isl_give isl_printer *isl_printer_print_double(__isl_take isl_printer *p,
+	double d);
+__isl_give isl_printer *isl_printer_print_int(__isl_take isl_printer *p, int i);
+__isl_give isl_printer *isl_printer_print_str(__isl_take isl_printer *p,
+	const char *s);
+
+__isl_give isl_printer *isl_printer_flush(__isl_take isl_printer *p);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/schedule.h b/polly/lib/External/isl/include/isl/schedule.h
new file mode 100644
index 00000000000..83b89d910c2
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/schedule.h
@@ -0,0 +1,93 @@
+#ifndef ISL_SCHEDULE_H
+#define ISL_SCHEDULE_H
+
+#include <isl/union_set_type.h>
+#include <isl/union_map_type.h>
+#include <isl/band.h>
+#include <isl/list.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_schedule_constraints;
+typedef struct isl_schedule_constraints isl_schedule_constraints;
+struct isl_schedule;
+typedef struct isl_schedule isl_schedule;
+
+int isl_options_set_schedule_max_coefficient(isl_ctx *ctx, int val);
+int isl_options_get_schedule_max_coefficient(isl_ctx *ctx);
+
+int isl_options_set_schedule_max_constant_term(isl_ctx *ctx, int val);
+int isl_options_get_schedule_max_constant_term(isl_ctx *ctx);
+
+int isl_options_set_schedule_maximize_band_depth(isl_ctx *ctx, int val);
+int isl_options_get_schedule_maximize_band_depth(isl_ctx *ctx);
+
+int isl_options_set_schedule_outer_coincidence(isl_ctx *ctx, int val);
+int isl_options_get_schedule_outer_coincidence(isl_ctx *ctx);
+
+int isl_options_set_schedule_split_scaled(isl_ctx *ctx, int val);
+int isl_options_get_schedule_split_scaled(isl_ctx *ctx);
+
+int isl_options_set_schedule_separate_components(isl_ctx *ctx, int val);
+int isl_options_get_schedule_separate_components(isl_ctx *ctx);
+
+#define		ISL_SCHEDULE_FUSE_MAX			0
+#define		ISL_SCHEDULE_FUSE_MIN			1
+int isl_options_set_schedule_fuse(isl_ctx *ctx, int val);
+int isl_options_get_schedule_fuse(isl_ctx *ctx);
+
+__isl_give isl_schedule_constraints *isl_schedule_constraints_copy(
+	__isl_keep isl_schedule_constraints *sc);
+__isl_give isl_schedule_constraints *isl_schedule_constraints_on_domain(
+	__isl_take isl_union_set *domain);
+__isl_give isl_schedule_constraints *isl_schedule_constraints_set_validity(
+	__isl_take isl_schedule_constraints *sc,
+	__isl_take isl_union_map *validity);
+__isl_give isl_schedule_constraints *isl_schedule_constraints_set_coincidence(
+	__isl_take isl_schedule_constraints *sc,
+	__isl_take isl_union_map *coincidence);
+__isl_give isl_schedule_constraints *isl_schedule_constraints_set_proximity(
+	__isl_take isl_schedule_constraints *sc,
+	__isl_take isl_union_map *proximity);
+__isl_give isl_schedule_constraints *
+isl_schedule_constraints_set_conditional_validity(
+	__isl_take isl_schedule_constraints *sc,
+	__isl_take isl_union_map *condition,
+	__isl_take isl_union_map *validity);
+__isl_null isl_schedule_constraints *isl_schedule_constraints_free(
+	__isl_take isl_schedule_constraints *sc);
+
+isl_ctx *isl_schedule_constraints_get_ctx(
+	__isl_keep isl_schedule_constraints *sc);
+
+void isl_schedule_constraints_dump(__isl_keep isl_schedule_constraints *sc);
+
+__isl_give isl_schedule *isl_schedule_constraints_compute_schedule(
+	__isl_take isl_schedule_constraints *sc);
+
+__isl_give isl_schedule *isl_union_set_compute_schedule(
+	__isl_take isl_union_set *domain,
+	__isl_take isl_union_map *validity,
+	__isl_take isl_union_map *proximity);
+__isl_null isl_schedule *isl_schedule_free(__isl_take isl_schedule *sched);
+__isl_give isl_union_map *isl_schedule_get_map(__isl_keep isl_schedule *sched);
+
+isl_ctx *isl_schedule_get_ctx(__isl_keep isl_schedule *sched);
+
+__isl_give isl_band_list *isl_schedule_get_band_forest(
+	__isl_keep isl_schedule *schedule);
+
+__isl_give isl_printer *isl_printer_print_schedule(__isl_take isl_printer *p,
+	__isl_keep isl_schedule *schedule);
+void isl_schedule_dump(__isl_keep isl_schedule *schedule);
+
+int isl_schedule_foreach_band(__isl_keep isl_schedule *sched,
+	int (*fn)(__isl_keep isl_band *band, void *user), void *user);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/set.h b/polly/lib/External/isl/include/isl/set.h
new file mode 100644
index 00000000000..73b6570621f
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/set.h
@@ -0,0 +1,540 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_SET_H
+#define ISL_SET_H
+
+#include <isl/map_type.h>
+#include <isl/aff_type.h>
+#include <isl/list.h>
+#include <isl/mat.h>
+#include <isl/point.h>
+#include <isl/local_space.h>
+#include <isl/val.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+unsigned isl_basic_set_n_dim(__isl_keep isl_basic_set *bset);
+unsigned isl_basic_set_n_param(__isl_keep isl_basic_set *bset);
+unsigned isl_basic_set_total_dim(const struct isl_basic_set *bset);
+unsigned isl_basic_set_dim(__isl_keep isl_basic_set *bset,
+				enum isl_dim_type type);
+
+unsigned isl_set_n_dim(__isl_keep isl_set *set);
+unsigned isl_set_n_param(__isl_keep isl_set *set);
+unsigned isl_set_dim(__isl_keep isl_set *set, enum isl_dim_type type);
+
+isl_ctx *isl_basic_set_get_ctx(__isl_keep isl_basic_set *bset);
+isl_ctx *isl_set_get_ctx(__isl_keep isl_set *set);
+__isl_give isl_space *isl_basic_set_get_space(__isl_keep isl_basic_set *bset);
+__isl_give isl_space *isl_set_get_space(__isl_keep isl_set *set);
+__isl_give isl_set *isl_set_reset_space(__isl_take isl_set *set,
+	__isl_take isl_space *dim);
+
+__isl_give isl_aff *isl_basic_set_get_div(__isl_keep isl_basic_set *bset,
+	int pos);
+
+__isl_give isl_local_space *isl_basic_set_get_local_space(
+	__isl_keep isl_basic_set *bset);
+
+const char *isl_basic_set_get_tuple_name(__isl_keep isl_basic_set *bset);
+int isl_set_has_tuple_name(__isl_keep isl_set *set);
+const char *isl_set_get_tuple_name(__isl_keep isl_set *set);
+__isl_give isl_basic_set *isl_basic_set_set_tuple_name(
+	__isl_take isl_basic_set *set, const char *s);
+__isl_give isl_set *isl_set_set_tuple_name(__isl_take isl_set *set,
+	const char *s);
+const char *isl_basic_set_get_dim_name(__isl_keep isl_basic_set *bset,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_basic_set *isl_basic_set_set_dim_name(
+	__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned pos, const char *s);
+int isl_set_has_dim_name(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos);
+const char *isl_set_get_dim_name(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_set *isl_set_set_dim_name(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, const char *s);
+
+__isl_give isl_id *isl_basic_set_get_dim_id(__isl_keep isl_basic_set *bset,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_basic_set *isl_basic_set_set_tuple_id(
+	__isl_take isl_basic_set *bset, __isl_take isl_id *id);
+__isl_give isl_set *isl_set_set_dim_id(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id);
+int isl_set_has_dim_id(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_id *isl_set_get_dim_id(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_set *isl_set_set_tuple_id(__isl_take isl_set *set,
+	__isl_take isl_id *id);
+__isl_give isl_set *isl_set_reset_tuple_id(__isl_take isl_set *set);
+int isl_set_has_tuple_id(__isl_keep isl_set *set);
+__isl_give isl_id *isl_set_get_tuple_id(__isl_keep isl_set *set);
+__isl_give isl_set *isl_set_reset_user(__isl_take isl_set *set);
+
+int isl_set_find_dim_by_id(__isl_keep isl_set *set, enum isl_dim_type type,
+	__isl_keep isl_id *id);
+int isl_set_find_dim_by_name(__isl_keep isl_set *set, enum isl_dim_type type,
+	const char *name);
+
+int isl_basic_set_is_rational(__isl_keep isl_basic_set *bset);
+
+struct isl_basic_set *isl_basic_set_alloc(struct isl_ctx *ctx,
+		unsigned nparam, unsigned dim, unsigned extra,
+		unsigned n_eq, unsigned n_ineq);
+struct isl_basic_set *isl_basic_set_extend(struct isl_basic_set *base,
+		unsigned nparam, unsigned dim, unsigned extra,
+		unsigned n_eq, unsigned n_ineq);
+struct isl_basic_set *isl_basic_set_extend_constraints(
+		struct isl_basic_set *base, unsigned n_eq, unsigned n_ineq);
+struct isl_basic_set *isl_basic_set_finalize(struct isl_basic_set *bset);
+__isl_null isl_basic_set *isl_basic_set_free(__isl_take isl_basic_set *bset);
+__isl_give isl_basic_set *isl_basic_set_copy(__isl_keep isl_basic_set *bset);
+struct isl_basic_set *isl_basic_set_dup(struct isl_basic_set *bset);
+__isl_give isl_basic_set *isl_basic_set_empty(__isl_take isl_space *dim);
+struct isl_basic_set *isl_basic_set_empty_like(struct isl_basic_set *bset);
+__isl_give isl_basic_set *isl_basic_set_universe(__isl_take isl_space *dim);
+__isl_give isl_basic_set *isl_basic_set_nat_universe(__isl_take isl_space *dim);
+struct isl_basic_set *isl_basic_set_universe_like(struct isl_basic_set *bset);
+__isl_give isl_basic_set *isl_basic_set_universe_like_set(
+	__isl_keep isl_set *model);
+__isl_give isl_basic_set *isl_basic_set_positive_orthant(
+	__isl_take isl_space *space);
+void isl_basic_set_print_internal(__isl_keep isl_basic_set *bset,
+				FILE *out, int indent);
+__isl_export
+__isl_give isl_basic_set *isl_basic_set_intersect(
+		__isl_take isl_basic_set *bset1,
+		__isl_take isl_basic_set *bset2);
+__isl_export
+__isl_give isl_basic_set *isl_basic_set_intersect_params(
+	__isl_take isl_basic_set *bset1, __isl_take isl_basic_set *bset2);
+__isl_export
+__isl_give isl_basic_set *isl_basic_set_apply(
+		__isl_take isl_basic_set *bset,
+		__isl_take isl_basic_map *bmap);
+__isl_give isl_basic_set *isl_basic_set_preimage_multi_aff(
+	__isl_take isl_basic_set *bset, __isl_take isl_multi_aff *ma);
+__isl_export
+__isl_give isl_basic_set *isl_basic_set_affine_hull(
+		__isl_take isl_basic_set *bset);
+__isl_give isl_basic_set *isl_basic_set_remove_dims(
+	__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_export
+__isl_give isl_basic_set *isl_basic_set_sample(__isl_take isl_basic_set *bset);
+struct isl_basic_set *isl_basic_set_simplify(struct isl_basic_set *bset);
+__isl_export
+__isl_give isl_basic_set *isl_basic_set_detect_equalities(
+						__isl_take isl_basic_set *bset);
+__isl_give isl_basic_set *isl_basic_set_remove_redundancies(
+	__isl_take isl_basic_set *bset);
+__isl_give isl_set *isl_set_remove_redundancies(__isl_take isl_set *set);
+__isl_give isl_basic_set *isl_basic_set_list_intersect(
+	__isl_take struct isl_basic_set_list *list);
+__isl_give isl_basic_set *isl_basic_set_list_product(
+	__isl_take struct isl_basic_set_list *list);
+
+__isl_give isl_basic_set *isl_basic_set_read_from_file(isl_ctx *ctx,
+	FILE *input);
+__isl_constructor
+__isl_give isl_basic_set *isl_basic_set_read_from_str(isl_ctx *ctx,
+	const char *str);
+__isl_give isl_set *isl_set_read_from_file(isl_ctx *ctx, FILE *input);
+__isl_constructor
+__isl_give isl_set *isl_set_read_from_str(isl_ctx *ctx, const char *str);
+void isl_basic_set_dump(__isl_keep isl_basic_set *bset);
+void isl_set_dump(__isl_keep isl_set *set);
+__isl_give isl_printer *isl_printer_print_basic_set(
+	__isl_take isl_printer *printer, __isl_keep isl_basic_set *bset);
+__isl_give isl_printer *isl_printer_print_set(__isl_take isl_printer *printer,
+	__isl_keep isl_set *map);
+void isl_basic_set_print(__isl_keep isl_basic_set *bset, FILE *out, int indent,
+	const char *prefix, const char *suffix, unsigned output_format);
+void isl_set_print(__isl_keep struct isl_set *set, FILE *out, int indent,
+	unsigned output_format);
+__isl_give isl_basic_set *isl_basic_set_fix_si(__isl_take isl_basic_set *bset,
+		enum isl_dim_type type, unsigned pos, int value);
+__isl_give isl_basic_set *isl_basic_set_fix_val(__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_val *v);
+__isl_give isl_set *isl_set_fix_si(__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned pos, int value);
+__isl_give isl_set *isl_set_lower_bound_si(__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned pos, int value);
+__isl_give isl_set *isl_set_lower_bound_val(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_val *value);
+__isl_give isl_set *isl_set_upper_bound_si(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, int value);
+__isl_give isl_set *isl_set_upper_bound_val(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_val *value);
+
+__isl_give isl_set *isl_set_equate(__isl_take isl_set *set,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2);
+
+struct isl_basic_set *isl_basic_set_from_underlying_set(
+	struct isl_basic_set *bset, struct isl_basic_set *like);
+struct isl_set *isl_set_from_underlying_set(
+	struct isl_set *set, struct isl_basic_set *like);
+struct isl_set *isl_set_to_underlying_set(struct isl_set *set);
+
+__isl_export
+int isl_basic_set_is_equal(
+		struct isl_basic_set *bset1, struct isl_basic_set *bset2);
+int isl_basic_set_is_disjoint(__isl_keep isl_basic_set *bset1,
+	__isl_keep isl_basic_set *bset2);
+
+__isl_give isl_set *isl_basic_set_partial_lexmin(
+		__isl_take isl_basic_set *bset, __isl_take isl_basic_set *dom,
+		__isl_give isl_set **empty);
+__isl_give isl_set *isl_basic_set_partial_lexmax(
+		__isl_take isl_basic_set *bset, __isl_take isl_basic_set *dom,
+		__isl_give isl_set **empty);
+__isl_give isl_set *isl_set_partial_lexmin(
+		__isl_take isl_set *set, __isl_take isl_set *dom,
+		__isl_give isl_set **empty);
+__isl_give isl_set *isl_set_partial_lexmax(
+		__isl_take isl_set *set, __isl_take isl_set *dom,
+		__isl_give isl_set **empty);
+__isl_export
+__isl_give isl_set *isl_basic_set_lexmin(__isl_take isl_basic_set *bset);
+__isl_export
+__isl_give isl_set *isl_basic_set_lexmax(__isl_take isl_basic_set *bset);
+__isl_export
+__isl_give isl_set *isl_set_lexmin(__isl_take isl_set *set);
+__isl_export
+__isl_give isl_set *isl_set_lexmax(__isl_take isl_set *set);
+__isl_give isl_pw_multi_aff *isl_basic_set_partial_lexmin_pw_multi_aff(
+	__isl_take isl_basic_set *bset, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty);
+__isl_give isl_pw_multi_aff *isl_basic_set_partial_lexmax_pw_multi_aff(
+	__isl_take isl_basic_set *bset, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty);
+__isl_give isl_pw_multi_aff *isl_set_lexmin_pw_multi_aff(
+	__isl_take isl_set *set);
+__isl_give isl_pw_multi_aff *isl_set_lexmax_pw_multi_aff(
+	__isl_take isl_set *set);
+
+__isl_export
+__isl_give isl_set *isl_basic_set_union(
+		__isl_take isl_basic_set *bset1,
+		__isl_take isl_basic_set *bset2);
+
+int isl_basic_set_compare_at(struct isl_basic_set *bset1,
+	struct isl_basic_set *bset2, int pos);
+int isl_set_follows_at(__isl_keep isl_set *set1,
+	__isl_keep isl_set *set2, int pos);
+
+__isl_give isl_basic_set *isl_basic_set_params(__isl_take isl_basic_set *bset);
+__isl_give isl_basic_set *isl_basic_set_from_params(
+	__isl_take isl_basic_set *bset);
+__isl_give isl_set *isl_set_params(__isl_take isl_set *set);
+__isl_give isl_set *isl_set_from_params(__isl_take isl_set *set);
+
+int isl_basic_set_dims_get_sign(__isl_keep isl_basic_set *bset,
+	enum isl_dim_type type, unsigned pos, unsigned n, int *signs);
+
+int isl_basic_set_is_universe(__isl_keep isl_basic_set *bset);
+int isl_basic_set_plain_is_empty(__isl_keep isl_basic_set *bset);
+ISL_DEPRECATED
+int isl_basic_set_fast_is_empty(__isl_keep isl_basic_set *bset);
+__isl_export
+int isl_basic_set_is_empty(__isl_keep isl_basic_set *bset);
+int isl_basic_set_is_bounded(__isl_keep isl_basic_set *bset);
+__isl_export
+int isl_basic_set_is_subset(__isl_keep isl_basic_set *bset1,
+	__isl_keep isl_basic_set *bset2);
+int isl_basic_set_plain_is_equal(__isl_keep isl_basic_set *bset1,
+	__isl_keep isl_basic_set *bset2);
+
+struct isl_set *isl_set_alloc(struct isl_ctx *ctx,
+		unsigned nparam, unsigned dim, int n, unsigned flags);
+struct isl_set *isl_set_extend(struct isl_set *base,
+		unsigned nparam, unsigned dim);
+__isl_give isl_set *isl_set_empty(__isl_take isl_space *dim);
+struct isl_set *isl_set_empty_like(struct isl_set *set);
+__isl_give isl_set *isl_set_universe(__isl_take isl_space *dim);
+__isl_give isl_set *isl_set_nat_universe(__isl_take isl_space *dim);
+__isl_give isl_set *isl_set_universe_like(__isl_keep isl_set *model);
+__isl_give isl_set *isl_set_add_basic_set(__isl_take isl_set *set,
+						__isl_take isl_basic_set *bset);
+struct isl_set *isl_set_finalize(struct isl_set *set);
+__isl_give isl_set *isl_set_copy(__isl_keep isl_set *set);
+__isl_null isl_set *isl_set_free(__isl_take isl_set *set);
+struct isl_set *isl_set_dup(struct isl_set *set);
+__isl_constructor
+__isl_give isl_set *isl_set_from_basic_set(__isl_take isl_basic_set *bset);
+__isl_export
+__isl_give isl_basic_set *isl_set_sample(__isl_take isl_set *set);
+__isl_give isl_point *isl_basic_set_sample_point(__isl_take isl_basic_set *bset);
+__isl_give isl_point *isl_set_sample_point(__isl_take isl_set *set);
+__isl_export
+__isl_give isl_set *isl_set_detect_equalities(__isl_take isl_set *set);
+__isl_export
+__isl_give isl_basic_set *isl_set_affine_hull(__isl_take isl_set *set);
+__isl_give isl_basic_set *isl_set_convex_hull(__isl_take isl_set *set);
+__isl_export
+__isl_give isl_basic_set *isl_set_polyhedral_hull(__isl_take isl_set *set);
+__isl_give isl_basic_set *isl_set_simple_hull(__isl_take isl_set *set);
+__isl_give isl_basic_set *isl_set_unshifted_simple_hull(
+	__isl_take isl_set *set);
+__isl_give isl_basic_set *isl_set_unshifted_simple_hull_from_set_list(
+	__isl_take isl_set *set, __isl_take isl_set_list *list);
+struct isl_basic_set *isl_set_bounded_simple_hull(struct isl_set *set);
+__isl_give isl_set *isl_set_recession_cone(__isl_take isl_set *set);
+
+struct isl_set *isl_set_union_disjoint(
+			struct isl_set *set1, struct isl_set *set2);
+__isl_export
+__isl_give isl_set *isl_set_union(
+		__isl_take isl_set *set1,
+		__isl_take isl_set *set2);
+__isl_give isl_set *isl_set_product(__isl_take isl_set *set1,
+	__isl_take isl_set *set2);
+__isl_give isl_basic_set *isl_basic_set_flat_product(
+	__isl_take isl_basic_set *bset1, __isl_take isl_basic_set *bset2);
+__isl_give isl_set *isl_set_flat_product(__isl_take isl_set *set1,
+	__isl_take isl_set *set2);
+__isl_export
+__isl_give isl_set *isl_set_intersect(
+		__isl_take isl_set *set1,
+		__isl_take isl_set *set2);
+__isl_export
+__isl_give isl_set *isl_set_intersect_params(__isl_take isl_set *set,
+		__isl_take isl_set *params);
+__isl_export
+__isl_give isl_set *isl_set_subtract(
+		__isl_take isl_set *set1,
+		__isl_take isl_set *set2);
+__isl_export
+__isl_give isl_set *isl_set_complement(__isl_take isl_set *set);
+__isl_export
+__isl_give isl_set *isl_set_apply(
+		__isl_take isl_set *set,
+		__isl_take isl_map *map);
+__isl_give isl_set *isl_set_preimage_multi_aff(__isl_take isl_set *set,
+	__isl_take isl_multi_aff *ma);
+__isl_give isl_set *isl_set_preimage_pw_multi_aff(__isl_take isl_set *set,
+	__isl_take isl_pw_multi_aff *pma);
+__isl_give isl_set *isl_set_preimage_multi_pw_aff(__isl_take isl_set *set,
+	__isl_take isl_multi_pw_aff *mpa);
+__isl_give isl_set *isl_set_fix_val(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_val *v);
+struct isl_set *isl_set_fix_dim_si(struct isl_set *set,
+		unsigned dim, int value);
+__isl_give isl_basic_set *isl_basic_set_insert_dims(
+	__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned pos, unsigned n);
+__isl_give isl_set *isl_set_insert_dims(__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned pos, unsigned n);
+__isl_give isl_basic_set *isl_basic_set_add_dims(__isl_take isl_basic_set *bset,
+		enum isl_dim_type type, unsigned n);
+/* deprecated */
+__isl_give isl_basic_set *isl_basic_set_add(__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned n);
+__isl_give isl_set *isl_set_add_dims(__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned n);
+__isl_give isl_basic_set *isl_basic_set_move_dims(__isl_take isl_basic_set *bset,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n);
+__isl_give isl_set *isl_set_move_dims(__isl_take isl_set *set,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n);
+__isl_give isl_basic_set *isl_basic_set_project_out(
+		__isl_take isl_basic_set *bset,
+		enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_set *isl_set_project_out(__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_basic_set *isl_basic_set_remove_divs(
+	__isl_take isl_basic_set *bset);
+__isl_give isl_basic_set *isl_basic_set_eliminate(
+	__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_set *isl_set_eliminate(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned first, unsigned n);
+struct isl_set *isl_set_eliminate_dims(struct isl_set *set,
+		unsigned first, unsigned n);
+__isl_give isl_set *isl_set_remove_dims(__isl_take isl_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_basic_set *isl_basic_set_remove_divs_involving_dims(
+	__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_set *isl_set_remove_divs_involving_dims(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_basic_set *isl_basic_set_remove_unknown_divs(
+	__isl_take isl_basic_set *bset);
+__isl_give isl_set *isl_set_remove_unknown_divs(__isl_take isl_set *set);
+__isl_give isl_set *isl_set_remove_divs(__isl_take isl_set *set);
+__isl_give isl_set *isl_set_split_dims(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+__isl_give isl_basic_set *isl_basic_set_drop_constraints_involving_dims(
+	__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_basic_set *isl_basic_set_drop_constraints_not_involving_dims(
+	__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_set *isl_set_drop_constraints_involving_dims(
+	__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+int isl_basic_set_involves_dims(__isl_keep isl_basic_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n);
+int isl_set_involves_dims(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+void isl_set_print_internal(__isl_keep isl_set *set, FILE *out, int indent);
+int isl_set_plain_is_empty(__isl_keep isl_set *set);
+ISL_DEPRECATED
+int isl_set_fast_is_empty(__isl_keep isl_set *set);
+int isl_set_plain_is_universe(__isl_keep isl_set *set);
+ISL_DEPRECATED
+int isl_set_fast_is_universe(__isl_keep isl_set *set);
+int isl_set_is_params(__isl_keep isl_set *set);
+__isl_export
+int isl_set_is_empty(__isl_keep isl_set *set);
+int isl_set_is_bounded(__isl_keep isl_set *set);
+__isl_export
+int isl_set_is_subset(__isl_keep isl_set *set1, __isl_keep isl_set *set2);
+__isl_export
+int isl_set_is_strict_subset(__isl_keep isl_set *set1, __isl_keep isl_set *set2);
+__isl_export
+int isl_set_is_equal(__isl_keep isl_set *set1, __isl_keep isl_set *set2);
+__isl_export
+int isl_set_is_disjoint(__isl_keep isl_set *set1, __isl_keep isl_set *set2);
+int isl_set_is_singleton(__isl_keep isl_set *set);
+int isl_set_is_box(__isl_keep isl_set *set);
+int isl_set_has_equal_space(__isl_keep isl_set *set1, __isl_keep isl_set *set2);
+
+__isl_give isl_set *isl_set_sum(__isl_take isl_set *set1,
+	__isl_take isl_set *set2);
+__isl_give isl_basic_set *isl_basic_set_neg(__isl_take isl_basic_set *bset);
+__isl_give isl_set *isl_set_neg(__isl_take isl_set *set);
+
+__isl_give isl_set *isl_set_make_disjoint(__isl_take isl_set *set);
+struct isl_set *isl_basic_set_compute_divs(struct isl_basic_set *bset);
+__isl_give isl_set *isl_set_compute_divs(__isl_take isl_set *set);
+__isl_give isl_set *isl_set_align_divs(__isl_take isl_set *set);
+
+struct isl_basic_set *isl_set_copy_basic_set(struct isl_set *set);
+struct isl_set *isl_set_drop_basic_set(struct isl_set *set,
+						struct isl_basic_set *bset);
+
+__isl_give isl_val *isl_set_plain_get_val_if_fixed(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos);
+int isl_set_dim_is_bounded(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos);
+int isl_set_dim_has_lower_bound(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos);
+int isl_set_dim_has_upper_bound(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos);
+int isl_set_dim_has_any_lower_bound(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos);
+int isl_set_dim_has_any_upper_bound(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos);
+
+__isl_export
+__isl_give isl_basic_set *isl_basic_set_gist(__isl_take isl_basic_set *bset,
+					    __isl_take isl_basic_set *context);
+__isl_give isl_set *isl_set_gist_basic_set(__isl_take isl_set *set,
+	__isl_take isl_basic_set *context);
+__isl_export
+__isl_give isl_set *isl_set_gist(__isl_take isl_set *set,
+	__isl_take isl_set *context);
+__isl_give isl_set *isl_set_gist_params(__isl_take isl_set *set,
+	__isl_take isl_set *context);
+int isl_set_dim_residue_class_val(__isl_keep isl_set *set,
+	int pos, __isl_give isl_val **modulo, __isl_give isl_val **residue);
+
+__isl_export
+__isl_give isl_set *isl_set_coalesce(__isl_take isl_set *set);
+
+int isl_set_plain_cmp(__isl_keep isl_set *set1, __isl_keep isl_set *set2);
+int isl_set_plain_is_equal(__isl_keep isl_set *set1, __isl_keep isl_set *set2);
+ISL_DEPRECATED
+int isl_set_fast_is_equal(__isl_keep isl_set *set1, __isl_keep isl_set *set2);
+int isl_set_plain_is_disjoint(__isl_keep isl_set *set1,
+	__isl_keep isl_set *set2);
+ISL_DEPRECATED
+int isl_set_fast_is_disjoint(__isl_keep isl_set *set1,
+	__isl_keep isl_set *set2);
+
+uint32_t isl_set_get_hash(struct isl_set *set);
+
+int isl_set_dim_is_unique(struct isl_set *set, unsigned dim);
+
+int isl_set_n_basic_set(__isl_keep isl_set *set);
+__isl_export
+int isl_set_foreach_basic_set(__isl_keep isl_set *set,
+	int (*fn)(__isl_take isl_basic_set *bset, void *user), void *user);
+
+int isl_set_foreach_point(__isl_keep isl_set *set,
+	int (*fn)(__isl_take isl_point *pnt, void *user), void *user);
+__isl_give isl_val *isl_set_count_val(__isl_keep isl_set *set);
+
+__isl_give isl_basic_set *isl_basic_set_from_point(__isl_take isl_point *pnt);
+__isl_give isl_set *isl_set_from_point(__isl_take isl_point *pnt);
+__isl_give isl_basic_set *isl_basic_set_box_from_points(
+	__isl_take isl_point *pnt1, __isl_take isl_point *pnt2);
+__isl_give isl_set *isl_set_box_from_points(__isl_take isl_point *pnt1,
+	__isl_take isl_point *pnt2);
+
+__isl_give isl_basic_set *isl_basic_set_lift(__isl_take isl_basic_set *bset);
+__isl_give isl_set *isl_set_lift(__isl_take isl_set *set);
+
+__isl_give isl_map *isl_set_lex_le_set(__isl_take isl_set *set1,
+	__isl_take isl_set *set2);
+__isl_give isl_map *isl_set_lex_lt_set(__isl_take isl_set *set1,
+	__isl_take isl_set *set2);
+__isl_give isl_map *isl_set_lex_ge_set(__isl_take isl_set *set1,
+	__isl_take isl_set *set2);
+__isl_give isl_map *isl_set_lex_gt_set(__isl_take isl_set *set1,
+	__isl_take isl_set *set2);
+
+int isl_set_size(__isl_keep isl_set *set);
+
+__isl_give isl_basic_set *isl_basic_set_align_params(
+	__isl_take isl_basic_set *bset, __isl_take isl_space *model);
+__isl_give isl_set *isl_set_align_params(__isl_take isl_set *set,
+	__isl_take isl_space *model);
+
+__isl_give isl_mat *isl_basic_set_equalities_matrix(
+	__isl_keep isl_basic_set *bset, enum isl_dim_type c1,
+	enum isl_dim_type c2, enum isl_dim_type c3, enum isl_dim_type c4);
+__isl_give isl_mat *isl_basic_set_inequalities_matrix(
+	__isl_keep isl_basic_set *bset, enum isl_dim_type c1,
+	enum isl_dim_type c2, enum isl_dim_type c3, enum isl_dim_type c4);
+__isl_give isl_basic_set *isl_basic_set_from_constraint_matrices(
+	__isl_take isl_space *dim,
+	__isl_take isl_mat *eq, __isl_take isl_mat *ineq, enum isl_dim_type c1,
+	enum isl_dim_type c2, enum isl_dim_type c3, enum isl_dim_type c4);
+
+__isl_give isl_mat *isl_basic_set_reduced_basis(__isl_keep isl_basic_set *bset);
+
+__isl_give isl_basic_set *isl_basic_set_coefficients(
+	__isl_take isl_basic_set *bset);
+__isl_give isl_basic_set *isl_set_coefficients(__isl_take isl_set *set);
+__isl_give isl_basic_set *isl_basic_set_solutions(
+	__isl_take isl_basic_set *bset);
+__isl_give isl_basic_set *isl_set_solutions(__isl_take isl_set *set);
+
+__isl_give isl_pw_aff *isl_set_dim_max(__isl_take isl_set *set, int pos);
+__isl_give isl_pw_aff *isl_set_dim_min(__isl_take isl_set *set, int pos);
+
+__isl_give char *isl_set_to_str(__isl_keep isl_set *set);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#include <isl/dim.h>
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/set_type.h b/polly/lib/External/isl/include/isl/set_type.h
new file mode 100644
index 00000000000..ce349e1b5d4
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/set_type.h
@@ -0,0 +1,6 @@
+#ifndef ISL_SET_TYPE_H
+#define ISL_SET_TYPE_H
+
+#include <isl/map_type.h>
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/space.h b/polly/lib/External/isl/include/isl/space.h
new file mode 100644
index 00000000000..d6fce404156
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/space.h
@@ -0,0 +1,173 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_SPACE_H
+#define ISL_SPACE_H
+
+#include <isl/ctx.h>
+#include <isl/id.h>
+#include <isl/printer.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_space;
+typedef struct isl_space isl_space;
+
+enum isl_dim_type {
+	isl_dim_cst,
+	isl_dim_param,
+	isl_dim_in,
+	isl_dim_out,
+	isl_dim_set = isl_dim_out,
+	isl_dim_div,
+	isl_dim_all
+};
+
+isl_ctx *isl_space_get_ctx(__isl_keep isl_space *dim);
+__isl_give isl_space *isl_space_alloc(isl_ctx *ctx,
+			unsigned nparam, unsigned n_in, unsigned n_out);
+__isl_give isl_space *isl_space_set_alloc(isl_ctx *ctx,
+			unsigned nparam, unsigned dim);
+__isl_give isl_space *isl_space_params_alloc(isl_ctx *ctx, unsigned nparam);
+__isl_give isl_space *isl_space_copy(__isl_keep isl_space *dim);
+__isl_null isl_space *isl_space_free(__isl_take isl_space *space);
+
+int isl_space_is_params(__isl_keep isl_space *space);
+int isl_space_is_set(__isl_keep isl_space *space);
+int isl_space_is_map(__isl_keep isl_space *space);
+
+__isl_give isl_space *isl_space_set_tuple_name(__isl_take isl_space *dim,
+	enum isl_dim_type type, const char *s);
+int isl_space_has_tuple_name(__isl_keep isl_space *space,
+	enum isl_dim_type type);
+const char *isl_space_get_tuple_name(__isl_keep isl_space *dim,
+				 enum isl_dim_type type);
+__isl_give isl_space *isl_space_set_tuple_id(__isl_take isl_space *dim,
+	enum isl_dim_type type, __isl_take isl_id *id);
+__isl_give isl_space *isl_space_reset_tuple_id(__isl_take isl_space *dim,
+	enum isl_dim_type type);
+int isl_space_has_tuple_id(__isl_keep isl_space *dim, enum isl_dim_type type);
+__isl_give isl_id *isl_space_get_tuple_id(__isl_keep isl_space *dim,
+	enum isl_dim_type type);
+__isl_give isl_space *isl_space_reset_user(__isl_take isl_space *space);
+
+__isl_give isl_space *isl_space_set_dim_id(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id);
+int isl_space_has_dim_id(__isl_keep isl_space *dim,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_id *isl_space_get_dim_id(__isl_keep isl_space *dim,
+	enum isl_dim_type type, unsigned pos);
+
+int isl_space_find_dim_by_id(__isl_keep isl_space *dim, enum isl_dim_type type,
+	__isl_keep isl_id *id);
+int isl_space_find_dim_by_name(__isl_keep isl_space *space,
+	enum isl_dim_type type, const char *name);
+
+int isl_space_has_dim_name(__isl_keep isl_space *space,
+	enum isl_dim_type type, unsigned pos);
+__isl_give isl_space *isl_space_set_dim_name(__isl_take isl_space *dim,
+				 enum isl_dim_type type, unsigned pos,
+				 __isl_keep const char *name);
+__isl_keep const char *isl_space_get_dim_name(__isl_keep isl_space *dim,
+				 enum isl_dim_type type, unsigned pos);
+
+__isl_give isl_space *isl_space_extend(__isl_take isl_space *dim,
+			unsigned nparam, unsigned n_in, unsigned n_out);
+__isl_give isl_space *isl_space_add_dims(__isl_take isl_space *dim, enum isl_dim_type type,
+		unsigned n);
+__isl_give isl_space *isl_space_move_dims(__isl_take isl_space *dim,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n);
+__isl_give isl_space *isl_space_insert_dims(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned pos, unsigned n);
+__isl_give isl_space *isl_space_join(__isl_take isl_space *left,
+	__isl_take isl_space *right);
+__isl_give isl_space *isl_space_product(__isl_take isl_space *left,
+	__isl_take isl_space *right);
+__isl_give isl_space *isl_space_domain_product(__isl_take isl_space *left,
+	__isl_take isl_space *right);
+__isl_give isl_space *isl_space_range_product(__isl_take isl_space *left,
+	__isl_take isl_space *right);
+__isl_give isl_space *isl_space_factor_domain(__isl_take isl_space *space);
+__isl_give isl_space *isl_space_factor_range(__isl_take isl_space *space);
+__isl_give isl_space *isl_space_domain_factor_domain(
+	__isl_take isl_space *space);
+__isl_give isl_space *isl_space_domain_factor_range(
+	__isl_take isl_space *space);
+__isl_give isl_space *isl_space_range_factor_domain(
+	__isl_take isl_space *space);
+__isl_give isl_space *isl_space_range_factor_range(
+	__isl_take isl_space *space);
+__isl_give isl_space *isl_space_map_from_set(__isl_take isl_space *dim);
+__isl_give isl_space *isl_space_map_from_domain_and_range(
+	__isl_take isl_space *domain, __isl_take isl_space *range);
+__isl_give isl_space *isl_space_reverse(__isl_take isl_space *dim);
+__isl_give isl_space *isl_space_drop_dims(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned first, unsigned num);
+__isl_give isl_space *isl_space_drop_inputs(__isl_take isl_space *dim,
+		unsigned first, unsigned n);
+__isl_give isl_space *isl_space_drop_outputs(__isl_take isl_space *dim,
+		unsigned first, unsigned n);
+__isl_give isl_space *isl_space_domain(__isl_take isl_space *dim);
+__isl_give isl_space *isl_space_from_domain(__isl_take isl_space *dim);
+__isl_give isl_space *isl_space_range(__isl_take isl_space *dim);
+__isl_give isl_space *isl_space_from_range(__isl_take isl_space *dim);
+__isl_give isl_space *isl_space_domain_map(__isl_take isl_space *space);
+__isl_give isl_space *isl_space_range_map(__isl_take isl_space *space);
+__isl_give isl_space *isl_space_params(__isl_take isl_space *space);
+__isl_give isl_space *isl_space_set_from_params(__isl_take isl_space *space);
+
+__isl_give isl_space *isl_space_align_params(__isl_take isl_space *dim1,
+	__isl_take isl_space *dim2);
+
+int isl_space_is_wrapping(__isl_keep isl_space *dim);
+int isl_space_domain_is_wrapping(__isl_keep isl_space *space);
+int isl_space_range_is_wrapping(__isl_keep isl_space *space);
+__isl_give isl_space *isl_space_wrap(__isl_take isl_space *dim);
+__isl_give isl_space *isl_space_unwrap(__isl_take isl_space *dim);
+
+int isl_space_can_zip(__isl_keep isl_space *dim);
+__isl_give isl_space *isl_space_zip(__isl_take isl_space *dim);
+
+int isl_space_can_curry(__isl_keep isl_space *space);
+__isl_give isl_space *isl_space_curry(__isl_take isl_space *space);
+
+int isl_space_can_uncurry(__isl_keep isl_space *space);
+__isl_give isl_space *isl_space_uncurry(__isl_take isl_space *space);
+
+int isl_space_is_domain(__isl_keep isl_space *space1,
+	__isl_keep isl_space *space2);
+int isl_space_is_range(__isl_keep isl_space *space1,
+	__isl_keep isl_space *space2);
+int isl_space_is_equal(__isl_keep isl_space *space1,
+	__isl_keep isl_space *space2);
+int isl_space_tuple_is_equal(__isl_keep isl_space *space1,
+	enum isl_dim_type type1, __isl_keep isl_space *space2,
+	enum isl_dim_type type2);
+int isl_space_match(__isl_keep isl_space *dim1, enum isl_dim_type dim1_type,
+	__isl_keep isl_space *dim2, enum isl_dim_type dim2_type);
+ISL_DEPRECATED
+int isl_space_tuple_match(__isl_keep isl_space *space1, enum isl_dim_type type1,
+	__isl_keep isl_space *space2, enum isl_dim_type type2);
+int isl_space_compatible(__isl_keep isl_space *dim1,
+	__isl_keep isl_space *dim2);
+unsigned isl_space_dim(__isl_keep isl_space *dim, enum isl_dim_type type);
+
+__isl_give char *isl_space_to_str(__isl_keep isl_space *space);
+__isl_give isl_printer *isl_printer_print_space(__isl_take isl_printer *p,
+	__isl_keep isl_space *dim);
+void isl_space_dump(__isl_keep isl_space *dim);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/stream.h b/polly/lib/External/isl/include/isl/stream.h
new file mode 100644
index 00000000000..42734996780
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/stream.h
@@ -0,0 +1,103 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_STREAM_H
+#define ISL_STREAM_H
+
+#include <stdio.h>
+#include <isl/hash.h>
+#include <isl/aff_type.h>
+#include <isl/obj.h>
+#include <isl/val.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+enum isl_token_type { ISL_TOKEN_ERROR = -1,
+			ISL_TOKEN_UNKNOWN = 256, ISL_TOKEN_VALUE,
+			ISL_TOKEN_IDENT, ISL_TOKEN_GE,
+			ISL_TOKEN_LE, ISL_TOKEN_GT, ISL_TOKEN_LT,
+			ISL_TOKEN_NE, ISL_TOKEN_EQ_EQ,
+			ISL_TOKEN_LEX_GE, ISL_TOKEN_LEX_LE,
+			ISL_TOKEN_LEX_GT, ISL_TOKEN_LEX_LT,
+			ISL_TOKEN_TO, ISL_TOKEN_AND,
+			ISL_TOKEN_OR, ISL_TOKEN_EXISTS, ISL_TOKEN_NOT,
+			ISL_TOKEN_DEF, ISL_TOKEN_INFTY, ISL_TOKEN_NAN,
+			ISL_TOKEN_MIN, ISL_TOKEN_MAX, ISL_TOKEN_RAT,
+			ISL_TOKEN_TRUE, ISL_TOKEN_FALSE,
+			ISL_TOKEN_CEILD, ISL_TOKEN_FLOORD, ISL_TOKEN_MOD,
+			ISL_TOKEN_STRING,
+			ISL_TOKEN_MAP, ISL_TOKEN_AFF,
+			ISL_TOKEN_CEIL, ISL_TOKEN_FLOOR,
+			ISL_TOKEN_IMPLIES,
+			ISL_TOKEN_LAST };
+
+struct isl_token;
+
+__isl_give isl_val *isl_token_get_val(isl_ctx *ctx, struct isl_token *tok);
+__isl_give char *isl_token_get_str(isl_ctx *ctx, struct isl_token *tok);
+int isl_token_get_type(struct isl_token *tok);
+void isl_token_free(struct isl_token *tok);
+
+struct isl_stream {
+	struct isl_ctx	*ctx;
+	FILE        	*file;
+	const char  	*str;
+	int	    	line;
+	int	    	col;
+	int	    	eof;
+
+	char	    	*buffer;
+	size_t	    	size;
+	size_t	    	len;
+	int	    	c;
+	int		un[5];
+	int		n_un;
+
+	struct isl_token	*tokens[5];
+	int	    	n_token;
+
+	struct isl_hash_table	*keywords;
+	enum isl_token_type	 next_type;
+};
+
+struct isl_stream* isl_stream_new_file(struct isl_ctx *ctx, FILE *file);
+struct isl_stream* isl_stream_new_str(struct isl_ctx *ctx, const char *str);
+void isl_stream_free(struct isl_stream *s);
+
+void isl_stream_error(struct isl_stream *s, struct isl_token *tok, char *msg);
+
+struct isl_token *isl_stream_next_token(struct isl_stream *s);
+struct isl_token *isl_stream_next_token_on_same_line(struct isl_stream *s);
+int isl_stream_next_token_is(struct isl_stream *s, int type);
+void isl_stream_push_token(struct isl_stream *s, struct isl_token *tok);
+void isl_stream_flush_tokens(struct isl_stream *s);
+int isl_stream_eat_if_available(struct isl_stream *s, int type);
+char *isl_stream_read_ident_if_available(struct isl_stream *s);
+int isl_stream_eat(struct isl_stream *s, int type);
+int isl_stream_is_empty(struct isl_stream *s);
+int isl_stream_skip_line(struct isl_stream *s);
+
+enum isl_token_type isl_stream_register_keyword(struct isl_stream *s,
+	const char *name);
+
+struct isl_obj isl_stream_read_obj(struct isl_stream *s);
+__isl_give isl_multi_aff *isl_stream_read_multi_aff(struct isl_stream *s);
+__isl_give isl_map *isl_stream_read_map(struct isl_stream *s);
+__isl_give isl_set *isl_stream_read_set(struct isl_stream *s);
+__isl_give isl_pw_qpolynomial *isl_stream_read_pw_qpolynomial(
+	struct isl_stream *s);
+__isl_give isl_union_map *isl_stream_read_union_map(struct isl_stream *s);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/union_map.h b/polly/lib/External/isl/include/isl/union_map.h
new file mode 100644
index 00000000000..2d4d94cd3b3
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/union_map.h
@@ -0,0 +1,249 @@
+#ifndef ISL_UNION_MAP_H
+#define ISL_UNION_MAP_H
+
+#include <isl/space.h>
+#include <isl/map_type.h>
+#include <isl/union_map_type.h>
+#include <isl/printer.h>
+#include <isl/val.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+unsigned isl_union_map_dim(__isl_keep isl_union_map *umap,
+	enum isl_dim_type type);
+int isl_union_map_involves_dims(__isl_keep isl_union_map *umap,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_id *isl_union_map_get_dim_id(__isl_keep isl_union_map *umap,
+	enum isl_dim_type type, unsigned pos);
+
+__isl_constructor
+__isl_give isl_union_map *isl_union_map_from_basic_map(
+	__isl_take isl_basic_map *bmap);
+__isl_constructor
+__isl_give isl_union_map *isl_union_map_from_map(__isl_take isl_map *map);
+__isl_give isl_union_map *isl_union_map_empty(__isl_take isl_space *dim);
+__isl_give isl_union_map *isl_union_map_copy(__isl_keep isl_union_map *umap);
+__isl_null isl_union_map *isl_union_map_free(__isl_take isl_union_map *umap);
+
+isl_ctx *isl_union_map_get_ctx(__isl_keep isl_union_map *umap);
+__isl_give isl_space *isl_union_map_get_space(__isl_keep isl_union_map *umap);
+
+__isl_give isl_union_map *isl_union_map_reset_user(
+	__isl_take isl_union_map *umap);
+
+int isl_union_map_find_dim_by_name(__isl_keep isl_union_map *umap,
+	enum isl_dim_type type, const char *name);
+
+__isl_give isl_union_map *isl_union_map_universe(
+	__isl_take isl_union_map *umap);
+__isl_give isl_set *isl_union_map_params(__isl_take isl_union_map *umap);
+__isl_give isl_union_set *isl_union_map_domain(__isl_take isl_union_map *umap);
+__isl_give isl_union_set *isl_union_map_range(__isl_take isl_union_map *umap);
+__isl_give isl_union_map *isl_union_map_domain_map(
+	__isl_take isl_union_map *umap);
+__isl_give isl_union_map *isl_union_map_range_map(
+	__isl_take isl_union_map *umap);
+__isl_give isl_union_map *isl_union_set_wrapped_domain_map(
+	__isl_take isl_union_set *uset);
+__isl_give isl_union_map *isl_union_map_from_domain(
+	__isl_take isl_union_set *uset);
+__isl_give isl_union_map *isl_union_map_from_range(
+	__isl_take isl_union_set *uset);
+
+__isl_export
+__isl_give isl_union_map *isl_union_map_affine_hull(
+	__isl_take isl_union_map *umap);
+__isl_export
+__isl_give isl_union_map *isl_union_map_polyhedral_hull(
+	__isl_take isl_union_map *umap);
+__isl_give isl_union_map *isl_union_map_simple_hull(
+	__isl_take isl_union_map *umap);
+__isl_export
+__isl_give isl_union_map *isl_union_map_coalesce(
+	__isl_take isl_union_map *umap);
+__isl_give isl_union_map *isl_union_map_compute_divs(
+	__isl_take isl_union_map *umap);
+__isl_export
+__isl_give isl_union_map *isl_union_map_lexmin(__isl_take isl_union_map *umap);
+__isl_export
+__isl_give isl_union_map *isl_union_map_lexmax(__isl_take isl_union_map *umap);
+
+__isl_give isl_union_map *isl_union_map_add_map(__isl_take isl_union_map *umap,
+	__isl_take isl_map *map);
+__isl_export
+__isl_give isl_union_map *isl_union_map_union(__isl_take isl_union_map *umap1,
+	__isl_take isl_union_map *umap2);
+__isl_export
+__isl_give isl_union_map *isl_union_map_subtract(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2);
+__isl_export
+__isl_give isl_union_map *isl_union_map_intersect(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2);
+__isl_export
+__isl_give isl_union_map *isl_union_map_intersect_params(
+	__isl_take isl_union_map *umap, __isl_take isl_set *set);
+__isl_give isl_union_map *isl_union_map_product(__isl_take isl_union_map *umap1,
+	__isl_take isl_union_map *umap2);
+__isl_give isl_union_map *isl_union_map_domain_product(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2);
+__isl_give isl_union_map *isl_union_map_range_product(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2);
+__isl_give isl_union_map *isl_union_map_flat_range_product(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2);
+__isl_give isl_union_map *isl_union_map_domain_factor_domain(
+	__isl_take isl_union_map *umap);
+__isl_give isl_union_map *isl_union_map_domain_factor_range(
+	__isl_take isl_union_map *umap);
+__isl_give isl_union_map *isl_union_map_range_factor_range(
+	__isl_take isl_union_map *umap);
+__isl_give isl_union_map *isl_union_map_factor_domain(
+	__isl_take isl_union_map *umap);
+__isl_give isl_union_map *isl_union_map_factor_range(
+	__isl_take isl_union_map *umap);
+__isl_export
+__isl_give isl_union_map *isl_union_map_gist(__isl_take isl_union_map *umap,
+	__isl_take isl_union_map *context);
+__isl_export
+__isl_give isl_union_map *isl_union_map_gist_params(
+	__isl_take isl_union_map *umap, __isl_take isl_set *set);
+__isl_export
+__isl_give isl_union_map *isl_union_map_gist_domain(
+	__isl_take isl_union_map *umap, __isl_take isl_union_set *uset);
+__isl_export
+__isl_give isl_union_map *isl_union_map_gist_range(
+	__isl_take isl_union_map *umap, __isl_take isl_union_set *uset);
+
+__isl_export
+__isl_give isl_union_map *isl_union_map_intersect_domain(
+	__isl_take isl_union_map *umap, __isl_take isl_union_set *uset);
+__isl_export
+__isl_give isl_union_map *isl_union_map_intersect_range(
+	__isl_take isl_union_map *umap, __isl_take isl_union_set *uset);
+
+__isl_export
+__isl_give isl_union_map *isl_union_map_subtract_domain(
+	__isl_take isl_union_map *umap, __isl_take isl_union_set *dom);
+__isl_export
+__isl_give isl_union_map *isl_union_map_subtract_range(
+	__isl_take isl_union_map *umap, __isl_take isl_union_set *dom);
+
+__isl_export
+__isl_give isl_union_map *isl_union_map_apply_domain(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2);
+__isl_export
+__isl_give isl_union_map *isl_union_map_apply_range(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2);
+__isl_give isl_union_map *isl_union_map_preimage_domain_multi_aff(
+	__isl_take isl_union_map *umap, __isl_take isl_multi_aff *ma);
+__isl_give isl_union_map *isl_union_map_preimage_range_multi_aff(
+	__isl_take isl_union_map *umap, __isl_take isl_multi_aff *ma);
+__isl_give isl_union_map *isl_union_map_preimage_domain_pw_multi_aff(
+	__isl_take isl_union_map *umap, __isl_take isl_pw_multi_aff *pma);
+__isl_give isl_union_map *isl_union_map_preimage_range_pw_multi_aff(
+	__isl_take isl_union_map *umap, __isl_take isl_pw_multi_aff *pma);
+__isl_give isl_union_map *isl_union_map_preimage_domain_multi_pw_aff(
+	__isl_take isl_union_map *umap, __isl_take isl_multi_pw_aff *mpa);
+__isl_give isl_union_map *isl_union_map_preimage_domain_union_pw_multi_aff(
+	__isl_take isl_union_map *umap,
+	__isl_take isl_union_pw_multi_aff *upma);
+__isl_give isl_union_map *isl_union_map_preimage_range_union_pw_multi_aff(
+	__isl_take isl_union_map *umap,
+	__isl_take isl_union_pw_multi_aff *upma);
+__isl_export
+__isl_give isl_union_map *isl_union_map_reverse(__isl_take isl_union_map *umap);
+__isl_give isl_union_map *isl_union_map_from_domain_and_range(
+	__isl_take isl_union_set *domain, __isl_take isl_union_set *range);
+
+__isl_export
+__isl_give isl_union_map *isl_union_map_detect_equalities(
+	__isl_take isl_union_map *umap);
+__isl_export
+__isl_give isl_union_set *isl_union_map_deltas(__isl_take isl_union_map *umap);
+__isl_give isl_union_map *isl_union_map_deltas_map(
+	__isl_take isl_union_map *umap);
+__isl_export
+__isl_give isl_union_map *isl_union_set_identity(__isl_take isl_union_set *uset);
+
+__isl_give isl_union_map *isl_union_map_project_out(
+	__isl_take isl_union_map *umap,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+__isl_export
+int isl_union_map_is_empty(__isl_keep isl_union_map *umap);
+__isl_export
+int isl_union_map_is_single_valued(__isl_keep isl_union_map *umap);
+int isl_union_map_plain_is_injective(__isl_keep isl_union_map *umap);
+__isl_export
+int isl_union_map_is_injective(__isl_keep isl_union_map *umap);
+__isl_export
+int isl_union_map_is_bijective(__isl_keep isl_union_map *umap);
+
+__isl_export
+int isl_union_map_is_subset(__isl_keep isl_union_map *umap1,
+	__isl_keep isl_union_map *umap2);
+__isl_export
+int isl_union_map_is_equal(__isl_keep isl_union_map *umap1,
+	__isl_keep isl_union_map *umap2);
+__isl_export
+int isl_union_map_is_strict_subset(__isl_keep isl_union_map *umap1,
+	__isl_keep isl_union_map *umap2);
+
+int isl_union_map_n_map(__isl_keep isl_union_map *umap);
+__isl_export
+int isl_union_map_foreach_map(__isl_keep isl_union_map *umap,
+	int (*fn)(__isl_take isl_map *map, void *user), void *user);
+__isl_give int isl_union_map_contains(__isl_keep isl_union_map *umap,
+	__isl_keep isl_space *dim);
+__isl_give isl_map *isl_union_map_extract_map(__isl_keep isl_union_map *umap,
+	__isl_take isl_space *dim);
+__isl_give isl_map *isl_map_from_union_map(__isl_take isl_union_map *umap);
+
+__isl_give isl_basic_map *isl_union_map_sample(__isl_take isl_union_map *umap);
+
+__isl_give isl_union_map *isl_union_map_fixed_power_val(
+	__isl_take isl_union_map *umap, __isl_take isl_val *exp);
+__isl_give isl_union_map *isl_union_map_power(__isl_take isl_union_map *umap,
+	int *exact);
+__isl_give isl_union_map *isl_union_map_transitive_closure(
+	__isl_take isl_union_map *umap, int *exact);
+
+__isl_give isl_union_map *isl_union_map_lex_lt_union_map(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2);
+__isl_give isl_union_map *isl_union_map_lex_le_union_map(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2);
+__isl_give isl_union_map *isl_union_map_lex_gt_union_map(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2);
+__isl_give isl_union_map *isl_union_map_lex_ge_union_map(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2);
+
+__isl_give isl_union_map *isl_union_map_read_from_file(isl_ctx *ctx,
+	FILE *input);
+__isl_constructor
+__isl_give isl_union_map *isl_union_map_read_from_str(isl_ctx *ctx,
+	const char *str);
+__isl_give char *isl_union_map_to_str(__isl_keep isl_union_map *umap);
+__isl_give isl_printer *isl_printer_print_union_map(__isl_take isl_printer *p,
+	__isl_keep isl_union_map *umap);
+void isl_union_map_dump(__isl_keep isl_union_map *umap);
+
+__isl_give isl_union_set *isl_union_map_wrap(__isl_take isl_union_map *umap);
+__isl_give isl_union_map *isl_union_set_unwrap(__isl_take isl_union_set *uset);
+
+__isl_give isl_union_map *isl_union_map_zip(__isl_take isl_union_map *umap);
+__isl_give isl_union_map *isl_union_map_curry(__isl_take isl_union_map *umap);
+__isl_give isl_union_map *isl_union_map_uncurry(__isl_take isl_union_map *umap);
+
+__isl_give isl_union_map *isl_union_map_align_params(
+	__isl_take isl_union_map *umap, __isl_take isl_space *model);
+__isl_give isl_union_set *isl_union_set_align_params(
+	__isl_take isl_union_set *uset, __isl_take isl_space *model);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#include <isl/dim.h>
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/union_map_type.h b/polly/lib/External/isl/include/isl/union_map_type.h
new file mode 100644
index 00000000000..e93f2cc7137
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/union_map_type.h
@@ -0,0 +1,21 @@
+#ifndef ISL_UNION_MAP_TYPE_H
+#define ISL_UNION_MAP_TYPE_H
+
+#include <isl/ctx.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct __isl_export isl_union_map;
+typedef struct isl_union_map isl_union_map;
+#ifndef isl_union_set
+struct __isl_export isl_union_set;
+typedef struct isl_union_set isl_union_set;
+#endif
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/union_set.h b/polly/lib/External/isl/include/isl/union_set.h
new file mode 100644
index 00000000000..616acb8011c
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/union_set.h
@@ -0,0 +1,147 @@
+#ifndef ISL_UNION_SET_H
+#define ISL_UNION_SET_H
+
+#include <isl/point.h>
+#include <isl/union_map.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+unsigned isl_union_set_dim(__isl_keep isl_union_set *uset,
+	enum isl_dim_type type);
+
+__isl_constructor
+__isl_give isl_union_set *isl_union_set_from_basic_set(
+	__isl_take isl_basic_set *bset);
+__isl_constructor
+__isl_give isl_union_set *isl_union_set_from_set(__isl_take isl_set *set);
+__isl_give isl_union_set *isl_union_set_empty(__isl_take isl_space *dim);
+__isl_give isl_union_set *isl_union_set_copy(__isl_keep isl_union_set *uset);
+__isl_null isl_union_set *isl_union_set_free(__isl_take isl_union_set *uset);
+
+isl_ctx *isl_union_set_get_ctx(__isl_keep isl_union_set *uset);
+__isl_give isl_space *isl_union_set_get_space(__isl_keep isl_union_set *uset);
+
+__isl_give isl_union_set *isl_union_set_reset_user(
+	__isl_take isl_union_set *uset);
+
+__isl_give isl_union_set *isl_union_set_universe(
+	__isl_take isl_union_set *uset);
+__isl_give isl_set *isl_union_set_params(__isl_take isl_union_set *uset);
+
+__isl_export
+__isl_give isl_union_set *isl_union_set_detect_equalities(
+	__isl_take isl_union_set *uset);
+__isl_export
+__isl_give isl_union_set *isl_union_set_affine_hull(
+	__isl_take isl_union_set *uset);
+__isl_export
+__isl_give isl_union_set *isl_union_set_polyhedral_hull(
+	__isl_take isl_union_set *uset);
+__isl_give isl_union_set *isl_union_set_simple_hull(
+	__isl_take isl_union_set *uset);
+__isl_export
+__isl_give isl_union_set *isl_union_set_coalesce(
+	__isl_take isl_union_set *uset);
+__isl_give isl_union_set *isl_union_set_compute_divs(
+	__isl_take isl_union_set *uset);
+__isl_export
+__isl_give isl_union_set *isl_union_set_lexmin(__isl_take isl_union_set *uset);
+__isl_export
+__isl_give isl_union_set *isl_union_set_lexmax(__isl_take isl_union_set *uset);
+
+__isl_give isl_union_set *isl_union_set_add_set(__isl_take isl_union_set *uset,
+	__isl_take isl_set *set);
+__isl_give isl_union_set *isl_union_set_union(__isl_take isl_union_set *uset1,
+	__isl_take isl_union_set *uset2);
+__isl_export
+__isl_give isl_union_set *isl_union_set_subtract(
+	__isl_take isl_union_set *uset1, __isl_take isl_union_set *uset2);
+__isl_export
+__isl_give isl_union_set *isl_union_set_intersect(
+	__isl_take isl_union_set *uset1, __isl_take isl_union_set *uset2);
+__isl_export
+__isl_give isl_union_set *isl_union_set_intersect_params(
+	__isl_take isl_union_set *uset, __isl_take isl_set *set);
+__isl_give isl_union_set *isl_union_set_product(__isl_take isl_union_set *uset1,
+	__isl_take isl_union_set *uset2);
+__isl_export
+__isl_give isl_union_set *isl_union_set_gist(__isl_take isl_union_set *uset,
+	__isl_take isl_union_set *context);
+__isl_export
+__isl_give isl_union_set *isl_union_set_gist_params(
+	__isl_take isl_union_set *uset, __isl_take isl_set *set);
+
+__isl_export
+__isl_give isl_union_set *isl_union_set_apply(
+	__isl_take isl_union_set *uset, __isl_take isl_union_map *umap);
+__isl_give isl_union_set *isl_union_set_preimage_multi_aff(
+	__isl_take isl_union_set *uset, __isl_take isl_multi_aff *ma);
+__isl_give isl_union_set *isl_union_set_preimage_pw_multi_aff(
+	__isl_take isl_union_set *uset, __isl_take isl_pw_multi_aff *pma);
+__isl_give isl_union_set *isl_union_set_preimage_union_pw_multi_aff(
+	__isl_take isl_union_set *uset,
+	__isl_take isl_union_pw_multi_aff *upma);
+
+int isl_union_set_is_params(__isl_keep isl_union_set *uset);
+__isl_export
+int isl_union_set_is_empty(__isl_keep isl_union_set *uset);
+
+__isl_export
+int isl_union_set_is_subset(__isl_keep isl_union_set *uset1,
+	__isl_keep isl_union_set *uset2);
+__isl_export
+int isl_union_set_is_equal(__isl_keep isl_union_set *uset1,
+	__isl_keep isl_union_set *uset2);
+__isl_export
+int isl_union_set_is_strict_subset(__isl_keep isl_union_set *uset1,
+	__isl_keep isl_union_set *uset2);
+
+int isl_union_set_n_set(__isl_keep isl_union_set *uset);
+__isl_export
+int isl_union_set_foreach_set(__isl_keep isl_union_set *uset,
+	int (*fn)(__isl_take isl_set *set, void *user), void *user);
+__isl_give int isl_union_set_contains(__isl_keep isl_union_set *uset,
+	__isl_keep isl_space *dim);
+__isl_give isl_set *isl_union_set_extract_set(__isl_keep isl_union_set *uset,
+	__isl_take isl_space *dim);
+__isl_give isl_set *isl_set_from_union_set(__isl_take isl_union_set *uset);
+int isl_union_set_foreach_point(__isl_keep isl_union_set *uset,
+	int (*fn)(__isl_take isl_point *pnt, void *user), void *user);
+
+__isl_give isl_basic_set *isl_union_set_sample(__isl_take isl_union_set *uset);
+
+__isl_give isl_union_set *isl_union_set_lift(__isl_take isl_union_set *uset);
+
+__isl_give isl_union_map *isl_union_set_lex_lt_union_set(
+	__isl_take isl_union_set *uset1, __isl_take isl_union_set *uset2);
+__isl_give isl_union_map *isl_union_set_lex_le_union_set(
+	__isl_take isl_union_set *uset1, __isl_take isl_union_set *uset2);
+__isl_give isl_union_map *isl_union_set_lex_gt_union_set(
+	__isl_take isl_union_set *uset1, __isl_take isl_union_set *uset2);
+__isl_give isl_union_map *isl_union_set_lex_ge_union_set(
+	__isl_take isl_union_set *uset1, __isl_take isl_union_set *uset2);
+
+__isl_give isl_union_set *isl_union_set_coefficients(
+	__isl_take isl_union_set *bset);
+__isl_give isl_union_set *isl_union_set_solutions(
+	__isl_take isl_union_set *bset);
+
+__isl_give isl_union_set *isl_union_set_read_from_file(isl_ctx *ctx,
+	FILE *input);
+__isl_constructor
+__isl_give isl_union_set *isl_union_set_read_from_str(isl_ctx *ctx,
+	const char *str);
+__isl_give char *isl_union_set_to_str(__isl_keep isl_union_set *uset);
+__isl_give isl_printer *isl_printer_print_union_set(__isl_take isl_printer *p,
+	__isl_keep isl_union_set *uset);
+void isl_union_set_dump(__isl_keep isl_union_set *uset);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#include <isl/dim.h>
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/union_set_type.h b/polly/lib/External/isl/include/isl/union_set_type.h
new file mode 100644
index 00000000000..86b2c317fa1
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/union_set_type.h
@@ -0,0 +1,6 @@
+#ifndef ISL_UNION_SET_TYPE_H
+#define ISL_UNION_SET_TYPE_H
+
+#include <isl/union_map_type.h>
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/val.h b/polly/lib/External/isl/include/isl/val.h
new file mode 100644
index 00000000000..238f0c282cf
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/val.h
@@ -0,0 +1,114 @@
+#ifndef ISL_VAL_H
+#define ISL_VAL_H
+
+#include <isl/ctx.h>
+#include <isl/list.h>
+#include <isl/multi.h>
+#include <isl/printer.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_val;
+typedef struct isl_val isl_val;
+
+ISL_DECLARE_LIST(val)
+
+struct isl_multi_val;
+typedef struct isl_multi_val isl_multi_val;
+
+ISL_DECLARE_MULTI(val)
+
+__isl_give isl_val *isl_val_zero(isl_ctx *ctx);
+__isl_give isl_val *isl_val_one(isl_ctx *ctx);
+__isl_give isl_val *isl_val_negone(isl_ctx *ctx);
+__isl_give isl_val *isl_val_nan(isl_ctx *ctx);
+__isl_give isl_val *isl_val_infty(isl_ctx *ctx);
+__isl_give isl_val *isl_val_neginfty(isl_ctx *ctx);
+__isl_give isl_val *isl_val_int_from_si(isl_ctx *ctx, long i);
+__isl_give isl_val *isl_val_int_from_ui(isl_ctx *ctx, unsigned long u);
+__isl_give isl_val *isl_val_int_from_chunks(isl_ctx *ctx, size_t n,
+	size_t size, const void *chunks);
+
+__isl_give isl_val *isl_val_copy(__isl_keep isl_val *v);
+__isl_null isl_val *isl_val_free(__isl_take isl_val *v);
+
+isl_ctx *isl_val_get_ctx(__isl_keep isl_val *val);
+long isl_val_get_num_si(__isl_keep isl_val *v);
+long isl_val_get_den_si(__isl_keep isl_val *v);
+double isl_val_get_d(__isl_keep isl_val *v);
+size_t isl_val_n_abs_num_chunks(__isl_keep isl_val *v, size_t size);
+int isl_val_get_abs_num_chunks(__isl_keep isl_val *v, size_t size,
+	void *chunks);
+
+__isl_give isl_val *isl_val_set_si(__isl_take isl_val *v, long i);
+
+__isl_give isl_val *isl_val_abs(__isl_take isl_val *v);
+__isl_give isl_val *isl_val_neg(__isl_take isl_val *v);
+__isl_give isl_val *isl_val_inv(__isl_take isl_val *v);
+__isl_give isl_val *isl_val_floor(__isl_take isl_val *v);
+__isl_give isl_val *isl_val_ceil(__isl_take isl_val *v);
+__isl_give isl_val *isl_val_trunc(__isl_take isl_val *v);
+__isl_give isl_val *isl_val_2exp(__isl_take isl_val *v);
+__isl_give isl_val *isl_val_min(__isl_take isl_val *v1, __isl_take isl_val *v2);
+__isl_give isl_val *isl_val_max(__isl_take isl_val *v1, __isl_take isl_val *v2);
+__isl_give isl_val *isl_val_add(__isl_take isl_val *v1, __isl_take isl_val *v2);
+__isl_give isl_val *isl_val_add_ui(__isl_take isl_val *v1, unsigned long v2);
+__isl_give isl_val *isl_val_sub(__isl_take isl_val *v1, __isl_take isl_val *v2);
+__isl_give isl_val *isl_val_sub_ui(__isl_take isl_val *v1, unsigned long v2);
+__isl_give isl_val *isl_val_mul(__isl_take isl_val *v1, __isl_take isl_val *v2);
+__isl_give isl_val *isl_val_mul_ui(__isl_take isl_val *v1, unsigned long v2);
+__isl_give isl_val *isl_val_div(__isl_take isl_val *v1, __isl_take isl_val *v2);
+__isl_give isl_val *isl_val_mod(__isl_take isl_val *v1, __isl_take isl_val *v2);
+__isl_give isl_val *isl_val_gcd(__isl_take isl_val *v1, __isl_take isl_val *v2);
+__isl_give isl_val *isl_val_gcdext(__isl_take isl_val *v1,
+	__isl_take isl_val *v2, __isl_give isl_val **x, __isl_give isl_val **y);
+
+int isl_val_sgn(__isl_keep isl_val *v);
+int isl_val_is_zero(__isl_keep isl_val *v);
+int isl_val_is_one(__isl_keep isl_val *v);
+int isl_val_is_negone(__isl_keep isl_val *v);
+int isl_val_is_nonneg(__isl_keep isl_val *v);
+int isl_val_is_nonpos(__isl_keep isl_val *v);
+int isl_val_is_pos(__isl_keep isl_val *v);
+int isl_val_is_neg(__isl_keep isl_val *v);
+int isl_val_is_int(__isl_keep isl_val *v);
+int isl_val_is_rat(__isl_keep isl_val *v);
+int isl_val_is_nan(__isl_keep isl_val *v);
+int isl_val_is_infty(__isl_keep isl_val *v);
+int isl_val_is_neginfty(__isl_keep isl_val *v);
+
+int isl_val_cmp_si(__isl_keep isl_val *v, long i);
+
+int isl_val_lt(__isl_keep isl_val *v1, __isl_keep isl_val *v2);
+int isl_val_le(__isl_keep isl_val *v1, __isl_keep isl_val *v2);
+int isl_val_gt(__isl_keep isl_val *v1, __isl_keep isl_val *v2);
+int isl_val_ge(__isl_keep isl_val *v1, __isl_keep isl_val *v2);
+int isl_val_eq(__isl_keep isl_val *v1, __isl_keep isl_val *v2);
+int isl_val_ne(__isl_keep isl_val *v1, __isl_keep isl_val *v2);
+int isl_val_abs_eq(__isl_keep isl_val *v1, __isl_keep isl_val *v2);
+
+int isl_val_is_divisible_by(__isl_keep isl_val *v1, __isl_keep isl_val *v2);
+
+__isl_give isl_val *isl_val_read_from_str(isl_ctx *ctx, const char *str);
+__isl_give isl_printer *isl_printer_print_val(__isl_take isl_printer *p,
+	__isl_keep isl_val *v);
+void isl_val_dump(__isl_keep isl_val *v);
+__isl_give char *isl_val_to_str(__isl_keep isl_val *v);
+
+__isl_give isl_multi_val *isl_multi_val_add_val(__isl_take isl_multi_val *mv,
+	__isl_take isl_val *v);
+__isl_give isl_multi_val *isl_multi_val_mod_val(__isl_take isl_multi_val *mv,
+	__isl_take isl_val *v);
+
+__isl_give isl_printer *isl_printer_print_multi_val(__isl_take isl_printer *p,
+	__isl_keep isl_multi_val *mv);
+void isl_multi_val_dump(__isl_keep isl_multi_val *mv);
+__isl_give char *isl_multi_val_to_str(__isl_keep isl_multi_val *mv);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/val_gmp.h b/polly/lib/External/isl/include/isl/val_gmp.h
new file mode 100644
index 00000000000..9558c1a1338
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/val_gmp.h
@@ -0,0 +1,21 @@
+#ifndef ISL_VAL_GMP_H
+#define ISL_VAL_GMP_H
+
+#include <gmp.h>
+#include <isl/val.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+__isl_give isl_val *isl_val_int_from_gmp(isl_ctx *ctx, mpz_t z);
+__isl_give isl_val *isl_val_from_gmp(isl_ctx *ctx,
+	const mpz_t n, const mpz_t d);
+int isl_val_get_num_gmp(__isl_keep isl_val *v, mpz_t z);
+int isl_val_get_den_gmp(__isl_keep isl_val *v, mpz_t z);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/vec.h b/polly/lib/External/isl/include/isl/vec.h
new file mode 100644
index 00000000000..d2fb48ccdb9
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/vec.h
@@ -0,0 +1,78 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_VEC_H
+#define ISL_VEC_H
+
+#include <stdio.h>
+
+#include <isl/ctx.h>
+#include <isl/val.h>
+#include <isl/printer.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_vec;
+typedef struct isl_vec isl_vec;
+
+__isl_give isl_vec *isl_vec_alloc(isl_ctx *ctx, unsigned size);
+__isl_give isl_vec *isl_vec_copy(__isl_keep isl_vec *vec);
+__isl_null isl_vec *isl_vec_free(__isl_take isl_vec *vec);
+
+isl_ctx *isl_vec_get_ctx(__isl_keep isl_vec *vec);
+
+int isl_vec_size(__isl_keep isl_vec *vec);
+__isl_give isl_val *isl_vec_get_element_val(__isl_keep isl_vec *vec, int pos);
+__isl_give isl_vec *isl_vec_set_element_si(__isl_take isl_vec *vec,
+	int pos, int v);
+__isl_give isl_vec *isl_vec_set_element_val(__isl_take isl_vec *vec,
+	int pos, __isl_take isl_val *v);
+
+int isl_vec_is_equal(__isl_keep isl_vec *vec1, __isl_keep isl_vec *vec2);
+int isl_vec_cmp_element(__isl_keep isl_vec *vec1, __isl_keep isl_vec *vec2,
+	int pos);
+
+void isl_vec_dump(__isl_keep isl_vec *vec);
+__isl_give isl_printer *isl_printer_print_vec(__isl_take isl_printer *printer,
+	__isl_keep isl_vec *vec);
+
+struct isl_vec *isl_vec_ceil(struct isl_vec *vec);
+struct isl_vec *isl_vec_normalize(struct isl_vec *vec);
+__isl_give isl_vec *isl_vec_set_si(__isl_take isl_vec *vec, int v);
+__isl_give isl_vec *isl_vec_set_val(__isl_take isl_vec *vec,
+	__isl_take isl_val *v);
+__isl_give isl_vec *isl_vec_clr(__isl_take isl_vec *vec);
+__isl_give isl_vec *isl_vec_neg(__isl_take isl_vec *vec);
+__isl_give isl_vec *isl_vec_add(__isl_take isl_vec *vec1,
+	__isl_take isl_vec *vec2);
+__isl_give isl_vec *isl_vec_extend(__isl_take isl_vec *vec, unsigned size);
+__isl_give isl_vec *isl_vec_zero_extend(__isl_take isl_vec *vec, unsigned size);
+__isl_give isl_vec *isl_vec_concat(__isl_take isl_vec *vec1,
+	__isl_take isl_vec *vec2);
+
+__isl_give isl_vec *isl_vec_sort(__isl_take isl_vec *vec);
+
+__isl_give isl_vec *isl_vec_read_from_file(isl_ctx *ctx, FILE *input);
+
+__isl_give isl_vec *isl_vec_drop_els(__isl_take isl_vec *vec,
+	unsigned pos, unsigned n);
+__isl_give isl_vec *isl_vec_insert_els(__isl_take isl_vec *vec,
+	unsigned pos, unsigned n);
+__isl_give isl_vec *isl_vec_insert_zero_els(__isl_take isl_vec *vec,
+	unsigned pos, unsigned n);
+__isl_give isl_vec *isl_vec_move_els(__isl_take isl_vec *vec,
+	unsigned dst_col, unsigned src_col, unsigned n);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/version.h b/polly/lib/External/isl/include/isl/version.h
new file mode 100644
index 00000000000..7f8f23d6945
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/version.h
@@ -0,0 +1,14 @@
+#ifndef ISL_VERSION_H
+#define ISL_VERSION_H
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+const char *isl_version(void);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/include/isl/vertices.h b/polly/lib/External/isl/include/isl/vertices.h
new file mode 100644
index 00000000000..7a4b353a5a9
--- /dev/null
+++ b/polly/lib/External/isl/include/isl/vertices.h
@@ -0,0 +1,47 @@
+#ifndef ISL_VERTICES_H
+#define ISL_VERTICES_H
+
+#include <isl/aff_type.h>
+#include <isl/set_type.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_external_vertex;
+typedef struct isl_external_vertex	isl_vertex;
+
+struct isl_cell;
+typedef struct isl_cell		isl_cell;
+
+struct isl_vertices;
+typedef struct isl_vertices	isl_vertices;
+
+isl_ctx *isl_vertex_get_ctx(__isl_keep isl_vertex *vertex);
+int isl_vertex_get_id(__isl_keep isl_vertex *vertex);
+__isl_give isl_basic_set *isl_vertex_get_domain(__isl_keep isl_vertex *vertex);
+__isl_give isl_multi_aff *isl_vertex_get_expr(__isl_keep isl_vertex *vertex);
+void isl_vertex_free(__isl_take isl_vertex *vertex);
+
+__isl_give isl_vertices *isl_basic_set_compute_vertices(
+	__isl_keep isl_basic_set *bset);
+isl_ctx *isl_vertices_get_ctx(__isl_keep isl_vertices *vertices);
+int isl_vertices_get_n_vertices(__isl_keep isl_vertices *vertices);
+int isl_vertices_foreach_vertex(__isl_keep isl_vertices *vertices,
+	int (*fn)(__isl_take isl_vertex *vertex, void *user), void *user);
+void isl_vertices_free(__isl_take isl_vertices *vertices);
+
+isl_ctx *isl_cell_get_ctx(__isl_keep isl_cell *cell);
+__isl_give isl_basic_set *isl_cell_get_domain(__isl_keep isl_cell *cell);
+int isl_cell_foreach_vertex(__isl_keep isl_cell *cell,
+	int (*fn)(__isl_take isl_vertex *vertex, void *user), void *user);
+void isl_cell_free(__isl_take isl_cell *cell);
+
+int isl_vertices_foreach_cell(__isl_keep isl_vertices *vertices,
+	int (*fn)(__isl_take isl_cell *cell, void *user), void *user);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/interface/Makefile.am b/polly/lib/External/isl/interface/Makefile.am
new file mode 100644
index 00000000000..0269e9113d0
--- /dev/null
+++ b/polly/lib/External/isl/interface/Makefile.am
@@ -0,0 +1,32 @@
+AUTOMAKE_OPTIONS = nostdinc
+
+noinst_PROGRAMS = extract_interface
+
+AM_CXXFLAGS = $(CLANG_CXXFLAGS)
+AM_LDFLAGS = $(CLANG_LDFLAGS)
+
+includes = -I$(top_builddir) -I$(top_srcdir) \
+	-I$(top_builddir)/include -I$(top_srcdir)/include
+
+extract_interface_CPPFLAGS = $(includes)
+extract_interface_SOURCES = \
+	python.h \
+	python.cc \
+	extract_interface.h \
+	extract_interface.cc
+extract_interface_LDADD = \
+	-lclangFrontend -lclangSerialization -lclangParse -lclangSema \
+	$(LIB_CLANG_EDIT) \
+	-lclangAnalysis -lclangAST -lclangLex -lclangBasic -lclangDriver \
+	$(CLANG_LIBS) $(CLANG_LDFLAGS)
+
+test: extract_interface
+	./extract_interface$(EXEEXT) $(includes) $(srcdir)/all.h
+
+isl.py: extract_interface isl.py.top
+	(cat $(srcdir)/isl.py.top; \
+		./extract_interface$(EXEEXT) $(includes) $(srcdir)/all.h) \
+			> isl.py
+
+dist-hook: isl.py
+	cp isl.py $(distdir)/
diff --git a/polly/lib/External/isl/interface/all.h b/polly/lib/External/isl/interface/all.h
new file mode 100644
index 00000000000..46634e2b8db
--- /dev/null
+++ b/polly/lib/External/isl/interface/all.h
@@ -0,0 +1,4 @@
+#include <isl/set.h>
+#include <isl/map.h>
+#include <isl/union_set.h>
+#include <isl/union_map.h>
diff --git a/polly/lib/External/isl/interface/extract_interface.cc b/polly/lib/External/isl/interface/extract_interface.cc
new file mode 100644
index 00000000000..fd83003329e
--- /dev/null
+++ b/polly/lib/External/isl/interface/extract_interface.cc
@@ -0,0 +1,388 @@
+/*
+ * Copyright 2011 Sven Verdoolaege. All rights reserved.
+ * 
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions
+ * are met:
+ * 
+ *    1. Redistributions of source code must retain the above copyright
+ *       notice, this list of conditions and the following disclaimer.
+ * 
+ *    2. Redistributions in binary form must reproduce the above
+ *       copyright notice, this list of conditions and the following
+ *       disclaimer in the documentation and/or other materials provided
+ *       with the distribution.
+ * 
+ * THIS SOFTWARE IS PROVIDED BY SVEN VERDOOLAEGE ''AS IS'' AND ANY
+ * EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+ * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
+ * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL SVEN VERDOOLAEGE OR
+ * CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
+ * EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+ * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA,
+ * OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
+ * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+ * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+ * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+ * 
+ * The views and conclusions contained in the software and documentation
+ * are those of the authors and should not be interpreted as
+ * representing official policies, either expressed or implied, of
+ * Sven Verdoolaege.
+ */ 
+
+#include "isl_config.h"
+
+#include <assert.h>
+#include <iostream>
+#ifdef HAVE_ADT_OWNINGPTR_H
+#include <llvm/ADT/OwningPtr.h>
+#else
+#include <memory>
+#endif
+#include <llvm/Support/raw_ostream.h>
+#include <llvm/Support/CommandLine.h>
+#include <llvm/Support/Host.h>
+#include <llvm/Support/ManagedStatic.h>
+#include <clang/AST/ASTContext.h>
+#include <clang/AST/ASTConsumer.h>
+#include <clang/Basic/FileSystemOptions.h>
+#include <clang/Basic/FileManager.h>
+#include <clang/Basic/TargetOptions.h>
+#include <clang/Basic/TargetInfo.h>
+#include <clang/Basic/Version.h>
+#include <clang/Driver/Compilation.h>
+#include <clang/Driver/Driver.h>
+#include <clang/Driver/Tool.h>
+#include <clang/Frontend/CompilerInstance.h>
+#include <clang/Frontend/CompilerInvocation.h>
+#ifdef HAVE_BASIC_DIAGNOSTICOPTIONS_H
+#include <clang/Basic/DiagnosticOptions.h>
+#else
+#include <clang/Frontend/DiagnosticOptions.h>
+#endif
+#include <clang/Frontend/TextDiagnosticPrinter.h>
+#include <clang/Frontend/Utils.h>
+#include <clang/Lex/HeaderSearch.h>
+#include <clang/Lex/Preprocessor.h>
+#include <clang/Parse/ParseAST.h>
+#include <clang/Sema/Sema.h>
+
+#include "extract_interface.h"
+#include "python.h"
+
+using namespace std;
+using namespace clang;
+using namespace clang::driver;
+
+#ifdef HAVE_ADT_OWNINGPTR_H
+#define unique_ptr	llvm::OwningPtr
+#endif
+
+static llvm::cl::opt<string> InputFilename(llvm::cl::Positional,
+			llvm::cl::Required, llvm::cl::desc("<input file>"));
+static llvm::cl::list<string> Includes("I",
+			llvm::cl::desc("Header search path"),
+			llvm::cl::value_desc("path"), llvm::cl::Prefix);
+
+static const char *ResourceDir =
+	CLANG_PREFIX "/lib/clang/" CLANG_VERSION_STRING;
+
+/* Does decl have an attribute of the following form?
+ *
+ *	__attribute__((annotate("name")))
+ */
+bool has_annotation(Decl *decl, const char *name)
+{
+	if (!decl->hasAttrs())
+		return false;
+
+	AttrVec attrs = decl->getAttrs();
+	for (AttrVec::const_iterator i = attrs.begin() ; i != attrs.end(); ++i) {
+		const AnnotateAttr *ann = dyn_cast<AnnotateAttr>(*i);
+		if (!ann)
+			continue;
+		if (ann->getAnnotation().str() == name)
+			return true;
+	}
+
+	return false;
+}
+
+/* Is decl marked as exported?
+ */
+static bool is_exported(Decl *decl)
+{
+	return has_annotation(decl, "isl_export");
+}
+
+/* Collect all types and functions that are annotated "isl_export"
+ * in "types" and "function".
+ *
+ * We currently only consider single declarations.
+ */
+struct MyASTConsumer : public ASTConsumer {
+	set<RecordDecl *> types;
+	set<FunctionDecl *> functions;
+
+	virtual HandleTopLevelDeclReturn HandleTopLevelDecl(DeclGroupRef D) {
+		Decl *decl;
+
+		if (!D.isSingleDecl())
+			return HandleTopLevelDeclContinue;
+		decl = D.getSingleDecl();
+		if (!is_exported(decl))
+			return HandleTopLevelDeclContinue;
+		switch (decl->getKind()) {
+		case Decl::Record:
+			types.insert(cast<RecordDecl>(decl));
+			break;
+		case Decl::Function:
+			functions.insert(cast<FunctionDecl>(decl));
+			break;
+		default:
+			break;
+		}
+		return HandleTopLevelDeclContinue;
+	}
+};
+
+#ifdef USE_ARRAYREF
+
+#ifdef HAVE_CXXISPRODUCTION
+static Driver *construct_driver(const char *binary, DiagnosticsEngine &Diags)
+{
+	return new Driver(binary, llvm::sys::getDefaultTargetTriple(),
+			    "", false, false, Diags);
+}
+#elif defined(HAVE_ISPRODUCTION)
+static Driver *construct_driver(const char *binary, DiagnosticsEngine &Diags)
+{
+	return new Driver(binary, llvm::sys::getDefaultTargetTriple(),
+			    "", false, Diags);
+}
+#elif defined(DRIVER_CTOR_TAKES_DEFAULTIMAGENAME)
+static Driver *construct_driver(const char *binary, DiagnosticsEngine &Diags)
+{
+	return new Driver(binary, llvm::sys::getDefaultTargetTriple(),
+			    "", Diags);
+}
+#else
+static Driver *construct_driver(const char *binary, DiagnosticsEngine &Diags)
+{
+	return new Driver(binary, llvm::sys::getDefaultTargetTriple(), Diags);
+}
+#endif
+
+/* Create a CompilerInvocation object that stores the command line
+ * arguments constructed by the driver.
+ * The arguments are mainly useful for setting up the system include
+ * paths on newer clangs and on some platforms.
+ */
+static CompilerInvocation *construct_invocation(const char *filename,
+	DiagnosticsEngine &Diags)
+{
+	const char *binary = CLANG_PREFIX"/bin/clang";
+	const unique_ptr<Driver> driver(construct_driver(binary, Diags));
+	std::vector<const char *> Argv;
+	Argv.push_back(binary);
+	Argv.push_back(filename);
+	const unique_ptr<Compilation> compilation(
+		driver->BuildCompilation(llvm::ArrayRef<const char *>(Argv)));
+	JobList &Jobs = compilation->getJobs();
+
+	Command *cmd = cast<Command>(*Jobs.begin());
+	if (strcmp(cmd->getCreator().getName(), "clang"))
+		return NULL;
+
+	const ArgStringList *args = &cmd->getArguments();
+
+	CompilerInvocation *invocation = new CompilerInvocation;
+	CompilerInvocation::CreateFromArgs(*invocation, args->data() + 1,
+						args->data() + args->size(),
+						Diags);
+	return invocation;
+}
+
+#else
+
+static CompilerInvocation *construct_invocation(const char *filename,
+	DiagnosticsEngine &Diags)
+{
+	return NULL;
+}
+
+#endif
+
+#ifdef HAVE_BASIC_DIAGNOSTICOPTIONS_H
+
+static TextDiagnosticPrinter *construct_printer(void)
+{
+	return new TextDiagnosticPrinter(llvm::errs(), new DiagnosticOptions());
+}
+
+#else
+
+static TextDiagnosticPrinter *construct_printer(void)
+{
+	DiagnosticOptions DO;
+	return new TextDiagnosticPrinter(llvm::errs(), DO);
+}
+
+#endif
+
+#ifdef CREATETARGETINFO_TAKES_SHARED_PTR
+
+static TargetInfo *create_target_info(CompilerInstance *Clang,
+	DiagnosticsEngine &Diags)
+{
+	shared_ptr<TargetOptions> TO = Clang->getInvocation().TargetOpts;
+	TO->Triple = llvm::sys::getDefaultTargetTriple();
+	return TargetInfo::CreateTargetInfo(Diags, TO);
+}
+
+#elif defined(CREATETARGETINFO_TAKES_POINTER)
+
+static TargetInfo *create_target_info(CompilerInstance *Clang,
+	DiagnosticsEngine &Diags)
+{
+	TargetOptions &TO = Clang->getTargetOpts();
+	TO.Triple = llvm::sys::getDefaultTargetTriple();
+	return TargetInfo::CreateTargetInfo(Diags, &TO);
+}
+
+#else
+
+static TargetInfo *create_target_info(CompilerInstance *Clang,
+	DiagnosticsEngine &Diags)
+{
+	TargetOptions &TO = Clang->getTargetOpts();
+	TO.Triple = llvm::sys::getDefaultTargetTriple();
+	return TargetInfo::CreateTargetInfo(Diags, TO);
+}
+
+#endif
+
+#ifdef CREATEDIAGNOSTICS_TAKES_ARG
+
+static void create_diagnostics(CompilerInstance *Clang)
+{
+	Clang->createDiagnostics(0, NULL, construct_printer());
+}
+
+#else
+
+static void create_diagnostics(CompilerInstance *Clang)
+{
+	Clang->createDiagnostics(construct_printer());
+}
+
+#endif
+
+#ifdef CREATEPREPROCESSOR_TAKES_TUKIND
+
+static void create_preprocessor(CompilerInstance *Clang)
+{
+	Clang->createPreprocessor(TU_Complete);
+}
+
+#else
+
+static void create_preprocessor(CompilerInstance *Clang)
+{
+	Clang->createPreprocessor();
+}
+
+#endif
+
+#ifdef ADDPATH_TAKES_4_ARGUMENTS
+
+void add_path(HeaderSearchOptions &HSO, string Path)
+{
+	HSO.AddPath(Path, frontend::Angled, false, false);
+}
+
+#else
+
+void add_path(HeaderSearchOptions &HSO, string Path)
+{
+	HSO.AddPath(Path, frontend::Angled, true, false, false);
+}
+
+#endif
+
+#ifdef HAVE_SETMAINFILEID
+
+static void create_main_file_id(SourceManager &SM, const FileEntry *file)
+{
+	SM.setMainFileID(SM.createFileID(file, SourceLocation(),
+					SrcMgr::C_User));
+}
+
+#else
+
+static void create_main_file_id(SourceManager &SM, const FileEntry *file)
+{
+	SM.createMainFileID(file);
+}
+
+#endif
+
+int main(int argc, char *argv[])
+{
+	llvm::cl::ParseCommandLineOptions(argc, argv);
+
+	CompilerInstance *Clang = new CompilerInstance();
+	create_diagnostics(Clang);
+	DiagnosticsEngine &Diags = Clang->getDiagnostics();
+	Diags.setSuppressSystemWarnings(true);
+	CompilerInvocation *invocation =
+		construct_invocation(InputFilename.c_str(), Diags);
+	if (invocation)
+		Clang->setInvocation(invocation);
+	Clang->createFileManager();
+	Clang->createSourceManager(Clang->getFileManager());
+	TargetInfo *target = create_target_info(Clang, Diags);
+	Clang->setTarget(target);
+	CompilerInvocation::setLangDefaults(Clang->getLangOpts(), IK_C,
+					    LangStandard::lang_unspecified);
+	HeaderSearchOptions &HSO = Clang->getHeaderSearchOpts();
+	LangOptions &LO = Clang->getLangOpts();
+	PreprocessorOptions &PO = Clang->getPreprocessorOpts();
+	HSO.ResourceDir = ResourceDir;
+
+	for (int i = 0; i < Includes.size(); ++i)
+		add_path(HSO, Includes[i]);
+
+	PO.addMacroDef("__isl_give=__attribute__((annotate(\"isl_give\")))");
+	PO.addMacroDef("__isl_keep=__attribute__((annotate(\"isl_keep\")))");
+	PO.addMacroDef("__isl_take=__attribute__((annotate(\"isl_take\")))");
+	PO.addMacroDef("__isl_export=__attribute__((annotate(\"isl_export\")))");
+	PO.addMacroDef("__isl_constructor=__attribute__((annotate(\"isl_constructor\"))) __attribute__((annotate(\"isl_export\")))");
+	PO.addMacroDef("__isl_subclass(super)=__attribute__((annotate(\"isl_subclass(\" #super \")\"))) __attribute__((annotate(\"isl_export\")))");
+
+	create_preprocessor(Clang);
+	Preprocessor &PP = Clang->getPreprocessor();
+
+	PP.getBuiltinInfo().InitializeBuiltins(PP.getIdentifierTable(), LO);
+
+	const FileEntry *file = Clang->getFileManager().getFile(InputFilename);
+	assert(file);
+	create_main_file_id(Clang->getSourceManager(), file);
+
+	Clang->createASTContext();
+	MyASTConsumer consumer;
+	Sema *sema = new Sema(PP, Clang->getASTContext(), consumer);
+
+	Diags.getClient()->BeginSourceFile(LO, &PP);
+	ParseAST(*sema);
+	Diags.getClient()->EndSourceFile();
+
+	generate_python(consumer.types, consumer.functions);
+
+	delete sema;
+	delete Clang;
+	llvm::llvm_shutdown();
+
+	return 0;
+}
diff --git a/polly/lib/External/isl/interface/extract_interface.h b/polly/lib/External/isl/interface/extract_interface.h
new file mode 100644
index 00000000000..b6788f1343d
--- /dev/null
+++ b/polly/lib/External/isl/interface/extract_interface.h
@@ -0,0 +1,3 @@
+#include <clang/AST/Decl.h>
+
+bool has_annotation(clang::Decl *decl, const char *name);
diff --git a/polly/lib/External/isl/interface/isl.py.top b/polly/lib/External/isl/interface/isl.py.top
new file mode 100644
index 00000000000..6d4ae49eb9b
--- /dev/null
+++ b/polly/lib/External/isl/interface/isl.py.top
@@ -0,0 +1,29 @@
+from ctypes import *
+
+isl = cdll.LoadLibrary("libisl.so")
+libc = cdll.LoadLibrary("libc.so.6")
+
+class Error(Exception):
+    pass
+
+class Context:
+    defaultInstance = None
+
+    def __init__(self):
+        ptr = isl.isl_ctx_alloc()
+        self.ptr = ptr
+
+    def __del__(self):
+        isl.isl_ctx_free(self)
+
+    def from_param(self):
+        return self.ptr
+
+    @staticmethod
+    def getDefaultInstance():
+        if Context.defaultInstance == None:
+            Context.defaultInstance = Context()
+        return Context.defaultInstance
+
+isl.isl_ctx_alloc.restype = c_void_p
+isl.isl_ctx_free.argtypes = [Context]
diff --git a/polly/lib/External/isl/interface/python.cc b/polly/lib/External/isl/interface/python.cc
new file mode 100644
index 00000000000..bb5f708028c
--- /dev/null
+++ b/polly/lib/External/isl/interface/python.cc
@@ -0,0 +1,519 @@
+/*
+ * Copyright 2011 Sven Verdoolaege. All rights reserved.
+ * 
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions
+ * are met:
+ * 
+ *    1. Redistributions of source code must retain the above copyright
+ *       notice, this list of conditions and the following disclaimer.
+ * 
+ *    2. Redistributions in binary form must reproduce the above
+ *       copyright notice, this list of conditions and the following
+ *       disclaimer in the documentation and/or other materials provided
+ *       with the distribution.
+ * 
+ * THIS SOFTWARE IS PROVIDED BY SVEN VERDOOLAEGE ''AS IS'' AND ANY
+ * EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+ * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
+ * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL SVEN VERDOOLAEGE OR
+ * CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
+ * EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+ * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA,
+ * OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
+ * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+ * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+ * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+ * 
+ * The views and conclusions contained in the software and documentation
+ * are those of the authors and should not be interpreted as
+ * representing official policies, either expressed or implied, of
+ * Sven Verdoolaege.
+ */ 
+
+#include "isl_config.h"
+
+#include <stdio.h>
+#include <iostream>
+#include <map>
+#include <clang/AST/Attr.h>
+#include "extract_interface.h"
+#include "python.h"
+
+/* Is the given type declaration marked as being a subtype of some other
+ * type?  If so, return that other type in "super".
+ */
+static bool is_subclass(RecordDecl *decl, string &super)
+{
+	if (!decl->hasAttrs())
+		return false;
+
+	string sub = "isl_subclass";
+	size_t len = sub.length();
+	AttrVec attrs = decl->getAttrs();
+	for (AttrVec::const_iterator i = attrs.begin() ; i != attrs.end(); ++i) {
+		const AnnotateAttr *ann = dyn_cast<AnnotateAttr>(*i);
+		if (!ann)
+			continue;
+		string s = ann->getAnnotation().str();
+		if (s.substr(0, len) == sub) {
+			super = s.substr(len + 1, s.length() - len  - 2);
+			return true;
+		}
+	}
+
+	return false;
+}
+
+/* Is decl marked as a constructor?
+ */
+static bool is_constructor(Decl *decl)
+{
+	return has_annotation(decl, "isl_constructor");
+}
+
+/* Is decl marked as consuming a reference?
+ */
+static bool takes(Decl *decl)
+{
+	return has_annotation(decl, "isl_take");
+}
+
+/* isl_class collects all constructors and methods for an isl "class".
+ * "name" is the name of the class.
+ * "type" is the declaration that introduces the type.
+ */
+struct isl_class {
+	string name;
+	RecordDecl *type;
+	set<FunctionDecl *> constructors;
+	set<FunctionDecl *> methods;
+
+	void print(map<string, isl_class> &classes, set<string> &done);
+	void print_constructor(FunctionDecl *method);
+	void print_method(FunctionDecl *method, bool subclass, string super);
+};
+
+/* Return the class that has a name that matches the initial part
+ * of the namd of function "fd".
+ */
+static isl_class &method2class(map<string, isl_class> &classes,
+	FunctionDecl *fd)
+{
+	string best;
+	map<string, isl_class>::iterator ci;
+	string name = fd->getNameAsString();
+
+	for (ci = classes.begin(); ci != classes.end(); ++ci) {
+		if (name.substr(0, ci->first.length()) == ci->first)
+			best = ci->first;
+	}
+
+	return classes[best];
+}
+
+/* Is "type" the type "isl_ctx *"?
+ */
+static bool is_isl_ctx(QualType type)
+{
+	if (!type->isPointerType())
+		return 0;
+	type = type->getPointeeType();
+	if (type.getAsString() != "isl_ctx")
+		return false;
+
+	return true;
+}
+
+/* Is the first argument of "fd" of type "isl_ctx *"?
+ */
+static bool first_arg_is_isl_ctx(FunctionDecl *fd)
+{
+	ParmVarDecl *param;
+
+	if (fd->getNumParams() < 1)
+		return false;
+
+	param = fd->getParamDecl(0);
+	return is_isl_ctx(param->getOriginalType());
+}
+
+/* Is "type" that of a pointer to an isl_* structure?
+ */
+static bool is_isl_type(QualType type)
+{
+	if (type->isPointerType()) {
+		string s = type->getPointeeType().getAsString();
+		return s.substr(0, 4) == "isl_";
+	}
+
+	return false;
+}
+
+/* Is "type" that of a pointer to a function?
+ */
+static bool is_callback(QualType type)
+{
+	if (!type->isPointerType())
+		return false;
+	type = type->getPointeeType();
+	return type->isFunctionType();
+}
+
+/* Is "type" that of "char *" of "const char *"?
+ */
+static bool is_string(QualType type)
+{
+	if (type->isPointerType()) {
+		string s = type->getPointeeType().getAsString();
+		return s == "const char" || s == "char";
+	}
+
+	return false;
+}
+
+/* Return the name of the type that "type" points to.
+ * The input "type" is assumed to be a pointer type.
+ */
+static string extract_type(QualType type)
+{
+	if (type->isPointerType())
+		return type->getPointeeType().getAsString();
+	assert(0);
+}
+
+/* Drop the "isl_" initial part of the type name "name".
+ */
+static string type2python(string name)
+{
+	return name.substr(4);
+}
+
+/* Construct a wrapper for a callback argument (at position "arg").
+ * Assign the wrapper to "cb".  We assume here that a function call
+ * has at most one callback argument.
+ *
+ * The wrapper converts the arguments of the callback to python types.
+ * If any exception is thrown, the wrapper keeps track of it in exc_info[0]
+ * and returns -1.  Otherwise the wrapper returns 0.
+ */
+static void print_callback(QualType type, int arg)
+{
+	const FunctionProtoType *fn = type->getAs<FunctionProtoType>();
+	unsigned n_arg = fn->getNumArgs();
+
+	printf("        exc_info = [None]\n");
+	printf("        fn = CFUNCTYPE(c_int");
+	for (int i = 0; i < n_arg - 1; ++i) {
+		QualType arg_type = fn->getArgType(i);
+		assert(is_isl_type(arg_type));
+		printf(", c_void_p");
+	}
+	printf(", c_void_p)\n");
+	printf("        def cb_func(");
+	for (int i = 0; i < n_arg; ++i) {
+		if (i)
+			printf(", ");
+		printf("cb_arg%d", i);
+	}
+	printf("):\n");
+	for (int i = 0; i < n_arg - 1; ++i) {
+		string arg_type;
+		arg_type = type2python(extract_type(fn->getArgType(i)));
+		printf("            cb_arg%d = %s(ctx=arg0.ctx, ptr=cb_arg%d)\n",
+			i, arg_type.c_str(), i);
+	}
+	printf("            try:\n");
+	printf("                arg%d(", arg);
+	for (int i = 0; i < n_arg - 1; ++i) {
+		if (i)
+			printf(", ");
+		printf("cb_arg%d", i);
+	}
+	printf(")\n");
+	printf("            except:\n");
+	printf("                import sys\n");
+	printf("                exc_info[0] = sys.exc_info()\n");
+	printf("                return -1\n");
+	printf("            return 0\n");
+	printf("        cb = fn(cb_func)\n");
+}
+
+/* Print a python method corresponding to the C function "method".
+ * "subclass" is set if the method belongs to a class that is a subclass
+ * of some other class ("super").
+ *
+ * If the function has a callback argument, then it also has a "user"
+ * argument.  Since Python has closures, there is no need for such
+ * a user argument in the Python interface, so we simply drop it.
+ * We also create a wrapper ("cb") for the callback.
+ *
+ * For each argument of the function that refers to an isl structure,
+ * including the object on which the method is called,
+ * we check if the corresponding actual argument is of the right type.
+ * If not, we try to convert it to the right type.
+ * It that doesn't work and if subclass is set, we try to convert self
+ * to the type of the superclass and call the corresponding method.
+ *
+ * If the function consumes a reference, then we pass it a copy of
+ * the actual argument.
+ */
+void isl_class::print_method(FunctionDecl *method, bool subclass, string super)
+{
+	string fullname = method->getName();
+	string cname = fullname.substr(name.length() + 1);
+	int num_params = method->getNumParams();
+	int drop_user = 0;
+
+	for (int i = 1; i < num_params; ++i) {
+		ParmVarDecl *param = method->getParamDecl(i);
+		QualType type = param->getOriginalType();
+		if (is_callback(type))
+			drop_user = 1;
+	}
+
+	printf("    def %s(arg0", cname.c_str());
+	for (int i = 1; i < num_params - drop_user; ++i)
+		printf(", arg%d", i);
+	printf("):\n");
+
+	for (int i = 0; i < num_params; ++i) {
+		ParmVarDecl *param = method->getParamDecl(i);
+		string type;
+		if (!is_isl_type(param->getOriginalType()))
+			continue;
+		type = type2python(extract_type(param->getOriginalType()));
+		printf("        try:\n");
+		printf("            if not arg%d.__class__ is %s:\n",
+			i, type.c_str());
+		printf("                arg%d = %s(arg%d)\n",
+			i, type.c_str(), i);
+		printf("        except:\n");
+		if (i > 0 && subclass) {
+			printf("            return %s(arg0).%s(",
+				type2python(super).c_str(), cname.c_str());
+			for (int i = 1; i < num_params - drop_user; ++i) {
+				if (i != 1)
+					printf(", ");
+				printf("arg%d", i);
+			}
+			printf(")\n");
+		} else
+			printf("            raise\n");
+	}
+	for (int i = 1; i < num_params; ++i) {
+		ParmVarDecl *param = method->getParamDecl(i);
+		QualType type = param->getOriginalType();
+		if (!is_callback(type))
+			continue;
+		print_callback(type->getPointeeType(), i);
+	}
+	printf("        res = isl.%s(", fullname.c_str());
+	if (takes(method->getParamDecl(0)))
+		printf("isl.%s_copy(arg0.ptr)", name.c_str());
+	else
+		printf("arg0.ptr");
+	for (int i = 1; i < num_params - drop_user; ++i) {
+		ParmVarDecl *param = method->getParamDecl(i);
+		QualType type = param->getOriginalType();
+		if (is_callback(type))
+			printf(", cb");
+		else if (takes(param)) {
+			string type_s = extract_type(type);
+			printf(", isl.%s_copy(arg%d.ptr)", type_s.c_str(), i);
+		} else
+			printf(", arg%d.ptr", i);
+	}
+	if (drop_user)
+		printf(", None");
+	printf(")\n");
+
+	if (is_isl_type(method->getReturnType())) {
+		string type;
+		type = type2python(extract_type(method->getReturnType()));
+		printf("        return %s(ctx=arg0.ctx, ptr=res)\n",
+			type.c_str());
+	} else {
+		if (drop_user) {
+			printf("        if exc_info[0] != None:\n");
+			printf("            raise exc_info[0][0], "
+				"exc_info[0][1], exc_info[0][2]\n");
+		}
+		printf("        return res\n");
+	}
+}
+
+/* Print part of the constructor for this isl_class.
+ *
+ * In particular, check if the actual arguments correspond to the
+ * formal arguments of "cons" and if so call "cons" and put the
+ * result in self.ptr and a reference to the default context in self.ctx.
+ *
+ * If the function consumes a reference, then we pass it a copy of
+ * the actual argument.
+ */
+void isl_class::print_constructor(FunctionDecl *cons)
+{
+	string fullname = cons->getName();
+	string cname = fullname.substr(name.length() + 1);
+	int num_params = cons->getNumParams();
+	int drop_ctx = first_arg_is_isl_ctx(cons);
+
+	printf("        if len(args) == %d", num_params - drop_ctx);
+	for (int i = drop_ctx; i < num_params; ++i) {
+		ParmVarDecl *param = cons->getParamDecl(i);
+		if (is_isl_type(param->getOriginalType())) {
+			string type;
+			type = extract_type(param->getOriginalType());
+			type = type2python(type);
+			printf(" and args[%d].__class__ is %s",
+				i - drop_ctx, type.c_str());
+		} else
+			printf(" and type(args[%d]) == str", i - drop_ctx);
+	}
+	printf(":\n");
+	printf("            self.ctx = Context.getDefaultInstance()\n");
+	printf("            self.ptr = isl.%s(", fullname.c_str());
+	if (drop_ctx)
+		printf("self.ctx");
+	for (int i = drop_ctx; i < num_params; ++i) {
+		ParmVarDecl *param = cons->getParamDecl(i);
+		if (i)
+			printf(", ");
+		if (is_isl_type(param->getOriginalType())) {
+			if (takes(param)) {
+				string type;
+				type = extract_type(param->getOriginalType());
+				printf("isl.%s_copy(args[%d].ptr)",
+					type.c_str(), i - drop_ctx);
+			} else
+				printf("args[%d].ptr", i - drop_ctx);
+		} else
+			printf("args[%d]", i - drop_ctx);
+	}
+	printf(")\n");
+	printf("            return\n");
+}
+
+/* Print out the definition of this isl_class.
+ *
+ * We first check if this isl_class is a subclass of some other class.
+ * If it is, we make sure the superclass is printed out first.
+ *
+ * Then we print a constructor with several cases, one for constructing
+ * a Python object from a return value and one for each function that
+ * was marked as a constructor.
+ *
+ * Next, we print out some common methods and the methods corresponding
+ * to functions that are not marked as constructors.
+ *
+ * Finally, we tell ctypes about the types of the arguments of the
+ * constructor functions and the return types of those function returning
+ * an isl object.
+ */
+void isl_class::print(map<string, isl_class> &classes, set<string> &done)
+{
+	string super;
+	string p_name = type2python(name);
+	set<FunctionDecl *>::iterator in;
+	bool subclass = is_subclass(type, super);
+
+	if (subclass && done.find(super) == done.end())
+		classes[super].print(classes, done);
+	done.insert(name);
+
+	printf("\n");
+	printf("class %s", p_name.c_str());
+	if (subclass)
+		printf("(%s)", type2python(super).c_str());
+	printf(":\n");
+	printf("    def __init__(self, *args, **keywords):\n");
+
+	printf("        if \"ptr\" in keywords:\n");
+	printf("            self.ctx = keywords[\"ctx\"]\n");
+	printf("            self.ptr = keywords[\"ptr\"]\n");
+	printf("            return\n");
+
+	for (in = constructors.begin(); in != constructors.end(); ++in)
+		print_constructor(*in);
+	printf("        raise Error\n");
+	printf("    def __del__(self):\n");
+	printf("        if hasattr(self, 'ptr'):\n");
+	printf("            isl.%s_free(self.ptr)\n", name.c_str());
+	printf("    def __str__(self):\n");
+	printf("        ptr = isl.%s_to_str(self.ptr)\n", name.c_str());
+	printf("        res = str(cast(ptr, c_char_p).value)\n");
+	printf("        libc.free(ptr)\n");
+	printf("        return res\n");
+	printf("    def __repr__(self):\n");
+	printf("        return 'isl.%s(\"%%s\")' %% str(self)\n", p_name.c_str());
+
+	for (in = methods.begin(); in != methods.end(); ++in)
+		print_method(*in, subclass, super);
+
+	printf("\n");
+	for (in = constructors.begin(); in != constructors.end(); ++in) {
+		string fullname = (*in)->getName();
+		printf("isl.%s.restype = c_void_p\n", fullname.c_str());
+		printf("isl.%s.argtypes = [", fullname.c_str());
+		for (int i = 0; i < (*in)->getNumParams(); ++i) {
+			ParmVarDecl *param = (*in)->getParamDecl(i);
+			QualType type = param->getOriginalType();
+			if (i)
+				printf(", ");
+			if (is_isl_ctx(type))
+				printf("Context");
+			else if (is_isl_type(type))
+				printf("c_void_p");
+			else if (is_string(type))
+				printf("c_char_p");
+			else
+				printf("c_int");
+		}
+		printf("]\n");
+	}
+	for (in = methods.begin(); in != methods.end(); ++in) {
+		string fullname = (*in)->getName();
+		if (is_isl_type((*in)->getReturnType()))
+			printf("isl.%s.restype = c_void_p\n", fullname.c_str());
+	}
+	printf("isl.%s_free.argtypes = [c_void_p]\n", name.c_str());
+	printf("isl.%s_to_str.argtypes = [c_void_p]\n", name.c_str());
+	printf("isl.%s_to_str.restype = POINTER(c_char)\n", name.c_str());
+}
+
+/* Generate a python interface based on the extracted types and functions.
+ * We first collect all functions that belong to a certain type,
+ * separating constructors from regular methods.
+ *
+ * Then we print out each class in turn.  If one of these is a subclass
+ * of some other class, it will make sure the superclass is printed out first.
+ */
+void generate_python(set<RecordDecl *> &types, set<FunctionDecl *> functions)
+{
+	map<string, isl_class> classes;
+	map<string, isl_class>::iterator ci;
+	set<string> done;
+
+	set<RecordDecl *>::iterator it;
+	for (it = types.begin(); it != types.end(); ++it) {
+		RecordDecl *decl = *it;
+		string name = decl->getName();
+		classes[name].name = name;
+		classes[name].type = decl;
+	}
+
+	set<FunctionDecl *>::iterator in;
+	for (in = functions.begin(); in != functions.end(); ++in) {
+		isl_class &c = method2class(classes, *in);
+		if (is_constructor(*in))
+			c.constructors.insert(*in);
+		else
+			c.methods.insert(*in);
+	}
+
+	for (ci = classes.begin(); ci != classes.end(); ++ci) {
+		if (done.find(ci->first) == done.end())
+			ci->second.print(classes, done);
+	}
+}
diff --git a/polly/lib/External/isl/interface/python.h b/polly/lib/External/isl/interface/python.h
new file mode 100644
index 00000000000..0f6e8ec5b2f
--- /dev/null
+++ b/polly/lib/External/isl/interface/python.h
@@ -0,0 +1,7 @@
+#include <set>
+#include <clang/AST/Decl.h>
+
+using namespace std;
+using namespace clang;
+
+void generate_python(set<RecordDecl *> &types, set<FunctionDecl *> functions);
diff --git a/polly/lib/External/isl/isl.py b/polly/lib/External/isl/isl.py
new file mode 100644
index 00000000000..6382f6384e7
--- /dev/null
+++ b/polly/lib/External/isl/isl.py
@@ -0,0 +1,100 @@
+import gdb
+import re
+
+# GDB Pretty Printers for most isl objects
+class IslObjectPrinter:
+	"""Print an isl object"""
+	def __init__ (self, val, type):
+                self.val = val
+                self.type = type
+
+	def to_string (self):
+                # Cast val to a void pointer to stop gdb using this pretty
+                # printer for the pointer which would lead to an infinite loop.
+                void_ptr = gdb.lookup_type('void').pointer()
+                value = str(self.val.cast(void_ptr))
+                printer = gdb.parse_and_eval("isl_printer_to_str(isl_"
+                                             + str(self.type)
+					     + "_get_ctx(" + value + "))")
+                printer = gdb.parse_and_eval("isl_printer_print_"
+                                             + str(self.type) + "("
+                                             + str(printer) + ", "
+                                             + value + ")")
+                string = gdb.parse_and_eval("(char*)isl_printer_get_str("
+                                            + str(printer) + ")")
+                gdb.parse_and_eval("isl_printer_free(" + str(printer) + ")")
+                return string
+
+	def display_hint (self):
+                return 'string'
+
+class IslIntPrinter:
+	"""Print an isl_int """
+	def __init__ (self, val):
+                self.val = val
+
+	def to_string (self):
+                # Cast val to a void pointer to stop gdb using this pretty
+                # printer for the pointer which would lead to an infinite loop.
+                void_ptr = gdb.lookup_type('void').pointer()
+                value = str(self.val.cast(void_ptr))
+
+                context = gdb.parse_and_eval("isl_ctx_alloc()")
+                printer = gdb.parse_and_eval("isl_printer_to_str("
+                                             + str(context) + ")")
+                printer = gdb.parse_and_eval("isl_printer_print_isl_int("
+                                             + str(printer) + ", "
+                                             + value + ")")
+                string = gdb.parse_and_eval("(char*)isl_printer_get_str("
+                                            + str(printer) + ")")
+                gdb.parse_and_eval("isl_printer_free(" + str(printer) + ")")
+                gdb.parse_and_eval("isl_ctx_free(" + str(context) + ")")
+                return string
+
+	def display_hint (self):
+                return 'string'
+
+class IslPrintCommand (gdb.Command):
+        """Print an isl value."""
+        def __init__ (self):
+                super (IslPrintCommand, self).__init__ ("islprint",
+                                                        gdb.COMMAND_OBSCURE)
+        def invoke (self, arg, from_tty):
+                arg = gdb.parse_and_eval(arg);
+                printer = str_lookup_function(arg)
+
+                if printer == None:
+                        print "No isl printer for this type"
+                        return
+
+                print printer.to_string()
+
+IslPrintCommand()
+
+def str_lookup_function (val):
+	if val.type.code != gdb.TYPE_CODE_PTR:
+		if str(val.type) == "isl_int":
+			return IslIntPrinter(val)
+                else:
+                        return None
+
+	lookup_tag = val.type.target()
+	regex = re.compile ("^isl_(.*)$")
+
+	if lookup_tag == None:
+		return None
+
+	m = regex.match (str(lookup_tag))
+
+	if m:
+                # Those types of printers defined in isl.
+                if m.group(1) in ["basic_set", "set", "union_set", "basic_map",
+                                  "map", "union_map", "qpolynomial",
+                                  "pw_qpolynomial", "pw_qpolynomial_fold",
+                                  "union_pw_qpolynomial",
+                                  "union_pw_qpolynomial_fold"]:
+                        return IslObjectPrinter(val, m.group(1))
+        return None
+
+# Do not register the pretty printer.
+# gdb.current_objfile().pretty_printers.append(str_lookup_function)
diff --git a/polly/lib/External/isl/isl_aff.c b/polly/lib/External/isl/isl_aff.c
new file mode 100644
index 00000000000..7c785047635
--- /dev/null
+++ b/polly/lib/External/isl/isl_aff.c
@@ -0,0 +1,6809 @@
+/*
+ * Copyright 2011      INRIA Saclay
+ * Copyright 2011      Sven Verdoolaege
+ * Copyright 2012-2014 Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+#include <isl_ctx_private.h>
+#define ISL_DIM_H
+#include <isl_map_private.h>
+#include <isl_union_map_private.h>
+#include <isl_aff_private.h>
+#include <isl_space_private.h>
+#include <isl_local_space_private.h>
+#include <isl_vec_private.h>
+#include <isl_mat_private.h>
+#include <isl/constraint.h>
+#include <isl_seq.h>
+#include <isl/set.h>
+#include <isl_val_private.h>
+#include <isl/deprecated/aff_int.h>
+#include <isl_config.h>
+
+#undef BASE
+#define BASE aff
+
+#include <isl_list_templ.c>
+
+#undef BASE
+#define BASE pw_aff
+
+#include <isl_list_templ.c>
+
+__isl_give isl_aff *isl_aff_alloc_vec(__isl_take isl_local_space *ls,
+	__isl_take isl_vec *v)
+{
+	isl_aff *aff;
+
+	if (!ls || !v)
+		goto error;
+
+	aff = isl_calloc_type(v->ctx, struct isl_aff);
+	if (!aff)
+		goto error;
+
+	aff->ref = 1;
+	aff->ls = ls;
+	aff->v = v;
+
+	return aff;
+error:
+	isl_local_space_free(ls);
+	isl_vec_free(v);
+	return NULL;
+}
+
+__isl_give isl_aff *isl_aff_alloc(__isl_take isl_local_space *ls)
+{
+	isl_ctx *ctx;
+	isl_vec *v;
+	unsigned total;
+
+	if (!ls)
+		return NULL;
+
+	ctx = isl_local_space_get_ctx(ls);
+	if (!isl_local_space_divs_known(ls))
+		isl_die(ctx, isl_error_invalid, "local space has unknown divs",
+			goto error);
+	if (!isl_local_space_is_set(ls))
+		isl_die(ctx, isl_error_invalid,
+			"domain of affine expression should be a set",
+			goto error);
+
+	total = isl_local_space_dim(ls, isl_dim_all);
+	v = isl_vec_alloc(ctx, 1 + 1 + total);
+	return isl_aff_alloc_vec(ls, v);
+error:
+	isl_local_space_free(ls);
+	return NULL;
+}
+
+__isl_give isl_aff *isl_aff_zero_on_domain(__isl_take isl_local_space *ls)
+{
+	isl_aff *aff;
+
+	aff = isl_aff_alloc(ls);
+	if (!aff)
+		return NULL;
+
+	isl_int_set_si(aff->v->el[0], 1);
+	isl_seq_clr(aff->v->el + 1, aff->v->size - 1);
+
+	return aff;
+}
+
+/* Return a piecewise affine expression defined on the specified domain
+ * that is equal to zero.
+ */
+__isl_give isl_pw_aff *isl_pw_aff_zero_on_domain(__isl_take isl_local_space *ls)
+{
+	return isl_pw_aff_from_aff(isl_aff_zero_on_domain(ls));
+}
+
+/* Return an affine expression defined on the specified domain
+ * that represents NaN.
+ */
+__isl_give isl_aff *isl_aff_nan_on_domain(__isl_take isl_local_space *ls)
+{
+	isl_aff *aff;
+
+	aff = isl_aff_alloc(ls);
+	if (!aff)
+		return NULL;
+
+	isl_seq_clr(aff->v->el, aff->v->size);
+
+	return aff;
+}
+
+/* Return a piecewise affine expression defined on the specified domain
+ * that represents NaN.
+ */
+__isl_give isl_pw_aff *isl_pw_aff_nan_on_domain(__isl_take isl_local_space *ls)
+{
+	return isl_pw_aff_from_aff(isl_aff_nan_on_domain(ls));
+}
+
+/* Return an affine expression that is equal to "val" on
+ * domain local space "ls".
+ */
+__isl_give isl_aff *isl_aff_val_on_domain(__isl_take isl_local_space *ls,
+	__isl_take isl_val *val)
+{
+	isl_aff *aff;
+
+	if (!ls || !val)
+		goto error;
+	if (!isl_val_is_rat(val))
+		isl_die(isl_val_get_ctx(val), isl_error_invalid,
+			"expecting rational value", goto error);
+
+	aff = isl_aff_alloc(isl_local_space_copy(ls));
+	if (!aff)
+		goto error;
+
+	isl_seq_clr(aff->v->el + 2, aff->v->size - 2);
+	isl_int_set(aff->v->el[1], val->n);
+	isl_int_set(aff->v->el[0], val->d);
+
+	isl_local_space_free(ls);
+	isl_val_free(val);
+	return aff;
+error:
+	isl_local_space_free(ls);
+	isl_val_free(val);
+	return NULL;
+}
+
+/* Return an affine expression that is equal to the specified dimension
+ * in "ls".
+ */
+__isl_give isl_aff *isl_aff_var_on_domain(__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos)
+{
+	isl_space *space;
+	isl_aff *aff;
+
+	if (!ls)
+		return NULL;
+
+	space = isl_local_space_get_space(ls);
+	if (!space)
+		goto error;
+	if (isl_space_is_map(space))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"expecting (parameter) set space", goto error);
+	if (pos >= isl_local_space_dim(ls, type))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"position out of bounds", goto error);
+
+	isl_space_free(space);
+	aff = isl_aff_alloc(ls);
+	if (!aff)
+		return NULL;
+
+	pos += isl_local_space_offset(aff->ls, type);
+
+	isl_int_set_si(aff->v->el[0], 1);
+	isl_seq_clr(aff->v->el + 1, aff->v->size - 1);
+	isl_int_set_si(aff->v->el[1 + pos], 1);
+
+	return aff;
+error:
+	isl_local_space_free(ls);
+	isl_space_free(space);
+	return NULL;
+}
+
+/* Return a piecewise affine expression that is equal to
+ * the specified dimension in "ls".
+ */
+__isl_give isl_pw_aff *isl_pw_aff_var_on_domain(__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos)
+{
+	return isl_pw_aff_from_aff(isl_aff_var_on_domain(ls, type, pos));
+}
+
+__isl_give isl_aff *isl_aff_copy(__isl_keep isl_aff *aff)
+{
+	if (!aff)
+		return NULL;
+
+	aff->ref++;
+	return aff;
+}
+
+__isl_give isl_aff *isl_aff_dup(__isl_keep isl_aff *aff)
+{
+	if (!aff)
+		return NULL;
+
+	return isl_aff_alloc_vec(isl_local_space_copy(aff->ls),
+				 isl_vec_copy(aff->v));
+}
+
+__isl_give isl_aff *isl_aff_cow(__isl_take isl_aff *aff)
+{
+	if (!aff)
+		return NULL;
+
+	if (aff->ref == 1)
+		return aff;
+	aff->ref--;
+	return isl_aff_dup(aff);
+}
+
+__isl_null isl_aff *isl_aff_free(__isl_take isl_aff *aff)
+{
+	if (!aff)
+		return NULL;
+
+	if (--aff->ref > 0)
+		return NULL;
+
+	isl_local_space_free(aff->ls);
+	isl_vec_free(aff->v);
+
+	free(aff);
+
+	return NULL;
+}
+
+isl_ctx *isl_aff_get_ctx(__isl_keep isl_aff *aff)
+{
+	return aff ? isl_local_space_get_ctx(aff->ls) : NULL;
+}
+
+/* Externally, an isl_aff has a map space, but internally, the
+ * ls field corresponds to the domain of that space.
+ */
+int isl_aff_dim(__isl_keep isl_aff *aff, enum isl_dim_type type)
+{
+	if (!aff)
+		return 0;
+	if (type == isl_dim_out)
+		return 1;
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+	return isl_local_space_dim(aff->ls, type);
+}
+
+/* Return the position of the dimension of the given type and name
+ * in "aff".
+ * Return -1 if no such dimension can be found.
+ */
+int isl_aff_find_dim_by_name(__isl_keep isl_aff *aff, enum isl_dim_type type,
+	const char *name)
+{
+	if (!aff)
+		return -1;
+	if (type == isl_dim_out)
+		return -1;
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+	return isl_local_space_find_dim_by_name(aff->ls, type, name);
+}
+
+__isl_give isl_space *isl_aff_get_domain_space(__isl_keep isl_aff *aff)
+{
+	return aff ? isl_local_space_get_space(aff->ls) : NULL;
+}
+
+__isl_give isl_space *isl_aff_get_space(__isl_keep isl_aff *aff)
+{
+	isl_space *space;
+	if (!aff)
+		return NULL;
+	space = isl_local_space_get_space(aff->ls);
+	space = isl_space_from_domain(space);
+	space = isl_space_add_dims(space, isl_dim_out, 1);
+	return space;
+}
+
+__isl_give isl_local_space *isl_aff_get_domain_local_space(
+	__isl_keep isl_aff *aff)
+{
+	return aff ? isl_local_space_copy(aff->ls) : NULL;
+}
+
+__isl_give isl_local_space *isl_aff_get_local_space(__isl_keep isl_aff *aff)
+{
+	isl_local_space *ls;
+	if (!aff)
+		return NULL;
+	ls = isl_local_space_copy(aff->ls);
+	ls = isl_local_space_from_domain(ls);
+	ls = isl_local_space_add_dims(ls, isl_dim_out, 1);
+	return ls;
+}
+
+/* Externally, an isl_aff has a map space, but internally, the
+ * ls field corresponds to the domain of that space.
+ */
+const char *isl_aff_get_dim_name(__isl_keep isl_aff *aff,
+	enum isl_dim_type type, unsigned pos)
+{
+	if (!aff)
+		return NULL;
+	if (type == isl_dim_out)
+		return NULL;
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+	return isl_local_space_get_dim_name(aff->ls, type, pos);
+}
+
+__isl_give isl_aff *isl_aff_reset_domain_space(__isl_take isl_aff *aff,
+	__isl_take isl_space *dim)
+{
+	aff = isl_aff_cow(aff);
+	if (!aff || !dim)
+		goto error;
+
+	aff->ls = isl_local_space_reset_space(aff->ls, dim);
+	if (!aff->ls)
+		return isl_aff_free(aff);
+
+	return aff;
+error:
+	isl_aff_free(aff);
+	isl_space_free(dim);
+	return NULL;
+}
+
+/* Reset the space of "aff".  This function is called from isl_pw_templ.c
+ * and doesn't know if the space of an element object is represented
+ * directly or through its domain.  It therefore passes along both.
+ */
+__isl_give isl_aff *isl_aff_reset_space_and_domain(__isl_take isl_aff *aff,
+	__isl_take isl_space *space, __isl_take isl_space *domain)
+{
+	isl_space_free(space);
+	return isl_aff_reset_domain_space(aff, domain);
+}
+
+/* Reorder the coefficients of the affine expression based
+ * on the given reodering.
+ * The reordering r is assumed to have been extended with the local
+ * variables.
+ */
+static __isl_give isl_vec *vec_reorder(__isl_take isl_vec *vec,
+	__isl_take isl_reordering *r, int n_div)
+{
+	isl_vec *res;
+	int i;
+
+	if (!vec || !r)
+		goto error;
+
+	res = isl_vec_alloc(vec->ctx,
+			    2 + isl_space_dim(r->dim, isl_dim_all) + n_div);
+	isl_seq_cpy(res->el, vec->el, 2);
+	isl_seq_clr(res->el + 2, res->size - 2);
+	for (i = 0; i < r->len; ++i)
+		isl_int_set(res->el[2 + r->pos[i]], vec->el[2 + i]);
+
+	isl_reordering_free(r);
+	isl_vec_free(vec);
+	return res;
+error:
+	isl_vec_free(vec);
+	isl_reordering_free(r);
+	return NULL;
+}
+
+/* Reorder the dimensions of the domain of "aff" according
+ * to the given reordering.
+ */
+__isl_give isl_aff *isl_aff_realign_domain(__isl_take isl_aff *aff,
+	__isl_take isl_reordering *r)
+{
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		goto error;
+
+	r = isl_reordering_extend(r, aff->ls->div->n_row);
+	aff->v = vec_reorder(aff->v, isl_reordering_copy(r),
+				aff->ls->div->n_row);
+	aff->ls = isl_local_space_realign(aff->ls, r);
+
+	if (!aff->v || !aff->ls)
+		return isl_aff_free(aff);
+
+	return aff;
+error:
+	isl_aff_free(aff);
+	isl_reordering_free(r);
+	return NULL;
+}
+
+__isl_give isl_aff *isl_aff_align_params(__isl_take isl_aff *aff,
+	__isl_take isl_space *model)
+{
+	if (!aff || !model)
+		goto error;
+
+	if (!isl_space_match(aff->ls->dim, isl_dim_param,
+			     model, isl_dim_param)) {
+		isl_reordering *exp;
+
+		model = isl_space_drop_dims(model, isl_dim_in,
+					0, isl_space_dim(model, isl_dim_in));
+		model = isl_space_drop_dims(model, isl_dim_out,
+					0, isl_space_dim(model, isl_dim_out));
+		exp = isl_parameter_alignment_reordering(aff->ls->dim, model);
+		exp = isl_reordering_extend_space(exp,
+					isl_aff_get_domain_space(aff));
+		aff = isl_aff_realign_domain(aff, exp);
+	}
+
+	isl_space_free(model);
+	return aff;
+error:
+	isl_space_free(model);
+	isl_aff_free(aff);
+	return NULL;
+}
+
+/* Is "aff" obviously equal to zero?
+ *
+ * If the denominator is zero, then "aff" is not equal to zero.
+ */
+int isl_aff_plain_is_zero(__isl_keep isl_aff *aff)
+{
+	if (!aff)
+		return -1;
+
+	if (isl_int_is_zero(aff->v->el[0]))
+		return 0;
+	return isl_seq_first_non_zero(aff->v->el + 1, aff->v->size - 1) < 0;
+}
+
+/* Does "aff" represent NaN?
+ */
+int isl_aff_is_nan(__isl_keep isl_aff *aff)
+{
+	if (!aff)
+		return -1;
+
+	return isl_seq_first_non_zero(aff->v->el, 2) < 0;
+}
+
+/* Does "pa" involve any NaNs?
+ */
+int isl_pw_aff_involves_nan(__isl_keep isl_pw_aff *pa)
+{
+	int i;
+
+	if (!pa)
+		return -1;
+	if (pa->n == 0)
+		return 0;
+
+	for (i = 0; i < pa->n; ++i) {
+		int is_nan = isl_aff_is_nan(pa->p[i].aff);
+		if (is_nan < 0 || is_nan)
+			return is_nan;
+	}
+
+	return 0;
+}
+
+/* Are "aff1" and "aff2" obviously equal?
+ *
+ * NaN is not equal to anything, not even to another NaN.
+ */
+int isl_aff_plain_is_equal(__isl_keep isl_aff *aff1, __isl_keep isl_aff *aff2)
+{
+	int equal;
+
+	if (!aff1 || !aff2)
+		return -1;
+
+	if (isl_aff_is_nan(aff1) || isl_aff_is_nan(aff2))
+		return 0;
+
+	equal = isl_local_space_is_equal(aff1->ls, aff2->ls);
+	if (equal < 0 || !equal)
+		return equal;
+
+	return isl_vec_is_equal(aff1->v, aff2->v);
+}
+
+/* Return the common denominator of "aff" in "v".
+ *
+ * We cannot return anything meaningful in case of a NaN.
+ */
+int isl_aff_get_denominator(__isl_keep isl_aff *aff, isl_int *v)
+{
+	if (!aff)
+		return -1;
+	if (isl_aff_is_nan(aff))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"cannot get denominator of NaN", return -1);
+	isl_int_set(*v, aff->v->el[0]);
+	return 0;
+}
+
+/* Return the common denominator of "aff".
+ */
+__isl_give isl_val *isl_aff_get_denominator_val(__isl_keep isl_aff *aff)
+{
+	isl_ctx *ctx;
+
+	if (!aff)
+		return NULL;
+
+	ctx = isl_aff_get_ctx(aff);
+	if (isl_aff_is_nan(aff))
+		return isl_val_nan(ctx);
+	return isl_val_int_from_isl_int(ctx, aff->v->el[0]);
+}
+
+/* Return the constant term of "aff" in "v".
+ *
+ * We cannot return anything meaningful in case of a NaN.
+ */
+int isl_aff_get_constant(__isl_keep isl_aff *aff, isl_int *v)
+{
+	if (!aff)
+		return -1;
+	if (isl_aff_is_nan(aff))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"cannot get constant term of NaN", return -1);
+	isl_int_set(*v, aff->v->el[1]);
+	return 0;
+}
+
+/* Return the constant term of "aff".
+ */
+__isl_give isl_val *isl_aff_get_constant_val(__isl_keep isl_aff *aff)
+{
+	isl_ctx *ctx;
+	isl_val *v;
+
+	if (!aff)
+		return NULL;
+
+	ctx = isl_aff_get_ctx(aff);
+	if (isl_aff_is_nan(aff))
+		return isl_val_nan(ctx);
+	v = isl_val_rat_from_isl_int(ctx, aff->v->el[1], aff->v->el[0]);
+	return isl_val_normalize(v);
+}
+
+/* Return the coefficient of the variable of type "type" at position "pos"
+ * of "aff" in "v".
+ *
+ * We cannot return anything meaningful in case of a NaN.
+ */
+int isl_aff_get_coefficient(__isl_keep isl_aff *aff,
+	enum isl_dim_type type, int pos, isl_int *v)
+{
+	if (!aff)
+		return -1;
+
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			return -1);
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	if (pos >= isl_local_space_dim(aff->ls, type))
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"position out of bounds", return -1);
+
+	if (isl_aff_is_nan(aff))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"cannot get coefficient of NaN", return -1);
+	pos += isl_local_space_offset(aff->ls, type);
+	isl_int_set(*v, aff->v->el[1 + pos]);
+
+	return 0;
+}
+
+/* Return the coefficient of the variable of type "type" at position "pos"
+ * of "aff".
+ */
+__isl_give isl_val *isl_aff_get_coefficient_val(__isl_keep isl_aff *aff,
+	enum isl_dim_type type, int pos)
+{
+	isl_ctx *ctx;
+	isl_val *v;
+
+	if (!aff)
+		return NULL;
+
+	ctx = isl_aff_get_ctx(aff);
+	if (type == isl_dim_out)
+		isl_die(ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			return NULL);
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	if (pos >= isl_local_space_dim(aff->ls, type))
+		isl_die(ctx, isl_error_invalid,
+			"position out of bounds", return NULL);
+
+	if (isl_aff_is_nan(aff))
+		return isl_val_nan(ctx);
+	pos += isl_local_space_offset(aff->ls, type);
+	v = isl_val_rat_from_isl_int(ctx, aff->v->el[1 + pos], aff->v->el[0]);
+	return isl_val_normalize(v);
+}
+
+/* Return the sign of the coefficient of the variable of type "type"
+ * at position "pos" of "aff".
+ */
+int isl_aff_coefficient_sgn(__isl_keep isl_aff *aff, enum isl_dim_type type,
+	int pos)
+{
+	isl_ctx *ctx;
+
+	if (!aff)
+		return 0;
+
+	ctx = isl_aff_get_ctx(aff);
+	if (type == isl_dim_out)
+		isl_die(ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			return 0);
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	if (pos >= isl_local_space_dim(aff->ls, type))
+		isl_die(ctx, isl_error_invalid,
+			"position out of bounds", return 0);
+
+	pos += isl_local_space_offset(aff->ls, type);
+	return isl_int_sgn(aff->v->el[1 + pos]);
+}
+
+/* Replace the denominator of "aff" by "v".
+ *
+ * A NaN is unaffected by this operation.
+ */
+__isl_give isl_aff *isl_aff_set_denominator(__isl_take isl_aff *aff, isl_int v)
+{
+	if (!aff)
+		return NULL;
+	if (isl_aff_is_nan(aff))
+		return aff;
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	isl_int_set(aff->v->el[0], v);
+
+	return aff;
+}
+
+/* Replace the numerator of the constant term of "aff" by "v".
+ *
+ * A NaN is unaffected by this operation.
+ */
+__isl_give isl_aff *isl_aff_set_constant(__isl_take isl_aff *aff, isl_int v)
+{
+	if (!aff)
+		return NULL;
+	if (isl_aff_is_nan(aff))
+		return aff;
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	isl_int_set(aff->v->el[1], v);
+
+	return aff;
+}
+
+/* Replace the constant term of "aff" by "v".
+ *
+ * A NaN is unaffected by this operation.
+ */
+__isl_give isl_aff *isl_aff_set_constant_val(__isl_take isl_aff *aff,
+	__isl_take isl_val *v)
+{
+	if (!aff || !v)
+		goto error;
+
+	if (isl_aff_is_nan(aff)) {
+		isl_val_free(v);
+		return aff;
+	}
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"expecting rational value", goto error);
+
+	if (isl_int_eq(aff->v->el[1], v->n) &&
+	    isl_int_eq(aff->v->el[0], v->d)) {
+		isl_val_free(v);
+		return aff;
+	}
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		goto error;
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		goto error;
+
+	if (isl_int_eq(aff->v->el[0], v->d)) {
+		isl_int_set(aff->v->el[1], v->n);
+	} else if (isl_int_is_one(v->d)) {
+		isl_int_mul(aff->v->el[1], aff->v->el[0], v->n);
+	} else {
+		isl_seq_scale(aff->v->el + 1,
+				aff->v->el + 1, v->d, aff->v->size - 1);
+		isl_int_mul(aff->v->el[1], aff->v->el[0], v->n);
+		isl_int_mul(aff->v->el[0], aff->v->el[0], v->d);
+		aff->v = isl_vec_normalize(aff->v);
+		if (!aff->v)
+			goto error;
+	}
+
+	isl_val_free(v);
+	return aff;
+error:
+	isl_aff_free(aff);
+	isl_val_free(v);
+	return NULL;
+}
+
+/* Add "v" to the constant term of "aff".
+ *
+ * A NaN is unaffected by this operation.
+ */
+__isl_give isl_aff *isl_aff_add_constant(__isl_take isl_aff *aff, isl_int v)
+{
+	if (isl_int_is_zero(v))
+		return aff;
+
+	if (!aff)
+		return NULL;
+	if (isl_aff_is_nan(aff))
+		return aff;
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	isl_int_addmul(aff->v->el[1], aff->v->el[0], v);
+
+	return aff;
+}
+
+/* Add "v" to the constant term of "aff".
+ *
+ * A NaN is unaffected by this operation.
+ */
+__isl_give isl_aff *isl_aff_add_constant_val(__isl_take isl_aff *aff,
+	__isl_take isl_val *v)
+{
+	if (!aff || !v)
+		goto error;
+
+	if (isl_aff_is_nan(aff) || isl_val_is_zero(v)) {
+		isl_val_free(v);
+		return aff;
+	}
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"expecting rational value", goto error);
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		goto error;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		goto error;
+
+	if (isl_int_is_one(v->d)) {
+		isl_int_addmul(aff->v->el[1], aff->v->el[0], v->n);
+	} else if (isl_int_eq(aff->v->el[0], v->d)) {
+		isl_int_add(aff->v->el[1], aff->v->el[1], v->n);
+		aff->v = isl_vec_normalize(aff->v);
+		if (!aff->v)
+			goto error;
+	} else {
+		isl_seq_scale(aff->v->el + 1,
+				aff->v->el + 1, v->d, aff->v->size - 1);
+		isl_int_addmul(aff->v->el[1], aff->v->el[0], v->n);
+		isl_int_mul(aff->v->el[0], aff->v->el[0], v->d);
+		aff->v = isl_vec_normalize(aff->v);
+		if (!aff->v)
+			goto error;
+	}
+
+	isl_val_free(v);
+	return aff;
+error:
+	isl_aff_free(aff);
+	isl_val_free(v);
+	return NULL;
+}
+
+__isl_give isl_aff *isl_aff_add_constant_si(__isl_take isl_aff *aff, int v)
+{
+	isl_int t;
+
+	isl_int_init(t);
+	isl_int_set_si(t, v);
+	aff = isl_aff_add_constant(aff, t);
+	isl_int_clear(t);
+
+	return aff;
+}
+
+/* Add "v" to the numerator of the constant term of "aff".
+ *
+ * A NaN is unaffected by this operation.
+ */
+__isl_give isl_aff *isl_aff_add_constant_num(__isl_take isl_aff *aff, isl_int v)
+{
+	if (isl_int_is_zero(v))
+		return aff;
+
+	if (!aff)
+		return NULL;
+	if (isl_aff_is_nan(aff))
+		return aff;
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	isl_int_add(aff->v->el[1], aff->v->el[1], v);
+
+	return aff;
+}
+
+/* Add "v" to the numerator of the constant term of "aff".
+ *
+ * A NaN is unaffected by this operation.
+ */
+__isl_give isl_aff *isl_aff_add_constant_num_si(__isl_take isl_aff *aff, int v)
+{
+	isl_int t;
+
+	if (v == 0)
+		return aff;
+
+	isl_int_init(t);
+	isl_int_set_si(t, v);
+	aff = isl_aff_add_constant_num(aff, t);
+	isl_int_clear(t);
+
+	return aff;
+}
+
+/* Replace the numerator of the constant term of "aff" by "v".
+ *
+ * A NaN is unaffected by this operation.
+ */
+__isl_give isl_aff *isl_aff_set_constant_si(__isl_take isl_aff *aff, int v)
+{
+	if (!aff)
+		return NULL;
+	if (isl_aff_is_nan(aff))
+		return aff;
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	isl_int_set_si(aff->v->el[1], v);
+
+	return aff;
+}
+
+/* Replace the numerator of the coefficient of the variable of type "type"
+ * at position "pos" of "aff" by "v".
+ *
+ * A NaN is unaffected by this operation.
+ */
+__isl_give isl_aff *isl_aff_set_coefficient(__isl_take isl_aff *aff,
+	enum isl_dim_type type, int pos, isl_int v)
+{
+	if (!aff)
+		return NULL;
+
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			return isl_aff_free(aff));
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	if (pos >= isl_local_space_dim(aff->ls, type))
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"position out of bounds", return isl_aff_free(aff));
+
+	if (isl_aff_is_nan(aff))
+		return aff;
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	pos += isl_local_space_offset(aff->ls, type);
+	isl_int_set(aff->v->el[1 + pos], v);
+
+	return aff;
+}
+
+/* Replace the numerator of the coefficient of the variable of type "type"
+ * at position "pos" of "aff" by "v".
+ *
+ * A NaN is unaffected by this operation.
+ */
+__isl_give isl_aff *isl_aff_set_coefficient_si(__isl_take isl_aff *aff,
+	enum isl_dim_type type, int pos, int v)
+{
+	if (!aff)
+		return NULL;
+
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			return isl_aff_free(aff));
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	if (pos < 0 || pos >= isl_local_space_dim(aff->ls, type))
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"position out of bounds", return isl_aff_free(aff));
+
+	if (isl_aff_is_nan(aff))
+		return aff;
+	pos += isl_local_space_offset(aff->ls, type);
+	if (isl_int_cmp_si(aff->v->el[1 + pos], v) == 0)
+		return aff;
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	isl_int_set_si(aff->v->el[1 + pos], v);
+
+	return aff;
+}
+
+/* Replace the coefficient of the variable of type "type" at position "pos"
+ * of "aff" by "v".
+ *
+ * A NaN is unaffected by this operation.
+ */
+__isl_give isl_aff *isl_aff_set_coefficient_val(__isl_take isl_aff *aff,
+	enum isl_dim_type type, int pos, __isl_take isl_val *v)
+{
+	if (!aff || !v)
+		goto error;
+
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			goto error);
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	if (pos >= isl_local_space_dim(aff->ls, type))
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"position out of bounds", goto error);
+
+	if (isl_aff_is_nan(aff)) {
+		isl_val_free(v);
+		return aff;
+	}
+	if (!isl_val_is_rat(v))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"expecting rational value", goto error);
+
+	pos += isl_local_space_offset(aff->ls, type);
+	if (isl_int_eq(aff->v->el[1 + pos], v->n) &&
+	    isl_int_eq(aff->v->el[0], v->d)) {
+		isl_val_free(v);
+		return aff;
+	}
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		goto error;
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		goto error;
+
+	if (isl_int_eq(aff->v->el[0], v->d)) {
+		isl_int_set(aff->v->el[1 + pos], v->n);
+	} else if (isl_int_is_one(v->d)) {
+		isl_int_mul(aff->v->el[1 + pos], aff->v->el[0], v->n);
+	} else {
+		isl_seq_scale(aff->v->el + 1,
+				aff->v->el + 1, v->d, aff->v->size - 1);
+		isl_int_mul(aff->v->el[1 + pos], aff->v->el[0], v->n);
+		isl_int_mul(aff->v->el[0], aff->v->el[0], v->d);
+		aff->v = isl_vec_normalize(aff->v);
+		if (!aff->v)
+			goto error;
+	}
+
+	isl_val_free(v);
+	return aff;
+error:
+	isl_aff_free(aff);
+	isl_val_free(v);
+	return NULL;
+}
+
+/* Add "v" to the coefficient of the variable of type "type"
+ * at position "pos" of "aff".
+ *
+ * A NaN is unaffected by this operation.
+ */
+__isl_give isl_aff *isl_aff_add_coefficient(__isl_take isl_aff *aff,
+	enum isl_dim_type type, int pos, isl_int v)
+{
+	if (!aff)
+		return NULL;
+
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			return isl_aff_free(aff));
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	if (pos >= isl_local_space_dim(aff->ls, type))
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"position out of bounds", return isl_aff_free(aff));
+
+	if (isl_aff_is_nan(aff))
+		return aff;
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	pos += isl_local_space_offset(aff->ls, type);
+	isl_int_addmul(aff->v->el[1 + pos], aff->v->el[0], v);
+
+	return aff;
+}
+
+/* Add "v" to the coefficient of the variable of type "type"
+ * at position "pos" of "aff".
+ *
+ * A NaN is unaffected by this operation.
+ */
+__isl_give isl_aff *isl_aff_add_coefficient_val(__isl_take isl_aff *aff,
+	enum isl_dim_type type, int pos, __isl_take isl_val *v)
+{
+	if (!aff || !v)
+		goto error;
+
+	if (isl_val_is_zero(v)) {
+		isl_val_free(v);
+		return aff;
+	}
+
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			goto error);
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	if (pos >= isl_local_space_dim(aff->ls, type))
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"position out of bounds", goto error);
+
+	if (isl_aff_is_nan(aff)) {
+		isl_val_free(v);
+		return aff;
+	}
+	if (!isl_val_is_rat(v))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"expecting rational value", goto error);
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		goto error;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		goto error;
+
+	pos += isl_local_space_offset(aff->ls, type);
+	if (isl_int_is_one(v->d)) {
+		isl_int_addmul(aff->v->el[1 + pos], aff->v->el[0], v->n);
+	} else if (isl_int_eq(aff->v->el[0], v->d)) {
+		isl_int_add(aff->v->el[1 + pos], aff->v->el[1 + pos], v->n);
+		aff->v = isl_vec_normalize(aff->v);
+		if (!aff->v)
+			goto error;
+	} else {
+		isl_seq_scale(aff->v->el + 1,
+				aff->v->el + 1, v->d, aff->v->size - 1);
+		isl_int_addmul(aff->v->el[1 + pos], aff->v->el[0], v->n);
+		isl_int_mul(aff->v->el[0], aff->v->el[0], v->d);
+		aff->v = isl_vec_normalize(aff->v);
+		if (!aff->v)
+			goto error;
+	}
+
+	isl_val_free(v);
+	return aff;
+error:
+	isl_aff_free(aff);
+	isl_val_free(v);
+	return NULL;
+}
+
+__isl_give isl_aff *isl_aff_add_coefficient_si(__isl_take isl_aff *aff,
+	enum isl_dim_type type, int pos, int v)
+{
+	isl_int t;
+
+	isl_int_init(t);
+	isl_int_set_si(t, v);
+	aff = isl_aff_add_coefficient(aff, type, pos, t);
+	isl_int_clear(t);
+
+	return aff;
+}
+
+__isl_give isl_aff *isl_aff_get_div(__isl_keep isl_aff *aff, int pos)
+{
+	if (!aff)
+		return NULL;
+
+	return isl_local_space_get_div(aff->ls, pos);
+}
+
+/* Return the negation of "aff".
+ *
+ * As a special case, -NaN = NaN.
+ */
+__isl_give isl_aff *isl_aff_neg(__isl_take isl_aff *aff)
+{
+	if (!aff)
+		return NULL;
+	if (isl_aff_is_nan(aff))
+		return aff;
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	isl_seq_neg(aff->v->el + 1, aff->v->el + 1, aff->v->size - 1);
+
+	return aff;
+}
+
+/* Remove divs from the local space that do not appear in the affine
+ * expression.
+ * We currently only remove divs at the end.
+ * Some intermediate divs may also not appear directly in the affine
+ * expression, but we would also need to check that no other divs are
+ * defined in terms of them.
+ */
+__isl_give isl_aff *isl_aff_remove_unused_divs( __isl_take isl_aff *aff)
+{
+	int pos;
+	int off;
+	int n;
+
+	if (!aff)
+		return NULL;
+
+	n = isl_local_space_dim(aff->ls, isl_dim_div);
+	off = isl_local_space_offset(aff->ls, isl_dim_div);
+
+	pos = isl_seq_last_non_zero(aff->v->el + 1 + off, n) + 1;
+	if (pos == n)
+		return aff;
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->ls = isl_local_space_drop_dims(aff->ls, isl_dim_div, pos, n - pos);
+	aff->v = isl_vec_drop_els(aff->v, 1 + off + pos, n - pos);
+	if (!aff->ls || !aff->v)
+		return isl_aff_free(aff);
+
+	return aff;
+}
+
+/* Given two affine expressions "p" of length p_len (including the
+ * denominator and the constant term) and "subs" of length subs_len,
+ * plug in "subs" for the variable at position "pos".
+ * The variables of "subs" and "p" are assumed to match up to subs_len,
+ * but "p" may have additional variables.
+ * "v" is an initialized isl_int that can be used internally.
+ *
+ * In particular, if "p" represents the expression
+ *
+ *	(a i + g)/m
+ *
+ * with i the variable at position "pos" and "subs" represents the expression
+ *
+ *	f/d
+ *
+ * then the result represents the expression
+ *
+ *	(a f + d g)/(m d)
+ *
+ */
+void isl_seq_substitute(isl_int *p, int pos, isl_int *subs,
+	int p_len, int subs_len, isl_int v)
+{
+	isl_int_set(v, p[1 + pos]);
+	isl_int_set_si(p[1 + pos], 0);
+	isl_seq_combine(p + 1, subs[0], p + 1, v, subs + 1, subs_len - 1);
+	isl_seq_scale(p + subs_len, p + subs_len, subs[0], p_len - subs_len);
+	isl_int_mul(p[0], p[0], subs[0]);
+}
+
+/* Look for any divs in the aff->ls with a denominator equal to one
+ * and plug them into the affine expression and any subsequent divs
+ * that may reference the div.
+ */
+static __isl_give isl_aff *plug_in_integral_divs(__isl_take isl_aff *aff)
+{
+	int i, n;
+	int len;
+	isl_int v;
+	isl_vec *vec;
+	isl_local_space *ls;
+	unsigned pos;
+
+	if (!aff)
+		return NULL;
+
+	n = isl_local_space_dim(aff->ls, isl_dim_div);
+	len = aff->v->size;
+	for (i = 0; i < n; ++i) {
+		if (!isl_int_is_one(aff->ls->div->row[i][0]))
+			continue;
+		ls = isl_local_space_copy(aff->ls);
+		ls = isl_local_space_substitute_seq(ls, isl_dim_div, i,
+				aff->ls->div->row[i], len, i + 1, n - (i + 1));
+		vec = isl_vec_copy(aff->v);
+		vec = isl_vec_cow(vec);
+		if (!ls || !vec)
+			goto error;
+
+		isl_int_init(v);
+
+		pos = isl_local_space_offset(aff->ls, isl_dim_div) + i;
+		isl_seq_substitute(vec->el, pos, aff->ls->div->row[i],
+					len, len, v);
+
+		isl_int_clear(v);
+
+		isl_vec_free(aff->v);
+		aff->v = vec;
+		isl_local_space_free(aff->ls);
+		aff->ls = ls;
+	}
+
+	return aff;
+error:
+	isl_vec_free(vec);
+	isl_local_space_free(ls);
+	return isl_aff_free(aff);
+}
+
+/* Look for any divs j that appear with a unit coefficient inside
+ * the definitions of other divs i and plug them into the definitions
+ * of the divs i.
+ *
+ * In particular, an expression of the form
+ *
+ *	floor((f(..) + floor(g(..)/n))/m)
+ *
+ * is simplified to
+ *
+ *	floor((n * f(..) + g(..))/(n * m))
+ *
+ * This simplification is correct because we can move the expression
+ * f(..) into the inner floor in the original expression to obtain
+ *
+ *	floor(floor((n * f(..) + g(..))/n)/m)
+ *
+ * from which we can derive the simplified expression.
+ */
+static __isl_give isl_aff *plug_in_unit_divs(__isl_take isl_aff *aff)
+{
+	int i, j, n;
+	int off;
+
+	if (!aff)
+		return NULL;
+
+	n = isl_local_space_dim(aff->ls, isl_dim_div);
+	off = isl_local_space_offset(aff->ls, isl_dim_div);
+	for (i = 1; i < n; ++i) {
+		for (j = 0; j < i; ++j) {
+			if (!isl_int_is_one(aff->ls->div->row[i][1 + off + j]))
+				continue;
+			aff->ls = isl_local_space_substitute_seq(aff->ls,
+				isl_dim_div, j, aff->ls->div->row[j],
+				aff->v->size, i, 1);
+			if (!aff->ls)
+				return isl_aff_free(aff);
+		}
+	}
+
+	return aff;
+}
+
+/* Swap divs "a" and "b" in "aff", which is assumed to be non-NULL.
+ *
+ * Even though this function is only called on isl_affs with a single
+ * reference, we are careful to only change aff->v and aff->ls together.
+ */
+static __isl_give isl_aff *swap_div(__isl_take isl_aff *aff, int a, int b)
+{
+	unsigned off = isl_local_space_offset(aff->ls, isl_dim_div);
+	isl_local_space *ls;
+	isl_vec *v;
+
+	ls = isl_local_space_copy(aff->ls);
+	ls = isl_local_space_swap_div(ls, a, b);
+	v = isl_vec_copy(aff->v);
+	v = isl_vec_cow(v);
+	if (!ls || !v)
+		goto error;
+
+	isl_int_swap(v->el[1 + off + a], v->el[1 + off + b]);
+	isl_vec_free(aff->v);
+	aff->v = v;
+	isl_local_space_free(aff->ls);
+	aff->ls = ls;
+
+	return aff;
+error:
+	isl_vec_free(v);
+	isl_local_space_free(ls);
+	return isl_aff_free(aff);
+}
+
+/* Merge divs "a" and "b" in "aff", which is assumed to be non-NULL.
+ *
+ * We currently do not actually remove div "b", but simply add its
+ * coefficient to that of "a" and then zero it out.
+ */
+static __isl_give isl_aff *merge_divs(__isl_take isl_aff *aff, int a, int b)
+{
+	unsigned off = isl_local_space_offset(aff->ls, isl_dim_div);
+
+	if (isl_int_is_zero(aff->v->el[1 + off + b]))
+		return aff;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	isl_int_add(aff->v->el[1 + off + a],
+		    aff->v->el[1 + off + a], aff->v->el[1 + off + b]);
+	isl_int_set_si(aff->v->el[1 + off + b], 0);
+
+	return aff;
+}
+
+/* Sort the divs in the local space of "aff" according to
+ * the comparison function "cmp_row" in isl_local_space.c,
+ * combining the coefficients of identical divs.
+ *
+ * Reordering divs does not change the semantics of "aff",
+ * so there is no need to call isl_aff_cow.
+ * Moreover, this function is currently only called on isl_affs
+ * with a single reference.
+ */
+static __isl_give isl_aff *sort_divs(__isl_take isl_aff *aff)
+{
+	int i, j, n;
+	unsigned off;
+
+	if (!aff)
+		return NULL;
+
+	off = isl_local_space_offset(aff->ls, isl_dim_div);
+	n = isl_aff_dim(aff, isl_dim_div);
+	for (i = 1; i < n; ++i) {
+		for (j = i - 1; j >= 0; --j) {
+			int cmp = isl_mat_cmp_div(aff->ls->div, j, j + 1);
+			if (cmp < 0)
+				break;
+			if (cmp == 0)
+				aff = merge_divs(aff, j, j + 1);
+			else
+				aff = swap_div(aff, j, j + 1);
+			if (!aff)
+				return NULL;
+		}
+	}
+
+	return aff;
+}
+
+/* Normalize the representation of "aff".
+ *
+ * This function should only be called of "new" isl_affs, i.e.,
+ * with only a single reference.  We therefore do not need to
+ * worry about affecting other instances.
+ */
+__isl_give isl_aff *isl_aff_normalize(__isl_take isl_aff *aff)
+{
+	if (!aff)
+		return NULL;
+	aff->v = isl_vec_normalize(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+	aff = plug_in_integral_divs(aff);
+	aff = plug_in_unit_divs(aff);
+	aff = sort_divs(aff);
+	aff = isl_aff_remove_unused_divs(aff);
+	return aff;
+}
+
+/* Given f, return floor(f).
+ * If f is an integer expression, then just return f.
+ * If f is a constant, then return the constant floor(f).
+ * Otherwise, if f = g/m, write g = q m + r,
+ * create a new div d = [r/m] and return the expression q + d.
+ * The coefficients in r are taken to lie between -m/2 and m/2.
+ *
+ * As a special case, floor(NaN) = NaN.
+ */
+__isl_give isl_aff *isl_aff_floor(__isl_take isl_aff *aff)
+{
+	int i;
+	int size;
+	isl_ctx *ctx;
+	isl_vec *div;
+
+	if (!aff)
+		return NULL;
+
+	if (isl_aff_is_nan(aff))
+		return aff;
+	if (isl_int_is_one(aff->v->el[0]))
+		return aff;
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	if (isl_aff_is_cst(aff)) {
+		isl_int_fdiv_q(aff->v->el[1], aff->v->el[1], aff->v->el[0]);
+		isl_int_set_si(aff->v->el[0], 1);
+		return aff;
+	}
+
+	div = isl_vec_copy(aff->v);
+	div = isl_vec_cow(div);
+	if (!div)
+		return isl_aff_free(aff);
+
+	ctx = isl_aff_get_ctx(aff);
+	isl_int_fdiv_q(aff->v->el[0], aff->v->el[0], ctx->two);
+	for (i = 1; i < aff->v->size; ++i) {
+		isl_int_fdiv_r(div->el[i], div->el[i], div->el[0]);
+		isl_int_fdiv_q(aff->v->el[i], aff->v->el[i], div->el[0]);
+		if (isl_int_gt(div->el[i], aff->v->el[0])) {
+			isl_int_sub(div->el[i], div->el[i], div->el[0]);
+			isl_int_add_ui(aff->v->el[i], aff->v->el[i], 1);
+		}
+	}
+
+	aff->ls = isl_local_space_add_div(aff->ls, div);
+	if (!aff->ls)
+		return isl_aff_free(aff);
+
+	size = aff->v->size;
+	aff->v = isl_vec_extend(aff->v, size + 1);
+	if (!aff->v)
+		return isl_aff_free(aff);
+	isl_int_set_si(aff->v->el[0], 1);
+	isl_int_set_si(aff->v->el[size], 1);
+
+	aff = isl_aff_normalize(aff);
+
+	return aff;
+}
+
+/* Compute
+ *
+ *	aff mod m = aff - m * floor(aff/m)
+ */
+__isl_give isl_aff *isl_aff_mod(__isl_take isl_aff *aff, isl_int m)
+{
+	isl_aff *res;
+
+	res = isl_aff_copy(aff);
+	aff = isl_aff_scale_down(aff, m);
+	aff = isl_aff_floor(aff);
+	aff = isl_aff_scale(aff, m);
+	res = isl_aff_sub(res, aff);
+
+	return res;
+}
+
+/* Compute
+ *
+ *	aff mod m = aff - m * floor(aff/m)
+ *
+ * with m an integer value.
+ */
+__isl_give isl_aff *isl_aff_mod_val(__isl_take isl_aff *aff,
+	__isl_take isl_val *m)
+{
+	isl_aff *res;
+
+	if (!aff || !m)
+		goto error;
+
+	if (!isl_val_is_int(m))
+		isl_die(isl_val_get_ctx(m), isl_error_invalid,
+			"expecting integer modulo", goto error);
+
+	res = isl_aff_copy(aff);
+	aff = isl_aff_scale_down_val(aff, isl_val_copy(m));
+	aff = isl_aff_floor(aff);
+	aff = isl_aff_scale_val(aff, m);
+	res = isl_aff_sub(res, aff);
+
+	return res;
+error:
+	isl_aff_free(aff);
+	isl_val_free(m);
+	return NULL;
+}
+
+/* Compute
+ *
+ *	pwaff mod m = pwaff - m * floor(pwaff/m)
+ */
+__isl_give isl_pw_aff *isl_pw_aff_mod(__isl_take isl_pw_aff *pwaff, isl_int m)
+{
+	isl_pw_aff *res;
+
+	res = isl_pw_aff_copy(pwaff);
+	pwaff = isl_pw_aff_scale_down(pwaff, m);
+	pwaff = isl_pw_aff_floor(pwaff);
+	pwaff = isl_pw_aff_scale(pwaff, m);
+	res = isl_pw_aff_sub(res, pwaff);
+
+	return res;
+}
+
+/* Compute
+ *
+ *	pa mod m = pa - m * floor(pa/m)
+ *
+ * with m an integer value.
+ */
+__isl_give isl_pw_aff *isl_pw_aff_mod_val(__isl_take isl_pw_aff *pa,
+	__isl_take isl_val *m)
+{
+	if (!pa || !m)
+		goto error;
+	if (!isl_val_is_int(m))
+		isl_die(isl_pw_aff_get_ctx(pa), isl_error_invalid,
+			"expecting integer modulo", goto error);
+	pa = isl_pw_aff_mod(pa, m->n);
+	isl_val_free(m);
+	return pa;
+error:
+	isl_pw_aff_free(pa);
+	isl_val_free(m);
+	return NULL;
+}
+
+/* Given f, return ceil(f).
+ * If f is an integer expression, then just return f.
+ * Otherwise, let f be the expression
+ *
+ *	e/m
+ *
+ * then return
+ *
+ *	floor((e + m - 1)/m)
+ *
+ * As a special case, ceil(NaN) = NaN.
+ */
+__isl_give isl_aff *isl_aff_ceil(__isl_take isl_aff *aff)
+{
+	if (!aff)
+		return NULL;
+
+	if (isl_aff_is_nan(aff))
+		return aff;
+	if (isl_int_is_one(aff->v->el[0]))
+		return aff;
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	isl_int_add(aff->v->el[1], aff->v->el[1], aff->v->el[0]);
+	isl_int_sub_ui(aff->v->el[1], aff->v->el[1], 1);
+	aff = isl_aff_floor(aff);
+
+	return aff;
+}
+
+/* Apply the expansion computed by isl_merge_divs.
+ * The expansion itself is given by "exp" while the resulting
+ * list of divs is given by "div".
+ */
+__isl_give isl_aff *isl_aff_expand_divs( __isl_take isl_aff *aff,
+	__isl_take isl_mat *div, int *exp)
+{
+	int i, j;
+	int old_n_div;
+	int new_n_div;
+	int offset;
+
+	aff = isl_aff_cow(aff);
+	if (!aff || !div)
+		goto error;
+
+	old_n_div = isl_local_space_dim(aff->ls, isl_dim_div);
+	new_n_div = isl_mat_rows(div);
+	if (new_n_div < old_n_div)
+		isl_die(isl_mat_get_ctx(div), isl_error_invalid,
+			"not an expansion", goto error);
+
+	aff->v = isl_vec_extend(aff->v, aff->v->size + new_n_div - old_n_div);
+	if (!aff->v)
+		goto error;
+
+	offset = 1 + isl_local_space_offset(aff->ls, isl_dim_div);
+	j = old_n_div - 1;
+	for (i = new_n_div - 1; i >= 0; --i) {
+		if (j >= 0 && exp[j] == i) {
+			if (i != j)
+				isl_int_swap(aff->v->el[offset + i],
+					     aff->v->el[offset + j]);
+			j--;
+		} else
+			isl_int_set_si(aff->v->el[offset + i], 0);
+	}
+
+	aff->ls = isl_local_space_replace_divs(aff->ls, isl_mat_copy(div));
+	if (!aff->ls)
+		goto error;
+	isl_mat_free(div);
+	return aff;
+error:
+	isl_aff_free(aff);
+	isl_mat_free(div);
+	return NULL;
+}
+
+/* Add two affine expressions that live in the same local space.
+ */
+static __isl_give isl_aff *add_expanded(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2)
+{
+	isl_int gcd, f;
+
+	aff1 = isl_aff_cow(aff1);
+	if (!aff1 || !aff2)
+		goto error;
+
+	aff1->v = isl_vec_cow(aff1->v);
+	if (!aff1->v)
+		goto error;
+
+	isl_int_init(gcd);
+	isl_int_init(f);
+	isl_int_gcd(gcd, aff1->v->el[0], aff2->v->el[0]);
+	isl_int_divexact(f, aff2->v->el[0], gcd);
+	isl_seq_scale(aff1->v->el + 1, aff1->v->el + 1, f, aff1->v->size - 1);
+	isl_int_divexact(f, aff1->v->el[0], gcd);
+	isl_seq_addmul(aff1->v->el + 1, f, aff2->v->el + 1, aff1->v->size - 1);
+	isl_int_divexact(f, aff2->v->el[0], gcd);
+	isl_int_mul(aff1->v->el[0], aff1->v->el[0], f);
+	isl_int_clear(f);
+	isl_int_clear(gcd);
+
+	isl_aff_free(aff2);
+	return aff1;
+error:
+	isl_aff_free(aff1);
+	isl_aff_free(aff2);
+	return NULL;
+}
+
+/* Return the sum of "aff1" and "aff2".
+ *
+ * If either of the two is NaN, then the result is NaN.
+ */
+__isl_give isl_aff *isl_aff_add(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2)
+{
+	isl_ctx *ctx;
+	int *exp1 = NULL;
+	int *exp2 = NULL;
+	isl_mat *div;
+	int n_div1, n_div2;
+
+	if (!aff1 || !aff2)
+		goto error;
+
+	ctx = isl_aff_get_ctx(aff1);
+	if (!isl_space_is_equal(aff1->ls->dim, aff2->ls->dim))
+		isl_die(ctx, isl_error_invalid,
+			"spaces don't match", goto error);
+
+	if (isl_aff_is_nan(aff1)) {
+		isl_aff_free(aff2);
+		return aff1;
+	}
+	if (isl_aff_is_nan(aff2)) {
+		isl_aff_free(aff1);
+		return aff2;
+	}
+
+	n_div1 = isl_aff_dim(aff1, isl_dim_div);
+	n_div2 = isl_aff_dim(aff2, isl_dim_div);
+	if (n_div1 == 0 && n_div2 == 0)
+		return add_expanded(aff1, aff2);
+
+	exp1 = isl_alloc_array(ctx, int, n_div1);
+	exp2 = isl_alloc_array(ctx, int, n_div2);
+	if ((n_div1 && !exp1) || (n_div2 && !exp2))
+		goto error;
+
+	div = isl_merge_divs(aff1->ls->div, aff2->ls->div, exp1, exp2);
+	aff1 = isl_aff_expand_divs(aff1, isl_mat_copy(div), exp1);
+	aff2 = isl_aff_expand_divs(aff2, div, exp2);
+	free(exp1);
+	free(exp2);
+
+	return add_expanded(aff1, aff2);
+error:
+	free(exp1);
+	free(exp2);
+	isl_aff_free(aff1);
+	isl_aff_free(aff2);
+	return NULL;
+}
+
+__isl_give isl_aff *isl_aff_sub(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2)
+{
+	return isl_aff_add(aff1, isl_aff_neg(aff2));
+}
+
+/* Return the result of scaling "aff" by a factor of "f".
+ *
+ * As a special case, f * NaN = NaN.
+ */
+__isl_give isl_aff *isl_aff_scale(__isl_take isl_aff *aff, isl_int f)
+{
+	isl_int gcd;
+
+	if (!aff)
+		return NULL;
+	if (isl_aff_is_nan(aff))
+		return aff;
+
+	if (isl_int_is_one(f))
+		return aff;
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	if (isl_int_is_pos(f) && isl_int_is_divisible_by(aff->v->el[0], f)) {
+		isl_int_divexact(aff->v->el[0], aff->v->el[0], f);
+		return aff;
+	}
+
+	isl_int_init(gcd);
+	isl_int_gcd(gcd, aff->v->el[0], f);
+	isl_int_divexact(aff->v->el[0], aff->v->el[0], gcd);
+	isl_int_divexact(gcd, f, gcd);
+	isl_seq_scale(aff->v->el + 1, aff->v->el + 1, gcd, aff->v->size - 1);
+	isl_int_clear(gcd);
+
+	return aff;
+}
+
+/* Multiple "aff" by "v".
+ */
+__isl_give isl_aff *isl_aff_scale_val(__isl_take isl_aff *aff,
+	__isl_take isl_val *v)
+{
+	if (!aff || !v)
+		goto error;
+
+	if (isl_val_is_one(v)) {
+		isl_val_free(v);
+		return aff;
+	}
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"expecting rational factor", goto error);
+
+	aff = isl_aff_scale(aff, v->n);
+	aff = isl_aff_scale_down(aff, v->d);
+
+	isl_val_free(v);
+	return aff;
+error:
+	isl_aff_free(aff);
+	isl_val_free(v);
+	return NULL;
+}
+
+/* Return the result of scaling "aff" down by a factor of "f".
+ *
+ * As a special case, NaN/f = NaN.
+ */
+__isl_give isl_aff *isl_aff_scale_down(__isl_take isl_aff *aff, isl_int f)
+{
+	isl_int gcd;
+
+	if (!aff)
+		return NULL;
+	if (isl_aff_is_nan(aff))
+		return aff;
+
+	if (isl_int_is_one(f))
+		return aff;
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	if (isl_int_is_zero(f))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"cannot scale down by zero", return isl_aff_free(aff));
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	isl_int_init(gcd);
+	isl_seq_gcd(aff->v->el + 1, aff->v->size - 1, &gcd);
+	isl_int_gcd(gcd, gcd, f);
+	isl_seq_scale_down(aff->v->el + 1, aff->v->el + 1, gcd, aff->v->size - 1);
+	isl_int_divexact(gcd, f, gcd);
+	isl_int_mul(aff->v->el[0], aff->v->el[0], gcd);
+	isl_int_clear(gcd);
+
+	return aff;
+}
+
+/* Divide "aff" by "v".
+ */
+__isl_give isl_aff *isl_aff_scale_down_val(__isl_take isl_aff *aff,
+	__isl_take isl_val *v)
+{
+	if (!aff || !v)
+		goto error;
+
+	if (isl_val_is_one(v)) {
+		isl_val_free(v);
+		return aff;
+	}
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"expecting rational factor", goto error);
+	if (!isl_val_is_pos(v))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"factor needs to be positive", goto error);
+
+	aff = isl_aff_scale(aff, v->d);
+	aff = isl_aff_scale_down(aff, v->n);
+
+	isl_val_free(v);
+	return aff;
+error:
+	isl_aff_free(aff);
+	isl_val_free(v);
+	return NULL;
+}
+
+__isl_give isl_aff *isl_aff_scale_down_ui(__isl_take isl_aff *aff, unsigned f)
+{
+	isl_int v;
+
+	if (f == 1)
+		return aff;
+
+	isl_int_init(v);
+	isl_int_set_ui(v, f);
+	aff = isl_aff_scale_down(aff, v);
+	isl_int_clear(v);
+
+	return aff;
+}
+
+__isl_give isl_aff *isl_aff_set_dim_name(__isl_take isl_aff *aff,
+	enum isl_dim_type type, unsigned pos, const char *s)
+{
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"cannot set name of output/set dimension",
+			return isl_aff_free(aff));
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+	aff->ls = isl_local_space_set_dim_name(aff->ls, type, pos, s);
+	if (!aff->ls)
+		return isl_aff_free(aff);
+
+	return aff;
+}
+
+__isl_give isl_aff *isl_aff_set_dim_id(__isl_take isl_aff *aff,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id)
+{
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		goto error;
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"cannot set name of output/set dimension",
+			goto error);
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+	aff->ls = isl_local_space_set_dim_id(aff->ls, type, pos, id);
+	if (!aff->ls)
+		return isl_aff_free(aff);
+
+	return aff;
+error:
+	isl_id_free(id);
+	isl_aff_free(aff);
+	return NULL;
+}
+
+/* Replace the identifier of the input tuple of "aff" by "id".
+ * type is currently required to be equal to isl_dim_in
+ */
+__isl_give isl_aff *isl_aff_set_tuple_id(__isl_take isl_aff *aff,
+	enum isl_dim_type type, __isl_take isl_id *id)
+{
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		goto error;
+	if (type != isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"cannot only set id of input tuple", goto error);
+	aff->ls = isl_local_space_set_tuple_id(aff->ls, isl_dim_set, id);
+	if (!aff->ls)
+		return isl_aff_free(aff);
+
+	return aff;
+error:
+	isl_id_free(id);
+	isl_aff_free(aff);
+	return NULL;
+}
+
+/* Exploit the equalities in "eq" to simplify the affine expression
+ * and the expressions of the integer divisions in the local space.
+ * The integer divisions in this local space are assumed to appear
+ * as regular dimensions in "eq".
+ */
+static __isl_give isl_aff *isl_aff_substitute_equalities_lifted(
+	__isl_take isl_aff *aff, __isl_take isl_basic_set *eq)
+{
+	int i, j;
+	unsigned total;
+	unsigned n_div;
+
+	if (!eq)
+		goto error;
+	if (eq->n_eq == 0) {
+		isl_basic_set_free(eq);
+		return aff;
+	}
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		goto error;
+
+	aff->ls = isl_local_space_substitute_equalities(aff->ls,
+							isl_basic_set_copy(eq));
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->ls || !aff->v)
+		goto error;
+
+	total = 1 + isl_space_dim(eq->dim, isl_dim_all);
+	n_div = eq->n_div;
+	for (i = 0; i < eq->n_eq; ++i) {
+		j = isl_seq_last_non_zero(eq->eq[i], total + n_div);
+		if (j < 0 || j == 0 || j >= total)
+			continue;
+
+		isl_seq_elim(aff->v->el + 1, eq->eq[i], j, total,
+				&aff->v->el[0]);
+	}
+
+	isl_basic_set_free(eq);
+	aff = isl_aff_normalize(aff);
+	return aff;
+error:
+	isl_basic_set_free(eq);
+	isl_aff_free(aff);
+	return NULL;
+}
+
+/* Exploit the equalities in "eq" to simplify the affine expression
+ * and the expressions of the integer divisions in the local space.
+ */
+static __isl_give isl_aff *isl_aff_substitute_equalities(
+	__isl_take isl_aff *aff, __isl_take isl_basic_set *eq)
+{
+	int n_div;
+
+	if (!aff || !eq)
+		goto error;
+	n_div = isl_local_space_dim(aff->ls, isl_dim_div);
+	if (n_div > 0)
+		eq = isl_basic_set_add_dims(eq, isl_dim_set, n_div);
+	return isl_aff_substitute_equalities_lifted(aff, eq);
+error:
+	isl_basic_set_free(eq);
+	isl_aff_free(aff);
+	return NULL;
+}
+
+/* Look for equalities among the variables shared by context and aff
+ * and the integer divisions of aff, if any.
+ * The equalities are then used to eliminate coefficients and/or integer
+ * divisions from aff.
+ */
+__isl_give isl_aff *isl_aff_gist(__isl_take isl_aff *aff,
+	__isl_take isl_set *context)
+{
+	isl_basic_set *hull;
+	int n_div;
+
+	if (!aff)
+		goto error;
+	n_div = isl_local_space_dim(aff->ls, isl_dim_div);
+	if (n_div > 0) {
+		isl_basic_set *bset;
+		isl_local_space *ls;
+		context = isl_set_add_dims(context, isl_dim_set, n_div);
+		ls = isl_aff_get_domain_local_space(aff);
+		bset = isl_basic_set_from_local_space(ls);
+		bset = isl_basic_set_lift(bset);
+		bset = isl_basic_set_flatten(bset);
+		context = isl_set_intersect(context,
+					    isl_set_from_basic_set(bset));
+	}
+
+	hull = isl_set_affine_hull(context);
+	return isl_aff_substitute_equalities_lifted(aff, hull);
+error:
+	isl_aff_free(aff);
+	isl_set_free(context);
+	return NULL;
+}
+
+__isl_give isl_aff *isl_aff_gist_params(__isl_take isl_aff *aff,
+	__isl_take isl_set *context)
+{
+	isl_set *dom_context = isl_set_universe(isl_aff_get_domain_space(aff));
+	dom_context = isl_set_intersect_params(dom_context, context);
+	return isl_aff_gist(aff, dom_context);
+}
+
+/* Return a basic set containing those elements in the space
+ * of aff where it is non-negative.
+ * If "rational" is set, then return a rational basic set.
+ *
+ * If "aff" is NaN, then it is not non-negative (it's not negative either).
+ */
+static __isl_give isl_basic_set *aff_nonneg_basic_set(
+	__isl_take isl_aff *aff, int rational)
+{
+	isl_constraint *ineq;
+	isl_basic_set *bset;
+
+	if (!aff)
+		return NULL;
+	if (isl_aff_is_nan(aff)) {
+		isl_space *space = isl_aff_get_domain_space(aff);
+		isl_aff_free(aff);
+		return isl_basic_set_empty(space);
+	}
+
+	ineq = isl_inequality_from_aff(aff);
+
+	bset = isl_basic_set_from_constraint(ineq);
+	if (rational)
+		bset = isl_basic_set_set_rational(bset);
+	bset = isl_basic_set_simplify(bset);
+	return bset;
+}
+
+/* Return a basic set containing those elements in the space
+ * of aff where it is non-negative.
+ */
+__isl_give isl_basic_set *isl_aff_nonneg_basic_set(__isl_take isl_aff *aff)
+{
+	return aff_nonneg_basic_set(aff, 0);
+}
+
+/* Return a basic set containing those elements in the domain space
+ * of aff where it is negative.
+ */
+__isl_give isl_basic_set *isl_aff_neg_basic_set(__isl_take isl_aff *aff)
+{
+	aff = isl_aff_neg(aff);
+	aff = isl_aff_add_constant_num_si(aff, -1);
+	return isl_aff_nonneg_basic_set(aff);
+}
+
+/* Return a basic set containing those elements in the space
+ * of aff where it is zero.
+ * If "rational" is set, then return a rational basic set.
+ *
+ * If "aff" is NaN, then it is not zero.
+ */
+static __isl_give isl_basic_set *aff_zero_basic_set(__isl_take isl_aff *aff,
+	int rational)
+{
+	isl_constraint *ineq;
+	isl_basic_set *bset;
+
+	if (!aff)
+		return NULL;
+	if (isl_aff_is_nan(aff)) {
+		isl_space *space = isl_aff_get_domain_space(aff);
+		isl_aff_free(aff);
+		return isl_basic_set_empty(space);
+	}
+
+	ineq = isl_equality_from_aff(aff);
+
+	bset = isl_basic_set_from_constraint(ineq);
+	if (rational)
+		bset = isl_basic_set_set_rational(bset);
+	bset = isl_basic_set_simplify(bset);
+	return bset;
+}
+
+/* Return a basic set containing those elements in the space
+ * of aff where it is zero.
+ */
+__isl_give isl_basic_set *isl_aff_zero_basic_set(__isl_take isl_aff *aff)
+{
+	return aff_zero_basic_set(aff, 0);
+}
+
+/* Return a basic set containing those elements in the shared space
+ * of aff1 and aff2 where aff1 is greater than or equal to aff2.
+ */
+__isl_give isl_basic_set *isl_aff_ge_basic_set(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2)
+{
+	aff1 = isl_aff_sub(aff1, aff2);
+
+	return isl_aff_nonneg_basic_set(aff1);
+}
+
+/* Return a basic set containing those elements in the shared space
+ * of aff1 and aff2 where aff1 is smaller than or equal to aff2.
+ */
+__isl_give isl_basic_set *isl_aff_le_basic_set(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2)
+{
+	return isl_aff_ge_basic_set(aff2, aff1);
+}
+
+__isl_give isl_aff *isl_aff_add_on_domain(__isl_keep isl_set *dom,
+	__isl_take isl_aff *aff1, __isl_take isl_aff *aff2)
+{
+	aff1 = isl_aff_add(aff1, aff2);
+	aff1 = isl_aff_gist(aff1, isl_set_copy(dom));
+	return aff1;
+}
+
+int isl_aff_is_empty(__isl_keep isl_aff *aff)
+{
+	if (!aff)
+		return -1;
+
+	return 0;
+}
+
+/* Check whether the given affine expression has non-zero coefficient
+ * for any dimension in the given range or if any of these dimensions
+ * appear with non-zero coefficients in any of the integer divisions
+ * involved in the affine expression.
+ */
+int isl_aff_involves_dims(__isl_keep isl_aff *aff,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+	isl_ctx *ctx;
+	int *active = NULL;
+	int involves = 0;
+
+	if (!aff)
+		return -1;
+	if (n == 0)
+		return 0;
+
+	ctx = isl_aff_get_ctx(aff);
+	if (first + n > isl_aff_dim(aff, type))
+		isl_die(ctx, isl_error_invalid,
+			"range out of bounds", return -1);
+
+	active = isl_local_space_get_active(aff->ls, aff->v->el + 2);
+	if (!active)
+		goto error;
+
+	first += isl_local_space_offset(aff->ls, type) - 1;
+	for (i = 0; i < n; ++i)
+		if (active[first + i]) {
+			involves = 1;
+			break;
+		}
+
+	free(active);
+
+	return involves;
+error:
+	free(active);
+	return -1;
+}
+
+__isl_give isl_aff *isl_aff_drop_dims(__isl_take isl_aff *aff,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	isl_ctx *ctx;
+
+	if (!aff)
+		return NULL;
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"cannot drop output/set dimension",
+			return isl_aff_free(aff));
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+	if (n == 0 && !isl_local_space_is_named_or_nested(aff->ls, type))
+		return aff;
+
+	ctx = isl_aff_get_ctx(aff);
+	if (first + n > isl_local_space_dim(aff->ls, type))
+		isl_die(ctx, isl_error_invalid, "range out of bounds",
+			return isl_aff_free(aff));
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->ls = isl_local_space_drop_dims(aff->ls, type, first, n);
+	if (!aff->ls)
+		return isl_aff_free(aff);
+
+	first += 1 + isl_local_space_offset(aff->ls, type);
+	aff->v = isl_vec_drop_els(aff->v, first, n);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	return aff;
+}
+
+/* Project the domain of the affine expression onto its parameter space.
+ * The affine expression may not involve any of the domain dimensions.
+ */
+__isl_give isl_aff *isl_aff_project_domain_on_params(__isl_take isl_aff *aff)
+{
+	isl_space *space;
+	unsigned n;
+	int involves;
+
+	n = isl_aff_dim(aff, isl_dim_in);
+	involves = isl_aff_involves_dims(aff, isl_dim_in, 0, n);
+	if (involves < 0)
+		return isl_aff_free(aff);
+	if (involves)
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+		    "affine expression involves some of the domain dimensions",
+		    return isl_aff_free(aff));
+	aff = isl_aff_drop_dims(aff, isl_dim_in, 0, n);
+	space = isl_aff_get_domain_space(aff);
+	space = isl_space_params(space);
+	aff = isl_aff_reset_domain_space(aff, space);
+	return aff;
+}
+
+__isl_give isl_aff *isl_aff_insert_dims(__isl_take isl_aff *aff,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	isl_ctx *ctx;
+
+	if (!aff)
+		return NULL;
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"cannot insert output/set dimensions",
+			return isl_aff_free(aff));
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+	if (n == 0 && !isl_local_space_is_named_or_nested(aff->ls, type))
+		return aff;
+
+	ctx = isl_aff_get_ctx(aff);
+	if (first > isl_local_space_dim(aff->ls, type))
+		isl_die(ctx, isl_error_invalid, "position out of bounds",
+			return isl_aff_free(aff));
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->ls = isl_local_space_insert_dims(aff->ls, type, first, n);
+	if (!aff->ls)
+		return isl_aff_free(aff);
+
+	first += 1 + isl_local_space_offset(aff->ls, type);
+	aff->v = isl_vec_insert_zero_els(aff->v, first, n);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	return aff;
+}
+
+__isl_give isl_aff *isl_aff_add_dims(__isl_take isl_aff *aff,
+	enum isl_dim_type type, unsigned n)
+{
+	unsigned pos;
+
+	pos = isl_aff_dim(aff, type);
+
+	return isl_aff_insert_dims(aff, type, pos, n);
+}
+
+__isl_give isl_pw_aff *isl_pw_aff_add_dims(__isl_take isl_pw_aff *pwaff,
+	enum isl_dim_type type, unsigned n)
+{
+	unsigned pos;
+
+	pos = isl_pw_aff_dim(pwaff, type);
+
+	return isl_pw_aff_insert_dims(pwaff, type, pos, n);
+}
+
+/* Move the "n" dimensions of "src_type" starting at "src_pos" of "aff"
+ * to dimensions of "dst_type" at "dst_pos".
+ *
+ * We only support moving input dimensions to parameters and vice versa.
+ */
+__isl_give isl_aff *isl_aff_move_dims(__isl_take isl_aff *aff,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n)
+{
+	unsigned g_dst_pos;
+	unsigned g_src_pos;
+
+	if (!aff)
+		return NULL;
+	if (n == 0 &&
+	    !isl_local_space_is_named_or_nested(aff->ls, src_type) &&
+	    !isl_local_space_is_named_or_nested(aff->ls, dst_type))
+		return aff;
+
+	if (dst_type == isl_dim_out || src_type == isl_dim_out)
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"cannot move output/set dimension", isl_aff_free(aff));
+	if (dst_type == isl_dim_div || src_type == isl_dim_div)
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"cannot move divs", isl_aff_free(aff));
+	if (dst_type == isl_dim_in)
+		dst_type = isl_dim_set;
+	if (src_type == isl_dim_in)
+		src_type = isl_dim_set;
+
+	if (src_pos + n > isl_local_space_dim(aff->ls, src_type))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"range out of bounds", isl_aff_free(aff));
+	if (dst_type == src_type)
+		isl_die(isl_aff_get_ctx(aff), isl_error_unsupported,
+			"moving dims within the same type not supported",
+			isl_aff_free(aff));
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	g_src_pos = 1 + isl_local_space_offset(aff->ls, src_type) + src_pos;
+	g_dst_pos = 1 + isl_local_space_offset(aff->ls, dst_type) + dst_pos;
+	if (dst_type > src_type)
+		g_dst_pos -= n;
+
+	aff->v = isl_vec_move_els(aff->v, g_dst_pos, g_src_pos, n);
+	aff->ls = isl_local_space_move_dims(aff->ls, dst_type, dst_pos,
+						src_type, src_pos, n);
+	if (!aff->v || !aff->ls)
+		return isl_aff_free(aff);
+
+	aff = sort_divs(aff);
+
+	return aff;
+}
+
+__isl_give isl_pw_aff *isl_pw_aff_from_aff(__isl_take isl_aff *aff)
+{
+	isl_set *dom = isl_set_universe(isl_aff_get_domain_space(aff));
+	return isl_pw_aff_alloc(dom, aff);
+}
+
+#undef PW
+#define PW isl_pw_aff
+#undef EL
+#define EL isl_aff
+#undef EL_IS_ZERO
+#define EL_IS_ZERO is_empty
+#undef ZERO
+#define ZERO empty
+#undef IS_ZERO
+#define IS_ZERO is_empty
+#undef FIELD
+#define FIELD aff
+#undef DEFAULT_IS_ZERO
+#define DEFAULT_IS_ZERO 0
+
+#define NO_EVAL
+#define NO_OPT
+#define NO_LIFT
+#define NO_MORPH
+
+#include <isl_pw_templ.c>
+
+static __isl_give isl_set *align_params_pw_pw_set_and(
+	__isl_take isl_pw_aff *pwaff1, __isl_take isl_pw_aff *pwaff2,
+	__isl_give isl_set *(*fn)(__isl_take isl_pw_aff *pwaff1,
+				    __isl_take isl_pw_aff *pwaff2))
+{
+	if (!pwaff1 || !pwaff2)
+		goto error;
+	if (isl_space_match(pwaff1->dim, isl_dim_param,
+			  pwaff2->dim, isl_dim_param))
+		return fn(pwaff1, pwaff2);
+	if (!isl_space_has_named_params(pwaff1->dim) ||
+	    !isl_space_has_named_params(pwaff2->dim))
+		isl_die(isl_pw_aff_get_ctx(pwaff1), isl_error_invalid,
+			"unaligned unnamed parameters", goto error);
+	pwaff1 = isl_pw_aff_align_params(pwaff1, isl_pw_aff_get_space(pwaff2));
+	pwaff2 = isl_pw_aff_align_params(pwaff2, isl_pw_aff_get_space(pwaff1));
+	return fn(pwaff1, pwaff2);
+error:
+	isl_pw_aff_free(pwaff1);
+	isl_pw_aff_free(pwaff2);
+	return NULL;
+}
+
+/* Compute a piecewise quasi-affine expression with a domain that
+ * is the union of those of pwaff1 and pwaff2 and such that on each
+ * cell, the quasi-affine expression is the better (according to cmp)
+ * of those of pwaff1 and pwaff2.  If only one of pwaff1 or pwaff2
+ * is defined on a given cell, then the associated expression
+ * is the defined one.
+ */
+static __isl_give isl_pw_aff *pw_aff_union_opt(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2,
+	__isl_give isl_basic_set *(*cmp)(__isl_take isl_aff *aff1,
+					__isl_take isl_aff *aff2))
+{
+	int i, j, n;
+	isl_pw_aff *res;
+	isl_ctx *ctx;
+	isl_set *set;
+
+	if (!pwaff1 || !pwaff2)
+		goto error;
+
+	ctx = isl_space_get_ctx(pwaff1->dim);
+	if (!isl_space_is_equal(pwaff1->dim, pwaff2->dim))
+		isl_die(ctx, isl_error_invalid,
+			"arguments should live in same space", goto error);
+
+	if (isl_pw_aff_is_empty(pwaff1)) {
+		isl_pw_aff_free(pwaff1);
+		return pwaff2;
+	}
+
+	if (isl_pw_aff_is_empty(pwaff2)) {
+		isl_pw_aff_free(pwaff2);
+		return pwaff1;
+	}
+
+	n = 2 * (pwaff1->n + 1) * (pwaff2->n + 1);
+	res = isl_pw_aff_alloc_size(isl_space_copy(pwaff1->dim), n);
+
+	for (i = 0; i < pwaff1->n; ++i) {
+		set = isl_set_copy(pwaff1->p[i].set);
+		for (j = 0; j < pwaff2->n; ++j) {
+			struct isl_set *common;
+			isl_set *better;
+
+			common = isl_set_intersect(
+					isl_set_copy(pwaff1->p[i].set),
+					isl_set_copy(pwaff2->p[j].set));
+			better = isl_set_from_basic_set(cmp(
+					isl_aff_copy(pwaff2->p[j].aff),
+					isl_aff_copy(pwaff1->p[i].aff)));
+			better = isl_set_intersect(common, better);
+			if (isl_set_plain_is_empty(better)) {
+				isl_set_free(better);
+				continue;
+			}
+			set = isl_set_subtract(set, isl_set_copy(better));
+
+			res = isl_pw_aff_add_piece(res, better,
+						isl_aff_copy(pwaff2->p[j].aff));
+		}
+		res = isl_pw_aff_add_piece(res, set,
+						isl_aff_copy(pwaff1->p[i].aff));
+	}
+
+	for (j = 0; j < pwaff2->n; ++j) {
+		set = isl_set_copy(pwaff2->p[j].set);
+		for (i = 0; i < pwaff1->n; ++i)
+			set = isl_set_subtract(set,
+					isl_set_copy(pwaff1->p[i].set));
+		res = isl_pw_aff_add_piece(res, set,
+						isl_aff_copy(pwaff2->p[j].aff));
+	}
+
+	isl_pw_aff_free(pwaff1);
+	isl_pw_aff_free(pwaff2);
+
+	return res;
+error:
+	isl_pw_aff_free(pwaff1);
+	isl_pw_aff_free(pwaff2);
+	return NULL;
+}
+
+/* Compute a piecewise quasi-affine expression with a domain that
+ * is the union of those of pwaff1 and pwaff2 and such that on each
+ * cell, the quasi-affine expression is the maximum of those of pwaff1
+ * and pwaff2.  If only one of pwaff1 or pwaff2 is defined on a given
+ * cell, then the associated expression is the defined one.
+ */
+static __isl_give isl_pw_aff *pw_aff_union_max(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return pw_aff_union_opt(pwaff1, pwaff2, &isl_aff_ge_basic_set);
+}
+
+__isl_give isl_pw_aff *isl_pw_aff_union_max(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return isl_pw_aff_align_params_pw_pw_and(pwaff1, pwaff2,
+							&pw_aff_union_max);
+}
+
+/* Compute a piecewise quasi-affine expression with a domain that
+ * is the union of those of pwaff1 and pwaff2 and such that on each
+ * cell, the quasi-affine expression is the minimum of those of pwaff1
+ * and pwaff2.  If only one of pwaff1 or pwaff2 is defined on a given
+ * cell, then the associated expression is the defined one.
+ */
+static __isl_give isl_pw_aff *pw_aff_union_min(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return pw_aff_union_opt(pwaff1, pwaff2, &isl_aff_le_basic_set);
+}
+
+__isl_give isl_pw_aff *isl_pw_aff_union_min(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return isl_pw_aff_align_params_pw_pw_and(pwaff1, pwaff2,
+							&pw_aff_union_min);
+}
+
+__isl_give isl_pw_aff *isl_pw_aff_union_opt(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2, int max)
+{
+	if (max)
+		return isl_pw_aff_union_max(pwaff1, pwaff2);
+	else
+		return isl_pw_aff_union_min(pwaff1, pwaff2);
+}
+
+/* Construct a map with as domain the domain of pwaff and
+ * one-dimensional range corresponding to the affine expressions.
+ */
+static __isl_give isl_map *map_from_pw_aff(__isl_take isl_pw_aff *pwaff)
+{
+	int i;
+	isl_space *dim;
+	isl_map *map;
+
+	if (!pwaff)
+		return NULL;
+
+	dim = isl_pw_aff_get_space(pwaff);
+	map = isl_map_empty(dim);
+
+	for (i = 0; i < pwaff->n; ++i) {
+		isl_basic_map *bmap;
+		isl_map *map_i;
+
+		bmap = isl_basic_map_from_aff(isl_aff_copy(pwaff->p[i].aff));
+		map_i = isl_map_from_basic_map(bmap);
+		map_i = isl_map_intersect_domain(map_i,
+						isl_set_copy(pwaff->p[i].set));
+		map = isl_map_union_disjoint(map, map_i);
+	}
+
+	isl_pw_aff_free(pwaff);
+
+	return map;
+}
+
+/* Construct a map with as domain the domain of pwaff and
+ * one-dimensional range corresponding to the affine expressions.
+ */
+__isl_give isl_map *isl_map_from_pw_aff(__isl_take isl_pw_aff *pwaff)
+{
+	if (!pwaff)
+		return NULL;
+	if (isl_space_is_set(pwaff->dim))
+		isl_die(isl_pw_aff_get_ctx(pwaff), isl_error_invalid,
+			"space of input is not a map", goto error);
+	return map_from_pw_aff(pwaff);
+error:
+	isl_pw_aff_free(pwaff);
+	return NULL;
+}
+
+/* Construct a one-dimensional set with as parameter domain
+ * the domain of pwaff and the single set dimension
+ * corresponding to the affine expressions.
+ */
+__isl_give isl_set *isl_set_from_pw_aff(__isl_take isl_pw_aff *pwaff)
+{
+	if (!pwaff)
+		return NULL;
+	if (!isl_space_is_set(pwaff->dim))
+		isl_die(isl_pw_aff_get_ctx(pwaff), isl_error_invalid,
+			"space of input is not a set", goto error);
+	return map_from_pw_aff(pwaff);
+error:
+	isl_pw_aff_free(pwaff);
+	return NULL;
+}
+
+/* Return a set containing those elements in the domain
+ * of pwaff where it is non-negative.
+ */
+__isl_give isl_set *isl_pw_aff_nonneg_set(__isl_take isl_pw_aff *pwaff)
+{
+	int i;
+	isl_set *set;
+
+	if (!pwaff)
+		return NULL;
+
+	set = isl_set_empty(isl_pw_aff_get_domain_space(pwaff));
+
+	for (i = 0; i < pwaff->n; ++i) {
+		isl_basic_set *bset;
+		isl_set *set_i;
+		int rational;
+
+		rational = isl_set_has_rational(pwaff->p[i].set);
+		bset = aff_nonneg_basic_set(isl_aff_copy(pwaff->p[i].aff),
+						rational);
+		set_i = isl_set_from_basic_set(bset);
+		set_i = isl_set_intersect(set_i, isl_set_copy(pwaff->p[i].set));
+		set = isl_set_union_disjoint(set, set_i);
+	}
+
+	isl_pw_aff_free(pwaff);
+
+	return set;
+}
+
+/* Return a set containing those elements in the domain
+ * of pwaff where it is zero (if complement is 0) or not zero
+ * (if complement is 1).
+ *
+ * The pieces with a NaN never belong to the result since
+ * NaN is neither zero nor non-zero.
+ */
+static __isl_give isl_set *pw_aff_zero_set(__isl_take isl_pw_aff *pwaff,
+	int complement)
+{
+	int i;
+	isl_set *set;
+
+	if (!pwaff)
+		return NULL;
+
+	set = isl_set_empty(isl_pw_aff_get_domain_space(pwaff));
+
+	for (i = 0; i < pwaff->n; ++i) {
+		isl_basic_set *bset;
+		isl_set *set_i, *zero;
+		int rational;
+
+		if (isl_aff_is_nan(pwaff->p[i].aff))
+			continue;
+
+		rational = isl_set_has_rational(pwaff->p[i].set);
+		bset = aff_zero_basic_set(isl_aff_copy(pwaff->p[i].aff),
+						rational);
+		zero = isl_set_from_basic_set(bset);
+		set_i = isl_set_copy(pwaff->p[i].set);
+		if (complement)
+			set_i = isl_set_subtract(set_i, zero);
+		else
+			set_i = isl_set_intersect(set_i, zero);
+		set = isl_set_union_disjoint(set, set_i);
+	}
+
+	isl_pw_aff_free(pwaff);
+
+	return set;
+}
+
+/* Return a set containing those elements in the domain
+ * of pwaff where it is zero.
+ */
+__isl_give isl_set *isl_pw_aff_zero_set(__isl_take isl_pw_aff *pwaff)
+{
+	return pw_aff_zero_set(pwaff, 0);
+}
+
+/* Return a set containing those elements in the domain
+ * of pwaff where it is not zero.
+ */
+__isl_give isl_set *isl_pw_aff_non_zero_set(__isl_take isl_pw_aff *pwaff)
+{
+	return pw_aff_zero_set(pwaff, 1);
+}
+
+/* Return a set containing those elements in the shared domain
+ * of pwaff1 and pwaff2 where pwaff1 is greater than (or equal) to pwaff2.
+ *
+ * We compute the difference on the shared domain and then construct
+ * the set of values where this difference is non-negative.
+ * If strict is set, we first subtract 1 from the difference.
+ * If equal is set, we only return the elements where pwaff1 and pwaff2
+ * are equal.
+ */
+static __isl_give isl_set *pw_aff_gte_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2, int strict, int equal)
+{
+	isl_set *set1, *set2;
+
+	set1 = isl_pw_aff_domain(isl_pw_aff_copy(pwaff1));
+	set2 = isl_pw_aff_domain(isl_pw_aff_copy(pwaff2));
+	set1 = isl_set_intersect(set1, set2);
+	pwaff1 = isl_pw_aff_intersect_domain(pwaff1, isl_set_copy(set1));
+	pwaff2 = isl_pw_aff_intersect_domain(pwaff2, isl_set_copy(set1));
+	pwaff1 = isl_pw_aff_add(pwaff1, isl_pw_aff_neg(pwaff2));
+
+	if (strict) {
+		isl_space *dim = isl_set_get_space(set1);
+		isl_aff *aff;
+		aff = isl_aff_zero_on_domain(isl_local_space_from_space(dim));
+		aff = isl_aff_add_constant_si(aff, -1);
+		pwaff1 = isl_pw_aff_add(pwaff1, isl_pw_aff_alloc(set1, aff));
+	} else
+		isl_set_free(set1);
+
+	if (equal)
+		return isl_pw_aff_zero_set(pwaff1);
+	return isl_pw_aff_nonneg_set(pwaff1);
+}
+
+/* Return a set containing those elements in the shared domain
+ * of pwaff1 and pwaff2 where pwaff1 is equal to pwaff2.
+ */
+static __isl_give isl_set *pw_aff_eq_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return pw_aff_gte_set(pwaff1, pwaff2, 0, 1);
+}
+
+__isl_give isl_set *isl_pw_aff_eq_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return align_params_pw_pw_set_and(pwaff1, pwaff2, &pw_aff_eq_set);
+}
+
+/* Return a set containing those elements in the shared domain
+ * of pwaff1 and pwaff2 where pwaff1 is greater than or equal to pwaff2.
+ */
+static __isl_give isl_set *pw_aff_ge_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return pw_aff_gte_set(pwaff1, pwaff2, 0, 0);
+}
+
+__isl_give isl_set *isl_pw_aff_ge_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return align_params_pw_pw_set_and(pwaff1, pwaff2, &pw_aff_ge_set);
+}
+
+/* Return a set containing those elements in the shared domain
+ * of pwaff1 and pwaff2 where pwaff1 is strictly greater than pwaff2.
+ */
+static __isl_give isl_set *pw_aff_gt_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return pw_aff_gte_set(pwaff1, pwaff2, 1, 0);
+}
+
+__isl_give isl_set *isl_pw_aff_gt_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return align_params_pw_pw_set_and(pwaff1, pwaff2, &pw_aff_gt_set);
+}
+
+__isl_give isl_set *isl_pw_aff_le_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return isl_pw_aff_ge_set(pwaff2, pwaff1);
+}
+
+__isl_give isl_set *isl_pw_aff_lt_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return isl_pw_aff_gt_set(pwaff2, pwaff1);
+}
+
+/* Return a set containing those elements in the shared domain
+ * of the elements of list1 and list2 where each element in list1
+ * has the relation specified by "fn" with each element in list2.
+ */
+static __isl_give isl_set *pw_aff_list_set(__isl_take isl_pw_aff_list *list1,
+	__isl_take isl_pw_aff_list *list2,
+	__isl_give isl_set *(*fn)(__isl_take isl_pw_aff *pwaff1,
+				    __isl_take isl_pw_aff *pwaff2))
+{
+	int i, j;
+	isl_ctx *ctx;
+	isl_set *set;
+
+	if (!list1 || !list2)
+		goto error;
+
+	ctx = isl_pw_aff_list_get_ctx(list1);
+	if (list1->n < 1 || list2->n < 1)
+		isl_die(ctx, isl_error_invalid,
+			"list should contain at least one element", goto error);
+
+	set = isl_set_universe(isl_pw_aff_get_domain_space(list1->p[0]));
+	for (i = 0; i < list1->n; ++i)
+		for (j = 0; j < list2->n; ++j) {
+			isl_set *set_ij;
+
+			set_ij = fn(isl_pw_aff_copy(list1->p[i]),
+				    isl_pw_aff_copy(list2->p[j]));
+			set = isl_set_intersect(set, set_ij);
+		}
+
+	isl_pw_aff_list_free(list1);
+	isl_pw_aff_list_free(list2);
+	return set;
+error:
+	isl_pw_aff_list_free(list1);
+	isl_pw_aff_list_free(list2);
+	return NULL;
+}
+
+/* Return a set containing those elements in the shared domain
+ * of the elements of list1 and list2 where each element in list1
+ * is equal to each element in list2.
+ */
+__isl_give isl_set *isl_pw_aff_list_eq_set(__isl_take isl_pw_aff_list *list1,
+	__isl_take isl_pw_aff_list *list2)
+{
+	return pw_aff_list_set(list1, list2, &isl_pw_aff_eq_set);
+}
+
+__isl_give isl_set *isl_pw_aff_list_ne_set(__isl_take isl_pw_aff_list *list1,
+	__isl_take isl_pw_aff_list *list2)
+{
+	return pw_aff_list_set(list1, list2, &isl_pw_aff_ne_set);
+}
+
+/* Return a set containing those elements in the shared domain
+ * of the elements of list1 and list2 where each element in list1
+ * is less than or equal to each element in list2.
+ */
+__isl_give isl_set *isl_pw_aff_list_le_set(__isl_take isl_pw_aff_list *list1,
+	__isl_take isl_pw_aff_list *list2)
+{
+	return pw_aff_list_set(list1, list2, &isl_pw_aff_le_set);
+}
+
+__isl_give isl_set *isl_pw_aff_list_lt_set(__isl_take isl_pw_aff_list *list1,
+	__isl_take isl_pw_aff_list *list2)
+{
+	return pw_aff_list_set(list1, list2, &isl_pw_aff_lt_set);
+}
+
+__isl_give isl_set *isl_pw_aff_list_ge_set(__isl_take isl_pw_aff_list *list1,
+	__isl_take isl_pw_aff_list *list2)
+{
+	return pw_aff_list_set(list1, list2, &isl_pw_aff_ge_set);
+}
+
+__isl_give isl_set *isl_pw_aff_list_gt_set(__isl_take isl_pw_aff_list *list1,
+	__isl_take isl_pw_aff_list *list2)
+{
+	return pw_aff_list_set(list1, list2, &isl_pw_aff_gt_set);
+}
+
+
+/* Return a set containing those elements in the shared domain
+ * of pwaff1 and pwaff2 where pwaff1 is not equal to pwaff2.
+ */
+static __isl_give isl_set *pw_aff_ne_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	isl_set *set_lt, *set_gt;
+
+	set_lt = isl_pw_aff_lt_set(isl_pw_aff_copy(pwaff1),
+				   isl_pw_aff_copy(pwaff2));
+	set_gt = isl_pw_aff_gt_set(pwaff1, pwaff2);
+	return isl_set_union_disjoint(set_lt, set_gt);
+}
+
+__isl_give isl_set *isl_pw_aff_ne_set(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return align_params_pw_pw_set_and(pwaff1, pwaff2, &pw_aff_ne_set);
+}
+
+__isl_give isl_pw_aff *isl_pw_aff_scale_down(__isl_take isl_pw_aff *pwaff,
+	isl_int v)
+{
+	int i;
+
+	if (isl_int_is_one(v))
+		return pwaff;
+	if (!isl_int_is_pos(v))
+		isl_die(isl_pw_aff_get_ctx(pwaff), isl_error_invalid,
+			"factor needs to be positive",
+			return isl_pw_aff_free(pwaff));
+	pwaff = isl_pw_aff_cow(pwaff);
+	if (!pwaff)
+		return NULL;
+	if (pwaff->n == 0)
+		return pwaff;
+
+	for (i = 0; i < pwaff->n; ++i) {
+		pwaff->p[i].aff = isl_aff_scale_down(pwaff->p[i].aff, v);
+		if (!pwaff->p[i].aff)
+			return isl_pw_aff_free(pwaff);
+	}
+
+	return pwaff;
+}
+
+__isl_give isl_pw_aff *isl_pw_aff_floor(__isl_take isl_pw_aff *pwaff)
+{
+	int i;
+
+	pwaff = isl_pw_aff_cow(pwaff);
+	if (!pwaff)
+		return NULL;
+	if (pwaff->n == 0)
+		return pwaff;
+
+	for (i = 0; i < pwaff->n; ++i) {
+		pwaff->p[i].aff = isl_aff_floor(pwaff->p[i].aff);
+		if (!pwaff->p[i].aff)
+			return isl_pw_aff_free(pwaff);
+	}
+
+	return pwaff;
+}
+
+__isl_give isl_pw_aff *isl_pw_aff_ceil(__isl_take isl_pw_aff *pwaff)
+{
+	int i;
+
+	pwaff = isl_pw_aff_cow(pwaff);
+	if (!pwaff)
+		return NULL;
+	if (pwaff->n == 0)
+		return pwaff;
+
+	for (i = 0; i < pwaff->n; ++i) {
+		pwaff->p[i].aff = isl_aff_ceil(pwaff->p[i].aff);
+		if (!pwaff->p[i].aff)
+			return isl_pw_aff_free(pwaff);
+	}
+
+	return pwaff;
+}
+
+/* Assuming that "cond1" and "cond2" are disjoint,
+ * return an affine expression that is equal to pwaff1 on cond1
+ * and to pwaff2 on cond2.
+ */
+static __isl_give isl_pw_aff *isl_pw_aff_select(
+	__isl_take isl_set *cond1, __isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_set *cond2, __isl_take isl_pw_aff *pwaff2)
+{
+	pwaff1 = isl_pw_aff_intersect_domain(pwaff1, cond1);
+	pwaff2 = isl_pw_aff_intersect_domain(pwaff2, cond2);
+
+	return isl_pw_aff_add_disjoint(pwaff1, pwaff2);
+}
+
+/* Return an affine expression that is equal to pwaff_true for elements
+ * where "cond" is non-zero and to pwaff_false for elements where "cond"
+ * is zero.
+ * That is, return cond ? pwaff_true : pwaff_false;
+ *
+ * If "cond" involves and NaN, then we conservatively return a NaN
+ * on its entire domain.  In principle, we could consider the pieces
+ * where it is NaN separately from those where it is not.
+ */
+__isl_give isl_pw_aff *isl_pw_aff_cond(__isl_take isl_pw_aff *cond,
+	__isl_take isl_pw_aff *pwaff_true, __isl_take isl_pw_aff *pwaff_false)
+{
+	isl_set *cond_true, *cond_false;
+
+	if (!cond)
+		goto error;
+	if (isl_pw_aff_involves_nan(cond)) {
+		isl_space *space = isl_pw_aff_get_domain_space(cond);
+		isl_local_space *ls = isl_local_space_from_space(space);
+		isl_pw_aff_free(cond);
+		isl_pw_aff_free(pwaff_true);
+		isl_pw_aff_free(pwaff_false);
+		return isl_pw_aff_nan_on_domain(ls);
+	}
+
+	cond_true = isl_pw_aff_non_zero_set(isl_pw_aff_copy(cond));
+	cond_false = isl_pw_aff_zero_set(cond);
+	return isl_pw_aff_select(cond_true, pwaff_true,
+				 cond_false, pwaff_false);
+error:
+	isl_pw_aff_free(cond);
+	isl_pw_aff_free(pwaff_true);
+	isl_pw_aff_free(pwaff_false);
+	return NULL;
+}
+
+int isl_aff_is_cst(__isl_keep isl_aff *aff)
+{
+	if (!aff)
+		return -1;
+
+	return isl_seq_first_non_zero(aff->v->el + 2, aff->v->size - 2) == -1;
+}
+
+/* Check whether pwaff is a piecewise constant.
+ */
+int isl_pw_aff_is_cst(__isl_keep isl_pw_aff *pwaff)
+{
+	int i;
+
+	if (!pwaff)
+		return -1;
+
+	for (i = 0; i < pwaff->n; ++i) {
+		int is_cst = isl_aff_is_cst(pwaff->p[i].aff);
+		if (is_cst < 0 || !is_cst)
+			return is_cst;
+	}
+
+	return 1;
+}
+
+/* Return the product of "aff1" and "aff2".
+ *
+ * If either of the two is NaN, then the result is NaN.
+ *
+ * Otherwise, at least one of "aff1" or "aff2" needs to be a constant.
+ */
+__isl_give isl_aff *isl_aff_mul(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2)
+{
+	if (!aff1 || !aff2)
+		goto error;
+
+	if (isl_aff_is_nan(aff1)) {
+		isl_aff_free(aff2);
+		return aff1;
+	}
+	if (isl_aff_is_nan(aff2)) {
+		isl_aff_free(aff1);
+		return aff2;
+	}
+
+	if (!isl_aff_is_cst(aff2) && isl_aff_is_cst(aff1))
+		return isl_aff_mul(aff2, aff1);
+
+	if (!isl_aff_is_cst(aff2))
+		isl_die(isl_aff_get_ctx(aff1), isl_error_invalid,
+			"at least one affine expression should be constant",
+			goto error);
+
+	aff1 = isl_aff_cow(aff1);
+	if (!aff1 || !aff2)
+		goto error;
+
+	aff1 = isl_aff_scale(aff1, aff2->v->el[1]);
+	aff1 = isl_aff_scale_down(aff1, aff2->v->el[0]);
+
+	isl_aff_free(aff2);
+	return aff1;
+error:
+	isl_aff_free(aff1);
+	isl_aff_free(aff2);
+	return NULL;
+}
+
+/* Divide "aff1" by "aff2", assuming "aff2" is a constant.
+ *
+ * If either of the two is NaN, then the result is NaN.
+ */
+__isl_give isl_aff *isl_aff_div(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2)
+{
+	int is_cst;
+	int neg;
+
+	if (!aff1 || !aff2)
+		goto error;
+
+	if (isl_aff_is_nan(aff1)) {
+		isl_aff_free(aff2);
+		return aff1;
+	}
+	if (isl_aff_is_nan(aff2)) {
+		isl_aff_free(aff1);
+		return aff2;
+	}
+
+	is_cst = isl_aff_is_cst(aff2);
+	if (is_cst < 0)
+		goto error;
+	if (!is_cst)
+		isl_die(isl_aff_get_ctx(aff2), isl_error_invalid,
+			"second argument should be a constant", goto error);
+
+	if (!aff2)
+		goto error;
+
+	neg = isl_int_is_neg(aff2->v->el[1]);
+	if (neg) {
+		isl_int_neg(aff2->v->el[0], aff2->v->el[0]);
+		isl_int_neg(aff2->v->el[1], aff2->v->el[1]);
+	}
+
+	aff1 = isl_aff_scale(aff1, aff2->v->el[0]);
+	aff1 = isl_aff_scale_down(aff1, aff2->v->el[1]);
+
+	if (neg) {
+		isl_int_neg(aff2->v->el[0], aff2->v->el[0]);
+		isl_int_neg(aff2->v->el[1], aff2->v->el[1]);
+	}
+
+	isl_aff_free(aff2);
+	return aff1;
+error:
+	isl_aff_free(aff1);
+	isl_aff_free(aff2);
+	return NULL;
+}
+
+static __isl_give isl_pw_aff *pw_aff_add(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return isl_pw_aff_on_shared_domain(pwaff1, pwaff2, &isl_aff_add);
+}
+
+__isl_give isl_pw_aff *isl_pw_aff_add(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return isl_pw_aff_align_params_pw_pw_and(pwaff1, pwaff2, &pw_aff_add);
+}
+
+__isl_give isl_pw_aff *isl_pw_aff_union_add(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return isl_pw_aff_union_add_(pwaff1, pwaff2);
+}
+
+static __isl_give isl_pw_aff *pw_aff_mul(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return isl_pw_aff_on_shared_domain(pwaff1, pwaff2, &isl_aff_mul);
+}
+
+__isl_give isl_pw_aff *isl_pw_aff_mul(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return isl_pw_aff_align_params_pw_pw_and(pwaff1, pwaff2, &pw_aff_mul);
+}
+
+static __isl_give isl_pw_aff *pw_aff_div(__isl_take isl_pw_aff *pa1,
+	__isl_take isl_pw_aff *pa2)
+{
+	return isl_pw_aff_on_shared_domain(pa1, pa2, &isl_aff_div);
+}
+
+/* Divide "pa1" by "pa2", assuming "pa2" is a piecewise constant.
+ */
+__isl_give isl_pw_aff *isl_pw_aff_div(__isl_take isl_pw_aff *pa1,
+	__isl_take isl_pw_aff *pa2)
+{
+	int is_cst;
+
+	is_cst = isl_pw_aff_is_cst(pa2);
+	if (is_cst < 0)
+		goto error;
+	if (!is_cst)
+		isl_die(isl_pw_aff_get_ctx(pa2), isl_error_invalid,
+			"second argument should be a piecewise constant",
+			goto error);
+	return isl_pw_aff_align_params_pw_pw_and(pa1, pa2, &pw_aff_div);
+error:
+	isl_pw_aff_free(pa1);
+	isl_pw_aff_free(pa2);
+	return NULL;
+}
+
+/* Compute the quotient of the integer division of "pa1" by "pa2"
+ * with rounding towards zero.
+ * "pa2" is assumed to be a piecewise constant.
+ *
+ * In particular, return
+ *
+ *	pa1 >= 0 ? floor(pa1/pa2) : ceil(pa1/pa2)
+ *
+ */
+__isl_give isl_pw_aff *isl_pw_aff_tdiv_q(__isl_take isl_pw_aff *pa1,
+	__isl_take isl_pw_aff *pa2)
+{
+	int is_cst;
+	isl_set *cond;
+	isl_pw_aff *f, *c;
+
+	is_cst = isl_pw_aff_is_cst(pa2);
+	if (is_cst < 0)
+		goto error;
+	if (!is_cst)
+		isl_die(isl_pw_aff_get_ctx(pa2), isl_error_invalid,
+			"second argument should be a piecewise constant",
+			goto error);
+
+	pa1 = isl_pw_aff_div(pa1, pa2);
+
+	cond = isl_pw_aff_nonneg_set(isl_pw_aff_copy(pa1));
+	f = isl_pw_aff_floor(isl_pw_aff_copy(pa1));
+	c = isl_pw_aff_ceil(pa1);
+	return isl_pw_aff_cond(isl_set_indicator_function(cond), f, c);
+error:
+	isl_pw_aff_free(pa1);
+	isl_pw_aff_free(pa2);
+	return NULL;
+}
+
+/* Compute the remainder of the integer division of "pa1" by "pa2"
+ * with rounding towards zero.
+ * "pa2" is assumed to be a piecewise constant.
+ *
+ * In particular, return
+ *
+ *	pa1 - pa2 * (pa1 >= 0 ? floor(pa1/pa2) : ceil(pa1/pa2))
+ *
+ */
+__isl_give isl_pw_aff *isl_pw_aff_tdiv_r(__isl_take isl_pw_aff *pa1,
+	__isl_take isl_pw_aff *pa2)
+{
+	int is_cst;
+	isl_pw_aff *res;
+
+	is_cst = isl_pw_aff_is_cst(pa2);
+	if (is_cst < 0)
+		goto error;
+	if (!is_cst)
+		isl_die(isl_pw_aff_get_ctx(pa2), isl_error_invalid,
+			"second argument should be a piecewise constant",
+			goto error);
+	res = isl_pw_aff_tdiv_q(isl_pw_aff_copy(pa1), isl_pw_aff_copy(pa2));
+	res = isl_pw_aff_mul(pa2, res);
+	res = isl_pw_aff_sub(pa1, res);
+	return res;
+error:
+	isl_pw_aff_free(pa1);
+	isl_pw_aff_free(pa2);
+	return NULL;
+}
+
+static __isl_give isl_pw_aff *pw_aff_min(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	isl_set *le;
+	isl_set *dom;
+
+	dom = isl_set_intersect(isl_pw_aff_domain(isl_pw_aff_copy(pwaff1)),
+				isl_pw_aff_domain(isl_pw_aff_copy(pwaff2)));
+	le = isl_pw_aff_le_set(isl_pw_aff_copy(pwaff1),
+				isl_pw_aff_copy(pwaff2));
+	dom = isl_set_subtract(dom, isl_set_copy(le));
+	return isl_pw_aff_select(le, pwaff1, dom, pwaff2);
+}
+
+__isl_give isl_pw_aff *isl_pw_aff_min(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return isl_pw_aff_align_params_pw_pw_and(pwaff1, pwaff2, &pw_aff_min);
+}
+
+static __isl_give isl_pw_aff *pw_aff_max(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	isl_set *ge;
+	isl_set *dom;
+
+	dom = isl_set_intersect(isl_pw_aff_domain(isl_pw_aff_copy(pwaff1)),
+				isl_pw_aff_domain(isl_pw_aff_copy(pwaff2)));
+	ge = isl_pw_aff_ge_set(isl_pw_aff_copy(pwaff1),
+				isl_pw_aff_copy(pwaff2));
+	dom = isl_set_subtract(dom, isl_set_copy(ge));
+	return isl_pw_aff_select(ge, pwaff1, dom, pwaff2);
+}
+
+__isl_give isl_pw_aff *isl_pw_aff_max(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return isl_pw_aff_align_params_pw_pw_and(pwaff1, pwaff2, &pw_aff_max);
+}
+
+static __isl_give isl_pw_aff *pw_aff_list_reduce(
+	__isl_take isl_pw_aff_list *list,
+	__isl_give isl_pw_aff *(*fn)(__isl_take isl_pw_aff *pwaff1,
+					__isl_take isl_pw_aff *pwaff2))
+{
+	int i;
+	isl_ctx *ctx;
+	isl_pw_aff *res;
+
+	if (!list)
+		return NULL;
+
+	ctx = isl_pw_aff_list_get_ctx(list);
+	if (list->n < 1)
+		isl_die(ctx, isl_error_invalid,
+			"list should contain at least one element", goto error);
+
+	res = isl_pw_aff_copy(list->p[0]);
+	for (i = 1; i < list->n; ++i)
+		res = fn(res, isl_pw_aff_copy(list->p[i]));
+
+	isl_pw_aff_list_free(list);
+	return res;
+error:
+	isl_pw_aff_list_free(list);
+	return NULL;
+}
+
+/* Return an isl_pw_aff that maps each element in the intersection of the
+ * domains of the elements of list to the minimal corresponding affine
+ * expression.
+ */
+__isl_give isl_pw_aff *isl_pw_aff_list_min(__isl_take isl_pw_aff_list *list)
+{
+	return pw_aff_list_reduce(list, &isl_pw_aff_min);
+}
+
+/* Return an isl_pw_aff that maps each element in the intersection of the
+ * domains of the elements of list to the maximal corresponding affine
+ * expression.
+ */
+__isl_give isl_pw_aff *isl_pw_aff_list_max(__isl_take isl_pw_aff_list *list)
+{
+	return pw_aff_list_reduce(list, &isl_pw_aff_max);
+}
+
+/* Mark the domains of "pwaff" as rational.
+ */
+__isl_give isl_pw_aff *isl_pw_aff_set_rational(__isl_take isl_pw_aff *pwaff)
+{
+	int i;
+
+	pwaff = isl_pw_aff_cow(pwaff);
+	if (!pwaff)
+		return NULL;
+	if (pwaff->n == 0)
+		return pwaff;
+
+	for (i = 0; i < pwaff->n; ++i) {
+		pwaff->p[i].set = isl_set_set_rational(pwaff->p[i].set);
+		if (!pwaff->p[i].set)
+			return isl_pw_aff_free(pwaff);
+	}
+
+	return pwaff;
+}
+
+/* Mark the domains of the elements of "list" as rational.
+ */
+__isl_give isl_pw_aff_list *isl_pw_aff_list_set_rational(
+	__isl_take isl_pw_aff_list *list)
+{
+	int i, n;
+
+	if (!list)
+		return NULL;
+	if (list->n == 0)
+		return list;
+
+	n = list->n;
+	for (i = 0; i < n; ++i) {
+		isl_pw_aff *pa;
+
+		pa = isl_pw_aff_list_get_pw_aff(list, i);
+		pa = isl_pw_aff_set_rational(pa);
+		list = isl_pw_aff_list_set_pw_aff(list, i, pa);
+	}
+
+	return list;
+}
+
+/* Do the parameters of "aff" match those of "space"?
+ */
+int isl_aff_matching_params(__isl_keep isl_aff *aff,
+	__isl_keep isl_space *space)
+{
+	isl_space *aff_space;
+	int match;
+
+	if (!aff || !space)
+		return -1;
+
+	aff_space = isl_aff_get_domain_space(aff);
+
+	match = isl_space_match(space, isl_dim_param, aff_space, isl_dim_param);
+
+	isl_space_free(aff_space);
+	return match;
+}
+
+/* Check that the domain space of "aff" matches "space".
+ *
+ * Return 0 on success and -1 on error.
+ */
+int isl_aff_check_match_domain_space(__isl_keep isl_aff *aff,
+	__isl_keep isl_space *space)
+{
+	isl_space *aff_space;
+	int match;
+
+	if (!aff || !space)
+		return -1;
+
+	aff_space = isl_aff_get_domain_space(aff);
+
+	match = isl_space_match(space, isl_dim_param, aff_space, isl_dim_param);
+	if (match < 0)
+		goto error;
+	if (!match)
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"parameters don't match", goto error);
+	match = isl_space_tuple_is_equal(space, isl_dim_in,
+					aff_space, isl_dim_set);
+	if (match < 0)
+		goto error;
+	if (!match)
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"domains don't match", goto error);
+	isl_space_free(aff_space);
+	return 0;
+error:
+	isl_space_free(aff_space);
+	return -1;
+}
+
+#undef BASE
+#define BASE aff
+#define NO_INTERSECT_DOMAIN
+#define NO_DOMAIN
+
+#include <isl_multi_templ.c>
+
+#undef NO_DOMAIN
+#undef NO_INTERSECT_DOMAIN
+
+/* Remove any internal structure of the domain of "ma".
+ * If there is any such internal structure in the input,
+ * then the name of the corresponding space is also removed.
+ */
+__isl_give isl_multi_aff *isl_multi_aff_flatten_domain(
+	__isl_take isl_multi_aff *ma)
+{
+	isl_space *space;
+
+	if (!ma)
+		return NULL;
+
+	if (!ma->space->nested[0])
+		return ma;
+
+	space = isl_multi_aff_get_space(ma);
+	space = isl_space_flatten_domain(space);
+	ma = isl_multi_aff_reset_space(ma, space);
+
+	return ma;
+}
+
+/* Given a map space, return an isl_multi_aff that maps a wrapped copy
+ * of the space to its domain.
+ */
+__isl_give isl_multi_aff *isl_multi_aff_domain_map(__isl_take isl_space *space)
+{
+	int i, n_in;
+	isl_local_space *ls;
+	isl_multi_aff *ma;
+
+	if (!space)
+		return NULL;
+	if (!isl_space_is_map(space))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"not a map space", goto error);
+
+	n_in = isl_space_dim(space, isl_dim_in);
+	space = isl_space_domain_map(space);
+
+	ma = isl_multi_aff_alloc(isl_space_copy(space));
+	if (n_in == 0) {
+		isl_space_free(space);
+		return ma;
+	}
+
+	space = isl_space_domain(space);
+	ls = isl_local_space_from_space(space);
+	for (i = 0; i < n_in; ++i) {
+		isl_aff *aff;
+
+		aff = isl_aff_var_on_domain(isl_local_space_copy(ls),
+						isl_dim_set, i);
+		ma = isl_multi_aff_set_aff(ma, i, aff);
+	}
+	isl_local_space_free(ls);
+	return ma;
+error:
+	isl_space_free(space);
+	return NULL;
+}
+
+/* Given a map space, return an isl_multi_aff that maps a wrapped copy
+ * of the space to its range.
+ */
+__isl_give isl_multi_aff *isl_multi_aff_range_map(__isl_take isl_space *space)
+{
+	int i, n_in, n_out;
+	isl_local_space *ls;
+	isl_multi_aff *ma;
+
+	if (!space)
+		return NULL;
+	if (!isl_space_is_map(space))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"not a map space", goto error);
+
+	n_in = isl_space_dim(space, isl_dim_in);
+	n_out = isl_space_dim(space, isl_dim_out);
+	space = isl_space_range_map(space);
+
+	ma = isl_multi_aff_alloc(isl_space_copy(space));
+	if (n_out == 0) {
+		isl_space_free(space);
+		return ma;
+	}
+
+	space = isl_space_domain(space);
+	ls = isl_local_space_from_space(space);
+	for (i = 0; i < n_out; ++i) {
+		isl_aff *aff;
+
+		aff = isl_aff_var_on_domain(isl_local_space_copy(ls),
+						isl_dim_set, n_in + i);
+		ma = isl_multi_aff_set_aff(ma, i, aff);
+	}
+	isl_local_space_free(ls);
+	return ma;
+error:
+	isl_space_free(space);
+	return NULL;
+}
+
+/* Given the space of a set and a range of set dimensions,
+ * construct an isl_multi_aff that projects out those dimensions.
+ */
+__isl_give isl_multi_aff *isl_multi_aff_project_out_map(
+	__isl_take isl_space *space, enum isl_dim_type type,
+	unsigned first, unsigned n)
+{
+	int i, dim;
+	isl_local_space *ls;
+	isl_multi_aff *ma;
+
+	if (!space)
+		return NULL;
+	if (!isl_space_is_set(space))
+		isl_die(isl_space_get_ctx(space), isl_error_unsupported,
+			"expecting set space", goto error);
+	if (type != isl_dim_set)
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"only set dimensions can be projected out", goto error);
+
+	dim = isl_space_dim(space, isl_dim_set);
+	if (first + n > dim)
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"range out of bounds", goto error);
+
+	space = isl_space_from_domain(space);
+	space = isl_space_add_dims(space, isl_dim_out, dim - n);
+
+	if (dim == n)
+		return isl_multi_aff_alloc(space);
+
+	ma = isl_multi_aff_alloc(isl_space_copy(space));
+	space = isl_space_domain(space);
+	ls = isl_local_space_from_space(space);
+
+	for (i = 0; i < first; ++i) {
+		isl_aff *aff;
+
+		aff = isl_aff_var_on_domain(isl_local_space_copy(ls),
+						isl_dim_set, i);
+		ma = isl_multi_aff_set_aff(ma, i, aff);
+	}
+
+	for (i = 0; i < dim - (first + n); ++i) {
+		isl_aff *aff;
+
+		aff = isl_aff_var_on_domain(isl_local_space_copy(ls),
+						isl_dim_set, first + n + i);
+		ma = isl_multi_aff_set_aff(ma, first + i, aff);
+	}
+
+	isl_local_space_free(ls);
+	return ma;
+error:
+	isl_space_free(space);
+	return NULL;
+}
+
+/* Given the space of a set and a range of set dimensions,
+ * construct an isl_pw_multi_aff that projects out those dimensions.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_project_out_map(
+	__isl_take isl_space *space, enum isl_dim_type type,
+	unsigned first, unsigned n)
+{
+	isl_multi_aff *ma;
+
+	ma = isl_multi_aff_project_out_map(space, type, first, n);
+	return isl_pw_multi_aff_from_multi_aff(ma);
+}
+
+/* Create an isl_pw_multi_aff with the given isl_multi_aff on a universe
+ * domain.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_multi_aff(
+	__isl_take isl_multi_aff *ma)
+{
+	isl_set *dom = isl_set_universe(isl_multi_aff_get_domain_space(ma));
+	return isl_pw_multi_aff_alloc(dom, ma);
+}
+
+/* Create a piecewise multi-affine expression in the given space that maps each
+ * input dimension to the corresponding output dimension.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_identity(
+	__isl_take isl_space *space)
+{
+	return isl_pw_multi_aff_from_multi_aff(isl_multi_aff_identity(space));
+}
+
+/* Add "ma2" to "ma1" and return the result.
+ *
+ * The parameters of "ma1" and "ma2" are assumed to have been aligned.
+ */
+static __isl_give isl_multi_aff *isl_multi_aff_add_aligned(
+	__isl_take isl_multi_aff *maff1, __isl_take isl_multi_aff *maff2)
+{
+	return isl_multi_aff_bin_op(maff1, maff2, &isl_aff_add);
+}
+
+/* Add "ma2" to "ma1" and return the result.
+ */
+__isl_give isl_multi_aff *isl_multi_aff_add(__isl_take isl_multi_aff *ma1,
+	__isl_take isl_multi_aff *ma2)
+{
+	return isl_multi_aff_align_params_multi_multi_and(ma1, ma2,
+						&isl_multi_aff_add_aligned);
+}
+
+/* Subtract "ma2" from "ma1" and return the result.
+ *
+ * The parameters of "ma1" and "ma2" are assumed to have been aligned.
+ */
+static __isl_give isl_multi_aff *isl_multi_aff_sub_aligned(
+	__isl_take isl_multi_aff *ma1, __isl_take isl_multi_aff *ma2)
+{
+	return isl_multi_aff_bin_op(ma1, ma2, &isl_aff_sub);
+}
+
+/* Subtract "ma2" from "ma1" and return the result.
+ */
+__isl_give isl_multi_aff *isl_multi_aff_sub(__isl_take isl_multi_aff *ma1,
+	__isl_take isl_multi_aff *ma2)
+{
+	return isl_multi_aff_align_params_multi_multi_and(ma1, ma2,
+						&isl_multi_aff_sub_aligned);
+}
+
+/* Exploit the equalities in "eq" to simplify the affine expressions.
+ */
+static __isl_give isl_multi_aff *isl_multi_aff_substitute_equalities(
+	__isl_take isl_multi_aff *maff, __isl_take isl_basic_set *eq)
+{
+	int i;
+
+	maff = isl_multi_aff_cow(maff);
+	if (!maff || !eq)
+		goto error;
+
+	for (i = 0; i < maff->n; ++i) {
+		maff->p[i] = isl_aff_substitute_equalities(maff->p[i],
+						    isl_basic_set_copy(eq));
+		if (!maff->p[i])
+			goto error;
+	}
+
+	isl_basic_set_free(eq);
+	return maff;
+error:
+	isl_basic_set_free(eq);
+	isl_multi_aff_free(maff);
+	return NULL;
+}
+
+/* Given f, return floor(f).
+ */
+__isl_give isl_multi_aff *isl_multi_aff_floor(__isl_take isl_multi_aff *ma)
+{
+	int i;
+
+	ma = isl_multi_aff_cow(ma);
+	if (!ma)
+		return NULL;
+
+	for (i = 0; i < ma->n; ++i) {
+		ma->p[i] = isl_aff_floor(ma->p[i]);
+		if (!ma->p[i])
+			return isl_multi_aff_free(ma);
+	}
+
+	return ma;
+}
+
+__isl_give isl_multi_aff *isl_multi_aff_scale(__isl_take isl_multi_aff *maff,
+	isl_int f)
+{
+	int i;
+
+	maff = isl_multi_aff_cow(maff);
+	if (!maff)
+		return NULL;
+
+	for (i = 0; i < maff->n; ++i) {
+		maff->p[i] = isl_aff_scale(maff->p[i], f);
+		if (!maff->p[i])
+			return isl_multi_aff_free(maff);
+	}
+
+	return maff;
+}
+
+__isl_give isl_multi_aff *isl_multi_aff_add_on_domain(__isl_keep isl_set *dom,
+	__isl_take isl_multi_aff *maff1, __isl_take isl_multi_aff *maff2)
+{
+	maff1 = isl_multi_aff_add(maff1, maff2);
+	maff1 = isl_multi_aff_gist(maff1, isl_set_copy(dom));
+	return maff1;
+}
+
+int isl_multi_aff_is_empty(__isl_keep isl_multi_aff *maff)
+{
+	if (!maff)
+		return -1;
+
+	return 0;
+}
+
+/* Return the set of domain elements where "ma1" is lexicographically
+ * smaller than or equal to "ma2".
+ */
+__isl_give isl_set *isl_multi_aff_lex_le_set(__isl_take isl_multi_aff *ma1,
+	__isl_take isl_multi_aff *ma2)
+{
+	return isl_multi_aff_lex_ge_set(ma2, ma1);
+}
+
+/* Return the set of domain elements where "ma1" is lexicographically
+ * greater than or equal to "ma2".
+ */
+__isl_give isl_set *isl_multi_aff_lex_ge_set(__isl_take isl_multi_aff *ma1,
+	__isl_take isl_multi_aff *ma2)
+{
+	isl_space *space;
+	isl_map *map1, *map2;
+	isl_map *map, *ge;
+
+	map1 = isl_map_from_multi_aff(ma1);
+	map2 = isl_map_from_multi_aff(ma2);
+	map = isl_map_range_product(map1, map2);
+	space = isl_space_range(isl_map_get_space(map));
+	space = isl_space_domain(isl_space_unwrap(space));
+	ge = isl_map_lex_ge(space);
+	map = isl_map_intersect_range(map, isl_map_wrap(ge));
+
+	return isl_map_domain(map);
+}
+
+#undef PW
+#define PW isl_pw_multi_aff
+#undef EL
+#define EL isl_multi_aff
+#undef EL_IS_ZERO
+#define EL_IS_ZERO is_empty
+#undef ZERO
+#define ZERO empty
+#undef IS_ZERO
+#define IS_ZERO is_empty
+#undef FIELD
+#define FIELD maff
+#undef DEFAULT_IS_ZERO
+#define DEFAULT_IS_ZERO 0
+
+#define NO_NEG
+#define NO_EVAL
+#define NO_OPT
+#define NO_INVOLVES_DIMS
+#define NO_INSERT_DIMS
+#define NO_LIFT
+#define NO_MORPH
+
+#include <isl_pw_templ.c>
+
+#undef UNION
+#define UNION isl_union_pw_multi_aff
+#undef PART
+#define PART isl_pw_multi_aff
+#undef PARTS
+#define PARTS pw_multi_aff
+#define ALIGN_DOMAIN
+
+#define NO_EVAL
+
+#include <isl_union_templ.c>
+
+/* Given a function "cmp" that returns the set of elements where
+ * "ma1" is "better" than "ma2", return the intersection of this
+ * set with "dom1" and "dom2".
+ */
+static __isl_give isl_set *shared_and_better(__isl_keep isl_set *dom1,
+	__isl_keep isl_set *dom2, __isl_keep isl_multi_aff *ma1,
+	__isl_keep isl_multi_aff *ma2,
+	__isl_give isl_set *(*cmp)(__isl_take isl_multi_aff *ma1,
+				    __isl_take isl_multi_aff *ma2))
+{
+	isl_set *common;
+	isl_set *better;
+	int is_empty;
+
+	common = isl_set_intersect(isl_set_copy(dom1), isl_set_copy(dom2));
+	is_empty = isl_set_plain_is_empty(common);
+	if (is_empty >= 0 && is_empty)
+		return common;
+	if (is_empty < 0)
+		return isl_set_free(common);
+	better = cmp(isl_multi_aff_copy(ma1), isl_multi_aff_copy(ma2));
+	better = isl_set_intersect(common, better);
+
+	return better;
+}
+
+/* Given a function "cmp" that returns the set of elements where
+ * "ma1" is "better" than "ma2", return a piecewise multi affine
+ * expression defined on the union of the definition domains
+ * of "pma1" and "pma2" that maps to the "best" of "pma1" and
+ * "pma2" on each cell.  If only one of the two input functions
+ * is defined on a given cell, then it is considered the best.
+ */
+static __isl_give isl_pw_multi_aff *pw_multi_aff_union_opt(
+	__isl_take isl_pw_multi_aff *pma1,
+	__isl_take isl_pw_multi_aff *pma2,
+	__isl_give isl_set *(*cmp)(__isl_take isl_multi_aff *ma1,
+				    __isl_take isl_multi_aff *ma2))
+{
+	int i, j, n;
+	isl_pw_multi_aff *res = NULL;
+	isl_ctx *ctx;
+	isl_set *set = NULL;
+
+	if (!pma1 || !pma2)
+		goto error;
+
+	ctx = isl_space_get_ctx(pma1->dim);
+	if (!isl_space_is_equal(pma1->dim, pma2->dim))
+		isl_die(ctx, isl_error_invalid,
+			"arguments should live in the same space", goto error);
+
+	if (isl_pw_multi_aff_is_empty(pma1)) {
+		isl_pw_multi_aff_free(pma1);
+		return pma2;
+	}
+
+	if (isl_pw_multi_aff_is_empty(pma2)) {
+		isl_pw_multi_aff_free(pma2);
+		return pma1;
+	}
+
+	n = 2 * (pma1->n + 1) * (pma2->n + 1);
+	res = isl_pw_multi_aff_alloc_size(isl_space_copy(pma1->dim), n);
+
+	for (i = 0; i < pma1->n; ++i) {
+		set = isl_set_copy(pma1->p[i].set);
+		for (j = 0; j < pma2->n; ++j) {
+			isl_set *better;
+			int is_empty;
+
+			better = shared_and_better(pma2->p[j].set,
+					pma1->p[i].set, pma2->p[j].maff,
+					pma1->p[i].maff, cmp);
+			is_empty = isl_set_plain_is_empty(better);
+			if (is_empty < 0 || is_empty) {
+				isl_set_free(better);
+				if (is_empty < 0)
+					goto error;
+				continue;
+			}
+			set = isl_set_subtract(set, isl_set_copy(better));
+
+			res = isl_pw_multi_aff_add_piece(res, better,
+					isl_multi_aff_copy(pma2->p[j].maff));
+		}
+		res = isl_pw_multi_aff_add_piece(res, set,
+					isl_multi_aff_copy(pma1->p[i].maff));
+	}
+
+	for (j = 0; j < pma2->n; ++j) {
+		set = isl_set_copy(pma2->p[j].set);
+		for (i = 0; i < pma1->n; ++i)
+			set = isl_set_subtract(set,
+					isl_set_copy(pma1->p[i].set));
+		res = isl_pw_multi_aff_add_piece(res, set,
+					isl_multi_aff_copy(pma2->p[j].maff));
+	}
+
+	isl_pw_multi_aff_free(pma1);
+	isl_pw_multi_aff_free(pma2);
+
+	return res;
+error:
+	isl_pw_multi_aff_free(pma1);
+	isl_pw_multi_aff_free(pma2);
+	isl_set_free(set);
+	return isl_pw_multi_aff_free(res);
+}
+
+static __isl_give isl_pw_multi_aff *pw_multi_aff_union_lexmax(
+	__isl_take isl_pw_multi_aff *pma1,
+	__isl_take isl_pw_multi_aff *pma2)
+{
+	return pw_multi_aff_union_opt(pma1, pma2, &isl_multi_aff_lex_ge_set);
+}
+
+/* Given two piecewise multi affine expressions, return a piecewise
+ * multi-affine expression defined on the union of the definition domains
+ * of the inputs that is equal to the lexicographic maximum of the two
+ * inputs on each cell.  If only one of the two inputs is defined on
+ * a given cell, then it is considered to be the maximum.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_lexmax(
+	__isl_take isl_pw_multi_aff *pma1,
+	__isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_align_params_pw_pw_and(pma1, pma2,
+						    &pw_multi_aff_union_lexmax);
+}
+
+static __isl_give isl_pw_multi_aff *pw_multi_aff_union_lexmin(
+	__isl_take isl_pw_multi_aff *pma1,
+	__isl_take isl_pw_multi_aff *pma2)
+{
+	return pw_multi_aff_union_opt(pma1, pma2, &isl_multi_aff_lex_le_set);
+}
+
+/* Given two piecewise multi affine expressions, return a piecewise
+ * multi-affine expression defined on the union of the definition domains
+ * of the inputs that is equal to the lexicographic minimum of the two
+ * inputs on each cell.  If only one of the two inputs is defined on
+ * a given cell, then it is considered to be the minimum.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_lexmin(
+	__isl_take isl_pw_multi_aff *pma1,
+	__isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_align_params_pw_pw_and(pma1, pma2,
+						    &pw_multi_aff_union_lexmin);
+}
+
+static __isl_give isl_pw_multi_aff *pw_multi_aff_add(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_on_shared_domain(pma1, pma2,
+						&isl_multi_aff_add);
+}
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_add(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_align_params_pw_pw_and(pma1, pma2,
+						&pw_multi_aff_add);
+}
+
+static __isl_give isl_pw_multi_aff *pw_multi_aff_sub(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_on_shared_domain(pma1, pma2,
+						&isl_multi_aff_sub);
+}
+
+/* Subtract "pma2" from "pma1" and return the result.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_sub(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_align_params_pw_pw_and(pma1, pma2,
+						&pw_multi_aff_sub);
+}
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_add(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_union_add_(pma1, pma2);
+}
+
+/* Compute the sum of "upma1" and "upma2" on the union of their domains,
+ * with the actual sum on the shared domain and
+ * the defined expression on the symmetric difference of the domains.
+ */
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_union_add(
+	__isl_take isl_union_pw_multi_aff *upma1,
+	__isl_take isl_union_pw_multi_aff *upma2)
+{
+	return isl_union_pw_multi_aff_union_add_(upma1, upma2);
+}
+
+/* Given two piecewise multi-affine expressions A -> B and C -> D,
+ * construct a piecewise multi-affine expression [A -> C] -> [B -> D].
+ */
+static __isl_give isl_pw_multi_aff *pw_multi_aff_product(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	int i, j, n;
+	isl_space *space;
+	isl_pw_multi_aff *res;
+
+	if (!pma1 || !pma2)
+		goto error;
+
+	n = pma1->n * pma2->n;
+	space = isl_space_product(isl_space_copy(pma1->dim),
+				  isl_space_copy(pma2->dim));
+	res = isl_pw_multi_aff_alloc_size(space, n);
+
+	for (i = 0; i < pma1->n; ++i) {
+		for (j = 0; j < pma2->n; ++j) {
+			isl_set *domain;
+			isl_multi_aff *ma;
+
+			domain = isl_set_product(isl_set_copy(pma1->p[i].set),
+						 isl_set_copy(pma2->p[j].set));
+			ma = isl_multi_aff_product(
+					isl_multi_aff_copy(pma1->p[i].maff),
+					isl_multi_aff_copy(pma2->p[j].maff));
+			res = isl_pw_multi_aff_add_piece(res, domain, ma);
+		}
+	}
+
+	isl_pw_multi_aff_free(pma1);
+	isl_pw_multi_aff_free(pma2);
+	return res;
+error:
+	isl_pw_multi_aff_free(pma1);
+	isl_pw_multi_aff_free(pma2);
+	return NULL;
+}
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_product(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_align_params_pw_pw_and(pma1, pma2,
+						&pw_multi_aff_product);
+}
+
+/* Construct a map mapping the domain of the piecewise multi-affine expression
+ * to its range, with each dimension in the range equated to the
+ * corresponding affine expression on its cell.
+ */
+__isl_give isl_map *isl_map_from_pw_multi_aff(__isl_take isl_pw_multi_aff *pma)
+{
+	int i;
+	isl_map *map;
+
+	if (!pma)
+		return NULL;
+
+	map = isl_map_empty(isl_pw_multi_aff_get_space(pma));
+
+	for (i = 0; i < pma->n; ++i) {
+		isl_multi_aff *maff;
+		isl_basic_map *bmap;
+		isl_map *map_i;
+
+		maff = isl_multi_aff_copy(pma->p[i].maff);
+		bmap = isl_basic_map_from_multi_aff(maff);
+		map_i = isl_map_from_basic_map(bmap);
+		map_i = isl_map_intersect_domain(map_i,
+						isl_set_copy(pma->p[i].set));
+		map = isl_map_union_disjoint(map, map_i);
+	}
+
+	isl_pw_multi_aff_free(pma);
+	return map;
+}
+
+__isl_give isl_set *isl_set_from_pw_multi_aff(__isl_take isl_pw_multi_aff *pma)
+{
+	if (!pma)
+		return NULL;
+
+	if (!isl_space_is_set(pma->dim))
+		isl_die(isl_pw_multi_aff_get_ctx(pma), isl_error_invalid,
+			"isl_pw_multi_aff cannot be converted into an isl_set",
+			goto error);
+
+	return isl_map_from_pw_multi_aff(pma);
+error:
+	isl_pw_multi_aff_free(pma);
+	return NULL;
+}
+
+/* Given a basic map with a single output dimension that is defined
+ * in terms of the parameters and input dimensions using an equality,
+ * extract an isl_aff that expresses the output dimension in terms
+ * of the parameters and input dimensions.
+ * Note that this expression may involve integer divisions defined
+ * in terms of parameters and input dimensions.
+ *
+ * This function shares some similarities with
+ * isl_basic_map_has_defining_equality and isl_constraint_get_bound.
+ */
+static __isl_give isl_aff *extract_isl_aff_from_basic_map(
+	__isl_take isl_basic_map *bmap)
+{
+	int eq;
+	unsigned offset;
+	unsigned n_div;
+	isl_local_space *ls;
+	isl_aff *aff;
+
+	if (!bmap)
+		return NULL;
+	if (isl_basic_map_dim(bmap, isl_dim_out) != 1)
+		isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
+			"basic map should have a single output dimension",
+			goto error);
+	eq = isl_basic_map_output_defining_equality(bmap, 0);
+	if (eq >= bmap->n_eq)
+		isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
+			"unable to find suitable equality", goto error);
+	ls = isl_basic_map_get_local_space(bmap);
+	aff = isl_aff_alloc(isl_local_space_domain(ls));
+	if (!aff)
+		goto error;
+	offset = isl_basic_map_offset(bmap, isl_dim_out);
+	n_div = isl_basic_map_dim(bmap, isl_dim_div);
+	if (isl_int_is_neg(bmap->eq[eq][offset])) {
+		isl_seq_cpy(aff->v->el + 1, bmap->eq[eq], offset);
+		isl_seq_cpy(aff->v->el + 1 + offset, bmap->eq[eq] + offset + 1,
+			    n_div);
+	} else {
+		isl_seq_neg(aff->v->el + 1, bmap->eq[eq], offset);
+		isl_seq_neg(aff->v->el + 1 + offset, bmap->eq[eq] + offset + 1,
+			    n_div);
+	}
+	isl_int_abs(aff->v->el[0], bmap->eq[eq][offset]);
+	isl_basic_map_free(bmap);
+
+	aff = isl_aff_remove_unused_divs(aff);
+	return aff;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Given a basic map where each output dimension is defined
+ * in terms of the parameters and input dimensions using an equality,
+ * extract an isl_multi_aff that expresses the output dimensions in terms
+ * of the parameters and input dimensions.
+ */
+static __isl_give isl_multi_aff *extract_isl_multi_aff_from_basic_map(
+	__isl_take isl_basic_map *bmap)
+{
+	int i;
+	unsigned n_out;
+	isl_multi_aff *ma;
+
+	if (!bmap)
+		return NULL;
+
+	ma = isl_multi_aff_alloc(isl_basic_map_get_space(bmap));
+	n_out = isl_basic_map_dim(bmap, isl_dim_out);
+
+	for (i = 0; i < n_out; ++i) {
+		isl_basic_map *bmap_i;
+		isl_aff *aff;
+
+		bmap_i = isl_basic_map_copy(bmap);
+		bmap_i = isl_basic_map_project_out(bmap_i, isl_dim_out,
+							i + 1, n_out - (1 + i));
+		bmap_i = isl_basic_map_project_out(bmap_i, isl_dim_out, 0, i);
+		aff = extract_isl_aff_from_basic_map(bmap_i);
+		ma = isl_multi_aff_set_aff(ma, i, aff);
+	}
+
+	isl_basic_map_free(bmap);
+
+	return ma;
+}
+
+/* Given a basic set where each set dimension is defined
+ * in terms of the parameters using an equality,
+ * extract an isl_multi_aff that expresses the set dimensions in terms
+ * of the parameters.
+ */
+__isl_give isl_multi_aff *isl_multi_aff_from_basic_set_equalities(
+	__isl_take isl_basic_set *bset)
+{
+	return extract_isl_multi_aff_from_basic_map(bset);
+}
+
+/* Create an isl_pw_multi_aff that is equivalent to
+ * isl_map_intersect_domain(isl_map_from_basic_map(bmap), domain).
+ * The given basic map is such that each output dimension is defined
+ * in terms of the parameters and input dimensions using an equality.
+ *
+ * Since some applications expect the result of isl_pw_multi_aff_from_map
+ * to only contain integer affine expressions, we compute the floor
+ * of the expression before returning.
+ */
+static __isl_give isl_pw_multi_aff *plain_pw_multi_aff_from_map(
+	__isl_take isl_set *domain, __isl_take isl_basic_map *bmap)
+{
+	isl_multi_aff *ma;
+
+	ma = extract_isl_multi_aff_from_basic_map(bmap);
+	ma = isl_multi_aff_floor(ma);
+	return isl_pw_multi_aff_alloc(domain, ma);
+}
+
+/* Try and create an isl_pw_multi_aff that is equivalent to the given isl_map.
+ * This obviously only works if the input "map" is single-valued.
+ * If so, we compute the lexicographic minimum of the image in the form
+ * of an isl_pw_multi_aff.  Since the image is unique, it is equal
+ * to its lexicographic minimum.
+ * If the input is not single-valued, we produce an error.
+ */
+static __isl_give isl_pw_multi_aff *pw_multi_aff_from_map_base(
+	__isl_take isl_map *map)
+{
+	int i;
+	int sv;
+	isl_pw_multi_aff *pma;
+
+	sv = isl_map_is_single_valued(map);
+	if (sv < 0)
+		goto error;
+	if (!sv)
+		isl_die(isl_map_get_ctx(map), isl_error_invalid,
+			"map is not single-valued", goto error);
+	map = isl_map_make_disjoint(map);
+	if (!map)
+		return NULL;
+
+	pma = isl_pw_multi_aff_empty(isl_map_get_space(map));
+
+	for (i = 0; i < map->n; ++i) {
+		isl_pw_multi_aff *pma_i;
+		isl_basic_map *bmap;
+		bmap = isl_basic_map_copy(map->p[i]);
+		pma_i = isl_basic_map_lexmin_pw_multi_aff(bmap);
+		pma = isl_pw_multi_aff_add_disjoint(pma, pma_i);
+	}
+
+	isl_map_free(map);
+	return pma;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Try and create an isl_pw_multi_aff that is equivalent to the given isl_map,
+ * taking into account that the output dimension at position "d"
+ * can be represented as
+ *
+ *	x = floor((e(...) + c1) / m)
+ *
+ * given that constraint "i" is of the form
+ *
+ *	e(...) + c1 - m x >= 0
+ *
+ *
+ * Let "map" be of the form
+ *
+ *	A -> B
+ *
+ * We construct a mapping
+ *
+ *	A -> [A -> x = floor(...)]
+ *
+ * apply that to the map, obtaining
+ *
+ *	[A -> x = floor(...)] -> B
+ *
+ * and equate dimension "d" to x.
+ * We then compute a isl_pw_multi_aff representation of the resulting map
+ * and plug in the mapping above.
+ */
+static __isl_give isl_pw_multi_aff *pw_multi_aff_from_map_div(
+	__isl_take isl_map *map, __isl_take isl_basic_map *hull, int d, int i)
+{
+	isl_ctx *ctx;
+	isl_space *space;
+	isl_local_space *ls;
+	isl_multi_aff *ma;
+	isl_aff *aff;
+	isl_vec *v;
+	isl_map *insert;
+	int offset;
+	int n;
+	int n_in;
+	isl_pw_multi_aff *pma;
+	int is_set;
+
+	is_set = isl_map_is_set(map);
+
+	offset = isl_basic_map_offset(hull, isl_dim_out);
+	ctx = isl_map_get_ctx(map);
+	space = isl_space_domain(isl_map_get_space(map));
+	n_in = isl_space_dim(space, isl_dim_set);
+	n = isl_space_dim(space, isl_dim_all);
+
+	v = isl_vec_alloc(ctx, 1 + 1 + n);
+	if (v) {
+		isl_int_neg(v->el[0], hull->ineq[i][offset + d]);
+		isl_seq_cpy(v->el + 1, hull->ineq[i], 1 + n);
+	}
+	isl_basic_map_free(hull);
+
+	ls = isl_local_space_from_space(isl_space_copy(space));
+	aff = isl_aff_alloc_vec(ls, v);
+	aff = isl_aff_floor(aff);
+	if (is_set) {
+		isl_space_free(space);
+		ma = isl_multi_aff_from_aff(aff);
+	} else {
+		ma = isl_multi_aff_identity(isl_space_map_from_set(space));
+		ma = isl_multi_aff_range_product(ma,
+						isl_multi_aff_from_aff(aff));
+	}
+
+	insert = isl_map_from_multi_aff(isl_multi_aff_copy(ma));
+	map = isl_map_apply_domain(map, insert);
+	map = isl_map_equate(map, isl_dim_in, n_in, isl_dim_out, d);
+	pma = isl_pw_multi_aff_from_map(map);
+	pma = isl_pw_multi_aff_pullback_multi_aff(pma, ma);
+
+	return pma;
+}
+
+/* Is constraint "c" of the form
+ *
+ *	e(...) + c1 - m x >= 0
+ *
+ * or
+ *
+ *	-e(...) + c2 + m x >= 0
+ *
+ * where m > 1 and e only depends on parameters and input dimemnsions?
+ *
+ * "offset" is the offset of the output dimensions
+ * "pos" is the position of output dimension x.
+ */
+static int is_potential_div_constraint(isl_int *c, int offset, int d, int total)
+{
+	if (isl_int_is_zero(c[offset + d]))
+		return 0;
+	if (isl_int_is_one(c[offset + d]))
+		return 0;
+	if (isl_int_is_negone(c[offset + d]))
+		return 0;
+	if (isl_seq_first_non_zero(c + offset, d) != -1)
+		return 0;
+	if (isl_seq_first_non_zero(c + offset + d + 1,
+				    total - (offset + d + 1)) != -1)
+		return 0;
+	return 1;
+}
+
+/* Try and create an isl_pw_multi_aff that is equivalent to the given isl_map.
+ *
+ * As a special case, we first check if there is any pair of constraints,
+ * shared by all the basic maps in "map" that force a given dimension
+ * to be equal to the floor of some affine combination of the input dimensions.
+ *
+ * In particular, if we can find two constraints
+ *
+ *	e(...) + c1 - m x >= 0		i.e.,		m x <= e(...) + c1
+ *
+ * and
+ *
+ *	-e(...) + c2 + m x >= 0		i.e.,		m x >= e(...) - c2
+ *
+ * where m > 1 and e only depends on parameters and input dimemnsions,
+ * and such that
+ *
+ *	c1 + c2 < m			i.e.,		-c2 >= c1 - (m - 1)
+ *
+ * then we know that we can take
+ *
+ *	x = floor((e(...) + c1) / m)
+ *
+ * without having to perform any computation.
+ *
+ * Note that we know that
+ *
+ *	c1 + c2 >= 1
+ *
+ * If c1 + c2 were 0, then we would have detected an equality during
+ * simplification.  If c1 + c2 were negative, then we would have detected
+ * a contradiction.
+ */
+static __isl_give isl_pw_multi_aff *pw_multi_aff_from_map_check_div(
+	__isl_take isl_map *map)
+{
+	int d, dim;
+	int i, j, n;
+	int offset, total;
+	isl_int sum;
+	isl_basic_map *hull;
+
+	hull = isl_map_unshifted_simple_hull(isl_map_copy(map));
+	if (!hull)
+		goto error;
+
+	isl_int_init(sum);
+	dim = isl_map_dim(map, isl_dim_out);
+	offset = isl_basic_map_offset(hull, isl_dim_out);
+	total = 1 + isl_basic_map_total_dim(hull);
+	n = hull->n_ineq;
+	for (d = 0; d < dim; ++d) {
+		for (i = 0; i < n; ++i) {
+			if (!is_potential_div_constraint(hull->ineq[i],
+							offset, d, total))
+				continue;
+			for (j = i + 1; j < n; ++j) {
+				if (!isl_seq_is_neg(hull->ineq[i] + 1,
+						hull->ineq[j] + 1, total - 1))
+					continue;
+				isl_int_add(sum, hull->ineq[i][0],
+						hull->ineq[j][0]);
+				if (isl_int_abs_lt(sum,
+						    hull->ineq[i][offset + d]))
+					break;
+
+			}
+			if (j >= n)
+				continue;
+			isl_int_clear(sum);
+			if (isl_int_is_pos(hull->ineq[j][offset + d]))
+				j = i;
+			return pw_multi_aff_from_map_div(map, hull, d, j);
+		}
+	}
+	isl_int_clear(sum);
+	isl_basic_map_free(hull);
+	return pw_multi_aff_from_map_base(map);
+error:
+	isl_map_free(map);
+	isl_basic_map_free(hull);
+	return NULL;
+}
+
+/* Given an affine expression
+ *
+ *	[A -> B] -> f(A,B)
+ *
+ * construct an isl_multi_aff
+ *
+ *	[A -> B] -> B'
+ *
+ * such that dimension "d" in B' is set to "aff" and the remaining
+ * dimensions are set equal to the corresponding dimensions in B.
+ * "n_in" is the dimension of the space A.
+ * "n_out" is the dimension of the space B.
+ *
+ * If "is_set" is set, then the affine expression is of the form
+ *
+ *	[B] -> f(B)
+ *
+ * and we construct an isl_multi_aff
+ *
+ *	B -> B'
+ */
+static __isl_give isl_multi_aff *range_map(__isl_take isl_aff *aff, int d,
+	unsigned n_in, unsigned n_out, int is_set)
+{
+	int i;
+	isl_multi_aff *ma;
+	isl_space *space, *space2;
+	isl_local_space *ls;
+
+	space = isl_aff_get_domain_space(aff);
+	ls = isl_local_space_from_space(isl_space_copy(space));
+	space2 = isl_space_copy(space);
+	if (!is_set)
+		space2 = isl_space_range(isl_space_unwrap(space2));
+	space = isl_space_map_from_domain_and_range(space, space2);
+	ma = isl_multi_aff_alloc(space);
+	ma = isl_multi_aff_set_aff(ma, d, aff);
+
+	for (i = 0; i < n_out; ++i) {
+		if (i == d)
+			continue;
+		aff = isl_aff_var_on_domain(isl_local_space_copy(ls),
+						isl_dim_set, n_in + i);
+		ma = isl_multi_aff_set_aff(ma, i, aff);
+	}
+
+	isl_local_space_free(ls);
+
+	return ma;
+}
+
+/* Try and create an isl_pw_multi_aff that is equivalent to the given isl_map,
+ * taking into account that the dimension at position "d" can be written as
+ *
+ *	x = m a + f(..)						(1)
+ *
+ * where m is equal to "gcd".
+ * "i" is the index of the equality in "hull" that defines f(..).
+ * In particular, the equality is of the form
+ *
+ *	f(..) - x + m g(existentials) = 0
+ *
+ * or
+ *
+ *	-f(..) + x + m g(existentials) = 0
+ *
+ * We basically plug (1) into "map", resulting in a map with "a"
+ * in the range instead of "x".  The corresponding isl_pw_multi_aff
+ * defining "a" is then plugged back into (1) to obtain a definition fro "x".
+ *
+ * Specifically, given the input map
+ *
+ *	A -> B
+ *
+ * We first wrap it into a set
+ *
+ *	[A -> B]
+ *
+ * and define (1) on top of the corresponding space, resulting in "aff".
+ * We use this to create an isl_multi_aff that maps the output position "d"
+ * from "a" to "x", leaving all other (intput and output) dimensions unchanged.
+ * We plug this into the wrapped map, unwrap the result and compute the
+ * corresponding isl_pw_multi_aff.
+ * The result is an expression
+ *
+ *	A -> T(A)
+ *
+ * We adjust that to
+ *
+ *	A -> [A -> T(A)]
+ *
+ * so that we can plug that into "aff", after extending the latter to
+ * a mapping
+ *
+ *	[A -> B] -> B'
+ *
+ *
+ * If "map" is actually a set, then there is no "A" space, meaning
+ * that we do not need to perform any wrapping, and that the result
+ * of the recursive call is of the form
+ *
+ *	[T]
+ *
+ * which is plugged into a mapping of the form
+ *
+ *	B -> B'
+ */
+static __isl_give isl_pw_multi_aff *pw_multi_aff_from_map_stride(
+	__isl_take isl_map *map, __isl_take isl_basic_map *hull, int d, int i,
+	isl_int gcd)
+{
+	isl_set *set;
+	isl_space *space;
+	isl_local_space *ls;
+	isl_aff *aff;
+	isl_multi_aff *ma;
+	isl_pw_multi_aff *pma, *id;
+	unsigned n_in;
+	unsigned o_out;
+	unsigned n_out;
+	int is_set;
+
+	is_set = isl_map_is_set(map);
+
+	n_in = isl_basic_map_dim(hull, isl_dim_in);
+	n_out = isl_basic_map_dim(hull, isl_dim_out);
+	o_out = isl_basic_map_offset(hull, isl_dim_out);
+
+	if (is_set)
+		set = map;
+	else
+		set = isl_map_wrap(map);
+	space = isl_space_map_from_set(isl_set_get_space(set));
+	ma = isl_multi_aff_identity(space);
+	ls = isl_local_space_from_space(isl_set_get_space(set));
+	aff = isl_aff_alloc(ls);
+	if (aff) {
+		isl_int_set_si(aff->v->el[0], 1);
+		if (isl_int_is_one(hull->eq[i][o_out + d]))
+			isl_seq_neg(aff->v->el + 1, hull->eq[i],
+				    aff->v->size - 1);
+		else
+			isl_seq_cpy(aff->v->el + 1, hull->eq[i],
+				    aff->v->size - 1);
+		isl_int_set(aff->v->el[1 + o_out + d], gcd);
+	}
+	ma = isl_multi_aff_set_aff(ma, n_in + d, isl_aff_copy(aff));
+	set = isl_set_preimage_multi_aff(set, ma);
+
+	ma = range_map(aff, d, n_in, n_out, is_set);
+
+	if (is_set)
+		map = set;
+	else
+		map = isl_set_unwrap(set);
+	pma = isl_pw_multi_aff_from_map(set);
+
+	if (!is_set) {
+		space = isl_pw_multi_aff_get_domain_space(pma);
+		space = isl_space_map_from_set(space);
+		id = isl_pw_multi_aff_identity(space);
+		pma = isl_pw_multi_aff_range_product(id, pma);
+	}
+	id = isl_pw_multi_aff_from_multi_aff(ma);
+	pma = isl_pw_multi_aff_pullback_pw_multi_aff(id, pma);
+
+	isl_basic_map_free(hull);
+	return pma;
+}
+
+/* Try and create an isl_pw_multi_aff that is equivalent to the given isl_map.
+ *
+ * As a special case, we first check if all output dimensions are uniquely
+ * defined in terms of the parameters and input dimensions over the entire
+ * domain.  If so, we extract the desired isl_pw_multi_aff directly
+ * from the affine hull of "map" and its domain.
+ *
+ * Otherwise, we check if any of the output dimensions is "strided".
+ * That is, we check if can be written as
+ *
+ *	x = m a + f(..)
+ *
+ * with m greater than 1, a some combination of existentiall quantified
+ * variables and f and expression in the parameters and input dimensions.
+ * If so, we remove the stride in pw_multi_aff_from_map_stride.
+ *
+ * Otherwise, we continue with pw_multi_aff_from_map_check_div for a further
+ * special case.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_map(__isl_take isl_map *map)
+{
+	int i, j;
+	int sv;
+	isl_basic_map *hull;
+	unsigned n_out;
+	unsigned o_out;
+	unsigned n_div;
+	unsigned o_div;
+	isl_int gcd;
+
+	if (!map)
+		return NULL;
+
+	hull = isl_map_affine_hull(isl_map_copy(map));
+	sv = isl_basic_map_plain_is_single_valued(hull);
+	if (sv >= 0 && sv)
+		return plain_pw_multi_aff_from_map(isl_map_domain(map), hull);
+	if (sv < 0)
+		hull = isl_basic_map_free(hull);
+	if (!hull)
+		goto error;
+
+	n_div = isl_basic_map_dim(hull, isl_dim_div);
+	o_div = isl_basic_map_offset(hull, isl_dim_div);
+
+	if (n_div == 0) {
+		isl_basic_map_free(hull);
+		return pw_multi_aff_from_map_check_div(map);
+	}
+
+	isl_int_init(gcd);
+
+	n_out = isl_basic_map_dim(hull, isl_dim_out);
+	o_out = isl_basic_map_offset(hull, isl_dim_out);
+
+	for (i = 0; i < n_out; ++i) {
+		for (j = 0; j < hull->n_eq; ++j) {
+			isl_int *eq = hull->eq[j];
+			isl_pw_multi_aff *res;
+
+			if (!isl_int_is_one(eq[o_out + i]) &&
+			    !isl_int_is_negone(eq[o_out + i]))
+				continue;
+			if (isl_seq_first_non_zero(eq + o_out, i) != -1)
+				continue;
+			if (isl_seq_first_non_zero(eq + o_out + i + 1,
+						    n_out - (i + 1)) != -1)
+				continue;
+			isl_seq_gcd(eq + o_div, n_div, &gcd);
+			if (isl_int_is_zero(gcd))
+				continue;
+			if (isl_int_is_one(gcd))
+				continue;
+
+			res = pw_multi_aff_from_map_stride(map, hull,
+								i, j, gcd);
+			isl_int_clear(gcd);
+			return res;
+		}
+	}
+
+	isl_int_clear(gcd);
+	isl_basic_map_free(hull);
+	return pw_multi_aff_from_map_check_div(map);
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_set(__isl_take isl_set *set)
+{
+	return isl_pw_multi_aff_from_map(set);
+}
+
+/* Convert "map" into an isl_pw_multi_aff (if possible) and
+ * add it to *user.
+ */
+static int pw_multi_aff_from_map(__isl_take isl_map *map, void *user)
+{
+	isl_union_pw_multi_aff **upma = user;
+	isl_pw_multi_aff *pma;
+
+	pma = isl_pw_multi_aff_from_map(map);
+	*upma = isl_union_pw_multi_aff_add_pw_multi_aff(*upma, pma);
+
+	return *upma ? 0 : -1;
+}
+
+/* Try and create an isl_union_pw_multi_aff that is equivalent
+ * to the given isl_union_map.
+ * The isl_union_map is required to be single-valued in each space.
+ * Otherwise, an error is produced.
+ */
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_from_union_map(
+	__isl_take isl_union_map *umap)
+{
+	isl_space *space;
+	isl_union_pw_multi_aff *upma;
+
+	space = isl_union_map_get_space(umap);
+	upma = isl_union_pw_multi_aff_empty(space);
+	if (isl_union_map_foreach_map(umap, &pw_multi_aff_from_map, &upma) < 0)
+		upma = isl_union_pw_multi_aff_free(upma);
+	isl_union_map_free(umap);
+
+	return upma;
+}
+
+/* Try and create an isl_union_pw_multi_aff that is equivalent
+ * to the given isl_union_set.
+ * The isl_union_set is required to be a singleton in each space.
+ * Otherwise, an error is produced.
+ */
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_from_union_set(
+	__isl_take isl_union_set *uset)
+{
+	return isl_union_pw_multi_aff_from_union_map(uset);
+}
+
+/* Return the piecewise affine expression "set ? 1 : 0".
+ */
+__isl_give isl_pw_aff *isl_set_indicator_function(__isl_take isl_set *set)
+{
+	isl_pw_aff *pa;
+	isl_space *space = isl_set_get_space(set);
+	isl_local_space *ls = isl_local_space_from_space(space);
+	isl_aff *zero = isl_aff_zero_on_domain(isl_local_space_copy(ls));
+	isl_aff *one = isl_aff_zero_on_domain(ls);
+
+	one = isl_aff_add_constant_si(one, 1);
+	pa = isl_pw_aff_alloc(isl_set_copy(set), one);
+	set = isl_set_complement(set);
+	pa = isl_pw_aff_add_disjoint(pa, isl_pw_aff_alloc(set, zero));
+
+	return pa;
+}
+
+/* Plug in "subs" for dimension "type", "pos" of "aff".
+ *
+ * Let i be the dimension to replace and let "subs" be of the form
+ *
+ *	f/d
+ *
+ * and "aff" of the form
+ *
+ *	(a i + g)/m
+ *
+ * The result is
+ *
+ *	(a f + d g')/(m d)
+ *
+ * where g' is the result of plugging in "subs" in each of the integer
+ * divisions in g.
+ */
+__isl_give isl_aff *isl_aff_substitute(__isl_take isl_aff *aff,
+	enum isl_dim_type type, unsigned pos, __isl_keep isl_aff *subs)
+{
+	isl_ctx *ctx;
+	isl_int v;
+
+	aff = isl_aff_cow(aff);
+	if (!aff || !subs)
+		return isl_aff_free(aff);
+
+	ctx = isl_aff_get_ctx(aff);
+	if (!isl_space_is_equal(aff->ls->dim, subs->ls->dim))
+		isl_die(ctx, isl_error_invalid,
+			"spaces don't match", return isl_aff_free(aff));
+	if (isl_local_space_dim(subs->ls, isl_dim_div) != 0)
+		isl_die(ctx, isl_error_unsupported,
+			"cannot handle divs yet", return isl_aff_free(aff));
+
+	aff->ls = isl_local_space_substitute(aff->ls, type, pos, subs);
+	if (!aff->ls)
+		return isl_aff_free(aff);
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	pos += isl_local_space_offset(aff->ls, type);
+
+	isl_int_init(v);
+	isl_seq_substitute(aff->v->el, pos, subs->v->el,
+			    aff->v->size, subs->v->size, v);
+	isl_int_clear(v);
+
+	return aff;
+}
+
+/* Plug in "subs" for dimension "type", "pos" in each of the affine
+ * expressions in "maff".
+ */
+__isl_give isl_multi_aff *isl_multi_aff_substitute(
+	__isl_take isl_multi_aff *maff, enum isl_dim_type type, unsigned pos,
+	__isl_keep isl_aff *subs)
+{
+	int i;
+
+	maff = isl_multi_aff_cow(maff);
+	if (!maff || !subs)
+		return isl_multi_aff_free(maff);
+
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	for (i = 0; i < maff->n; ++i) {
+		maff->p[i] = isl_aff_substitute(maff->p[i], type, pos, subs);
+		if (!maff->p[i])
+			return isl_multi_aff_free(maff);
+	}
+
+	return maff;
+}
+
+/* Plug in "subs" for dimension "type", "pos" of "pma".
+ *
+ * pma is of the form
+ *
+ *	A_i(v) -> M_i(v)
+ *
+ * while subs is of the form
+ *
+ *	v' = B_j(v) -> S_j
+ *
+ * Each pair i,j such that C_ij = A_i \cap B_i is non-empty
+ * has a contribution in the result, in particular
+ *
+ *	C_ij(S_j) -> M_i(S_j)
+ *
+ * Note that plugging in S_j in C_ij may also result in an empty set
+ * and this contribution should simply be discarded.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_substitute(
+	__isl_take isl_pw_multi_aff *pma, enum isl_dim_type type, unsigned pos,
+	__isl_keep isl_pw_aff *subs)
+{
+	int i, j, n;
+	isl_pw_multi_aff *res;
+
+	if (!pma || !subs)
+		return isl_pw_multi_aff_free(pma);
+
+	n = pma->n * subs->n;
+	res = isl_pw_multi_aff_alloc_size(isl_space_copy(pma->dim), n);
+
+	for (i = 0; i < pma->n; ++i) {
+		for (j = 0; j < subs->n; ++j) {
+			isl_set *common;
+			isl_multi_aff *res_ij;
+			int empty;
+
+			common = isl_set_intersect(
+					isl_set_copy(pma->p[i].set),
+					isl_set_copy(subs->p[j].set));
+			common = isl_set_substitute(common,
+					type, pos, subs->p[j].aff);
+			empty = isl_set_plain_is_empty(common);
+			if (empty < 0 || empty) {
+				isl_set_free(common);
+				if (empty < 0)
+					goto error;
+				continue;
+			}
+
+			res_ij = isl_multi_aff_substitute(
+					isl_multi_aff_copy(pma->p[i].maff),
+					type, pos, subs->p[j].aff);
+
+			res = isl_pw_multi_aff_add_piece(res, common, res_ij);
+		}
+	}
+
+	isl_pw_multi_aff_free(pma);
+	return res;
+error:
+	isl_pw_multi_aff_free(pma);
+	isl_pw_multi_aff_free(res);
+	return NULL;
+}
+
+/* Compute the preimage of a range of dimensions in the affine expression "src"
+ * under "ma" and put the result in "dst".  The number of dimensions in "src"
+ * that precede the range is given by "n_before".  The number of dimensions
+ * in the range is given by the number of output dimensions of "ma".
+ * The number of dimensions that follow the range is given by "n_after".
+ * If "has_denom" is set (to one),
+ * then "src" and "dst" have an extra initial denominator.
+ * "n_div_ma" is the number of existentials in "ma"
+ * "n_div_bset" is the number of existentials in "src"
+ * The resulting "dst" (which is assumed to have been allocated by
+ * the caller) contains coefficients for both sets of existentials,
+ * first those in "ma" and then those in "src".
+ * f, c1, c2 and g are temporary objects that have been initialized
+ * by the caller.
+ *
+ * Let src represent the expression
+ *
+ *	(a(p) + f_u u + b v + f_w w + c(divs))/d
+ *
+ * and let ma represent the expressions
+ *
+ *	v_i = (r_i(p) + s_i(y) + t_i(divs'))/m_i
+ *
+ * We start out with the following expression for dst:
+ *
+ *	(a(p) + f_u u + 0 y + f_w w + 0 divs' + c(divs) + f \sum_i b_i v_i)/d
+ *
+ * with the multiplication factor f initially equal to 1
+ * and f \sum_i b_i v_i kept separately.
+ * For each x_i that we substitute, we multiply the numerator
+ * (and denominator) of dst by c_1 = m_i and add the numerator
+ * of the x_i expression multiplied by c_2 = f b_i,
+ * after removing the common factors of c_1 and c_2.
+ * The multiplication factor f also needs to be multiplied by c_1
+ * for the next x_j, j > i.
+ */
+void isl_seq_preimage(isl_int *dst, isl_int *src,
+	__isl_keep isl_multi_aff *ma, int n_before, int n_after,
+	int n_div_ma, int n_div_bmap,
+	isl_int f, isl_int c1, isl_int c2, isl_int g, int has_denom)
+{
+	int i;
+	int n_param, n_in, n_out;
+	int o_dst, o_src;
+
+	n_param = isl_multi_aff_dim(ma, isl_dim_param);
+	n_in = isl_multi_aff_dim(ma, isl_dim_in);
+	n_out = isl_multi_aff_dim(ma, isl_dim_out);
+
+	isl_seq_cpy(dst, src, has_denom + 1 + n_param + n_before);
+	o_dst = o_src = has_denom + 1 + n_param + n_before;
+	isl_seq_clr(dst + o_dst, n_in);
+	o_dst += n_in;
+	o_src += n_out;
+	isl_seq_cpy(dst + o_dst, src + o_src, n_after);
+	o_dst += n_after;
+	o_src += n_after;
+	isl_seq_clr(dst + o_dst, n_div_ma);
+	o_dst += n_div_ma;
+	isl_seq_cpy(dst + o_dst, src + o_src, n_div_bmap);
+
+	isl_int_set_si(f, 1);
+
+	for (i = 0; i < n_out; ++i) {
+		int offset = has_denom + 1 + n_param + n_before + i;
+
+		if (isl_int_is_zero(src[offset]))
+			continue;
+		isl_int_set(c1, ma->p[i]->v->el[0]);
+		isl_int_mul(c2, f, src[offset]);
+		isl_int_gcd(g, c1, c2);
+		isl_int_divexact(c1, c1, g);
+		isl_int_divexact(c2, c2, g);
+
+		isl_int_mul(f, f, c1);
+		o_dst = has_denom;
+		o_src = 1;
+		isl_seq_combine(dst + o_dst, c1, dst + o_dst,
+				c2, ma->p[i]->v->el + o_src, 1 + n_param);
+		o_dst += 1 + n_param;
+		o_src += 1 + n_param;
+		isl_seq_scale(dst + o_dst, dst + o_dst, c1, n_before);
+		o_dst += n_before;
+		isl_seq_combine(dst + o_dst, c1, dst + o_dst,
+				c2, ma->p[i]->v->el + o_src, n_in);
+		o_dst += n_in;
+		o_src += n_in;
+		isl_seq_scale(dst + o_dst, dst + o_dst, c1, n_after);
+		o_dst += n_after;
+		isl_seq_combine(dst + o_dst, c1, dst + o_dst,
+				c2, ma->p[i]->v->el + o_src, n_div_ma);
+		o_dst += n_div_ma;
+		o_src += n_div_ma;
+		isl_seq_scale(dst + o_dst, dst + o_dst, c1, n_div_bmap);
+		if (has_denom)
+			isl_int_mul(dst[0], dst[0], c1);
+	}
+}
+
+/* Compute the pullback of "aff" by the function represented by "ma".
+ * In other words, plug in "ma" in "aff".  The result is an affine expression
+ * defined over the domain space of "ma".
+ *
+ * If "aff" is represented by
+ *
+ *	(a(p) + b x + c(divs))/d
+ *
+ * and ma is represented by
+ *
+ *	x = D(p) + F(y) + G(divs')
+ *
+ * then the result is
+ *
+ *	(a(p) + b D(p) + b F(y) + b G(divs') + c(divs))/d
+ *
+ * The divs in the local space of the input are similarly adjusted
+ * through a call to isl_local_space_preimage_multi_aff.
+ */
+__isl_give isl_aff *isl_aff_pullback_multi_aff(__isl_take isl_aff *aff,
+	__isl_take isl_multi_aff *ma)
+{
+	isl_aff *res = NULL;
+	isl_local_space *ls;
+	int n_div_aff, n_div_ma;
+	isl_int f, c1, c2, g;
+
+	ma = isl_multi_aff_align_divs(ma);
+	if (!aff || !ma)
+		goto error;
+
+	n_div_aff = isl_aff_dim(aff, isl_dim_div);
+	n_div_ma = ma->n ? isl_aff_dim(ma->p[0], isl_dim_div) : 0;
+
+	ls = isl_aff_get_domain_local_space(aff);
+	ls = isl_local_space_preimage_multi_aff(ls, isl_multi_aff_copy(ma));
+	res = isl_aff_alloc(ls);
+	if (!res)
+		goto error;
+
+	isl_int_init(f);
+	isl_int_init(c1);
+	isl_int_init(c2);
+	isl_int_init(g);
+
+	isl_seq_preimage(res->v->el, aff->v->el, ma, 0, 0, n_div_ma, n_div_aff,
+			f, c1, c2, g, 1);
+
+	isl_int_clear(f);
+	isl_int_clear(c1);
+	isl_int_clear(c2);
+	isl_int_clear(g);
+
+	isl_aff_free(aff);
+	isl_multi_aff_free(ma);
+	res = isl_aff_normalize(res);
+	return res;
+error:
+	isl_aff_free(aff);
+	isl_multi_aff_free(ma);
+	isl_aff_free(res);
+	return NULL;
+}
+
+/* Compute the pullback of "aff1" by the function represented by "aff2".
+ * In other words, plug in "aff2" in "aff1".  The result is an affine expression
+ * defined over the domain space of "aff1".
+ *
+ * The domain of "aff1" should match the range of "aff2", which means
+ * that it should be single-dimensional.
+ */
+__isl_give isl_aff *isl_aff_pullback_aff(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2)
+{
+	isl_multi_aff *ma;
+
+	ma = isl_multi_aff_from_aff(aff2);
+	return isl_aff_pullback_multi_aff(aff1, ma);
+}
+
+/* Compute the pullback of "ma1" by the function represented by "ma2".
+ * In other words, plug in "ma2" in "ma1".
+ *
+ * The parameters of "ma1" and "ma2" are assumed to have been aligned.
+ */
+static __isl_give isl_multi_aff *isl_multi_aff_pullback_multi_aff_aligned(
+	__isl_take isl_multi_aff *ma1, __isl_take isl_multi_aff *ma2)
+{
+	int i;
+	isl_space *space = NULL;
+
+	ma2 = isl_multi_aff_align_divs(ma2);
+	ma1 = isl_multi_aff_cow(ma1);
+	if (!ma1 || !ma2)
+		goto error;
+
+	space = isl_space_join(isl_multi_aff_get_space(ma2),
+				isl_multi_aff_get_space(ma1));
+
+	for (i = 0; i < ma1->n; ++i) {
+		ma1->p[i] = isl_aff_pullback_multi_aff(ma1->p[i],
+						    isl_multi_aff_copy(ma2));
+		if (!ma1->p[i])
+			goto error;
+	}
+
+	ma1 = isl_multi_aff_reset_space(ma1, space);
+	isl_multi_aff_free(ma2);
+	return ma1;
+error:
+	isl_space_free(space);
+	isl_multi_aff_free(ma2);
+	isl_multi_aff_free(ma1);
+	return NULL;
+}
+
+/* Compute the pullback of "ma1" by the function represented by "ma2".
+ * In other words, plug in "ma2" in "ma1".
+ */
+__isl_give isl_multi_aff *isl_multi_aff_pullback_multi_aff(
+	__isl_take isl_multi_aff *ma1, __isl_take isl_multi_aff *ma2)
+{
+	return isl_multi_aff_align_params_multi_multi_and(ma1, ma2,
+				&isl_multi_aff_pullback_multi_aff_aligned);
+}
+
+/* Extend the local space of "dst" to include the divs
+ * in the local space of "src".
+ */
+__isl_give isl_aff *isl_aff_align_divs(__isl_take isl_aff *dst,
+	__isl_keep isl_aff *src)
+{
+	isl_ctx *ctx;
+	int *exp1 = NULL;
+	int *exp2 = NULL;
+	isl_mat *div;
+
+	if (!src || !dst)
+		return isl_aff_free(dst);
+
+	ctx = isl_aff_get_ctx(src);
+	if (!isl_space_is_equal(src->ls->dim, dst->ls->dim))
+		isl_die(ctx, isl_error_invalid,
+			"spaces don't match", goto error);
+
+	if (src->ls->div->n_row == 0)
+		return dst;
+
+	exp1 = isl_alloc_array(ctx, int, src->ls->div->n_row);
+	exp2 = isl_alloc_array(ctx, int, dst->ls->div->n_row);
+	if (!exp1 || (dst->ls->div->n_row && !exp2))
+		goto error;
+
+	div = isl_merge_divs(src->ls->div, dst->ls->div, exp1, exp2);
+	dst = isl_aff_expand_divs(dst, div, exp2);
+	free(exp1);
+	free(exp2);
+
+	return dst;
+error:
+	free(exp1);
+	free(exp2);
+	return isl_aff_free(dst);
+}
+
+/* Adjust the local spaces of the affine expressions in "maff"
+ * such that they all have the save divs.
+ */
+__isl_give isl_multi_aff *isl_multi_aff_align_divs(
+	__isl_take isl_multi_aff *maff)
+{
+	int i;
+
+	if (!maff)
+		return NULL;
+	if (maff->n == 0)
+		return maff;
+	maff = isl_multi_aff_cow(maff);
+	if (!maff)
+		return NULL;
+
+	for (i = 1; i < maff->n; ++i)
+		maff->p[0] = isl_aff_align_divs(maff->p[0], maff->p[i]);
+	for (i = 1; i < maff->n; ++i) {
+		maff->p[i] = isl_aff_align_divs(maff->p[i], maff->p[0]);
+		if (!maff->p[i])
+			return isl_multi_aff_free(maff);
+	}
+
+	return maff;
+}
+
+__isl_give isl_aff *isl_aff_lift(__isl_take isl_aff *aff)
+{
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->ls = isl_local_space_lift(aff->ls);
+	if (!aff->ls)
+		return isl_aff_free(aff);
+
+	return aff;
+}
+
+/* Lift "maff" to a space with extra dimensions such that the result
+ * has no more existentially quantified variables.
+ * If "ls" is not NULL, then *ls is assigned the local space that lies
+ * at the basis of the lifting applied to "maff".
+ */
+__isl_give isl_multi_aff *isl_multi_aff_lift(__isl_take isl_multi_aff *maff,
+	__isl_give isl_local_space **ls)
+{
+	int i;
+	isl_space *space;
+	unsigned n_div;
+
+	if (ls)
+		*ls = NULL;
+
+	if (!maff)
+		return NULL;
+
+	if (maff->n == 0) {
+		if (ls) {
+			isl_space *space = isl_multi_aff_get_domain_space(maff);
+			*ls = isl_local_space_from_space(space);
+			if (!*ls)
+				return isl_multi_aff_free(maff);
+		}
+		return maff;
+	}
+
+	maff = isl_multi_aff_cow(maff);
+	maff = isl_multi_aff_align_divs(maff);
+	if (!maff)
+		return NULL;
+
+	n_div = isl_aff_dim(maff->p[0], isl_dim_div);
+	space = isl_multi_aff_get_space(maff);
+	space = isl_space_lift(isl_space_domain(space), n_div);
+	space = isl_space_extend_domain_with_range(space,
+						isl_multi_aff_get_space(maff));
+	if (!space)
+		return isl_multi_aff_free(maff);
+	isl_space_free(maff->space);
+	maff->space = space;
+
+	if (ls) {
+		*ls = isl_aff_get_domain_local_space(maff->p[0]);
+		if (!*ls)
+			return isl_multi_aff_free(maff);
+	}
+
+	for (i = 0; i < maff->n; ++i) {
+		maff->p[i] = isl_aff_lift(maff->p[i]);
+		if (!maff->p[i])
+			goto error;
+	}
+
+	return maff;
+error:
+	if (ls)
+		isl_local_space_free(*ls);
+	return isl_multi_aff_free(maff);
+}
+
+
+/* Extract an isl_pw_aff corresponding to output dimension "pos" of "pma".
+ */
+__isl_give isl_pw_aff *isl_pw_multi_aff_get_pw_aff(
+	__isl_keep isl_pw_multi_aff *pma, int pos)
+{
+	int i;
+	int n_out;
+	isl_space *space;
+	isl_pw_aff *pa;
+
+	if (!pma)
+		return NULL;
+
+	n_out = isl_pw_multi_aff_dim(pma, isl_dim_out);
+	if (pos < 0 || pos >= n_out)
+		isl_die(isl_pw_multi_aff_get_ctx(pma), isl_error_invalid,
+			"index out of bounds", return NULL);
+
+	space = isl_pw_multi_aff_get_space(pma);
+	space = isl_space_drop_dims(space, isl_dim_out,
+				    pos + 1, n_out - pos - 1);
+	space = isl_space_drop_dims(space, isl_dim_out, 0, pos);
+
+	pa = isl_pw_aff_alloc_size(space, pma->n);
+	for (i = 0; i < pma->n; ++i) {
+		isl_aff *aff;
+		aff = isl_multi_aff_get_aff(pma->p[i].maff, pos);
+		pa = isl_pw_aff_add_piece(pa, isl_set_copy(pma->p[i].set), aff);
+	}
+
+	return pa;
+}
+
+/* Return an isl_pw_multi_aff with the given "set" as domain and
+ * an unnamed zero-dimensional range.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_domain(
+	__isl_take isl_set *set)
+{
+	isl_multi_aff *ma;
+	isl_space *space;
+
+	space = isl_set_get_space(set);
+	space = isl_space_from_domain(space);
+	ma = isl_multi_aff_zero(space);
+	return isl_pw_multi_aff_alloc(set, ma);
+}
+
+/* Add an isl_pw_multi_aff with the given "set" as domain and
+ * an unnamed zero-dimensional range to *user.
+ */
+static int add_pw_multi_aff_from_domain(__isl_take isl_set *set, void *user)
+{
+	isl_union_pw_multi_aff **upma = user;
+	isl_pw_multi_aff *pma;
+
+	pma = isl_pw_multi_aff_from_domain(set);
+	*upma = isl_union_pw_multi_aff_add_pw_multi_aff(*upma, pma);
+
+	return 0;
+}
+
+/* Return an isl_union_pw_multi_aff with the given "uset" as domain and
+ * an unnamed zero-dimensional range.
+ */
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_from_domain(
+	__isl_take isl_union_set *uset)
+{
+	isl_space *space;
+	isl_union_pw_multi_aff *upma;
+
+	if (!uset)
+		return NULL;
+
+	space = isl_union_set_get_space(uset);
+	upma = isl_union_pw_multi_aff_empty(space);
+
+	if (isl_union_set_foreach_set(uset,
+				    &add_pw_multi_aff_from_domain, &upma) < 0)
+		goto error;
+
+	isl_union_set_free(uset);
+	return upma;
+error:
+	isl_union_set_free(uset);
+	isl_union_pw_multi_aff_free(upma);
+	return NULL;
+}
+
+/* Convert "pma" to an isl_map and add it to *umap.
+ */
+static int map_from_pw_multi_aff(__isl_take isl_pw_multi_aff *pma, void *user)
+{
+	isl_union_map **umap = user;
+	isl_map *map;
+
+	map = isl_map_from_pw_multi_aff(pma);
+	*umap = isl_union_map_add_map(*umap, map);
+
+	return 0;
+}
+
+/* Construct a union map mapping the domain of the union
+ * piecewise multi-affine expression to its range, with each dimension
+ * in the range equated to the corresponding affine expression on its cell.
+ */
+__isl_give isl_union_map *isl_union_map_from_union_pw_multi_aff(
+	__isl_take isl_union_pw_multi_aff *upma)
+{
+	isl_space *space;
+	isl_union_map *umap;
+
+	if (!upma)
+		return NULL;
+
+	space = isl_union_pw_multi_aff_get_space(upma);
+	umap = isl_union_map_empty(space);
+
+	if (isl_union_pw_multi_aff_foreach_pw_multi_aff(upma,
+					&map_from_pw_multi_aff, &umap) < 0)
+		goto error;
+
+	isl_union_pw_multi_aff_free(upma);
+	return umap;
+error:
+	isl_union_pw_multi_aff_free(upma);
+	isl_union_map_free(umap);
+	return NULL;
+}
+
+/* Local data for bin_entry and the callback "fn".
+ */
+struct isl_union_pw_multi_aff_bin_data {
+	isl_union_pw_multi_aff *upma2;
+	isl_union_pw_multi_aff *res;
+	isl_pw_multi_aff *pma;
+	int (*fn)(void **entry, void *user);
+};
+
+/* Given an isl_pw_multi_aff from upma1, store it in data->pma
+ * and call data->fn for each isl_pw_multi_aff in data->upma2.
+ */
+static int bin_entry(void **entry, void *user)
+{
+	struct isl_union_pw_multi_aff_bin_data *data = user;
+	isl_pw_multi_aff *pma = *entry;
+
+	data->pma = pma;
+	if (isl_hash_table_foreach(data->upma2->space->ctx, &data->upma2->table,
+				   data->fn, data) < 0)
+		return -1;
+
+	return 0;
+}
+
+/* Call "fn" on each pair of isl_pw_multi_affs in "upma1" and "upma2".
+ * The isl_pw_multi_aff from upma1 is stored in data->pma (where data is
+ * passed as user field) and the isl_pw_multi_aff from upma2 is available
+ * as *entry.  The callback should adjust data->res if desired.
+ */
+static __isl_give isl_union_pw_multi_aff *bin_op(
+	__isl_take isl_union_pw_multi_aff *upma1,
+	__isl_take isl_union_pw_multi_aff *upma2,
+	int (*fn)(void **entry, void *user))
+{
+	isl_space *space;
+	struct isl_union_pw_multi_aff_bin_data data = { NULL, NULL, NULL, fn };
+
+	space = isl_union_pw_multi_aff_get_space(upma2);
+	upma1 = isl_union_pw_multi_aff_align_params(upma1, space);
+	space = isl_union_pw_multi_aff_get_space(upma1);
+	upma2 = isl_union_pw_multi_aff_align_params(upma2, space);
+
+	if (!upma1 || !upma2)
+		goto error;
+
+	data.upma2 = upma2;
+	data.res = isl_union_pw_multi_aff_alloc(isl_space_copy(upma1->space),
+				       upma1->table.n);
+	if (isl_hash_table_foreach(upma1->space->ctx, &upma1->table,
+				   &bin_entry, &data) < 0)
+		goto error;
+
+	isl_union_pw_multi_aff_free(upma1);
+	isl_union_pw_multi_aff_free(upma2);
+	return data.res;
+error:
+	isl_union_pw_multi_aff_free(upma1);
+	isl_union_pw_multi_aff_free(upma2);
+	isl_union_pw_multi_aff_free(data.res);
+	return NULL;
+}
+
+/* Given two aligned isl_pw_multi_affs A -> B and C -> D,
+ * construct an isl_pw_multi_aff (A * C) -> [B -> D].
+ */
+static __isl_give isl_pw_multi_aff *pw_multi_aff_range_product(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	isl_space *space;
+
+	space = isl_space_range_product(isl_pw_multi_aff_get_space(pma1),
+					isl_pw_multi_aff_get_space(pma2));
+	return isl_pw_multi_aff_on_shared_domain_in(pma1, pma2, space,
+					    &isl_multi_aff_range_product);
+}
+
+/* Given two isl_pw_multi_affs A -> B and C -> D,
+ * construct an isl_pw_multi_aff (A * C) -> [B -> D].
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_range_product(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_align_params_pw_pw_and(pma1, pma2,
+					    &pw_multi_aff_range_product);
+}
+
+/* Given two aligned isl_pw_multi_affs A -> B and C -> D,
+ * construct an isl_pw_multi_aff (A * C) -> (B, D).
+ */
+static __isl_give isl_pw_multi_aff *pw_multi_aff_flat_range_product(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	isl_space *space;
+
+	space = isl_space_range_product(isl_pw_multi_aff_get_space(pma1),
+					isl_pw_multi_aff_get_space(pma2));
+	space = isl_space_flatten_range(space);
+	return isl_pw_multi_aff_on_shared_domain_in(pma1, pma2, space,
+					    &isl_multi_aff_flat_range_product);
+}
+
+/* Given two isl_pw_multi_affs A -> B and C -> D,
+ * construct an isl_pw_multi_aff (A * C) -> (B, D).
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_flat_range_product(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_align_params_pw_pw_and(pma1, pma2,
+					    &pw_multi_aff_flat_range_product);
+}
+
+/* If data->pma and *entry have the same domain space, then compute
+ * their flat range product and the result to data->res.
+ */
+static int flat_range_product_entry(void **entry, void *user)
+{
+	struct isl_union_pw_multi_aff_bin_data *data = user;
+	isl_pw_multi_aff *pma2 = *entry;
+
+	if (!isl_space_tuple_is_equal(data->pma->dim, isl_dim_in,
+				 pma2->dim, isl_dim_in))
+		return 0;
+
+	pma2 = isl_pw_multi_aff_flat_range_product(
+					isl_pw_multi_aff_copy(data->pma),
+					isl_pw_multi_aff_copy(pma2));
+
+	data->res = isl_union_pw_multi_aff_add_pw_multi_aff(data->res, pma2);
+
+	return 0;
+}
+
+/* Given two isl_union_pw_multi_affs A -> B and C -> D,
+ * construct an isl_union_pw_multi_aff (A * C) -> (B, D).
+ */
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_flat_range_product(
+	__isl_take isl_union_pw_multi_aff *upma1,
+	__isl_take isl_union_pw_multi_aff *upma2)
+{
+	return bin_op(upma1, upma2, &flat_range_product_entry);
+}
+
+/* Replace the affine expressions at position "pos" in "pma" by "pa".
+ * The parameters are assumed to have been aligned.
+ *
+ * The implementation essentially performs an isl_pw_*_on_shared_domain,
+ * except that it works on two different isl_pw_* types.
+ */
+static __isl_give isl_pw_multi_aff *pw_multi_aff_set_pw_aff(
+	__isl_take isl_pw_multi_aff *pma, unsigned pos,
+	__isl_take isl_pw_aff *pa)
+{
+	int i, j, n;
+	isl_pw_multi_aff *res = NULL;
+
+	if (!pma || !pa)
+		goto error;
+
+	if (!isl_space_tuple_is_equal(pma->dim, isl_dim_in,
+					pa->dim, isl_dim_in))
+		isl_die(isl_pw_multi_aff_get_ctx(pma), isl_error_invalid,
+			"domains don't match", goto error);
+	if (pos >= isl_pw_multi_aff_dim(pma, isl_dim_out))
+		isl_die(isl_pw_multi_aff_get_ctx(pma), isl_error_invalid,
+			"index out of bounds", goto error);
+
+	n = pma->n * pa->n;
+	res = isl_pw_multi_aff_alloc_size(isl_pw_multi_aff_get_space(pma), n);
+
+	for (i = 0; i < pma->n; ++i) {
+		for (j = 0; j < pa->n; ++j) {
+			isl_set *common;
+			isl_multi_aff *res_ij;
+			int empty;
+
+			common = isl_set_intersect(isl_set_copy(pma->p[i].set),
+						   isl_set_copy(pa->p[j].set));
+			empty = isl_set_plain_is_empty(common);
+			if (empty < 0 || empty) {
+				isl_set_free(common);
+				if (empty < 0)
+					goto error;
+				continue;
+			}
+
+			res_ij = isl_multi_aff_set_aff(
+					isl_multi_aff_copy(pma->p[i].maff), pos,
+					isl_aff_copy(pa->p[j].aff));
+			res_ij = isl_multi_aff_gist(res_ij,
+					isl_set_copy(common));
+
+			res = isl_pw_multi_aff_add_piece(res, common, res_ij);
+		}
+	}
+
+	isl_pw_multi_aff_free(pma);
+	isl_pw_aff_free(pa);
+	return res;
+error:
+	isl_pw_multi_aff_free(pma);
+	isl_pw_aff_free(pa);
+	return isl_pw_multi_aff_free(res);
+}
+
+/* Replace the affine expressions at position "pos" in "pma" by "pa".
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_set_pw_aff(
+	__isl_take isl_pw_multi_aff *pma, unsigned pos,
+	__isl_take isl_pw_aff *pa)
+{
+	if (!pma || !pa)
+		goto error;
+	if (isl_space_match(pma->dim, isl_dim_param, pa->dim, isl_dim_param))
+		return pw_multi_aff_set_pw_aff(pma, pos, pa);
+	if (!isl_space_has_named_params(pma->dim) ||
+	    !isl_space_has_named_params(pa->dim))
+		isl_die(isl_pw_multi_aff_get_ctx(pma), isl_error_invalid,
+			"unaligned unnamed parameters", goto error);
+	pma = isl_pw_multi_aff_align_params(pma, isl_pw_aff_get_space(pa));
+	pa = isl_pw_aff_align_params(pa, isl_pw_multi_aff_get_space(pma));
+	return pw_multi_aff_set_pw_aff(pma, pos, pa);
+error:
+	isl_pw_multi_aff_free(pma);
+	isl_pw_aff_free(pa);
+	return NULL;
+}
+
+/* Do the parameters of "pa" match those of "space"?
+ */
+int isl_pw_aff_matching_params(__isl_keep isl_pw_aff *pa,
+	__isl_keep isl_space *space)
+{
+	isl_space *pa_space;
+	int match;
+
+	if (!pa || !space)
+		return -1;
+
+	pa_space = isl_pw_aff_get_space(pa);
+
+	match = isl_space_match(space, isl_dim_param, pa_space, isl_dim_param);
+
+	isl_space_free(pa_space);
+	return match;
+}
+
+/* Check that the domain space of "pa" matches "space".
+ *
+ * Return 0 on success and -1 on error.
+ */
+int isl_pw_aff_check_match_domain_space(__isl_keep isl_pw_aff *pa,
+	__isl_keep isl_space *space)
+{
+	isl_space *pa_space;
+	int match;
+
+	if (!pa || !space)
+		return -1;
+
+	pa_space = isl_pw_aff_get_space(pa);
+
+	match = isl_space_match(space, isl_dim_param, pa_space, isl_dim_param);
+	if (match < 0)
+		goto error;
+	if (!match)
+		isl_die(isl_pw_aff_get_ctx(pa), isl_error_invalid,
+			"parameters don't match", goto error);
+	match = isl_space_tuple_is_equal(space, isl_dim_in,
+					pa_space, isl_dim_in);
+	if (match < 0)
+		goto error;
+	if (!match)
+		isl_die(isl_pw_aff_get_ctx(pa), isl_error_invalid,
+			"domains don't match", goto error);
+	isl_space_free(pa_space);
+	return 0;
+error:
+	isl_space_free(pa_space);
+	return -1;
+}
+
+#undef BASE
+#define BASE pw_aff
+
+#include <isl_multi_templ.c>
+
+/* Scale the elements of "pma" by the corresponding elements of "mv".
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_scale_multi_val(
+	__isl_take isl_pw_multi_aff *pma, __isl_take isl_multi_val *mv)
+{
+	int i;
+
+	pma = isl_pw_multi_aff_cow(pma);
+	if (!pma || !mv)
+		goto error;
+	if (!isl_space_tuple_is_equal(pma->dim, isl_dim_out,
+					mv->space, isl_dim_set))
+		isl_die(isl_pw_multi_aff_get_ctx(pma), isl_error_invalid,
+			"spaces don't match", goto error);
+	if (!isl_space_match(pma->dim, isl_dim_param,
+					mv->space, isl_dim_param)) {
+		pma = isl_pw_multi_aff_align_params(pma,
+					    isl_multi_val_get_space(mv));
+		mv = isl_multi_val_align_params(mv,
+					    isl_pw_multi_aff_get_space(pma));
+		if (!pma || !mv)
+			goto error;
+	}
+
+	for (i = 0; i < pma->n; ++i) {
+		pma->p[i].maff = isl_multi_aff_scale_multi_val(pma->p[i].maff,
+							isl_multi_val_copy(mv));
+		if (!pma->p[i].maff)
+			goto error;
+	}
+
+	isl_multi_val_free(mv);
+	return pma;
+error:
+	isl_multi_val_free(mv);
+	isl_pw_multi_aff_free(pma);
+	return NULL;
+}
+
+/* Internal data structure for isl_union_pw_multi_aff_scale_multi_val.
+ * mv contains the mv argument.
+ * res collects the results.
+ */
+struct isl_union_pw_multi_aff_scale_multi_val_data {
+	isl_multi_val *mv;
+	isl_union_pw_multi_aff *res;
+};
+
+/* This function is called for each entry of an isl_union_pw_multi_aff.
+ * If the space of the entry matches that of data->mv,
+ * then apply isl_pw_multi_aff_scale_multi_val and add the result
+ * to data->res.
+ */
+static int union_pw_multi_aff_scale_multi_val_entry(void **entry, void *user)
+{
+	struct isl_union_pw_multi_aff_scale_multi_val_data *data = user;
+	isl_pw_multi_aff *pma = *entry;
+
+	if (!pma)
+		return -1;
+	if (!isl_space_tuple_is_equal(pma->dim, isl_dim_out,
+				    data->mv->space, isl_dim_set))
+		return 0;
+
+	pma = isl_pw_multi_aff_copy(pma);
+	pma = isl_pw_multi_aff_scale_multi_val(pma,
+						isl_multi_val_copy(data->mv));
+	data->res = isl_union_pw_multi_aff_add_pw_multi_aff(data->res, pma);
+	if (!data->res)
+		return -1;
+
+	return 0;
+}
+
+/* Scale the elements of "upma" by the corresponding elements of "mv",
+ * for those entries that match the space of "mv".
+ */
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_scale_multi_val(
+	__isl_take isl_union_pw_multi_aff *upma, __isl_take isl_multi_val *mv)
+{
+	struct isl_union_pw_multi_aff_scale_multi_val_data data;
+
+	upma = isl_union_pw_multi_aff_align_params(upma,
+						isl_multi_val_get_space(mv));
+	mv = isl_multi_val_align_params(mv,
+					isl_union_pw_multi_aff_get_space(upma));
+	if (!upma || !mv)
+		goto error;
+
+	data.mv = mv;
+	data.res = isl_union_pw_multi_aff_alloc(isl_space_copy(upma->space),
+						upma->table.n);
+	if (isl_hash_table_foreach(upma->space->ctx, &upma->table,
+		       &union_pw_multi_aff_scale_multi_val_entry, &data) < 0)
+		goto error;
+
+	isl_multi_val_free(mv);
+	isl_union_pw_multi_aff_free(upma);
+	return data.res;
+error:
+	isl_multi_val_free(mv);
+	isl_union_pw_multi_aff_free(upma);
+	return NULL;
+}
+
+/* Construct and return a piecewise multi affine expression
+ * in the given space with value zero in each of the output dimensions and
+ * a universe domain.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_zero(__isl_take isl_space *space)
+{
+	return isl_pw_multi_aff_from_multi_aff(isl_multi_aff_zero(space));
+}
+
+/* Construct and return a piecewise multi affine expression
+ * that is equal to the given piecewise affine expression.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_pw_aff(
+	__isl_take isl_pw_aff *pa)
+{
+	int i;
+	isl_space *space;
+	isl_pw_multi_aff *pma;
+
+	if (!pa)
+		return NULL;
+
+	space = isl_pw_aff_get_space(pa);
+	pma = isl_pw_multi_aff_alloc_size(space, pa->n);
+
+	for (i = 0; i < pa->n; ++i) {
+		isl_set *set;
+		isl_multi_aff *ma;
+
+		set = isl_set_copy(pa->p[i].set);
+		ma = isl_multi_aff_from_aff(isl_aff_copy(pa->p[i].aff));
+		pma = isl_pw_multi_aff_add_piece(pma, set, ma);
+	}
+
+	isl_pw_aff_free(pa);
+	return pma;
+}
+
+/* Construct a set or map mapping the shared (parameter) domain
+ * of the piecewise affine expressions to the range of "mpa"
+ * with each dimension in the range equated to the
+ * corresponding piecewise affine expression.
+ */
+static __isl_give isl_map *map_from_multi_pw_aff(
+	__isl_take isl_multi_pw_aff *mpa)
+{
+	int i;
+	isl_space *space;
+	isl_map *map;
+
+	if (!mpa)
+		return NULL;
+
+	if (isl_space_dim(mpa->space, isl_dim_out) != mpa->n)
+		isl_die(isl_multi_pw_aff_get_ctx(mpa), isl_error_internal,
+			"invalid space", goto error);
+
+	space = isl_multi_pw_aff_get_domain_space(mpa);
+	map = isl_map_universe(isl_space_from_domain(space));
+
+	for (i = 0; i < mpa->n; ++i) {
+		isl_pw_aff *pa;
+		isl_map *map_i;
+
+		pa = isl_pw_aff_copy(mpa->p[i]);
+		map_i = map_from_pw_aff(pa);
+
+		map = isl_map_flat_range_product(map, map_i);
+	}
+
+	map = isl_map_reset_space(map, isl_multi_pw_aff_get_space(mpa));
+
+	isl_multi_pw_aff_free(mpa);
+	return map;
+error:
+	isl_multi_pw_aff_free(mpa);
+	return NULL;
+}
+
+/* Construct a map mapping the shared domain
+ * of the piecewise affine expressions to the range of "mpa"
+ * with each dimension in the range equated to the
+ * corresponding piecewise affine expression.
+ */
+__isl_give isl_map *isl_map_from_multi_pw_aff(__isl_take isl_multi_pw_aff *mpa)
+{
+	if (!mpa)
+		return NULL;
+	if (isl_space_is_set(mpa->space))
+		isl_die(isl_multi_pw_aff_get_ctx(mpa), isl_error_internal,
+			"space of input is not a map", goto error);
+
+	return map_from_multi_pw_aff(mpa);
+error:
+	isl_multi_pw_aff_free(mpa);
+	return NULL;
+}
+
+/* Construct a set mapping the shared parameter domain
+ * of the piecewise affine expressions to the space of "mpa"
+ * with each dimension in the range equated to the
+ * corresponding piecewise affine expression.
+ */
+__isl_give isl_set *isl_set_from_multi_pw_aff(__isl_take isl_multi_pw_aff *mpa)
+{
+	if (!mpa)
+		return NULL;
+	if (!isl_space_is_set(mpa->space))
+		isl_die(isl_multi_pw_aff_get_ctx(mpa), isl_error_internal,
+			"space of input is not a set", goto error);
+
+	return map_from_multi_pw_aff(mpa);
+error:
+	isl_multi_pw_aff_free(mpa);
+	return NULL;
+}
+
+/* Construct and return a piecewise multi affine expression
+ * that is equal to the given multi piecewise affine expression
+ * on the shared domain of the piecewise affine expressions.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_multi_pw_aff(
+	__isl_take isl_multi_pw_aff *mpa)
+{
+	int i;
+	isl_space *space;
+	isl_pw_aff *pa;
+	isl_pw_multi_aff *pma;
+
+	if (!mpa)
+		return NULL;
+
+	space = isl_multi_pw_aff_get_space(mpa);
+
+	if (mpa->n == 0) {
+		isl_multi_pw_aff_free(mpa);
+		return isl_pw_multi_aff_zero(space);
+	}
+
+	pa = isl_multi_pw_aff_get_pw_aff(mpa, 0);
+	pma = isl_pw_multi_aff_from_pw_aff(pa);
+
+	for (i = 1; i < mpa->n; ++i) {
+		isl_pw_multi_aff *pma_i;
+
+		pa = isl_multi_pw_aff_get_pw_aff(mpa, i);
+		pma_i = isl_pw_multi_aff_from_pw_aff(pa);
+		pma = isl_pw_multi_aff_range_product(pma, pma_i);
+	}
+
+	pma = isl_pw_multi_aff_reset_space(pma, space);
+
+	isl_multi_pw_aff_free(mpa);
+	return pma;
+}
+
+/* Construct and return a multi piecewise affine expression
+ * that is equal to the given multi affine expression.
+ */
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_from_multi_aff(
+	__isl_take isl_multi_aff *ma)
+{
+	int i, n;
+	isl_multi_pw_aff *mpa;
+
+	if (!ma)
+		return NULL;
+
+	n = isl_multi_aff_dim(ma, isl_dim_out);
+	mpa = isl_multi_pw_aff_alloc(isl_multi_aff_get_space(ma));
+
+	for (i = 0; i < n; ++i) {
+		isl_pw_aff *pa;
+
+		pa = isl_pw_aff_from_aff(isl_multi_aff_get_aff(ma, i));
+		mpa = isl_multi_pw_aff_set_pw_aff(mpa, i, pa);
+	}
+
+	isl_multi_aff_free(ma);
+	return mpa;
+}
+
+/* Construct and return a multi piecewise affine expression
+ * that is equal to the given piecewise multi affine expression.
+ */
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_from_pw_multi_aff(
+	__isl_take isl_pw_multi_aff *pma)
+{
+	int i, n;
+	isl_space *space;
+	isl_multi_pw_aff *mpa;
+
+	if (!pma)
+		return NULL;
+
+	n = isl_pw_multi_aff_dim(pma, isl_dim_out);
+	space = isl_pw_multi_aff_get_space(pma);
+	mpa = isl_multi_pw_aff_alloc(space);
+
+	for (i = 0; i < n; ++i) {
+		isl_pw_aff *pa;
+
+		pa = isl_pw_multi_aff_get_pw_aff(pma, i);
+		mpa = isl_multi_pw_aff_set_pw_aff(mpa, i, pa);
+	}
+
+	isl_pw_multi_aff_free(pma);
+	return mpa;
+}
+
+/* Do "pa1" and "pa2" represent the same function?
+ *
+ * We first check if they are obviously equal.
+ * If not, we convert them to maps and check if those are equal.
+ */
+int isl_pw_aff_is_equal(__isl_keep isl_pw_aff *pa1, __isl_keep isl_pw_aff *pa2)
+{
+	int equal;
+	isl_map *map1, *map2;
+
+	if (!pa1 || !pa2)
+		return -1;
+
+	equal = isl_pw_aff_plain_is_equal(pa1, pa2);
+	if (equal < 0 || equal)
+		return equal;
+
+	map1 = map_from_pw_aff(isl_pw_aff_copy(pa1));
+	map2 = map_from_pw_aff(isl_pw_aff_copy(pa2));
+	equal = isl_map_is_equal(map1, map2);
+	isl_map_free(map1);
+	isl_map_free(map2);
+
+	return equal;
+}
+
+/* Do "mpa1" and "mpa2" represent the same function?
+ *
+ * Note that we cannot convert the entire isl_multi_pw_aff
+ * to a map because the domains of the piecewise affine expressions
+ * may not be the same.
+ */
+int isl_multi_pw_aff_is_equal(__isl_keep isl_multi_pw_aff *mpa1,
+	__isl_keep isl_multi_pw_aff *mpa2)
+{
+	int i;
+	int equal;
+
+	if (!mpa1 || !mpa2)
+		return -1;
+
+	if (!isl_space_match(mpa1->space, isl_dim_param,
+			     mpa2->space, isl_dim_param)) {
+		if (!isl_space_has_named_params(mpa1->space))
+			return 0;
+		if (!isl_space_has_named_params(mpa2->space))
+			return 0;
+		mpa1 = isl_multi_pw_aff_copy(mpa1);
+		mpa2 = isl_multi_pw_aff_copy(mpa2);
+		mpa1 = isl_multi_pw_aff_align_params(mpa1,
+					    isl_multi_pw_aff_get_space(mpa2));
+		mpa2 = isl_multi_pw_aff_align_params(mpa2,
+					    isl_multi_pw_aff_get_space(mpa1));
+		equal = isl_multi_pw_aff_is_equal(mpa1, mpa2);
+		isl_multi_pw_aff_free(mpa1);
+		isl_multi_pw_aff_free(mpa2);
+		return equal;
+	}
+
+	equal = isl_space_is_equal(mpa1->space, mpa2->space);
+	if (equal < 0 || !equal)
+		return equal;
+
+	for (i = 0; i < mpa1->n; ++i) {
+		equal = isl_pw_aff_is_equal(mpa1->p[i], mpa2->p[i]);
+		if (equal < 0 || !equal)
+			return equal;
+	}
+
+	return 1;
+}
+
+/* Coalesce the elements of "mpa".
+ *
+ * Note that such coalescing does not change the meaning of "mpa"
+ * so there is no need to cow.  We do need to be careful not to
+ * destroy any other copies of "mpa" in case of failure.
+ */
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_coalesce(
+	__isl_take isl_multi_pw_aff *mpa)
+{
+	int i;
+
+	if (!mpa)
+		return NULL;
+
+	for (i = 0; i < mpa->n; ++i) {
+		isl_pw_aff *pa = isl_pw_aff_copy(mpa->p[i]);
+		pa = isl_pw_aff_coalesce(pa);
+		if (!pa)
+			return isl_multi_pw_aff_free(mpa);
+		isl_pw_aff_free(mpa->p[i]);
+		mpa->p[i] = pa;
+	}
+
+	return mpa;
+}
+
+/* Compute the pullback of "mpa" by the function represented by "ma".
+ * In other words, plug in "ma" in "mpa".
+ *
+ * The parameters of "mpa" and "ma" are assumed to have been aligned.
+ */
+static __isl_give isl_multi_pw_aff *isl_multi_pw_aff_pullback_multi_aff_aligned(
+	__isl_take isl_multi_pw_aff *mpa, __isl_take isl_multi_aff *ma)
+{
+	int i;
+	isl_space *space = NULL;
+
+	mpa = isl_multi_pw_aff_cow(mpa);
+	if (!mpa || !ma)
+		goto error;
+
+	space = isl_space_join(isl_multi_aff_get_space(ma),
+				isl_multi_pw_aff_get_space(mpa));
+	if (!space)
+		goto error;
+
+	for (i = 0; i < mpa->n; ++i) {
+		mpa->p[i] = isl_pw_aff_pullback_multi_aff(mpa->p[i],
+						    isl_multi_aff_copy(ma));
+		if (!mpa->p[i])
+			goto error;
+	}
+
+	isl_multi_aff_free(ma);
+	isl_space_free(mpa->space);
+	mpa->space = space;
+	return mpa;
+error:
+	isl_space_free(space);
+	isl_multi_pw_aff_free(mpa);
+	isl_multi_aff_free(ma);
+	return NULL;
+}
+
+/* Compute the pullback of "mpa" by the function represented by "ma".
+ * In other words, plug in "ma" in "mpa".
+ */
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_pullback_multi_aff(
+	__isl_take isl_multi_pw_aff *mpa, __isl_take isl_multi_aff *ma)
+{
+	if (!mpa || !ma)
+		goto error;
+	if (isl_space_match(mpa->space, isl_dim_param,
+			    ma->space, isl_dim_param))
+		return isl_multi_pw_aff_pullback_multi_aff_aligned(mpa, ma);
+	mpa = isl_multi_pw_aff_align_params(mpa, isl_multi_aff_get_space(ma));
+	ma = isl_multi_aff_align_params(ma, isl_multi_pw_aff_get_space(mpa));
+	return isl_multi_pw_aff_pullback_multi_aff_aligned(mpa, ma);
+error:
+	isl_multi_pw_aff_free(mpa);
+	isl_multi_aff_free(ma);
+	return NULL;
+}
+
+/* Compute the pullback of "mpa" by the function represented by "pma".
+ * In other words, plug in "pma" in "mpa".
+ *
+ * The parameters of "mpa" and "mpa" are assumed to have been aligned.
+ */
+static __isl_give isl_multi_pw_aff *
+isl_multi_pw_aff_pullback_pw_multi_aff_aligned(
+	__isl_take isl_multi_pw_aff *mpa, __isl_take isl_pw_multi_aff *pma)
+{
+	int i;
+	isl_space *space = NULL;
+
+	mpa = isl_multi_pw_aff_cow(mpa);
+	if (!mpa || !pma)
+		goto error;
+
+	space = isl_space_join(isl_pw_multi_aff_get_space(pma),
+				isl_multi_pw_aff_get_space(mpa));
+
+	for (i = 0; i < mpa->n; ++i) {
+		mpa->p[i] = isl_pw_aff_pullback_pw_multi_aff_aligned(mpa->p[i],
+						    isl_pw_multi_aff_copy(pma));
+		if (!mpa->p[i])
+			goto error;
+	}
+
+	isl_pw_multi_aff_free(pma);
+	isl_space_free(mpa->space);
+	mpa->space = space;
+	return mpa;
+error:
+	isl_space_free(space);
+	isl_multi_pw_aff_free(mpa);
+	isl_pw_multi_aff_free(pma);
+	return NULL;
+}
+
+/* Compute the pullback of "mpa" by the function represented by "pma".
+ * In other words, plug in "pma" in "mpa".
+ */
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_pullback_pw_multi_aff(
+	__isl_take isl_multi_pw_aff *mpa, __isl_take isl_pw_multi_aff *pma)
+{
+	if (!mpa || !pma)
+		goto error;
+	if (isl_space_match(mpa->space, isl_dim_param, pma->dim, isl_dim_param))
+		return isl_multi_pw_aff_pullback_pw_multi_aff_aligned(mpa, pma);
+	mpa = isl_multi_pw_aff_align_params(mpa,
+					    isl_pw_multi_aff_get_space(pma));
+	pma = isl_pw_multi_aff_align_params(pma,
+					    isl_multi_pw_aff_get_space(mpa));
+	return isl_multi_pw_aff_pullback_pw_multi_aff_aligned(mpa, pma);
+error:
+	isl_multi_pw_aff_free(mpa);
+	isl_pw_multi_aff_free(pma);
+	return NULL;
+}
+
+/* Apply "aff" to "mpa".  The range of "mpa" needs to be compatible
+ * with the domain of "aff".  The domain of the result is the same
+ * as that of "mpa".
+ * "mpa" and "aff" are assumed to have been aligned.
+ *
+ * We first extract the parametric constant from "aff", defined
+ * over the correct domain.
+ * Then we add the appropriate combinations of the members of "mpa".
+ * Finally, we add the integer divisions through recursive calls.
+ */
+static __isl_give isl_pw_aff *isl_multi_pw_aff_apply_aff_aligned(
+	__isl_take isl_multi_pw_aff *mpa, __isl_take isl_aff *aff)
+{
+	int i, n_param, n_in, n_div;
+	isl_space *space;
+	isl_val *v;
+	isl_pw_aff *pa;
+	isl_aff *tmp;
+
+	n_param = isl_aff_dim(aff, isl_dim_param);
+	n_in = isl_aff_dim(aff, isl_dim_in);
+	n_div = isl_aff_dim(aff, isl_dim_div);
+
+	space = isl_space_domain(isl_multi_pw_aff_get_space(mpa));
+	tmp = isl_aff_copy(aff);
+	tmp = isl_aff_drop_dims(tmp, isl_dim_div, 0, n_div);
+	tmp = isl_aff_drop_dims(tmp, isl_dim_in, 0, n_in);
+	tmp = isl_aff_add_dims(tmp, isl_dim_in,
+				isl_space_dim(space, isl_dim_set));
+	tmp = isl_aff_reset_domain_space(tmp, space);
+	pa = isl_pw_aff_from_aff(tmp);
+
+	for (i = 0; i < n_in; ++i) {
+		isl_pw_aff *pa_i;
+
+		if (!isl_aff_involves_dims(aff, isl_dim_in, i, 1))
+			continue;
+		v = isl_aff_get_coefficient_val(aff, isl_dim_in, i);
+		pa_i = isl_multi_pw_aff_get_pw_aff(mpa, i);
+		pa_i = isl_pw_aff_scale_val(pa_i, v);
+		pa = isl_pw_aff_add(pa, pa_i);
+	}
+
+	for (i = 0; i < n_div; ++i) {
+		isl_aff *div;
+		isl_pw_aff *pa_i;
+
+		if (!isl_aff_involves_dims(aff, isl_dim_div, i, 1))
+			continue;
+		div = isl_aff_get_div(aff, i);
+		pa_i = isl_multi_pw_aff_apply_aff_aligned(
+					    isl_multi_pw_aff_copy(mpa), div);
+		pa_i = isl_pw_aff_floor(pa_i);
+		v = isl_aff_get_coefficient_val(aff, isl_dim_div, i);
+		pa_i = isl_pw_aff_scale_val(pa_i, v);
+		pa = isl_pw_aff_add(pa, pa_i);
+	}
+
+	isl_multi_pw_aff_free(mpa);
+	isl_aff_free(aff);
+
+	return pa;
+}
+
+/* Apply "aff" to "mpa".  The range of "mpa" needs to be compatible
+ * with the domain of "aff".  The domain of the result is the same
+ * as that of "mpa".
+ */
+__isl_give isl_pw_aff *isl_multi_pw_aff_apply_aff(
+	__isl_take isl_multi_pw_aff *mpa, __isl_take isl_aff *aff)
+{
+	if (!aff || !mpa)
+		goto error;
+	if (isl_space_match(aff->ls->dim, isl_dim_param,
+				mpa->space, isl_dim_param))
+		return isl_multi_pw_aff_apply_aff_aligned(mpa, aff);
+
+	aff = isl_aff_align_params(aff, isl_multi_pw_aff_get_space(mpa));
+	mpa = isl_multi_pw_aff_align_params(mpa, isl_aff_get_space(aff));
+
+	return isl_multi_pw_aff_apply_aff_aligned(mpa, aff);
+error:
+	isl_aff_free(aff);
+	isl_multi_pw_aff_free(mpa);
+	return NULL;
+}
+
+/* Apply "pa" to "mpa".  The range of "mpa" needs to be compatible
+ * with the domain of "pa".  The domain of the result is the same
+ * as that of "mpa".
+ * "mpa" and "pa" are assumed to have been aligned.
+ *
+ * We consider each piece in turn.  Note that the domains of the
+ * pieces are assumed to be disjoint and they remain disjoint
+ * after taking the preimage (over the same function).
+ */
+static __isl_give isl_pw_aff *isl_multi_pw_aff_apply_pw_aff_aligned(
+	__isl_take isl_multi_pw_aff *mpa, __isl_take isl_pw_aff *pa)
+{
+	isl_space *space;
+	isl_pw_aff *res;
+	int i;
+
+	if (!mpa || !pa)
+		goto error;
+
+	space = isl_space_join(isl_multi_pw_aff_get_space(mpa),
+				isl_pw_aff_get_space(pa));
+	res = isl_pw_aff_empty(space);
+
+	for (i = 0; i < pa->n; ++i) {
+		isl_pw_aff *pa_i;
+		isl_set *domain;
+
+		pa_i = isl_multi_pw_aff_apply_aff_aligned(
+					isl_multi_pw_aff_copy(mpa),
+					isl_aff_copy(pa->p[i].aff));
+		domain = isl_set_copy(pa->p[i].set);
+		domain = isl_set_preimage_multi_pw_aff(domain,
+					isl_multi_pw_aff_copy(mpa));
+		pa_i = isl_pw_aff_intersect_domain(pa_i, domain);
+		res = isl_pw_aff_add_disjoint(res, pa_i);
+	}
+
+	isl_pw_aff_free(pa);
+	isl_multi_pw_aff_free(mpa);
+	return res;
+error:
+	isl_pw_aff_free(pa);
+	isl_multi_pw_aff_free(mpa);
+	return NULL;
+}
+
+/* Apply "pa" to "mpa".  The range of "mpa" needs to be compatible
+ * with the domain of "pa".  The domain of the result is the same
+ * as that of "mpa".
+ */
+__isl_give isl_pw_aff *isl_multi_pw_aff_apply_pw_aff(
+	__isl_take isl_multi_pw_aff *mpa, __isl_take isl_pw_aff *pa)
+{
+	if (!pa || !mpa)
+		goto error;
+	if (isl_space_match(pa->dim, isl_dim_param, mpa->space, isl_dim_param))
+		return isl_multi_pw_aff_apply_pw_aff_aligned(mpa, pa);
+
+	pa = isl_pw_aff_align_params(pa, isl_multi_pw_aff_get_space(mpa));
+	mpa = isl_multi_pw_aff_align_params(mpa, isl_pw_aff_get_space(pa));
+
+	return isl_multi_pw_aff_apply_pw_aff_aligned(mpa, pa);
+error:
+	isl_pw_aff_free(pa);
+	isl_multi_pw_aff_free(mpa);
+	return NULL;
+}
+
+/* Compute the pullback of "pa" by the function represented by "mpa".
+ * In other words, plug in "mpa" in "pa".
+ * "pa" and "mpa" are assumed to have been aligned.
+ *
+ * The pullback is computed by applying "pa" to "mpa".
+ */
+static __isl_give isl_pw_aff *isl_pw_aff_pullback_multi_pw_aff_aligned(
+	__isl_take isl_pw_aff *pa, __isl_take isl_multi_pw_aff *mpa)
+{
+	return isl_multi_pw_aff_apply_pw_aff_aligned(mpa, pa);
+}
+
+/* Compute the pullback of "pa" by the function represented by "mpa".
+ * In other words, plug in "mpa" in "pa".
+ *
+ * The pullback is computed by applying "pa" to "mpa".
+ */
+__isl_give isl_pw_aff *isl_pw_aff_pullback_multi_pw_aff(
+	__isl_take isl_pw_aff *pa, __isl_take isl_multi_pw_aff *mpa)
+{
+	return isl_multi_pw_aff_apply_pw_aff(mpa, pa);
+}
+
+/* Compute the pullback of "mpa1" by the function represented by "mpa2".
+ * In other words, plug in "mpa2" in "mpa1".
+ *
+ * The parameters of "mpa1" and "mpa2" are assumed to have been aligned.
+ *
+ * We pullback each member of "mpa1" in turn.
+ */
+static __isl_give isl_multi_pw_aff *
+isl_multi_pw_aff_pullback_multi_pw_aff_aligned(
+	__isl_take isl_multi_pw_aff *mpa1, __isl_take isl_multi_pw_aff *mpa2)
+{
+	int i;
+	isl_space *space = NULL;
+
+	mpa1 = isl_multi_pw_aff_cow(mpa1);
+	if (!mpa1 || !mpa2)
+		goto error;
+
+	space = isl_space_join(isl_multi_pw_aff_get_space(mpa2),
+				isl_multi_pw_aff_get_space(mpa1));
+
+	for (i = 0; i < mpa1->n; ++i) {
+		mpa1->p[i] = isl_pw_aff_pullback_multi_pw_aff_aligned(
+				mpa1->p[i], isl_multi_pw_aff_copy(mpa2));
+		if (!mpa1->p[i])
+			goto error;
+	}
+
+	mpa1 = isl_multi_pw_aff_reset_space(mpa1, space);
+
+	isl_multi_pw_aff_free(mpa2);
+	return mpa1;
+error:
+	isl_space_free(space);
+	isl_multi_pw_aff_free(mpa1);
+	isl_multi_pw_aff_free(mpa2);
+	return NULL;
+}
+
+/* Compute the pullback of "mpa1" by the function represented by "mpa2".
+ * In other words, plug in "mpa2" in "mpa1".
+ */
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_pullback_multi_pw_aff(
+	__isl_take isl_multi_pw_aff *mpa1, __isl_take isl_multi_pw_aff *mpa2)
+{
+	return isl_multi_pw_aff_align_params_multi_multi_and(mpa1, mpa2,
+			&isl_multi_pw_aff_pullback_multi_pw_aff_aligned);
+}
+
+/* Compare two isl_affs.
+ *
+ * Return -1 if "aff1" is "smaller" than "aff2", 1 if "aff1" is "greater"
+ * than "aff2" and 0 if they are equal.
+ *
+ * The order is fairly arbitrary.  We do consider expressions that only involve
+ * earlier dimensions as "smaller".
+ */
+int isl_aff_plain_cmp(__isl_keep isl_aff *aff1, __isl_keep isl_aff *aff2)
+{
+	int cmp;
+	int last1, last2;
+
+	if (aff1 == aff2)
+		return 0;
+
+	if (!aff1)
+		return -1;
+	if (!aff2)
+		return 1;
+
+	cmp = isl_local_space_cmp(aff1->ls, aff2->ls);
+	if (cmp != 0)
+		return cmp;
+
+	last1 = isl_seq_last_non_zero(aff1->v->el + 1, aff1->v->size - 1);
+	last2 = isl_seq_last_non_zero(aff2->v->el + 1, aff1->v->size - 1);
+	if (last1 != last2)
+		return last1 - last2;
+
+	return isl_seq_cmp(aff1->v->el, aff2->v->el, aff1->v->size);
+}
+
+/* Compare two isl_pw_affs.
+ *
+ * Return -1 if "pa1" is "smaller" than "pa2", 1 if "pa1" is "greater"
+ * than "pa2" and 0 if they are equal.
+ *
+ * The order is fairly arbitrary.  We do consider expressions that only involve
+ * earlier dimensions as "smaller".
+ */
+int isl_pw_aff_plain_cmp(__isl_keep isl_pw_aff *pa1,
+	__isl_keep isl_pw_aff *pa2)
+{
+	int i;
+	int cmp;
+
+	if (pa1 == pa2)
+		return 0;
+
+	if (!pa1)
+		return -1;
+	if (!pa2)
+		return 1;
+
+	cmp = isl_space_cmp(pa1->dim, pa2->dim);
+	if (cmp != 0)
+		return cmp;
+
+	if (pa1->n != pa2->n)
+		return pa1->n - pa2->n;
+
+	for (i = 0; i < pa1->n; ++i) {
+		cmp = isl_set_plain_cmp(pa1->p[i].set, pa2->p[i].set);
+		if (cmp != 0)
+			return cmp;
+		cmp = isl_aff_plain_cmp(pa1->p[i].aff, pa2->p[i].aff);
+		if (cmp != 0)
+			return cmp;
+	}
+
+	return 0;
+}
+
+/* Return a piecewise affine expression that is equal to "v" on "domain".
+ */
+__isl_give isl_pw_aff *isl_pw_aff_val_on_domain(__isl_take isl_set *domain,
+	__isl_take isl_val *v)
+{
+	isl_space *space;
+	isl_local_space *ls;
+	isl_aff *aff;
+
+	space = isl_set_get_space(domain);
+	ls = isl_local_space_from_space(space);
+	aff = isl_aff_val_on_domain(ls, v);
+
+	return isl_pw_aff_alloc(domain, aff);
+}
+
+/* Return a multi affine expression that is equal to "mv" on domain
+ * space "space".
+ */
+__isl_give isl_multi_aff *isl_multi_aff_multi_val_on_space(
+	__isl_take isl_space *space, __isl_take isl_multi_val *mv)
+{
+	int i, n;
+	isl_space *space2;
+	isl_local_space *ls;
+	isl_multi_aff *ma;
+
+	if (!space || !mv)
+		goto error;
+
+	n = isl_multi_val_dim(mv, isl_dim_set);
+	space2 = isl_multi_val_get_space(mv);
+	space2 = isl_space_align_params(space2, isl_space_copy(space));
+	space = isl_space_align_params(space, isl_space_copy(space2));
+	space = isl_space_map_from_domain_and_range(space, space2);
+	ma = isl_multi_aff_alloc(isl_space_copy(space));
+	ls = isl_local_space_from_space(isl_space_domain(space));
+	for (i = 0; i < n; ++i) {
+		isl_val *v;
+		isl_aff *aff;
+
+		v = isl_multi_val_get_val(mv, i);
+		aff = isl_aff_val_on_domain(isl_local_space_copy(ls), v);
+		ma = isl_multi_aff_set_aff(ma, i, aff);
+	}
+	isl_local_space_free(ls);
+
+	isl_multi_val_free(mv);
+	return ma;
+error:
+	isl_space_free(space);
+	isl_multi_val_free(mv);
+	return NULL;
+}
+
+/* Return a piecewise multi-affine expression
+ * that is equal to "mv" on "domain".
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_multi_val_on_domain(
+	__isl_take isl_set *domain, __isl_take isl_multi_val *mv)
+{
+	isl_space *space;
+	isl_multi_aff *ma;
+
+	space = isl_set_get_space(domain);
+	ma = isl_multi_aff_multi_val_on_space(space, mv);
+
+	return isl_pw_multi_aff_alloc(domain, ma);
+}
+
+/* Internal data structure for isl_union_pw_multi_aff_multi_val_on_domain.
+ * mv is the value that should be attained on each domain set
+ * res collects the results
+ */
+struct isl_union_pw_multi_aff_multi_val_on_domain_data {
+	isl_multi_val *mv;
+	isl_union_pw_multi_aff *res;
+};
+
+/* Create an isl_pw_multi_aff equal to data->mv on "domain"
+ * and add it to data->res.
+ */
+static int pw_multi_aff_multi_val_on_domain(__isl_take isl_set *domain,
+	void *user)
+{
+	struct isl_union_pw_multi_aff_multi_val_on_domain_data *data = user;
+	isl_pw_multi_aff *pma;
+	isl_multi_val *mv;
+
+	mv = isl_multi_val_copy(data->mv);
+	pma = isl_pw_multi_aff_multi_val_on_domain(domain, mv);
+	data->res = isl_union_pw_multi_aff_add_pw_multi_aff(data->res, pma);
+
+	return data->res ? 0 : -1;
+}
+
+/* Return a union piecewise multi-affine expression
+ * that is equal to "mv" on "domain".
+ */
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_multi_val_on_domain(
+	__isl_take isl_union_set *domain, __isl_take isl_multi_val *mv)
+{
+	struct isl_union_pw_multi_aff_multi_val_on_domain_data data;
+	isl_space *space;
+
+	space = isl_union_set_get_space(domain);
+	data.res = isl_union_pw_multi_aff_empty(space);
+	data.mv = mv;
+	if (isl_union_set_foreach_set(domain,
+			&pw_multi_aff_multi_val_on_domain, &data) < 0)
+		data.res = isl_union_pw_multi_aff_free(data.res);
+	isl_union_set_free(domain);
+	isl_multi_val_free(mv);
+	return data.res;
+}
diff --git a/polly/lib/External/isl/isl_aff_private.h b/polly/lib/External/isl/isl_aff_private.h
new file mode 100644
index 00000000000..bb959f4bcb3
--- /dev/null
+++ b/polly/lib/External/isl/isl_aff_private.h
@@ -0,0 +1,158 @@
+#ifndef ISL_AFF_PRIVATE_H
+#define ISL_AFF_PRIVATE_H
+
+#include <isl/aff.h>
+#include <isl/vec.h>
+#include <isl/mat.h>
+#include <isl/local_space.h>
+#include <isl_int.h>
+#include <isl_reordering.h>
+
+/* ls represents the domain space.
+ *
+ * If the first two elements of "v" (the denominator and the constant term)
+ * are zero, then the isl_aff represents NaN.
+ */
+struct isl_aff {
+	int ref;
+
+	isl_local_space	*ls;
+	isl_vec		*v;
+};
+
+#undef EL
+#define EL isl_aff
+
+#include <isl_list_templ.h>
+
+struct isl_pw_aff_piece {
+	struct isl_set *set;
+	struct isl_aff *aff;
+};
+
+struct isl_pw_aff {
+	int ref;
+
+	isl_space *dim;
+
+	int n;
+
+	size_t size;
+	struct isl_pw_aff_piece p[1];
+};
+
+#undef EL
+#define EL isl_pw_aff
+
+#include <isl_list_templ.h>
+
+struct isl_pw_multi_aff_piece {
+	isl_set *set;
+	isl_multi_aff *maff;
+};
+
+struct isl_pw_multi_aff {
+	int ref;
+
+	isl_space *dim;
+
+	int n;
+
+	size_t size;
+	struct isl_pw_multi_aff_piece p[1];
+};
+
+__isl_give isl_aff *isl_aff_alloc_vec(__isl_take isl_local_space *ls,
+	__isl_take isl_vec *v);
+__isl_give isl_aff *isl_aff_alloc(__isl_take isl_local_space *ls);
+
+__isl_give isl_aff *isl_aff_reset_space_and_domain(__isl_take isl_aff *aff,
+	__isl_take isl_space *space, __isl_take isl_space *domain);
+__isl_give isl_aff *isl_aff_reset_domain_space(__isl_take isl_aff *aff,
+	__isl_take isl_space *dim);
+__isl_give isl_aff *isl_aff_realign_domain(__isl_take isl_aff *aff,
+	__isl_take isl_reordering *r);
+
+int isl_aff_get_constant(__isl_keep isl_aff *aff, isl_int *v);
+__isl_give isl_aff *isl_aff_set_constant(__isl_take isl_aff *aff, isl_int v);
+__isl_give isl_aff *isl_aff_set_coefficient(__isl_take isl_aff *aff,
+	enum isl_dim_type type, int pos, isl_int v);
+__isl_give isl_aff *isl_aff_add_constant(__isl_take isl_aff *aff, isl_int v);
+
+__isl_give isl_aff *isl_aff_normalize(__isl_take isl_aff *aff);
+
+__isl_give isl_aff *isl_aff_expand_divs( __isl_take isl_aff *aff,
+	__isl_take isl_mat *div, int *exp);
+
+__isl_give isl_pw_aff *isl_pw_aff_alloc_size(__isl_take isl_space *space,
+	int n);
+__isl_give isl_pw_aff *isl_pw_aff_reset_space(__isl_take isl_pw_aff *pwaff,
+	__isl_take isl_space *dim);
+__isl_give isl_pw_aff *isl_pw_aff_reset_domain_space(
+	__isl_take isl_pw_aff *pwaff, __isl_take isl_space *space);
+__isl_give isl_pw_aff *isl_pw_aff_add_disjoint(
+	__isl_take isl_pw_aff *pwaff1, __isl_take isl_pw_aff *pwaff2);
+
+__isl_give isl_pw_aff *isl_pw_aff_union_opt(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2, int max);
+
+__isl_give isl_pw_aff *isl_pw_aff_set_rational(__isl_take isl_pw_aff *pwaff);
+__isl_give isl_pw_aff_list *isl_pw_aff_list_set_rational(
+	__isl_take isl_pw_aff_list *list);
+
+__isl_give isl_pw_aff *isl_pw_aff_scale(__isl_take isl_pw_aff *pwaff,
+	isl_int f);
+__isl_give isl_pw_aff *isl_pw_aff_scale_down(__isl_take isl_pw_aff *pwaff,
+	isl_int f);
+
+int isl_aff_matching_params(__isl_keep isl_aff *aff,
+	__isl_keep isl_space *space);
+int isl_aff_check_match_domain_space(__isl_keep isl_aff *aff,
+	__isl_keep isl_space *space);
+
+#undef BASE
+#define BASE aff
+
+#include <isl_multi_templ.h>
+
+__isl_give isl_multi_aff *isl_multi_aff_dup(__isl_keep isl_multi_aff *multi);
+
+__isl_give isl_multi_aff *isl_multi_aff_align_divs(
+	__isl_take isl_multi_aff *maff);
+
+__isl_give isl_multi_aff *isl_multi_aff_from_basic_set_equalities(
+	__isl_take isl_basic_set *bset);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_reset_domain_space(
+	__isl_take isl_pw_multi_aff *pwmaff, __isl_take isl_space *space);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_reset_space(
+	__isl_take isl_pw_multi_aff *pwmaff, __isl_take isl_space *space);
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_add_disjoint(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_project_out(
+	__isl_take isl_pw_multi_aff *pma,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+void isl_seq_substitute(isl_int *p, int pos, isl_int *subs,
+	int p_len, int subs_len, isl_int v);
+void isl_seq_preimage(isl_int *dst, isl_int *src,
+	__isl_keep isl_multi_aff *ma, int n_before, int n_after,
+	int n_div_ma, int n_div_bmap,
+	isl_int f, isl_int c1, isl_int c2, isl_int g, int has_denom);
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_substitute(
+	__isl_take isl_pw_multi_aff *pma, enum isl_dim_type type, unsigned pos,
+	__isl_keep isl_pw_aff *subs);
+
+int isl_pw_aff_matching_params(__isl_keep isl_pw_aff *pa,
+	__isl_keep isl_space *space);
+int isl_pw_aff_check_match_domain_space(__isl_keep isl_pw_aff *pa,
+	__isl_keep isl_space *space);
+
+#undef BASE
+#define BASE pw_aff
+
+#include <isl_multi_templ.h>
+
+#endif
diff --git a/polly/lib/External/isl/isl_affine_hull.c b/polly/lib/External/isl/isl_affine_hull.c
new file mode 100644
index 00000000000..81352817071
--- /dev/null
+++ b/polly/lib/External/isl/isl_affine_hull.c
@@ -0,0 +1,1405 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2010      INRIA Saclay
+ * Copyright 2012      Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France
+ * and Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl_seq.h>
+#include <isl/set.h>
+#include <isl/lp.h>
+#include <isl/map.h>
+#include "isl_equalities.h"
+#include "isl_sample.h"
+#include "isl_tab.h"
+#include <isl_mat_private.h>
+#include <isl_vec_private.h>
+
+struct isl_basic_map *isl_basic_map_implicit_equalities(
+						struct isl_basic_map *bmap)
+{
+	struct isl_tab *tab;
+
+	if (!bmap)
+		return bmap;
+
+	bmap = isl_basic_map_gauss(bmap, NULL);
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
+		return bmap;
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_IMPLICIT))
+		return bmap;
+	if (bmap->n_ineq <= 1)
+		return bmap;
+
+	tab = isl_tab_from_basic_map(bmap, 0);
+	if (isl_tab_detect_implicit_equalities(tab) < 0)
+		goto error;
+	bmap = isl_basic_map_update_from_tab(bmap, tab);
+	isl_tab_free(tab);
+	bmap = isl_basic_map_gauss(bmap, NULL);
+	ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
+	return bmap;
+error:
+	isl_tab_free(tab);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_implicit_equalities(
+						struct isl_basic_set *bset)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_implicit_equalities((struct isl_basic_map*)bset);
+}
+
+struct isl_map *isl_map_implicit_equalities(struct isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return map;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_implicit_equalities(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Make eq[row][col] of both bmaps equal so we can add the row
+ * add the column to the common matrix.
+ * Note that because of the echelon form, the columns of row row
+ * after column col are zero.
+ */
+static void set_common_multiple(
+	struct isl_basic_set *bset1, struct isl_basic_set *bset2,
+	unsigned row, unsigned col)
+{
+	isl_int m, c;
+
+	if (isl_int_eq(bset1->eq[row][col], bset2->eq[row][col]))
+		return;
+
+	isl_int_init(c);
+	isl_int_init(m);
+	isl_int_lcm(m, bset1->eq[row][col], bset2->eq[row][col]);
+	isl_int_divexact(c, m, bset1->eq[row][col]);
+	isl_seq_scale(bset1->eq[row], bset1->eq[row], c, col+1);
+	isl_int_divexact(c, m, bset2->eq[row][col]);
+	isl_seq_scale(bset2->eq[row], bset2->eq[row], c, col+1);
+	isl_int_clear(c);
+	isl_int_clear(m);
+}
+
+/* Delete a given equality, moving all the following equalities one up.
+ */
+static void delete_row(struct isl_basic_set *bset, unsigned row)
+{
+	isl_int *t;
+	int r;
+
+	t = bset->eq[row];
+	bset->n_eq--;
+	for (r = row; r < bset->n_eq; ++r)
+		bset->eq[r] = bset->eq[r+1];
+	bset->eq[bset->n_eq] = t;
+}
+
+/* Make first row entries in column col of bset1 identical to
+ * those of bset2, using the fact that entry bset1->eq[row][col]=a
+ * is non-zero.  Initially, these elements of bset1 are all zero.
+ * For each row i < row, we set
+ *		A[i] = a * A[i] + B[i][col] * A[row]
+ *		B[i] = a * B[i]
+ * so that
+ *		A[i][col] = B[i][col] = a * old(B[i][col])
+ */
+static void construct_column(
+	struct isl_basic_set *bset1, struct isl_basic_set *bset2,
+	unsigned row, unsigned col)
+{
+	int r;
+	isl_int a;
+	isl_int b;
+	unsigned total;
+
+	isl_int_init(a);
+	isl_int_init(b);
+	total = 1 + isl_basic_set_n_dim(bset1);
+	for (r = 0; r < row; ++r) {
+		if (isl_int_is_zero(bset2->eq[r][col]))
+			continue;
+		isl_int_gcd(b, bset2->eq[r][col], bset1->eq[row][col]);
+		isl_int_divexact(a, bset1->eq[row][col], b);
+		isl_int_divexact(b, bset2->eq[r][col], b);
+		isl_seq_combine(bset1->eq[r], a, bset1->eq[r],
+					      b, bset1->eq[row], total);
+		isl_seq_scale(bset2->eq[r], bset2->eq[r], a, total);
+	}
+	isl_int_clear(a);
+	isl_int_clear(b);
+	delete_row(bset1, row);
+}
+
+/* Make first row entries in column col of bset1 identical to
+ * those of bset2, using only these entries of the two matrices.
+ * Let t be the last row with different entries.
+ * For each row i < t, we set
+ *	A[i] = (A[t][col]-B[t][col]) * A[i] + (B[i][col]-A[i][col) * A[t]
+ *	B[i] = (A[t][col]-B[t][col]) * B[i] + (B[i][col]-A[i][col) * B[t]
+ * so that
+ *	A[i][col] = B[i][col] = old(A[t][col]*B[i][col]-A[i][col]*B[t][col])
+ */
+static int transform_column(
+	struct isl_basic_set *bset1, struct isl_basic_set *bset2,
+	unsigned row, unsigned col)
+{
+	int i, t;
+	isl_int a, b, g;
+	unsigned total;
+
+	for (t = row-1; t >= 0; --t)
+		if (isl_int_ne(bset1->eq[t][col], bset2->eq[t][col]))
+			break;
+	if (t < 0)
+		return 0;
+
+	total = 1 + isl_basic_set_n_dim(bset1);
+	isl_int_init(a);
+	isl_int_init(b);
+	isl_int_init(g);
+	isl_int_sub(b, bset1->eq[t][col], bset2->eq[t][col]);
+	for (i = 0; i < t; ++i) {
+		isl_int_sub(a, bset2->eq[i][col], bset1->eq[i][col]);
+		isl_int_gcd(g, a, b);
+		isl_int_divexact(a, a, g);
+		isl_int_divexact(g, b, g);
+		isl_seq_combine(bset1->eq[i], g, bset1->eq[i], a, bset1->eq[t],
+				total);
+		isl_seq_combine(bset2->eq[i], g, bset2->eq[i], a, bset2->eq[t],
+				total);
+	}
+	isl_int_clear(a);
+	isl_int_clear(b);
+	isl_int_clear(g);
+	delete_row(bset1, t);
+	delete_row(bset2, t);
+	return 1;
+}
+
+/* The implementation is based on Section 5.2 of Michael Karr,
+ * "Affine Relationships Among Variables of a Program",
+ * except that the echelon form we use starts from the last column
+ * and that we are dealing with integer coefficients.
+ */
+static struct isl_basic_set *affine_hull(
+	struct isl_basic_set *bset1, struct isl_basic_set *bset2)
+{
+	unsigned total;
+	int col;
+	int row;
+
+	if (!bset1 || !bset2)
+		goto error;
+
+	total = 1 + isl_basic_set_n_dim(bset1);
+
+	row = 0;
+	for (col = total-1; col >= 0; --col) {
+		int is_zero1 = row >= bset1->n_eq ||
+			isl_int_is_zero(bset1->eq[row][col]);
+		int is_zero2 = row >= bset2->n_eq ||
+			isl_int_is_zero(bset2->eq[row][col]);
+		if (!is_zero1 && !is_zero2) {
+			set_common_multiple(bset1, bset2, row, col);
+			++row;
+		} else if (!is_zero1 && is_zero2) {
+			construct_column(bset1, bset2, row, col);
+		} else if (is_zero1 && !is_zero2) {
+			construct_column(bset2, bset1, row, col);
+		} else {
+			if (transform_column(bset1, bset2, row, col))
+				--row;
+		}
+	}
+	isl_assert(bset1->ctx, row == bset1->n_eq, goto error);
+	isl_basic_set_free(bset2);
+	bset1 = isl_basic_set_normalize_constraints(bset1);
+	return bset1;
+error:
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return NULL;
+}
+
+/* Find an integer point in the set represented by "tab"
+ * that lies outside of the equality "eq" e(x) = 0.
+ * If "up" is true, look for a point satisfying e(x) - 1 >= 0.
+ * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1).
+ * The point, if found, is returned.
+ * If no point can be found, a zero-length vector is returned.
+ *
+ * Before solving an ILP problem, we first check if simply
+ * adding the normal of the constraint to one of the known
+ * integer points in the basic set represented by "tab"
+ * yields another point inside the basic set.
+ *
+ * The caller of this function ensures that the tableau is bounded or
+ * that tab->basis and tab->n_unbounded have been set appropriately.
+ */
+static struct isl_vec *outside_point(struct isl_tab *tab, isl_int *eq, int up)
+{
+	struct isl_ctx *ctx;
+	struct isl_vec *sample = NULL;
+	struct isl_tab_undo *snap;
+	unsigned dim;
+
+	if (!tab)
+		return NULL;
+	ctx = tab->mat->ctx;
+
+	dim = tab->n_var;
+	sample = isl_vec_alloc(ctx, 1 + dim);
+	if (!sample)
+		return NULL;
+	isl_int_set_si(sample->el[0], 1);
+	isl_seq_combine(sample->el + 1,
+		ctx->one, tab->bmap->sample->el + 1,
+		up ? ctx->one : ctx->negone, eq + 1, dim);
+	if (isl_basic_map_contains(tab->bmap, sample))
+		return sample;
+	isl_vec_free(sample);
+	sample = NULL;
+
+	snap = isl_tab_snap(tab);
+
+	if (!up)
+		isl_seq_neg(eq, eq, 1 + dim);
+	isl_int_sub_ui(eq[0], eq[0], 1);
+
+	if (isl_tab_extend_cons(tab, 1) < 0)
+		goto error;
+	if (isl_tab_add_ineq(tab, eq) < 0)
+		goto error;
+
+	sample = isl_tab_sample(tab);
+
+	isl_int_add_ui(eq[0], eq[0], 1);
+	if (!up)
+		isl_seq_neg(eq, eq, 1 + dim);
+
+	if (sample && isl_tab_rollback(tab, snap) < 0)
+		goto error;
+
+	return sample;
+error:
+	isl_vec_free(sample);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_recession_cone(struct isl_basic_set *bset)
+{
+	int i;
+
+	bset = isl_basic_set_cow(bset);
+	if (!bset)
+		return NULL;
+	isl_assert(bset->ctx, bset->n_div == 0, goto error);
+
+	for (i = 0; i < bset->n_eq; ++i)
+		isl_int_set_si(bset->eq[i][0], 0);
+
+	for (i = 0; i < bset->n_ineq; ++i)
+		isl_int_set_si(bset->ineq[i][0], 0);
+
+	ISL_F_CLR(bset, ISL_BASIC_SET_NO_IMPLICIT);
+	return isl_basic_set_implicit_equalities(bset);
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_recession_cone(__isl_take isl_set *set)
+{
+	int i;
+
+	if (!set)
+		return NULL;
+	if (set->n == 0)
+		return set;
+
+	set = isl_set_remove_divs(set);
+	set = isl_set_cow(set);
+	if (!set)
+		return NULL;
+
+	for (i = 0; i < set->n; ++i) {
+		set->p[i] = isl_basic_set_recession_cone(set->p[i]);
+		if (!set->p[i])
+			goto error;
+	}
+
+	return set;
+error:
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Move "sample" to a point that is one up (or down) from the original
+ * point in dimension "pos".
+ */
+static void adjacent_point(__isl_keep isl_vec *sample, int pos, int up)
+{
+	if (up)
+		isl_int_add_ui(sample->el[1 + pos], sample->el[1 + pos], 1);
+	else
+		isl_int_sub_ui(sample->el[1 + pos], sample->el[1 + pos], 1);
+}
+
+/* Check if any points that are adjacent to "sample" also belong to "bset".
+ * If so, add them to "hull" and return the updated hull.
+ *
+ * Before checking whether and adjacent point belongs to "bset", we first
+ * check whether it already belongs to "hull" as this test is typically
+ * much cheaper.
+ */
+static __isl_give isl_basic_set *add_adjacent_points(
+	__isl_take isl_basic_set *hull, __isl_take isl_vec *sample,
+	__isl_keep isl_basic_set *bset)
+{
+	int i, up;
+	int dim;
+
+	if (!sample)
+		goto error;
+
+	dim = isl_basic_set_dim(hull, isl_dim_set);
+
+	for (i = 0; i < dim; ++i) {
+		for (up = 0; up <= 1; ++up) {
+			int contains;
+			isl_basic_set *point;
+
+			adjacent_point(sample, i, up);
+			contains = isl_basic_set_contains(hull, sample);
+			if (contains < 0)
+				goto error;
+			if (contains) {
+				adjacent_point(sample, i, !up);
+				continue;
+			}
+			contains = isl_basic_set_contains(bset, sample);
+			if (contains < 0)
+				goto error;
+			if (contains) {
+				point = isl_basic_set_from_vec(
+							isl_vec_copy(sample));
+				hull = affine_hull(hull, point);
+			}
+			adjacent_point(sample, i, !up);
+			if (contains)
+				break;
+		}
+	}
+
+	isl_vec_free(sample);
+
+	return hull;
+error:
+	isl_vec_free(sample);
+	isl_basic_set_free(hull);
+	return NULL;
+}
+
+/* Extend an initial (under-)approximation of the affine hull of basic
+ * set represented by the tableau "tab"
+ * by looking for points that do not satisfy one of the equalities
+ * in the current approximation and adding them to that approximation
+ * until no such points can be found any more.
+ *
+ * The caller of this function ensures that "tab" is bounded or
+ * that tab->basis and tab->n_unbounded have been set appropriately.
+ *
+ * "bset" may be either NULL or the basic set represented by "tab".
+ * If "bset" is not NULL, we check for any point we find if any
+ * of its adjacent points also belong to "bset".
+ */
+static __isl_give isl_basic_set *extend_affine_hull(struct isl_tab *tab,
+	__isl_take isl_basic_set *hull, __isl_keep isl_basic_set *bset)
+{
+	int i, j;
+	unsigned dim;
+
+	if (!tab || !hull)
+		goto error;
+
+	dim = tab->n_var;
+
+	if (isl_tab_extend_cons(tab, 2 * dim + 1) < 0)
+		goto error;
+
+	for (i = 0; i < dim; ++i) {
+		struct isl_vec *sample;
+		struct isl_basic_set *point;
+		for (j = 0; j < hull->n_eq; ++j) {
+			sample = outside_point(tab, hull->eq[j], 1);
+			if (!sample)
+				goto error;
+			if (sample->size > 0)
+				break;
+			isl_vec_free(sample);
+			sample = outside_point(tab, hull->eq[j], 0);
+			if (!sample)
+				goto error;
+			if (sample->size > 0)
+				break;
+			isl_vec_free(sample);
+
+			if (isl_tab_add_eq(tab, hull->eq[j]) < 0)
+				goto error;
+		}
+		if (j == hull->n_eq)
+			break;
+		if (tab->samples &&
+		    isl_tab_add_sample(tab, isl_vec_copy(sample)) < 0)
+			hull = isl_basic_set_free(hull);
+		if (bset)
+			hull = add_adjacent_points(hull, isl_vec_copy(sample),
+						    bset);
+		point = isl_basic_set_from_vec(sample);
+		hull = affine_hull(hull, point);
+		if (!hull)
+			return NULL;
+	}
+
+	return hull;
+error:
+	isl_basic_set_free(hull);
+	return NULL;
+}
+
+/* Drop all constraints in bmap that involve any of the dimensions
+ * first to first+n-1.
+ */
+static __isl_give isl_basic_map *isl_basic_map_drop_constraints_involving(
+	__isl_take isl_basic_map *bmap, unsigned first, unsigned n)
+{
+	int i;
+
+	if (n == 0)
+		return bmap;
+
+	bmap = isl_basic_map_cow(bmap);
+
+	if (!bmap)
+		return NULL;
+
+	for (i = bmap->n_eq - 1; i >= 0; --i) {
+		if (isl_seq_first_non_zero(bmap->eq[i] + 1 + first, n) == -1)
+			continue;
+		isl_basic_map_drop_equality(bmap, i);
+	}
+
+	for (i = bmap->n_ineq - 1; i >= 0; --i) {
+		if (isl_seq_first_non_zero(bmap->ineq[i] + 1 + first, n) == -1)
+			continue;
+		isl_basic_map_drop_inequality(bmap, i);
+	}
+
+	bmap = isl_basic_map_add_known_div_constraints(bmap);
+	return bmap;
+}
+
+/* Drop all constraints in bset that involve any of the dimensions
+ * first to first+n-1.
+ */
+__isl_give isl_basic_set *isl_basic_set_drop_constraints_involving(
+	__isl_take isl_basic_set *bset, unsigned first, unsigned n)
+{
+	return isl_basic_map_drop_constraints_involving(bset, first, n);
+}
+
+/* Drop all constraints in bmap that do not involve any of the dimensions
+ * first to first + n - 1 of the given type.
+ */
+__isl_give isl_basic_map *isl_basic_map_drop_constraints_not_involving_dims(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+	unsigned dim;
+
+	if (n == 0) {
+		isl_space *space = isl_basic_map_get_space(bmap);
+		isl_basic_map_free(bmap);
+		return isl_basic_map_universe(space);
+	}
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+
+	dim = isl_basic_map_dim(bmap, type);
+	if (first + n > dim || first + n < first)
+		isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
+			"index out of bounds", return isl_basic_map_free(bmap));
+
+	first += isl_basic_map_offset(bmap, type) - 1;
+
+	for (i = bmap->n_eq - 1; i >= 0; --i) {
+		if (isl_seq_first_non_zero(bmap->eq[i] + 1 + first, n) != -1)
+			continue;
+		isl_basic_map_drop_equality(bmap, i);
+	}
+
+	for (i = bmap->n_ineq - 1; i >= 0; --i) {
+		if (isl_seq_first_non_zero(bmap->ineq[i] + 1 + first, n) != -1)
+			continue;
+		isl_basic_map_drop_inequality(bmap, i);
+	}
+
+	bmap = isl_basic_map_add_known_div_constraints(bmap);
+	return bmap;
+}
+
+/* Drop all constraints in bset that do not involve any of the dimensions
+ * first to first + n - 1 of the given type.
+ */
+__isl_give isl_basic_set *isl_basic_set_drop_constraints_not_involving_dims(
+	__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return isl_basic_map_drop_constraints_not_involving_dims(bset,
+							    type, first, n);
+}
+
+/* Drop all constraints in bmap that involve any of the dimensions
+ * first to first + n - 1 of the given type.
+ */
+__isl_give isl_basic_map *isl_basic_map_drop_constraints_involving_dims(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	unsigned dim;
+
+	if (!bmap)
+		return NULL;
+	if (n == 0)
+		return bmap;
+
+	dim = isl_basic_map_dim(bmap, type);
+	if (first + n > dim || first + n < first)
+		isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
+			"index out of bounds", return isl_basic_map_free(bmap));
+
+	bmap = isl_basic_map_remove_divs_involving_dims(bmap, type, first, n);
+	first += isl_basic_map_offset(bmap, type) - 1;
+	return isl_basic_map_drop_constraints_involving(bmap, first, n);
+}
+
+/* Drop all constraints in bset that involve any of the dimensions
+ * first to first + n - 1 of the given type.
+ */
+__isl_give isl_basic_set *isl_basic_set_drop_constraints_involving_dims(
+	__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return isl_basic_map_drop_constraints_involving_dims(bset,
+							    type, first, n);
+}
+
+/* Drop all constraints in map that involve any of the dimensions
+ * first to first + n - 1 of the given type.
+ */
+__isl_give isl_map *isl_map_drop_constraints_involving_dims(
+	__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+	unsigned dim;
+
+	if (!map)
+		return NULL;
+	if (n == 0)
+		return map;
+
+	dim = isl_map_dim(map, type);
+	if (first + n > dim || first + n < first)
+		isl_die(isl_map_get_ctx(map), isl_error_invalid,
+			"index out of bounds", return isl_map_free(map));
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_drop_constraints_involving_dims(
+						    map->p[i], type, first, n);
+		if (!map->p[i])
+			return isl_map_free(map);
+	}
+
+	return map;
+}
+
+/* Drop all constraints in set that involve any of the dimensions
+ * first to first + n - 1 of the given type.
+ */
+__isl_give isl_set *isl_set_drop_constraints_involving_dims(
+	__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return isl_map_drop_constraints_involving_dims(set, type, first, n);
+}
+
+/* Construct an initial underapproximatino of the hull of "bset"
+ * from "sample" and any of its adjacent points that also belong to "bset".
+ */
+static __isl_give isl_basic_set *initialize_hull(__isl_keep isl_basic_set *bset,
+	__isl_take isl_vec *sample)
+{
+	isl_basic_set *hull;
+
+	hull = isl_basic_set_from_vec(isl_vec_copy(sample));
+	hull = add_adjacent_points(hull, sample, bset);
+
+	return hull;
+}
+
+/* Look for all equalities satisfied by the integer points in bset,
+ * which is assumed to be bounded.
+ *
+ * The equalities are obtained by successively looking for
+ * a point that is affinely independent of the points found so far.
+ * In particular, for each equality satisfied by the points so far,
+ * we check if there is any point on a hyperplane parallel to the
+ * corresponding hyperplane shifted by at least one (in either direction).
+ */
+static struct isl_basic_set *uset_affine_hull_bounded(struct isl_basic_set *bset)
+{
+	struct isl_vec *sample = NULL;
+	struct isl_basic_set *hull;
+	struct isl_tab *tab = NULL;
+	unsigned dim;
+
+	if (isl_basic_set_plain_is_empty(bset))
+		return bset;
+
+	dim = isl_basic_set_n_dim(bset);
+
+	if (bset->sample && bset->sample->size == 1 + dim) {
+		int contains = isl_basic_set_contains(bset, bset->sample);
+		if (contains < 0)
+			goto error;
+		if (contains) {
+			if (dim == 0)
+				return bset;
+			sample = isl_vec_copy(bset->sample);
+		} else {
+			isl_vec_free(bset->sample);
+			bset->sample = NULL;
+		}
+	}
+
+	tab = isl_tab_from_basic_set(bset, 1);
+	if (!tab)
+		goto error;
+	if (tab->empty) {
+		isl_tab_free(tab);
+		isl_vec_free(sample);
+		return isl_basic_set_set_to_empty(bset);
+	}
+
+	if (!sample) {
+		struct isl_tab_undo *snap;
+		snap = isl_tab_snap(tab);
+		sample = isl_tab_sample(tab);
+		if (isl_tab_rollback(tab, snap) < 0)
+			goto error;
+		isl_vec_free(tab->bmap->sample);
+		tab->bmap->sample = isl_vec_copy(sample);
+	}
+
+	if (!sample)
+		goto error;
+	if (sample->size == 0) {
+		isl_tab_free(tab);
+		isl_vec_free(sample);
+		return isl_basic_set_set_to_empty(bset);
+	}
+
+	hull = initialize_hull(bset, sample);
+
+	hull = extend_affine_hull(tab, hull, bset);
+	isl_basic_set_free(bset);
+	isl_tab_free(tab);
+
+	return hull;
+error:
+	isl_vec_free(sample);
+	isl_tab_free(tab);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Given an unbounded tableau and an integer point satisfying the tableau,
+ * construct an initial affine hull containing the recession cone
+ * shifted to the given point.
+ *
+ * The unbounded directions are taken from the last rows of the basis,
+ * which is assumed to have been initialized appropriately.
+ */
+static __isl_give isl_basic_set *initial_hull(struct isl_tab *tab,
+	__isl_take isl_vec *vec)
+{
+	int i;
+	int k;
+	struct isl_basic_set *bset = NULL;
+	struct isl_ctx *ctx;
+	unsigned dim;
+
+	if (!vec || !tab)
+		return NULL;
+	ctx = vec->ctx;
+	isl_assert(ctx, vec->size != 0, goto error);
+
+	bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
+	if (!bset)
+		goto error;
+	dim = isl_basic_set_n_dim(bset) - tab->n_unbounded;
+	for (i = 0; i < dim; ++i) {
+		k = isl_basic_set_alloc_equality(bset);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(bset->eq[k] + 1, tab->basis->row[1 + i] + 1,
+			    vec->size - 1);
+		isl_seq_inner_product(bset->eq[k] + 1, vec->el +1,
+				      vec->size - 1, &bset->eq[k][0]);
+		isl_int_neg(bset->eq[k][0], bset->eq[k][0]);
+	}
+	bset->sample = vec;
+	bset = isl_basic_set_gauss(bset, NULL);
+
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	isl_vec_free(vec);
+	return NULL;
+}
+
+/* Given a tableau of a set and a tableau of the corresponding
+ * recession cone, detect and add all equalities to the tableau.
+ * If the tableau is bounded, then we can simply keep the
+ * tableau in its state after the return from extend_affine_hull.
+ * However, if the tableau is unbounded, then
+ * isl_tab_set_initial_basis_with_cone will add some additional
+ * constraints to the tableau that have to be removed again.
+ * In this case, we therefore rollback to the state before
+ * any constraints were added and then add the equalities back in.
+ */
+struct isl_tab *isl_tab_detect_equalities(struct isl_tab *tab,
+	struct isl_tab *tab_cone)
+{
+	int j;
+	struct isl_vec *sample;
+	struct isl_basic_set *hull = NULL;
+	struct isl_tab_undo *snap;
+
+	if (!tab || !tab_cone)
+		goto error;
+
+	snap = isl_tab_snap(tab);
+
+	isl_mat_free(tab->basis);
+	tab->basis = NULL;
+
+	isl_assert(tab->mat->ctx, tab->bmap, goto error);
+	isl_assert(tab->mat->ctx, tab->samples, goto error);
+	isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, goto error);
+	isl_assert(tab->mat->ctx, tab->n_sample > tab->n_outside, goto error);
+
+	if (isl_tab_set_initial_basis_with_cone(tab, tab_cone) < 0)
+		goto error;
+
+	sample = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_var);
+	if (!sample)
+		goto error;
+
+	isl_seq_cpy(sample->el, tab->samples->row[tab->n_outside], sample->size);
+
+	isl_vec_free(tab->bmap->sample);
+	tab->bmap->sample = isl_vec_copy(sample);
+
+	if (tab->n_unbounded == 0)
+		hull = isl_basic_set_from_vec(isl_vec_copy(sample));
+	else
+		hull = initial_hull(tab, isl_vec_copy(sample));
+
+	for (j = tab->n_outside + 1; j < tab->n_sample; ++j) {
+		isl_seq_cpy(sample->el, tab->samples->row[j], sample->size);
+		hull = affine_hull(hull,
+				isl_basic_set_from_vec(isl_vec_copy(sample)));
+	}
+
+	isl_vec_free(sample);
+
+	hull = extend_affine_hull(tab, hull, NULL);
+	if (!hull)
+		goto error;
+
+	if (tab->n_unbounded == 0) {
+		isl_basic_set_free(hull);
+		return tab;
+	}
+
+	if (isl_tab_rollback(tab, snap) < 0)
+		goto error;
+
+	if (hull->n_eq > tab->n_zero) {
+		for (j = 0; j < hull->n_eq; ++j) {
+			isl_seq_normalize(tab->mat->ctx, hull->eq[j], 1 + tab->n_var);
+			if (isl_tab_add_eq(tab, hull->eq[j]) < 0)
+				goto error;
+		}
+	}
+
+	isl_basic_set_free(hull);
+
+	return tab;
+error:
+	isl_basic_set_free(hull);
+	isl_tab_free(tab);
+	return NULL;
+}
+
+/* Compute the affine hull of "bset", where "cone" is the recession cone
+ * of "bset".
+ *
+ * We first compute a unimodular transformation that puts the unbounded
+ * directions in the last dimensions.  In particular, we take a transformation
+ * that maps all equalities to equalities (in HNF) on the first dimensions.
+ * Let x be the original dimensions and y the transformed, with y_1 bounded
+ * and y_2 unbounded.
+ *
+ *	       [ y_1 ]			[ y_1 ]   [ Q_1 ]
+ *	x = U  [ y_2 ]			[ y_2 ] = [ Q_2 ] x
+ *
+ * Let's call the input basic set S.  We compute S' = preimage(S, U)
+ * and drop the final dimensions including any constraints involving them.
+ * This results in set S''.
+ * Then we compute the affine hull A'' of S''.
+ * Let F y_1 >= g be the constraint system of A''.  In the transformed
+ * space the y_2 are unbounded, so we can add them back without any constraints,
+ * resulting in
+ *
+ *		        [ y_1 ]
+ *		[ F 0 ] [ y_2 ] >= g
+ * or
+ *		        [ Q_1 ]
+ *		[ F 0 ] [ Q_2 ] x >= g
+ * or
+ *		F Q_1 x >= g
+ *
+ * The affine hull in the original space is then obtained as
+ * A = preimage(A'', Q_1).
+ */
+static struct isl_basic_set *affine_hull_with_cone(struct isl_basic_set *bset,
+	struct isl_basic_set *cone)
+{
+	unsigned total;
+	unsigned cone_dim;
+	struct isl_basic_set *hull;
+	struct isl_mat *M, *U, *Q;
+
+	if (!bset || !cone)
+		goto error;
+
+	total = isl_basic_set_total_dim(cone);
+	cone_dim = total - cone->n_eq;
+
+	M = isl_mat_sub_alloc6(bset->ctx, cone->eq, 0, cone->n_eq, 1, total);
+	M = isl_mat_left_hermite(M, 0, &U, &Q);
+	if (!M)
+		goto error;
+	isl_mat_free(M);
+
+	U = isl_mat_lin_to_aff(U);
+	bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
+
+	bset = isl_basic_set_drop_constraints_involving(bset, total - cone_dim,
+							cone_dim);
+	bset = isl_basic_set_drop_dims(bset, total - cone_dim, cone_dim);
+
+	Q = isl_mat_lin_to_aff(Q);
+	Q = isl_mat_drop_rows(Q, 1 + total - cone_dim, cone_dim);
+
+	if (bset && bset->sample && bset->sample->size == 1 + total)
+		bset->sample = isl_mat_vec_product(isl_mat_copy(Q), bset->sample);
+
+	hull = uset_affine_hull_bounded(bset);
+
+	if (!hull) {
+		isl_mat_free(Q);
+		isl_mat_free(U);
+	} else {
+		struct isl_vec *sample = isl_vec_copy(hull->sample);
+		U = isl_mat_drop_cols(U, 1 + total - cone_dim, cone_dim);
+		if (sample && sample->size > 0)
+			sample = isl_mat_vec_product(U, sample);
+		else
+			isl_mat_free(U);
+		hull = isl_basic_set_preimage(hull, Q);
+		if (hull) {
+			isl_vec_free(hull->sample);
+			hull->sample = sample;
+		} else
+			isl_vec_free(sample);
+	}
+
+	isl_basic_set_free(cone);
+
+	return hull;
+error:
+	isl_basic_set_free(bset);
+	isl_basic_set_free(cone);
+	return NULL;
+}
+
+/* Look for all equalities satisfied by the integer points in bset,
+ * which is assumed not to have any explicit equalities.
+ *
+ * The equalities are obtained by successively looking for
+ * a point that is affinely independent of the points found so far.
+ * In particular, for each equality satisfied by the points so far,
+ * we check if there is any point on a hyperplane parallel to the
+ * corresponding hyperplane shifted by at least one (in either direction).
+ *
+ * Before looking for any outside points, we first compute the recession
+ * cone.  The directions of this recession cone will always be part
+ * of the affine hull, so there is no need for looking for any points
+ * in these directions.
+ * In particular, if the recession cone is full-dimensional, then
+ * the affine hull is simply the whole universe.
+ */
+static struct isl_basic_set *uset_affine_hull(struct isl_basic_set *bset)
+{
+	struct isl_basic_set *cone;
+
+	if (isl_basic_set_plain_is_empty(bset))
+		return bset;
+
+	cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));
+	if (!cone)
+		goto error;
+	if (cone->n_eq == 0) {
+		struct isl_basic_set *hull;
+		isl_basic_set_free(cone);
+		hull = isl_basic_set_universe_like(bset);
+		isl_basic_set_free(bset);
+		return hull;
+	}
+
+	if (cone->n_eq < isl_basic_set_total_dim(cone))
+		return affine_hull_with_cone(bset, cone);
+
+	isl_basic_set_free(cone);
+	return uset_affine_hull_bounded(bset);
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Look for all equalities satisfied by the integer points in bmap
+ * that are independent of the equalities already explicitly available
+ * in bmap.
+ *
+ * We first remove all equalities already explicitly available,
+ * then look for additional equalities in the reduced space
+ * and then transform the result to the original space.
+ * The original equalities are _not_ added to this set.  This is
+ * the responsibility of the calling function.
+ * The resulting basic set has all meaning about the dimensions removed.
+ * In particular, dimensions that correspond to existential variables
+ * in bmap and that are found to be fixed are not removed.
+ */
+static struct isl_basic_set *equalities_in_underlying_set(
+						struct isl_basic_map *bmap)
+{
+	struct isl_mat *T1 = NULL;
+	struct isl_mat *T2 = NULL;
+	struct isl_basic_set *bset = NULL;
+	struct isl_basic_set *hull = NULL;
+
+	bset = isl_basic_map_underlying_set(bmap);
+	if (!bset)
+		return NULL;
+	if (bset->n_eq)
+		bset = isl_basic_set_remove_equalities(bset, &T1, &T2);
+	if (!bset)
+		goto error;
+
+	hull = uset_affine_hull(bset);
+	if (!T2)
+		return hull;
+
+	if (!hull) {
+		isl_mat_free(T1);
+		isl_mat_free(T2);
+	} else {
+		struct isl_vec *sample = isl_vec_copy(hull->sample);
+		if (sample && sample->size > 0)
+			sample = isl_mat_vec_product(T1, sample);
+		else
+			isl_mat_free(T1);
+		hull = isl_basic_set_preimage(hull, T2);
+		if (hull) {
+			isl_vec_free(hull->sample);
+			hull->sample = sample;
+		} else
+			isl_vec_free(sample);
+	}
+
+	return hull;
+error:
+	isl_mat_free(T1);
+	isl_mat_free(T2);
+	isl_basic_set_free(bset);
+	isl_basic_set_free(hull);
+	return NULL;
+}
+
+/* Detect and make explicit all equalities satisfied by the (integer)
+ * points in bmap.
+ */
+struct isl_basic_map *isl_basic_map_detect_equalities(
+						struct isl_basic_map *bmap)
+{
+	int i, j;
+	struct isl_basic_set *hull = NULL;
+
+	if (!bmap)
+		return NULL;
+	if (bmap->n_ineq == 0)
+		return bmap;
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
+		return bmap;
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_ALL_EQUALITIES))
+		return bmap;
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
+		return isl_basic_map_implicit_equalities(bmap);
+
+	hull = equalities_in_underlying_set(isl_basic_map_copy(bmap));
+	if (!hull)
+		goto error;
+	if (ISL_F_ISSET(hull, ISL_BASIC_SET_EMPTY)) {
+		isl_basic_set_free(hull);
+		return isl_basic_map_set_to_empty(bmap);
+	}
+	bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim), 0,
+					hull->n_eq, 0);
+	for (i = 0; i < hull->n_eq; ++i) {
+		j = isl_basic_map_alloc_equality(bmap);
+		if (j < 0)
+			goto error;
+		isl_seq_cpy(bmap->eq[j], hull->eq[i],
+				1 + isl_basic_set_total_dim(hull));
+	}
+	isl_vec_free(bmap->sample);
+	bmap->sample = isl_vec_copy(hull->sample);
+	isl_basic_set_free(hull);
+	ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT | ISL_BASIC_MAP_ALL_EQUALITIES);
+	bmap = isl_basic_map_simplify(bmap);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_set_free(hull);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_detect_equalities(
+						__isl_take isl_basic_set *bset)
+{
+	return (isl_basic_set *)
+		isl_basic_map_detect_equalities((isl_basic_map *)bset);
+}
+
+__isl_give isl_map *isl_map_detect_equalities(__isl_take isl_map *map)
+{
+	return isl_map_inline_foreach_basic_map(map,
+					    &isl_basic_map_detect_equalities);
+}
+
+__isl_give isl_set *isl_set_detect_equalities(__isl_take isl_set *set)
+{
+	return (isl_set *)isl_map_detect_equalities((isl_map *)set);
+}
+
+/* After computing the rational affine hull (by detecting the implicit
+ * equalities), we compute the additional equalities satisfied by
+ * the integer points (if any) and add the original equalities back in.
+ */
+struct isl_basic_map *isl_basic_map_affine_hull(struct isl_basic_map *bmap)
+{
+	bmap = isl_basic_map_detect_equalities(bmap);
+	bmap = isl_basic_map_cow(bmap);
+	if (bmap)
+		isl_basic_map_free_inequality(bmap, bmap->n_ineq);
+	bmap = isl_basic_map_finalize(bmap);
+	return bmap;
+}
+
+struct isl_basic_set *isl_basic_set_affine_hull(struct isl_basic_set *bset)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_affine_hull((struct isl_basic_map *)bset);
+}
+
+/* Given a rational affine matrix "M", add stride constraints to "bmap"
+ * that ensure that
+ *
+ *		M(x)
+ *
+ * is an integer vector.  The variables x include all the variables
+ * of "bmap" except the unknown divs.
+ *
+ * If d is the common denominator of M, then we need to impose that
+ *
+ *		d M(x) = 0 	mod d
+ *
+ * or
+ *
+ *		exists alpha : d M(x) = d alpha
+ *
+ * This function is similar to add_strides in isl_morph.c
+ */
+static __isl_give isl_basic_map *add_strides(__isl_take isl_basic_map *bmap,
+	__isl_keep isl_mat *M, int n_known)
+{
+	int i, div, k;
+	isl_int gcd;
+
+	if (isl_int_is_one(M->row[0][0]))
+		return bmap;
+
+	bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim),
+					M->n_row - 1, M->n_row - 1, 0);
+
+	isl_int_init(gcd);
+	for (i = 1; i < M->n_row; ++i) {
+		isl_seq_gcd(M->row[i], M->n_col, &gcd);
+		if (isl_int_is_divisible_by(gcd, M->row[0][0]))
+			continue;
+		div = isl_basic_map_alloc_div(bmap);
+		if (div < 0)
+			goto error;
+		isl_int_set_si(bmap->div[div][0], 0);
+		k = isl_basic_map_alloc_equality(bmap);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(bmap->eq[k], M->row[i], M->n_col);
+		isl_seq_clr(bmap->eq[k] + M->n_col, bmap->n_div - n_known);
+		isl_int_set(bmap->eq[k][M->n_col - n_known + div],
+			    M->row[0][0]);
+	}
+	isl_int_clear(gcd);
+
+	return bmap;
+error:
+	isl_int_clear(gcd);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* If there are any equalities that involve (multiple) unknown divs,
+ * then extract the stride information encoded by those equalities
+ * and make it explicitly available in "bmap".
+ *
+ * We first sort the divs so that the unknown divs appear last and
+ * then we count how many equalities involve these divs.
+ *
+ * Let these equalities be of the form
+ *
+ *		A(x) + B y = 0
+ *
+ * where y represents the unknown divs and x the remaining variables.
+ * Let [H 0] be the Hermite Normal Form of B, i.e.,
+ *
+ *		B = [H 0] Q
+ *
+ * Then x is a solution of the equalities iff
+ *
+ *		H^-1 A(x) (= - [I 0] Q y)
+ *
+ * is an integer vector.  Let d be the common denominator of H^-1.
+ * We impose
+ *
+ *		d H^-1 A(x) = d alpha
+ *
+ * in add_strides, with alpha fresh existentially quantified variables.
+ */
+static __isl_give isl_basic_map *isl_basic_map_make_strides_explicit(
+	__isl_take isl_basic_map *bmap)
+{
+	int known;
+	int n_known;
+	int n, n_col;
+	int total;
+	isl_ctx *ctx;
+	isl_mat *A, *B, *M;
+
+	known = isl_basic_map_divs_known(bmap);
+	if (known < 0)
+		return isl_basic_map_free(bmap);
+	if (known)
+		return bmap;
+	bmap = isl_basic_map_sort_divs(bmap);
+	bmap = isl_basic_map_gauss(bmap, NULL);
+	if (!bmap)
+		return NULL;
+
+	for (n_known = 0; n_known < bmap->n_div; ++n_known)
+		if (isl_int_is_zero(bmap->div[n_known][0]))
+			break;
+	ctx = isl_basic_map_get_ctx(bmap);
+	total = isl_space_dim(bmap->dim, isl_dim_all);
+	for (n = 0; n < bmap->n_eq; ++n)
+		if (isl_seq_first_non_zero(bmap->eq[n] + 1 + total + n_known,
+					    bmap->n_div - n_known) == -1)
+			break;
+	if (n == 0)
+		return bmap;
+	B = isl_mat_sub_alloc6(ctx, bmap->eq, 0, n, 0, 1 + total + n_known);
+	n_col = bmap->n_div - n_known;
+	A = isl_mat_sub_alloc6(ctx, bmap->eq, 0, n, 1 + total + n_known, n_col);
+	A = isl_mat_left_hermite(A, 0, NULL, NULL);
+	A = isl_mat_drop_cols(A, n, n_col - n);
+	A = isl_mat_lin_to_aff(A);
+	A = isl_mat_right_inverse(A);
+	B = isl_mat_insert_zero_rows(B, 0, 1);
+	B = isl_mat_set_element_si(B, 0, 0, 1);
+	M = isl_mat_product(A, B);
+	if (!M)
+		return isl_basic_map_free(bmap);
+	bmap = add_strides(bmap, M, n_known);
+	bmap = isl_basic_map_gauss(bmap, NULL);
+	isl_mat_free(M);
+
+	return bmap;
+}
+
+/* Compute the affine hull of each basic map in "map" separately
+ * and make all stride information explicit so that we can remove
+ * all unknown divs without losing this information.
+ * The result is also guaranteed to be gaussed.
+ *
+ * In simple cases where a div is determined by an equality,
+ * calling isl_basic_map_gauss is enough to make the stride information
+ * explicit, as it will derive an explicit representation for the div
+ * from the equality.  If, however, the stride information
+ * is encoded through multiple unknown divs then we need to make
+ * some extra effort in isl_basic_map_make_strides_explicit.
+ */
+static __isl_give isl_map *isl_map_local_affine_hull(__isl_take isl_map *map)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_affine_hull(map->p[i]);
+		map->p[i] = isl_basic_map_gauss(map->p[i], NULL);
+		map->p[i] = isl_basic_map_make_strides_explicit(map->p[i]);
+		if (!map->p[i])
+			return isl_map_free(map);
+	}
+
+	return map;
+}
+
+static __isl_give isl_set *isl_set_local_affine_hull(__isl_take isl_set *set)
+{
+	return isl_map_local_affine_hull(set);
+}
+
+/* Compute the affine hull of "map".
+ *
+ * We first compute the affine hull of each basic map separately.
+ * Then we align the divs and recompute the affine hulls of the basic
+ * maps since some of them may now have extra divs.
+ * In order to avoid performing parametric integer programming to
+ * compute explicit expressions for the divs, possible leading to
+ * an explosion in the number of basic maps, we first drop all unknown
+ * divs before aligning the divs.  Note that isl_map_local_affine_hull tries
+ * to make sure that all stride information is explicitly available
+ * in terms of known divs.  This involves calling isl_basic_set_gauss,
+ * which is also needed because affine_hull assumes its input has been gaussed,
+ * while isl_map_affine_hull may be called on input that has not been gaussed,
+ * in particular from initial_facet_constraint.
+ * Similarly, align_divs may reorder some divs so that we need to
+ * gauss the result again.
+ * Finally, we combine the individual affine hulls into a single
+ * affine hull.
+ */
+__isl_give isl_basic_map *isl_map_affine_hull(__isl_take isl_map *map)
+{
+	struct isl_basic_map *model = NULL;
+	struct isl_basic_map *hull = NULL;
+	struct isl_set *set;
+	isl_basic_set *bset;
+
+	map = isl_map_detect_equalities(map);
+	map = isl_map_local_affine_hull(map);
+	map = isl_map_remove_empty_parts(map);
+	map = isl_map_remove_unknown_divs(map);
+	map = isl_map_align_divs(map);
+
+	if (!map)
+		return NULL;
+
+	if (map->n == 0) {
+		hull = isl_basic_map_empty_like_map(map);
+		isl_map_free(map);
+		return hull;
+	}
+
+	model = isl_basic_map_copy(map->p[0]);
+	set = isl_map_underlying_set(map);
+	set = isl_set_cow(set);
+	set = isl_set_local_affine_hull(set);
+	if (!set)
+		goto error;
+
+	while (set->n > 1)
+		set->p[0] = affine_hull(set->p[0], set->p[--set->n]);
+
+	bset = isl_basic_set_copy(set->p[0]);
+	hull = isl_basic_map_overlying_set(bset, model);
+	isl_set_free(set);
+	hull = isl_basic_map_simplify(hull);
+	return isl_basic_map_finalize(hull);
+error:
+	isl_basic_map_free(model);
+	isl_set_free(set);
+	return NULL;
+}
+
+struct isl_basic_set *isl_set_affine_hull(struct isl_set *set)
+{
+	return (struct isl_basic_set *)
+		isl_map_affine_hull((struct isl_map *)set);
+}
diff --git a/polly/lib/External/isl/isl_arg.c b/polly/lib/External/isl/isl_arg.c
new file mode 100644
index 00000000000..fefb67d5c46
--- /dev/null
+++ b/polly/lib/External/isl/isl_arg.c
@@ -0,0 +1,1308 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+
+#include <isl/arg.h>
+#include <isl/ctx.h>
+
+static struct isl_arg help_arg[] = {
+ISL_ARG_PHANTOM_BOOL('h', "help", NULL, "print this help, then exit")
+};
+
+static void set_default_choice(struct isl_arg *arg, void *opt)
+{
+	*(unsigned *)(((char *)opt) + arg->offset) = arg->u.choice.default_value;
+}
+
+static void set_default_flags(struct isl_arg *arg, void *opt)
+{
+	*(unsigned *)(((char *)opt) + arg->offset) = arg->u.flags.default_value;
+}
+
+static void set_default_bool(struct isl_arg *arg, void *opt)
+{
+	if (arg->offset == (size_t) -1)
+		return;
+	*(unsigned *)(((char *)opt) + arg->offset) = arg->u.b.default_value;
+}
+
+static void set_default_child(struct isl_arg *arg, void *opt)
+{
+	void *child;
+
+	if (arg->offset == (size_t) -1)
+		child = opt;
+	else {
+		child = calloc(1, arg->u.child.child->options_size);
+		*(void **)(((char *)opt) + arg->offset) = child;
+	}
+
+	if (child)
+		isl_args_set_defaults(arg->u.child.child, child);
+}
+
+static void set_default_user(struct isl_arg *arg, void *opt)
+{
+	arg->u.user.init(((char *)opt) + arg->offset);
+}
+
+static void set_default_int(struct isl_arg *arg, void *opt)
+{
+	*(int *)(((char *)opt) + arg->offset) = arg->u.i.default_value;
+}
+
+static void set_default_long(struct isl_arg *arg, void *opt)
+{
+	*(long *)(((char *)opt) + arg->offset) = arg->u.l.default_value;
+}
+
+static void set_default_ulong(struct isl_arg *arg, void *opt)
+{
+	*(unsigned long *)(((char *)opt) + arg->offset) = arg->u.ul.default_value;
+}
+
+static void set_default_str(struct isl_arg *arg, void *opt)
+{
+	const char *str = NULL;
+	if (arg->u.str.default_value)
+		str = strdup(arg->u.str.default_value);
+	*(const char **)(((char *)opt) + arg->offset) = str;
+}
+
+static void set_default_str_list(struct isl_arg *arg, void *opt)
+{
+	*(const char ***)(((char *) opt) + arg->offset) = NULL;
+	*(int *)(((char *) opt) + arg->u.str_list.offset_n) = 0;
+}
+
+void isl_args_set_defaults(struct isl_args *args, void *opt)
+{
+	int i;
+
+	for (i = 0; args->args[i].type != isl_arg_end; ++i) {
+		switch (args->args[i].type) {
+		case isl_arg_choice:
+			set_default_choice(&args->args[i], opt);
+			break;
+		case isl_arg_flags:
+			set_default_flags(&args->args[i], opt);
+			break;
+		case isl_arg_bool:
+			set_default_bool(&args->args[i], opt);
+			break;
+		case isl_arg_child:
+			set_default_child(&args->args[i], opt);
+			break;
+		case isl_arg_user:
+			set_default_user(&args->args[i], opt);
+			break;
+		case isl_arg_int:
+			set_default_int(&args->args[i], opt);
+			break;
+		case isl_arg_long:
+			set_default_long(&args->args[i], opt);
+			break;
+		case isl_arg_ulong:
+			set_default_ulong(&args->args[i], opt);
+			break;
+		case isl_arg_arg:
+		case isl_arg_str:
+			set_default_str(&args->args[i], opt);
+			break;
+		case isl_arg_str_list:
+			set_default_str_list(&args->args[i], opt);
+			break;
+		case isl_arg_alias:
+		case isl_arg_footer:
+		case isl_arg_version:
+		case isl_arg_end:
+			break;
+		}
+	}
+}
+
+static void free_args(struct isl_arg *arg, void *opt);
+
+static void free_child(struct isl_arg *arg, void *opt)
+{
+	if (arg->offset == (size_t) -1)
+		free_args(arg->u.child.child->args, opt);
+	else
+		isl_args_free(arg->u.child.child,
+			    *(void **)(((char *)opt) + arg->offset));
+}
+
+static void free_str_list(struct isl_arg *arg, void *opt)
+{
+	int i;
+	int n = *(int *)(((char *) opt) + arg->u.str_list.offset_n);
+	char **list = *(char ***)(((char *) opt) + arg->offset);
+
+	for (i = 0; i < n; ++i)
+		free(list[i]);
+	free(list);
+}
+
+static void free_user(struct isl_arg *arg, void *opt)
+{
+	if (arg->u.user.clear)
+		arg->u.user.clear(((char *)opt) + arg->offset);
+}
+
+static void free_args(struct isl_arg *arg, void *opt)
+{
+	int i;
+
+	for (i = 0; arg[i].type != isl_arg_end; ++i) {
+		switch (arg[i].type) {
+		case isl_arg_child:
+			free_child(&arg[i], opt);
+			break;
+		case isl_arg_arg:
+		case isl_arg_str:
+			free(*(char **)(((char *)opt) + arg[i].offset));
+			break;
+		case isl_arg_str_list:
+			free_str_list(&arg[i], opt);
+			break;
+		case isl_arg_user:
+			free_user(&arg[i], opt);
+			break;
+		case isl_arg_alias:
+		case isl_arg_bool:
+		case isl_arg_choice:
+		case isl_arg_flags:
+		case isl_arg_int:
+		case isl_arg_long:
+		case isl_arg_ulong:
+		case isl_arg_version:
+		case isl_arg_footer:
+		case isl_arg_end:
+			break;
+		}
+	}
+}
+
+void isl_args_free(struct isl_args *args, void *opt)
+{
+	if (!opt)
+		return;
+
+	free_args(args->args, opt);
+
+	free(opt);
+}
+
+/* Data structure for collecting the prefixes of ancestor nodes.
+ *
+ * n is the number of prefixes.
+ * prefix[i] for i < n is a prefix of an ancestor.
+ * len[i] for i < n is the length of prefix[i].
+ */
+struct isl_prefixes {
+	int n;
+	const char *prefix[10];
+	size_t len[10];
+};
+
+/* Add "prefix" to the list of prefixes and return the updated
+ * number of prefixes.
+ */
+static int add_prefix(struct isl_prefixes *prefixes, const char *prefix)
+{
+	int n = prefixes->n;
+
+	if (!prefix)
+		return n;
+
+	if (prefixes->n >= 10) {
+		fprintf(stderr, "too many prefixes\n");
+		exit(EXIT_FAILURE);
+	}
+	prefixes->len[prefixes->n] = strlen(prefix);
+	prefixes->prefix[prefixes->n] = prefix;
+	prefixes->n++;
+
+	return n;
+}
+
+/* Drop all prefixes starting at "first".
+ */
+static void drop_prefix(struct isl_prefixes *prefixes, int first)
+{
+	prefixes->n = first;
+}
+
+/* Print the prefixes in "prefixes".
+ */
+static int print_prefixes(struct isl_prefixes *prefixes)
+{
+	int i;
+	int len = 0;
+
+	if (!prefixes)
+		return 0;
+
+	for (i = 0; i < prefixes->n; ++i) {
+		printf("%s-", prefixes->prefix[i]);
+		len += strlen(prefixes->prefix[i]) + 1;
+	}
+
+	return len;
+}
+
+/* Check if "name" starts with one or more of the prefixes in "prefixes",
+ * starting at *first.  If so, advance the pointer beyond the prefixes
+ * and return the updated pointer.  Additionally, update *first to
+ * the index after the last prefix found.
+ */
+static const char *skip_prefixes(const char *name,
+	struct isl_prefixes *prefixes, int *first)
+{
+	int i;
+
+	for (i = first ? *first : 0; i < prefixes->n; ++i) {
+		size_t len = prefixes->len[i];
+		const char *prefix = prefixes->prefix[i];
+		if (strncmp(name, prefix, len) == 0 && name[len] == '-') {
+			name += len + 1;
+			if (first)
+				*first = i + 1;
+		}
+	}
+
+	return name;
+}
+
+static int print_arg_help(struct isl_arg *decl, struct isl_prefixes *prefixes,
+	int no)
+{
+	int len = 0;
+
+	if (!decl->long_name) {
+		printf("  -%c", decl->short_name);
+		return 4;
+	}
+
+	if (decl->short_name) {
+		printf("  -%c, --", decl->short_name);
+		len += 8;
+	} else if (decl->flags & ISL_ARG_SINGLE_DASH) {
+		printf("  -");
+		len += 3;
+	} else {
+		printf("      --");
+		len += 8;
+	}
+
+	if (no) {
+		printf("no-");
+		len += 3;
+	}
+	len += print_prefixes(prefixes);
+	printf("%s", decl->long_name);
+	len += strlen(decl->long_name);
+
+	while ((++decl)->type == isl_arg_alias) {
+		printf(", --");
+		len += 4;
+		if (no) {
+			printf("no-");
+			len += 3;
+		}
+		printf("%s", decl->long_name);
+		len += strlen(decl->long_name);
+	}
+
+	return len;
+}
+
+const void *isl_memrchr(const void *s, int c, size_t n)
+{
+	const char *p = s;
+	while (n-- > 0)
+		if (p[n] == c)
+			return p + n;
+	return NULL;
+}
+
+static int wrap_msg(const char *s, int indent, int pos)
+{
+	int len;
+	int wrap_len = 75 - indent;
+
+	if (pos + 1 >= indent)
+		printf("\n%*s", indent, "");
+	else
+		printf("%*s", indent - pos, "");
+
+	len = strlen(s);
+	while (len > wrap_len) {
+		const char *space = isl_memrchr(s, ' ', wrap_len);
+		int l;
+
+		if (!space)
+			space = strchr(s + wrap_len, ' ');
+		if (!space)
+			break;
+		l = space - s;
+		printf("%.*s", l, s);
+		s = space + 1;
+		len -= l + 1;
+		printf("\n%*s", indent, "");
+	}
+
+	printf("%s", s);
+	return len;
+}
+
+static int print_help_msg(struct isl_arg *decl, int pos)
+{
+	if (!decl->help_msg)
+		return pos;
+
+	return wrap_msg(decl->help_msg, 30, pos);
+}
+
+static void print_default(struct isl_arg *decl, const char *def, int pos)
+{
+	const char *default_prefix = "[default: ";
+	const char *default_suffix = "]";
+	int len;
+
+	len = strlen(default_prefix) + strlen(def) + strlen(default_suffix);
+
+	if (!decl->help_msg) {
+		if (pos >= 29)
+			printf("\n%30s", "");
+		else
+			printf("%*s", 30 - pos, "");
+		pos = 0;
+	} else {
+		if (pos + len >= 48)
+			printf("\n%30s", "");
+		else
+			printf(" ");
+	}
+	printf("%s%s%s", default_prefix, def, default_suffix);
+}
+
+static void print_default_choice(struct isl_arg *decl, void *opt, int pos)
+{
+	int i;
+	const char *s = "none";
+	unsigned *p;
+
+	p = (unsigned *)(((char *) opt) + decl->offset);
+	for (i = 0; decl->u.choice.choice[i].name; ++i)
+		if (decl->u.choice.choice[i].value == *p) {
+			s = decl->u.choice.choice[i].name;
+			break;
+		}
+
+	print_default(decl, s, pos);
+}
+
+static void print_choice_help(struct isl_arg *decl,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	int i;
+	int pos;
+
+	pos = print_arg_help(decl, prefixes, 0);
+	printf("=");
+	pos++;
+
+	for (i = 0; decl->u.choice.choice[i].name; ++i) {
+		if (i) {
+			printf("|");
+			pos++;
+		}
+		printf("%s", decl->u.choice.choice[i].name);
+		pos += strlen(decl->u.choice.choice[i].name);
+	}
+
+	pos = print_help_msg(decl, pos);
+	print_default_choice(decl, opt, pos);
+
+	printf("\n");
+}
+
+static void print_default_flags(struct isl_arg *decl, void *opt, int pos)
+{
+	int i, first;
+	const char *default_prefix = "[default: ";
+	const char *default_suffix = "]";
+	int len = strlen(default_prefix) + strlen(default_suffix);
+	unsigned *p;
+
+	p = (unsigned *)(((char *) opt) + decl->offset);
+	for (i = 0; decl->u.flags.flags[i].name; ++i)
+		if ((*p & decl->u.flags.flags[i].mask) ==
+		     decl->u.flags.flags[i].value)
+			len += strlen(decl->u.flags.flags[i].name);
+
+	if (!decl->help_msg) {
+		if (pos >= 29)
+			printf("\n%30s", "");
+		else
+			printf("%*s", 30 - pos, "");
+		pos = 0;
+	} else {
+		if (pos + len >= 48)
+			printf("\n%30s", "");
+		else
+			printf(" ");
+	}
+	printf("%s", default_prefix);
+
+	for (first = 1, i = 0; decl->u.flags.flags[i].name; ++i)
+		if ((*p & decl->u.flags.flags[i].mask) ==
+		     decl->u.flags.flags[i].value) {
+			if (!first)
+				printf(",");
+			printf("%s", decl->u.flags.flags[i].name);
+			first = 0;
+		}
+
+	printf("%s", default_suffix);
+}
+
+static void print_flags_help(struct isl_arg *decl,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	int i, j;
+	int pos;
+
+	pos = print_arg_help(decl, prefixes, 0);
+	printf("=");
+	pos++;
+
+	for (i = 0; decl->u.flags.flags[i].name; ++i) {
+		if (i) {
+			printf(",");
+			pos++;
+		}
+		for (j = i;
+		     decl->u.flags.flags[j].mask == decl->u.flags.flags[i].mask;
+		     ++j) {
+			if (j != i) {
+				printf("|");
+				pos++;
+			}
+			printf("%s", decl->u.flags.flags[j].name);
+			pos += strlen(decl->u.flags.flags[j].name);
+		}
+		i = j - 1;
+	}
+
+	pos = print_help_msg(decl, pos);
+	print_default_flags(decl, opt, pos);
+
+	printf("\n");
+}
+
+static void print_bool_help(struct isl_arg *decl,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	int pos;
+	unsigned *p = opt ? (unsigned *)(((char *) opt) + decl->offset) : NULL;
+	int no = p ? *p == 1 : 0;
+	pos = print_arg_help(decl, prefixes, no);
+	pos = print_help_msg(decl, pos);
+	if (decl->offset != (size_t) -1)
+		print_default(decl, no ? "yes" : "no", pos);
+	printf("\n");
+}
+
+static int print_argument_name(struct isl_arg *decl, const char *name, int pos)
+{
+	printf("%c<%s>", decl->long_name ? '=' : ' ', name);
+	return pos + 3 + strlen(name);
+}
+
+static void print_int_help(struct isl_arg *decl,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	int pos;
+	char val[20];
+	int *p = (int *)(((char *) opt) + decl->offset);
+	pos = print_arg_help(decl, prefixes, 0);
+	pos = print_argument_name(decl, decl->argument_name, pos);
+	pos = print_help_msg(decl, pos);
+	snprintf(val, sizeof(val), "%d", *p);
+	print_default(decl, val, pos);
+	printf("\n");
+}
+
+static void print_long_help(struct isl_arg *decl,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	int pos;
+	long *p = (long *)(((char *) opt) + decl->offset);
+	pos = print_arg_help(decl, prefixes, 0);
+	if (*p != decl->u.l.default_selected) {
+		printf("[");
+		pos++;
+	}
+	printf("=long");
+	pos += 5;
+	if (*p != decl->u.l.default_selected) {
+		printf("]");
+		pos++;
+	}
+	print_help_msg(decl, pos);
+	printf("\n");
+}
+
+static void print_ulong_help(struct isl_arg *decl,
+	struct isl_prefixes *prefixes)
+{
+	int pos;
+	pos = print_arg_help(decl, prefixes, 0);
+	printf("=ulong");
+	pos += 6;
+	print_help_msg(decl, pos);
+	printf("\n");
+}
+
+static void print_str_help(struct isl_arg *decl,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	int pos;
+	const char *a = decl->argument_name ? decl->argument_name : "string";
+	const char **p = (const char **)(((char *) opt) + decl->offset);
+	pos = print_arg_help(decl, prefixes, 0);
+	pos = print_argument_name(decl, a, pos);
+	pos = print_help_msg(decl, pos);
+	if (*p)
+		print_default(decl, *p, pos);
+	printf("\n");
+}
+
+static void print_str_list_help(struct isl_arg *decl,
+	struct isl_prefixes *prefixes)
+{
+	int pos;
+	const char *a = decl->argument_name ? decl->argument_name : "string";
+	pos = print_arg_help(decl, prefixes, 0);
+	pos = print_argument_name(decl, a, pos);
+	pos = print_help_msg(decl, pos);
+	printf("\n");
+}
+
+static void print_help(struct isl_arg *arg,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	int i;
+	int any = 0;
+
+	for (i = 0; arg[i].type != isl_arg_end; ++i) {
+		if (arg[i].flags & ISL_ARG_HIDDEN)
+			continue;
+		switch (arg[i].type) {
+		case isl_arg_flags:
+			print_flags_help(&arg[i], prefixes, opt);
+			any = 1;
+			break;
+		case isl_arg_choice:
+			print_choice_help(&arg[i], prefixes, opt);
+			any = 1;
+			break;
+		case isl_arg_bool:
+			print_bool_help(&arg[i], prefixes, opt);
+			any = 1;
+			break;
+		case isl_arg_int:
+			print_int_help(&arg[i], prefixes, opt);
+			any = 1;
+			break;
+		case isl_arg_long:
+			print_long_help(&arg[i], prefixes, opt);
+			any = 1;
+			break;
+		case isl_arg_ulong:
+			print_ulong_help(&arg[i], prefixes);
+			any = 1;
+			break;
+		case isl_arg_str:
+			print_str_help(&arg[i], prefixes, opt);
+			any = 1;
+			break;
+		case isl_arg_str_list:
+			print_str_list_help(&arg[i], prefixes);
+			any = 1;
+			break;
+		case isl_arg_alias:
+		case isl_arg_version:
+		case isl_arg_arg:
+		case isl_arg_footer:
+		case isl_arg_child:
+		case isl_arg_user:
+		case isl_arg_end:
+			break;
+		}
+	}
+
+	for (i = 0; arg[i].type != isl_arg_end; ++i) {
+		void *child;
+		int first;
+
+		if (arg[i].type != isl_arg_child)
+			continue;
+		if (arg[i].flags & ISL_ARG_HIDDEN)
+			continue;
+
+		if (any)
+			printf("\n");
+		if (arg[i].help_msg)
+			printf(" %s\n", arg[i].help_msg);
+		if (arg[i].offset == (size_t) -1)
+			child = opt;
+		else
+			child = *(void **)(((char *) opt) + arg[i].offset);
+		first = add_prefix(prefixes, arg[i].long_name);
+		print_help(arg[i].u.child.child->args, prefixes, child);
+		drop_prefix(prefixes, first);
+		any = 1;
+	}
+}
+
+static const char *prog_name(const char *prog)
+{
+	const char *slash;
+
+	slash = strrchr(prog, '/');
+	if (slash)
+		prog = slash + 1;
+	if (strncmp(prog, "lt-", 3) == 0)
+		prog += 3;
+
+	return prog;
+}
+
+static int any_version(struct isl_arg *decl)
+{
+	int i;
+
+	for (i = 0; decl[i].type != isl_arg_end; ++i) {
+		switch (decl[i].type) {
+		case isl_arg_version:
+			return 1;
+		case isl_arg_child:
+			if (any_version(decl[i].u.child.child->args))
+				return 1;
+			break;
+		default:
+			break;
+		}
+	}
+
+	return 0;
+}
+
+static void print_help_and_exit(struct isl_arg *arg, const char *prog,
+	void *opt)
+{
+	int i;
+	struct isl_prefixes prefixes = { 0 };
+
+	printf("Usage: %s [OPTION...]", prog_name(prog));
+
+	for (i = 0; arg[i].type != isl_arg_end; ++i)
+		if (arg[i].type == isl_arg_arg)
+			printf(" %s", arg[i].argument_name);
+
+	printf("\n\n");
+
+	print_help(arg, &prefixes, opt);
+	printf("\n");
+	if (any_version(arg))
+		printf("  -V, --version\n");
+	print_bool_help(help_arg, NULL, NULL);
+
+	for (i = 0; arg[i].type != isl_arg_end; ++i) {
+		if (arg[i].type != isl_arg_footer)
+			continue;
+		wrap_msg(arg[i].help_msg, 0, 0);
+		printf("\n");
+	}
+
+	exit(0);
+}
+
+static int match_long_name(struct isl_arg *decl,
+	const char *start, const char *end)
+{
+	do {
+		if (end - start == strlen(decl->long_name) &&
+		    !strncmp(start, decl->long_name, end - start))
+			return 1;
+	} while ((++decl)->type == isl_arg_alias);
+
+	return 0;
+}
+
+static const char *skip_dash_dash(struct isl_arg *decl, const char *arg)
+{
+	if (!strncmp(arg, "--", 2))
+		return arg + 2;
+	if ((decl->flags & ISL_ARG_SINGLE_DASH) && arg[0] == '-')
+		return arg + 1;
+	return NULL;
+}
+
+static const char *skip_name(struct isl_arg *decl, const char *arg,
+	struct isl_prefixes *prefixes, int need_argument, int *has_argument)
+{
+	const char *equal;
+	const char *name;
+	const char *end;
+
+	if (arg[0] == '-' && arg[1] && arg[1] == decl->short_name) {
+		if (need_argument && !arg[2])
+			return NULL;
+		if (has_argument)
+			*has_argument = arg[2] != '\0';
+		return arg + 2;
+	}
+	if (!decl->long_name)
+		return NULL;
+
+	name = skip_dash_dash(decl, arg);
+	if (!name)
+		return NULL;
+
+	equal = strchr(name, '=');
+	if (need_argument && !equal)
+		return NULL;
+
+	if (has_argument)
+		*has_argument = !!equal;
+	end = equal ? equal : name + strlen(name);
+
+	name = skip_prefixes(name, prefixes, NULL);
+
+	if (!match_long_name(decl, name, end))
+		return NULL;
+
+	return equal ? equal + 1 : end;
+}
+
+static int parse_choice_option(struct isl_arg *decl, char **arg,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	int i;
+	int has_argument;
+	const char *choice;
+
+	choice = skip_name(decl, arg[0], prefixes, 0, &has_argument);
+	if (!choice)
+		return 0;
+
+	if (!has_argument && (!arg[1] || arg[1][0] == '-')) {
+		unsigned u = decl->u.choice.default_selected;
+		*(unsigned *)(((char *)opt) + decl->offset) = u;
+		if (decl->u.choice.set)
+			decl->u.choice.set(opt, u);
+
+		return 1;
+	}
+
+	if (!has_argument)
+		choice = arg[1];
+
+	for (i = 0; decl->u.choice.choice[i].name; ++i) {
+		unsigned u;
+
+		if (strcmp(choice, decl->u.choice.choice[i].name))
+			continue;
+
+		u = decl->u.choice.choice[i].value;
+		*(unsigned *)(((char *)opt) + decl->offset) = u;
+		if (decl->u.choice.set)
+			decl->u.choice.set(opt, u);
+
+		return has_argument ? 1 : 2;
+	}
+
+	return 0;
+}
+
+static int set_flag(struct isl_arg *decl, unsigned *val, const char *flag,
+	size_t len)
+{
+	int i;
+
+	for (i = 0; decl->u.flags.flags[i].name; ++i) {
+		if (strncmp(flag, decl->u.flags.flags[i].name, len))
+			continue;
+
+		*val &= ~decl->u.flags.flags[i].mask;
+		*val |= decl->u.flags.flags[i].value;
+
+		return 1;
+	}
+
+	return 0;
+}
+
+static int parse_flags_option(struct isl_arg *decl, char **arg,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	int has_argument;
+	const char *flags;
+	const char *comma;
+	unsigned val;
+
+	flags = skip_name(decl, arg[0], prefixes, 0, &has_argument);
+	if (!flags)
+		return 0;
+
+	if (!has_argument && !arg[1])
+		return 0;
+
+	if (!has_argument)
+		flags = arg[1];
+
+	val = 0;
+
+	while ((comma = strchr(flags, ',')) != NULL) {
+		if (!set_flag(decl, &val, flags, comma - flags))
+			return 0;
+		flags = comma + 1;
+	}
+	if (!set_flag(decl, &val, flags, strlen(flags)))
+		return 0;
+
+	*(unsigned *)(((char *)opt) + decl->offset) = val;
+
+	return has_argument ? 1 : 2;
+}
+
+static int parse_bool_option(struct isl_arg *decl, char **arg,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	const char *name;
+	unsigned *p = (unsigned *)(((char *)opt) + decl->offset);
+	int next_prefix;
+
+	if (skip_name(decl, arg[0], prefixes, 0, NULL)) {
+		if ((decl->flags & ISL_ARG_BOOL_ARG) && arg[1]) {
+			char *endptr;
+			int val = strtol(arg[1], &endptr, 0);
+			if (*endptr == '\0' && (val == 0 || val == 1)) {
+				if (decl->offset != (size_t) -1)
+					*p = val;
+				if (decl->u.b.set)
+					decl->u.b.set(opt, val);
+				return 2;
+			}
+		}
+		if (decl->offset != (size_t) -1)
+			*p = 1;
+		if (decl->u.b.set)
+			decl->u.b.set(opt, 1);
+
+		return 1;
+	}
+
+	if (!decl->long_name)
+		return 0;
+
+	name = skip_dash_dash(decl, arg[0]);
+	if (!name)
+		return 0;
+
+	next_prefix = 0;
+	name = skip_prefixes(name, prefixes, &next_prefix);
+
+	if (strncmp(name, "no-", 3))
+		return 0;
+	name += 3;
+
+	name = skip_prefixes(name, prefixes, &next_prefix);
+
+	if (match_long_name(decl, name, name + strlen(name))) {
+		if (decl->offset != (size_t) -1)
+			*p = 0;
+		if (decl->u.b.set)
+			decl->u.b.set(opt, 0);
+
+		return 1;
+	}
+
+	return 0;
+}
+
+static int parse_str_option(struct isl_arg *decl, char **arg,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	int has_argument;
+	const char *s;
+	char **p = (char **)(((char *)opt) + decl->offset);
+
+	s = skip_name(decl, arg[0], prefixes, 0, &has_argument);
+	if (!s)
+		return 0;
+
+	if (has_argument) {
+		free(*p);
+		*p = strdup(s);
+		return 1;
+	}
+
+	if (arg[1]) {
+		free(*p);
+		*p = strdup(arg[1]);
+		return 2;
+	}
+
+	return 0;
+}
+
+static int isl_arg_str_list_append(struct isl_arg *decl, void *opt,
+	const char *s)
+{
+	int *n = (int *)(((char *) opt) + decl->u.str_list.offset_n);
+	char **list = *(char ***)(((char *) opt) + decl->offset);
+
+	list = realloc(list, (*n + 1) * sizeof(char *));
+	if (!list)
+		return -1;
+	*(char ***)(((char *) opt) + decl->offset) = list;
+	list[*n] = strdup(s);
+	(*n)++;
+	return 0;
+}
+
+static int parse_str_list_option(struct isl_arg *decl, char **arg,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	int has_argument;
+	const char *s;
+
+	s = skip_name(decl, arg[0], prefixes, 0, &has_argument);
+	if (!s)
+		return 0;
+
+	if (has_argument) {
+		isl_arg_str_list_append(decl, opt, s);
+		return 1;
+	}
+
+	if (arg[1]) {
+		isl_arg_str_list_append(decl, opt, arg[1]);
+		return 2;
+	}
+
+	return 0;
+}
+
+static int parse_int_option(struct isl_arg *decl, char **arg,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	int has_argument;
+	const char *val;
+	char *endptr;
+	int *p = (int *)(((char *)opt) + decl->offset);
+
+	val = skip_name(decl, arg[0], prefixes, 0, &has_argument);
+	if (!val)
+		return 0;
+
+	if (has_argument) {
+		*p = atoi(val);
+		return 1;
+	}
+
+	if (arg[1]) {
+		int i = strtol(arg[1], &endptr, 0);
+		if (*endptr == '\0') {
+			*p = i;
+			return 2;
+		}
+	}
+
+	return 0;
+}
+
+static int parse_long_option(struct isl_arg *decl, char **arg,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	int has_argument;
+	const char *val;
+	char *endptr;
+	long *p = (long *)(((char *)opt) + decl->offset);
+
+	val = skip_name(decl, arg[0], prefixes, 0, &has_argument);
+	if (!val)
+		return 0;
+
+	if (has_argument) {
+		long l = strtol(val, NULL, 0);
+		*p = l;
+		if (decl->u.l.set)
+			decl->u.l.set(opt, l);
+		return 1;
+	}
+
+	if (arg[1]) {
+		long l = strtol(arg[1], &endptr, 0);
+		if (*endptr == '\0') {
+			*p = l;
+			if (decl->u.l.set)
+				decl->u.l.set(opt, l);
+			return 2;
+		}
+	}
+
+	if (decl->u.l.default_value != decl->u.l.default_selected) {
+		*p = decl->u.l.default_selected;
+		if (decl->u.l.set)
+			decl->u.l.set(opt, decl->u.l.default_selected);
+		return 1;
+	}
+
+	return 0;
+}
+
+static int parse_ulong_option(struct isl_arg *decl, char **arg,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	int has_argument;
+	const char *val;
+	char *endptr;
+	unsigned long *p = (unsigned long *)(((char *)opt) + decl->offset);
+
+	val = skip_name(decl, arg[0], prefixes, 0, &has_argument);
+	if (!val)
+		return 0;
+
+	if (has_argument) {
+		*p = strtoul(val, NULL, 0);
+		return 1;
+	}
+
+	if (arg[1]) {
+		unsigned long ul = strtoul(arg[1], &endptr, 0);
+		if (*endptr == '\0') {
+			*p = ul;
+			return 2;
+		}
+	}
+
+	return 0;
+}
+
+static int parse_option(struct isl_arg *decl, char **arg,
+	struct isl_prefixes *prefixes, void *opt);
+
+static int parse_child_option(struct isl_arg *decl, char **arg,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	void *child;
+	int first, parsed;
+
+	if (decl->offset == (size_t) -1)
+		child = opt;
+	else
+		child = *(void **)(((char *)opt) + decl->offset);
+
+	first = add_prefix(prefixes, decl->long_name);
+	parsed = parse_option(decl->u.child.child->args, arg, prefixes, child);
+	drop_prefix(prefixes, first);
+
+	return parsed;
+}
+
+static int parse_option(struct isl_arg *decl, char **arg,
+	struct isl_prefixes *prefixes, void *opt)
+{
+	int i;
+
+	for (i = 0; decl[i].type != isl_arg_end; ++i) {
+		int parsed = 0;
+		switch (decl[i].type) {
+		case isl_arg_choice:
+			parsed = parse_choice_option(&decl[i], arg,
+							prefixes, opt);
+			break;
+		case isl_arg_flags:
+			parsed = parse_flags_option(&decl[i], arg,
+							prefixes, opt);
+			break;
+		case isl_arg_int:
+			parsed = parse_int_option(&decl[i], arg, prefixes, opt);
+			break;
+		case isl_arg_long:
+			parsed = parse_long_option(&decl[i], arg,
+							prefixes, opt);
+			break;
+		case isl_arg_ulong:
+			parsed = parse_ulong_option(&decl[i], arg,
+							prefixes, opt);
+			break;
+		case isl_arg_bool:
+			parsed = parse_bool_option(&decl[i], arg,
+							prefixes, opt);
+			break;
+		case isl_arg_str:
+			parsed = parse_str_option(&decl[i], arg, prefixes, opt);
+			break;
+		case isl_arg_str_list:
+			parsed = parse_str_list_option(&decl[i], arg, prefixes,
+							opt);
+			break;
+		case isl_arg_child:
+			parsed = parse_child_option(&decl[i], arg,
+							prefixes, opt);
+			break;
+		case isl_arg_alias:
+		case isl_arg_arg:
+		case isl_arg_footer:
+		case isl_arg_user:
+		case isl_arg_version:
+		case isl_arg_end:
+			break;
+		}
+		if (parsed)
+			return parsed;
+	}
+
+	return 0;
+}
+
+static void print_version(struct isl_arg *decl)
+{
+	int i;
+
+	for (i = 0; decl[i].type != isl_arg_end; ++i) {
+		switch (decl[i].type) {
+		case isl_arg_version:
+			decl[i].u.version.print_version();
+			break;
+		case isl_arg_child:
+			print_version(decl[i].u.child.child->args);
+			break;
+		default:
+			break;
+		}
+	}
+}
+
+static void print_version_and_exit(struct isl_arg *decl)
+{
+	print_version(decl);
+
+	exit(0);
+}
+
+static int drop_argument(int argc, char **argv, int drop, int n)
+{
+	for (; drop + n < argc; ++drop)
+		argv[drop] = argv[drop + n];
+
+	return argc - n;
+}
+
+static int n_arg(struct isl_arg *arg)
+{
+	int i;
+	int n_arg = 0;
+
+	for (i = 0; arg[i].type != isl_arg_end; ++i)
+		if (arg[i].type == isl_arg_arg)
+			n_arg++;
+
+	return n_arg;
+}
+
+static int next_arg(struct isl_arg *arg, int a)
+{
+	for (++a; arg[a].type != isl_arg_end; ++a)
+		if (arg[a].type == isl_arg_arg)
+			return a;
+
+	return -1;
+}
+
+/* Unless ISL_ARG_SKIP_HELP is set, check if "arg" is
+ * equal to "--help" and if so call print_help_and_exit.
+ */
+static void check_help(struct isl_args *args, char *arg, char *prog, void *opt,
+	unsigned flags)
+{
+	if (ISL_FL_ISSET(flags, ISL_ARG_SKIP_HELP))
+		return;
+
+	if (strcmp(arg, "--help") == 0)
+		print_help_and_exit(args->args, prog, opt);
+}
+
+int isl_args_parse(struct isl_args *args, int argc, char **argv, void *opt,
+	unsigned flags)
+{
+	int a = -1;
+	int skip = 0;
+	int i;
+	int n;
+	struct isl_prefixes prefixes = { 0 };
+
+	n = n_arg(args->args);
+
+	for (i = 1; i < argc; ++i) {
+		if ((strcmp(argv[i], "--version") == 0 ||
+		     strcmp(argv[i], "-V") == 0) && any_version(args->args))
+			print_version_and_exit(args->args);
+	}
+
+	while (argc > 1 + skip) {
+		int parsed;
+		if (argv[1 + skip][0] != '-') {
+			a = next_arg(args->args, a);
+			if (a >= 0) {
+				char **p;
+				p = (char **)(((char *)opt)+args->args[a].offset);
+				free(*p);
+				*p = strdup(argv[1 + skip]);
+				argc = drop_argument(argc, argv, 1 + skip, 1);
+				--n;
+			} else if (ISL_FL_ISSET(flags, ISL_ARG_ALL)) {
+				fprintf(stderr, "%s: extra argument: %s\n",
+					    prog_name(argv[0]), argv[1 + skip]);
+				exit(-1);
+			} else
+				++skip;
+			continue;
+		}
+		check_help(args, argv[1 + skip], argv[0], opt, flags);
+		parsed = parse_option(args->args, &argv[1 + skip],
+					&prefixes, opt);
+		if (parsed)
+			argc = drop_argument(argc, argv, 1 + skip, parsed);
+		else if (ISL_FL_ISSET(flags, ISL_ARG_ALL)) {
+			fprintf(stderr, "%s: unrecognized option: %s\n",
+					prog_name(argv[0]), argv[1 + skip]);
+			exit(-1);
+		} else
+			++skip;
+	}
+
+	if (n > 0) {
+		fprintf(stderr, "%s: expecting %d more argument(s)\n",
+				prog_name(argv[0]), n);
+		exit(-1);
+	}
+
+	return argc;
+}
diff --git a/polly/lib/External/isl/isl_ast.c b/polly/lib/External/isl/isl_ast.c
new file mode 100644
index 00000000000..0d77162cbc9
--- /dev/null
+++ b/polly/lib/External/isl/isl_ast.c
@@ -0,0 +1,2069 @@
+/*
+ * Copyright 2012-2013 Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege,
+ * Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+#include <isl_ast_private.h>
+
+#undef BASE
+#define BASE ast_expr
+
+#include <isl_list_templ.c>
+
+#undef BASE
+#define BASE ast_node
+
+#include <isl_list_templ.c>
+
+isl_ctx *isl_ast_print_options_get_ctx(
+	__isl_keep isl_ast_print_options *options)
+{
+	return options ? options->ctx : NULL;
+}
+
+__isl_give isl_ast_print_options *isl_ast_print_options_alloc(isl_ctx *ctx)
+{
+	isl_ast_print_options *options;
+
+	options = isl_calloc_type(ctx, isl_ast_print_options);
+	if (!options)
+		return NULL;
+
+	options->ctx = ctx;
+	isl_ctx_ref(ctx);
+	options->ref = 1;
+
+	return options;
+}
+
+__isl_give isl_ast_print_options *isl_ast_print_options_dup(
+	__isl_keep isl_ast_print_options *options)
+{
+	isl_ctx *ctx;
+	isl_ast_print_options *dup;
+
+	if (!options)
+		return NULL;
+
+	ctx = isl_ast_print_options_get_ctx(options);
+	dup = isl_ast_print_options_alloc(ctx);
+	if (!dup)
+		return NULL;
+
+	dup->print_for = options->print_for;
+	dup->print_for_user = options->print_for_user;
+	dup->print_user = options->print_user;
+	dup->print_user_user = options->print_user_user;
+
+	return dup;
+}
+
+__isl_give isl_ast_print_options *isl_ast_print_options_cow(
+	__isl_take isl_ast_print_options *options)
+{
+	if (!options)
+		return NULL;
+
+	if (options->ref == 1)
+		return options;
+	options->ref--;
+	return isl_ast_print_options_dup(options);
+}
+
+__isl_give isl_ast_print_options *isl_ast_print_options_copy(
+	__isl_keep isl_ast_print_options *options)
+{
+	if (!options)
+		return NULL;
+
+	options->ref++;
+	return options;
+}
+
+__isl_null isl_ast_print_options *isl_ast_print_options_free(
+	__isl_take isl_ast_print_options *options)
+{
+	if (!options)
+		return NULL;
+
+	if (--options->ref > 0)
+		return NULL;
+
+	isl_ctx_deref(options->ctx);
+
+	free(options);
+	return NULL;
+}
+
+/* Set the print_user callback of "options" to "print_user".
+ *
+ * If this callback is set, then it used to print user nodes in the AST.
+ * Otherwise, the expression associated to the user node is printed.
+ */
+__isl_give isl_ast_print_options *isl_ast_print_options_set_print_user(
+	__isl_take isl_ast_print_options *options,
+	__isl_give isl_printer *(*print_user)(__isl_take isl_printer *p,
+		__isl_take isl_ast_print_options *options,
+		__isl_keep isl_ast_node *node, void *user),
+	void *user)
+{
+	options = isl_ast_print_options_cow(options);
+	if (!options)
+		return NULL;
+
+	options->print_user = print_user;
+	options->print_user_user = user;
+
+	return options;
+}
+
+/* Set the print_for callback of "options" to "print_for".
+ *
+ * If this callback is set, then it used to print for nodes in the AST.
+ */
+__isl_give isl_ast_print_options *isl_ast_print_options_set_print_for(
+	__isl_take isl_ast_print_options *options,
+	__isl_give isl_printer *(*print_for)(__isl_take isl_printer *p,
+		__isl_take isl_ast_print_options *options,
+		__isl_keep isl_ast_node *node, void *user),
+	void *user)
+{
+	options = isl_ast_print_options_cow(options);
+	if (!options)
+		return NULL;
+
+	options->print_for = print_for;
+	options->print_for_user = user;
+
+	return options;
+}
+
+__isl_give isl_ast_expr *isl_ast_expr_copy(__isl_keep isl_ast_expr *expr)
+{
+	if (!expr)
+		return NULL;
+
+	expr->ref++;
+	return expr;
+}
+
+__isl_give isl_ast_expr *isl_ast_expr_dup(__isl_keep isl_ast_expr *expr)
+{
+	int i;
+	isl_ctx *ctx;
+	isl_ast_expr *dup;
+
+	if (!expr)
+		return NULL;
+
+	ctx = isl_ast_expr_get_ctx(expr);
+	switch (expr->type) {
+	case isl_ast_expr_int:
+		dup = isl_ast_expr_from_val(isl_val_copy(expr->u.v));
+		break;
+	case isl_ast_expr_id:
+		dup = isl_ast_expr_from_id(isl_id_copy(expr->u.id));
+		break;
+	case isl_ast_expr_op:
+		dup = isl_ast_expr_alloc_op(ctx,
+					    expr->u.op.op, expr->u.op.n_arg);
+		if (!dup)
+			return NULL;
+		for (i = 0; i < expr->u.op.n_arg; ++i)
+			dup->u.op.args[i] =
+				isl_ast_expr_copy(expr->u.op.args[i]);
+		break;
+	case isl_ast_expr_error:
+		dup = NULL;
+	}
+
+	if (!dup)
+		return NULL;
+
+	return dup;
+}
+
+__isl_give isl_ast_expr *isl_ast_expr_cow(__isl_take isl_ast_expr *expr)
+{
+	if (!expr)
+		return NULL;
+
+	if (expr->ref == 1)
+		return expr;
+	expr->ref--;
+	return isl_ast_expr_dup(expr);
+}
+
+__isl_null isl_ast_expr *isl_ast_expr_free(__isl_take isl_ast_expr *expr)
+{
+	int i;
+
+	if (!expr)
+		return NULL;
+
+	if (--expr->ref > 0)
+		return NULL;
+
+	isl_ctx_deref(expr->ctx);
+
+	switch (expr->type) {
+	case isl_ast_expr_int:
+		isl_val_free(expr->u.v);
+		break;
+	case isl_ast_expr_id:
+		isl_id_free(expr->u.id);
+		break;
+	case isl_ast_expr_op:
+		if (expr->u.op.args)
+			for (i = 0; i < expr->u.op.n_arg; ++i)
+				isl_ast_expr_free(expr->u.op.args[i]);
+		free(expr->u.op.args);
+		break;
+	case isl_ast_expr_error:
+		break;
+	}
+
+	free(expr);
+	return NULL;
+}
+
+isl_ctx *isl_ast_expr_get_ctx(__isl_keep isl_ast_expr *expr)
+{
+	return expr ? expr->ctx : NULL;
+}
+
+enum isl_ast_expr_type isl_ast_expr_get_type(__isl_keep isl_ast_expr *expr)
+{
+	return expr ? expr->type : isl_ast_expr_error;
+}
+
+/* Return the integer value represented by "expr".
+ */
+__isl_give isl_val *isl_ast_expr_get_val(__isl_keep isl_ast_expr *expr)
+{
+	if (!expr)
+		return NULL;
+	if (expr->type != isl_ast_expr_int)
+		isl_die(isl_ast_expr_get_ctx(expr), isl_error_invalid,
+			"expression not an int", return NULL);
+	return isl_val_copy(expr->u.v);
+}
+
+__isl_give isl_id *isl_ast_expr_get_id(__isl_keep isl_ast_expr *expr)
+{
+	if (!expr)
+		return NULL;
+	if (expr->type != isl_ast_expr_id)
+		isl_die(isl_ast_expr_get_ctx(expr), isl_error_invalid,
+			"expression not an identifier", return NULL);
+
+	return isl_id_copy(expr->u.id);
+}
+
+enum isl_ast_op_type isl_ast_expr_get_op_type(__isl_keep isl_ast_expr *expr)
+{
+	if (!expr)
+		return isl_ast_op_error;
+	if (expr->type != isl_ast_expr_op)
+		isl_die(isl_ast_expr_get_ctx(expr), isl_error_invalid,
+			"expression not an operation", return isl_ast_op_error);
+	return expr->u.op.op;
+}
+
+int isl_ast_expr_get_op_n_arg(__isl_keep isl_ast_expr *expr)
+{
+	if (!expr)
+		return -1;
+	if (expr->type != isl_ast_expr_op)
+		isl_die(isl_ast_expr_get_ctx(expr), isl_error_invalid,
+			"expression not an operation", return -1);
+	return expr->u.op.n_arg;
+}
+
+__isl_give isl_ast_expr *isl_ast_expr_get_op_arg(__isl_keep isl_ast_expr *expr,
+	int pos)
+{
+	if (!expr)
+		return NULL;
+	if (expr->type != isl_ast_expr_op)
+		isl_die(isl_ast_expr_get_ctx(expr), isl_error_invalid,
+			"expression not an operation", return NULL);
+	if (pos < 0 || pos >= expr->u.op.n_arg)
+		isl_die(isl_ast_expr_get_ctx(expr), isl_error_invalid,
+			"index out of bounds", return NULL);
+
+	return isl_ast_expr_copy(expr->u.op.args[pos]);
+}
+
+/* Replace the argument at position "pos" of "expr" by "arg".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_set_op_arg(__isl_take isl_ast_expr *expr,
+	int pos, __isl_take isl_ast_expr *arg)
+{
+	expr = isl_ast_expr_cow(expr);
+	if (!expr || !arg)
+		goto error;
+	if (expr->type != isl_ast_expr_op)
+		isl_die(isl_ast_expr_get_ctx(expr), isl_error_invalid,
+			"expression not an operation", goto error);
+	if (pos < 0 || pos >= expr->u.op.n_arg)
+		isl_die(isl_ast_expr_get_ctx(expr), isl_error_invalid,
+			"index out of bounds", goto error);
+
+	isl_ast_expr_free(expr->u.op.args[pos]);
+	expr->u.op.args[pos] = arg;
+
+	return expr;
+error:
+	isl_ast_expr_free(arg);
+	return isl_ast_expr_free(expr);
+}
+
+/* Is "expr1" equal to "expr2"?
+ */
+int isl_ast_expr_is_equal(__isl_keep isl_ast_expr *expr1,
+	__isl_keep isl_ast_expr *expr2)
+{
+	int i;
+
+	if (!expr1 || !expr2)
+		return -1;
+
+	if (expr1 == expr2)
+		return 1;
+	if (expr1->type != expr2->type)
+		return 0;
+	switch (expr1->type) {
+	case isl_ast_expr_int:
+		return isl_val_eq(expr1->u.v, expr2->u.v);
+	case isl_ast_expr_id:
+		return expr1->u.id == expr2->u.id;
+	case isl_ast_expr_op:
+		if (expr1->u.op.op != expr2->u.op.op)
+			return 0;
+		if (expr1->u.op.n_arg != expr2->u.op.n_arg)
+			return 0;
+		for (i = 0; i < expr1->u.op.n_arg; ++i) {
+			int equal;
+			equal = isl_ast_expr_is_equal(expr1->u.op.args[i],
+							expr2->u.op.args[i]);
+				return 0;
+			if (equal < 0 || !equal)
+				return equal;
+		}
+		return 1;
+	case isl_ast_expr_error:
+		return -1;
+	}
+}
+
+/* Create a new operation expression of operation type "op",
+ * with "n_arg" as yet unspecified arguments.
+ */
+__isl_give isl_ast_expr *isl_ast_expr_alloc_op(isl_ctx *ctx,
+	enum isl_ast_op_type op, int n_arg)
+{
+	isl_ast_expr *expr;
+
+	expr = isl_calloc_type(ctx, isl_ast_expr);
+	if (!expr)
+		return NULL;
+
+	expr->ctx = ctx;
+	isl_ctx_ref(ctx);
+	expr->ref = 1;
+	expr->type = isl_ast_expr_op;
+	expr->u.op.op = op;
+	expr->u.op.n_arg = n_arg;
+	expr->u.op.args = isl_calloc_array(ctx, isl_ast_expr *, n_arg);
+
+	if (n_arg && !expr->u.op.args)
+		return isl_ast_expr_free(expr);
+
+	return expr;
+}
+
+/* Create a new id expression representing "id".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_from_id(__isl_take isl_id *id)
+{
+	isl_ctx *ctx;
+	isl_ast_expr *expr;
+
+	if (!id)
+		return NULL;
+
+	ctx = isl_id_get_ctx(id);
+	expr = isl_calloc_type(ctx, isl_ast_expr);
+	if (!expr)
+		goto error;
+
+	expr->ctx = ctx;
+	isl_ctx_ref(ctx);
+	expr->ref = 1;
+	expr->type = isl_ast_expr_id;
+	expr->u.id = id;
+
+	return expr;
+error:
+	isl_id_free(id);
+	return NULL;
+}
+
+/* Create a new integer expression representing "i".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_alloc_int_si(isl_ctx *ctx, int i)
+{
+	isl_ast_expr *expr;
+
+	expr = isl_calloc_type(ctx, isl_ast_expr);
+	if (!expr)
+		return NULL;
+
+	expr->ctx = ctx;
+	isl_ctx_ref(ctx);
+	expr->ref = 1;
+	expr->type = isl_ast_expr_int;
+	expr->u.v = isl_val_int_from_si(ctx, i);
+	if (!expr->u.v)
+		return isl_ast_expr_free(expr);
+
+	return expr;
+}
+
+/* Create a new integer expression representing "v".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_from_val(__isl_take isl_val *v)
+{
+	isl_ctx *ctx;
+	isl_ast_expr *expr;
+
+	if (!v)
+		return NULL;
+	if (!isl_val_is_int(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting integer value", goto error);
+
+	ctx = isl_val_get_ctx(v);
+	expr = isl_calloc_type(ctx, isl_ast_expr);
+	if (!expr)
+		goto error;
+
+	expr->ctx = ctx;
+	isl_ctx_ref(ctx);
+	expr->ref = 1;
+	expr->type = isl_ast_expr_int;
+	expr->u.v = v;
+
+	return expr;
+error:
+	isl_val_free(v);
+	return NULL;
+}
+
+/* Create an expression representing the unary operation "type" applied to
+ * "arg".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_alloc_unary(enum isl_ast_op_type type,
+	__isl_take isl_ast_expr *arg)
+{
+	isl_ctx *ctx;
+	isl_ast_expr *expr = NULL;
+
+	if (!arg)
+		return NULL;
+
+	ctx = isl_ast_expr_get_ctx(arg);
+	expr = isl_ast_expr_alloc_op(ctx, type, 1);
+	if (!expr)
+		goto error;
+
+	expr->u.op.args[0] = arg;
+
+	return expr;
+error:
+	isl_ast_expr_free(arg);
+	return NULL;
+}
+
+/* Create an expression representing the negation of "arg".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_neg(__isl_take isl_ast_expr *arg)
+{
+	return isl_ast_expr_alloc_unary(isl_ast_op_minus, arg);
+}
+
+/* Create an expression representing the address of "expr".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_address_of(__isl_take isl_ast_expr *expr)
+{
+	if (!expr)
+		return NULL;
+
+	if (isl_ast_expr_get_type(expr) != isl_ast_expr_op ||
+	    isl_ast_expr_get_op_type(expr) != isl_ast_op_access)
+		isl_die(isl_ast_expr_get_ctx(expr), isl_error_invalid,
+			"can only take address of access expressions",
+			return isl_ast_expr_free(expr));
+
+	return isl_ast_expr_alloc_unary(isl_ast_op_address_of, expr);
+}
+
+/* Create an expression representing the binary operation "type"
+ * applied to "expr1" and "expr2".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_alloc_binary(enum isl_ast_op_type type,
+	__isl_take isl_ast_expr *expr1, __isl_take isl_ast_expr *expr2)
+{
+	isl_ctx *ctx;
+	isl_ast_expr *expr = NULL;
+
+	if (!expr1 || !expr2)
+		goto error;
+
+	ctx = isl_ast_expr_get_ctx(expr1);
+	expr = isl_ast_expr_alloc_op(ctx, type, 2);
+	if (!expr)
+		goto error;
+
+	expr->u.op.args[0] = expr1;
+	expr->u.op.args[1] = expr2;
+
+	return expr;
+error:
+	isl_ast_expr_free(expr1);
+	isl_ast_expr_free(expr2);
+	return NULL;
+}
+
+/* Create an expression representing the sum of "expr1" and "expr2".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_add(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2)
+{
+	return isl_ast_expr_alloc_binary(isl_ast_op_add, expr1, expr2);
+}
+
+/* Create an expression representing the difference of "expr1" and "expr2".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_sub(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2)
+{
+	return isl_ast_expr_alloc_binary(isl_ast_op_sub, expr1, expr2);
+}
+
+/* Create an expression representing the product of "expr1" and "expr2".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_mul(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2)
+{
+	return isl_ast_expr_alloc_binary(isl_ast_op_mul, expr1, expr2);
+}
+
+/* Create an expression representing the quotient of "expr1" and "expr2".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_div(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2)
+{
+	return isl_ast_expr_alloc_binary(isl_ast_op_div, expr1, expr2);
+}
+
+/* Create an expression representing the conjunction of "expr1" and "expr2".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_and(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2)
+{
+	return isl_ast_expr_alloc_binary(isl_ast_op_and, expr1, expr2);
+}
+
+/* Create an expression representing the disjunction of "expr1" and "expr2".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_or(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2)
+{
+	return isl_ast_expr_alloc_binary(isl_ast_op_or, expr1, expr2);
+}
+
+/* Create an expression representing "expr1" less than or equal to "expr2".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_le(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2)
+{
+	return isl_ast_expr_alloc_binary(isl_ast_op_le, expr1, expr2);
+}
+
+/* Create an expression representing "expr1" less than "expr2".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_lt(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2)
+{
+	return isl_ast_expr_alloc_binary(isl_ast_op_lt, expr1, expr2);
+}
+
+/* Create an expression representing "expr1" greater than or equal to "expr2".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_ge(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2)
+{
+	return isl_ast_expr_alloc_binary(isl_ast_op_ge, expr1, expr2);
+}
+
+/* Create an expression representing "expr1" greater than "expr2".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_gt(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2)
+{
+	return isl_ast_expr_alloc_binary(isl_ast_op_gt, expr1, expr2);
+}
+
+/* Create an expression representing "expr1" equal to "expr2".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_eq(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2)
+{
+	return isl_ast_expr_alloc_binary(isl_ast_op_eq, expr1, expr2);
+}
+
+/* Create an expression representing an access to "array" with index
+ * expressions "indices".
+ */
+__isl_give isl_ast_expr *isl_ast_expr_access(__isl_take isl_ast_expr *array,
+	__isl_take isl_ast_expr_list *indices)
+{
+	int i, n;
+	isl_ctx *ctx;
+	isl_ast_expr *access = NULL;
+
+	if (!array || !indices)
+		goto error;
+
+	ctx = isl_ast_expr_get_ctx(array);
+	n = isl_ast_expr_list_n_ast_expr(indices);
+	access = isl_ast_expr_alloc_op(ctx, isl_ast_op_access, 1 + n);
+	if (!access)
+		goto error;
+	for (i = 0; i < n; ++i) {
+		isl_ast_expr *index;
+		index = isl_ast_expr_list_get_ast_expr(indices, i);
+		access->u.op.args[1 + i] = index;
+		if (!index)
+			goto error;
+	}
+	access->u.op.args[0] = array;
+
+	isl_ast_expr_list_free(indices);
+	return access;
+error:
+	isl_ast_expr_free(array);
+	isl_ast_expr_list_free(indices);
+	isl_ast_expr_free(access);
+	return NULL;
+}
+
+/* For each subexpression of "expr" of type isl_ast_expr_id,
+ * if it appears in "id2expr", then replace it by the corresponding
+ * expression.
+ */
+__isl_give isl_ast_expr *isl_ast_expr_substitute_ids(
+	__isl_take isl_ast_expr *expr, __isl_take isl_id_to_ast_expr *id2expr)
+{
+	int i;
+	isl_id *id;
+
+	if (!expr || !id2expr)
+		goto error;
+
+	switch (expr->type) {
+	case isl_ast_expr_int:
+		break;
+	case isl_ast_expr_id:
+		if (!isl_id_to_ast_expr_has(id2expr, expr->u.id))
+			break;
+		id = isl_id_copy(expr->u.id);
+		isl_ast_expr_free(expr);
+		expr = isl_id_to_ast_expr_get(id2expr, id);
+		break;
+	case isl_ast_expr_op:
+		for (i = 0; i < expr->u.op.n_arg; ++i) {
+			isl_ast_expr *arg;
+			arg = isl_ast_expr_copy(expr->u.op.args[i]);
+			arg = isl_ast_expr_substitute_ids(arg,
+					    isl_id_to_ast_expr_copy(id2expr));
+			if (arg == expr->u.op.args[i]) {
+				isl_ast_expr_free(arg);
+				continue;
+			}
+			if (!arg)
+				expr = isl_ast_expr_free(expr);
+			expr = isl_ast_expr_cow(expr);
+			if (!expr) {
+				isl_ast_expr_free(arg);
+				break;
+			}
+			isl_ast_expr_free(expr->u.op.args[i]);
+			expr->u.op.args[i] = arg;
+		}
+		break;
+	case isl_ast_expr_error:
+		expr = isl_ast_expr_free(expr);
+		break;
+	}
+
+	isl_id_to_ast_expr_free(id2expr);
+	return expr;
+error:
+	isl_ast_expr_free(expr);
+	isl_id_to_ast_expr_free(id2expr);
+	return NULL;
+}
+
+isl_ctx *isl_ast_node_get_ctx(__isl_keep isl_ast_node *node)
+{
+	return node ? node->ctx : NULL;
+}
+
+enum isl_ast_node_type isl_ast_node_get_type(__isl_keep isl_ast_node *node)
+{
+	return node ? node->type : isl_ast_node_error;
+}
+
+__isl_give isl_ast_node *isl_ast_node_alloc(isl_ctx *ctx,
+	enum isl_ast_node_type type)
+{
+	isl_ast_node *node;
+
+	node = isl_calloc_type(ctx, isl_ast_node);
+	if (!node)
+		return NULL;
+
+	node->ctx = ctx;
+	isl_ctx_ref(ctx);
+	node->ref = 1;
+	node->type = type;
+
+	return node;
+}
+
+/* Create an if node with the given guard.
+ *
+ * The then body needs to be filled in later.
+ */
+__isl_give isl_ast_node *isl_ast_node_alloc_if(__isl_take isl_ast_expr *guard)
+{
+	isl_ast_node *node;
+
+	if (!guard)
+		return NULL;
+
+	node = isl_ast_node_alloc(isl_ast_expr_get_ctx(guard), isl_ast_node_if);
+	if (!node)
+		goto error;
+	node->u.i.guard = guard;
+
+	return node;
+error:
+	isl_ast_expr_free(guard);
+	return NULL;
+}
+
+/* Create a for node with the given iterator.
+ *
+ * The remaining fields need to be filled in later.
+ */
+__isl_give isl_ast_node *isl_ast_node_alloc_for(__isl_take isl_id *id)
+{
+	isl_ast_node *node;
+	isl_ctx *ctx;
+
+	if (!id)
+		return NULL;
+
+	ctx = isl_id_get_ctx(id);
+	node = isl_ast_node_alloc(ctx, isl_ast_node_for);
+	if (!node)
+		goto error;
+
+	node->u.f.iterator = isl_ast_expr_from_id(id);
+	if (!node->u.f.iterator)
+		return isl_ast_node_free(node);
+
+	return node;
+error:
+	isl_id_free(id);
+	return NULL;
+}
+
+/* Create a user node evaluating "expr".
+ */
+__isl_give isl_ast_node *isl_ast_node_alloc_user(__isl_take isl_ast_expr *expr)
+{
+	isl_ctx *ctx;
+	isl_ast_node *node;
+
+	if (!expr)
+		return NULL;
+
+	ctx = isl_ast_expr_get_ctx(expr);
+	node = isl_ast_node_alloc(ctx, isl_ast_node_user);
+	if (!node)
+		goto error;
+
+	node->u.e.expr = expr;
+
+	return node;
+error:
+	isl_ast_expr_free(expr);
+	return NULL;
+}
+
+/* Create a block node with the given children.
+ */
+__isl_give isl_ast_node *isl_ast_node_alloc_block(
+	__isl_take isl_ast_node_list *list)
+{
+	isl_ast_node *node;
+	isl_ctx *ctx;
+
+	if (!list)
+		return NULL;
+
+	ctx = isl_ast_node_list_get_ctx(list);
+	node = isl_ast_node_alloc(ctx, isl_ast_node_block);
+	if (!node)
+		goto error;
+
+	node->u.b.children = list;
+
+	return node;
+error:
+	isl_ast_node_list_free(list);
+	return NULL;
+}
+
+/* Represent the given list of nodes as a single node, either by
+ * extract the node from a single element list or by creating
+ * a block node with the list of nodes as children.
+ */
+__isl_give isl_ast_node *isl_ast_node_from_ast_node_list(
+	__isl_take isl_ast_node_list *list)
+{
+	isl_ast_node *node;
+
+	if (isl_ast_node_list_n_ast_node(list) != 1)
+		return isl_ast_node_alloc_block(list);
+
+	node = isl_ast_node_list_get_ast_node(list, 0);
+	isl_ast_node_list_free(list);
+
+	return node;
+}
+
+__isl_give isl_ast_node *isl_ast_node_copy(__isl_keep isl_ast_node *node)
+{
+	if (!node)
+		return NULL;
+
+	node->ref++;
+	return node;
+}
+
+__isl_give isl_ast_node *isl_ast_node_dup(__isl_keep isl_ast_node *node)
+{
+	isl_ast_node *dup;
+
+	if (!node)
+		return NULL;
+
+	dup = isl_ast_node_alloc(isl_ast_node_get_ctx(node), node->type);
+	if (!dup)
+		return NULL;
+
+	switch (node->type) {
+	case isl_ast_node_if:
+		dup->u.i.guard = isl_ast_expr_copy(node->u.i.guard);
+		dup->u.i.then = isl_ast_node_copy(node->u.i.then);
+		dup->u.i.else_node = isl_ast_node_copy(node->u.i.else_node);
+		if (!dup->u.i.guard  || !dup->u.i.then ||
+		    (node->u.i.else_node && !dup->u.i.else_node))
+			return isl_ast_node_free(dup);
+		break;
+	case isl_ast_node_for:
+		dup->u.f.iterator = isl_ast_expr_copy(node->u.f.iterator);
+		dup->u.f.init = isl_ast_expr_copy(node->u.f.init);
+		dup->u.f.cond = isl_ast_expr_copy(node->u.f.cond);
+		dup->u.f.inc = isl_ast_expr_copy(node->u.f.inc);
+		dup->u.f.body = isl_ast_node_copy(node->u.f.body);
+		if (!dup->u.f.iterator || !dup->u.f.init || !dup->u.f.cond ||
+		    !dup->u.f.inc || !dup->u.f.body)
+			return isl_ast_node_free(dup);
+		break;
+	case isl_ast_node_block:
+		dup->u.b.children = isl_ast_node_list_copy(node->u.b.children);
+		if (!dup->u.b.children)
+			return isl_ast_node_free(dup);
+		break;
+	case isl_ast_node_user:
+		dup->u.e.expr = isl_ast_expr_copy(node->u.e.expr);
+		if (!dup->u.e.expr)
+			return isl_ast_node_free(dup);
+		break;
+	case isl_ast_node_error:
+		break;
+	}
+
+	return dup;
+}
+
+__isl_give isl_ast_node *isl_ast_node_cow(__isl_take isl_ast_node *node)
+{
+	if (!node)
+		return NULL;
+
+	if (node->ref == 1)
+		return node;
+	node->ref--;
+	return isl_ast_node_dup(node);
+}
+
+__isl_null isl_ast_node *isl_ast_node_free(__isl_take isl_ast_node *node)
+{
+	if (!node)
+		return NULL;
+
+	if (--node->ref > 0)
+		return NULL;
+
+	switch (node->type) {
+	case isl_ast_node_if:
+		isl_ast_expr_free(node->u.i.guard);
+		isl_ast_node_free(node->u.i.then);
+		isl_ast_node_free(node->u.i.else_node);
+		break;
+	case isl_ast_node_for:
+		isl_ast_expr_free(node->u.f.iterator);
+		isl_ast_expr_free(node->u.f.init);
+		isl_ast_expr_free(node->u.f.cond);
+		isl_ast_expr_free(node->u.f.inc);
+		isl_ast_node_free(node->u.f.body);
+		break;
+	case isl_ast_node_block:
+		isl_ast_node_list_free(node->u.b.children);
+		break;
+	case isl_ast_node_user:
+		isl_ast_expr_free(node->u.e.expr);
+		break;
+	case isl_ast_node_error:
+		break;
+	}
+
+	isl_id_free(node->annotation);
+	isl_ctx_deref(node->ctx);
+	free(node);
+
+	return NULL;
+}
+
+/* Replace the body of the for node "node" by "body".
+ */
+__isl_give isl_ast_node *isl_ast_node_for_set_body(
+	__isl_take isl_ast_node *node, __isl_take isl_ast_node *body)
+{
+	node = isl_ast_node_cow(node);
+	if (!node || !body)
+		goto error;
+	if (node->type != isl_ast_node_for)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not a for node", goto error);
+
+	isl_ast_node_free(node->u.f.body);
+	node->u.f.body = body;
+
+	return node;
+error:
+	isl_ast_node_free(node);
+	isl_ast_node_free(body);
+	return NULL;
+}
+
+__isl_give isl_ast_node *isl_ast_node_for_get_body(
+	__isl_keep isl_ast_node *node)
+{
+	if (!node)
+		return NULL;
+	if (node->type != isl_ast_node_for)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not a for node", return NULL);
+	return isl_ast_node_copy(node->u.f.body);
+}
+
+/* Mark the given for node as being degenerate.
+ */
+__isl_give isl_ast_node *isl_ast_node_for_mark_degenerate(
+	__isl_take isl_ast_node *node)
+{
+	node = isl_ast_node_cow(node);
+	if (!node)
+		return NULL;
+	node->u.f.degenerate = 1;
+	return node;
+}
+
+int isl_ast_node_for_is_degenerate(__isl_keep isl_ast_node *node)
+{
+	if (!node)
+		return -1;
+	if (node->type != isl_ast_node_for)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not a for node", return -1);
+	return node->u.f.degenerate;
+}
+
+__isl_give isl_ast_expr *isl_ast_node_for_get_iterator(
+	__isl_keep isl_ast_node *node)
+{
+	if (!node)
+		return NULL;
+	if (node->type != isl_ast_node_for)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not a for node", return NULL);
+	return isl_ast_expr_copy(node->u.f.iterator);
+}
+
+__isl_give isl_ast_expr *isl_ast_node_for_get_init(
+	__isl_keep isl_ast_node *node)
+{
+	if (!node)
+		return NULL;
+	if (node->type != isl_ast_node_for)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not a for node", return NULL);
+	return isl_ast_expr_copy(node->u.f.init);
+}
+
+/* Return the condition expression of the given for node.
+ *
+ * If the for node is degenerate, then the condition is not explicitly
+ * stored in the node.  Instead, it is constructed as
+ *
+ *	iterator <= init
+ */
+__isl_give isl_ast_expr *isl_ast_node_for_get_cond(
+	__isl_keep isl_ast_node *node)
+{
+	if (!node)
+		return NULL;
+	if (node->type != isl_ast_node_for)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not a for node", return NULL);
+	if (!node->u.f.degenerate)
+		return isl_ast_expr_copy(node->u.f.cond);
+
+	return isl_ast_expr_alloc_binary(isl_ast_op_le,
+				isl_ast_expr_copy(node->u.f.iterator),
+				isl_ast_expr_copy(node->u.f.init));
+}
+
+/* Return the increment of the given for node.
+ *
+ * If the for node is degenerate, then the increment is not explicitly
+ * stored in the node.  We simply return "1".
+ */
+__isl_give isl_ast_expr *isl_ast_node_for_get_inc(
+	__isl_keep isl_ast_node *node)
+{
+	if (!node)
+		return NULL;
+	if (node->type != isl_ast_node_for)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not a for node", return NULL);
+	if (!node->u.f.degenerate)
+		return isl_ast_expr_copy(node->u.f.inc);
+	return isl_ast_expr_alloc_int_si(isl_ast_node_get_ctx(node), 1);
+}
+
+/* Replace the then branch of the if node "node" by "child".
+ */
+__isl_give isl_ast_node *isl_ast_node_if_set_then(
+	__isl_take isl_ast_node *node, __isl_take isl_ast_node *child)
+{
+	node = isl_ast_node_cow(node);
+	if (!node || !child)
+		goto error;
+	if (node->type != isl_ast_node_if)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not an if node", goto error);
+
+	isl_ast_node_free(node->u.i.then);
+	node->u.i.then = child;
+
+	return node;
+error:
+	isl_ast_node_free(node);
+	isl_ast_node_free(child);
+	return NULL;
+}
+
+__isl_give isl_ast_node *isl_ast_node_if_get_then(
+	__isl_keep isl_ast_node *node)
+{
+	if (!node)
+		return NULL;
+	if (node->type != isl_ast_node_if)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not an if node", return NULL);
+	return isl_ast_node_copy(node->u.i.then);
+}
+
+int isl_ast_node_if_has_else(
+	__isl_keep isl_ast_node *node)
+{
+	if (!node)
+		return -1;
+	if (node->type != isl_ast_node_if)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not an if node", return -1);
+	return node->u.i.else_node != NULL;
+}
+
+__isl_give isl_ast_node *isl_ast_node_if_get_else(
+	__isl_keep isl_ast_node *node)
+{
+	if (!node)
+		return NULL;
+	if (node->type != isl_ast_node_if)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not an if node", return NULL);
+	return isl_ast_node_copy(node->u.i.else_node);
+}
+
+__isl_give isl_ast_expr *isl_ast_node_if_get_cond(
+	__isl_keep isl_ast_node *node)
+{
+	if (!node)
+		return NULL;
+	if (node->type != isl_ast_node_if)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not a guard node", return NULL);
+	return isl_ast_expr_copy(node->u.i.guard);
+}
+
+__isl_give isl_ast_node_list *isl_ast_node_block_get_children(
+	__isl_keep isl_ast_node *node)
+{
+	if (!node)
+		return NULL;
+	if (node->type != isl_ast_node_block)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not a block node", return NULL);
+	return isl_ast_node_list_copy(node->u.b.children);
+}
+
+__isl_give isl_ast_expr *isl_ast_node_user_get_expr(
+	__isl_keep isl_ast_node *node)
+{
+	if (!node)
+		return NULL;
+	if (node->type != isl_ast_node_user)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not a user node", return NULL);
+
+	return isl_ast_expr_copy(node->u.e.expr);
+}
+
+__isl_give isl_id *isl_ast_node_get_annotation(__isl_keep isl_ast_node *node)
+{
+	return node ? isl_id_copy(node->annotation) : NULL;
+}
+
+/* Replace node->annotation by "annotation".
+ */
+__isl_give isl_ast_node *isl_ast_node_set_annotation(
+	__isl_take isl_ast_node *node, __isl_take isl_id *annotation)
+{
+	node = isl_ast_node_cow(node);
+	if (!node || !annotation)
+		goto error;
+
+	isl_id_free(node->annotation);
+	node->annotation = annotation;
+
+	return node;
+error:
+	isl_id_free(annotation);
+	return isl_ast_node_free(node);
+}
+
+/* Textual C representation of the various operators.
+ */
+static char *op_str[] = {
+	[isl_ast_op_and] = "&&",
+	[isl_ast_op_and_then] = "&&",
+	[isl_ast_op_or] = "||",
+	[isl_ast_op_or_else] = "||",
+	[isl_ast_op_max] = "max",
+	[isl_ast_op_min] = "min",
+	[isl_ast_op_minus] = "-",
+	[isl_ast_op_add] = "+",
+	[isl_ast_op_sub] = "-",
+	[isl_ast_op_mul] = "*",
+	[isl_ast_op_pdiv_q] = "/",
+	[isl_ast_op_pdiv_r] = "%",
+	[isl_ast_op_zdiv_r] = "%",
+	[isl_ast_op_div] = "/",
+	[isl_ast_op_eq] = "==",
+	[isl_ast_op_le] = "<=",
+	[isl_ast_op_ge] = ">=",
+	[isl_ast_op_lt] = "<",
+	[isl_ast_op_gt] = ">",
+	[isl_ast_op_member] = ".",
+	[isl_ast_op_address_of] = "&"
+};
+
+/* Precedence in C of the various operators.
+ * Based on http://en.wikipedia.org/wiki/Operators_in_C_and_C++
+ * Lowest value means highest precedence.
+ */
+static int op_prec[] = {
+	[isl_ast_op_and] = 13,
+	[isl_ast_op_and_then] = 13,
+	[isl_ast_op_or] = 14,
+	[isl_ast_op_or_else] = 14,
+	[isl_ast_op_max] = 2,
+	[isl_ast_op_min] = 2,
+	[isl_ast_op_minus] = 3,
+	[isl_ast_op_add] = 6,
+	[isl_ast_op_sub] = 6,
+	[isl_ast_op_mul] = 5,
+	[isl_ast_op_div] = 5,
+	[isl_ast_op_fdiv_q] = 2,
+	[isl_ast_op_pdiv_q] = 5,
+	[isl_ast_op_pdiv_r] = 5,
+	[isl_ast_op_zdiv_r] = 5,
+	[isl_ast_op_cond] = 15,
+	[isl_ast_op_select] = 15,
+	[isl_ast_op_eq] = 9,
+	[isl_ast_op_le] = 8,
+	[isl_ast_op_ge] = 8,
+	[isl_ast_op_lt] = 8,
+	[isl_ast_op_gt] = 8,
+	[isl_ast_op_call] = 2,
+	[isl_ast_op_access] = 2,
+	[isl_ast_op_member] = 2,
+	[isl_ast_op_address_of] = 3
+};
+
+/* Is the operator left-to-right associative?
+ */
+static int op_left[] = {
+	[isl_ast_op_and] = 1,
+	[isl_ast_op_and_then] = 1,
+	[isl_ast_op_or] = 1,
+	[isl_ast_op_or_else] = 1,
+	[isl_ast_op_max] = 1,
+	[isl_ast_op_min] = 1,
+	[isl_ast_op_minus] = 0,
+	[isl_ast_op_add] = 1,
+	[isl_ast_op_sub] = 1,
+	[isl_ast_op_mul] = 1,
+	[isl_ast_op_div] = 1,
+	[isl_ast_op_fdiv_q] = 1,
+	[isl_ast_op_pdiv_q] = 1,
+	[isl_ast_op_pdiv_r] = 1,
+	[isl_ast_op_zdiv_r] = 1,
+	[isl_ast_op_cond] = 0,
+	[isl_ast_op_select] = 0,
+	[isl_ast_op_eq] = 1,
+	[isl_ast_op_le] = 1,
+	[isl_ast_op_ge] = 1,
+	[isl_ast_op_lt] = 1,
+	[isl_ast_op_gt] = 1,
+	[isl_ast_op_call] = 1,
+	[isl_ast_op_access] = 1,
+	[isl_ast_op_member] = 1,
+	[isl_ast_op_address_of] = 0
+};
+
+static int is_and(enum isl_ast_op_type op)
+{
+	return op == isl_ast_op_and || op == isl_ast_op_and_then;
+}
+
+static int is_or(enum isl_ast_op_type op)
+{
+	return op == isl_ast_op_or || op == isl_ast_op_or_else;
+}
+
+static int is_add_sub(enum isl_ast_op_type op)
+{
+	return op == isl_ast_op_add || op == isl_ast_op_sub;
+}
+
+static int is_div_mod(enum isl_ast_op_type op)
+{
+	return op == isl_ast_op_div ||
+	       op == isl_ast_op_pdiv_r ||
+	       op == isl_ast_op_zdiv_r;
+}
+
+/* Do we need/want parentheses around "expr" as a subexpression of
+ * an "op" operation?  If "left" is set, then "expr" is the left-most
+ * operand.
+ *
+ * We only need parentheses if "expr" represents an operation.
+ *
+ * If op has a higher precedence than expr->u.op.op, then we need
+ * parentheses.
+ * If op and expr->u.op.op have the same precedence, but the operations
+ * are performed in an order that is different from the associativity,
+ * then we need parentheses.
+ *
+ * An and inside an or technically does not require parentheses,
+ * but some compilers complain about that, so we add them anyway.
+ *
+ * Computations such as "a / b * c" and "a % b + c" can be somewhat
+ * difficult to read, so we add parentheses for those as well.
+ */
+static int sub_expr_need_parens(enum isl_ast_op_type op,
+	__isl_keep isl_ast_expr *expr, int left)
+{
+	if (expr->type != isl_ast_expr_op)
+		return 0;
+
+	if (op_prec[expr->u.op.op] > op_prec[op])
+		return 1;
+	if (op_prec[expr->u.op.op] == op_prec[op] && left != op_left[op])
+		return 1;
+
+	if (is_or(op) && is_and(expr->u.op.op))
+		return 1;
+	if (op == isl_ast_op_mul && expr->u.op.op != isl_ast_op_mul &&
+	    op_prec[expr->u.op.op] == op_prec[op])
+		return 1;
+	if (is_add_sub(op) && is_div_mod(expr->u.op.op))
+		return 1;
+
+	return 0;
+}
+
+/* Print "expr" as a subexpression of an "op" operation.
+ * If "left" is set, then "expr" is the left-most operand.
+ */
+static __isl_give isl_printer *print_sub_expr(__isl_take isl_printer *p,
+	enum isl_ast_op_type op, __isl_keep isl_ast_expr *expr, int left)
+{
+	int need_parens;
+
+	need_parens = sub_expr_need_parens(op, expr, left);
+
+	if (need_parens)
+		p = isl_printer_print_str(p, "(");
+	p = isl_printer_print_ast_expr(p, expr);
+	if (need_parens)
+		p = isl_printer_print_str(p, ")");
+	return p;
+}
+
+/* Print a min or max reduction "expr".
+ */
+static __isl_give isl_printer *print_min_max(__isl_take isl_printer *p,
+	__isl_keep isl_ast_expr *expr)
+{
+	int i = 0;
+
+	for (i = 1; i < expr->u.op.n_arg; ++i) {
+		p = isl_printer_print_str(p, op_str[expr->u.op.op]);
+		p = isl_printer_print_str(p, "(");
+	}
+	p = isl_printer_print_ast_expr(p, expr->u.op.args[0]);
+	for (i = 1; i < expr->u.op.n_arg; ++i) {
+		p = isl_printer_print_str(p, ", ");
+		p = isl_printer_print_ast_expr(p, expr->u.op.args[i]);
+		p = isl_printer_print_str(p, ")");
+	}
+
+	return p;
+}
+
+/* Print a function call "expr".
+ *
+ * The first argument represents the function to be called.
+ */
+static __isl_give isl_printer *print_call(__isl_take isl_printer *p,
+	__isl_keep isl_ast_expr *expr)
+{
+	int i = 0;
+
+	p = isl_printer_print_ast_expr(p, expr->u.op.args[0]);
+	p = isl_printer_print_str(p, "(");
+	for (i = 1; i < expr->u.op.n_arg; ++i) {
+		if (i != 1)
+			p = isl_printer_print_str(p, ", ");
+		p = isl_printer_print_ast_expr(p, expr->u.op.args[i]);
+	}
+	p = isl_printer_print_str(p, ")");
+
+	return p;
+}
+
+/* Print an array access "expr".
+ *
+ * The first argument represents the array being accessed.
+ */
+static __isl_give isl_printer *print_access(__isl_take isl_printer *p,
+	__isl_keep isl_ast_expr *expr)
+{
+	int i = 0;
+
+	p = isl_printer_print_ast_expr(p, expr->u.op.args[0]);
+	for (i = 1; i < expr->u.op.n_arg; ++i) {
+		p = isl_printer_print_str(p, "[");
+		p = isl_printer_print_ast_expr(p, expr->u.op.args[i]);
+		p = isl_printer_print_str(p, "]");
+	}
+
+	return p;
+}
+
+/* Print "expr" to "p".
+ *
+ * If we are printing in isl format, then we also print an indication
+ * of the size of the expression (if it was computed).
+ */
+__isl_give isl_printer *isl_printer_print_ast_expr(__isl_take isl_printer *p,
+	__isl_keep isl_ast_expr *expr)
+{
+	if (!p)
+		return NULL;
+	if (!expr)
+		return isl_printer_free(p);
+
+	switch (expr->type) {
+	case isl_ast_expr_op:
+		if (expr->u.op.op == isl_ast_op_call) {
+			p = print_call(p, expr);
+			break;
+		}
+		if (expr->u.op.op == isl_ast_op_access) {
+			p = print_access(p, expr);
+			break;
+		}
+		if (expr->u.op.n_arg == 1) {
+			p = isl_printer_print_str(p, op_str[expr->u.op.op]);
+			p = print_sub_expr(p, expr->u.op.op,
+						expr->u.op.args[0], 0);
+			break;
+		}
+		if (expr->u.op.op == isl_ast_op_fdiv_q) {
+			p = isl_printer_print_str(p, "floord(");
+			p = isl_printer_print_ast_expr(p, expr->u.op.args[0]);
+			p = isl_printer_print_str(p, ", ");
+			p = isl_printer_print_ast_expr(p, expr->u.op.args[1]);
+			p = isl_printer_print_str(p, ")");
+			break;
+		}
+		if (expr->u.op.op == isl_ast_op_max ||
+		    expr->u.op.op == isl_ast_op_min) {
+			p = print_min_max(p, expr);
+			break;
+		}
+		if (expr->u.op.op == isl_ast_op_cond ||
+		    expr->u.op.op == isl_ast_op_select) {
+			p = isl_printer_print_ast_expr(p, expr->u.op.args[0]);
+			p = isl_printer_print_str(p, " ? ");
+			p = isl_printer_print_ast_expr(p, expr->u.op.args[1]);
+			p = isl_printer_print_str(p, " : ");
+			p = isl_printer_print_ast_expr(p, expr->u.op.args[2]);
+			break;
+		}
+		if (expr->u.op.n_arg != 2)
+			isl_die(isl_printer_get_ctx(p), isl_error_internal,
+				"operation should have two arguments",
+				goto error);
+		p = print_sub_expr(p, expr->u.op.op, expr->u.op.args[0], 1);
+		if (expr->u.op.op != isl_ast_op_member)
+			p = isl_printer_print_str(p, " ");
+		p = isl_printer_print_str(p, op_str[expr->u.op.op]);
+		if (expr->u.op.op != isl_ast_op_member)
+			p = isl_printer_print_str(p, " ");
+		p = print_sub_expr(p, expr->u.op.op, expr->u.op.args[1], 0);
+		break;
+	case isl_ast_expr_id:
+		p = isl_printer_print_str(p, isl_id_get_name(expr->u.id));
+		break;
+	case isl_ast_expr_int:
+		p = isl_printer_print_val(p, expr->u.v);
+		break;
+	case isl_ast_expr_error:
+		break;
+	}
+
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+/* Print "node" to "p" in "isl format".
+ */
+static __isl_give isl_printer *print_ast_node_isl(__isl_take isl_printer *p,
+	__isl_keep isl_ast_node *node)
+{
+	p = isl_printer_print_str(p, "(");
+	switch (node->type) {
+	case isl_ast_node_for:
+		if (node->u.f.degenerate) {
+			p = isl_printer_print_ast_expr(p, node->u.f.init);
+		} else {
+			p = isl_printer_print_str(p, "init: ");
+			p = isl_printer_print_ast_expr(p, node->u.f.init);
+			p = isl_printer_print_str(p, ", ");
+			p = isl_printer_print_str(p, "cond: ");
+			p = isl_printer_print_ast_expr(p, node->u.f.cond);
+			p = isl_printer_print_str(p, ", ");
+			p = isl_printer_print_str(p, "inc: ");
+			p = isl_printer_print_ast_expr(p, node->u.f.inc);
+		}
+		if (node->u.f.body) {
+			p = isl_printer_print_str(p, ", ");
+			p = isl_printer_print_str(p, "body: ");
+			p = isl_printer_print_ast_node(p, node->u.f.body);
+		}
+		break;
+	case isl_ast_node_user:
+		p = isl_printer_print_ast_expr(p, node->u.e.expr);
+		break;
+	case isl_ast_node_if:
+		p = isl_printer_print_str(p, "guard: ");
+		p = isl_printer_print_ast_expr(p, node->u.i.guard);
+		if (node->u.i.then) {
+			p = isl_printer_print_str(p, ", ");
+			p = isl_printer_print_str(p, "then: ");
+			p = isl_printer_print_ast_node(p, node->u.i.then);
+		}
+		if (node->u.i.else_node) {
+			p = isl_printer_print_str(p, ", ");
+			p = isl_printer_print_str(p, "else: ");
+			p = isl_printer_print_ast_node(p, node->u.i.else_node);
+		}
+		break;
+	case isl_ast_node_block:
+		p = isl_printer_print_ast_node_list(p, node->u.b.children);
+		break;
+	default:
+		break;
+	}
+	p = isl_printer_print_str(p, ")");
+	return p;
+}
+
+/* Do we need to print a block around the body "node" of a for or if node?
+ *
+ * If the node is a block, then we need to print a block.
+ * Also if the node is a degenerate for then we will print it as
+ * an assignment followed by the body of the for loop, so we need a block
+ * as well.
+ * If the node is an if node with an else, then we print a block
+ * to avoid spurious dangling else warnings emitted by some compilers.
+ * If the ast_always_print_block option has been set, then we print a block.
+ */
+static int need_block(__isl_keep isl_ast_node *node)
+{
+	isl_ctx *ctx;
+
+	if (node->type == isl_ast_node_block)
+		return 1;
+	if (node->type == isl_ast_node_for && node->u.f.degenerate)
+		return 1;
+	if (node->type == isl_ast_node_if && node->u.i.else_node)
+		return 1;
+
+	ctx = isl_ast_node_get_ctx(node);
+	return isl_options_get_ast_always_print_block(ctx);
+}
+
+static __isl_give isl_printer *print_ast_node_c(__isl_take isl_printer *p,
+	__isl_keep isl_ast_node *node,
+	__isl_keep isl_ast_print_options *options, int in_block, int in_list);
+static __isl_give isl_printer *print_if_c(__isl_take isl_printer *p,
+	__isl_keep isl_ast_node *node,
+	__isl_keep isl_ast_print_options *options, int new_line);
+
+/* Print the body "node" of a for or if node.
+ * If "else_node" is set, then it is printed as well.
+ *
+ * We first check if we need to print out a block.
+ * We always print out a block if there is an else node to make
+ * sure that the else node is matched to the correct if node.
+ *
+ * If the else node is itself an if, then we print it as
+ *
+ *	} else if (..)
+ *
+ * Otherwise the else node is printed as
+ *
+ *	} else
+ *	  node
+ */
+static __isl_give isl_printer *print_body_c(__isl_take isl_printer *p,
+	__isl_keep isl_ast_node *node, __isl_keep isl_ast_node *else_node,
+	__isl_keep isl_ast_print_options *options)
+{
+	if (!node)
+		return isl_printer_free(p);
+
+	if (!else_node && !need_block(node)) {
+		p = isl_printer_end_line(p);
+		p = isl_printer_indent(p, 2);
+		p = isl_ast_node_print(node, p,
+					isl_ast_print_options_copy(options));
+		p = isl_printer_indent(p, -2);
+		return p;
+	}
+
+	p = isl_printer_print_str(p, " {");
+	p = isl_printer_end_line(p);
+	p = isl_printer_indent(p, 2);
+	p = print_ast_node_c(p, node, options, 1, 0);
+	p = isl_printer_indent(p, -2);
+	p = isl_printer_start_line(p);
+	p = isl_printer_print_str(p, "}");
+	if (else_node) {
+		if (else_node->type == isl_ast_node_if) {
+			p = isl_printer_print_str(p, " else ");
+			p = print_if_c(p, else_node, options, 0);
+		} else {
+			p = isl_printer_print_str(p, " else");
+			p = print_body_c(p, else_node, NULL, options);
+		}
+	} else
+		p = isl_printer_end_line(p);
+
+	return p;
+}
+
+/* Print the start of a compound statement.
+ */
+static __isl_give isl_printer *start_block(__isl_take isl_printer *p)
+{
+	p = isl_printer_start_line(p);
+	p = isl_printer_print_str(p, "{");
+	p = isl_printer_end_line(p);
+	p = isl_printer_indent(p, 2);
+
+	return p;
+}
+
+/* Print the end of a compound statement.
+ */
+static __isl_give isl_printer *end_block(__isl_take isl_printer *p)
+{
+	p = isl_printer_indent(p, -2);
+	p = isl_printer_start_line(p);
+	p = isl_printer_print_str(p, "}");
+	p = isl_printer_end_line(p);
+
+	return p;
+}
+
+/* Print the for node "node".
+ *
+ * If the for node is degenerate, it is printed as
+ *
+ *	type iterator = init;
+ *	body
+ *
+ * Otherwise, it is printed as
+ *
+ *	for (type iterator = init; cond; iterator += inc)
+ *		body
+ *
+ * "in_block" is set if we are currently inside a block.
+ * "in_list" is set if the current node is not alone in the block.
+ * If we are not in a block or if the current not is not alone in the block
+ * then we print a block around a degenerate for loop such that the variable
+ * declaration will not conflict with any potential other declaration
+ * of the same variable.
+ */
+static __isl_give isl_printer *print_for_c(__isl_take isl_printer *p,
+	__isl_keep isl_ast_node *node,
+	__isl_keep isl_ast_print_options *options, int in_block, int in_list)
+{
+	isl_id *id;
+	const char *name;
+	const char *type;
+
+	type = isl_options_get_ast_iterator_type(isl_printer_get_ctx(p));
+	if (!node->u.f.degenerate) {
+		id = isl_ast_expr_get_id(node->u.f.iterator);
+		name = isl_id_get_name(id);
+		isl_id_free(id);
+		p = isl_printer_start_line(p);
+		p = isl_printer_print_str(p, "for (");
+		p = isl_printer_print_str(p, type);
+		p = isl_printer_print_str(p, " ");
+		p = isl_printer_print_str(p, name);
+		p = isl_printer_print_str(p, " = ");
+		p = isl_printer_print_ast_expr(p, node->u.f.init);
+		p = isl_printer_print_str(p, "; ");
+		p = isl_printer_print_ast_expr(p, node->u.f.cond);
+		p = isl_printer_print_str(p, "; ");
+		p = isl_printer_print_str(p, name);
+		p = isl_printer_print_str(p, " += ");
+		p = isl_printer_print_ast_expr(p, node->u.f.inc);
+		p = isl_printer_print_str(p, ")");
+		p = print_body_c(p, node->u.f.body, NULL, options);
+	} else {
+		id = isl_ast_expr_get_id(node->u.f.iterator);
+		name = isl_id_get_name(id);
+		isl_id_free(id);
+		if (!in_block || in_list)
+			p = start_block(p);
+		p = isl_printer_start_line(p);
+		p = isl_printer_print_str(p, type);
+		p = isl_printer_print_str(p, " ");
+		p = isl_printer_print_str(p, name);
+		p = isl_printer_print_str(p, " = ");
+		p = isl_printer_print_ast_expr(p, node->u.f.init);
+		p = isl_printer_print_str(p, ";");
+		p = isl_printer_end_line(p);
+		p = print_ast_node_c(p, node->u.f.body, options, 1, 0);
+		if (!in_block || in_list)
+			p = end_block(p);
+	}
+
+	return p;
+}
+
+/* Print the if node "node".
+ * If "new_line" is set then the if node should be printed on a new line.
+ */
+static __isl_give isl_printer *print_if_c(__isl_take isl_printer *p,
+	__isl_keep isl_ast_node *node,
+	__isl_keep isl_ast_print_options *options, int new_line)
+{
+	if (new_line)
+		p = isl_printer_start_line(p);
+	p = isl_printer_print_str(p, "if (");
+	p = isl_printer_print_ast_expr(p, node->u.i.guard);
+	p = isl_printer_print_str(p, ")");
+	p = print_body_c(p, node->u.i.then, node->u.i.else_node, options);
+
+	return p;
+}
+
+/* Print the "node" to "p".
+ *
+ * "in_block" is set if we are currently inside a block.
+ * If so, we do not print a block around the children of a block node.
+ * We do this to avoid an extra block around the body of a degenerate
+ * for node.
+ *
+ * "in_list" is set if the current node is not alone in the block.
+ */
+static __isl_give isl_printer *print_ast_node_c(__isl_take isl_printer *p,
+	__isl_keep isl_ast_node *node,
+	__isl_keep isl_ast_print_options *options, int in_block, int in_list)
+{
+	switch (node->type) {
+	case isl_ast_node_for:
+		if (options->print_for)
+			return options->print_for(p,
+					isl_ast_print_options_copy(options),
+					node, options->print_for_user);
+		p = print_for_c(p, node, options, in_block, in_list);
+		break;
+	case isl_ast_node_if:
+		p = print_if_c(p, node, options, 1);
+		break;
+	case isl_ast_node_block:
+		if (!in_block)
+			p = start_block(p);
+		p = isl_ast_node_list_print(node->u.b.children, p, options);
+		if (!in_block)
+			p = end_block(p);
+		break;
+	case isl_ast_node_user:
+		if (options->print_user)
+			return options->print_user(p,
+					isl_ast_print_options_copy(options),
+					node, options->print_user_user);
+		p = isl_printer_start_line(p);
+		p = isl_printer_print_ast_expr(p, node->u.e.expr);
+		p = isl_printer_print_str(p, ";");
+		p = isl_printer_end_line(p);
+		break;
+	case isl_ast_node_error:
+		break;
+	}
+	return p;
+}
+
+/* Print the for node "node" to "p".
+ */
+__isl_give isl_printer *isl_ast_node_for_print(__isl_keep isl_ast_node *node,
+	__isl_take isl_printer *p, __isl_take isl_ast_print_options *options)
+{
+	if (!node || !options)
+		goto error;
+	if (node->type != isl_ast_node_for)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not a for node", goto error);
+	p = print_for_c(p, node, options, 0, 0);
+	isl_ast_print_options_free(options);
+	return p;
+error:
+	isl_ast_print_options_free(options);
+	isl_printer_free(p);
+	return NULL;
+}
+
+/* Print the if node "node" to "p".
+ */
+__isl_give isl_printer *isl_ast_node_if_print(__isl_keep isl_ast_node *node,
+	__isl_take isl_printer *p, __isl_take isl_ast_print_options *options)
+{
+	if (!node || !options)
+		goto error;
+	if (node->type != isl_ast_node_if)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_invalid,
+			"not an if node", goto error);
+	p = print_if_c(p, node, options, 1);
+	isl_ast_print_options_free(options);
+	return p;
+error:
+	isl_ast_print_options_free(options);
+	isl_printer_free(p);
+	return NULL;
+}
+
+/* Print "node" to "p".
+ */
+__isl_give isl_printer *isl_ast_node_print(__isl_keep isl_ast_node *node,
+	__isl_take isl_printer *p, __isl_take isl_ast_print_options *options)
+{
+	if (!options || !node)
+		goto error;
+	p = print_ast_node_c(p, node, options, 0, 0);
+	isl_ast_print_options_free(options);
+	return p;
+error:
+	isl_ast_print_options_free(options);
+	isl_printer_free(p);
+	return NULL;
+}
+
+/* Print "node" to "p".
+ */
+__isl_give isl_printer *isl_printer_print_ast_node(__isl_take isl_printer *p,
+	__isl_keep isl_ast_node *node)
+{
+	int format;
+	isl_ast_print_options *options;
+
+	if (!p)
+		return NULL;
+
+	format = isl_printer_get_output_format(p);
+	switch (format) {
+	case ISL_FORMAT_ISL:
+		p = print_ast_node_isl(p, node);
+		break;
+	case ISL_FORMAT_C:
+		options = isl_ast_print_options_alloc(isl_printer_get_ctx(p));
+		p = isl_ast_node_print(node, p, options);
+		break;
+	default:
+		isl_die(isl_printer_get_ctx(p), isl_error_unsupported,
+			"output format not supported for ast_node",
+			return isl_printer_free(p));
+	}
+
+	return p;
+}
+
+/* Print the list of nodes "list" to "p".
+ */
+__isl_give isl_printer *isl_ast_node_list_print(
+	__isl_keep isl_ast_node_list *list, __isl_take isl_printer *p,
+	__isl_keep isl_ast_print_options *options)
+{
+	int i;
+
+	if (!p || !list || !options)
+		return isl_printer_free(p);
+
+	for (i = 0; i < list->n; ++i)
+		p = print_ast_node_c(p, list->p[i], options, 1, 1);
+
+	return p;
+}
+
+#define ISL_AST_MACRO_FLOORD	(1 << 0)
+#define ISL_AST_MACRO_MIN	(1 << 1)
+#define ISL_AST_MACRO_MAX	(1 << 2)
+#define ISL_AST_MACRO_ALL	(ISL_AST_MACRO_FLOORD | \
+				 ISL_AST_MACRO_MIN | \
+				 ISL_AST_MACRO_MAX)
+
+/* If "expr" contains an isl_ast_op_min, isl_ast_op_max or isl_ast_op_fdiv_q
+ * then set the corresponding bit in "macros".
+ */
+static int ast_expr_required_macros(__isl_keep isl_ast_expr *expr, int macros)
+{
+	int i;
+
+	if (macros == ISL_AST_MACRO_ALL)
+		return macros;
+
+	if (expr->type != isl_ast_expr_op)
+		return macros;
+
+	if (expr->u.op.op == isl_ast_op_min)
+		macros |= ISL_AST_MACRO_MIN;
+	if (expr->u.op.op == isl_ast_op_max)
+		macros |= ISL_AST_MACRO_MAX;
+	if (expr->u.op.op == isl_ast_op_fdiv_q)
+		macros |= ISL_AST_MACRO_FLOORD;
+
+	for (i = 0; i < expr->u.op.n_arg; ++i)
+		macros = ast_expr_required_macros(expr->u.op.args[i], macros);
+
+	return macros;
+}
+
+static int ast_node_list_required_macros(__isl_keep isl_ast_node_list *list,
+	int macros);
+
+/* If "node" contains an isl_ast_op_min, isl_ast_op_max or isl_ast_op_fdiv_q
+ * then set the corresponding bit in "macros".
+ */
+static int ast_node_required_macros(__isl_keep isl_ast_node *node, int macros)
+{
+	if (macros == ISL_AST_MACRO_ALL)
+		return macros;
+
+	switch (node->type) {
+	case isl_ast_node_for:
+		macros = ast_expr_required_macros(node->u.f.init, macros);
+		if (!node->u.f.degenerate) {
+			macros = ast_expr_required_macros(node->u.f.cond,
+								macros);
+			macros = ast_expr_required_macros(node->u.f.inc,
+								macros);
+		}
+		macros = ast_node_required_macros(node->u.f.body, macros);
+		break;
+	case isl_ast_node_if:
+		macros = ast_expr_required_macros(node->u.i.guard, macros);
+		macros = ast_node_required_macros(node->u.i.then, macros);
+		if (node->u.i.else_node)
+			macros = ast_node_required_macros(node->u.i.else_node,
+								macros);
+		break;
+	case isl_ast_node_block:
+		macros = ast_node_list_required_macros(node->u.b.children,
+							macros);
+		break;
+	case isl_ast_node_user:
+		macros = ast_expr_required_macros(node->u.e.expr, macros);
+		break;
+	case isl_ast_node_error:
+		break;
+	}
+
+	return macros;
+}
+
+/* If "list" contains an isl_ast_op_min, isl_ast_op_max or isl_ast_op_fdiv_q
+ * then set the corresponding bit in "macros".
+ */
+static int ast_node_list_required_macros(__isl_keep isl_ast_node_list *list,
+	int macros)
+{
+	int i;
+
+	for (i = 0; i < list->n; ++i)
+		macros = ast_node_required_macros(list->p[i], macros);
+
+	return macros;
+}
+
+/* Print a macro definition for the operator "type".
+ */
+__isl_give isl_printer *isl_ast_op_type_print_macro(
+	enum isl_ast_op_type type, __isl_take isl_printer *p)
+{
+	switch (type) {
+	case isl_ast_op_min:
+		p = isl_printer_start_line(p);
+		p = isl_printer_print_str(p,
+			"#define min(x,y)    ((x) < (y) ? (x) : (y))");
+		p = isl_printer_end_line(p);
+		break;
+	case isl_ast_op_max:
+		p = isl_printer_start_line(p);
+		p = isl_printer_print_str(p,
+			"#define max(x,y)    ((x) > (y) ? (x) : (y))");
+		p = isl_printer_end_line(p);
+		break;
+	case isl_ast_op_fdiv_q:
+		p = isl_printer_start_line(p);
+		p = isl_printer_print_str(p,
+			"#define floord(n,d) "
+			"(((n)<0) ? -((-(n)+(d)-1)/(d)) : (n)/(d))");
+		p = isl_printer_end_line(p);
+		break;
+	default:
+		break;
+	}
+
+	return p;
+}
+
+/* Call "fn" for each type of operation that appears in "node"
+ * and that requires a macro definition.
+ */
+int isl_ast_node_foreach_ast_op_type(__isl_keep isl_ast_node *node,
+	int (*fn)(enum isl_ast_op_type type, void *user), void *user)
+{
+	int macros;
+
+	if (!node)
+		return -1;
+
+	macros = ast_node_required_macros(node, 0);
+
+	if (macros & ISL_AST_MACRO_MIN && fn(isl_ast_op_min, user) < 0)
+		return -1;
+	if (macros & ISL_AST_MACRO_MAX && fn(isl_ast_op_max, user) < 0)
+		return -1;
+	if (macros & ISL_AST_MACRO_FLOORD && fn(isl_ast_op_fdiv_q, user) < 0)
+		return -1;
+
+	return 0;
+}
+
+static int ast_op_type_print_macro(enum isl_ast_op_type type, void *user)
+{
+	isl_printer **p = user;
+
+	*p = isl_ast_op_type_print_macro(type, *p);
+
+	return 0;
+}
+
+/* Print macro definitions for all the macros used in the result
+ * of printing "node.
+ */
+__isl_give isl_printer *isl_ast_node_print_macros(
+	__isl_keep isl_ast_node *node, __isl_take isl_printer *p)
+{
+	if (isl_ast_node_foreach_ast_op_type(node,
+					    &ast_op_type_print_macro, &p) < 0)
+		return isl_printer_free(p);
+	return p;
+}
diff --git a/polly/lib/External/isl/isl_ast_build.c b/polly/lib/External/isl/isl_ast_build.c
new file mode 100644
index 00000000000..61576bb0a51
--- /dev/null
+++ b/polly/lib/External/isl/isl_ast_build.c
@@ -0,0 +1,2162 @@
+/*
+ * Copyright 2012      Ecole Normale Superieure
+ * Copyright 2014      INRIA Rocquencourt
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege,
+ * Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
+ * B.P. 105 - 78153 Le Chesnay, France
+ */
+
+#include <isl/map.h>
+#include <isl/aff.h>
+#include <isl/map.h>
+#include <isl_ast_build_private.h>
+#include <isl_ast_private.h>
+
+/* Construct a map that isolates the current dimension.
+ *
+ * Essentially, the current dimension of "set" is moved to the single output
+ * dimension in the result, with the current dimension in the domain replaced
+ * by an unconstrained variable.
+ */
+__isl_give isl_map *isl_ast_build_map_to_iterator(
+	__isl_keep isl_ast_build *build, __isl_take isl_set *set)
+{
+	isl_map *map;
+
+	map = isl_map_from_domain(set);
+	map = isl_map_add_dims(map, isl_dim_out, 1);
+
+	if (!build)
+		return isl_map_free(map);
+
+	map = isl_map_equate(map, isl_dim_in, build->depth, isl_dim_out, 0);
+	map = isl_map_eliminate(map, isl_dim_in, build->depth, 1);
+
+	return map;
+}
+
+/* Initialize the information derived during the AST generation to default
+ * values for a schedule domain in "space".
+ *
+ * We also check that the remaining fields are not NULL so that
+ * the calling functions don't have to perform this test.
+ */
+static __isl_give isl_ast_build *isl_ast_build_init_derived(
+	__isl_take isl_ast_build *build, __isl_take isl_space *space)
+{
+	isl_ctx *ctx;
+	isl_vec *strides;
+
+	build = isl_ast_build_cow(build);
+	if (!build || !build->domain)
+		goto error;
+
+	ctx = isl_ast_build_get_ctx(build);
+	strides = isl_vec_alloc(ctx, isl_space_dim(space, isl_dim_set));
+	strides = isl_vec_set_si(strides, 1);
+
+	isl_vec_free(build->strides);
+	build->strides = strides;
+
+	space = isl_space_map_from_set(space);
+	isl_multi_aff_free(build->offsets);
+	build->offsets = isl_multi_aff_zero(isl_space_copy(space));
+	isl_multi_aff_free(build->values);
+	build->values = isl_multi_aff_identity(space);
+
+	if (!build->iterators || !build->domain || !build->generated ||
+	    !build->pending || !build->values ||
+	    !build->strides || !build->offsets || !build->options)
+		return isl_ast_build_free(build);
+
+	return build;
+error:
+	isl_space_free(space);
+	return isl_ast_build_free(build);
+}
+
+/* Return an isl_id called "c%d", with "%d" set to "i".
+ * If an isl_id with such a name already appears among the parameters
+ * in build->domain, then adjust the name to "c%d_%d".
+ */
+static __isl_give isl_id *generate_name(isl_ctx *ctx, int i,
+	__isl_keep isl_ast_build *build)
+{
+	int j;
+	char name[16];
+	isl_set *dom = build->domain;
+
+	snprintf(name, sizeof(name), "c%d", i);
+	j = 0;
+	while (isl_set_find_dim_by_name(dom, isl_dim_param, name) >= 0)
+		snprintf(name, sizeof(name), "c%d_%d", i, j++);
+	return isl_id_alloc(ctx, name, NULL);
+}
+
+/* Create an isl_ast_build with "set" as domain.
+ *
+ * The input set is usually a parameter domain, but we currently allow it to
+ * be any kind of set.  We set the domain of the returned isl_ast_build
+ * to "set" and initialize all the other fields to default values.
+ */
+__isl_give isl_ast_build *isl_ast_build_from_context(__isl_take isl_set *set)
+{
+	int i, n;
+	isl_ctx *ctx;
+	isl_space *space;
+	isl_ast_build *build;
+
+	set = isl_set_compute_divs(set);
+	if (!set)
+		return NULL;
+
+	ctx = isl_set_get_ctx(set);
+
+	build = isl_calloc_type(ctx, isl_ast_build);
+	if (!build)
+		goto error;
+
+	build->ref = 1;
+	build->domain = set;
+	build->generated = isl_set_copy(build->domain);
+	build->pending = isl_set_universe(isl_set_get_space(build->domain));
+	build->options = isl_union_map_empty(isl_space_params_alloc(ctx, 0));
+	n = isl_set_dim(set, isl_dim_set);
+	build->depth = n;
+	build->iterators = isl_id_list_alloc(ctx, n);
+	for (i = 0; i < n; ++i) {
+		isl_id *id;
+		if (isl_set_has_dim_id(set, isl_dim_set, i))
+			id = isl_set_get_dim_id(set, isl_dim_set, i);
+		else
+			id = generate_name(ctx, i, build);
+		build->iterators = isl_id_list_add(build->iterators, id);
+	}
+	space = isl_set_get_space(set);
+	if (isl_space_is_params(space))
+		space = isl_space_set_from_params(space);
+
+	return isl_ast_build_init_derived(build, space);
+error:
+	isl_set_free(set);
+	return NULL;
+}
+
+__isl_give isl_ast_build *isl_ast_build_copy(__isl_keep isl_ast_build *build)
+{
+	if (!build)
+		return NULL;
+
+	build->ref++;
+	return build;
+}
+
+__isl_give isl_ast_build *isl_ast_build_dup(__isl_keep isl_ast_build *build)
+{
+	isl_ctx *ctx;
+	isl_ast_build *dup;
+
+	if (!build)
+		return NULL;
+
+	ctx = isl_ast_build_get_ctx(build);
+	dup = isl_calloc_type(ctx, isl_ast_build);
+	if (!dup)
+		return NULL;
+
+	dup->ref = 1;
+	dup->outer_pos = build->outer_pos;
+	dup->depth = build->depth;
+	dup->iterators = isl_id_list_copy(build->iterators);
+	dup->domain = isl_set_copy(build->domain);
+	dup->generated = isl_set_copy(build->generated);
+	dup->pending = isl_set_copy(build->pending);
+	dup->values = isl_multi_aff_copy(build->values);
+	dup->value = isl_pw_aff_copy(build->value);
+	dup->strides = isl_vec_copy(build->strides);
+	dup->offsets = isl_multi_aff_copy(build->offsets);
+	dup->executed = isl_union_map_copy(build->executed);
+	dup->single_valued = build->single_valued;
+	dup->options = isl_union_map_copy(build->options);
+	dup->at_each_domain = build->at_each_domain;
+	dup->at_each_domain_user = build->at_each_domain_user;
+	dup->before_each_for = build->before_each_for;
+	dup->before_each_for_user = build->before_each_for_user;
+	dup->after_each_for = build->after_each_for;
+	dup->after_each_for_user = build->after_each_for_user;
+	dup->create_leaf = build->create_leaf;
+	dup->create_leaf_user = build->create_leaf_user;
+
+	if (!dup->iterators || !dup->domain || !dup->generated ||
+	    !dup->pending || !dup->values ||
+	    !dup->strides || !dup->offsets || !dup->options ||
+	    (build->executed && !dup->executed) ||
+	    (build->value && !dup->value))
+		return isl_ast_build_free(dup);
+
+	return dup;
+}
+
+/* Align the parameters of "build" to those of "model", introducing
+ * additional parameters if needed.
+ */
+__isl_give isl_ast_build *isl_ast_build_align_params(
+	__isl_take isl_ast_build *build, __isl_take isl_space *model)
+{
+	build = isl_ast_build_cow(build);
+	if (!build)
+		goto error;
+
+	build->domain = isl_set_align_params(build->domain,
+						isl_space_copy(model));
+	build->generated = isl_set_align_params(build->generated,
+						isl_space_copy(model));
+	build->pending = isl_set_align_params(build->pending,
+						isl_space_copy(model));
+	build->values = isl_multi_aff_align_params(build->values,
+						isl_space_copy(model));
+	build->offsets = isl_multi_aff_align_params(build->offsets,
+						isl_space_copy(model));
+	build->options = isl_union_map_align_params(build->options,
+						isl_space_copy(model));
+	isl_space_free(model);
+
+	if (!build->domain || !build->values || !build->offsets ||
+	    !build->options)
+		return isl_ast_build_free(build);
+
+	return build;
+error:
+	isl_space_free(model);
+	return NULL;
+}
+
+__isl_give isl_ast_build *isl_ast_build_cow(__isl_take isl_ast_build *build)
+{
+	if (!build)
+		return NULL;
+
+	if (build->ref == 1)
+		return build;
+	build->ref--;
+	return isl_ast_build_dup(build);
+}
+
+__isl_null isl_ast_build *isl_ast_build_free(
+	__isl_take isl_ast_build *build)
+{
+	if (!build)
+		return NULL;
+
+	if (--build->ref > 0)
+		return NULL;
+
+	isl_id_list_free(build->iterators);
+	isl_set_free(build->domain);
+	isl_set_free(build->generated);
+	isl_set_free(build->pending);
+	isl_multi_aff_free(build->values);
+	isl_pw_aff_free(build->value);
+	isl_vec_free(build->strides);
+	isl_multi_aff_free(build->offsets);
+	isl_multi_aff_free(build->schedule_map);
+	isl_union_map_free(build->executed);
+	isl_union_map_free(build->options);
+
+	free(build);
+
+	return NULL;
+}
+
+isl_ctx *isl_ast_build_get_ctx(__isl_keep isl_ast_build *build)
+{
+	return build ? isl_set_get_ctx(build->domain) : NULL;
+}
+
+/* Replace build->options by "options".
+ */
+__isl_give isl_ast_build *isl_ast_build_set_options(
+	__isl_take isl_ast_build *build, __isl_take isl_union_map *options)
+{
+	build = isl_ast_build_cow(build);
+
+	if (!build || !options)
+		goto error;
+
+	isl_union_map_free(build->options);
+	build->options = options;
+
+	return build;
+error:
+	isl_union_map_free(options);
+	return isl_ast_build_free(build);
+}
+
+/* Set the iterators for the next code generation.
+ *
+ * If we still have some iterators left from the previous code generation
+ * (if any) or if iterators have already been set by a previous
+ * call to this function, then we remove them first.
+ */
+__isl_give isl_ast_build *isl_ast_build_set_iterators(
+	__isl_take isl_ast_build *build, __isl_take isl_id_list *iterators)
+{
+	int dim, n_it;
+
+	build = isl_ast_build_cow(build);
+	if (!build)
+		goto error;
+
+	dim = isl_set_dim(build->domain, isl_dim_set);
+	n_it = isl_id_list_n_id(build->iterators);
+	if (n_it < dim)
+		isl_die(isl_ast_build_get_ctx(build), isl_error_internal,
+			"isl_ast_build in inconsistent state", goto error);
+	if (n_it > dim)
+		build->iterators = isl_id_list_drop(build->iterators,
+							dim, n_it - dim);
+	build->iterators = isl_id_list_concat(build->iterators, iterators);
+	if (!build->iterators)
+		return isl_ast_build_free(build);
+
+	return build;
+error:
+	isl_id_list_free(iterators);
+	return isl_ast_build_free(build);
+}
+
+/* Set the "at_each_domain" callback of "build" to "fn".
+ */
+__isl_give isl_ast_build *isl_ast_build_set_at_each_domain(
+	__isl_take isl_ast_build *build,
+	__isl_give isl_ast_node *(*fn)(__isl_take isl_ast_node *node,
+		__isl_keep isl_ast_build *build, void *user), void *user)
+{
+	build = isl_ast_build_cow(build);
+
+	if (!build)
+		return NULL;
+
+	build->at_each_domain = fn;
+	build->at_each_domain_user = user;
+
+	return build;
+}
+
+/* Set the "before_each_for" callback of "build" to "fn".
+ */
+__isl_give isl_ast_build *isl_ast_build_set_before_each_for(
+	__isl_take isl_ast_build *build,
+	__isl_give isl_id *(*fn)(__isl_keep isl_ast_build *build,
+		void *user), void *user)
+{
+	build = isl_ast_build_cow(build);
+
+	if (!build)
+		return NULL;
+
+	build->before_each_for = fn;
+	build->before_each_for_user = user;
+
+	return build;
+}
+
+/* Set the "after_each_for" callback of "build" to "fn".
+ */
+__isl_give isl_ast_build *isl_ast_build_set_after_each_for(
+	__isl_take isl_ast_build *build,
+	__isl_give isl_ast_node *(*fn)(__isl_take isl_ast_node *node,
+		__isl_keep isl_ast_build *build, void *user), void *user)
+{
+	build = isl_ast_build_cow(build);
+
+	if (!build)
+		return NULL;
+
+	build->after_each_for = fn;
+	build->after_each_for_user = user;
+
+	return build;
+}
+
+/* Set the "create_leaf" callback of "build" to "fn".
+ */
+__isl_give isl_ast_build *isl_ast_build_set_create_leaf(
+	__isl_take isl_ast_build *build,
+	__isl_give isl_ast_node *(*fn)(__isl_take isl_ast_build *build,
+		void *user), void *user)
+{
+	build = isl_ast_build_cow(build);
+
+	if (!build)
+		return NULL;
+
+	build->create_leaf = fn;
+	build->create_leaf_user = user;
+
+	return build;
+}
+
+/* Clear all information that is specific to this code generation
+ * and that is (probably) not meaningful to any nested code generation.
+ */
+__isl_give isl_ast_build *isl_ast_build_clear_local_info(
+	__isl_take isl_ast_build *build)
+{
+	isl_space *space;
+
+	build = isl_ast_build_cow(build);
+	if (!build)
+		return NULL;
+
+	space = isl_union_map_get_space(build->options);
+	isl_union_map_free(build->options);
+	build->options = isl_union_map_empty(space);
+
+	build->at_each_domain = NULL;
+	build->at_each_domain_user = NULL;
+	build->before_each_for = NULL;
+	build->before_each_for_user = NULL;
+	build->after_each_for = NULL;
+	build->after_each_for_user = NULL;
+	build->create_leaf = NULL;
+	build->create_leaf_user = NULL;
+
+	if (!build->options)
+		return isl_ast_build_free(build);
+
+	return build;
+}
+
+/* Have any loops been eliminated?
+ * That is, do any of the original schedule dimensions have a fixed
+ * value that has been substituted?
+ */
+static int any_eliminated(isl_ast_build *build)
+{
+	int i;
+
+	for (i = 0; i < build->depth; ++i)
+		if (isl_ast_build_has_affine_value(build, i))
+			return 1;
+
+	return 0;
+}
+
+/* Clear build->schedule_map.
+ * This function should be called whenever anything that might affect
+ * the result of isl_ast_build_get_schedule_map_multi_aff changes.
+ * In particular, it should be called when the depth is changed or
+ * when an iterator is determined to have a fixed value.
+ */
+static void isl_ast_build_reset_schedule_map(__isl_keep isl_ast_build *build)
+{
+	if (!build)
+		return;
+	isl_multi_aff_free(build->schedule_map);
+	build->schedule_map = NULL;
+}
+
+/* Do we need a (non-trivial) schedule map?
+ * That is, is the internal schedule space different from
+ * the external schedule space?
+ *
+ * The internal and external schedule spaces are only the same
+ * if code has been generated for the entire schedule and if none
+ * of the loops have been eliminated.
+ */
+__isl_give int isl_ast_build_need_schedule_map(__isl_keep isl_ast_build *build)
+{
+	int dim;
+
+	if (!build)
+		return -1;
+
+	dim = isl_set_dim(build->domain, isl_dim_set);
+	return build->depth != dim || any_eliminated(build);
+}
+
+/* Return a mapping from the internal schedule space to the external
+ * schedule space in the form of an isl_multi_aff.
+ * The internal schedule space originally corresponds to that of the
+ * input schedule.  This may change during the code generation if
+ * if isl_ast_build_insert_dim is ever called.
+ * The external schedule space corresponds to the
+ * loops that have been generated.
+ *
+ * Currently, the only difference between the internal schedule domain
+ * and the external schedule domain is that some dimensions are projected
+ * out in the external schedule domain.  In particular, the dimensions
+ * for which no code has been generated yet and the dimensions that correspond
+ * to eliminated loops.
+ *
+ * We cache a copy of the schedule_map in build->schedule_map.
+ * The cache is cleared through isl_ast_build_reset_schedule_map
+ * whenever anything changes that might affect the result of this function.
+ */
+__isl_give isl_multi_aff *isl_ast_build_get_schedule_map_multi_aff(
+	__isl_keep isl_ast_build *build)
+{
+	isl_space *space;
+	isl_multi_aff *ma;
+
+	if (!build)
+		return NULL;
+	if (build->schedule_map)
+		return isl_multi_aff_copy(build->schedule_map);
+
+	space = isl_ast_build_get_space(build, 1);
+	space = isl_space_map_from_set(space);
+	ma = isl_multi_aff_identity(space);
+	if (isl_ast_build_need_schedule_map(build)) {
+		int i;
+		int dim = isl_set_dim(build->domain, isl_dim_set);
+		ma = isl_multi_aff_drop_dims(ma, isl_dim_out,
+					build->depth, dim - build->depth);
+		for (i = build->depth - 1; i >= 0; --i)
+			if (isl_ast_build_has_affine_value(build, i))
+				ma = isl_multi_aff_drop_dims(ma,
+							isl_dim_out, i, 1);
+	}
+
+	build->schedule_map = ma;
+	return isl_multi_aff_copy(build->schedule_map);
+}
+
+/* Return a mapping from the internal schedule space to the external
+ * schedule space in the form of an isl_map.
+ */
+__isl_give isl_map *isl_ast_build_get_schedule_map(
+	__isl_keep isl_ast_build *build)
+{
+	isl_multi_aff *ma;
+
+	ma = isl_ast_build_get_schedule_map_multi_aff(build);
+	return isl_map_from_multi_aff(ma);
+}
+
+/* Return the position of the dimension in build->domain for which
+ * an AST node is currently being generated.
+ */
+int isl_ast_build_get_depth(__isl_keep isl_ast_build *build)
+{
+	return build ? build->depth : -1;
+}
+
+/* Prepare for generating code for the next level.
+ * In particular, increase the depth and reset any information
+ * that is local to the current depth.
+ */
+__isl_give isl_ast_build *isl_ast_build_increase_depth(
+	__isl_take isl_ast_build *build)
+{
+	build = isl_ast_build_cow(build);
+	if (!build)
+		return NULL;
+	build->depth++;
+	isl_ast_build_reset_schedule_map(build);
+	build->value = isl_pw_aff_free(build->value);
+	return build;
+}
+
+void isl_ast_build_dump(__isl_keep isl_ast_build *build)
+{
+	if (!build)
+		return;
+
+	fprintf(stderr, "domain: ");
+	isl_set_dump(build->domain);
+	fprintf(stderr, "generated: ");
+	isl_set_dump(build->generated);
+	fprintf(stderr, "pending: ");
+	isl_set_dump(build->pending);
+	fprintf(stderr, "iterators: ");
+	isl_id_list_dump(build->iterators);
+	fprintf(stderr, "values: ");
+	isl_multi_aff_dump(build->values);
+	if (build->value) {
+		fprintf(stderr, "value: ");
+		isl_pw_aff_dump(build->value);
+	}
+	fprintf(stderr, "strides: ");
+	isl_vec_dump(build->strides);
+	fprintf(stderr, "offsets: ");
+	isl_multi_aff_dump(build->offsets);
+}
+
+/* Initialize "build" for AST construction in schedule space "space"
+ * in the case that build->domain is a parameter set.
+ *
+ * build->iterators is assumed to have been updated already.
+ */
+static __isl_give isl_ast_build *isl_ast_build_init(
+	__isl_take isl_ast_build *build, __isl_take isl_space *space)
+{
+	isl_set *set;
+
+	build = isl_ast_build_cow(build);
+	if (!build)
+		goto error;
+
+	set = isl_set_universe(isl_space_copy(space));
+	build->domain = isl_set_intersect_params(isl_set_copy(set),
+						    build->domain);
+	build->pending = isl_set_intersect_params(isl_set_copy(set),
+						    build->pending);
+	build->generated = isl_set_intersect_params(set, build->generated);
+
+	return isl_ast_build_init_derived(build, space);
+error:
+	isl_ast_build_free(build);
+	isl_space_free(space);
+	return NULL;
+}
+
+/* Assign "aff" to *user and return -1, effectively extracting
+ * the first (and presumably only) affine expression in the isl_pw_aff
+ * on which this function is used.
+ */
+static int extract_single_piece(__isl_take isl_set *set,
+	__isl_take isl_aff *aff, void *user)
+{
+	isl_aff **p = user;
+
+	*p = aff;
+	isl_set_free(set);
+
+	return -1;
+}
+
+/* Intersect "set" with the stride constraint of "build", if any.
+ */
+static __isl_give isl_set *intersect_stride_constraint(__isl_take isl_set *set,
+	__isl_keep isl_ast_build *build)
+{
+	isl_set *stride;
+
+	if (!build)
+		return isl_set_free(set);
+	if (!isl_ast_build_has_stride(build, build->depth))
+		return set;
+
+	stride = isl_ast_build_get_stride_constraint(build);
+	return isl_set_intersect(set, stride);
+}
+
+/* Check if the given bounds on the current dimension (together with
+ * the stride constraint, if any) imply that
+ * this current dimension attains only a single value (in terms of
+ * parameters and outer dimensions).
+ * If so, we record it in build->value.
+ * If, moreover, this value can be represented as a single affine expression,
+ * then we also update build->values, effectively marking the current
+ * dimension as "eliminated".
+ *
+ * When computing the gist of the fixed value that can be represented
+ * as a single affine expression, it is important to only take into
+ * account the domain constraints in the original AST build and
+ * not the domain of the affine expression itself.
+ * Otherwise, a [i/3] is changed into a i/3 because we know that i
+ * is a multiple of 3, but then we end up not expressing anywhere
+ * in the context that i is a multiple of 3.
+ */
+static __isl_give isl_ast_build *update_values(
+	__isl_take isl_ast_build *build, __isl_take isl_basic_set *bounds)
+{
+	int sv;
+	isl_pw_multi_aff *pma;
+	isl_aff *aff = NULL;
+	isl_map *it_map;
+	isl_set *set;
+
+	set = isl_set_from_basic_set(bounds);
+	set = isl_set_intersect(set, isl_set_copy(build->domain));
+	set = intersect_stride_constraint(set, build);
+	it_map = isl_ast_build_map_to_iterator(build, set);
+
+	sv = isl_map_is_single_valued(it_map);
+	if (sv < 0)
+		build = isl_ast_build_free(build);
+	if (!build || !sv) {
+		isl_map_free(it_map);
+		return build;
+	}
+
+	pma = isl_pw_multi_aff_from_map(it_map);
+	build->value = isl_pw_multi_aff_get_pw_aff(pma, 0);
+	build->value = isl_ast_build_compute_gist_pw_aff(build, build->value);
+	build->value = isl_pw_aff_coalesce(build->value);
+	isl_pw_multi_aff_free(pma);
+
+	if (!build->value)
+		return isl_ast_build_free(build);
+
+	if (isl_pw_aff_n_piece(build->value) != 1)
+		return build;
+
+	isl_pw_aff_foreach_piece(build->value, &extract_single_piece, &aff);
+
+	build->values = isl_multi_aff_set_aff(build->values, build->depth, aff);
+	if (!build->values)
+		return isl_ast_build_free(build);
+	isl_ast_build_reset_schedule_map(build);
+	return build;
+}
+
+/* Update the AST build based on the given loop bounds for
+ * the current dimension and the stride information available in the build.
+ *
+ * We first make sure that the bounds do not refer to any iterators
+ * that have already been eliminated.
+ * Then, we check if the bounds imply that the current iterator
+ * has a fixed value.
+ * If they do and if this fixed value can be expressed as a single
+ * affine expression, we eliminate the iterators from the bounds.
+ * Note that we cannot simply plug in this single value using
+ * isl_basic_set_preimage_multi_aff as the single value may only
+ * be defined on a subset of the domain.  Plugging in the value
+ * would restrict the build domain to this subset, while this
+ * restriction may not be reflected in the generated code.
+ * Finally, we intersect build->domain with the updated bounds.
+ * We also add the stride constraint unless we have been able
+ * to find a fixed value expressed as a single affine expression.
+ *
+ * Note that the check for a fixed value in update_values requires
+ * us to intersect the bounds with the current build domain.
+ * When we intersect build->domain with the updated bounds in
+ * the final step, we make sure that these updated bounds have
+ * not been intersected with the old build->domain.
+ * Otherwise, we would indirectly intersect the build domain with itself,
+ * which can lead to inefficiencies, in particular if the build domain
+ * contains any unknown divs.
+ */
+__isl_give isl_ast_build *isl_ast_build_set_loop_bounds(
+	__isl_take isl_ast_build *build, __isl_take isl_basic_set *bounds)
+{
+	isl_set *set;
+
+	build = isl_ast_build_cow(build);
+	if (!build)
+		goto error;
+
+	bounds = isl_basic_set_preimage_multi_aff(bounds,
+					isl_multi_aff_copy(build->values));
+	build = update_values(build, isl_basic_set_copy(bounds));
+	if (!build)
+		goto error;
+	set = isl_set_from_basic_set(isl_basic_set_copy(bounds));
+	if (isl_ast_build_has_affine_value(build, build->depth)) {
+		set = isl_set_eliminate(set, isl_dim_set, build->depth, 1);
+		set = isl_set_compute_divs(set);
+		build->pending = isl_set_intersect(build->pending,
+							isl_set_copy(set));
+		build->domain = isl_set_intersect(build->domain, set);
+	} else {
+		isl_basic_set *generated, *pending;
+
+		pending = isl_basic_set_copy(bounds);
+		pending = isl_basic_set_drop_constraints_involving_dims(pending,
+					isl_dim_set, build->depth, 1);
+		build->pending = isl_set_intersect(build->pending,
+					isl_set_from_basic_set(pending));
+		generated = isl_basic_set_copy(bounds);
+		generated = isl_basic_set_drop_constraints_not_involving_dims(
+				    generated, isl_dim_set, build->depth, 1);
+		build->generated = isl_set_intersect(build->generated,
+					isl_set_from_basic_set(generated));
+		build->domain = isl_set_intersect(build->domain, set);
+		build = isl_ast_build_include_stride(build);
+		if (!build)
+			goto error;
+	}
+	isl_basic_set_free(bounds);
+
+	if (!build->domain || !build->pending || !build->generated)
+		return isl_ast_build_free(build);
+
+	return build;
+error:
+	isl_ast_build_free(build);
+	isl_basic_set_free(bounds);
+	return NULL;
+}
+
+/* Intersect build->domain with "set", where "set" is specified
+ * in terms of the internal schedule domain.
+ */
+static __isl_give isl_ast_build *isl_ast_build_restrict_internal(
+	__isl_take isl_ast_build *build, __isl_take isl_set *set)
+{
+	build = isl_ast_build_cow(build);
+	if (!build)
+		goto error;
+
+	set = isl_set_compute_divs(set);
+	build->domain = isl_set_intersect(build->domain, set);
+	build->domain = isl_set_coalesce(build->domain);
+
+	if (!build->domain)
+		return isl_ast_build_free(build);
+
+	return build;
+error:
+	isl_ast_build_free(build);
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Intersect build->generated and build->domain with "set",
+ * where "set" is specified in terms of the internal schedule domain.
+ */
+__isl_give isl_ast_build *isl_ast_build_restrict_generated(
+	__isl_take isl_ast_build *build, __isl_take isl_set *set)
+{
+	set = isl_set_compute_divs(set);
+	build = isl_ast_build_restrict_internal(build, isl_set_copy(set));
+	build = isl_ast_build_cow(build);
+	if (!build)
+		goto error;
+
+	build->generated = isl_set_intersect(build->generated, set);
+	build->generated = isl_set_coalesce(build->generated);
+
+	if (!build->generated)
+		return isl_ast_build_free(build);
+
+	return build;
+error:
+	isl_ast_build_free(build);
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Replace the set of pending constraints by "guard", which is then
+ * no longer considered as pending.
+ * That is, add "guard" to the generated constraints and clear all pending
+ * constraints, making the domain equal to the generated constraints.
+ */
+__isl_give isl_ast_build *isl_ast_build_replace_pending_by_guard(
+	__isl_take isl_ast_build *build, __isl_take isl_set *guard)
+{
+	build = isl_ast_build_restrict_generated(build, guard);
+	build = isl_ast_build_cow(build);
+	if (!build)
+		return NULL;
+
+	isl_set_free(build->domain);
+	build->domain = isl_set_copy(build->generated);
+	isl_set_free(build->pending);
+	build->pending = isl_set_universe(isl_set_get_space(build->domain));
+
+	if (!build->pending)
+		return isl_ast_build_free(build);
+
+	return build;
+}
+
+/* Intersect build->pending and build->domain with "set",
+ * where "set" is specified in terms of the internal schedule domain.
+ */
+__isl_give isl_ast_build *isl_ast_build_restrict_pending(
+	__isl_take isl_ast_build *build, __isl_take isl_set *set)
+{
+	set = isl_set_compute_divs(set);
+	build = isl_ast_build_restrict_internal(build, isl_set_copy(set));
+	build = isl_ast_build_cow(build);
+	if (!build)
+		goto error;
+
+	build->pending = isl_set_intersect(build->pending, set);
+	build->pending = isl_set_coalesce(build->pending);
+
+	if (!build->pending)
+		return isl_ast_build_free(build);
+
+	return build;
+error:
+	isl_ast_build_free(build);
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Intersect build->domain with "set", where "set" is specified
+ * in terms of the external schedule domain.
+ */
+__isl_give isl_ast_build *isl_ast_build_restrict(
+	__isl_take isl_ast_build *build, __isl_take isl_set *set)
+{
+	if (isl_set_is_params(set))
+		return isl_ast_build_restrict_generated(build, set);
+
+	if (isl_ast_build_need_schedule_map(build)) {
+		isl_multi_aff *ma;
+		ma = isl_ast_build_get_schedule_map_multi_aff(build);
+		set = isl_set_preimage_multi_aff(set, ma);
+	}
+	return isl_ast_build_restrict_generated(build, set);
+}
+
+/* Replace build->executed by "executed".
+ */
+__isl_give isl_ast_build *isl_ast_build_set_executed(
+	__isl_take isl_ast_build *build, __isl_take isl_union_map *executed)
+{
+	build = isl_ast_build_cow(build);
+	if (!build)
+		goto error;
+
+	isl_union_map_free(build->executed);
+	build->executed = executed;
+
+	return build;
+error:
+	isl_ast_build_free(build);
+	isl_union_map_free(executed);
+	return NULL;
+}
+
+/* Return a copy of the current schedule domain.
+ */
+__isl_give isl_set *isl_ast_build_get_domain(__isl_keep isl_ast_build *build)
+{
+	return build ? isl_set_copy(build->domain) : NULL;
+}
+
+/* Return a copy of the set of pending constraints.
+ */
+__isl_give isl_set *isl_ast_build_get_pending(
+	__isl_keep isl_ast_build *build)
+{
+	return build ? isl_set_copy(build->pending) : NULL;
+}
+
+/* Return a copy of the set of generated constraints.
+ */
+__isl_give isl_set *isl_ast_build_get_generated(
+	__isl_keep isl_ast_build *build)
+{
+	return build ? isl_set_copy(build->generated) : NULL;
+}
+
+/* Return the number of variables of the given type
+ * in the (internal) schedule space.
+ */
+unsigned isl_ast_build_dim(__isl_keep isl_ast_build *build,
+	enum isl_dim_type type)
+{
+	if (!build)
+		return 0;
+	return isl_set_dim(build->domain, type);
+}
+
+/* Return the (schedule) space of "build".
+ *
+ * If "internal" is set, then this space is the space of the internal
+ * representation of the entire schedule, including those parts for
+ * which no code has been generated yet.
+ *
+ * If "internal" is not set, then this space is the external representation
+ * of the loops generated so far.
+ */
+__isl_give isl_space *isl_ast_build_get_space(__isl_keep isl_ast_build *build,
+	int internal)
+{
+	int i;
+	int dim;
+	isl_space *space;
+
+	if (!build)
+		return NULL;
+
+	space = isl_set_get_space(build->domain);
+	if (internal)
+		return space;
+
+	if (!isl_ast_build_need_schedule_map(build))
+		return space;
+
+	dim = isl_set_dim(build->domain, isl_dim_set);
+	space = isl_space_drop_dims(space, isl_dim_set,
+				    build->depth, dim - build->depth);
+	for (i = build->depth - 1; i >= 0; --i)
+		if (isl_ast_build_has_affine_value(build, i))
+			space = isl_space_drop_dims(space, isl_dim_set, i, 1);
+
+	return space;
+}
+
+/* Return the external representation of the schedule space of "build",
+ * i.e., a space with a dimension for each loop generated so far,
+ * with the names of the dimensions set to the loop iterators.
+ */
+__isl_give isl_space *isl_ast_build_get_schedule_space(
+	__isl_keep isl_ast_build *build)
+{
+	isl_space *space;
+	int i, skip;
+
+	if (!build)
+		return NULL;
+
+	space = isl_ast_build_get_space(build, 0);
+
+	skip = 0;
+	for (i = 0; i < build->depth; ++i) {
+		isl_id *id;
+
+		if (isl_ast_build_has_affine_value(build, i)) {
+			skip++;
+			continue;
+		}
+
+		id = isl_ast_build_get_iterator_id(build, i);
+		space = isl_space_set_dim_id(space, isl_dim_set, i - skip, id);
+	}
+
+	return space;
+}
+
+/* Return the current schedule, as stored in build->executed, in terms
+ * of the external schedule domain.
+ */
+__isl_give isl_union_map *isl_ast_build_get_schedule(
+	__isl_keep isl_ast_build *build)
+{
+	isl_union_map *executed;
+	isl_union_map *schedule;
+
+	if (!build)
+		return NULL;
+
+	executed = isl_union_map_copy(build->executed);
+	if (isl_ast_build_need_schedule_map(build)) {
+		isl_map *proj = isl_ast_build_get_schedule_map(build);
+		executed = isl_union_map_apply_domain(executed,
+					isl_union_map_from_map(proj));
+	}
+	schedule = isl_union_map_reverse(executed);
+
+	return schedule;
+}
+
+/* Return the iterator attached to the internal schedule dimension "pos".
+ */
+__isl_give isl_id *isl_ast_build_get_iterator_id(
+	__isl_keep isl_ast_build *build, int pos)
+{
+	if (!build)
+		return NULL;
+
+	return isl_id_list_get_id(build->iterators, pos);
+}
+
+/* Set the stride and offset of the current dimension to the given
+ * value and expression.
+ *
+ * If we had already found a stride before, then the two strides
+ * are combined into a single stride.
+ *
+ * In particular, if the new stride information is of the form
+ *
+ *	i = f + s (...)
+ *
+ * and the old stride information is of the form
+ *
+ *	i = f2 + s2 (...)
+ *
+ * then we compute the extended gcd of s and s2
+ *
+ *	a s + b s2 = g,
+ *
+ * with g = gcd(s,s2), multiply the first equation with t1 = b s2/g
+ * and the second with t2 = a s1/g.
+ * This results in
+ *
+ *	i = (b s2 + a s1)/g i = t1 f + t2 f2 + (s s2)/g (...)
+ *
+ * so that t1 f + t2 f2 is the combined offset and (s s2)/g = lcm(s,s2)
+ * is the combined stride.
+ */
+static __isl_give isl_ast_build *set_stride(__isl_take isl_ast_build *build,
+	__isl_take isl_val *stride, __isl_take isl_aff *offset)
+{
+	int pos;
+
+	build = isl_ast_build_cow(build);
+	if (!build || !stride || !offset)
+		goto error;
+
+	pos = build->depth;
+
+	if (isl_ast_build_has_stride(build, pos)) {
+		isl_val *stride2, *a, *b, *g;
+		isl_aff *offset2;
+
+		stride2 = isl_vec_get_element_val(build->strides, pos);
+		g = isl_val_gcdext(isl_val_copy(stride), isl_val_copy(stride2),
+					&a, &b);
+		a = isl_val_mul(a, isl_val_copy(stride));
+		a = isl_val_div(a, isl_val_copy(g));
+		stride2 = isl_val_div(stride2, g);
+		b = isl_val_mul(b, isl_val_copy(stride2));
+		stride = isl_val_mul(stride, stride2);
+
+		offset2 = isl_multi_aff_get_aff(build->offsets, pos);
+		offset2 = isl_aff_scale_val(offset2, a);
+		offset = isl_aff_scale_val(offset, b);
+		offset = isl_aff_add(offset, offset2);
+	}
+
+	build->strides = isl_vec_set_element_val(build->strides, pos, stride);
+	build->offsets = isl_multi_aff_set_aff(build->offsets, pos, offset);
+	if (!build->strides || !build->offsets)
+		return isl_ast_build_free(build);
+
+	return build;
+error:
+	isl_val_free(stride);
+	isl_aff_free(offset);
+	return isl_ast_build_free(build);
+}
+
+/* Return a set expressing the stride constraint at the current depth.
+ *
+ * In particular, if the current iterator (i) is known to attain values
+ *
+ *	f + s a
+ *
+ * where f is the offset and s is the stride, then the returned set
+ * expresses the constraint
+ *
+ *	(f - i) mod s = 0
+ */
+__isl_give isl_set *isl_ast_build_get_stride_constraint(
+	__isl_keep isl_ast_build *build)
+{
+	isl_aff *aff;
+	isl_set *set;
+	isl_val *stride;
+	int pos;
+
+	if (!build)
+		return NULL;
+
+	pos = build->depth;
+
+	if (!isl_ast_build_has_stride(build, pos))
+		return isl_set_universe(isl_ast_build_get_space(build, 1));
+
+	stride = isl_ast_build_get_stride(build, pos);
+	aff = isl_ast_build_get_offset(build, pos);
+	aff = isl_aff_add_coefficient_si(aff, isl_dim_in, pos, -1);
+	aff = isl_aff_mod_val(aff, stride);
+	set = isl_set_from_basic_set(isl_aff_zero_basic_set(aff));
+
+	return set;
+}
+
+/* Return the expansion implied by the stride and offset at the current
+ * depth.
+ *
+ * That is, return the mapping
+ *
+ *	[i_0, ..., i_{d-1}, i_d, i_{d+1}, ...]
+ *		-> [i_0, ..., i_{d-1}, s * i_d + offset(i),  i_{d+1}, ...]
+ *
+ * where s is the stride at the current depth d and offset(i) is
+ * the corresponding offset.
+ */
+__isl_give isl_multi_aff *isl_ast_build_get_stride_expansion(
+	__isl_keep isl_ast_build *build)
+{
+	isl_space *space;
+	isl_multi_aff *ma;
+	int pos;
+	isl_aff *aff, *offset;
+	isl_val *stride;
+
+	if (!build)
+		return NULL;
+
+	pos = isl_ast_build_get_depth(build);
+	space = isl_ast_build_get_space(build, 1);
+	space = isl_space_map_from_set(space);
+	ma = isl_multi_aff_identity(space);
+
+	if (!isl_ast_build_has_stride(build, pos))
+		return ma;
+
+	offset = isl_ast_build_get_offset(build, pos);
+	stride = isl_ast_build_get_stride(build, pos);
+	aff = isl_multi_aff_get_aff(ma, pos);
+	aff = isl_aff_scale_val(aff, stride);
+	aff = isl_aff_add(aff, offset);
+	ma = isl_multi_aff_set_aff(ma, pos, aff);
+
+	return ma;
+}
+
+/* Add constraints corresponding to any previously detected
+ * stride on the current dimension to build->domain.
+ */
+__isl_give isl_ast_build *isl_ast_build_include_stride(
+	__isl_take isl_ast_build *build)
+{
+	isl_set *set;
+
+	if (!build)
+		return NULL;
+	if (!isl_ast_build_has_stride(build, build->depth))
+		return build;
+	build = isl_ast_build_cow(build);
+	if (!build)
+		return NULL;
+
+	set = isl_ast_build_get_stride_constraint(build);
+
+	build->domain = isl_set_intersect(build->domain, isl_set_copy(set));
+	build->generated = isl_set_intersect(build->generated, set);
+	if (!build->domain || !build->generated)
+		return isl_ast_build_free(build);
+
+	return build;
+}
+
+/* Information used inside detect_stride.
+ *
+ * "build" may be updated by detect_stride to include stride information.
+ * "pos" is equal to build->depth.
+ */
+struct isl_detect_stride_data {
+	isl_ast_build *build;
+	int pos;
+};
+
+/* Check if constraint "c" imposes any stride on dimension data->pos
+ * and, if so, update the stride information in data->build.
+ *
+ * In order to impose a stride on the dimension, "c" needs to be an equality
+ * and it needs to involve the dimension.  Note that "c" may also be
+ * a div constraint and thus an inequality that we cannot use.
+ *
+ * Let c be of the form
+ *
+ *	h(p) + g * v * i + g * stride * f(alpha) = 0
+ *
+ * with h(p) an expression in terms of the parameters and outer dimensions
+ * and f(alpha) an expression in terms of the existentially quantified
+ * variables.  Note that the inner dimensions have been eliminated so
+ * they do not appear in "c".
+ *
+ * If "stride" is not zero and not one, then it represents a non-trivial stride
+ * on "i".  We compute a and b such that
+ *
+ *	a v + b stride = 1
+ *
+ * We have
+ *
+ *	g v i = -h(p) + g stride f(alpha)
+ *
+ *	a g v i = -a h(p) + g stride f(alpha)
+ *
+ *	a g v i + b g stride i = -a h(p) + g stride * (...)
+ *
+ *	g i = -a h(p) + g stride * (...)
+ *
+ *	i = -a h(p)/g + stride * (...)
+ *
+ * The expression "-a h(p)/g" can therefore be used as offset.
+ */
+static int detect_stride(__isl_take isl_constraint *c, void *user)
+{
+	struct isl_detect_stride_data *data = user;
+	int i, n_div;
+	isl_ctx *ctx;
+	isl_val *v, *stride, *m;
+
+	if (!isl_constraint_is_equality(c) ||
+	    !isl_constraint_involves_dims(c, isl_dim_set, data->pos, 1)) {
+		isl_constraint_free(c);
+		return 0;
+	}
+
+	ctx = isl_constraint_get_ctx(c);
+	stride = isl_val_zero(ctx);
+	n_div = isl_constraint_dim(c, isl_dim_div);
+	for (i = 0; i < n_div; ++i) {
+		v = isl_constraint_get_coefficient_val(c, isl_dim_div, i);
+		stride = isl_val_gcd(stride, v);
+	}
+
+	v = isl_constraint_get_coefficient_val(c, isl_dim_set, data->pos);
+	m = isl_val_gcd(isl_val_copy(stride), isl_val_copy(v));
+	stride = isl_val_div(stride, isl_val_copy(m));
+	v = isl_val_div(v, isl_val_copy(m));
+
+	if (!isl_val_is_zero(stride) && !isl_val_is_one(stride)) {
+		isl_aff *aff;
+		isl_val *gcd, *a, *b;
+
+		gcd = isl_val_gcdext(v, isl_val_copy(stride), &a, &b);
+		isl_val_free(gcd);
+		isl_val_free(b);
+
+		aff = isl_constraint_get_aff(c);
+		for (i = 0; i < n_div; ++i)
+			aff = isl_aff_set_coefficient_si(aff,
+							 isl_dim_div, i, 0);
+		aff = isl_aff_set_coefficient_si(aff, isl_dim_in, data->pos, 0);
+		a = isl_val_neg(a);
+		aff = isl_aff_scale_val(aff, a);
+		aff = isl_aff_scale_down_val(aff, m);
+		data->build = set_stride(data->build, stride, aff);
+	} else {
+		isl_val_free(stride);
+		isl_val_free(m);
+		isl_val_free(v);
+	}
+
+	isl_constraint_free(c);
+	return 0;
+}
+
+/* Check if the constraints in "set" imply any stride on the current
+ * dimension and, if so, record the stride information in "build"
+ * and return the updated "build".
+ *
+ * We compute the affine hull and then check if any of the constraints
+ * in the hull imposes any stride on the current dimension.
+ *
+ * We assume that inner dimensions have been eliminated from "set"
+ * by the caller.  This is needed because the common stride
+ * may be imposed by different inner dimensions on different parts of
+ * the domain.
+ */
+__isl_give isl_ast_build *isl_ast_build_detect_strides(
+	__isl_take isl_ast_build *build, __isl_take isl_set *set)
+{
+	isl_basic_set *hull;
+	struct isl_detect_stride_data data;
+
+	if (!build)
+		goto error;
+
+	data.build = build;
+	data.pos = isl_ast_build_get_depth(build);
+	hull = isl_set_affine_hull(set);
+
+	if (isl_basic_set_foreach_constraint(hull, &detect_stride, &data) < 0)
+		data.build = isl_ast_build_free(data.build);
+
+	isl_basic_set_free(hull);
+	return data.build;
+error:
+	isl_set_free(set);
+	return NULL;
+}
+
+struct isl_ast_build_involves_data {
+	int depth;
+	int involves;
+};
+
+/* Check if "map" involves the input dimension data->depth.
+ */
+static int involves_depth(__isl_take isl_map *map, void *user)
+{
+	struct isl_ast_build_involves_data *data = user;
+
+	data->involves = isl_map_involves_dims(map, isl_dim_in, data->depth, 1);
+	isl_map_free(map);
+
+	if (data->involves < 0 || data->involves)
+		return -1;
+	return 0;
+}
+
+/* Do any options depend on the value of the dimension at the current depth?
+ */
+int isl_ast_build_options_involve_depth(__isl_keep isl_ast_build *build)
+{
+	struct isl_ast_build_involves_data data;
+
+	if (!build)
+		return -1;
+
+	data.depth = build->depth;
+	data.involves = 0;
+
+	if (isl_union_map_foreach_map(build->options,
+					&involves_depth, &data) < 0) {
+		if (data.involves < 0 || !data.involves)
+			return -1;
+	}
+
+	return data.involves;
+}
+
+/* Construct the map
+ *
+ *	{ [i] -> [i] : i < pos; [i] -> [i + 1] : i >= pos }
+ *
+ * with "space" the parameter space of the constructed map.
+ */
+static __isl_give isl_map *construct_insertion_map(__isl_take isl_space *space,
+	int pos)
+{
+	isl_constraint *c;
+	isl_basic_map *bmap1, *bmap2;
+
+	space = isl_space_set_from_params(space);
+	space = isl_space_add_dims(space, isl_dim_set, 1);
+	space = isl_space_map_from_set(space);
+	c = isl_equality_alloc(isl_local_space_from_space(space));
+	c = isl_constraint_set_coefficient_si(c, isl_dim_in, 0, 1);
+	c = isl_constraint_set_coefficient_si(c, isl_dim_out, 0, -1);
+	bmap1 = isl_basic_map_from_constraint(isl_constraint_copy(c));
+	c = isl_constraint_set_constant_si(c, 1);
+	bmap2 = isl_basic_map_from_constraint(c);
+
+	bmap1 = isl_basic_map_upper_bound_si(bmap1, isl_dim_in, 0, pos - 1);
+	bmap2 = isl_basic_map_lower_bound_si(bmap2, isl_dim_in, 0, pos);
+
+	return isl_basic_map_union(bmap1, bmap2);
+}
+
+static const char *option_str[] = {
+	[atomic] = "atomic",
+	[unroll] = "unroll",
+	[separate] = "separate"
+};
+
+/* Update the "options" to reflect the insertion of a dimension
+ * at position "pos" in the schedule domain space.
+ * "space" is the original domain space before the insertion and
+ * may be named and/or structured.
+ *
+ * The (relevant) input options all have "space" as domain, which
+ * has to be mapped to the extended space.
+ * The values of the ranges also refer to the schedule domain positions
+ * and they therefore also need to be adjusted.  In particular, values
+ * smaller than pos do not need to change, while values greater than or
+ * equal to pos need to be incremented.
+ * That is, we need to apply the following map.
+ *
+ *	{ atomic[i] -> atomic[i] : i < pos; [i] -> [i + 1] : i >= pos;
+ *	  unroll[i] -> unroll[i] : i < pos; [i] -> [i + 1] : i >= pos;
+ *	  separate[i] -> separate[i] : i < pos; [i] -> [i + 1] : i >= pos;
+ *	  separation_class[[i] -> [c]]
+ *		-> separation_class[[i] -> [c]] : i < pos;
+ *	  separation_class[[i] -> [c]]
+ *		-> separation_class[[i + 1] -> [c]] : i >= pos }
+ */
+static __isl_give isl_union_map *options_insert_dim(
+	__isl_take isl_union_map *options, __isl_take isl_space *space, int pos)
+{
+	isl_map *map;
+	isl_union_map *insertion;
+	enum isl_ast_build_domain_type type;
+	const char *name = "separation_class";
+
+	space = isl_space_map_from_set(space);
+	map = isl_map_identity(space);
+	map = isl_map_insert_dims(map, isl_dim_out, pos, 1);
+	options = isl_union_map_apply_domain(options,
+						isl_union_map_from_map(map));
+
+	if (!options)
+		return NULL;
+
+	map = construct_insertion_map(isl_union_map_get_space(options), pos);
+
+	insertion = isl_union_map_empty(isl_union_map_get_space(options));
+
+	for (type = atomic; type <= separate; ++type) {
+		isl_map *map_type = isl_map_copy(map);
+		const char *name = option_str[type];
+		map_type = isl_map_set_tuple_name(map_type, isl_dim_in, name);
+		map_type = isl_map_set_tuple_name(map_type, isl_dim_out, name);
+		insertion = isl_union_map_add_map(insertion, map_type);
+	}
+
+	map = isl_map_product(map, isl_map_identity(isl_map_get_space(map)));
+	map = isl_map_set_tuple_name(map, isl_dim_in, name);
+	map = isl_map_set_tuple_name(map, isl_dim_out, name);
+	insertion = isl_union_map_add_map(insertion, map);
+
+	options = isl_union_map_apply_range(options, insertion);
+
+	return options;
+}
+
+/* Insert a single dimension in the schedule domain at position "pos".
+ * The new dimension is given an isl_id with the empty string as name.
+ *
+ * The main difficulty is updating build->options to reflect the
+ * extra dimension.  This is handled in options_insert_dim.
+ *
+ * Note that because of the dimension manipulations, the resulting
+ * schedule domain space will always be unnamed and unstructured.
+ * However, the original schedule domain space may be named and/or
+ * structured, so we have to take this possibility into account
+ * while performing the transformations.
+ */
+__isl_give isl_ast_build *isl_ast_build_insert_dim(
+	__isl_take isl_ast_build *build, int pos)
+{
+	isl_ctx *ctx;
+	isl_space *space, *ma_space;
+	isl_id *id;
+	isl_multi_aff *ma;
+
+	build = isl_ast_build_cow(build);
+	if (!build)
+		return NULL;
+
+	ctx = isl_ast_build_get_ctx(build);
+	id = isl_id_alloc(ctx, "", NULL);
+	space = isl_ast_build_get_space(build, 1);
+	build->iterators = isl_id_list_insert(build->iterators, pos, id);
+	build->domain = isl_set_insert_dims(build->domain,
+						isl_dim_set, pos, 1);
+	build->generated = isl_set_insert_dims(build->generated,
+						isl_dim_set, pos, 1);
+	build->pending = isl_set_insert_dims(build->pending,
+						isl_dim_set, pos, 1);
+	build->strides = isl_vec_insert_els(build->strides, pos, 1);
+	build->strides = isl_vec_set_element_si(build->strides, pos, 1);
+	ma_space = isl_space_params(isl_multi_aff_get_space(build->offsets));
+	ma_space = isl_space_set_from_params(ma_space);
+	ma_space = isl_space_add_dims(ma_space, isl_dim_set, 1);
+	ma_space = isl_space_map_from_set(ma_space);
+	ma = isl_multi_aff_zero(isl_space_copy(ma_space));
+	build->offsets = isl_multi_aff_splice(build->offsets, pos, pos, ma);
+	ma = isl_multi_aff_identity(ma_space);
+	build->values = isl_multi_aff_splice(build->values, pos, pos, ma);
+	build->options = options_insert_dim(build->options, space, pos);
+
+	if (!build->iterators || !build->domain || !build->generated ||
+	    !build->pending || !build->values ||
+	    !build->strides || !build->offsets || !build->options)
+		return isl_ast_build_free(build);
+
+	return build;
+}
+
+/* Scale down the current dimension by a factor of "m".
+ * "umap" is an isl_union_map that implements the scaling down.
+ * That is, it is of the form
+ *
+ *	{ [.... i ....] -> [.... i' ....] : i = m i' }
+ *
+ * This function is called right after the strides have been
+ * detected, but before any constraints on the current dimension
+ * have been included in build->domain.
+ * We therefore only need to update stride, offset and the options.
+ */
+__isl_give isl_ast_build *isl_ast_build_scale_down(
+	__isl_take isl_ast_build *build, __isl_take isl_val *m,
+	__isl_take isl_union_map *umap)
+{
+	isl_aff *aff;
+	isl_val *v;
+	int depth;
+
+	build = isl_ast_build_cow(build);
+	if (!build || !umap || !m)
+		goto error;
+
+	depth = build->depth;
+
+	v = isl_vec_get_element_val(build->strides, depth);
+	v = isl_val_div(v, isl_val_copy(m));
+	build->strides = isl_vec_set_element_val(build->strides, depth, v);
+
+	aff = isl_multi_aff_get_aff(build->offsets, depth);
+	aff = isl_aff_scale_down_val(aff, m);
+	build->offsets = isl_multi_aff_set_aff(build->offsets, depth, aff);
+	build->options = isl_union_map_apply_domain(build->options, umap);
+	if (!build->strides || !build->offsets || !build->options)
+		return isl_ast_build_free(build);
+
+	return build;
+error:
+	isl_val_free(m);
+	isl_union_map_free(umap);
+	return isl_ast_build_free(build);
+}
+
+/* Return a list of "n" isl_ids called "c%d", with "%d" starting at "first".
+ * If an isl_id with such a name already appears among the parameters
+ * in build->domain, then adjust the name to "c%d_%d".
+ */
+static __isl_give isl_id_list *generate_names(isl_ctx *ctx, int n, int first,
+	__isl_keep isl_ast_build *build)
+{
+	int i;
+	isl_id_list *names;
+
+	names = isl_id_list_alloc(ctx, n);
+	for (i = 0; i < n; ++i) {
+		isl_id *id;
+
+		id = generate_name(ctx, first + i, build);
+		names = isl_id_list_add(names, id);
+	}
+
+	return names;
+}
+
+/* Embed "options" into the given isl_ast_build space.
+ *
+ * This function is called from within a nested call to
+ * isl_ast_build_ast_from_schedule.
+ * "options" refers to the additional schedule,
+ * while space refers to both the space of the outer isl_ast_build and
+ * that of the additional schedule.
+ * Specifically, space is of the form
+ *
+ *	[I -> S]
+ *
+ * while options lives in the space(s)
+ *
+ *	S -> *
+ *
+ * We compute
+ *
+ *	[I -> S] -> S
+ *
+ * and compose this with options, to obtain the new options
+ * living in the space(s)
+ *
+ *	[I -> S] -> *
+ */
+static __isl_give isl_union_map *embed_options(
+	__isl_take isl_union_map *options, __isl_take isl_space *space)
+{
+	isl_map *map;
+
+	map = isl_map_universe(isl_space_unwrap(space));
+	map = isl_map_range_map(map);
+
+	options = isl_union_map_apply_range(
+				isl_union_map_from_map(map), options);
+
+	return options;
+}
+
+/* Update "build" for use in a (possibly nested) code generation.  That is,
+ * extend "build" from an AST build on some domain O to an AST build
+ * on domain [O -> S], with S corresponding to "space".
+ * If the original domain is a parameter domain, then the new domain is
+ * simply S.
+ * "iterators" is a list of iterators for S, but the number of elements
+ * may be smaller or greater than the number of set dimensions of S.
+ * If "keep_iterators" is set, then any extra ids in build->iterators
+ * are reused for S.  Otherwise, these extra ids are dropped.
+ *
+ * We first update build->outer_pos to the current depth.
+ * This depth is zero in case this is the outermost code generation.
+ *
+ * We then add additional ids such that the number of iterators is at least
+ * equal to the dimension of the new build domain.
+ *
+ * If the original domain is parametric, then we are constructing
+ * an isl_ast_build for the outer code generation and we pass control
+ * to isl_ast_build_init.
+ *
+ * Otherwise, we adjust the fields of "build" to include "space".
+ */
+__isl_give isl_ast_build *isl_ast_build_product(
+	__isl_take isl_ast_build *build, __isl_take isl_space *space)
+{
+	isl_ctx *ctx;
+	isl_vec *strides;
+	isl_set *set;
+	isl_multi_aff *embedding;
+	int dim, n_it;
+
+	build = isl_ast_build_cow(build);
+	if (!build)
+		goto error;
+
+	build->outer_pos = build->depth;
+
+	ctx = isl_ast_build_get_ctx(build);
+	dim = isl_set_dim(build->domain, isl_dim_set);
+	dim += isl_space_dim(space, isl_dim_set);
+	n_it = isl_id_list_n_id(build->iterators);
+	if (n_it < dim) {
+		isl_id_list *l;
+		l = generate_names(ctx, dim - n_it, n_it, build);
+		build->iterators = isl_id_list_concat(build->iterators, l);
+	}
+
+	if (isl_set_is_params(build->domain))
+		return isl_ast_build_init(build, space);
+
+	set = isl_set_universe(isl_space_copy(space));
+	build->domain = isl_set_product(build->domain, isl_set_copy(set));
+	build->pending = isl_set_product(build->pending, isl_set_copy(set));
+	build->generated = isl_set_product(build->generated, set);
+
+	strides = isl_vec_alloc(ctx, isl_space_dim(space, isl_dim_set));
+	strides = isl_vec_set_si(strides, 1);
+	build->strides = isl_vec_concat(build->strides, strides);
+
+	space = isl_space_map_from_set(space);
+	build->offsets = isl_multi_aff_align_params(build->offsets,
+						    isl_space_copy(space));
+	build->offsets = isl_multi_aff_product(build->offsets,
+				isl_multi_aff_zero(isl_space_copy(space)));
+	build->values = isl_multi_aff_align_params(build->values,
+						    isl_space_copy(space));
+	embedding = isl_multi_aff_identity(space);
+	build->values = isl_multi_aff_product(build->values, embedding);
+
+	space = isl_ast_build_get_space(build, 1);
+	build->options = embed_options(build->options, space);
+
+	if (!build->iterators || !build->domain || !build->generated ||
+	    !build->pending || !build->values ||
+	    !build->strides || !build->offsets || !build->options)
+		return isl_ast_build_free(build);
+
+	return build;
+error:
+	isl_ast_build_free(build);
+	isl_space_free(space);
+	return NULL;
+}
+
+/* Does "aff" only attain non-negative values over build->domain?
+ * That is, does it not attain any negative values?
+ */
+int isl_ast_build_aff_is_nonneg(__isl_keep isl_ast_build *build,
+	__isl_keep isl_aff *aff)
+{
+	isl_set *test;
+	int empty;
+
+	if (!build)
+		return -1;
+
+	aff = isl_aff_copy(aff);
+	test = isl_set_from_basic_set(isl_aff_neg_basic_set(aff));
+	test = isl_set_intersect(test, isl_set_copy(build->domain));
+	empty = isl_set_is_empty(test);
+	isl_set_free(test);
+
+	return empty;
+}
+
+/* Does the dimension at (internal) position "pos" have a non-trivial stride?
+ */
+int isl_ast_build_has_stride(__isl_keep isl_ast_build *build, int pos)
+{
+	isl_val *v;
+	int has_stride;
+
+	if (!build)
+		return -1;
+
+	v = isl_vec_get_element_val(build->strides, pos);
+	if (!v)
+		return -1;
+	has_stride = !isl_val_is_one(v);
+	isl_val_free(v);
+
+	return has_stride;
+}
+
+/* Given that the dimension at position "pos" takes on values
+ *
+ *	f + s a
+ *
+ * with a an integer, return s through *stride.
+ */
+__isl_give isl_val *isl_ast_build_get_stride(__isl_keep isl_ast_build *build,
+	int pos)
+{
+	if (!build)
+		return NULL;
+
+	return isl_vec_get_element_val(build->strides, pos);
+}
+
+/* Given that the dimension at position "pos" takes on values
+ *
+ *	f + s a
+ *
+ * with a an integer, return f.
+ */
+__isl_give isl_aff *isl_ast_build_get_offset(
+	__isl_keep isl_ast_build *build, int pos)
+{
+	if (!build)
+		return NULL;
+
+	return isl_multi_aff_get_aff(build->offsets, pos);
+}
+
+/* Is the dimension at position "pos" known to attain only a single
+ * value that, moreover, can be described by a single affine expression
+ * in terms of the outer dimensions and parameters?
+ *
+ * If not, then the correponding affine expression in build->values
+ * is set to be equal to the same input dimension.
+ * Otherwise, it is set to the requested expression in terms of
+ * outer dimensions and parameters.
+ */
+int isl_ast_build_has_affine_value(__isl_keep isl_ast_build *build,
+	int pos)
+{
+	isl_aff *aff;
+	int involves;
+
+	if (!build)
+		return -1;
+
+	aff = isl_multi_aff_get_aff(build->values, pos);
+	involves = isl_aff_involves_dims(aff, isl_dim_in, pos, 1);
+	isl_aff_free(aff);
+
+	if (involves < 0)
+		return -1;
+
+	return !involves;
+}
+
+/* Plug in the known values (fixed affine expressions in terms of
+ * parameters and outer loop iterators) of all loop iterators
+ * in the domain of "umap".
+ *
+ * We simply precompose "umap" with build->values.
+ */
+__isl_give isl_union_map *isl_ast_build_substitute_values_union_map_domain(
+	__isl_keep isl_ast_build *build, __isl_take isl_union_map *umap)
+{
+	isl_multi_aff *values;
+
+	if (!build)
+		return isl_union_map_free(umap);
+
+	values = isl_multi_aff_copy(build->values);
+	umap = isl_union_map_preimage_domain_multi_aff(umap, values);
+
+	return umap;
+}
+
+/* Is the current dimension known to attain only a single value?
+ */
+int isl_ast_build_has_value(__isl_keep isl_ast_build *build)
+{
+	if (!build)
+		return -1;
+
+	return build->value != NULL;
+}
+
+/* Simplify the basic set "bset" based on what we know about
+ * the iterators of already generated loops.
+ *
+ * "bset" is assumed to live in the (internal) schedule domain.
+ */
+__isl_give isl_basic_set *isl_ast_build_compute_gist_basic_set(
+	__isl_keep isl_ast_build *build, __isl_take isl_basic_set *bset)
+{
+	if (!build)
+		goto error;
+
+	bset = isl_basic_set_preimage_multi_aff(bset,
+					isl_multi_aff_copy(build->values));
+	bset = isl_basic_set_gist(bset,
+			isl_set_simple_hull(isl_set_copy(build->domain)));
+
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Simplify the set "set" based on what we know about
+ * the iterators of already generated loops.
+ *
+ * "set" is assumed to live in the (internal) schedule domain.
+ */
+__isl_give isl_set *isl_ast_build_compute_gist(
+	__isl_keep isl_ast_build *build, __isl_take isl_set *set)
+{
+	if (!build)
+		goto error;
+
+	set = isl_set_preimage_multi_aff(set,
+					isl_multi_aff_copy(build->values));
+	set = isl_set_gist(set, isl_set_copy(build->domain));
+
+	return set;
+error:
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Include information about what we know about the iterators of
+ * already generated loops to "set".
+ *
+ * We currently only plug in the known affine values of outer loop
+ * iterators.
+ * In principle we could also introduce equalities or even other
+ * constraints implied by the intersection of "set" and build->domain.
+ */
+__isl_give isl_set *isl_ast_build_specialize(__isl_keep isl_ast_build *build,
+	__isl_take isl_set *set)
+{
+	if (!build)
+		return isl_set_free(set);
+
+	return isl_set_preimage_multi_aff(set,
+					isl_multi_aff_copy(build->values));
+}
+
+/* Simplify the map "map" based on what we know about
+ * the iterators of already generated loops.
+ *
+ * The domain of "map" is assumed to live in the (internal) schedule domain.
+ */
+__isl_give isl_map *isl_ast_build_compute_gist_map_domain(
+	__isl_keep isl_ast_build *build, __isl_take isl_map *map)
+{
+	if (!build)
+		goto error;
+
+	map = isl_map_gist_domain(map, isl_set_copy(build->domain));
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Simplify the affine expression "aff" based on what we know about
+ * the iterators of already generated loops.
+ *
+ * The domain of "aff" is assumed to live in the (internal) schedule domain.
+ */
+__isl_give isl_aff *isl_ast_build_compute_gist_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_aff *aff)
+{
+	if (!build)
+		goto error;
+
+	aff = isl_aff_gist(aff, isl_set_copy(build->domain));
+
+	return aff;
+error:
+	isl_aff_free(aff);
+	return NULL;
+}
+
+/* Simplify the piecewise affine expression "aff" based on what we know about
+ * the iterators of already generated loops.
+ *
+ * The domain of "pa" is assumed to live in the (internal) schedule domain.
+ */
+__isl_give isl_pw_aff *isl_ast_build_compute_gist_pw_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_pw_aff *pa)
+{
+	if (!build)
+		goto error;
+
+	if (!isl_set_is_params(build->domain))
+		pa = isl_pw_aff_pullback_multi_aff(pa,
+					isl_multi_aff_copy(build->values));
+	pa = isl_pw_aff_gist(pa, isl_set_copy(build->domain));
+
+	return pa;
+error:
+	isl_pw_aff_free(pa);
+	return NULL;
+}
+
+/* Simplify the piecewise multi-affine expression "aff" based on what
+ * we know about the iterators of already generated loops.
+ *
+ * The domain of "pma" is assumed to live in the (internal) schedule domain.
+ */
+__isl_give isl_pw_multi_aff *isl_ast_build_compute_gist_pw_multi_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_pw_multi_aff *pma)
+{
+	if (!build)
+		goto error;
+
+	pma = isl_pw_multi_aff_pullback_multi_aff(pma,
+					isl_multi_aff_copy(build->values));
+	pma = isl_pw_multi_aff_gist(pma, isl_set_copy(build->domain));
+
+	return pma;
+error:
+	isl_pw_multi_aff_free(pma);
+	return NULL;
+}
+
+/* Extract the schedule domain of the given type from build->options
+ * at the current depth.
+ *
+ * In particular, find the subset of build->options that is of
+ * the following form
+ *
+ *	schedule_domain -> type[depth]
+ *
+ * and return the corresponding domain, after eliminating inner dimensions
+ * and divs that depend on the current dimension.
+ *
+ * Note that the domain of build->options has been reformulated
+ * in terms of the internal build space in embed_options,
+ * but the position is still that within the current code generation.
+ */
+__isl_give isl_set *isl_ast_build_get_option_domain(
+	__isl_keep isl_ast_build *build,
+	enum isl_ast_build_domain_type type)
+{
+	const char *name;
+	isl_space *space;
+	isl_map *option;
+	isl_set *domain;
+	int local_pos;
+
+	if (!build)
+		return NULL;
+
+	name = option_str[type];
+	local_pos = build->depth - build->outer_pos;
+
+	space = isl_ast_build_get_space(build, 1);
+	space = isl_space_from_domain(space);
+	space = isl_space_add_dims(space, isl_dim_out, 1);
+	space = isl_space_set_tuple_name(space, isl_dim_out, name);
+
+	option = isl_union_map_extract_map(build->options, space);
+	option = isl_map_fix_si(option, isl_dim_out, 0, local_pos);
+
+	domain = isl_map_domain(option);
+	domain = isl_ast_build_eliminate(build, domain);
+
+	return domain;
+}
+
+/* Extract the separation class mapping at the current depth.
+ *
+ * In particular, find and return the subset of build->options that is of
+ * the following form
+ *
+ *	schedule_domain -> separation_class[[depth] -> [class]]
+ *
+ * The caller is expected to eliminate inner dimensions from the domain.
+ *
+ * Note that the domain of build->options has been reformulated
+ * in terms of the internal build space in embed_options,
+ * but the position is still that within the current code generation.
+ */
+__isl_give isl_map *isl_ast_build_get_separation_class(
+	__isl_keep isl_ast_build *build)
+{
+	isl_ctx *ctx;
+	isl_space *space_sep, *space;
+	isl_map *res;
+	int local_pos;
+
+	if (!build)
+		return NULL;
+
+	local_pos = build->depth - build->outer_pos;
+	ctx = isl_ast_build_get_ctx(build);
+	space_sep = isl_space_alloc(ctx, 0, 1, 1);
+	space_sep = isl_space_wrap(space_sep);
+	space_sep = isl_space_set_tuple_name(space_sep, isl_dim_set,
+						"separation_class");
+	space = isl_ast_build_get_space(build, 1);
+	space_sep = isl_space_align_params(space_sep, isl_space_copy(space));
+	space = isl_space_map_from_domain_and_range(space, space_sep);
+
+	res = isl_union_map_extract_map(build->options, space);
+	res = isl_map_fix_si(res, isl_dim_out, 0, local_pos);
+	res = isl_map_coalesce(res);
+
+	return res;
+}
+
+/* Eliminate dimensions inner to the current dimension.
+ */
+__isl_give isl_set *isl_ast_build_eliminate_inner(
+	__isl_keep isl_ast_build *build, __isl_take isl_set *set)
+{
+	int dim;
+	int depth;
+
+	if (!build)
+		return isl_set_free(set);
+
+	dim = isl_set_dim(set, isl_dim_set);
+	depth = build->depth;
+	set = isl_set_detect_equalities(set);
+	set = isl_set_eliminate(set, isl_dim_set, depth + 1, dim - (depth + 1));
+
+	return set;
+}
+
+/* Eliminate unknown divs and divs that depend on the current dimension.
+ *
+ * Note that during the elimination of unknown divs, we may discover
+ * an explicit representation of some other unknown divs, which may
+ * depend on the current dimension.  We therefore need to eliminate
+ * unknown divs first.
+ */
+__isl_give isl_set *isl_ast_build_eliminate_divs(
+	__isl_keep isl_ast_build *build, __isl_take isl_set *set)
+{
+	int depth;
+
+	if (!build)
+		return isl_set_free(set);
+
+	set = isl_set_remove_unknown_divs(set);
+	depth = build->depth;
+	set = isl_set_remove_divs_involving_dims(set, isl_dim_set, depth, 1);
+
+	return set;
+}
+
+/* Eliminate dimensions inner to the current dimension as well as
+ * unknown divs and divs that depend on the current dimension.
+ * The result then consists only of constraints that are independent
+ * of the current dimension and upper and lower bounds on the current
+ * dimension.
+ */
+__isl_give isl_set *isl_ast_build_eliminate(
+	__isl_keep isl_ast_build *build, __isl_take isl_set *domain)
+{
+	domain = isl_ast_build_eliminate_inner(build, domain);
+	domain = isl_ast_build_eliminate_divs(build, domain);
+	return domain;
+}
+
+/* Replace build->single_valued by "sv".
+ */
+__isl_give isl_ast_build *isl_ast_build_set_single_valued(
+	__isl_take isl_ast_build *build, int sv)
+{
+	if (!build)
+		return build;
+	if (build->single_valued == sv)
+		return build;
+	build = isl_ast_build_cow(build);
+	if (!build)
+		return build;
+	build->single_valued = sv;
+
+	return build;
+}
diff --git a/polly/lib/External/isl/isl_ast_build_expr.c b/polly/lib/External/isl/isl_ast_build_expr.c
new file mode 100644
index 00000000000..2c3bfa3a6f1
--- /dev/null
+++ b/polly/lib/External/isl/isl_ast_build_expr.c
@@ -0,0 +1,1779 @@
+/*
+ * Copyright 2012-2014 Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege,
+ * Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+#include <isl/ilp.h>
+#include <isl_ast_build_expr.h>
+#include <isl_ast_private.h>
+#include <isl_ast_build_private.h>
+
+/* Compute the "opposite" of the (numerator of the) argument of a div
+ * with denonimator "d".
+ *
+ * In particular, compute
+ *
+ *	-aff + (d - 1)
+ */
+static __isl_give isl_aff *oppose_div_arg(__isl_take isl_aff *aff,
+	__isl_take isl_val *d)
+{
+	aff = isl_aff_neg(aff);
+	aff = isl_aff_add_constant_val(aff, d);
+	aff = isl_aff_add_constant_si(aff, -1);
+
+	return aff;
+}
+
+/* Create an isl_ast_expr evaluating the div at position "pos" in "ls".
+ * The result is simplified in terms of build->domain.
+ *
+ * *change_sign is set by this function if the sign of
+ * the expression has changed.
+ * "ls" is known to be non-NULL.
+ *
+ * Let the div be of the form floor(e/d).
+ * If the ast_build_prefer_pdiv option is set then we check if "e"
+ * is non-negative, so that we can generate
+ *
+ *	(pdiv_q, expr(e), expr(d))
+ *
+ * instead of
+ *
+ *	(fdiv_q, expr(e), expr(d))
+ *
+ * If the ast_build_prefer_pdiv option is set and
+ * if "e" is not non-negative, then we check if "-e + d - 1" is non-negative.
+ * If so, we can rewrite
+ *
+ *	floor(e/d) = -ceil(-e/d) = -floor((-e + d - 1)/d)
+ *
+ * and still use pdiv_q.
+ */
+static __isl_give isl_ast_expr *var_div(int *change_sign,
+	__isl_keep isl_local_space *ls,
+	int pos, __isl_keep isl_ast_build *build)
+{
+	isl_ctx *ctx = isl_local_space_get_ctx(ls);
+	isl_aff *aff;
+	isl_ast_expr *num, *den;
+	isl_val *d;
+	enum isl_ast_op_type type;
+
+	aff = isl_local_space_get_div(ls, pos);
+	d = isl_aff_get_denominator_val(aff);
+	aff = isl_aff_scale_val(aff, isl_val_copy(d));
+	den = isl_ast_expr_from_val(isl_val_copy(d));
+
+	type = isl_ast_op_fdiv_q;
+	if (isl_options_get_ast_build_prefer_pdiv(ctx)) {
+		int non_neg = isl_ast_build_aff_is_nonneg(build, aff);
+		if (non_neg >= 0 && !non_neg) {
+			isl_aff *opp = oppose_div_arg(isl_aff_copy(aff),
+							isl_val_copy(d));
+			non_neg = isl_ast_build_aff_is_nonneg(build, opp);
+			if (non_neg >= 0 && non_neg) {
+				*change_sign = 1;
+				isl_aff_free(aff);
+				aff = opp;
+			} else
+				isl_aff_free(opp);
+		}
+		if (non_neg < 0)
+			aff = isl_aff_free(aff);
+		else if (non_neg)
+			type = isl_ast_op_pdiv_q;
+	}
+
+	isl_val_free(d);
+	num = isl_ast_expr_from_aff(aff, build);
+	return isl_ast_expr_alloc_binary(type, num, den);
+}
+
+/* Create an isl_ast_expr evaluating the specified dimension of "ls".
+ * The result is simplified in terms of build->domain.
+ *
+ * *change_sign is set by this function if the sign of
+ * the expression has changed.
+ *
+ * The isl_ast_expr is constructed based on the type of the dimension.
+ * - divs are constructed by var_div
+ * - set variables are constructed from the iterator isl_ids in "build"
+ * - parameters are constructed from the isl_ids in "ls"
+ */
+static __isl_give isl_ast_expr *var(int *change_sign,
+	__isl_keep isl_local_space *ls,
+	enum isl_dim_type type, int pos, __isl_keep isl_ast_build *build)
+{
+	isl_ctx *ctx = isl_local_space_get_ctx(ls);
+	isl_id *id;
+
+	if (type == isl_dim_div)
+		return var_div(change_sign, ls, pos, build);
+
+	if (type == isl_dim_set) {
+		id = isl_ast_build_get_iterator_id(build, pos);
+		return isl_ast_expr_from_id(id);
+	}
+
+	if (!isl_local_space_has_dim_id(ls, type, pos))
+		isl_die(ctx, isl_error_internal, "unnamed dimension",
+			return NULL);
+	id = isl_local_space_get_dim_id(ls, type, pos);
+	return isl_ast_expr_from_id(id);
+}
+
+/* Does "expr" represent the zero integer?
+ */
+static int ast_expr_is_zero(__isl_keep isl_ast_expr *expr)
+{
+	if (!expr)
+		return -1;
+	if (expr->type != isl_ast_expr_int)
+		return 0;
+	return isl_val_is_zero(expr->u.v);
+}
+
+/* Create an expression representing the sum of "expr1" and "expr2",
+ * provided neither of the two expressions is identically zero.
+ */
+static __isl_give isl_ast_expr *ast_expr_add(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2)
+{
+	if (!expr1 || !expr2)
+		goto error;
+
+	if (ast_expr_is_zero(expr1)) {
+		isl_ast_expr_free(expr1);
+		return expr2;
+	}
+
+	if (ast_expr_is_zero(expr2)) {
+		isl_ast_expr_free(expr2);
+		return expr1;
+	}
+
+	return isl_ast_expr_add(expr1, expr2);
+error:
+	isl_ast_expr_free(expr1);
+	isl_ast_expr_free(expr2);
+	return NULL;
+}
+
+/* Subtract expr2 from expr1.
+ *
+ * If expr2 is zero, we simply return expr1.
+ * If expr1 is zero, we return
+ *
+ *	(isl_ast_op_minus, expr2)
+ *
+ * Otherwise, we return
+ *
+ *	(isl_ast_op_sub, expr1, expr2)
+ */
+static __isl_give isl_ast_expr *ast_expr_sub(__isl_take isl_ast_expr *expr1,
+	__isl_take isl_ast_expr *expr2)
+{
+	if (!expr1 || !expr2)
+		goto error;
+
+	if (ast_expr_is_zero(expr2)) {
+		isl_ast_expr_free(expr2);
+		return expr1;
+	}
+
+	if (ast_expr_is_zero(expr1)) {
+		isl_ast_expr_free(expr1);
+		return isl_ast_expr_neg(expr2);
+	}
+
+	return isl_ast_expr_sub(expr1, expr2);
+error:
+	isl_ast_expr_free(expr1);
+	isl_ast_expr_free(expr2);
+	return NULL;
+}
+
+/* Return an isl_ast_expr that represents
+ *
+ *	v * (aff mod d)
+ *
+ * v is assumed to be non-negative.
+ * The result is simplified in terms of build->domain.
+ */
+static __isl_give isl_ast_expr *isl_ast_expr_mod(__isl_keep isl_val *v,
+	__isl_keep isl_aff *aff, __isl_keep isl_val *d,
+	__isl_keep isl_ast_build *build)
+{
+	isl_ctx *ctx;
+	isl_ast_expr *expr;
+	isl_ast_expr *c;
+
+	if (!aff)
+		return NULL;
+
+	ctx = isl_aff_get_ctx(aff);
+	expr = isl_ast_expr_from_aff(isl_aff_copy(aff), build);
+
+	c = isl_ast_expr_from_val(isl_val_copy(d));
+	expr = isl_ast_expr_alloc_binary(isl_ast_op_pdiv_r, expr, c);
+
+	if (!isl_val_is_one(v)) {
+		c = isl_ast_expr_from_val(isl_val_copy(v));
+		expr = isl_ast_expr_mul(c, expr);
+	}
+
+	return expr;
+}
+
+/* Create an isl_ast_expr that scales "expr" by "v".
+ *
+ * If v is 1, we simply return expr.
+ * If v is -1, we return
+ *
+ *	(isl_ast_op_minus, expr)
+ *
+ * Otherwise, we return
+ *
+ *	(isl_ast_op_mul, expr(v), expr)
+ */
+static __isl_give isl_ast_expr *scale(__isl_take isl_ast_expr *expr,
+	__isl_take isl_val *v)
+{
+	isl_ast_expr *c;
+
+	if (!expr || !v)
+		goto error;
+	if (isl_val_is_one(v)) {
+		isl_val_free(v);
+		return expr;
+	}
+
+	if (isl_val_is_negone(v)) {
+		isl_val_free(v);
+		expr = isl_ast_expr_neg(expr);
+	} else {
+		c = isl_ast_expr_from_val(v);
+		expr = isl_ast_expr_mul(c, expr);
+	}
+
+	return expr;
+error:
+	isl_val_free(v);
+	isl_ast_expr_free(expr);
+	return NULL;
+}
+
+/* Add an expression for "*v" times the specified dimension of "ls"
+ * to expr.
+ *
+ * Let e be the expression for the specified dimension,
+ * multiplied by the absolute value of "*v".
+ * If "*v" is negative, we create
+ *
+ *	(isl_ast_op_sub, expr, e)
+ *
+ * except when expr is trivially zero, in which case we create
+ *
+ *	(isl_ast_op_minus, e)
+ *
+ * instead.
+ *
+ * If "*v" is positive, we simply create
+ *
+ *	(isl_ast_op_add, expr, e)
+ *
+ */
+static __isl_give isl_ast_expr *isl_ast_expr_add_term(
+	__isl_take isl_ast_expr *expr,
+	__isl_keep isl_local_space *ls, enum isl_dim_type type, int pos,
+	__isl_take isl_val *v, __isl_keep isl_ast_build *build)
+{
+	isl_ast_expr *term;
+	int change_sign;
+
+	if (!expr)
+		return NULL;
+
+	change_sign = 0;
+	term = var(&change_sign, ls, type, pos, build);
+	if (change_sign)
+		v = isl_val_neg(v);
+
+	if (isl_val_is_neg(v) && !ast_expr_is_zero(expr)) {
+		v = isl_val_neg(v);
+		term = scale(term, v);
+		return ast_expr_sub(expr, term);
+	} else {
+		term = scale(term, v);
+		return ast_expr_add(expr, term);
+	}
+}
+
+/* Add an expression for "v" to expr.
+ */
+static __isl_give isl_ast_expr *isl_ast_expr_add_int(
+	__isl_take isl_ast_expr *expr, __isl_take isl_val *v)
+{
+	isl_ctx *ctx;
+	isl_ast_expr *expr_int;
+
+	if (!expr || !v)
+		goto error;
+
+	if (isl_val_is_zero(v)) {
+		isl_val_free(v);
+		return expr;
+	}
+
+	ctx = isl_ast_expr_get_ctx(expr);
+	if (isl_val_is_neg(v) && !ast_expr_is_zero(expr)) {
+		v = isl_val_neg(v);
+		expr_int = isl_ast_expr_from_val(v);
+		return ast_expr_sub(expr, expr_int);
+	} else {
+		expr_int = isl_ast_expr_from_val(v);
+		return ast_expr_add(expr, expr_int);
+	}
+error:
+	isl_ast_expr_free(expr);
+	isl_val_free(v);
+	return NULL;
+}
+
+/* Internal data structure used inside extract_modulos.
+ *
+ * If any modulo expressions are detected in "aff", then the
+ * expression is removed from "aff" and added to either "pos" or "neg"
+ * depending on the sign of the coefficient of the modulo expression
+ * inside "aff".
+ *
+ * "add" is an expression that needs to be added to "aff" at the end of
+ * the computation.  It is NULL as long as no modulos have been extracted.
+ *
+ * "i" is the position in "aff" of the div under investigation
+ * "v" is the coefficient in "aff" of the div
+ * "div" is the argument of the div, with the denominator removed
+ * "d" is the original denominator of the argument of the div
+ *
+ * "nonneg" is an affine expression that is non-negative over "build"
+ * and that can be used to extract a modulo expression from "div".
+ * In particular, if "sign" is 1, then the coefficients of "nonneg"
+ * are equal to those of "div" modulo "d".  If "sign" is -1, then
+ * the coefficients of "nonneg" are opposite to those of "div" modulo "d".
+ * If "sign" is 0, then no such affine expression has been found (yet).
+ */
+struct isl_extract_mod_data {
+	isl_ast_build *build;
+	isl_aff *aff;
+
+	isl_ast_expr *pos;
+	isl_ast_expr *neg;
+
+	isl_aff *add;
+
+	int i;
+	isl_val *v;
+	isl_val *d;
+	isl_aff *div;
+
+	isl_aff *nonneg;
+	int sign;
+};
+
+/* Given that data->v * div_i in data->aff is equal to
+ *
+ *	f * (term - (arg mod d))
+ *
+ * with data->d * f = data->v, add
+ *
+ *	f * term
+ *
+ * to data->add and
+ *
+ *	abs(f) * (arg mod d)
+ *
+ * to data->neg or data->pos depending on the sign of -f.
+ */
+static int extract_term_and_mod(struct isl_extract_mod_data *data,
+	__isl_take isl_aff *term, __isl_take isl_aff *arg)
+{
+	isl_ast_expr *expr;
+	int s;
+
+	data->v = isl_val_div(data->v, isl_val_copy(data->d));
+	s = isl_val_sgn(data->v);
+	data->v = isl_val_abs(data->v);
+	expr = isl_ast_expr_mod(data->v, arg, data->d, data->build);
+	isl_aff_free(arg);
+	if (s > 0)
+		data->neg = ast_expr_add(data->neg, expr);
+	else
+		data->pos = ast_expr_add(data->pos, expr);
+	data->aff = isl_aff_set_coefficient_si(data->aff,
+						isl_dim_div, data->i, 0);
+	if (s < 0)
+		data->v = isl_val_neg(data->v);
+	term = isl_aff_scale_val(data->div, isl_val_copy(data->v));
+
+	if (!data->add)
+		data->add = term;
+	else
+		data->add = isl_aff_add(data->add, term);
+	if (!data->add)
+		return -1;
+
+	return 0;
+}
+
+/* Given that data->v * div_i in data->aff is of the form
+ *
+ *	f * d * floor(div/d)
+ *
+ * with div nonnegative on data->build, rewrite it as
+ *
+ *	f * (div - (div mod d)) = f * div - f * (div mod d)
+ *
+ * and add
+ *
+ *	f * div
+ *
+ * to data->add and
+ *
+ *	abs(f) * (div mod d)
+ *
+ * to data->neg or data->pos depending on the sign of -f.
+ */
+static int extract_mod(struct isl_extract_mod_data *data)
+{
+	return extract_term_and_mod(data, isl_aff_copy(data->div),
+			isl_aff_copy(data->div));
+}
+
+/* Given that data->v * div_i in data->aff is of the form
+ *
+ *	f * d * floor(div/d)					(1)
+ *
+ * check if div is non-negative on data->build and, if so,
+ * extract the corresponding modulo from data->aff.
+ * If not, then check if
+ *
+ *	-div + d - 1
+ *
+ * is non-negative on data->build.  If so, replace (1) by
+ *
+ *	-f * d * floor((-div + d - 1)/d)
+ *
+ * and extract the corresponding modulo from data->aff.
+ *
+ * This function may modify data->div.
+ */
+static int extract_nonneg_mod(struct isl_extract_mod_data *data)
+{
+	int mod;
+
+	mod = isl_ast_build_aff_is_nonneg(data->build, data->div);
+	if (mod < 0)
+		goto error;
+	if (mod)
+		return extract_mod(data);
+
+	data->div = oppose_div_arg(data->div, isl_val_copy(data->d));
+	mod = isl_ast_build_aff_is_nonneg(data->build, data->div);
+	if (mod < 0)
+		goto error;
+	if (mod) {
+		data->v = isl_val_neg(data->v);
+		return extract_mod(data);
+	}
+
+	return 0;
+error:
+	data->aff = isl_aff_free(data->aff);
+	return -1;
+}
+
+/* Is the affine expression of constraint "c" "simpler" than data->nonneg
+ * for use in extracting a modulo expression?
+ *
+ * We currently only consider the constant term of the affine expression.
+ * In particular, we prefer the affine expression with the smallest constant
+ * term.
+ * This means that if there are two constraints, say x >= 0 and -x + 10 >= 0,
+ * then we would pick x >= 0
+ *
+ * More detailed heuristics could be used if it turns out that there is a need.
+ */
+static int mod_constraint_is_simpler(struct isl_extract_mod_data *data,
+	__isl_keep isl_constraint *c)
+{
+	isl_val *v1, *v2;
+	int simpler;
+
+	if (!data->nonneg)
+		return 1;
+
+	v1 = isl_val_abs(isl_constraint_get_constant_val(c));
+	v2 = isl_val_abs(isl_aff_get_constant_val(data->nonneg));
+	simpler = isl_val_lt(v1, v2);
+	isl_val_free(v1);
+	isl_val_free(v2);
+
+	return simpler;
+}
+
+/* Check if the coefficients of "c" are either equal or opposite to those
+ * of data->div modulo data->d.  If so, and if "c" is "simpler" than
+ * data->nonneg, then replace data->nonneg by the affine expression of "c"
+ * and set data->sign accordingly.
+ *
+ * Both "c" and data->div are assumed not to involve any integer divisions.
+ *
+ * Before we start the actual comparison, we first quickly check if
+ * "c" and data->div have the same non-zero coefficients.
+ * If not, then we assume that "c" is not of the desired form.
+ * Note that while the coefficients of data->div can be reasonably expected
+ * not to involve any coefficients that are multiples of d, "c" may
+ * very well involve such coefficients.  This means that we may actually
+ * miss some cases.
+ */
+static int check_parallel_or_opposite(__isl_take isl_constraint *c, void *user)
+{
+	struct isl_extract_mod_data *data = user;
+	enum isl_dim_type c_type[2] = { isl_dim_param, isl_dim_set };
+	enum isl_dim_type a_type[2] = { isl_dim_param, isl_dim_in };
+	int i, t;
+	int n[2];
+	int parallel = 1, opposite = 1;
+
+	for (t = 0; t < 2; ++t) {
+		n[t] = isl_constraint_dim(c, c_type[t]);
+		for (i = 0; i < n[t]; ++i) {
+			int a, b;
+
+			a = isl_constraint_involves_dims(c, c_type[t], i, 1);
+			b = isl_aff_involves_dims(data->div, a_type[t], i, 1);
+			if (a != b)
+				parallel = opposite = 0;
+		}
+	}
+
+	for (t = 0; t < 2; ++t) {
+		for (i = 0; i < n[t]; ++i) {
+			isl_val *v1, *v2;
+
+			if (!parallel && !opposite)
+				break;
+			v1 = isl_constraint_get_coefficient_val(c,
+								c_type[t], i);
+			v2 = isl_aff_get_coefficient_val(data->div,
+								a_type[t], i);
+			if (parallel) {
+				v1 = isl_val_sub(v1, isl_val_copy(v2));
+				parallel = isl_val_is_divisible_by(v1, data->d);
+				v1 = isl_val_add(v1, isl_val_copy(v2));
+			}
+			if (opposite) {
+				v1 = isl_val_add(v1, isl_val_copy(v2));
+				opposite = isl_val_is_divisible_by(v1, data->d);
+			}
+			isl_val_free(v1);
+			isl_val_free(v2);
+		}
+	}
+
+	if ((parallel || opposite) && mod_constraint_is_simpler(data, c)) {
+		isl_aff_free(data->nonneg);
+		data->nonneg = isl_constraint_get_aff(c);
+		data->sign = parallel ? 1 : -1;
+	}
+
+	isl_constraint_free(c);
+
+	if (data->sign != 0 && data->nonneg == NULL)
+		return -1;
+
+	return 0;
+}
+
+/* Given that data->v * div_i in data->aff is of the form
+ *
+ *	f * d * floor(div/d)					(1)
+ *
+ * see if we can find an expression div' that is non-negative over data->build
+ * and that is related to div through
+ *
+ *	div' = div + d * e
+ *
+ * or
+ *
+ *	div' = -div + d - 1 + d * e
+ *
+ * with e some affine expression.
+ * If so, we write (1) as
+ *
+ *	f * div + f * (div' mod d)
+ *
+ * or
+ *
+ *	-f * (-div + d - 1) - f * (div' mod d)
+ *
+ * exploiting (in the second case) the fact that
+ *
+ *	f * d * floor(div/d) =	-f * d * floor((-div + d - 1)/d)
+ *
+ *
+ * We first try to find an appropriate expression for div'
+ * from the constraints of data->build->domain (which is therefore
+ * guaranteed to be non-negative on data->build), where we remove
+ * any integer divisions from the constraints and skip this step
+ * if "div" itself involves any integer divisions.
+ * If we cannot find an appropriate expression this way, then
+ * we pass control to extract_nonneg_mod where check
+ * if div or "-div + d -1" themselves happen to be
+ * non-negative on data->build.
+ *
+ * While looking for an appropriate constraint in data->build->domain,
+ * we ignore the constant term, so after finding such a constraint,
+ * we still need to fix up the constant term.
+ * In particular, if a is the constant term of "div"
+ * (or d - 1 - the constant term of "div" if data->sign < 0)
+ * and b is the constant term of the constraint, then we need to find
+ * a non-negative constant c such that
+ *
+ *	b + c \equiv a	mod d
+ *
+ * We therefore take
+ *
+ *	c = (a - b) mod d
+ *
+ * and add it to b to obtain the constant term of div'.
+ * If this constant term is "too negative", then we add an appropriate
+ * multiple of d to make it positive.
+ *
+ *
+ * Note that the above is a only a very simple heuristic for finding an
+ * appropriate expression.  We could try a bit harder by also considering
+ * sums of constraints that involve disjoint sets of variables or
+ * we could consider arbitrary linear combinations of constraints,
+ * although that could potentially be much more expensive as it involves
+ * the solution of an LP problem.
+ *
+ * In particular, if v_i is a column vector representing constraint i,
+ * w represents div and e_i is the i-th unit vector, then we are looking
+ * for a solution of the constraints
+ *
+ *	\sum_i lambda_i v_i = w + \sum_i alpha_i d e_i
+ *
+ * with \lambda_i >= 0 and alpha_i of unrestricted sign.
+ * If we are not just interested in a non-negative expression, but
+ * also in one with a minimal range, then we don't just want
+ * c = \sum_i lambda_i v_i to be non-negative over the domain,
+ * but also beta - c = \sum_i mu_i v_i, where beta is a scalar
+ * that we want to minimize and we now also have to take into account
+ * the constant terms of the constraints.
+ * Alternatively, we could first compute the dual of the domain
+ * and plug in the constraints on the coefficients.
+ */
+static int try_extract_mod(struct isl_extract_mod_data *data)
+{
+	isl_basic_set *hull;
+	isl_val *v1, *v2;
+	int r;
+
+	if (!data->build)
+		goto error;
+
+	int n = isl_aff_dim(data->div, isl_dim_div);
+
+	if (isl_aff_involves_dims(data->div, isl_dim_div, 0, n))
+		return extract_nonneg_mod(data);
+
+	hull = isl_set_simple_hull(isl_set_copy(data->build->domain));
+	hull = isl_basic_set_remove_divs(hull);
+	data->sign = 0;
+	data->nonneg = NULL;
+	r = isl_basic_set_foreach_constraint(hull, &check_parallel_or_opposite,
+					data);
+	isl_basic_set_free(hull);
+
+	if (!data->sign || r < 0) {
+		isl_aff_free(data->nonneg);
+		if (r < 0)
+			goto error;
+		return extract_nonneg_mod(data);
+	}
+
+	v1 = isl_aff_get_constant_val(data->div);
+	v2 = isl_aff_get_constant_val(data->nonneg);
+	if (data->sign < 0) {
+		v1 = isl_val_neg(v1);
+		v1 = isl_val_add(v1, isl_val_copy(data->d));
+		v1 = isl_val_sub_ui(v1, 1);
+	}
+	v1 = isl_val_sub(v1, isl_val_copy(v2));
+	v1 = isl_val_mod(v1, isl_val_copy(data->d));
+	v1 = isl_val_add(v1, v2);
+	v2 = isl_val_div(isl_val_copy(v1), isl_val_copy(data->d));
+	v2 = isl_val_ceil(v2);
+	if (isl_val_is_neg(v2)) {
+		v2 = isl_val_mul(v2, isl_val_copy(data->d));
+		v1 = isl_val_sub(v1, isl_val_copy(v2));
+	}
+	data->nonneg = isl_aff_set_constant_val(data->nonneg, v1);
+	isl_val_free(v2);
+
+	if (data->sign < 0) {
+		data->div = oppose_div_arg(data->div, isl_val_copy(data->d));
+		data->v = isl_val_neg(data->v);
+	}
+
+	return extract_term_and_mod(data,
+				    isl_aff_copy(data->div), data->nonneg);
+error:
+	data->aff = isl_aff_free(data->aff);
+	return -1;
+}
+
+/* Check if "data->aff" involves any (implicit) modulo computations based
+ * on div "data->i".
+ * If so, remove them from aff and add expressions corresponding
+ * to those modulo computations to data->pos and/or data->neg.
+ *
+ * "aff" is assumed to be an integer affine expression.
+ *
+ * In particular, check if (v * div_j) is of the form
+ *
+ *	f * m * floor(a / m)
+ *
+ * and, if so, rewrite it as
+ *
+ *	f * (a - (a mod m)) = f * a - f * (a mod m)
+ *
+ * and extract out -f * (a mod m).
+ * In particular, if f > 0, we add (f * (a mod m)) to *neg.
+ * If f < 0, we add ((-f) * (a mod m)) to *pos.
+ *
+ * Note that in order to represent "a mod m" as
+ *
+ *	(isl_ast_op_pdiv_r, a, m)
+ *
+ * we need to make sure that a is non-negative.
+ * If not, we check if "-a + m - 1" is non-negative.
+ * If so, we can rewrite
+ *
+ *	floor(a/m) = -ceil(-a/m) = -floor((-a + m - 1)/m)
+ *
+ * and still extract a modulo.
+ */
+static int extract_modulo(struct isl_extract_mod_data *data)
+{
+	data->div = isl_aff_get_div(data->aff, data->i);
+	data->d = isl_aff_get_denominator_val(data->div);
+	if (isl_val_is_divisible_by(data->v, data->d)) {
+		data->div = isl_aff_scale_val(data->div, isl_val_copy(data->d));
+		if (try_extract_mod(data) < 0)
+			data->aff = isl_aff_free(data->aff);
+	}
+	isl_aff_free(data->div);
+	isl_val_free(data->d);
+	return 0;
+}
+
+/* Check if "aff" involves any (implicit) modulo computations.
+ * If so, remove them from aff and add expressions corresponding
+ * to those modulo computations to *pos and/or *neg.
+ * We only do this if the option ast_build_prefer_pdiv is set.
+ *
+ * "aff" is assumed to be an integer affine expression.
+ *
+ * A modulo expression is of the form
+ *
+ *	a mod m = a - m * floor(a / m)
+ *
+ * To detect them in aff, we look for terms of the form
+ *
+ *	f * m * floor(a / m)
+ *
+ * rewrite them as
+ *
+ *	f * (a - (a mod m)) = f * a - f * (a mod m)
+ *
+ * and extract out -f * (a mod m).
+ * In particular, if f > 0, we add (f * (a mod m)) to *neg.
+ * If f < 0, we add ((-f) * (a mod m)) to *pos.
+ */
+static __isl_give isl_aff *extract_modulos(__isl_take isl_aff *aff,
+	__isl_keep isl_ast_expr **pos, __isl_keep isl_ast_expr **neg,
+	__isl_keep isl_ast_build *build)
+{
+	struct isl_extract_mod_data data = { build, aff, *pos, *neg };
+	isl_ctx *ctx;
+	int n;
+
+	if (!aff)
+		return NULL;
+
+	ctx = isl_aff_get_ctx(aff);
+	if (!isl_options_get_ast_build_prefer_pdiv(ctx))
+		return aff;
+
+	n = isl_aff_dim(data.aff, isl_dim_div);
+	for (data.i = 0; data.i < n; ++data.i) {
+		data.v = isl_aff_get_coefficient_val(data.aff,
+							isl_dim_div, data.i);
+		if (!data.v)
+			return isl_aff_free(aff);
+		if (isl_val_is_zero(data.v) ||
+		    isl_val_is_one(data.v) || isl_val_is_negone(data.v)) {
+			isl_val_free(data.v);
+			continue;
+		}
+		if (extract_modulo(&data) < 0)
+			data.aff = isl_aff_free(data.aff);
+		isl_val_free(data.v);
+		if (!data.aff)
+			break;
+	}
+
+	if (data.add)
+		data.aff = isl_aff_add(data.aff, data.add);
+
+	*pos = data.pos;
+	*neg = data.neg;
+	return data.aff;
+}
+
+/* Check if aff involves any non-integer coefficients.
+ * If so, split aff into
+ *
+ *	aff = aff1 + (aff2 / d)
+ *
+ * with both aff1 and aff2 having only integer coefficients.
+ * Return aff1 and add (aff2 / d) to *expr.
+ */
+static __isl_give isl_aff *extract_rational(__isl_take isl_aff *aff,
+	__isl_keep isl_ast_expr **expr, __isl_keep isl_ast_build *build)
+{
+	int i, j, n;
+	isl_aff *rat = NULL;
+	isl_local_space *ls = NULL;
+	isl_ast_expr *rat_expr;
+	isl_val *v, *d;
+	enum isl_dim_type t[] = { isl_dim_param, isl_dim_in, isl_dim_div };
+	enum isl_dim_type l[] = { isl_dim_param, isl_dim_set, isl_dim_div };
+
+	if (!aff)
+		return NULL;
+	d = isl_aff_get_denominator_val(aff);
+	if (!d)
+		goto error;
+	if (isl_val_is_one(d)) {
+		isl_val_free(d);
+		return aff;
+	}
+
+	aff = isl_aff_scale_val(aff, isl_val_copy(d));
+
+	ls = isl_aff_get_domain_local_space(aff);
+	rat = isl_aff_zero_on_domain(isl_local_space_copy(ls));
+
+	for (i = 0; i < 3; ++i) {
+		n = isl_aff_dim(aff, t[i]);
+		for (j = 0; j < n; ++j) {
+			isl_aff *rat_j;
+
+			v = isl_aff_get_coefficient_val(aff, t[i], j);
+			if (!v)
+				goto error;
+			if (isl_val_is_divisible_by(v, d)) {
+				isl_val_free(v);
+				continue;
+			}
+			rat_j = isl_aff_var_on_domain(isl_local_space_copy(ls),
+							l[i], j);
+			rat_j = isl_aff_scale_val(rat_j, v);
+			rat = isl_aff_add(rat, rat_j);
+		}
+	}
+
+	v = isl_aff_get_constant_val(aff);
+	if (isl_val_is_divisible_by(v, d)) {
+		isl_val_free(v);
+	} else {
+		isl_aff *rat_0;
+
+		rat_0 = isl_aff_val_on_domain(isl_local_space_copy(ls), v);
+		rat = isl_aff_add(rat, rat_0);
+	}
+
+	isl_local_space_free(ls);
+
+	aff = isl_aff_sub(aff, isl_aff_copy(rat));
+	aff = isl_aff_scale_down_val(aff, isl_val_copy(d));
+
+	rat_expr = isl_ast_expr_from_aff(rat, build);
+	rat_expr = isl_ast_expr_div(rat_expr, isl_ast_expr_from_val(d));
+	*expr = ast_expr_add(*expr, rat_expr);
+
+	return aff;
+error:
+	isl_aff_free(rat);
+	isl_local_space_free(ls);
+	isl_aff_free(aff);
+	isl_val_free(d);
+	return NULL;
+}
+
+/* Construct an isl_ast_expr that evaluates the affine expression "aff",
+ * The result is simplified in terms of build->domain.
+ *
+ * We first extract hidden modulo computations from the affine expression
+ * and then add terms for each variable with a non-zero coefficient.
+ * Finally, if the affine expression has a non-trivial denominator,
+ * we divide the resulting isl_ast_expr by this denominator.
+ */
+__isl_give isl_ast_expr *isl_ast_expr_from_aff(__isl_take isl_aff *aff,
+	__isl_keep isl_ast_build *build)
+{
+	int i, j;
+	int n;
+	isl_val *v;
+	isl_ctx *ctx = isl_aff_get_ctx(aff);
+	isl_ast_expr *expr, *expr_neg;
+	enum isl_dim_type t[] = { isl_dim_param, isl_dim_in, isl_dim_div };
+	enum isl_dim_type l[] = { isl_dim_param, isl_dim_set, isl_dim_div };
+	isl_local_space *ls;
+
+	if (!aff)
+		return NULL;
+
+	expr = isl_ast_expr_alloc_int_si(ctx, 0);
+	expr_neg = isl_ast_expr_alloc_int_si(ctx, 0);
+
+	aff = extract_rational(aff, &expr, build);
+
+	aff = extract_modulos(aff, &expr, &expr_neg, build);
+	expr = ast_expr_sub(expr, expr_neg);
+
+	ls = isl_aff_get_domain_local_space(aff);
+
+	for (i = 0; i < 3; ++i) {
+		n = isl_aff_dim(aff, t[i]);
+		for (j = 0; j < n; ++j) {
+			v = isl_aff_get_coefficient_val(aff, t[i], j);
+			if (!v)
+				expr = isl_ast_expr_free(expr);
+			if (isl_val_is_zero(v)) {
+				isl_val_free(v);
+				continue;
+			}
+			expr = isl_ast_expr_add_term(expr,
+							ls, l[i], j, v, build);
+		}
+	}
+
+	v = isl_aff_get_constant_val(aff);
+	expr = isl_ast_expr_add_int(expr, v);
+
+	isl_local_space_free(ls);
+	isl_aff_free(aff);
+	return expr;
+}
+
+/* Add terms to "expr" for each variable in "aff" with a coefficient
+ * with sign equal to "sign".
+ * The result is simplified in terms of build->domain.
+ */
+static __isl_give isl_ast_expr *add_signed_terms(__isl_take isl_ast_expr *expr,
+	__isl_keep isl_aff *aff, int sign, __isl_keep isl_ast_build *build)
+{
+	int i, j;
+	isl_val *v;
+	enum isl_dim_type t[] = { isl_dim_param, isl_dim_in, isl_dim_div };
+	enum isl_dim_type l[] = { isl_dim_param, isl_dim_set, isl_dim_div };
+	isl_local_space *ls;
+
+	ls = isl_aff_get_domain_local_space(aff);
+
+	for (i = 0; i < 3; ++i) {
+		int n = isl_aff_dim(aff, t[i]);
+		for (j = 0; j < n; ++j) {
+			v = isl_aff_get_coefficient_val(aff, t[i], j);
+			if (sign * isl_val_sgn(v) <= 0) {
+				isl_val_free(v);
+				continue;
+			}
+			v = isl_val_abs(v);
+			expr = isl_ast_expr_add_term(expr,
+						ls, l[i], j, v, build);
+		}
+	}
+
+	isl_local_space_free(ls);
+
+	return expr;
+}
+
+/* Should the constant term "v" be considered positive?
+ *
+ * A positive constant will be added to "pos" by the caller,
+ * while a negative constant will be added to "neg".
+ * If either "pos" or "neg" is exactly zero, then we prefer
+ * to add the constant "v" to that side, irrespective of the sign of "v".
+ * This results in slightly shorter expressions and may reduce the risk
+ * of overflows.
+ */
+static int constant_is_considered_positive(__isl_keep isl_val *v,
+	__isl_keep isl_ast_expr *pos, __isl_keep isl_ast_expr *neg)
+{
+	if (ast_expr_is_zero(pos))
+		return 1;
+	if (ast_expr_is_zero(neg))
+		return 0;
+	return isl_val_is_pos(v);
+}
+
+/* Check if the equality
+ *
+ *	aff = 0
+ *
+ * represents a stride constraint on the integer division "pos".
+ *
+ * In particular, if the integer division "pos" is equal to
+ *
+ *	floor(e/d)
+ *
+ * then check if aff is equal to
+ *
+ *	e - d floor(e/d)
+ *
+ * or its opposite.
+ *
+ * If so, the equality is exactly
+ *
+ *	e mod d = 0
+ *
+ * Note that in principle we could also accept
+ *
+ *	e - d floor(e'/d)
+ *
+ * where e and e' differ by a constant.
+ */
+static int is_stride_constraint(__isl_keep isl_aff *aff, int pos)
+{
+	isl_aff *div;
+	isl_val *c, *d;
+	int eq;
+
+	div = isl_aff_get_div(aff, pos);
+	c = isl_aff_get_coefficient_val(aff, isl_dim_div, pos);
+	d = isl_aff_get_denominator_val(div);
+	eq = isl_val_abs_eq(c, d);
+	if (eq >= 0 && eq) {
+		aff = isl_aff_copy(aff);
+		aff = isl_aff_set_coefficient_si(aff, isl_dim_div, pos, 0);
+		div = isl_aff_scale_val(div, d);
+		if (isl_val_is_pos(c))
+			div = isl_aff_neg(div);
+		eq = isl_aff_plain_is_equal(div, aff);
+		isl_aff_free(aff);
+	} else
+		isl_val_free(d);
+	isl_val_free(c);
+	isl_aff_free(div);
+
+	return eq;
+}
+
+/* Are all coefficients of "aff" (zero or) negative?
+ */
+static int all_negative_coefficients(__isl_keep isl_aff *aff)
+{
+	int i, n;
+
+	if (!aff)
+		return 0;
+
+	n = isl_aff_dim(aff, isl_dim_param);
+	for (i = 0; i < n; ++i)
+		if (isl_aff_coefficient_sgn(aff, isl_dim_param, i) > 0)
+			return 0;
+
+	n = isl_aff_dim(aff, isl_dim_in);
+	for (i = 0; i < n; ++i)
+		if (isl_aff_coefficient_sgn(aff, isl_dim_in, i) > 0)
+			return 0;
+
+	return 1;
+}
+
+/* Give an equality of the form
+ *
+ *	aff = e - d floor(e/d) = 0
+ *
+ * or
+ *
+ *	aff = -e + d floor(e/d) = 0
+ *
+ * with the integer division "pos" equal to floor(e/d),
+ * construct the AST expression
+ *
+ *	(isl_ast_op_eq, (isl_ast_op_zdiv_r, expr(e), expr(d)), expr(0))
+ *
+ * If e only has negative coefficients, then construct
+ *
+ *	(isl_ast_op_eq, (isl_ast_op_zdiv_r, expr(-e), expr(d)), expr(0))
+ *
+ * instead.
+ */
+static __isl_give isl_ast_expr *extract_stride_constraint(
+	__isl_take isl_aff *aff, int pos, __isl_keep isl_ast_build *build)
+{
+	isl_ctx *ctx;
+	isl_val *c;
+	isl_ast_expr *expr, *cst;
+
+	if (!aff)
+		return NULL;
+
+	ctx = isl_aff_get_ctx(aff);
+
+	c = isl_aff_get_coefficient_val(aff, isl_dim_div, pos);
+	aff = isl_aff_set_coefficient_si(aff, isl_dim_div, pos, 0);
+
+	if (all_negative_coefficients(aff))
+		aff = isl_aff_neg(aff);
+
+	cst = isl_ast_expr_from_val(isl_val_abs(c));
+	expr = isl_ast_expr_from_aff(aff, build);
+
+	expr = isl_ast_expr_alloc_binary(isl_ast_op_zdiv_r, expr, cst);
+	cst = isl_ast_expr_alloc_int_si(ctx, 0);
+	expr = isl_ast_expr_alloc_binary(isl_ast_op_eq, expr, cst);
+
+	return expr;
+}
+
+/* Construct an isl_ast_expr that evaluates the condition "constraint",
+ * The result is simplified in terms of build->domain.
+ *
+ * We first check if the constraint is an equality of the form
+ *
+ *	e - d floor(e/d) = 0
+ *
+ * i.e.,
+ *
+ *	e mod d = 0
+ *
+ * If so, we convert it to
+ *
+ *	(isl_ast_op_eq, (isl_ast_op_zdiv_r, expr(e), expr(d)), expr(0))
+ *
+ * Otherwise, let the constraint by either "a >= 0" or "a == 0".
+ * We first extract hidden modulo computations from "a"
+ * and then collect all the terms with a positive coefficient in cons_pos
+ * and the terms with a negative coefficient in cons_neg.
+ *
+ * The result is then of the form
+ *
+ *	(isl_ast_op_ge, expr(pos), expr(-neg)))
+ *
+ * or
+ *
+ *	(isl_ast_op_eq, expr(pos), expr(-neg)))
+ *
+ * However, if the first expression is an integer constant (and the second
+ * is not), then we swap the two expressions.  This ensures that we construct,
+ * e.g., "i <= 5" rather than "5 >= i".
+ *
+ * Furthermore, is there are no terms with positive coefficients (or no terms
+ * with negative coefficients), then the constant term is added to "pos"
+ * (or "neg"), ignoring the sign of the constant term.
+ */
+static __isl_give isl_ast_expr *isl_ast_expr_from_constraint(
+	__isl_take isl_constraint *constraint, __isl_keep isl_ast_build *build)
+{
+	int i, n;
+	isl_ctx *ctx;
+	isl_ast_expr *expr_pos;
+	isl_ast_expr *expr_neg;
+	isl_ast_expr *expr;
+	isl_aff *aff;
+	isl_val *v;
+	int eq;
+	enum isl_ast_op_type type;
+
+	if (!constraint)
+		return NULL;
+
+	aff = isl_constraint_get_aff(constraint);
+	eq = isl_constraint_is_equality(constraint);
+	isl_constraint_free(constraint);
+
+	n = isl_aff_dim(aff, isl_dim_div);
+	if (eq && n > 0)
+		for (i = 0; i < n; ++i) {
+			int is_stride;
+			is_stride = is_stride_constraint(aff, i);
+			if (is_stride < 0)
+				goto error;
+			if (is_stride)
+				return extract_stride_constraint(aff, i, build);
+		}
+
+	ctx = isl_aff_get_ctx(aff);
+	expr_pos = isl_ast_expr_alloc_int_si(ctx, 0);
+	expr_neg = isl_ast_expr_alloc_int_si(ctx, 0);
+
+	aff = extract_modulos(aff, &expr_pos, &expr_neg, build);
+
+	expr_pos = add_signed_terms(expr_pos, aff, 1, build);
+	expr_neg = add_signed_terms(expr_neg, aff, -1, build);
+
+	v = isl_aff_get_constant_val(aff);
+	if (constant_is_considered_positive(v, expr_pos, expr_neg)) {
+		expr_pos = isl_ast_expr_add_int(expr_pos, v);
+	} else {
+		v = isl_val_neg(v);
+		expr_neg = isl_ast_expr_add_int(expr_neg, v);
+	}
+
+	if (isl_ast_expr_get_type(expr_pos) == isl_ast_expr_int &&
+	    isl_ast_expr_get_type(expr_neg) != isl_ast_expr_int) {
+		type = eq ? isl_ast_op_eq : isl_ast_op_le;
+		expr = isl_ast_expr_alloc_binary(type, expr_neg, expr_pos);
+	} else {
+		type = eq ? isl_ast_op_eq : isl_ast_op_ge;
+		expr = isl_ast_expr_alloc_binary(type, expr_pos, expr_neg);
+	}
+
+	isl_aff_free(aff);
+	return expr;
+error:
+	isl_aff_free(aff);
+	return NULL;
+}
+
+/* Wrapper around isl_constraint_cmp_last_non_zero for use
+ * as a callback to isl_constraint_list_sort.
+ * If isl_constraint_cmp_last_non_zero cannot tell the constraints
+ * apart, then use isl_constraint_plain_cmp instead.
+ */
+static int cmp_constraint(__isl_keep isl_constraint *a,
+	__isl_keep isl_constraint *b, void *user)
+{
+	int cmp;
+
+	cmp = isl_constraint_cmp_last_non_zero(a, b);
+	if (cmp != 0)
+		return cmp;
+	return isl_constraint_plain_cmp(a, b);
+}
+
+/* Construct an isl_ast_expr that evaluates the conditions defining "bset".
+ * The result is simplified in terms of build->domain.
+ *
+ * If "bset" is not bounded by any constraint, then we contruct
+ * the expression "1", i.e., "true".
+ *
+ * Otherwise, we sort the constraints, putting constraints that involve
+ * integer divisions after those that do not, and construct an "and"
+ * of the ast expressions of the individual constraints.
+ *
+ * Each constraint is added to the generated constraints of the build
+ * after it has been converted to an AST expression so that it can be used
+ * to simplify the following constraints.  This may change the truth value
+ * of subsequent constraints that do not satisfy the earlier constraints,
+ * but this does not affect the outcome of the conjunction as it is
+ * only true if all the conjuncts are true (no matter in what order
+ * they are evaluated).  In particular, the constraints that do not
+ * involve integer divisions may serve to simplify some constraints
+ * that do involve integer divisions.
+ */
+__isl_give isl_ast_expr *isl_ast_build_expr_from_basic_set(
+	 __isl_keep isl_ast_build *build, __isl_take isl_basic_set *bset)
+{
+	int i, n;
+	isl_constraint *c;
+	isl_constraint_list *list;
+	isl_ast_expr *res;
+	isl_set *set;
+
+	list = isl_basic_set_get_constraint_list(bset);
+	isl_basic_set_free(bset);
+	list = isl_constraint_list_sort(list, &cmp_constraint, NULL);
+	if (!list)
+		return NULL;
+	n = isl_constraint_list_n_constraint(list);
+	if (n == 0) {
+		isl_ctx *ctx = isl_basic_set_get_ctx(bset);
+		isl_constraint_list_free(list);
+		return isl_ast_expr_alloc_int_si(ctx, 1);
+	}
+
+	build = isl_ast_build_copy(build);
+
+	c = isl_constraint_list_get_constraint(list, 0);
+	bset = isl_basic_set_from_constraint(isl_constraint_copy(c));
+	set = isl_set_from_basic_set(bset);
+	res = isl_ast_expr_from_constraint(c, build);
+	build = isl_ast_build_restrict_generated(build, set);
+
+	for (i = 1; i < n; ++i) {
+		isl_ast_expr *expr;
+
+		c = isl_constraint_list_get_constraint(list, i);
+		bset = isl_basic_set_from_constraint(isl_constraint_copy(c));
+		set = isl_set_from_basic_set(bset);
+		expr = isl_ast_expr_from_constraint(c, build);
+		build = isl_ast_build_restrict_generated(build, set);
+		res = isl_ast_expr_and(res, expr);
+	}
+
+	isl_constraint_list_free(list);
+	isl_ast_build_free(build);
+	return res;
+}
+
+struct isl_expr_from_set_data {
+	isl_ast_build *build;
+	int first;
+	isl_ast_expr *res;
+};
+
+/* Construct an isl_ast_expr that evaluates the conditions defining "bset"
+ * and add it to data->res.
+ * The result is simplified in terms of data->build->domain.
+ */
+static int expr_from_set(__isl_take isl_basic_set *bset, void *user)
+{
+	struct isl_expr_from_set_data *data = user;
+	isl_ast_expr *expr;
+
+	expr = isl_ast_build_expr_from_basic_set(data->build, bset);
+	if (data->first)
+		data->res = expr;
+	else
+		data->res = isl_ast_expr_or(data->res, expr);
+
+	data->first = 0;
+
+	if (!data->res)
+		return -1;
+	return 0;
+}
+
+/* Construct an isl_ast_expr that evaluates the conditions defining "set".
+ * The result is simplified in terms of build->domain.
+ *
+ * If "set" is an (obviously) empty set, then return the expression "0".
+ */
+__isl_give isl_ast_expr *isl_ast_build_expr_from_set(
+	__isl_keep isl_ast_build *build, __isl_take isl_set *set)
+{
+	struct isl_expr_from_set_data data = { build, 1, NULL };
+
+	if (isl_set_foreach_basic_set(set, &expr_from_set, &data) < 0)
+		data.res = isl_ast_expr_free(data.res);
+	else if (data.first) {
+		isl_ctx *ctx = isl_ast_build_get_ctx(build);
+		data.res = isl_ast_expr_from_val(isl_val_zero(ctx));
+	}
+
+	isl_set_free(set);
+	return data.res;
+}
+
+struct isl_from_pw_aff_data {
+	isl_ast_build *build;
+	int n;
+	isl_ast_expr **next;
+	isl_set *dom;
+};
+
+/* This function is called during the construction of an isl_ast_expr
+ * that evaluates an isl_pw_aff.
+ * Adjust data->next to take into account this piece.
+ *
+ * data->n is the number of pairs of set and aff to go.
+ * data->dom is the domain of the entire isl_pw_aff.
+ *
+ * If this is the last pair, then data->next is set to evaluate aff
+ * and the domain is ignored.
+ * Otherwise, data->next is set to a select operation that selects
+ * an isl_ast_expr correponding to "aff" on "set" and to an expression
+ * that will be filled in by later calls otherwise.
+ *
+ * In both cases, the constraints of "set" are added to the generated
+ * constraints of the build such that they can be exploited to simplify
+ * the AST expression constructed from "aff".
+ */
+static int ast_expr_from_pw_aff(__isl_take isl_set *set,
+	__isl_take isl_aff *aff, void *user)
+{
+	struct isl_from_pw_aff_data *data = user;
+	isl_ctx *ctx;
+	isl_ast_build *build;
+
+	ctx = isl_set_get_ctx(set);
+	data->n--;
+	if (data->n == 0) {
+		build = isl_ast_build_copy(data->build);
+		build = isl_ast_build_restrict_generated(build, set);
+		*data->next = isl_ast_expr_from_aff(aff, build);
+		isl_ast_build_free(build);
+		if (!*data->next)
+			return -1;
+	} else {
+		isl_ast_expr *ternary, *arg;
+		isl_set *gist;
+
+		ternary = isl_ast_expr_alloc_op(ctx, isl_ast_op_select, 3);
+		gist = isl_set_gist(isl_set_copy(set), isl_set_copy(data->dom));
+		arg = isl_ast_build_expr_from_set(data->build, gist);
+		ternary = isl_ast_expr_set_op_arg(ternary, 0, arg);
+		build = isl_ast_build_copy(data->build);
+		build = isl_ast_build_restrict_generated(build, set);
+		arg = isl_ast_expr_from_aff(aff, build);
+		isl_ast_build_free(build);
+		ternary = isl_ast_expr_set_op_arg(ternary, 1, arg);
+		if (!ternary)
+			return -1;
+
+		*data->next = ternary;
+		data->next = &ternary->u.op.args[2];
+	}
+
+	return 0;
+}
+
+/* Construct an isl_ast_expr that evaluates "pa".
+ * The result is simplified in terms of build->domain.
+ *
+ * The domain of "pa" lives in the internal schedule space.
+ */
+__isl_give isl_ast_expr *isl_ast_build_expr_from_pw_aff_internal(
+	__isl_keep isl_ast_build *build, __isl_take isl_pw_aff *pa)
+{
+	struct isl_from_pw_aff_data data;
+	isl_ast_expr *res = NULL;
+
+	pa = isl_ast_build_compute_gist_pw_aff(build, pa);
+	pa = isl_pw_aff_coalesce(pa);
+	if (!pa)
+		return NULL;
+
+	data.build = build;
+	data.n = isl_pw_aff_n_piece(pa);
+	data.next = &res;
+	data.dom = isl_pw_aff_domain(isl_pw_aff_copy(pa));
+
+	if (isl_pw_aff_foreach_piece(pa, &ast_expr_from_pw_aff, &data) < 0)
+		res = isl_ast_expr_free(res);
+	else if (!res)
+		isl_die(isl_pw_aff_get_ctx(pa), isl_error_invalid,
+			"cannot handle void expression", res = NULL);
+
+	isl_pw_aff_free(pa);
+	isl_set_free(data.dom);
+	return res;
+}
+
+/* Construct an isl_ast_expr that evaluates "pa".
+ * The result is simplified in terms of build->domain.
+ *
+ * The domain of "pa" lives in the external schedule space.
+ */
+__isl_give isl_ast_expr *isl_ast_build_expr_from_pw_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_pw_aff *pa)
+{
+	isl_ast_expr *expr;
+
+	if (isl_ast_build_need_schedule_map(build)) {
+		isl_multi_aff *ma;
+		ma = isl_ast_build_get_schedule_map_multi_aff(build);
+		pa = isl_pw_aff_pullback_multi_aff(pa, ma);
+	}
+	expr = isl_ast_build_expr_from_pw_aff_internal(build, pa);
+	return expr;
+}
+
+/* Set the ids of the input dimensions of "mpa" to the iterator ids
+ * of "build".
+ *
+ * The domain of "mpa" is assumed to live in the internal schedule domain.
+ */
+static __isl_give isl_multi_pw_aff *set_iterator_names(
+	__isl_keep isl_ast_build *build, __isl_take isl_multi_pw_aff *mpa)
+{
+	int i, n;
+
+	n = isl_multi_pw_aff_dim(mpa, isl_dim_in);
+	for (i = 0; i < n; ++i) {
+		isl_id *id;
+
+		id = isl_ast_build_get_iterator_id(build, i);
+		mpa = isl_multi_pw_aff_set_dim_id(mpa, isl_dim_in, i, id);
+	}
+
+	return mpa;
+}
+
+/* Construct an isl_ast_expr of type "type" with as first argument "arg0" and
+ * the remaining arguments derived from "mpa".
+ * That is, construct a call or access expression that calls/accesses "arg0"
+ * with arguments/indices specified by "mpa".
+ */
+static __isl_give isl_ast_expr *isl_ast_build_with_arguments(
+	__isl_keep isl_ast_build *build, enum isl_ast_op_type type,
+	__isl_take isl_ast_expr *arg0, __isl_take isl_multi_pw_aff *mpa)
+{
+	int i, n;
+	isl_ctx *ctx;
+	isl_ast_expr *expr;
+
+	ctx = isl_ast_build_get_ctx(build);
+
+	n = isl_multi_pw_aff_dim(mpa, isl_dim_out);
+	expr = isl_ast_expr_alloc_op(ctx, type, 1 + n);
+	expr = isl_ast_expr_set_op_arg(expr, 0, arg0);
+	for (i = 0; i < n; ++i) {
+		isl_pw_aff *pa;
+		isl_ast_expr *arg;
+
+		pa = isl_multi_pw_aff_get_pw_aff(mpa, i);
+		arg = isl_ast_build_expr_from_pw_aff_internal(build, pa);
+		expr = isl_ast_expr_set_op_arg(expr, 1 + i, arg);
+	}
+
+	isl_multi_pw_aff_free(mpa);
+	return expr;
+}
+
+static __isl_give isl_ast_expr *isl_ast_build_from_multi_pw_aff_internal(
+	__isl_keep isl_ast_build *build, enum isl_ast_op_type type,
+	__isl_take isl_multi_pw_aff *mpa);
+
+/* Construct an isl_ast_expr that accesses the member specified by "mpa".
+ * The range of "mpa" is assumed to be wrapped relation.
+ * The domain of this wrapped relation specifies the structure being
+ * accessed, while the range of this wrapped relation spacifies the
+ * member of the structure being accessed.
+ *
+ * The domain of "mpa" is assumed to live in the internal schedule domain.
+ */
+static __isl_give isl_ast_expr *isl_ast_build_from_multi_pw_aff_member(
+	__isl_keep isl_ast_build *build, __isl_take isl_multi_pw_aff *mpa)
+{
+	isl_id *id;
+	isl_multi_pw_aff *domain;
+	isl_ast_expr *domain_expr, *expr;
+	enum isl_ast_op_type type = isl_ast_op_access;
+
+	domain = isl_multi_pw_aff_copy(mpa);
+	domain = isl_multi_pw_aff_range_factor_domain(domain);
+	domain_expr = isl_ast_build_from_multi_pw_aff_internal(build,
+								type, domain);
+	mpa = isl_multi_pw_aff_range_factor_range(mpa);
+	if (!isl_multi_pw_aff_has_tuple_id(mpa, isl_dim_out))
+		isl_die(isl_ast_build_get_ctx(build), isl_error_invalid,
+			"missing field name", goto error);
+	id = isl_multi_pw_aff_get_tuple_id(mpa, isl_dim_out);
+	expr = isl_ast_expr_from_id(id);
+	expr = isl_ast_expr_alloc_binary(isl_ast_op_member, domain_expr, expr);
+	return isl_ast_build_with_arguments(build, type, expr, mpa);
+error:
+	isl_multi_pw_aff_free(mpa);
+	return NULL;
+}
+
+/* Construct an isl_ast_expr of type "type" that calls or accesses
+ * the element specified by "mpa".
+ * The first argument is obtained from the output tuple name.
+ * The remaining arguments are given by the piecewise affine expressions.
+ *
+ * If the range of "mpa" is a mapped relation, then we assume it
+ * represents an access to a member of a structure.
+ *
+ * The domain of "mpa" is assumed to live in the internal schedule domain.
+ */
+static __isl_give isl_ast_expr *isl_ast_build_from_multi_pw_aff_internal(
+	__isl_keep isl_ast_build *build, enum isl_ast_op_type type,
+	__isl_take isl_multi_pw_aff *mpa)
+{
+	isl_ctx *ctx;
+	isl_id *id;
+	isl_ast_expr *expr;
+
+	if (!mpa)
+		goto error;
+
+	if (type == isl_ast_op_access &&
+	    isl_multi_pw_aff_range_is_wrapping(mpa))
+		return isl_ast_build_from_multi_pw_aff_member(build, mpa);
+
+	mpa = set_iterator_names(build, mpa);
+	if (!build || !mpa)
+		goto error;
+
+	ctx = isl_ast_build_get_ctx(build);
+
+	if (isl_multi_pw_aff_has_tuple_id(mpa, isl_dim_out))
+		id = isl_multi_pw_aff_get_tuple_id(mpa, isl_dim_out);
+	else
+		id = isl_id_alloc(ctx, "", NULL);
+
+	expr = isl_ast_expr_from_id(id);
+	return isl_ast_build_with_arguments(build, type, expr, mpa);
+error:
+	isl_multi_pw_aff_free(mpa);
+	return NULL;
+}
+
+/* Construct an isl_ast_expr of type "type" that calls or accesses
+ * the element specified by "pma".
+ * The first argument is obtained from the output tuple name.
+ * The remaining arguments are given by the piecewise affine expressions.
+ *
+ * The domain of "pma" is assumed to live in the internal schedule domain.
+ */
+static __isl_give isl_ast_expr *isl_ast_build_from_pw_multi_aff_internal(
+	__isl_keep isl_ast_build *build, enum isl_ast_op_type type,
+	__isl_take isl_pw_multi_aff *pma)
+{
+	isl_multi_pw_aff *mpa;
+
+	mpa = isl_multi_pw_aff_from_pw_multi_aff(pma);
+	return isl_ast_build_from_multi_pw_aff_internal(build, type, mpa);
+}
+
+/* Construct an isl_ast_expr of type "type" that calls or accesses
+ * the element specified by "mpa".
+ * The first argument is obtained from the output tuple name.
+ * The remaining arguments are given by the piecewise affine expressions.
+ *
+ * The domain of "mpa" is assumed to live in the external schedule domain.
+ */
+static __isl_give isl_ast_expr *isl_ast_build_from_multi_pw_aff(
+	__isl_keep isl_ast_build *build, enum isl_ast_op_type type,
+	__isl_take isl_multi_pw_aff *mpa)
+{
+	int is_domain;
+	isl_ast_expr *expr;
+	isl_space *space_build, *space_mpa;
+
+	space_build = isl_ast_build_get_space(build, 0);
+	space_mpa = isl_multi_pw_aff_get_space(mpa);
+	is_domain = isl_space_tuple_is_equal(space_build, isl_dim_set,
+					space_mpa, isl_dim_in);
+	isl_space_free(space_build);
+	isl_space_free(space_mpa);
+	if (is_domain < 0)
+		goto error;
+	if (!is_domain)
+		isl_die(isl_ast_build_get_ctx(build), isl_error_invalid,
+			"spaces don't match", goto error);
+
+	if (isl_ast_build_need_schedule_map(build)) {
+		isl_multi_aff *ma;
+		ma = isl_ast_build_get_schedule_map_multi_aff(build);
+		mpa = isl_multi_pw_aff_pullback_multi_aff(mpa, ma);
+	}
+
+	expr = isl_ast_build_from_multi_pw_aff_internal(build, type, mpa);
+	return expr;
+error:
+	isl_multi_pw_aff_free(mpa);
+	return NULL;
+}
+
+/* Construct an isl_ast_expr that calls the domain element specified by "mpa".
+ * The name of the function is obtained from the output tuple name.
+ * The arguments are given by the piecewise affine expressions.
+ *
+ * The domain of "mpa" is assumed to live in the external schedule domain.
+ */
+__isl_give isl_ast_expr *isl_ast_build_call_from_multi_pw_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_multi_pw_aff *mpa)
+{
+	return isl_ast_build_from_multi_pw_aff(build, isl_ast_op_call, mpa);
+}
+
+/* Construct an isl_ast_expr that accesses the array element specified by "mpa".
+ * The name of the array is obtained from the output tuple name.
+ * The index expressions are given by the piecewise affine expressions.
+ *
+ * The domain of "mpa" is assumed to live in the external schedule domain.
+ */
+__isl_give isl_ast_expr *isl_ast_build_access_from_multi_pw_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_multi_pw_aff *mpa)
+{
+	return isl_ast_build_from_multi_pw_aff(build, isl_ast_op_access, mpa);
+}
+
+/* Construct an isl_ast_expr of type "type" that calls or accesses
+ * the element specified by "pma".
+ * The first argument is obtained from the output tuple name.
+ * The remaining arguments are given by the piecewise affine expressions.
+ *
+ * The domain of "pma" is assumed to live in the external schedule domain.
+ */
+static __isl_give isl_ast_expr *isl_ast_build_from_pw_multi_aff(
+	__isl_keep isl_ast_build *build, enum isl_ast_op_type type,
+	__isl_take isl_pw_multi_aff *pma)
+{
+	isl_multi_pw_aff *mpa;
+
+	mpa = isl_multi_pw_aff_from_pw_multi_aff(pma);
+	return isl_ast_build_from_multi_pw_aff(build, type, mpa);
+}
+
+/* Construct an isl_ast_expr that calls the domain element specified by "pma".
+ * The name of the function is obtained from the output tuple name.
+ * The arguments are given by the piecewise affine expressions.
+ *
+ * The domain of "pma" is assumed to live in the external schedule domain.
+ */
+__isl_give isl_ast_expr *isl_ast_build_call_from_pw_multi_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_pw_multi_aff *pma)
+{
+	return isl_ast_build_from_pw_multi_aff(build, isl_ast_op_call, pma);
+}
+
+/* Construct an isl_ast_expr that accesses the array element specified by "pma".
+ * The name of the array is obtained from the output tuple name.
+ * The index expressions are given by the piecewise affine expressions.
+ *
+ * The domain of "pma" is assumed to live in the external schedule domain.
+ */
+__isl_give isl_ast_expr *isl_ast_build_access_from_pw_multi_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_pw_multi_aff *pma)
+{
+	return isl_ast_build_from_pw_multi_aff(build, isl_ast_op_access, pma);
+}
+
+/* Construct an isl_ast_expr that calls the domain element
+ * specified by "executed".
+ *
+ * "executed" is assumed to be single-valued, with a domain that lives
+ * in the internal schedule space.
+ */
+__isl_give isl_ast_node *isl_ast_build_call_from_executed(
+	__isl_keep isl_ast_build *build, __isl_take isl_map *executed)
+{
+	isl_pw_multi_aff *iteration;
+	isl_ast_expr *expr;
+
+	iteration = isl_pw_multi_aff_from_map(executed);
+	iteration = isl_ast_build_compute_gist_pw_multi_aff(build, iteration);
+	iteration = isl_pw_multi_aff_intersect_domain(iteration,
+					isl_ast_build_get_domain(build));
+	expr = isl_ast_build_from_pw_multi_aff_internal(build, isl_ast_op_call,
+							iteration);
+	return isl_ast_node_alloc_user(expr);
+}
diff --git a/polly/lib/External/isl/isl_ast_build_expr.h b/polly/lib/External/isl/isl_ast_build_expr.h
new file mode 100644
index 00000000000..bf3af04cb06
--- /dev/null
+++ b/polly/lib/External/isl/isl_ast_build_expr.h
@@ -0,0 +1,22 @@
+#ifndef ISL_AST_BUILD_EXPR_PRIVATE_H
+#define ISL_AST_BUILD_EXPR_PRIVATE_H
+
+#include <isl/ast.h>
+#include <isl/ast_build.h>
+
+__isl_give isl_ast_expr *isl_ast_build_expr_from_basic_set(
+	 __isl_keep isl_ast_build *build, __isl_take isl_basic_set *bset);
+__isl_give isl_ast_expr *isl_ast_build_expr_from_set(
+	__isl_keep isl_ast_build *build, __isl_take isl_set *set);
+
+__isl_give isl_ast_expr *isl_ast_build_expr_from_pw_aff_internal(
+	__isl_keep isl_ast_build *build, __isl_take isl_pw_aff *pa);
+__isl_give isl_ast_expr *isl_ast_expr_from_aff(__isl_take isl_aff *aff,
+	__isl_keep isl_ast_build *build);
+__isl_give isl_ast_expr *isl_ast_expr_set_op_arg(__isl_take isl_ast_expr *expr,
+	int pos, __isl_take isl_ast_expr *arg);
+
+__isl_give isl_ast_node *isl_ast_build_call_from_executed(
+	__isl_keep isl_ast_build *build, __isl_take isl_map *executed);
+
+#endif
diff --git a/polly/lib/External/isl/isl_ast_build_private.h b/polly/lib/External/isl/isl_ast_build_private.h
new file mode 100644
index 00000000000..62c66b8263c
--- /dev/null
+++ b/polly/lib/External/isl/isl_ast_build_private.h
@@ -0,0 +1,267 @@
+#ifndef ISL_AST_BUILD_PRIVATE_H
+#define ISL_AST_BUILD_PRIVATE_H
+
+#include <isl/aff.h>
+#include <isl/ast.h>
+#include <isl/ast_build.h>
+#include <isl/set.h>
+#include <isl/list.h>
+
+enum isl_ast_build_domain_type {
+	atomic,
+	unroll,
+	separate
+};
+
+/* An isl_ast_build represents the context in which AST is being
+ * generated.  That is, it (mostly) contains information about outer
+ * loops that can be used to simplify inner loops.
+ *
+ * "domain" represents constraints on the internal schedule domain,
+ * corresponding to the context of the AST generation and the constraints
+ * implied by the loops that have already been generated.
+ * When an isl_ast_build is first created, outside any AST generation,
+ * the domain is typically a parameter set.  It is only when a AST
+ * generation phase is initiated that the domain of the isl_ast_build
+ * is changed to refer to the internal schedule domain.
+ * The domain then lives in a space of the form
+ *
+ *	S
+ *
+ *  or
+ *
+ *	[O -> S]
+ *
+ * O represents the loops generated in outer AST generations.
+ * S represents the loops (both generated and to be generated)
+ * of the current AST generation.
+ * Both include eliminated loops.
+ * "domain" is expected not to have any unknown divs because
+ * it is used as the context argument in a call to isl_basic_set_gist
+ * in isl_ast_build_compute_gist_basic_set.
+ *
+ * "depth" is equal to the number of loops that have already
+ * been generated (including those in outer AST generations).
+ * "outer_pos" is equal to the number of loops in outer AST generations.
+ *
+ * "generated" is a superset of "domain" corresponding to those
+ * constraints that were either given by the user or that have
+ * effectively been generated (as bounds on a for loop).
+ *
+ * "pending" is a superset of "domain" corresponding to the constraints
+ * that still need to be generated (as guards), but that may end up
+ * not getting generated if they are implied by any constraints
+ * enforced by inner loops.
+ *
+ * "strides" contains the stride of each loop.  The number of elements
+ * is equal to the number of dimensions in "domain".
+ * "offsets" constains the offsets of strided loops.  If s is the stride
+ * for a given dimension and f is the corresponding offset, then the
+ * dimension takes on values
+ *
+ *	f + s a
+ *
+ * with a an integer.  For non-strided loops, the offset is zero.
+ *
+ * "iterators" contains the loop iterators of both generated and
+ * to be generated loops.  The number of elements is at least as
+ * large as the dimension of the internal schedule domain.  The
+ * number may be larger, in which case the additional ids can be
+ * used in a nested AST generation should the schedule be non-injective.
+ *
+ * "values" lives in the space
+ *
+ *	[O -> S] -> [O -> S]		(or S -> S)
+ *
+ * and expresses (if possible) loop iterators in terms of parameters
+ * and outer loop iterators.  If the value of a given loop iterator
+ * cannot be expressed as an affine expression (either because the iterator
+ * attains multiple values or because the single value is a piecewise
+ * affine expression), then it is expressed in "values" as being equal
+ * to itself.
+ *
+ * "value" is the value of the loop iterator at the current depth.
+ * It is NULL if it has not been computed yet or if the value of the
+ * given loop iterator cannot be expressed as a piecewise affine expression
+ * (because the iterator attains multiple values).
+ *
+ * "schedule_map" maps the internal schedule domain to the external schedule
+ * domain.  It may be NULL if it hasn't been computed yet.
+ * See isl_ast_build_get_schedule_map_multi_aff.
+ *
+ * The "create_leaf" callback is called for every leaf in the generated AST.
+ * The callback is responsible for creating the node to be placed at those
+ * leaves.  If this callback is not set, then isl will generated user
+ * nodes with call expressions corresponding to an element of the domain.
+ *
+ * The "at_each_domain" callback is called on every node created to represent
+ * an element of the domain.  Each of these nodes is a user node
+ * with as expression a call expression.
+ *
+ * The "before_each_for" callback is called on each for node before
+ * its children have been created.
+ *
+ * The "after_each_for" callback is called on each for node after
+ * its children have been created.
+ *
+ * "executed" contains the inverse schedule at this point
+ * of the AST generation.
+ * It is currently only used in isl_ast_build_get_schedule, which is
+ * in turn only used by user code from within a callback.
+ * The value is set right before we may be calling such a callback.
+ *
+ * "single_valued" is set if the current inverse schedule (which may or may
+ * not be stored in "executed") is known to be single valued, specifically
+ * an inverse schedule that was not (appeared not to be) single valued
+ * is extended to a single valued inverse schedule.  This is mainly used
+ * to avoid an infinite recursion when we fail to detect later on that
+ * the extended inverse schedule is single valued.
+ */
+struct isl_ast_build {
+	int ref;
+
+	int outer_pos;
+	int depth;
+
+	isl_id_list *iterators;
+
+	isl_set *domain;
+	isl_set *generated;
+	isl_set *pending;
+	isl_multi_aff *values;
+
+	isl_pw_aff *value;
+
+	isl_vec *strides;
+	isl_multi_aff *offsets;
+
+	isl_multi_aff *schedule_map;
+
+	isl_union_map *options;
+
+	__isl_give isl_ast_node *(*at_each_domain)(
+		__isl_take isl_ast_node *node,
+		__isl_keep isl_ast_build *build, void *user);
+	void *at_each_domain_user;
+
+	__isl_give isl_id *(*before_each_for)(
+		__isl_keep isl_ast_build *context, void *user);
+	void *before_each_for_user;
+	__isl_give isl_ast_node *(*after_each_for)(
+		__isl_take isl_ast_node *node,
+		__isl_keep isl_ast_build *context, void *user);
+	void *after_each_for_user;
+
+	__isl_give isl_ast_node *(*create_leaf)(
+		__isl_take isl_ast_build *build, void *user);
+	void *create_leaf_user;
+
+	isl_union_map *executed;
+	int single_valued;
+};
+
+__isl_give isl_ast_build *isl_ast_build_clear_local_info(
+	__isl_take isl_ast_build *build);
+__isl_give isl_ast_build *isl_ast_build_increase_depth(
+	__isl_take isl_ast_build *build);
+int isl_ast_build_get_depth(__isl_keep isl_ast_build *build);
+unsigned isl_ast_build_dim(__isl_keep isl_ast_build *build,
+	enum isl_dim_type type);
+__isl_give isl_space *isl_ast_build_get_space(
+	__isl_keep isl_ast_build *build, int internal);
+__isl_give isl_ast_build *isl_ast_build_align_params(
+	__isl_take isl_ast_build *build, __isl_take isl_space *model);
+__isl_give isl_ast_build *isl_ast_build_cow(
+	__isl_take isl_ast_build *build);
+__isl_give isl_ast_build *isl_ast_build_insert_dim(
+	__isl_take isl_ast_build *build, int pos);
+__isl_give isl_ast_build *isl_ast_build_scale_down(
+	__isl_take isl_ast_build *build, __isl_take isl_val *m,
+	__isl_take isl_union_map *umap);
+__isl_give isl_ast_build *isl_ast_build_product(
+	__isl_take isl_ast_build *build, __isl_take isl_space *embedding);
+__isl_give isl_ast_build *isl_ast_build_set_loop_bounds(
+	__isl_take isl_ast_build *build, __isl_take isl_basic_set *bounds);
+__isl_give isl_ast_build *isl_ast_build_detect_strides(
+	__isl_take isl_ast_build *build, __isl_take isl_set *set);
+__isl_give isl_ast_build *isl_ast_build_include_stride(
+	__isl_take isl_ast_build *build);
+__isl_give isl_ast_build *isl_ast_build_set_executed(
+	__isl_take isl_ast_build *build,
+	__isl_take isl_union_map *executed);
+__isl_give isl_ast_build *isl_ast_build_set_single_valued(
+	__isl_take isl_ast_build *build, int sv);
+__isl_give isl_set *isl_ast_build_get_domain(
+	__isl_keep isl_ast_build *build);
+__isl_give isl_set *isl_ast_build_get_pending(
+	__isl_keep isl_ast_build *build);
+__isl_give isl_set *isl_ast_build_get_generated(
+	__isl_keep isl_ast_build *build);
+__isl_give isl_ast_build *isl_ast_build_restrict_generated(
+	__isl_take isl_ast_build *build, __isl_take isl_set *set);
+__isl_give isl_ast_build *isl_ast_build_replace_pending_by_guard(
+	__isl_take isl_ast_build *build, __isl_take isl_set *guard);
+__isl_give isl_ast_build *isl_ast_build_restrict_pending(
+	__isl_take isl_ast_build *build, __isl_take isl_set *set);
+__isl_give int isl_ast_build_need_schedule_map(
+	__isl_keep isl_ast_build *build);
+__isl_give isl_multi_aff *isl_ast_build_get_schedule_map_multi_aff(
+	__isl_keep isl_ast_build *build);
+__isl_give isl_map *isl_ast_build_get_schedule_map(
+	__isl_keep isl_ast_build *build);
+int isl_ast_build_has_affine_value(__isl_keep isl_ast_build *build, int pos);
+int isl_ast_build_has_value(__isl_keep isl_ast_build *build);
+__isl_give isl_id *isl_ast_build_get_iterator_id(
+	__isl_keep isl_ast_build *build, int pos);
+
+__isl_give isl_basic_set *isl_ast_build_compute_gist_basic_set(
+	__isl_keep isl_ast_build *build, __isl_take isl_basic_set *bset);
+__isl_give isl_set *isl_ast_build_specialize(__isl_keep isl_ast_build *build,
+	__isl_take isl_set *set);
+__isl_give isl_set *isl_ast_build_compute_gist(
+	__isl_keep isl_ast_build *build, __isl_take isl_set *set);
+__isl_give isl_map *isl_ast_build_compute_gist_map_domain(
+	__isl_keep isl_ast_build *build, __isl_take isl_map *map);
+__isl_give isl_aff *isl_ast_build_compute_gist_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_aff *aff);
+__isl_give isl_pw_aff *isl_ast_build_compute_gist_pw_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_pw_aff *pa);
+__isl_give isl_pw_multi_aff *isl_ast_build_compute_gist_pw_multi_aff(
+	__isl_keep isl_ast_build *build, __isl_take isl_pw_multi_aff *pma);
+
+__isl_give isl_union_map *isl_ast_build_substitute_values_union_map_domain(
+	__isl_keep isl_ast_build *build, __isl_take isl_union_map *umap);
+
+int isl_ast_build_aff_is_nonneg(__isl_keep isl_ast_build *build,
+	__isl_keep isl_aff *aff);
+
+int isl_ast_build_has_stride(__isl_keep isl_ast_build *build, int pos);
+__isl_give isl_aff *isl_ast_build_get_offset(__isl_keep isl_ast_build *build,
+	int pos);
+__isl_give isl_val *isl_ast_build_get_stride(__isl_keep isl_ast_build *build,
+	int pos);
+__isl_give isl_set *isl_ast_build_get_stride_constraint(
+	__isl_keep isl_ast_build *build);
+__isl_give isl_multi_aff *isl_ast_build_get_stride_expansion(
+	__isl_keep isl_ast_build *build);
+
+void isl_ast_build_dump(__isl_keep isl_ast_build *build);
+
+__isl_give isl_set *isl_ast_build_get_option_domain(
+	__isl_keep isl_ast_build *build,
+	enum isl_ast_build_domain_type type);
+__isl_give isl_map *isl_ast_build_get_separation_class(
+	__isl_keep isl_ast_build *build);
+__isl_give isl_set *isl_ast_build_eliminate(
+	__isl_keep isl_ast_build *build, __isl_take isl_set *domain);
+__isl_give isl_set *isl_ast_build_eliminate_inner(
+	__isl_keep isl_ast_build *build, __isl_take isl_set *set);
+__isl_give isl_set *isl_ast_build_eliminate_divs(
+	__isl_keep isl_ast_build *build, __isl_take isl_set *set);
+
+__isl_give isl_map *isl_ast_build_map_to_iterator(
+	__isl_keep isl_ast_build *build, __isl_take isl_set *set);
+
+int isl_ast_build_options_involve_depth(__isl_keep isl_ast_build *build);
+
+#endif
diff --git a/polly/lib/External/isl/isl_ast_codegen.c b/polly/lib/External/isl/isl_ast_codegen.c
new file mode 100644
index 00000000000..679d2612d69
--- /dev/null
+++ b/polly/lib/External/isl/isl_ast_codegen.c
@@ -0,0 +1,3921 @@
+/*
+ * Copyright 2012-2014 Ecole Normale Superieure
+ * Copyright 2014      INRIA Rocquencourt
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege,
+ * Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
+ * B.P. 105 - 78153 Le Chesnay, France
+ */
+
+#include <limits.h>
+#include <isl/aff.h>
+#include <isl/set.h>
+#include <isl/ilp.h>
+#include <isl/union_map.h>
+#include <isl_sort.h>
+#include <isl_tarjan.h>
+#include <isl_ast_private.h>
+#include <isl_ast_build_expr.h>
+#include <isl_ast_build_private.h>
+#include <isl_ast_graft_private.h>
+
+/* Data used in generate_domain.
+ *
+ * "build" is the input build.
+ * "list" collects the results.
+ */
+struct isl_generate_domain_data {
+	isl_ast_build *build;
+
+	isl_ast_graft_list *list;
+};
+
+static __isl_give isl_ast_graft_list *generate_next_level(
+	__isl_take isl_union_map *executed,
+	__isl_take isl_ast_build *build);
+static __isl_give isl_ast_graft_list *generate_code(
+	__isl_take isl_union_map *executed, __isl_take isl_ast_build *build,
+	int internal);
+
+/* Generate an AST for a single domain based on
+ * the (non single valued) inverse schedule "executed".
+ *
+ * We extend the schedule with the iteration domain
+ * and continue generating through a call to generate_code.
+ *
+ * In particular, if executed has the form
+ *
+ *	S -> D
+ *
+ * then we continue generating code on
+ *
+ *	[S -> D] -> D
+ *
+ * The extended inverse schedule is clearly single valued
+ * ensuring that the nested generate_code will not reach this function,
+ * but will instead create calls to all elements of D that need
+ * to be executed from the current schedule domain.
+ */
+static int generate_non_single_valued(__isl_take isl_map *executed,
+	struct isl_generate_domain_data *data)
+{
+	isl_map *identity;
+	isl_ast_build *build;
+	isl_ast_graft_list *list;
+
+	build = isl_ast_build_copy(data->build);
+
+	identity = isl_set_identity(isl_map_range(isl_map_copy(executed)));
+	executed = isl_map_domain_product(executed, identity);
+	build = isl_ast_build_set_single_valued(build, 1);
+
+	list = generate_code(isl_union_map_from_map(executed), build, 1);
+
+	data->list = isl_ast_graft_list_concat(data->list, list);
+
+	return 0;
+}
+
+/* Call the at_each_domain callback, if requested by the user,
+ * after recording the current inverse schedule in the build.
+ */
+static __isl_give isl_ast_graft *at_each_domain(__isl_take isl_ast_graft *graft,
+	__isl_keep isl_map *executed, __isl_keep isl_ast_build *build)
+{
+	if (!graft || !build)
+		return isl_ast_graft_free(graft);
+	if (!build->at_each_domain)
+		return graft;
+
+	build = isl_ast_build_copy(build);
+	build = isl_ast_build_set_executed(build,
+			isl_union_map_from_map(isl_map_copy(executed)));
+	if (!build)
+		return isl_ast_graft_free(graft);
+
+	graft->node = build->at_each_domain(graft->node,
+					build, build->at_each_domain_user);
+	isl_ast_build_free(build);
+
+	if (!graft->node)
+		graft = isl_ast_graft_free(graft);
+
+	return graft;
+}
+
+/* Generate an AST for a single domain based on
+ * the inverse schedule "executed" and add it to data->list.
+ *
+ * If there is more than one domain element associated to the current
+ * schedule "time", then we need to continue the generation process
+ * in generate_non_single_valued.
+ * Note that the inverse schedule being single-valued may depend
+ * on constraints that are only available in the original context
+ * domain specified by the user.  We therefore first introduce
+ * some of the constraints of data->build->domain.  In particular,
+ * we intersect with a single-disjunct approximation of this set.
+ * We perform this approximation to avoid further splitting up
+ * the executed relation, possibly introducing a disjunctive guard
+ * on the statement.
+ *
+ * On the other hand, we only perform the test after having taken the gist
+ * of the domain as the resulting map is the one from which the call
+ * expression is constructed.  Using this map to construct the call
+ * expression usually yields simpler results.
+ * Because we perform the single-valuedness test on the gisted map,
+ * we may in rare cases fail to recognize that the inverse schedule
+ * is single-valued.  This becomes problematic if this happens
+ * from the recursive call through generate_non_single_valued
+ * as we would then end up in an infinite recursion.
+ * We therefore check if we are inside a call to generate_non_single_valued
+ * and revert to the ungisted map if the gisted map turns out not to be
+ * single-valued.
+ *
+ * Otherwise, we generate a call expression for the single executed
+ * domain element and put a guard around it based on the (simplified)
+ * domain of "executed".
+ *
+ * At this stage, any pending constraints in the build can no longer
+ * be simplified with respect to any enforced constraints since
+ * the call node does not have any enforced constraints.
+ * We therefore turn all pending constraints into guards
+ * (after simplifying them with respect to the already generated
+ * constraints) and add them to both the generated constraints
+ * and the guard of the constructed graft.  This guard will ensure
+ * that the constraints are effectively generated.
+ *
+ * If the user has set an at_each_domain callback, it is called
+ * on the constructed call expression node.
+ */
+static int generate_domain(__isl_take isl_map *executed, void *user)
+{
+	struct isl_generate_domain_data *data = user;
+	isl_ast_build *build;
+	isl_ast_graft *graft;
+	isl_ast_graft_list *list;
+	isl_set *guard, *domain;
+	isl_map *map = NULL;
+	int empty, sv;
+
+	domain = isl_ast_build_get_domain(data->build);
+	domain = isl_set_from_basic_set(isl_set_simple_hull(domain));
+	executed = isl_map_intersect_domain(executed, domain);
+	empty = isl_map_is_empty(executed);
+	if (empty < 0)
+		goto error;
+	if (empty) {
+		isl_map_free(executed);
+		return 0;
+	}
+
+	executed = isl_map_coalesce(executed);
+	map = isl_map_copy(executed);
+	map = isl_ast_build_compute_gist_map_domain(data->build, map);
+	sv = isl_map_is_single_valued(map);
+	if (sv < 0)
+		goto error;
+	if (!sv) {
+		isl_map_free(map);
+		if (data->build->single_valued)
+			map = isl_map_copy(executed);
+		else
+			return generate_non_single_valued(executed, data);
+	}
+	guard = isl_map_domain(isl_map_copy(map));
+	guard = isl_set_compute_divs(guard);
+	guard = isl_set_intersect(guard,
+				    isl_ast_build_get_pending(data->build));
+	guard = isl_set_coalesce(guard);
+	guard = isl_ast_build_specialize(data->build, guard);
+	guard = isl_set_gist(guard, isl_ast_build_get_generated(data->build));
+
+	build = isl_ast_build_copy(data->build);
+	build = isl_ast_build_replace_pending_by_guard(build,
+							isl_set_copy(guard));
+	graft = isl_ast_graft_alloc_domain(map, build);
+	graft = at_each_domain(graft, executed, build);
+	isl_ast_build_free(build);
+	isl_map_free(executed);
+	graft = isl_ast_graft_add_guard(graft, guard, data->build);
+
+	list = isl_ast_graft_list_from_ast_graft(graft);
+	data->list = isl_ast_graft_list_concat(data->list, list);
+
+	return 0;
+error:
+	isl_map_free(map);
+	isl_map_free(executed);
+	return -1;
+}
+
+/* Call build->create_leaf to a create "leaf" node in the AST,
+ * encapsulate the result in an isl_ast_graft and return the result
+ * as a 1-element list.
+ *
+ * Note that the node returned by the user may be an entire tree.
+ *
+ * Since the node itself cannot enforce any constraints, we turn
+ * all pending constraints into guards and add them to the resulting
+ * graft to ensure that they will be generated.
+ *
+ * Before we pass control to the user, we first clear some information
+ * from the build that is (presumbably) only meaningful
+ * for the current code generation.
+ * This includes the create_leaf callback itself, so we make a copy
+ * of the build first.
+ */
+static __isl_give isl_ast_graft_list *call_create_leaf(
+	__isl_take isl_union_map *executed, __isl_take isl_ast_build *build)
+{
+	isl_set *guard;
+	isl_ast_node *node;
+	isl_ast_graft *graft;
+	isl_ast_build *user_build;
+
+	guard = isl_ast_build_get_pending(build);
+	user_build = isl_ast_build_copy(build);
+	user_build = isl_ast_build_replace_pending_by_guard(user_build,
+							isl_set_copy(guard));
+	user_build = isl_ast_build_set_executed(user_build, executed);
+	user_build = isl_ast_build_clear_local_info(user_build);
+	if (!user_build)
+		node = NULL;
+	else
+		node = build->create_leaf(user_build, build->create_leaf_user);
+	graft = isl_ast_graft_alloc(node, build);
+	graft = isl_ast_graft_add_guard(graft, guard, build);
+	isl_ast_build_free(build);
+	return isl_ast_graft_list_from_ast_graft(graft);
+}
+
+/* Generate an AST after having handled the complete schedule
+ * of this call to the code generator.
+ *
+ * If the user has specified a create_leaf callback, control
+ * is passed to the user in call_create_leaf.
+ *
+ * Otherwise, we generate one or more calls for each individual
+ * domain in generate_domain.
+ */
+static __isl_give isl_ast_graft_list *generate_inner_level(
+	__isl_take isl_union_map *executed, __isl_take isl_ast_build *build)
+{
+	isl_ctx *ctx;
+	struct isl_generate_domain_data data = { build };
+
+	if (!build || !executed)
+		goto error;
+
+	if (build->create_leaf)
+		return call_create_leaf(executed, build);
+
+	ctx = isl_union_map_get_ctx(executed);
+	data.list = isl_ast_graft_list_alloc(ctx, 0);
+	if (isl_union_map_foreach_map(executed, &generate_domain, &data) < 0)
+		data.list = isl_ast_graft_list_free(data.list);
+
+	if (0)
+error:		data.list = NULL;
+	isl_ast_build_free(build);
+	isl_union_map_free(executed);
+	return data.list;
+}
+
+/* Call the before_each_for callback, if requested by the user.
+ */
+static __isl_give isl_ast_node *before_each_for(__isl_take isl_ast_node *node,
+	__isl_keep isl_ast_build *build)
+{
+	isl_id *id;
+
+	if (!node || !build)
+		return isl_ast_node_free(node);
+	if (!build->before_each_for)
+		return node;
+	id = build->before_each_for(build, build->before_each_for_user);
+	node = isl_ast_node_set_annotation(node, id);
+	return node;
+}
+
+/* Call the after_each_for callback, if requested by the user.
+ */
+static __isl_give isl_ast_graft *after_each_for(__isl_take isl_ast_graft *graft,
+	__isl_keep isl_ast_build *build)
+{
+	if (!graft || !build)
+		return isl_ast_graft_free(graft);
+	if (!build->after_each_for)
+		return graft;
+	graft->node = build->after_each_for(graft->node, build,
+						build->after_each_for_user);
+	if (!graft->node)
+		return isl_ast_graft_free(graft);
+	return graft;
+}
+
+/* Plug in all the know values of the current and outer dimensions
+ * in the domain of "executed".  In principle, we only need to plug
+ * in the known value of the current dimension since the values of
+ * outer dimensions have been plugged in already.
+ * However, it turns out to be easier to just plug in all known values.
+ */
+static __isl_give isl_union_map *plug_in_values(
+	__isl_take isl_union_map *executed, __isl_keep isl_ast_build *build)
+{
+	return isl_ast_build_substitute_values_union_map_domain(build,
+								    executed);
+}
+
+/* Check if the constraint "c" is a lower bound on dimension "pos",
+ * an upper bound, or independent of dimension "pos".
+ */
+static int constraint_type(isl_constraint *c, int pos)
+{
+	if (isl_constraint_is_lower_bound(c, isl_dim_set, pos))
+		return 1;
+	if (isl_constraint_is_upper_bound(c, isl_dim_set, pos))
+		return 2;
+	return 0;
+}
+
+/* Compare the types of the constraints "a" and "b",
+ * resulting in constraints that are independent of "depth"
+ * to be sorted before the lower bounds on "depth", which in
+ * turn are sorted before the upper bounds on "depth".
+ */
+static int cmp_constraint(__isl_keep isl_constraint *a,
+	__isl_keep isl_constraint *b, void *user)
+{
+	int *depth = user;
+	int t1 = constraint_type(a, *depth);
+	int t2 = constraint_type(b, *depth);
+
+	return t1 - t2;
+}
+
+/* Extract a lower bound on dimension "pos" from constraint "c".
+ *
+ * If the constraint is of the form
+ *
+ *	a x + f(...) >= 0
+ *
+ * then we essentially return
+ *
+ *	l = ceil(-f(...)/a)
+ *
+ * However, if the current dimension is strided, then we need to make
+ * sure that the lower bound we construct is of the form
+ *
+ *	f + s a
+ *
+ * with f the offset and s the stride.
+ * We therefore compute
+ *
+ *	f + s * ceil((l - f)/s)
+ */
+static __isl_give isl_aff *lower_bound(__isl_keep isl_constraint *c,
+	int pos, __isl_keep isl_ast_build *build)
+{
+	isl_aff *aff;
+
+	aff = isl_constraint_get_bound(c, isl_dim_set, pos);
+	aff = isl_aff_ceil(aff);
+
+	if (isl_ast_build_has_stride(build, pos)) {
+		isl_aff *offset;
+		isl_val *stride;
+
+		offset = isl_ast_build_get_offset(build, pos);
+		stride = isl_ast_build_get_stride(build, pos);
+
+		aff = isl_aff_sub(aff, isl_aff_copy(offset));
+		aff = isl_aff_scale_down_val(aff, isl_val_copy(stride));
+		aff = isl_aff_ceil(aff);
+		aff = isl_aff_scale_val(aff, stride);
+		aff = isl_aff_add(aff, offset);
+	}
+
+	aff = isl_ast_build_compute_gist_aff(build, aff);
+
+	return aff;
+}
+
+/* Return the exact lower bound (or upper bound if "upper" is set)
+ * of "domain" as a piecewise affine expression.
+ *
+ * If we are computing a lower bound (of a strided dimension), then
+ * we need to make sure it is of the form
+ *
+ *	f + s a
+ *
+ * where f is the offset and s is the stride.
+ * We therefore need to include the stride constraint before computing
+ * the minimum.
+ */
+static __isl_give isl_pw_aff *exact_bound(__isl_keep isl_set *domain,
+	__isl_keep isl_ast_build *build, int upper)
+{
+	isl_set *stride;
+	isl_map *it_map;
+	isl_pw_aff *pa;
+	isl_pw_multi_aff *pma;
+
+	domain = isl_set_copy(domain);
+	if (!upper) {
+		stride = isl_ast_build_get_stride_constraint(build);
+		domain = isl_set_intersect(domain, stride);
+	}
+	it_map = isl_ast_build_map_to_iterator(build, domain);
+	if (upper)
+		pma = isl_map_lexmax_pw_multi_aff(it_map);
+	else
+		pma = isl_map_lexmin_pw_multi_aff(it_map);
+	pa = isl_pw_multi_aff_get_pw_aff(pma, 0);
+	isl_pw_multi_aff_free(pma);
+	pa = isl_ast_build_compute_gist_pw_aff(build, pa);
+	pa = isl_pw_aff_coalesce(pa);
+
+	return pa;
+}
+
+/* Callback for sorting the isl_pw_aff_list passed to reduce_list and
+ * remove_redundant_lower_bounds.
+ */
+static int reduce_list_cmp(__isl_keep isl_pw_aff *a, __isl_keep isl_pw_aff *b,
+	void *user)
+{
+	return isl_pw_aff_plain_cmp(a, b);
+}
+
+/* Given a list of lower bounds "list", remove those that are redundant
+ * with respect to the other bounds in "list" and the domain of "build".
+ *
+ * We first sort the bounds in the same way as they would be sorted
+ * by set_for_node_expressions so that we can try and remove the last
+ * bounds first.
+ *
+ * For a lower bound to be effective, there needs to be at least
+ * one domain element for which it is larger than all other lower bounds.
+ * For each lower bound we therefore intersect the domain with
+ * the conditions that it is larger than all other bounds and
+ * check whether the result is empty.  If so, the bound can be removed.
+ */
+static __isl_give isl_pw_aff_list *remove_redundant_lower_bounds(
+	__isl_take isl_pw_aff_list *list, __isl_keep isl_ast_build *build)
+{
+	int i, j, n;
+	isl_set *domain;
+
+	list = isl_pw_aff_list_sort(list, &reduce_list_cmp, NULL);
+	if (!list)
+		return NULL;
+
+	n = isl_pw_aff_list_n_pw_aff(list);
+	if (n <= 1)
+		return list;
+
+	domain = isl_ast_build_get_domain(build);
+
+	for (i = n - 1; i >= 0; --i) {
+		isl_pw_aff *pa_i;
+		isl_set *domain_i;
+		int empty;
+
+		domain_i = isl_set_copy(domain);
+		pa_i = isl_pw_aff_list_get_pw_aff(list, i);
+
+		for (j = 0; j < n; ++j) {
+			isl_pw_aff *pa_j;
+			isl_set *better;
+
+			if (j == i)
+				continue;
+
+			pa_j = isl_pw_aff_list_get_pw_aff(list, j);
+			better = isl_pw_aff_gt_set(isl_pw_aff_copy(pa_i), pa_j);
+			domain_i = isl_set_intersect(domain_i, better);
+		}
+
+		empty = isl_set_is_empty(domain_i);
+
+		isl_set_free(domain_i);
+		isl_pw_aff_free(pa_i);
+
+		if (empty < 0)
+			goto error;
+		if (!empty)
+			continue;
+		list = isl_pw_aff_list_drop(list, i, 1);
+		n--;
+	}
+
+	isl_set_free(domain);
+
+	return list;
+error:
+	isl_set_free(domain);
+	return isl_pw_aff_list_free(list);
+}
+
+/* Extract a lower bound on dimension "pos" from each constraint
+ * in "constraints" and return the list of lower bounds.
+ * If "constraints" has zero elements, then we extract a lower bound
+ * from "domain" instead.
+ *
+ * If the current dimension is strided, then the lower bound
+ * is adjusted by lower_bound to match the stride information.
+ * This modification may make one or more lower bounds redundant
+ * with respect to the other lower bounds.  We therefore check
+ * for this condition and remove the redundant lower bounds.
+ */
+static __isl_give isl_pw_aff_list *lower_bounds(
+	__isl_keep isl_constraint_list *constraints, int pos,
+	__isl_keep isl_set *domain, __isl_keep isl_ast_build *build)
+{
+	isl_ctx *ctx;
+	isl_pw_aff_list *list;
+	int i, n;
+
+	if (!build)
+		return NULL;
+
+	n = isl_constraint_list_n_constraint(constraints);
+	if (n == 0) {
+		isl_pw_aff *pa;
+		pa = exact_bound(domain, build, 0);
+		return isl_pw_aff_list_from_pw_aff(pa);
+	}
+
+	ctx = isl_ast_build_get_ctx(build);
+	list = isl_pw_aff_list_alloc(ctx,n);
+
+	for (i = 0; i < n; ++i) {
+		isl_aff *aff;
+		isl_constraint *c;
+
+		c = isl_constraint_list_get_constraint(constraints, i);
+		aff = lower_bound(c, pos, build);
+		isl_constraint_free(c);
+		list = isl_pw_aff_list_add(list, isl_pw_aff_from_aff(aff));
+	}
+
+	if (isl_ast_build_has_stride(build, pos))
+		list = remove_redundant_lower_bounds(list, build);
+
+	return list;
+}
+
+/* Extract an upper bound on dimension "pos" from each constraint
+ * in "constraints" and return the list of upper bounds.
+ * If "constraints" has zero elements, then we extract an upper bound
+ * from "domain" instead.
+ */
+static __isl_give isl_pw_aff_list *upper_bounds(
+	__isl_keep isl_constraint_list *constraints, int pos,
+	__isl_keep isl_set *domain, __isl_keep isl_ast_build *build)
+{
+	isl_ctx *ctx;
+	isl_pw_aff_list *list;
+	int i, n;
+
+	n = isl_constraint_list_n_constraint(constraints);
+	if (n == 0) {
+		isl_pw_aff *pa;
+		pa = exact_bound(domain, build, 1);
+		return isl_pw_aff_list_from_pw_aff(pa);
+	}
+
+	ctx = isl_ast_build_get_ctx(build);
+	list = isl_pw_aff_list_alloc(ctx,n);
+
+	for (i = 0; i < n; ++i) {
+		isl_aff *aff;
+		isl_constraint *c;
+
+		c = isl_constraint_list_get_constraint(constraints, i);
+		aff = isl_constraint_get_bound(c, isl_dim_set, pos);
+		isl_constraint_free(c);
+		aff = isl_aff_floor(aff);
+		list = isl_pw_aff_list_add(list, isl_pw_aff_from_aff(aff));
+	}
+
+	return list;
+}
+
+/* Return an isl_ast_expr that performs the reduction of type "type"
+ * on AST expressions corresponding to the elements in "list".
+ *
+ * The list is assumed to contain at least one element.
+ * If the list contains exactly one element, then the returned isl_ast_expr
+ * simply computes that affine expression.
+ * If the list contains more than one element, then we sort it
+ * using a fairly abitrary but hopefully reasonably stable order.
+ */
+static __isl_give isl_ast_expr *reduce_list(enum isl_ast_op_type type,
+	__isl_keep isl_pw_aff_list *list, __isl_keep isl_ast_build *build)
+{
+	int i, n;
+	isl_ctx *ctx;
+	isl_ast_expr *expr;
+
+	if (!list)
+		return NULL;
+
+	n = isl_pw_aff_list_n_pw_aff(list);
+
+	if (n == 1)
+		return isl_ast_build_expr_from_pw_aff_internal(build,
+				isl_pw_aff_list_get_pw_aff(list, 0));
+
+	ctx = isl_pw_aff_list_get_ctx(list);
+	expr = isl_ast_expr_alloc_op(ctx, type, n);
+	if (!expr)
+		return NULL;
+
+	list = isl_pw_aff_list_copy(list);
+	list = isl_pw_aff_list_sort(list, &reduce_list_cmp, NULL);
+	if (!list)
+		return isl_ast_expr_free(expr);
+
+	for (i = 0; i < n; ++i) {
+		isl_ast_expr *expr_i;
+
+		expr_i = isl_ast_build_expr_from_pw_aff_internal(build,
+				isl_pw_aff_list_get_pw_aff(list, i));
+		if (!expr_i)
+			goto error;
+		expr->u.op.args[i] = expr_i;
+	}
+
+	isl_pw_aff_list_free(list);
+	return expr;
+error:
+	isl_pw_aff_list_free(list);
+	isl_ast_expr_free(expr);
+	return NULL;
+}
+
+/* Add guards implied by the "generated constraints",
+ * but not (necessarily) enforced by the generated AST to "guard".
+ * In particular, if there is any stride constraints,
+ * then add the guard implied by those constraints.
+ * If we have generated a degenerate loop, then add the guard
+ * implied by "bounds" on the outer dimensions, i.e., the guard
+ * that ensures that the single value actually exists.
+ */
+static __isl_give isl_set *add_implied_guards(__isl_take isl_set *guard,
+	int degenerate, __isl_keep isl_basic_set *bounds,
+	__isl_keep isl_ast_build *build)
+{
+	int depth, has_stride;
+	isl_set *dom;
+
+	depth = isl_ast_build_get_depth(build);
+	has_stride = isl_ast_build_has_stride(build, depth);
+	if (!has_stride && !degenerate)
+		return guard;
+
+	if (degenerate) {
+		bounds = isl_basic_set_copy(bounds);
+		bounds = isl_basic_set_drop_constraints_not_involving_dims(
+					bounds, isl_dim_set, depth, 1);
+		dom = isl_set_from_basic_set(bounds);
+		dom = isl_set_eliminate(dom, isl_dim_set, depth, 1);
+		dom = isl_ast_build_compute_gist(build, dom);
+		guard = isl_set_intersect(guard, dom);
+	}
+
+	if (has_stride) {
+		dom = isl_ast_build_get_stride_constraint(build);
+		dom = isl_set_eliminate(dom, isl_dim_set, depth, 1);
+		dom = isl_ast_build_compute_gist(build, dom);
+		guard = isl_set_intersect(guard, dom);
+	}
+
+	return guard;
+}
+
+/* Update "graft" based on "sub_build" for the degenerate case.
+ *
+ * "build" is the build in which graft->node was created
+ * "sub_build" contains information about the current level itself,
+ * including the single value attained.
+ *
+ * We set the initialization part of the for loop to the single
+ * value attained by the current dimension.
+ * The increment and condition are not strictly needed as the are known
+ * to be "1" and "iterator <= value" respectively.
+ */
+static __isl_give isl_ast_graft *refine_degenerate(
+	__isl_take isl_ast_graft *graft, __isl_keep isl_ast_build *build,
+	__isl_keep isl_ast_build *sub_build)
+{
+	isl_pw_aff *value;
+
+	if (!graft || !sub_build)
+		return isl_ast_graft_free(graft);
+
+	value = isl_pw_aff_copy(sub_build->value);
+
+	graft->node->u.f.init = isl_ast_build_expr_from_pw_aff_internal(build,
+						value);
+	if (!graft->node->u.f.init)
+		return isl_ast_graft_free(graft);
+
+	return graft;
+}
+
+/* Return the intersection of constraints in "list" as a set.
+ */
+static __isl_give isl_set *intersect_constraints(
+	__isl_keep isl_constraint_list *list)
+{
+	int i, n;
+	isl_basic_set *bset;
+
+	n = isl_constraint_list_n_constraint(list);
+	if (n < 1)
+		isl_die(isl_constraint_list_get_ctx(list), isl_error_internal,
+			"expecting at least one constraint", return NULL);
+
+	bset = isl_basic_set_from_constraint(
+				isl_constraint_list_get_constraint(list, 0));
+	for (i = 1; i < n; ++i) {
+		isl_basic_set *bset_i;
+
+		bset_i = isl_basic_set_from_constraint(
+				isl_constraint_list_get_constraint(list, i));
+		bset = isl_basic_set_intersect(bset, bset_i);
+	}
+
+	return isl_set_from_basic_set(bset);
+}
+
+/* Compute the constraints on the outer dimensions enforced by
+ * graft->node and add those constraints to graft->enforced,
+ * in case the upper bound is expressed as a set "upper".
+ *
+ * In particular, if l(...) is a lower bound in "lower", and
+ *
+ *	-a i + f(...) >= 0		or	a i <= f(...)
+ *
+ * is an upper bound ocnstraint on the current dimension i,
+ * then the for loop enforces the constraint
+ *
+ *	-a l(...) + f(...) >= 0		or	a l(...) <= f(...)
+ *
+ * We therefore simply take each lower bound in turn, plug it into
+ * the upper bounds and compute the intersection over all lower bounds.
+ *
+ * If a lower bound is a rational expression, then
+ * isl_basic_set_preimage_multi_aff will force this rational
+ * expression to have only integer values.  However, the loop
+ * itself does not enforce this integrality constraint.  We therefore
+ * use the ceil of the lower bounds instead of the lower bounds themselves.
+ * Other constraints will make sure that the for loop is only executed
+ * when each of the lower bounds attains an integral value.
+ * In particular, potentially rational values only occur in
+ * lower_bound if the offset is a (seemingly) rational expression,
+ * but then outer conditions will make sure that this rational expression
+ * only attains integer values.
+ */
+static __isl_give isl_ast_graft *set_enforced_from_set(
+	__isl_take isl_ast_graft *graft,
+	__isl_keep isl_pw_aff_list *lower, int pos, __isl_keep isl_set *upper)
+{
+	isl_space *space;
+	isl_basic_set *enforced;
+	isl_pw_multi_aff *pma;
+	int i, n;
+
+	if (!graft || !lower)
+		return isl_ast_graft_free(graft);
+
+	space = isl_set_get_space(upper);
+	enforced = isl_basic_set_universe(isl_space_copy(space));
+
+	space = isl_space_map_from_set(space);
+	pma = isl_pw_multi_aff_identity(space);
+
+	n = isl_pw_aff_list_n_pw_aff(lower);
+	for (i = 0; i < n; ++i) {
+		isl_pw_aff *pa;
+		isl_set *enforced_i;
+		isl_basic_set *hull;
+		isl_pw_multi_aff *pma_i;
+
+		pa = isl_pw_aff_list_get_pw_aff(lower, i);
+		pa = isl_pw_aff_ceil(pa);
+		pma_i = isl_pw_multi_aff_copy(pma);
+		pma_i = isl_pw_multi_aff_set_pw_aff(pma_i, pos, pa);
+		enforced_i = isl_set_copy(upper);
+		enforced_i = isl_set_preimage_pw_multi_aff(enforced_i, pma_i);
+		hull = isl_set_simple_hull(enforced_i);
+		enforced = isl_basic_set_intersect(enforced, hull);
+	}
+
+	isl_pw_multi_aff_free(pma);
+
+	graft = isl_ast_graft_enforce(graft, enforced);
+
+	return graft;
+}
+
+/* Compute the constraints on the outer dimensions enforced by
+ * graft->node and add those constraints to graft->enforced,
+ * in case the upper bound is expressed as
+ * a list of affine expressions "upper".
+ *
+ * The enforced condition is that each lower bound expression is less
+ * than or equal to each upper bound expression.
+ */
+static __isl_give isl_ast_graft *set_enforced_from_list(
+	__isl_take isl_ast_graft *graft,
+	__isl_keep isl_pw_aff_list *lower, __isl_keep isl_pw_aff_list *upper)
+{
+	isl_set *cond;
+	isl_basic_set *enforced;
+
+	lower = isl_pw_aff_list_copy(lower);
+	upper = isl_pw_aff_list_copy(upper);
+	cond = isl_pw_aff_list_le_set(lower, upper);
+	enforced = isl_set_simple_hull(cond);
+	graft = isl_ast_graft_enforce(graft, enforced);
+
+	return graft;
+}
+
+/* Does "aff" have a negative constant term?
+ */
+static int aff_constant_is_negative(__isl_take isl_set *set,
+	__isl_take isl_aff *aff, void *user)
+{
+	int *neg = user;
+	isl_val *v;
+
+	v = isl_aff_get_constant_val(aff);
+	*neg = isl_val_is_neg(v);
+	isl_val_free(v);
+	isl_set_free(set);
+	isl_aff_free(aff);
+
+	return *neg ? 0 : -1;
+}
+
+/* Does "pa" have a negative constant term over its entire domain?
+ */
+static int pw_aff_constant_is_negative(__isl_take isl_pw_aff *pa, void *user)
+{
+	int r;
+	int *neg = user;
+
+	r = isl_pw_aff_foreach_piece(pa, &aff_constant_is_negative, user);
+	isl_pw_aff_free(pa);
+
+	return *neg ? 0 : -1;
+}
+
+/* Does each element in "list" have a negative constant term?
+ *
+ * The callback terminates the iteration as soon an element has been
+ * found that does not have a negative constant term.
+ */
+static int list_constant_is_negative(__isl_keep isl_pw_aff_list *list)
+{
+	int neg = 1;
+
+	if (isl_pw_aff_list_foreach(list,
+				&pw_aff_constant_is_negative, &neg) < 0 && neg)
+		return -1;
+
+	return neg;
+}
+
+/* Add 1 to each of the elements in "list", where each of these elements
+ * is defined over the internal schedule space of "build".
+ */
+static __isl_give isl_pw_aff_list *list_add_one(
+	__isl_take isl_pw_aff_list *list, __isl_keep isl_ast_build *build)
+{
+	int i, n;
+	isl_space *space;
+	isl_aff *aff;
+	isl_pw_aff *one;
+
+	space = isl_ast_build_get_space(build, 1);
+	aff = isl_aff_zero_on_domain(isl_local_space_from_space(space));
+	aff = isl_aff_add_constant_si(aff, 1);
+	one = isl_pw_aff_from_aff(aff);
+
+	n = isl_pw_aff_list_n_pw_aff(list);
+	for (i = 0; i < n; ++i) {
+		isl_pw_aff *pa;
+		pa = isl_pw_aff_list_get_pw_aff(list, i);
+		pa = isl_pw_aff_add(pa, isl_pw_aff_copy(one));
+		list = isl_pw_aff_list_set_pw_aff(list, i, pa);
+	}
+
+	isl_pw_aff_free(one);
+
+	return list;
+}
+
+/* Set the condition part of the for node graft->node in case
+ * the upper bound is represented as a list of piecewise affine expressions.
+ *
+ * In particular, set the condition to
+ *
+ *	iterator <= min(list of upper bounds)
+ *
+ * If each of the upper bounds has a negative constant term, then
+ * set the condition to
+ *
+ *	iterator < min(list of (upper bound + 1)s)
+ *
+ */
+static __isl_give isl_ast_graft *set_for_cond_from_list(
+	__isl_take isl_ast_graft *graft, __isl_keep isl_pw_aff_list *list,
+	__isl_keep isl_ast_build *build)
+{
+	int neg;
+	isl_ast_expr *bound, *iterator, *cond;
+	enum isl_ast_op_type type = isl_ast_op_le;
+
+	if (!graft || !list)
+		return isl_ast_graft_free(graft);
+
+	neg = list_constant_is_negative(list);
+	if (neg < 0)
+		return isl_ast_graft_free(graft);
+	list = isl_pw_aff_list_copy(list);
+	if (neg) {
+		list = list_add_one(list, build);
+		type = isl_ast_op_lt;
+	}
+
+	bound = reduce_list(isl_ast_op_min, list, build);
+	iterator = isl_ast_expr_copy(graft->node->u.f.iterator);
+	cond = isl_ast_expr_alloc_binary(type, iterator, bound);
+	graft->node->u.f.cond = cond;
+
+	isl_pw_aff_list_free(list);
+	if (!graft->node->u.f.cond)
+		return isl_ast_graft_free(graft);
+	return graft;
+}
+
+/* Set the condition part of the for node graft->node in case
+ * the upper bound is represented as a set.
+ */
+static __isl_give isl_ast_graft *set_for_cond_from_set(
+	__isl_take isl_ast_graft *graft, __isl_keep isl_set *set,
+	__isl_keep isl_ast_build *build)
+{
+	isl_ast_expr *cond;
+
+	if (!graft)
+		return NULL;
+
+	cond = isl_ast_build_expr_from_set(build, isl_set_copy(set));
+	graft->node->u.f.cond = cond;
+	if (!graft->node->u.f.cond)
+		return isl_ast_graft_free(graft);
+	return graft;
+}
+
+/* Construct an isl_ast_expr for the increment (i.e., stride) of
+ * the current dimension.
+ */
+static __isl_give isl_ast_expr *for_inc(__isl_keep isl_ast_build *build)
+{
+	int depth;
+	isl_val *v;
+	isl_ctx *ctx;
+
+	if (!build)
+		return NULL;
+	ctx = isl_ast_build_get_ctx(build);
+	depth = isl_ast_build_get_depth(build);
+
+	if (!isl_ast_build_has_stride(build, depth))
+		return isl_ast_expr_alloc_int_si(ctx, 1);
+
+	v = isl_ast_build_get_stride(build, depth);
+	return isl_ast_expr_from_val(v);
+}
+
+/* Should we express the loop condition as
+ *
+ *	iterator <= min(list of upper bounds)
+ *
+ * or as a conjunction of constraints?
+ *
+ * The first is constructed from a list of upper bounds.
+ * The second is constructed from a set.
+ *
+ * If there are no upper bounds in "constraints", then this could mean
+ * that "domain" simply doesn't have an upper bound or that we didn't
+ * pick any upper bound.  In the first case, we want to generate the
+ * loop condition as a(n empty) conjunction of constraints
+ * In the second case, we will compute
+ * a single upper bound from "domain" and so we use the list form.
+ *
+ * If there are upper bounds in "constraints",
+ * then we use the list form iff the atomic_upper_bound option is set.
+ */
+static int use_upper_bound_list(isl_ctx *ctx, int n_upper,
+	__isl_keep isl_set *domain, int depth)
+{
+	if (n_upper > 0)
+		return isl_options_get_ast_build_atomic_upper_bound(ctx);
+	else
+		return isl_set_dim_has_upper_bound(domain, isl_dim_set, depth);
+}
+
+/* Fill in the expressions of the for node in graft->node.
+ *
+ * In particular,
+ * - set the initialization part of the loop to the maximum of the lower bounds
+ * - extract the increment from the stride of the current dimension
+ * - construct the for condition either based on a list of upper bounds
+ *	or on a set of upper bound constraints.
+ */
+static __isl_give isl_ast_graft *set_for_node_expressions(
+	__isl_take isl_ast_graft *graft, __isl_keep isl_pw_aff_list *lower,
+	int use_list, __isl_keep isl_pw_aff_list *upper_list,
+	__isl_keep isl_set *upper_set, __isl_keep isl_ast_build *build)
+{
+	isl_ast_node *node;
+
+	if (!graft)
+		return NULL;
+
+	build = isl_ast_build_copy(build);
+
+	node = graft->node;
+	node->u.f.init = reduce_list(isl_ast_op_max, lower, build);
+	node->u.f.inc = for_inc(build);
+
+	if (use_list)
+		graft = set_for_cond_from_list(graft, upper_list, build);
+	else
+		graft = set_for_cond_from_set(graft, upper_set, build);
+
+	isl_ast_build_free(build);
+
+	if (!node->u.f.iterator || !node->u.f.init ||
+	    !node->u.f.cond || !node->u.f.inc)
+		return isl_ast_graft_free(graft);
+
+	return graft;
+}
+
+/* Update "graft" based on "bounds" and "domain" for the generic,
+ * non-degenerate, case.
+ *
+ * "c_lower" and "c_upper" contain the lower and upper bounds
+ * that the loop node should express.
+ * "domain" is the subset of the intersection of the constraints
+ * for which some code is executed.
+ *
+ * There may be zero lower bounds or zero upper bounds in "constraints"
+ * in case the list of constraints was created
+ * based on the atomic option or based on separation with explicit bounds.
+ * In that case, we use "domain" to derive lower and/or upper bounds.
+ *
+ * We first compute a list of one or more lower bounds.
+ *
+ * Then we decide if we want to express the condition as
+ *
+ *	iterator <= min(list of upper bounds)
+ *
+ * or as a conjunction of constraints.
+ *
+ * The set of enforced constraints is then computed either based on
+ * a list of upper bounds or on a set of upper bound constraints.
+ * We do not compute any enforced constraints if we were forced
+ * to compute a lower or upper bound using exact_bound.  The domains
+ * of the resulting expressions may imply some bounds on outer dimensions
+ * that we do not want to appear in the enforced constraints since
+ * they are not actually enforced by the corresponding code.
+ *
+ * Finally, we fill in the expressions of the for node.
+ */
+static __isl_give isl_ast_graft *refine_generic_bounds(
+	__isl_take isl_ast_graft *graft,
+	__isl_take isl_constraint_list *c_lower,
+	__isl_take isl_constraint_list *c_upper,
+	__isl_keep isl_set *domain, __isl_keep isl_ast_build *build)
+{
+	int depth;
+	isl_ctx *ctx;
+	isl_pw_aff_list *lower;
+	int use_list;
+	isl_set *upper_set = NULL;
+	isl_pw_aff_list *upper_list = NULL;
+	int n_lower, n_upper;
+
+	if (!graft || !c_lower || !c_upper || !build)
+		goto error;
+
+	depth = isl_ast_build_get_depth(build);
+	ctx = isl_ast_graft_get_ctx(graft);
+
+	n_lower = isl_constraint_list_n_constraint(c_lower);
+	n_upper = isl_constraint_list_n_constraint(c_upper);
+
+	use_list = use_upper_bound_list(ctx, n_upper, domain, depth);
+
+	lower = lower_bounds(c_lower, depth, domain, build);
+
+	if (use_list)
+		upper_list = upper_bounds(c_upper, depth, domain, build);
+	else if (n_upper > 0)
+		upper_set = intersect_constraints(c_upper);
+	else
+		upper_set = isl_set_universe(isl_set_get_space(domain));
+
+	if (n_lower == 0 || n_upper == 0)
+		;
+	else if (use_list)
+		graft = set_enforced_from_list(graft, lower, upper_list);
+	else
+		graft = set_enforced_from_set(graft, lower, depth, upper_set);
+
+	graft = set_for_node_expressions(graft, lower, use_list, upper_list,
+					upper_set, build);
+
+	isl_pw_aff_list_free(lower);
+	isl_pw_aff_list_free(upper_list);
+	isl_set_free(upper_set);
+	isl_constraint_list_free(c_lower);
+	isl_constraint_list_free(c_upper);
+
+	return graft;
+error:
+	isl_constraint_list_free(c_lower);
+	isl_constraint_list_free(c_upper);
+	return isl_ast_graft_free(graft);
+}
+
+/* Internal data structure used inside count_constraints to keep
+ * track of the number of constraints that are independent of dimension "pos",
+ * the lower bounds in "pos" and the upper bounds in "pos".
+ */
+struct isl_ast_count_constraints_data {
+	int pos;
+
+	int n_indep;
+	int n_lower;
+	int n_upper;
+};
+
+/* Increment data->n_indep, data->lower or data->upper depending
+ * on whether "c" is independenct of dimensions data->pos,
+ * a lower bound or an upper bound.
+ */
+static int count_constraints(__isl_take isl_constraint *c, void *user)
+{
+	struct isl_ast_count_constraints_data *data = user;
+
+	if (isl_constraint_is_lower_bound(c, isl_dim_set, data->pos))
+		data->n_lower++;
+	else if (isl_constraint_is_upper_bound(c, isl_dim_set, data->pos))
+		data->n_upper++;
+	else
+		data->n_indep++;
+
+	isl_constraint_free(c);
+
+	return 0;
+}
+
+/* Update "graft" based on "bounds" and "domain" for the generic,
+ * non-degenerate, case.
+ *
+ * "list" respresent the list of bounds that need to be encoded by
+ * the for loop.  Only the constraints that involve the iterator
+ * are relevant here.  The other constraints are taken care of by
+ * the caller and are included in the generated constraints of "build".
+ * "domain" is the subset of the intersection of the constraints
+ * for which some code is executed.
+ * "build" is the build in which graft->node was created.
+ *
+ * We separate lower bounds, upper bounds and constraints that
+ * are independent of the loop iterator.
+ *
+ * The actual for loop bounds are generated in refine_generic_bounds.
+ */
+static __isl_give isl_ast_graft *refine_generic_split(
+	__isl_take isl_ast_graft *graft, __isl_take isl_constraint_list *list,
+	__isl_keep isl_set *domain, __isl_keep isl_ast_build *build)
+{
+	struct isl_ast_count_constraints_data data;
+	isl_constraint_list *lower;
+	isl_constraint_list *upper;
+
+	if (!list)
+		return isl_ast_graft_free(graft);
+
+	data.pos = isl_ast_build_get_depth(build);
+
+	list = isl_constraint_list_sort(list, &cmp_constraint, &data.pos);
+	if (!list)
+		return isl_ast_graft_free(graft);
+
+	data.n_indep = data.n_lower = data.n_upper = 0;
+	if (isl_constraint_list_foreach(list, &count_constraints, &data) < 0) {
+		isl_constraint_list_free(list);
+		return isl_ast_graft_free(graft);
+	}
+
+	lower = isl_constraint_list_drop(list, 0, data.n_indep);
+	upper = isl_constraint_list_copy(lower);
+	lower = isl_constraint_list_drop(lower, data.n_lower, data.n_upper);
+	upper = isl_constraint_list_drop(upper, 0, data.n_lower);
+
+	return refine_generic_bounds(graft, lower, upper, domain, build);
+}
+
+/* Update "graft" based on "bounds" and "domain" for the generic,
+ * non-degenerate, case.
+ *
+ * "bounds" respresent the bounds that need to be encoded by
+ * the for loop (or a guard around the for loop).
+ * "domain" is the subset of "bounds" for which some code is executed.
+ * "build" is the build in which graft->node was created.
+ *
+ * We break up "bounds" into a list of constraints and continue with
+ * refine_generic_split.
+ */
+static __isl_give isl_ast_graft *refine_generic(
+	__isl_take isl_ast_graft *graft,
+	__isl_keep isl_basic_set *bounds, __isl_keep isl_set *domain,
+	__isl_keep isl_ast_build *build)
+{
+	isl_constraint_list *list;
+
+	if (!build || !graft)
+		return isl_ast_graft_free(graft);
+
+	list = isl_basic_set_get_constraint_list(bounds);
+
+	graft = refine_generic_split(graft, list, domain, build);
+
+	return graft;
+}
+
+/* Create a for node for the current level.
+ *
+ * Mark the for node degenerate if "degenerate" is set.
+ */
+static __isl_give isl_ast_node *create_for(__isl_keep isl_ast_build *build,
+	int degenerate)
+{
+	int depth;
+	isl_id *id;
+	isl_ast_node *node;
+
+	if (!build)
+		return NULL;
+
+	depth = isl_ast_build_get_depth(build);
+	id = isl_ast_build_get_iterator_id(build, depth);
+	node = isl_ast_node_alloc_for(id);
+	if (degenerate)
+		node = isl_ast_node_for_mark_degenerate(node);
+
+	return node;
+}
+
+/* If the ast_build_exploit_nested_bounds option is set, then return
+ * the constraints enforced by all elements in "list".
+ * Otherwise, return the universe.
+ */
+static __isl_give isl_basic_set *extract_shared_enforced(
+	__isl_keep isl_ast_graft_list *list, __isl_keep isl_ast_build *build)
+{
+	isl_ctx *ctx;
+	isl_space *space;
+
+	if (!list)
+		return NULL;
+
+	ctx = isl_ast_graft_list_get_ctx(list);
+	if (isl_options_get_ast_build_exploit_nested_bounds(ctx))
+		return isl_ast_graft_list_extract_shared_enforced(list, build);
+
+	space = isl_ast_build_get_space(build, 1);
+	return isl_basic_set_universe(space);
+}
+
+/* Return the pending constraints of "build" that are not already taken
+ * care of (by a combination of "enforced" and the generated constraints
+ * of "build").
+ */
+static __isl_give isl_set *extract_pending(__isl_keep isl_ast_build *build,
+	__isl_keep isl_basic_set *enforced)
+{
+	isl_set *guard, *context;
+
+	guard = isl_ast_build_get_pending(build);
+	context = isl_set_from_basic_set(isl_basic_set_copy(enforced));
+	context = isl_set_intersect(context,
+					isl_ast_build_get_generated(build));
+	return isl_set_gist(guard, context);
+}
+
+/* Create an AST node for the current dimension based on
+ * the schedule domain "bounds" and return the node encapsulated
+ * in an isl_ast_graft.
+ *
+ * "executed" is the current inverse schedule, taking into account
+ * the bounds in "bounds"
+ * "domain" is the domain of "executed", with inner dimensions projected out.
+ * It may be a strict subset of "bounds" in case "bounds" was created
+ * based on the atomic option or based on separation with explicit bounds.
+ *
+ * "domain" may satisfy additional equalities that result
+ * from intersecting "executed" with "bounds" in add_node.
+ * It may also satisfy some global constraints that were dropped out because
+ * we performed separation with explicit bounds.
+ * The very first step is then to copy these constraints to "bounds".
+ *
+ * Since we may be calling before_each_for and after_each_for
+ * callbacks, we record the current inverse schedule in the build.
+ *
+ * We consider three builds,
+ * "build" is the one in which the current level is created,
+ * "body_build" is the build in which the next level is created,
+ * "sub_build" is essentially the same as "body_build", except that
+ * the depth has not been increased yet.
+ *
+ * "build" already contains information (in strides and offsets)
+ * about the strides at the current level, but this information is not
+ * reflected in the build->domain.
+ * We first add this information and the "bounds" to the sub_build->domain.
+ * isl_ast_build_set_loop_bounds adds the stride information and
+ * checks whether the current dimension attains
+ * only a single value and whether this single value can be represented using
+ * a single affine expression.
+ * In the first case, the current level is considered "degenerate".
+ * In the second, sub-case, the current level is considered "eliminated".
+ * Eliminated levels don't need to be reflected in the AST since we can
+ * simply plug in the affine expression.  For degenerate, but non-eliminated,
+ * levels, we do introduce a for node, but mark is as degenerate so that
+ * it can be printed as an assignment of the single value to the loop
+ * "iterator".
+ *
+ * If the current level is eliminated, we explicitly plug in the value
+ * for the current level found by isl_ast_build_set_loop_bounds in the
+ * inverse schedule.  This ensures that if we are working on a slice
+ * of the domain based on information available in the inverse schedule
+ * and the build domain, that then this information is also reflected
+ * in the inverse schedule.  This operation also eliminates the current
+ * dimension from the inverse schedule making sure no inner dimensions depend
+ * on the current dimension.  Otherwise, we create a for node, marking
+ * it degenerate if appropriate.  The initial for node is still incomplete
+ * and will be completed in either refine_degenerate or refine_generic.
+ *
+ * We then generate a sequence of grafts for the next level,
+ * create a surrounding graft for the current level and insert
+ * the for node we created (if the current level is not eliminated).
+ * Before creating a graft for the current level, we first extract
+ * hoistable constraints from the child guards and combine them
+ * with the pending constraints in the build.  These constraints
+ * are used to simplify the child guards and then added to the guard
+ * of the current graft to ensure that they will be generated.
+ * If the hoisted guard is a disjunction, then we use it directly
+ * to gist the guards on the children before intersect it with the
+ * pending constraints.  We do so because this disjunction is typically
+ * identical to the guards on the children such that these guards
+ * can be effectively removed completely.  After the intersection,
+ * the gist operation would have a harder time figuring this out.
+ *
+ * Finally, we set the bounds of the for loop in either
+ * refine_degenerate or refine_generic.
+ * We do so in a context where the pending constraints of the build
+ * have been replaced by the guard of the current graft.
+ */
+static __isl_give isl_ast_graft *create_node_scaled(
+	__isl_take isl_union_map *executed,
+	__isl_take isl_basic_set *bounds, __isl_take isl_set *domain,
+	__isl_take isl_ast_build *build)
+{
+	int depth;
+	int degenerate, eliminated;
+	isl_basic_set *hull;
+	isl_basic_set *enforced;
+	isl_set *guard, *hoisted;
+	isl_ast_node *node = NULL;
+	isl_ast_graft *graft;
+	isl_ast_graft_list *children;
+	isl_ast_build *sub_build;
+	isl_ast_build *body_build;
+
+	domain = isl_ast_build_eliminate_divs(build, domain);
+	domain = isl_set_detect_equalities(domain);
+	hull = isl_set_unshifted_simple_hull(isl_set_copy(domain));
+	bounds = isl_basic_set_intersect(bounds, hull);
+	build = isl_ast_build_set_executed(build, isl_union_map_copy(executed));
+
+	depth = isl_ast_build_get_depth(build);
+	sub_build = isl_ast_build_copy(build);
+	sub_build = isl_ast_build_set_loop_bounds(sub_build,
+						isl_basic_set_copy(bounds));
+	degenerate = isl_ast_build_has_value(sub_build);
+	eliminated = isl_ast_build_has_affine_value(sub_build, depth);
+	if (degenerate < 0 || eliminated < 0)
+		executed = isl_union_map_free(executed);
+	if (eliminated)
+		executed = plug_in_values(executed, sub_build);
+	else
+		node = create_for(build, degenerate);
+
+	body_build = isl_ast_build_copy(sub_build);
+	body_build = isl_ast_build_increase_depth(body_build);
+	if (!eliminated)
+		node = before_each_for(node, body_build);
+	children = generate_next_level(executed,
+				    isl_ast_build_copy(body_build));
+
+	enforced = extract_shared_enforced(children, build);
+	guard = extract_pending(sub_build, enforced);
+	hoisted = isl_ast_graft_list_extract_hoistable_guard(children, build);
+	if (isl_set_n_basic_set(hoisted) > 1)
+		children = isl_ast_graft_list_gist_guards(children,
+						    isl_set_copy(hoisted));
+	guard = isl_set_intersect(guard, hoisted);
+	if (!eliminated)
+		guard = add_implied_guards(guard, degenerate, bounds, build);
+
+	graft = isl_ast_graft_alloc_from_children(children,
+			    isl_set_copy(guard), enforced, build, sub_build);
+
+	if (!degenerate)
+		bounds = isl_ast_build_compute_gist_basic_set(build, bounds);
+	if (!eliminated) {
+		isl_ast_build *for_build;
+
+		graft = isl_ast_graft_insert_for(graft, node);
+		for_build = isl_ast_build_copy(build);
+		for_build = isl_ast_build_replace_pending_by_guard(for_build,
+							isl_set_copy(guard));
+		if (degenerate)
+			graft = refine_degenerate(graft, for_build, sub_build);
+		else
+			graft = refine_generic(graft, bounds,
+					domain, for_build);
+		isl_ast_build_free(for_build);
+	}
+	isl_set_free(guard);
+	if (!eliminated)
+		graft = after_each_for(graft, body_build);
+
+	isl_ast_build_free(body_build);
+	isl_ast_build_free(sub_build);
+	isl_ast_build_free(build);
+	isl_basic_set_free(bounds);
+	isl_set_free(domain);
+
+	return graft;
+}
+
+/* Internal data structure for checking if all constraints involving
+ * the input dimension "depth" are such that the other coefficients
+ * are multiples of "m", reducing "m" if they are not.
+ * If "m" is reduced all the way down to "1", then the check has failed
+ * and we break out of the iteration.
+ */
+struct isl_check_scaled_data {
+	int depth;
+	isl_val *m;
+};
+
+/* If constraint "c" involves the input dimension data->depth,
+ * then make sure that all the other coefficients are multiples of data->m,
+ * reducing data->m if needed.
+ * Break out of the iteration if data->m has become equal to "1".
+ */
+static int constraint_check_scaled(__isl_take isl_constraint *c, void *user)
+{
+	struct isl_check_scaled_data *data = user;
+	int i, j, n;
+	enum isl_dim_type t[] = { isl_dim_param, isl_dim_in, isl_dim_out,
+				    isl_dim_div };
+
+	if (!isl_constraint_involves_dims(c, isl_dim_in, data->depth, 1)) {
+		isl_constraint_free(c);
+		return 0;
+	}
+
+	for (i = 0; i < 4; ++i) {
+		n = isl_constraint_dim(c, t[i]);
+		for (j = 0; j < n; ++j) {
+			isl_val *d;
+
+			if (t[i] == isl_dim_in && j == data->depth)
+				continue;
+			if (!isl_constraint_involves_dims(c, t[i], j, 1))
+				continue;
+			d = isl_constraint_get_coefficient_val(c, t[i], j);
+			data->m = isl_val_gcd(data->m, d);
+			if (isl_val_is_one(data->m))
+				break;
+		}
+		if (j < n)
+			break;
+	}
+
+	isl_constraint_free(c);
+
+	return i < 4 ? -1 : 0;
+}
+
+/* For each constraint of "bmap" that involves the input dimension data->depth,
+ * make sure that all the other coefficients are multiples of data->m,
+ * reducing data->m if needed.
+ * Break out of the iteration if data->m has become equal to "1".
+ */
+static int basic_map_check_scaled(__isl_take isl_basic_map *bmap, void *user)
+{
+	int r;
+
+	r = isl_basic_map_foreach_constraint(bmap,
+						&constraint_check_scaled, user);
+	isl_basic_map_free(bmap);
+
+	return r;
+}
+
+/* For each constraint of "map" that involves the input dimension data->depth,
+ * make sure that all the other coefficients are multiples of data->m,
+ * reducing data->m if needed.
+ * Break out of the iteration if data->m has become equal to "1".
+ */
+static int map_check_scaled(__isl_take isl_map *map, void *user)
+{
+	int r;
+
+	r = isl_map_foreach_basic_map(map, &basic_map_check_scaled, user);
+	isl_map_free(map);
+
+	return r;
+}
+
+/* Create an AST node for the current dimension based on
+ * the schedule domain "bounds" and return the node encapsulated
+ * in an isl_ast_graft.
+ *
+ * "executed" is the current inverse schedule, taking into account
+ * the bounds in "bounds"
+ * "domain" is the domain of "executed", with inner dimensions projected out.
+ *
+ *
+ * Before moving on to the actual AST node construction in create_node_scaled,
+ * we first check if the current dimension is strided and if we can scale
+ * down this stride.  Note that we only do this if the ast_build_scale_strides
+ * option is set.
+ *
+ * In particular, let the current dimension take on values
+ *
+ *	f + s a
+ *
+ * with a an integer.  We check if we can find an integer m that (obviously)
+ * divides both f and s.
+ *
+ * If so, we check if the current dimension only appears in constraints
+ * where the coefficients of the other variables are multiples of m.
+ * We perform this extra check to avoid the risk of introducing
+ * divisions by scaling down the current dimension.
+ *
+ * If so, we scale the current dimension down by a factor of m.
+ * That is, we plug in
+ *
+ *	i = m i'							(1)
+ *
+ * Note that in principle we could always scale down strided loops
+ * by plugging in
+ *
+ *	i = f + s i'
+ *
+ * but this may result in i' taking on larger values than the original i,
+ * due to the shift by "f".
+ * By constrast, the scaling in (1) can only reduce the (absolute) value "i".
+ */
+static __isl_give isl_ast_graft *create_node(__isl_take isl_union_map *executed,
+	__isl_take isl_basic_set *bounds, __isl_take isl_set *domain,
+	__isl_take isl_ast_build *build)
+{
+	struct isl_check_scaled_data data;
+	isl_ctx *ctx;
+	isl_aff *offset;
+	isl_val *d;
+
+	ctx = isl_ast_build_get_ctx(build);
+	if (!isl_options_get_ast_build_scale_strides(ctx))
+		return create_node_scaled(executed, bounds, domain, build);
+
+	data.depth = isl_ast_build_get_depth(build);
+	if (!isl_ast_build_has_stride(build, data.depth))
+		return create_node_scaled(executed, bounds, domain, build);
+
+	offset = isl_ast_build_get_offset(build, data.depth);
+	data.m = isl_ast_build_get_stride(build, data.depth);
+	if (!data.m)
+		offset = isl_aff_free(offset);
+	offset = isl_aff_scale_down_val(offset, isl_val_copy(data.m));
+	d = isl_aff_get_denominator_val(offset);
+	if (!d)
+		executed = isl_union_map_free(executed);
+
+	if (executed && isl_val_is_divisible_by(data.m, d))
+		data.m = isl_val_div(data.m, d);
+	else {
+		data.m = isl_val_set_si(data.m, 1);
+		isl_val_free(d);
+	}
+
+	if (!isl_val_is_one(data.m)) {
+		if (isl_union_map_foreach_map(executed, &map_check_scaled,
+						&data) < 0 &&
+		    !isl_val_is_one(data.m))
+			executed = isl_union_map_free(executed);
+	}
+
+	if (!isl_val_is_one(data.m)) {
+		isl_space *space;
+		isl_multi_aff *ma;
+		isl_aff *aff;
+		isl_map *map;
+		isl_union_map *umap;
+
+		space = isl_ast_build_get_space(build, 1);
+		space = isl_space_map_from_set(space);
+		ma = isl_multi_aff_identity(space);
+		aff = isl_multi_aff_get_aff(ma, data.depth);
+		aff = isl_aff_scale_val(aff, isl_val_copy(data.m));
+		ma = isl_multi_aff_set_aff(ma, data.depth, aff);
+
+		bounds = isl_basic_set_preimage_multi_aff(bounds,
+						isl_multi_aff_copy(ma));
+		domain = isl_set_preimage_multi_aff(domain,
+						isl_multi_aff_copy(ma));
+		map = isl_map_reverse(isl_map_from_multi_aff(ma));
+		umap = isl_union_map_from_map(map);
+		executed = isl_union_map_apply_domain(executed,
+						isl_union_map_copy(umap));
+		build = isl_ast_build_scale_down(build, isl_val_copy(data.m),
+						umap);
+	}
+	isl_aff_free(offset);
+	isl_val_free(data.m);
+
+	return create_node_scaled(executed, bounds, domain, build);
+}
+
+/* Add the basic set to the list that "user" points to.
+ */
+static int collect_basic_set(__isl_take isl_basic_set *bset, void *user)
+{
+	isl_basic_set_list **list = user;
+
+	*list = isl_basic_set_list_add(*list, bset);
+
+	return 0;
+}
+
+/* Extract the basic sets of "set" and collect them in an isl_basic_set_list.
+ */
+static __isl_give isl_basic_set_list *isl_basic_set_list_from_set(
+	__isl_take isl_set *set)
+{
+	int n;
+	isl_ctx *ctx;
+	isl_basic_set_list *list;
+
+	if (!set)
+		return NULL;
+
+	ctx = isl_set_get_ctx(set);
+
+	n = isl_set_n_basic_set(set);
+	list = isl_basic_set_list_alloc(ctx, n);
+	if (isl_set_foreach_basic_set(set, &collect_basic_set, &list) < 0)
+		list = isl_basic_set_list_free(list);
+
+	isl_set_free(set);
+	return list;
+}
+
+/* Generate code for the schedule domain "bounds"
+ * and add the result to "list".
+ *
+ * We mainly detect strides here and check if the bounds do not
+ * conflict with the current build domain
+ * and then pass over control to create_node.
+ *
+ * "bounds" reflects the bounds on the current dimension and possibly
+ * some extra conditions on outer dimensions.
+ * It does not, however, include any divs involving the current dimension,
+ * so it does not capture any stride constraints.
+ * We therefore need to compute that part of the schedule domain that
+ * intersects with "bounds" and derive the strides from the result.
+ */
+static __isl_give isl_ast_graft_list *add_node(
+	__isl_take isl_ast_graft_list *list, __isl_take isl_union_map *executed,
+	__isl_take isl_basic_set *bounds, __isl_take isl_ast_build *build)
+{
+	isl_ast_graft *graft;
+	isl_set *domain = NULL;
+	isl_union_set *uset;
+	int empty, disjoint;
+
+	uset = isl_union_set_from_basic_set(isl_basic_set_copy(bounds));
+	executed = isl_union_map_intersect_domain(executed, uset);
+	empty = isl_union_map_is_empty(executed);
+	if (empty < 0)
+		goto error;
+	if (empty)
+		goto done;
+
+	uset = isl_union_map_domain(isl_union_map_copy(executed));
+	domain = isl_set_from_union_set(uset);
+	domain = isl_ast_build_specialize(build, domain);
+
+	domain = isl_set_compute_divs(domain);
+	domain = isl_ast_build_eliminate_inner(build, domain);
+	disjoint = isl_set_is_disjoint(domain, build->domain);
+	if (disjoint < 0)
+		goto error;
+	if (disjoint)
+		goto done;
+
+	build = isl_ast_build_detect_strides(build, isl_set_copy(domain));
+
+	graft = create_node(executed, bounds, domain,
+				isl_ast_build_copy(build));
+	list = isl_ast_graft_list_add(list, graft);
+	isl_ast_build_free(build);
+	return list;
+error:
+	list = isl_ast_graft_list_free(list);
+done:
+	isl_set_free(domain);
+	isl_basic_set_free(bounds);
+	isl_union_map_free(executed);
+	isl_ast_build_free(build);
+	return list;
+}
+
+/* Does any element of i follow or coincide with any element of j
+ * at the current depth for equal values of the outer dimensions?
+ */
+static int domain_follows_at_depth(__isl_keep isl_basic_set *i,
+	__isl_keep isl_basic_set *j, void *user)
+{
+	int depth = *(int *) user;
+	isl_basic_map *test;
+	int empty;
+	int l;
+
+	test = isl_basic_map_from_domain_and_range(isl_basic_set_copy(i),
+						    isl_basic_set_copy(j));
+	for (l = 0; l < depth; ++l)
+		test = isl_basic_map_equate(test, isl_dim_in, l,
+						isl_dim_out, l);
+	test = isl_basic_map_order_ge(test, isl_dim_in, depth,
+					isl_dim_out, depth);
+	empty = isl_basic_map_is_empty(test);
+	isl_basic_map_free(test);
+
+	return empty < 0 ? -1 : !empty;
+}
+
+/* Split up each element of "list" into a part that is related to "bset"
+ * according to "gt" and a part that is not.
+ * Return a list that consist of "bset" and all the pieces.
+ */
+static __isl_give isl_basic_set_list *add_split_on(
+	__isl_take isl_basic_set_list *list, __isl_take isl_basic_set *bset,
+	__isl_keep isl_basic_map *gt)
+{
+	int i, n;
+	isl_basic_set_list *res;
+
+	if (!list)
+		bset = isl_basic_set_free(bset);
+
+	gt = isl_basic_map_copy(gt);
+	gt = isl_basic_map_intersect_domain(gt, isl_basic_set_copy(bset));
+	n = isl_basic_set_list_n_basic_set(list);
+	res = isl_basic_set_list_from_basic_set(bset);
+	for (i = 0; res && i < n; ++i) {
+		isl_basic_set *bset;
+		isl_set *set1, *set2;
+		isl_basic_map *bmap;
+		int empty;
+
+		bset = isl_basic_set_list_get_basic_set(list, i);
+		bmap = isl_basic_map_copy(gt);
+		bmap = isl_basic_map_intersect_range(bmap, bset);
+		bset = isl_basic_map_range(bmap);
+		empty = isl_basic_set_is_empty(bset);
+		if (empty < 0)
+			res = isl_basic_set_list_free(res);
+		if (empty)  {
+			isl_basic_set_free(bset);
+			bset = isl_basic_set_list_get_basic_set(list, i);
+			res = isl_basic_set_list_add(res, bset);
+			continue;
+		}
+
+		res = isl_basic_set_list_add(res, isl_basic_set_copy(bset));
+		set1 = isl_set_from_basic_set(bset);
+		bset = isl_basic_set_list_get_basic_set(list, i);
+		set2 = isl_set_from_basic_set(bset);
+		set1 = isl_set_subtract(set2, set1);
+		set1 = isl_set_make_disjoint(set1);
+
+		res = isl_basic_set_list_concat(res,
+					    isl_basic_set_list_from_set(set1));
+	}
+	isl_basic_map_free(gt);
+	isl_basic_set_list_free(list);
+	return res;
+}
+
+static __isl_give isl_ast_graft_list *generate_sorted_domains(
+	__isl_keep isl_basic_set_list *domain_list,
+	__isl_keep isl_union_map *executed,
+	__isl_keep isl_ast_build *build);
+
+/* Internal data structure for add_nodes.
+ *
+ * "executed" and "build" are extra arguments to be passed to add_node.
+ * "list" collects the results.
+ */
+struct isl_add_nodes_data {
+	isl_union_map *executed;
+	isl_ast_build *build;
+
+	isl_ast_graft_list *list;
+};
+
+/* Generate code for the schedule domains in "scc"
+ * and add the results to "list".
+ *
+ * The domains in "scc" form a strongly connected component in the ordering.
+ * If the number of domains in "scc" is larger than 1, then this means
+ * that we cannot determine a valid ordering for the domains in the component.
+ * This should be fairly rare because the individual domains
+ * have been made disjoint first.
+ * The problem is that the domains may be integrally disjoint but not
+ * rationally disjoint.  For example, we may have domains
+ *
+ *	{ [i,i] : 0 <= i <= 1 }		and	{ [i,1-i] : 0 <= i <= 1 }
+ *
+ * These two domains have an empty intersection, but their rational
+ * relaxations do intersect.  It is impossible to order these domains
+ * in the second dimension because the first should be ordered before
+ * the second for outer dimension equal to 0, while it should be ordered
+ * after for outer dimension equal to 1.
+ *
+ * This may happen in particular in case of unrolling since the domain
+ * of each slice is replaced by its simple hull.
+ *
+ * For each basic set i in "scc" and for each of the following basic sets j,
+ * we split off that part of the basic set i that shares the outer dimensions
+ * with j and lies before j in the current dimension.
+ * We collect all the pieces in a new list that replaces "scc".
+ *
+ * While the elements in "scc" should be disjoint, we double-check
+ * this property to avoid running into an infinite recursion in case
+ * they intersect due to some internal error.
+ */
+static int add_nodes(__isl_take isl_basic_set_list *scc, void *user)
+{
+	struct isl_add_nodes_data *data = user;
+	int i, n, depth;
+	isl_basic_set *bset, *first;
+	isl_basic_set_list *list;
+	isl_space *space;
+	isl_basic_map *gt;
+
+	n = isl_basic_set_list_n_basic_set(scc);
+	bset = isl_basic_set_list_get_basic_set(scc, 0);
+	if (n == 1) {
+		isl_basic_set_list_free(scc);
+		data->list = add_node(data->list,
+				isl_union_map_copy(data->executed), bset,
+				isl_ast_build_copy(data->build));
+		return data->list ? 0 : -1;
+	}
+
+	depth = isl_ast_build_get_depth(data->build);
+	space = isl_basic_set_get_space(bset);
+	space = isl_space_map_from_set(space);
+	gt = isl_basic_map_universe(space);
+	for (i = 0; i < depth; ++i)
+		gt = isl_basic_map_equate(gt, isl_dim_in, i, isl_dim_out, i);
+	gt = isl_basic_map_order_gt(gt, isl_dim_in, depth, isl_dim_out, depth);
+
+	first = isl_basic_set_copy(bset);
+	list = isl_basic_set_list_from_basic_set(bset);
+	for (i = 1; i < n; ++i) {
+		int disjoint;
+
+		bset = isl_basic_set_list_get_basic_set(scc, i);
+
+		disjoint = isl_basic_set_is_disjoint(bset, first);
+		if (disjoint < 0)
+			list = isl_basic_set_list_free(list);
+		else if (!disjoint)
+			isl_die(isl_basic_set_list_get_ctx(scc),
+				isl_error_internal,
+				"basic sets in scc are assumed to be disjoint",
+				list = isl_basic_set_list_free(list));
+
+		list = add_split_on(list, bset, gt);
+	}
+	isl_basic_set_free(first);
+	isl_basic_map_free(gt);
+	isl_basic_set_list_free(scc);
+	scc = list;
+	data->list = isl_ast_graft_list_concat(data->list,
+		    generate_sorted_domains(scc, data->executed, data->build));
+	isl_basic_set_list_free(scc);
+
+	return data->list ? 0 : -1;
+}
+
+/* Sort the domains in "domain_list" according to the execution order
+ * at the current depth (for equal values of the outer dimensions),
+ * generate code for each of them, collecting the results in a list.
+ * If no code is generated (because the intersection of the inverse schedule
+ * with the domains turns out to be empty), then an empty list is returned.
+ *
+ * The caller is responsible for ensuring that the basic sets in "domain_list"
+ * are pair-wise disjoint.  It can, however, in principle happen that
+ * two basic sets should be ordered one way for one value of the outer
+ * dimensions and the other way for some other value of the outer dimensions.
+ * We therefore play safe and look for strongly connected components.
+ * The function add_nodes takes care of handling non-trivial components.
+ */
+static __isl_give isl_ast_graft_list *generate_sorted_domains(
+	__isl_keep isl_basic_set_list *domain_list,
+	__isl_keep isl_union_map *executed, __isl_keep isl_ast_build *build)
+{
+	isl_ctx *ctx;
+	struct isl_add_nodes_data data;
+	int depth;
+	int n;
+
+	if (!domain_list)
+		return NULL;
+
+	ctx = isl_basic_set_list_get_ctx(domain_list);
+	n = isl_basic_set_list_n_basic_set(domain_list);
+	data.list = isl_ast_graft_list_alloc(ctx, n);
+	if (n == 0)
+		return data.list;
+	if (n == 1)
+		return add_node(data.list, isl_union_map_copy(executed),
+			isl_basic_set_list_get_basic_set(domain_list, 0),
+			isl_ast_build_copy(build));
+
+	depth = isl_ast_build_get_depth(build);
+	data.executed = executed;
+	data.build = build;
+	if (isl_basic_set_list_foreach_scc(domain_list,
+					&domain_follows_at_depth, &depth,
+					&add_nodes, &data) < 0)
+		data.list = isl_ast_graft_list_free(data.list);
+
+	return data.list;
+}
+
+/* Do i and j share any values for the outer dimensions?
+ */
+static int shared_outer(__isl_keep isl_basic_set *i,
+	__isl_keep isl_basic_set *j, void *user)
+{
+	int depth = *(int *) user;
+	isl_basic_map *test;
+	int empty;
+	int l;
+
+	test = isl_basic_map_from_domain_and_range(isl_basic_set_copy(i),
+						    isl_basic_set_copy(j));
+	for (l = 0; l < depth; ++l)
+		test = isl_basic_map_equate(test, isl_dim_in, l,
+						isl_dim_out, l);
+	empty = isl_basic_map_is_empty(test);
+	isl_basic_map_free(test);
+
+	return empty < 0 ? -1 : !empty;
+}
+
+/* Internal data structure for generate_sorted_domains_wrap.
+ *
+ * "n" is the total number of basic sets
+ * "executed" and "build" are extra arguments to be passed
+ *	to generate_sorted_domains.
+ *
+ * "single" is set to 1 by generate_sorted_domains_wrap if there
+ * is only a single component.
+ * "list" collects the results.
+ */
+struct isl_ast_generate_parallel_domains_data {
+	int n;
+	isl_union_map *executed;
+	isl_ast_build *build;
+
+	int single;
+	isl_ast_graft_list *list;
+};
+
+/* Call generate_sorted_domains on "scc", fuse the result into a list
+ * with either zero or one graft and collect the these single element
+ * lists into data->list.
+ *
+ * If there is only one component, i.e., if the number of basic sets
+ * in the current component is equal to the total number of basic sets,
+ * then data->single is set to 1 and the result of generate_sorted_domains
+ * is not fused.
+ */
+static int generate_sorted_domains_wrap(__isl_take isl_basic_set_list *scc,
+	void *user)
+{
+	struct isl_ast_generate_parallel_domains_data *data = user;
+	isl_ast_graft_list *list;
+
+	list = generate_sorted_domains(scc, data->executed, data->build);
+	data->single = isl_basic_set_list_n_basic_set(scc) == data->n;
+	if (!data->single)
+		list = isl_ast_graft_list_fuse(list, data->build);
+	if (!data->list)
+		data->list = list;
+	else
+		data->list = isl_ast_graft_list_concat(data->list, list);
+
+	isl_basic_set_list_free(scc);
+	if (!data->list)
+		return -1;
+
+	return 0;
+}
+
+/* Look for any (weakly connected) components in the "domain_list"
+ * of domains that share some values of the outer dimensions.
+ * That is, domains in different components do not share any values
+ * of the outer dimensions.  This means that these components
+ * can be freely reordered.
+ * Within each of the components, we sort the domains according
+ * to the execution order at the current depth.
+ *
+ * If there is more than one component, then generate_sorted_domains_wrap
+ * fuses the result of each call to generate_sorted_domains
+ * into a list with either zero or one graft and collects these (at most)
+ * single element lists into a bigger list. This means that the elements of the
+ * final list can be freely reordered.  In particular, we sort them
+ * according to an arbitrary but fixed ordering to ease merging of
+ * graft lists from different components.
+ */
+static __isl_give isl_ast_graft_list *generate_parallel_domains(
+	__isl_keep isl_basic_set_list *domain_list,
+	__isl_keep isl_union_map *executed, __isl_keep isl_ast_build *build)
+{
+	int depth;
+	struct isl_ast_generate_parallel_domains_data data;
+
+	if (!domain_list)
+		return NULL;
+
+	data.n = isl_basic_set_list_n_basic_set(domain_list);
+	if (data.n <= 1)
+		return generate_sorted_domains(domain_list, executed, build);
+
+	depth = isl_ast_build_get_depth(build);
+	data.list = NULL;
+	data.executed = executed;
+	data.build = build;
+	data.single = 0;
+	if (isl_basic_set_list_foreach_scc(domain_list, &shared_outer, &depth,
+					    &generate_sorted_domains_wrap,
+					    &data) < 0)
+		data.list = isl_ast_graft_list_free(data.list);
+
+	if (!data.single)
+		data.list = isl_ast_graft_list_sort_guard(data.list);
+
+	return data.list;
+}
+
+/* Internal data for separate_domain.
+ *
+ * "explicit" is set if we only want to use explicit bounds.
+ *
+ * "domain" collects the separated domains.
+ */
+struct isl_separate_domain_data {
+	isl_ast_build *build;
+	int explicit;
+	isl_set *domain;
+};
+
+/* Extract implicit bounds on the current dimension for the executed "map".
+ *
+ * The domain of "map" may involve inner dimensions, so we
+ * need to eliminate them.
+ */
+static __isl_give isl_set *implicit_bounds(__isl_take isl_map *map,
+	__isl_keep isl_ast_build *build)
+{
+	isl_set *domain;
+
+	domain = isl_map_domain(map);
+	domain = isl_ast_build_eliminate(build, domain);
+
+	return domain;
+}
+
+/* Extract explicit bounds on the current dimension for the executed "map".
+ *
+ * Rather than eliminating the inner dimensions as in implicit_bounds,
+ * we simply drop any constraints involving those inner dimensions.
+ * The idea is that most bounds that are implied by constraints on the
+ * inner dimensions will be enforced by for loops and not by explicit guards.
+ * There is then no need to separate along those bounds.
+ */
+static __isl_give isl_set *explicit_bounds(__isl_take isl_map *map,
+	__isl_keep isl_ast_build *build)
+{
+	isl_set *domain;
+	int depth, dim;
+
+	dim = isl_map_dim(map, isl_dim_out);
+	map = isl_map_drop_constraints_involving_dims(map, isl_dim_out, 0, dim);
+
+	domain = isl_map_domain(map);
+	depth = isl_ast_build_get_depth(build);
+	dim = isl_set_dim(domain, isl_dim_set);
+	domain = isl_set_detect_equalities(domain);
+	domain = isl_set_drop_constraints_involving_dims(domain,
+				isl_dim_set, depth + 1, dim - (depth + 1));
+	domain = isl_set_remove_divs_involving_dims(domain,
+				isl_dim_set, depth, 1);
+	domain = isl_set_remove_unknown_divs(domain);
+
+	return domain;
+}
+
+/* Split data->domain into pieces that intersect with the range of "map"
+ * and pieces that do not intersect with the range of "map"
+ * and then add that part of the range of "map" that does not intersect
+ * with data->domain.
+ */
+static int separate_domain(__isl_take isl_map *map, void *user)
+{
+	struct isl_separate_domain_data *data = user;
+	isl_set *domain;
+	isl_set *d1, *d2;
+
+	if (data->explicit)
+		domain = explicit_bounds(map, data->build);
+	else
+		domain = implicit_bounds(map, data->build);
+
+	domain = isl_set_coalesce(domain);
+	domain = isl_set_make_disjoint(domain);
+	d1 = isl_set_subtract(isl_set_copy(domain), isl_set_copy(data->domain));
+	d2 = isl_set_subtract(isl_set_copy(data->domain), isl_set_copy(domain));
+	data->domain = isl_set_intersect(data->domain, domain);
+	data->domain = isl_set_union(data->domain, d1);
+	data->domain = isl_set_union(data->domain, d2);
+
+	return 0;
+}
+
+/* Separate the schedule domains of "executed".
+ *
+ * That is, break up the domain of "executed" into basic sets,
+ * such that for each basic set S, every element in S is associated with
+ * the same domain spaces.
+ *
+ * "space" is the (single) domain space of "executed".
+ */
+static __isl_give isl_set *separate_schedule_domains(
+	__isl_take isl_space *space, __isl_take isl_union_map *executed,
+	__isl_keep isl_ast_build *build)
+{
+	struct isl_separate_domain_data data = { build };
+	isl_ctx *ctx;
+
+	ctx = isl_ast_build_get_ctx(build);
+	data.explicit = isl_options_get_ast_build_separation_bounds(ctx) ==
+				    ISL_AST_BUILD_SEPARATION_BOUNDS_EXPLICIT;
+	data.domain = isl_set_empty(space);
+	if (isl_union_map_foreach_map(executed, &separate_domain, &data) < 0)
+		data.domain = isl_set_free(data.domain);
+
+	isl_union_map_free(executed);
+	return data.domain;
+}
+
+/* Temporary data used during the search for a lower bound for unrolling.
+ *
+ * "domain" is the original set for which to find a lower bound
+ * "depth" is the dimension for which to find a lower boudn
+ *
+ * "lower" is the best lower bound found so far.  It is NULL if we have not
+ * found any yet.
+ * "n" is the corresponding size.  If lower is NULL, then the value of n
+ * is undefined.
+ */
+struct isl_find_unroll_data {
+	isl_set *domain;
+	int depth;
+
+	isl_aff *lower;
+	int *n;
+};
+
+/* Check if we can use "c" as a lower bound and if it is better than
+ * any previously found lower bound.
+ *
+ * If "c" does not involve the dimension at the current depth,
+ * then we cannot use it.
+ * Otherwise, let "c" be of the form
+ *
+ *	i >= f(j)/a
+ *
+ * We compute the maximal value of
+ *
+ *	-ceil(f(j)/a)) + i + 1
+ *
+ * over the domain.  If there is such a value "n", then we know
+ *
+ *	-ceil(f(j)/a)) + i + 1 <= n
+ *
+ * or
+ *
+ *	i < ceil(f(j)/a)) + n
+ *
+ * meaning that we can use ceil(f(j)/a)) as a lower bound for unrolling.
+ * We just need to check if we have found any lower bound before and
+ * if the new lower bound is better (smaller n) than the previously found
+ * lower bounds.
+ */
+static int update_unrolling_lower_bound(struct isl_find_unroll_data *data,
+	__isl_keep isl_constraint *c)
+{
+	isl_aff *aff, *lower;
+	isl_val *max;
+
+	if (!isl_constraint_is_lower_bound(c, isl_dim_set, data->depth))
+		return 0;
+
+	lower = isl_constraint_get_bound(c, isl_dim_set, data->depth);
+	lower = isl_aff_ceil(lower);
+	aff = isl_aff_copy(lower);
+	aff = isl_aff_neg(aff);
+	aff = isl_aff_add_coefficient_si(aff, isl_dim_in, data->depth, 1);
+	aff = isl_aff_add_constant_si(aff, 1);
+	max = isl_set_max_val(data->domain, aff);
+	isl_aff_free(aff);
+
+	if (!max)
+		goto error;
+	if (isl_val_is_infty(max)) {
+		isl_val_free(max);
+		isl_aff_free(lower);
+		return 0;
+	}
+
+	if (isl_val_cmp_si(max, INT_MAX) <= 0 &&
+	    (!data->lower || isl_val_cmp_si(max, *data->n) < 0)) {
+		isl_aff_free(data->lower);
+		data->lower = lower;
+		*data->n = isl_val_get_num_si(max);
+	} else
+		isl_aff_free(lower);
+	isl_val_free(max);
+
+	return 1;
+error:
+	isl_aff_free(lower);
+	return -1;
+}
+
+/* Check if we can use "c" as a lower bound and if it is better than
+ * any previously found lower bound.
+ */
+static int constraint_find_unroll(__isl_take isl_constraint *c, void *user)
+{
+	struct isl_find_unroll_data *data;
+	int r;
+
+	data = (struct isl_find_unroll_data *) user;
+	r = update_unrolling_lower_bound(data, c);
+	isl_constraint_free(c);
+
+	return r;
+}
+
+/* Look for a lower bound l(i) on the dimension at "depth"
+ * and a size n such that "domain" is a subset of
+ *
+ *	{ [i] : l(i) <= i_d < l(i) + n }
+ *
+ * where d is "depth" and l(i) depends only on earlier dimensions.
+ * Furthermore, try and find a lower bound such that n is as small as possible.
+ * In particular, "n" needs to be finite.
+ *
+ * Inner dimensions have been eliminated from "domain" by the caller.
+ *
+ * We first construct a collection of lower bounds on the input set
+ * by computing its simple hull.  We then iterate through them,
+ * discarding those that we cannot use (either because they do not
+ * involve the dimension at "depth" or because they have no corresponding
+ * upper bound, meaning that "n" would be unbounded) and pick out the
+ * best from the remaining ones.
+ *
+ * If we cannot find a suitable lower bound, then we consider that
+ * to be an error.
+ */
+static __isl_give isl_aff *find_unroll_lower_bound(__isl_keep isl_set *domain,
+	int depth, int *n)
+{
+	struct isl_find_unroll_data data = { domain, depth, NULL, n };
+	isl_basic_set *hull;
+
+	hull = isl_set_simple_hull(isl_set_copy(domain));
+
+	if (isl_basic_set_foreach_constraint(hull,
+					    &constraint_find_unroll, &data) < 0)
+		goto error;
+
+	isl_basic_set_free(hull);
+
+	if (!data.lower)
+		isl_die(isl_set_get_ctx(domain), isl_error_invalid,
+			"cannot find lower bound for unrolling", return NULL);
+
+	return data.lower;
+error:
+	isl_basic_set_free(hull);
+	return isl_aff_free(data.lower);
+}
+
+/* Return the constraint
+ *
+ *	i_"depth" = aff + offset
+ */
+static __isl_give isl_constraint *at_offset(int depth, __isl_keep isl_aff *aff,
+	int offset)
+{
+	aff = isl_aff_copy(aff);
+	aff = isl_aff_add_coefficient_si(aff, isl_dim_in, depth, -1);
+	aff = isl_aff_add_constant_si(aff, offset);
+	return isl_equality_from_aff(aff);
+}
+
+/* Data structure for storing the results and the intermediate objects
+ * of compute_domains.
+ *
+ * "list" is the main result of the function and contains a list
+ * of disjoint basic sets for which code should be generated.
+ *
+ * "executed" and "build" are inputs to compute_domains.
+ * "schedule_domain" is the domain of "executed".
+ *
+ * "option" constains the domains at the current depth that should by
+ * atomic, separated or unrolled.  These domains are as specified by
+ * the user, except that inner dimensions have been eliminated and
+ * that they have been made pair-wise disjoint.
+ *
+ * "sep_class" contains the user-specified split into separation classes
+ * specialized to the current depth.
+ * "done" contains the union of the separation domains that have already
+ * been handled.
+ */
+struct isl_codegen_domains {
+	isl_basic_set_list *list;
+
+	isl_union_map *executed;
+	isl_ast_build *build;
+	isl_set *schedule_domain;
+
+	isl_set *option[3];
+
+	isl_map *sep_class;
+	isl_set *done;
+};
+
+/* Extend domains->list with a list of basic sets, one for each value
+ * of the current dimension in "domain" and remove the corresponding
+ * sets from the class domain.  Return the updated class domain.
+ * The divs that involve the current dimension have not been projected out
+ * from this domain.
+ *
+ * Since we are going to be iterating over the individual values,
+ * we first check if there are any strides on the current dimension.
+ * If there is, we rewrite the current dimension i as
+ *
+ *		i = stride i' + offset
+ *
+ * and then iterate over individual values of i' instead.
+ *
+ * We then look for a lower bound on i' and a size such that the domain
+ * is a subset of
+ *
+ *	{ [j,i'] : l(j) <= i' < l(j) + n }
+ *
+ * and then take slices of the domain at values of i'
+ * between l(j) and l(j) + n - 1.
+ *
+ * We compute the unshifted simple hull of each slice to ensure that
+ * we have a single basic set per offset.  The slicing constraint
+ * may get simplified away before the unshifted simple hull is taken
+ * and may therefore in some rare cases disappear from the result.
+ * We therefore explicitly add the constraint back after computing
+ * the unshifted simple hull to ensure that the basic sets
+ * remain disjoint.  The constraints that are dropped by taking the hull
+ * will be taken into account at the next level, as in the case of the
+ * atomic option.
+ *
+ * Finally, we map i' back to i and add each basic set to the list.
+ * Since we may have dropped some constraints, we intersect with
+ * the class domain again to ensure that each element in the list
+ * is disjoint from the other class domains.
+ */
+static __isl_give isl_set *do_unroll(struct isl_codegen_domains *domains,
+	__isl_take isl_set *domain, __isl_take isl_set *class_domain)
+{
+	int i, n;
+	int depth;
+	isl_ctx *ctx;
+	isl_aff *lower;
+	isl_multi_aff *expansion;
+	isl_basic_map *bmap;
+	isl_set *unroll_domain;
+	isl_ast_build *build;
+
+	if (!domain)
+		return isl_set_free(class_domain);
+
+	ctx = isl_set_get_ctx(domain);
+	depth = isl_ast_build_get_depth(domains->build);
+	build = isl_ast_build_copy(domains->build);
+	domain = isl_ast_build_eliminate_inner(build, domain);
+	domain = isl_set_intersect(domain, isl_ast_build_get_domain(build));
+	build = isl_ast_build_detect_strides(build, isl_set_copy(domain));
+	expansion = isl_ast_build_get_stride_expansion(build);
+
+	domain = isl_set_preimage_multi_aff(domain,
+					    isl_multi_aff_copy(expansion));
+	domain = isl_ast_build_eliminate_divs(build, domain);
+
+	isl_ast_build_free(build);
+
+	lower = find_unroll_lower_bound(domain, depth, &n);
+	if (!lower)
+		class_domain = isl_set_free(class_domain);
+
+	bmap = isl_basic_map_from_multi_aff(expansion);
+
+	unroll_domain = isl_set_empty(isl_set_get_space(domain));
+
+	for (i = 0; class_domain && i < n; ++i) {
+		isl_set *set;
+		isl_basic_set *bset;
+		isl_constraint *slice;
+		isl_basic_set_list *list;
+
+		slice = at_offset(depth, lower, i);
+		set = isl_set_copy(domain);
+		set = isl_set_add_constraint(set, isl_constraint_copy(slice));
+		bset = isl_set_unshifted_simple_hull(set);
+		bset = isl_basic_set_add_constraint(bset, slice);
+		bset = isl_basic_set_apply(bset, isl_basic_map_copy(bmap));
+		set = isl_set_from_basic_set(bset);
+		unroll_domain = isl_set_union(unroll_domain, isl_set_copy(set));
+		set = isl_set_intersect(set, isl_set_copy(class_domain));
+		set = isl_set_make_disjoint(set);
+		list = isl_basic_set_list_from_set(set);
+		domains->list = isl_basic_set_list_concat(domains->list, list);
+	}
+
+	class_domain = isl_set_subtract(class_domain, unroll_domain);
+
+	isl_aff_free(lower);
+	isl_set_free(domain);
+	isl_basic_map_free(bmap);
+
+	return class_domain;
+}
+
+/* Add domains to domains->list for each individual value of the current
+ * dimension, for that part of the schedule domain that lies in the
+ * intersection of the option domain and the class domain.
+ * Remove the corresponding sets from the class domain and
+ * return the updated class domain.
+ *
+ * We first break up the unroll option domain into individual pieces
+ * and then handle each of them separately.  The unroll option domain
+ * has been made disjoint in compute_domains_init_options,
+ *
+ * Note that we actively want to combine different pieces of the
+ * schedule domain that have the same value at the current dimension.
+ * We therefore need to break up the unroll option domain before
+ * intersecting with class and schedule domain, hoping that the
+ * unroll option domain specified by the user is relatively simple.
+ */
+static __isl_give isl_set *compute_unroll_domains(
+	struct isl_codegen_domains *domains, __isl_take isl_set *class_domain)
+{
+	isl_set *unroll_domain;
+	isl_basic_set_list *unroll_list;
+	int i, n;
+	int empty;
+
+	empty = isl_set_is_empty(domains->option[unroll]);
+	if (empty < 0)
+		return isl_set_free(class_domain);
+	if (empty)
+		return class_domain;
+
+	unroll_domain = isl_set_copy(domains->option[unroll]);
+	unroll_list = isl_basic_set_list_from_set(unroll_domain);
+
+	n = isl_basic_set_list_n_basic_set(unroll_list);
+	for (i = 0; i < n; ++i) {
+		isl_basic_set *bset;
+
+		bset = isl_basic_set_list_get_basic_set(unroll_list, i);
+		unroll_domain = isl_set_from_basic_set(bset);
+		unroll_domain = isl_set_intersect(unroll_domain,
+						    isl_set_copy(class_domain));
+		unroll_domain = isl_set_intersect(unroll_domain,
+					isl_set_copy(domains->schedule_domain));
+
+		empty = isl_set_is_empty(unroll_domain);
+		if (empty >= 0 && empty) {
+			isl_set_free(unroll_domain);
+			continue;
+		}
+
+		class_domain = do_unroll(domains, unroll_domain, class_domain);
+	}
+
+	isl_basic_set_list_free(unroll_list);
+
+	return class_domain;
+}
+
+/* Try and construct a single basic set that includes the intersection of
+ * the schedule domain, the atomic option domain and the class domain.
+ * Add the resulting basic set(s) to domains->list and remove them
+ * from class_domain.  Return the updated class domain.
+ *
+ * We construct a single domain rather than trying to combine
+ * the schedule domains of individual domains because we are working
+ * within a single component so that non-overlapping schedule domains
+ * should already have been separated.
+ * We do however need to make sure that this single domains is a subset
+ * of the class domain so that it would not intersect with any other
+ * class domains.  This means that we may end up splitting up the atomic
+ * domain in case separation classes are being used.
+ *
+ * "domain" is the intersection of the schedule domain and the class domain,
+ * with inner dimensions projected out.
+ */
+static __isl_give isl_set *compute_atomic_domain(
+	struct isl_codegen_domains *domains, __isl_take isl_set *class_domain)
+{
+	isl_basic_set *bset;
+	isl_basic_set_list *list;
+	isl_set *domain, *atomic_domain;
+	int empty;
+
+	domain = isl_set_copy(domains->option[atomic]);
+	domain = isl_set_intersect(domain, isl_set_copy(class_domain));
+	domain = isl_set_intersect(domain,
+				isl_set_copy(domains->schedule_domain));
+	empty = isl_set_is_empty(domain);
+	if (empty < 0)
+		class_domain = isl_set_free(class_domain);
+	if (empty) {
+		isl_set_free(domain);
+		return class_domain;
+	}
+
+	domain = isl_ast_build_eliminate(domains->build, domain);
+	domain = isl_set_coalesce(domain);
+	bset = isl_set_unshifted_simple_hull(domain);
+	domain = isl_set_from_basic_set(bset);
+	atomic_domain = isl_set_copy(domain);
+	domain = isl_set_intersect(domain, isl_set_copy(class_domain));
+	class_domain = isl_set_subtract(class_domain, atomic_domain);
+	domain = isl_set_make_disjoint(domain);
+	list = isl_basic_set_list_from_set(domain);
+	domains->list = isl_basic_set_list_concat(domains->list, list);
+
+	return class_domain;
+}
+
+/* Split up the schedule domain into uniform basic sets,
+ * in the sense that each element in a basic set is associated to
+ * elements of the same domains, and add the result to domains->list.
+ * Do this for that part of the schedule domain that lies in the
+ * intersection of "class_domain" and the separate option domain.
+ *
+ * "class_domain" may or may not include the constraints
+ * of the schedule domain, but this does not make a difference
+ * since we are going to intersect it with the domain of the inverse schedule.
+ * If it includes schedule domain constraints, then they may involve
+ * inner dimensions, but we will eliminate them in separation_domain.
+ */
+static int compute_separate_domain(struct isl_codegen_domains *domains,
+	__isl_keep isl_set *class_domain)
+{
+	isl_space *space;
+	isl_set *domain;
+	isl_union_map *executed;
+	isl_basic_set_list *list;
+	int empty;
+
+	domain = isl_set_copy(domains->option[separate]);
+	domain = isl_set_intersect(domain, isl_set_copy(class_domain));
+	executed = isl_union_map_copy(domains->executed);
+	executed = isl_union_map_intersect_domain(executed,
+				    isl_union_set_from_set(domain));
+	empty = isl_union_map_is_empty(executed);
+	if (empty < 0 || empty) {
+		isl_union_map_free(executed);
+		return empty < 0 ? -1 : 0;
+	}
+
+	space = isl_set_get_space(class_domain);
+	domain = separate_schedule_domains(space, executed, domains->build);
+
+	list = isl_basic_set_list_from_set(domain);
+	domains->list = isl_basic_set_list_concat(domains->list, list);
+
+	return 0;
+}
+
+/* Split up the domain at the current depth into disjoint
+ * basic sets for which code should be generated separately
+ * for the given separation class domain.
+ *
+ * If any separation classes have been defined, then "class_domain"
+ * is the domain of the current class and does not refer to inner dimensions.
+ * Otherwise, "class_domain" is the universe domain.
+ *
+ * We first make sure that the class domain is disjoint from
+ * previously considered class domains.
+ *
+ * The separate domains can be computed directly from the "class_domain".
+ *
+ * The unroll, atomic and remainder domains need the constraints
+ * from the schedule domain.
+ *
+ * For unrolling, the actual schedule domain is needed (with divs that
+ * may refer to the current dimension) so that stride detection can be
+ * performed.
+ *
+ * For atomic and remainder domains, inner dimensions and divs involving
+ * the current dimensions should be eliminated.
+ * In case we are working within a separation class, we need to intersect
+ * the result with the current "class_domain" to ensure that the domains
+ * are disjoint from those generated from other class domains.
+ *
+ * The domain that has been made atomic may be larger than specified
+ * by the user since it needs to be representable as a single basic set.
+ * This possibly larger domain is removed from class_domain by
+ * compute_atomic_domain.  It is computed first so that the extended domain
+ * would not overlap with any domains computed before.
+ * Similary, the unrolled domains may have some constraints removed and
+ * may therefore also be larger than specified by the user.
+ *
+ * If anything is left after handling separate, unroll and atomic,
+ * we split it up into basic sets and append the basic sets to domains->list.
+ */
+static int compute_partial_domains(struct isl_codegen_domains *domains,
+	__isl_take isl_set *class_domain)
+{
+	isl_basic_set_list *list;
+	isl_set *domain;
+
+	class_domain = isl_set_subtract(class_domain,
+					isl_set_copy(domains->done));
+	domains->done = isl_set_union(domains->done,
+					isl_set_copy(class_domain));
+
+	class_domain = compute_atomic_domain(domains, class_domain);
+	class_domain = compute_unroll_domains(domains, class_domain);
+
+	domain = isl_set_copy(class_domain);
+
+	if (compute_separate_domain(domains, domain) < 0)
+		goto error;
+	domain = isl_set_subtract(domain,
+				    isl_set_copy(domains->option[separate]));
+
+	domain = isl_set_intersect(domain,
+				isl_set_copy(domains->schedule_domain));
+
+	domain = isl_ast_build_eliminate(domains->build, domain);
+	domain = isl_set_intersect(domain, isl_set_copy(class_domain));
+
+	domain = isl_set_coalesce(domain);
+	domain = isl_set_make_disjoint(domain);
+
+	list = isl_basic_set_list_from_set(domain);
+	domains->list = isl_basic_set_list_concat(domains->list, list);
+
+	isl_set_free(class_domain);
+
+	return 0;
+error:
+	isl_set_free(domain);
+	isl_set_free(class_domain);
+	return -1;
+}
+
+/* Split up the domain at the current depth into disjoint
+ * basic sets for which code should be generated separately
+ * for the separation class identified by "pnt".
+ *
+ * We extract the corresponding class domain from domains->sep_class,
+ * eliminate inner dimensions and pass control to compute_partial_domains.
+ */
+static int compute_class_domains(__isl_take isl_point *pnt, void *user)
+{
+	struct isl_codegen_domains *domains = user;
+	isl_set *class_set;
+	isl_set *domain;
+	int disjoint;
+
+	class_set = isl_set_from_point(pnt);
+	domain = isl_map_domain(isl_map_intersect_range(
+				isl_map_copy(domains->sep_class), class_set));
+	domain = isl_ast_build_compute_gist(domains->build, domain);
+	domain = isl_ast_build_eliminate(domains->build, domain);
+
+	disjoint = isl_set_plain_is_disjoint(domain, domains->schedule_domain);
+	if (disjoint < 0)
+		return -1;
+	if (disjoint) {
+		isl_set_free(domain);
+		return 0;
+	}
+
+	return compute_partial_domains(domains, domain);
+}
+
+/* Extract the domains at the current depth that should be atomic,
+ * separated or unrolled and store them in option.
+ *
+ * The domains specified by the user might overlap, so we make
+ * them disjoint by subtracting earlier domains from later domains.
+ */
+static void compute_domains_init_options(isl_set *option[3],
+	__isl_keep isl_ast_build *build)
+{
+	enum isl_ast_build_domain_type type, type2;
+
+	for (type = atomic; type <= separate; ++type) {
+		option[type] = isl_ast_build_get_option_domain(build, type);
+		for (type2 = atomic; type2 < type; ++type2)
+			option[type] = isl_set_subtract(option[type],
+						isl_set_copy(option[type2]));
+	}
+
+	option[unroll] = isl_set_coalesce(option[unroll]);
+	option[unroll] = isl_set_make_disjoint(option[unroll]);
+}
+
+/* Split up the domain at the current depth into disjoint
+ * basic sets for which code should be generated separately,
+ * based on the user-specified options.
+ * Return the list of disjoint basic sets.
+ *
+ * There are three kinds of domains that we need to keep track of.
+ * - the "schedule domain" is the domain of "executed"
+ * - the "class domain" is the domain corresponding to the currrent
+ *	separation class
+ * - the "option domain" is the domain corresponding to one of the options
+ *	atomic, unroll or separate
+ *
+ * We first consider the individial values of the separation classes
+ * and split up the domain for each of them separately.
+ * Finally, we consider the remainder.  If no separation classes were
+ * specified, then we call compute_partial_domains with the universe
+ * "class_domain".  Otherwise, we take the "schedule_domain" as "class_domain",
+ * with inner dimensions removed.  We do this because we want to
+ * avoid computing the complement of the class domains (i.e., the difference
+ * between the universe and domains->done).
+ */
+static __isl_give isl_basic_set_list *compute_domains(
+	__isl_keep isl_union_map *executed, __isl_keep isl_ast_build *build)
+{
+	struct isl_codegen_domains domains;
+	isl_ctx *ctx;
+	isl_set *domain;
+	isl_union_set *schedule_domain;
+	isl_set *classes;
+	isl_space *space;
+	int n_param;
+	enum isl_ast_build_domain_type type;
+	int empty;
+
+	if (!executed)
+		return NULL;
+
+	ctx = isl_union_map_get_ctx(executed);
+	domains.list = isl_basic_set_list_alloc(ctx, 0);
+
+	schedule_domain = isl_union_map_domain(isl_union_map_copy(executed));
+	domain = isl_set_from_union_set(schedule_domain);
+
+	compute_domains_init_options(domains.option, build);
+
+	domains.sep_class = isl_ast_build_get_separation_class(build);
+	classes = isl_map_range(isl_map_copy(domains.sep_class));
+	n_param = isl_set_dim(classes, isl_dim_param);
+	classes = isl_set_project_out(classes, isl_dim_param, 0, n_param);
+
+	space = isl_set_get_space(domain);
+	domains.build = build;
+	domains.schedule_domain = isl_set_copy(domain);
+	domains.executed = executed;
+	domains.done = isl_set_empty(space);
+
+	if (isl_set_foreach_point(classes, &compute_class_domains, &domains) < 0)
+		domains.list = isl_basic_set_list_free(domains.list);
+	isl_set_free(classes);
+
+	empty = isl_set_is_empty(domains.done);
+	if (empty < 0) {
+		domains.list = isl_basic_set_list_free(domains.list);
+		domain = isl_set_free(domain);
+	} else if (empty) {
+		isl_set_free(domain);
+		domain = isl_set_universe(isl_set_get_space(domains.done));
+	} else {
+		domain = isl_ast_build_eliminate(build, domain);
+	}
+	if (compute_partial_domains(&domains, domain) < 0)
+		domains.list = isl_basic_set_list_free(domains.list);
+
+	isl_set_free(domains.schedule_domain);
+	isl_set_free(domains.done);
+	isl_map_free(domains.sep_class);
+	for (type = atomic; type <= separate; ++type)
+		isl_set_free(domains.option[type]);
+
+	return domains.list;
+}
+
+/* Generate code for a single component, after shifting (if any)
+ * has been applied.
+ *
+ * We first split up the domain at the current depth into disjoint
+ * basic sets based on the user-specified options.
+ * Then we generated code for each of them and concatenate the results.
+ */
+static __isl_give isl_ast_graft_list *generate_shifted_component(
+	__isl_take isl_union_map *executed, __isl_take isl_ast_build *build)
+{
+	isl_basic_set_list *domain_list;
+	isl_ast_graft_list *list = NULL;
+
+	domain_list = compute_domains(executed, build);
+	list = generate_parallel_domains(domain_list, executed, build);
+
+	isl_basic_set_list_free(domain_list);
+	isl_union_map_free(executed);
+	isl_ast_build_free(build);
+
+	return list;
+}
+
+struct isl_set_map_pair {
+	isl_set *set;
+	isl_map *map;
+};
+
+/* Given an array "domain" of isl_set_map_pairs and an array "order"
+ * of indices into the "domain" array,
+ * return the union of the "map" fields of the elements
+ * indexed by the first "n" elements of "order".
+ */
+static __isl_give isl_union_map *construct_component_executed(
+	struct isl_set_map_pair *domain, int *order, int n)
+{
+	int i;
+	isl_map *map;
+	isl_union_map *executed;
+
+	map = isl_map_copy(domain[order[0]].map);
+	executed = isl_union_map_from_map(map);
+	for (i = 1; i < n; ++i) {
+		map = isl_map_copy(domain[order[i]].map);
+		executed = isl_union_map_add_map(executed, map);
+	}
+
+	return executed;
+}
+
+/* Generate code for a single component, after shifting (if any)
+ * has been applied.
+ *
+ * The component inverse schedule is specified as the "map" fields
+ * of the elements of "domain" indexed by the first "n" elements of "order".
+ */
+static __isl_give isl_ast_graft_list *generate_shifted_component_from_list(
+	struct isl_set_map_pair *domain, int *order, int n,
+	__isl_take isl_ast_build *build)
+{
+	isl_union_map *executed;
+
+	executed = construct_component_executed(domain, order, n);
+	return generate_shifted_component(executed, build);
+}
+
+/* Does set dimension "pos" of "set" have an obviously fixed value?
+ */
+static int dim_is_fixed(__isl_keep isl_set *set, int pos)
+{
+	int fixed;
+	isl_val *v;
+
+	v = isl_set_plain_get_val_if_fixed(set, isl_dim_set, pos);
+	if (!v)
+		return -1;
+	fixed = !isl_val_is_nan(v);
+	isl_val_free(v);
+
+	return fixed;
+}
+
+/* Given an array "domain" of isl_set_map_pairs and an array "order"
+ * of indices into the "domain" array,
+ * do all (except for at most one) of the "set" field of the elements
+ * indexed by the first "n" elements of "order" have a fixed value
+ * at position "depth"?
+ */
+static int at_most_one_non_fixed(struct isl_set_map_pair *domain,
+	int *order, int n, int depth)
+{
+	int i;
+	int non_fixed = -1;
+
+	for (i = 0; i < n; ++i) {
+		int f;
+
+		f = dim_is_fixed(domain[order[i]].set, depth);
+		if (f < 0)
+			return -1;
+		if (f)
+			continue;
+		if (non_fixed >= 0)
+			return 0;
+		non_fixed = i;
+	}
+
+	return 1;
+}
+
+/* Given an array "domain" of isl_set_map_pairs and an array "order"
+ * of indices into the "domain" array,
+ * eliminate the inner dimensions from the "set" field of the elements
+ * indexed by the first "n" elements of "order", provided the current
+ * dimension does not have a fixed value.
+ *
+ * Return the index of the first element in "order" with a corresponding
+ * "set" field that does not have an (obviously) fixed value.
+ */
+static int eliminate_non_fixed(struct isl_set_map_pair *domain,
+	int *order, int n, int depth, __isl_keep isl_ast_build *build)
+{
+	int i;
+	int base = -1;
+
+	for (i = n - 1; i >= 0; --i) {
+		int f;
+		f = dim_is_fixed(domain[order[i]].set, depth);
+		if (f < 0)
+			return -1;
+		if (f)
+			continue;
+		domain[order[i]].set = isl_ast_build_eliminate_inner(build,
+							domain[order[i]].set);
+		base = i;
+	}
+
+	return base;
+}
+
+/* Given an array "domain" of isl_set_map_pairs and an array "order"
+ * of indices into the "domain" array,
+ * find the element of "domain" (amongst those indexed by the first "n"
+ * elements of "order") with the "set" field that has the smallest
+ * value for the current iterator.
+ *
+ * Note that the domain with the smallest value may depend on the parameters
+ * and/or outer loop dimension.  Since the result of this function is only
+ * used as heuristic, we only make a reasonable attempt at finding the best
+ * domain, one that should work in case a single domain provides the smallest
+ * value for the current dimension over all values of the parameters
+ * and outer dimensions.
+ *
+ * In particular, we compute the smallest value of the first domain
+ * and replace it by that of any later domain if that later domain
+ * has a smallest value that is smaller for at least some value
+ * of the parameters and outer dimensions.
+ */
+static int first_offset(struct isl_set_map_pair *domain, int *order, int n,
+	__isl_keep isl_ast_build *build)
+{
+	int i;
+	isl_map *min_first;
+	int first = 0;
+
+	min_first = isl_ast_build_map_to_iterator(build,
+					isl_set_copy(domain[order[0]].set));
+	min_first = isl_map_lexmin(min_first);
+
+	for (i = 1; i < n; ++i) {
+		isl_map *min, *test;
+		int empty;
+
+		min = isl_ast_build_map_to_iterator(build,
+					isl_set_copy(domain[order[i]].set));
+		min = isl_map_lexmin(min);
+		test = isl_map_copy(min);
+		test = isl_map_apply_domain(isl_map_copy(min_first), test);
+		test = isl_map_order_lt(test, isl_dim_in, 0, isl_dim_out, 0);
+		empty = isl_map_is_empty(test);
+		isl_map_free(test);
+		if (empty >= 0 && !empty) {
+			isl_map_free(min_first);
+			first = i;
+			min_first = min;
+		} else
+			isl_map_free(min);
+
+		if (empty < 0)
+			break;
+	}
+
+	isl_map_free(min_first);
+
+	return i < n ? -1 : first;
+}
+
+/* Construct a shifted inverse schedule based on the original inverse schedule,
+ * the stride and the offset.
+ *
+ * The original inverse schedule is specified as the "map" fields
+ * of the elements of "domain" indexed by the first "n" elements of "order".
+ *
+ * "stride" and "offset" are such that the difference
+ * between the values of the current dimension of domain "i"
+ * and the values of the current dimension for some reference domain are
+ * equal to
+ *
+ *	stride * integer + offset[i]
+ *
+ * Moreover, 0 <= offset[i] < stride.
+ *
+ * For each domain, we create a map
+ *
+ *	{ [..., j, ...] -> [..., j - offset[i], offset[i], ....] }
+ *
+ * where j refers to the current dimension and the other dimensions are
+ * unchanged, and apply this map to the original schedule domain.
+ *
+ * For example, for the original schedule
+ *
+ *	{ A[i] -> [2i]: 0 <= i < 10; B[i] -> [2i+1] : 0 <= i < 10 }
+ *
+ * and assuming the offset is 0 for the A domain and 1 for the B domain,
+ * we apply the mapping
+ *
+ *	{ [j] -> [j, 0] }
+ *
+ * to the schedule of the "A" domain and the mapping
+ *
+ *	{ [j - 1] -> [j, 1] }
+ *
+ * to the schedule of the "B" domain.
+ *
+ *
+ * Note that after the transformation, the differences between pairs
+ * of values of the current dimension over all domains are multiples
+ * of stride and that we have therefore exposed the stride.
+ *
+ *
+ * To see that the mapping preserves the lexicographic order,
+ * first note that each of the individual maps above preserves the order.
+ * If the value of the current iterator is j1 in one domain and j2 in another,
+ * then if j1 = j2, we know that the same map is applied to both domains
+ * and the order is preserved.
+ * Otherwise, let us assume, without loss of generality, that j1 < j2.
+ * If c1 >= c2 (with c1 and c2 the corresponding offsets), then
+ *
+ *	j1 - c1 < j2 - c2
+ *
+ * and the order is preserved.
+ * If c1 < c2, then we know
+ *
+ *	0 <= c2 - c1 < s
+ *
+ * We also have
+ *
+ *	j2 - j1 = n * s + r
+ *
+ * with n >= 0 and 0 <= r < s.
+ * In other words, r = c2 - c1.
+ * If n > 0, then
+ *
+ *	j1 - c1 < j2 - c2
+ *
+ * If n = 0, then
+ *
+ *	j1 - c1 = j2 - c2
+ *
+ * and so
+ *
+ *	(j1 - c1, c1) << (j2 - c2, c2)
+ *
+ * with "<<" the lexicographic order, proving that the order is preserved
+ * in all cases.
+ */
+static __isl_give isl_union_map *contruct_shifted_executed(
+	struct isl_set_map_pair *domain, int *order, int n,
+	__isl_keep isl_val *stride, __isl_keep isl_multi_val *offset,
+	__isl_take isl_ast_build *build)
+{
+	int i;
+	isl_union_map *executed;
+	isl_space *space;
+	isl_map *map;
+	int depth;
+	isl_constraint *c;
+
+	depth = isl_ast_build_get_depth(build);
+	space = isl_ast_build_get_space(build, 1);
+	executed = isl_union_map_empty(isl_space_copy(space));
+	space = isl_space_map_from_set(space);
+	map = isl_map_identity(isl_space_copy(space));
+	map = isl_map_eliminate(map, isl_dim_out, depth, 1);
+	map = isl_map_insert_dims(map, isl_dim_out, depth + 1, 1);
+	space = isl_space_insert_dims(space, isl_dim_out, depth + 1, 1);
+
+	c = isl_equality_alloc(isl_local_space_from_space(space));
+	c = isl_constraint_set_coefficient_si(c, isl_dim_in, depth, 1);
+	c = isl_constraint_set_coefficient_si(c, isl_dim_out, depth, -1);
+
+	for (i = 0; i < n; ++i) {
+		isl_map *map_i;
+		isl_val *v;
+
+		v = isl_multi_val_get_val(offset, i);
+		if (!v)
+			break;
+		map_i = isl_map_copy(map);
+		map_i = isl_map_fix_val(map_i, isl_dim_out, depth + 1,
+					isl_val_copy(v));
+		v = isl_val_neg(v);
+		c = isl_constraint_set_constant_val(c, v);
+		map_i = isl_map_add_constraint(map_i, isl_constraint_copy(c));
+
+		map_i = isl_map_apply_domain(isl_map_copy(domain[order[i]].map),
+						map_i);
+		executed = isl_union_map_add_map(executed, map_i);
+	}
+
+	isl_constraint_free(c);
+	isl_map_free(map);
+
+	if (i < n)
+		executed = isl_union_map_free(executed);
+
+	return executed;
+}
+
+/* Generate code for a single component, after exposing the stride,
+ * given that the schedule domain is "shifted strided".
+ *
+ * The component inverse schedule is specified as the "map" fields
+ * of the elements of "domain" indexed by the first "n" elements of "order".
+ *
+ * The schedule domain being "shifted strided" means that the differences
+ * between the values of the current dimension of domain "i"
+ * and the values of the current dimension for some reference domain are
+ * equal to
+ *
+ *	stride * integer + offset[i]
+ *
+ * We first look for the domain with the "smallest" value for the current
+ * dimension and adjust the offsets such that the offset of the "smallest"
+ * domain is equal to zero.  The other offsets are reduced modulo stride.
+ *
+ * Based on this information, we construct a new inverse schedule in
+ * contruct_shifted_executed that exposes the stride.
+ * Since this involves the introduction of a new schedule dimension,
+ * the build needs to be changed accodingly.
+ * After computing the AST, the newly introduced dimension needs
+ * to be removed again from the list of grafts.  We do this by plugging
+ * in a mapping that represents the new schedule domain in terms of the
+ * old schedule domain.
+ */
+static __isl_give isl_ast_graft_list *generate_shift_component(
+	struct isl_set_map_pair *domain, int *order, int n,
+	__isl_keep isl_val *stride, __isl_keep isl_multi_val *offset,
+	__isl_take isl_ast_build *build)
+{
+	isl_ast_graft_list *list;
+	int first;
+	int depth;
+	isl_ctx *ctx;
+	isl_val *val;
+	isl_multi_val *mv;
+	isl_space *space;
+	isl_multi_aff *ma, *zero;
+	isl_union_map *executed;
+
+	ctx = isl_ast_build_get_ctx(build);
+	depth = isl_ast_build_get_depth(build);
+
+	first = first_offset(domain, order, n, build);
+	if (first < 0)
+		goto error;
+
+	mv = isl_multi_val_copy(offset);
+	val = isl_multi_val_get_val(offset, first);
+	val = isl_val_neg(val);
+	mv = isl_multi_val_add_val(mv, val);
+	mv = isl_multi_val_mod_val(mv, isl_val_copy(stride));
+
+	executed = contruct_shifted_executed(domain, order, n, stride, mv,
+						build);
+	space = isl_ast_build_get_space(build, 1);
+	space = isl_space_map_from_set(space);
+	ma = isl_multi_aff_identity(isl_space_copy(space));
+	space = isl_space_from_domain(isl_space_domain(space));
+	space = isl_space_add_dims(space, isl_dim_out, 1);
+	zero = isl_multi_aff_zero(space);
+	ma = isl_multi_aff_range_splice(ma, depth + 1, zero);
+	build = isl_ast_build_insert_dim(build, depth + 1);
+	list = generate_shifted_component(executed, build);
+
+	list = isl_ast_graft_list_preimage_multi_aff(list, ma);
+
+	isl_multi_val_free(mv);
+
+	return list;
+error:
+	isl_ast_build_free(build);
+	return NULL;
+}
+
+/* Generate code for a single component.
+ *
+ * The component inverse schedule is specified as the "map" fields
+ * of the elements of "domain" indexed by the first "n" elements of "order".
+ *
+ * This function may modify the "set" fields of "domain".
+ *
+ * Before proceeding with the actual code generation for the component,
+ * we first check if there are any "shifted" strides, meaning that
+ * the schedule domains of the individual domains are all strided,
+ * but that they have different offsets, resulting in the union
+ * of schedule domains not being strided anymore.
+ *
+ * The simplest example is the schedule
+ *
+ *	{ A[i] -> [2i]: 0 <= i < 10; B[i] -> [2i+1] : 0 <= i < 10 }
+ *
+ * Both schedule domains are strided, but their union is not.
+ * This function detects such cases and then rewrites the schedule to
+ *
+ *	{ A[i] -> [2i, 0]: 0 <= i < 10; B[i] -> [2i, 1] : 0 <= i < 10 }
+ *
+ * In the new schedule, the schedule domains have the same offset (modulo
+ * the stride), ensuring that the union of schedule domains is also strided.
+ *
+ *
+ * If there is only a single domain in the component, then there is
+ * nothing to do.   Similarly, if the current schedule dimension has
+ * a fixed value for almost all domains then there is nothing to be done.
+ * In particular, we need at least two domains where the current schedule
+ * dimension does not have a fixed value.
+ * Finally, if any of the options refer to the current schedule dimension,
+ * then we bail out as well.  It would be possible to reformulate the options
+ * in terms of the new schedule domain, but that would introduce constraints
+ * that separate the domains in the options and that is something we would
+ * like to avoid.
+ *
+ *
+ * To see if there is any shifted stride, we look at the differences
+ * between the values of the current dimension in pairs of domains
+ * for equal values of outer dimensions.  These differences should be
+ * of the form
+ *
+ *	m x + r
+ *
+ * with "m" the stride and "r" a constant.  Note that we cannot perform
+ * this analysis on individual domains as the lower bound in each domain
+ * may depend on parameters or outer dimensions and so the current dimension
+ * itself may not have a fixed remainder on division by the stride.
+ *
+ * In particular, we compare the first domain that does not have an
+ * obviously fixed value for the current dimension to itself and all
+ * other domains and collect the offsets and the gcd of the strides.
+ * If the gcd becomes one, then we failed to find shifted strides.
+ * If the gcd is zero, then the differences were all fixed, meaning
+ * that some domains had non-obviously fixed values for the current dimension.
+ * If all the offsets are the same (for those domains that do not have
+ * an obviously fixed value for the current dimension), then we do not
+ * apply the transformation.
+ * If none of the domains were skipped, then there is nothing to do.
+ * If some of them were skipped, then if we apply separation, the schedule
+ * domain should get split in pieces with a (non-shifted) stride.
+ *
+ * Otherwise, we apply a shift to expose the stride in
+ * generate_shift_component.
+ */
+static __isl_give isl_ast_graft_list *generate_component(
+	struct isl_set_map_pair *domain, int *order, int n,
+	__isl_take isl_ast_build *build)
+{
+	int i, d;
+	int depth;
+	isl_ctx *ctx;
+	isl_map *map;
+	isl_set *deltas;
+	isl_val *gcd = NULL;
+	isl_multi_val *mv;
+	int fixed, skip;
+	int base;
+	isl_ast_graft_list *list;
+	int res = 0;
+
+	depth = isl_ast_build_get_depth(build);
+
+	skip = n == 1;
+	if (skip >= 0 && !skip)
+		skip = at_most_one_non_fixed(domain, order, n, depth);
+	if (skip >= 0 && !skip)
+		skip = isl_ast_build_options_involve_depth(build);
+	if (skip < 0)
+		goto error;
+	if (skip)
+		return generate_shifted_component_from_list(domain,
+							    order, n, build);
+
+	base = eliminate_non_fixed(domain, order, n, depth, build);
+	if (base < 0)
+		goto error;
+
+	ctx = isl_ast_build_get_ctx(build);
+
+	mv = isl_multi_val_zero(isl_space_set_alloc(ctx, 0, n));
+
+	fixed = 1;
+	for (i = 0; i < n; ++i) {
+		isl_val *r, *m;
+
+		map = isl_map_from_domain_and_range(
+					isl_set_copy(domain[order[base]].set),
+					isl_set_copy(domain[order[i]].set));
+		for (d = 0; d < depth; ++d)
+			map = isl_map_equate(map, isl_dim_in, d,
+						    isl_dim_out, d);
+		deltas = isl_map_deltas(map);
+		res = isl_set_dim_residue_class_val(deltas, depth, &m, &r);
+		isl_set_free(deltas);
+		if (res < 0)
+			break;
+
+		if (i == 0)
+			gcd = m;
+		else
+			gcd = isl_val_gcd(gcd, m);
+		if (isl_val_is_one(gcd)) {
+			isl_val_free(r);
+			break;
+		}
+		mv = isl_multi_val_set_val(mv, i, r);
+
+		res = dim_is_fixed(domain[order[i]].set, depth);
+		if (res < 0)
+			break;
+		if (res)
+			continue;
+
+		if (fixed && i > base) {
+			isl_val *a, *b;
+			a = isl_multi_val_get_val(mv, i);
+			b = isl_multi_val_get_val(mv, base);
+			if (isl_val_ne(a, b))
+				fixed = 0;
+			isl_val_free(a);
+			isl_val_free(b);
+		}
+	}
+
+	if (res < 0 || !gcd) {
+		isl_ast_build_free(build);
+		list = NULL;
+	} else if (i < n || fixed || isl_val_is_zero(gcd)) {
+		list = generate_shifted_component_from_list(domain,
+							    order, n, build);
+	} else {
+		list = generate_shift_component(domain, order, n, gcd, mv,
+						build);
+	}
+
+	isl_val_free(gcd);
+	isl_multi_val_free(mv);
+
+	return list;
+error:
+	isl_ast_build_free(build);
+	return NULL;
+}
+
+/* Store both "map" itself and its domain in the
+ * structure pointed to by *next and advance to the next array element.
+ */
+static int extract_domain(__isl_take isl_map *map, void *user)
+{
+	struct isl_set_map_pair **next = user;
+
+	(*next)->map = isl_map_copy(map);
+	(*next)->set = isl_map_domain(map);
+	(*next)++;
+
+	return 0;
+}
+
+/* Internal data for any_scheduled_after.
+ *
+ * "depth" is the number of loops that have already been generated
+ * "group_coscheduled" is a local copy of options->ast_build_group_coscheduled
+ * "domain" is an array of set-map pairs corresponding to the different
+ * iteration domains.  The set is the schedule domain, i.e., the domain
+ * of the inverse schedule, while the map is the inverse schedule itself.
+ */
+struct isl_any_scheduled_after_data {
+	int depth;
+	int group_coscheduled;
+	struct isl_set_map_pair *domain;
+};
+
+/* Is any element of domain "i" scheduled after any element of domain "j"
+ * (for a common iteration of the first data->depth loops)?
+ *
+ * data->domain[i].set contains the domain of the inverse schedule
+ * for domain "i", i.e., elements in the schedule domain.
+ *
+ * If data->group_coscheduled is set, then we also return 1 if there
+ * is any pair of elements in the two domains that are scheduled together.
+ */
+static int any_scheduled_after(int i, int j, void *user)
+{
+	struct isl_any_scheduled_after_data *data = user;
+	int dim = isl_set_dim(data->domain[i].set, isl_dim_set);
+	int pos;
+
+	for (pos = data->depth; pos < dim; ++pos) {
+		int follows;
+
+		follows = isl_set_follows_at(data->domain[i].set,
+						data->domain[j].set, pos);
+
+		if (follows < -1)
+			return -1;
+		if (follows > 0)
+			return 1;
+		if (follows < 0)
+			return 0;
+	}
+
+	return data->group_coscheduled;
+}
+
+/* Look for independent components at the current depth and generate code
+ * for each component separately.  The resulting lists of grafts are
+ * merged in an attempt to combine grafts with identical guards.
+ *
+ * Code for two domains can be generated separately if all the elements
+ * of one domain are scheduled before (or together with) all the elements
+ * of the other domain.  We therefore consider the graph with as nodes
+ * the domains and an edge between two nodes if any element of the first
+ * node is scheduled after any element of the second node.
+ * If the ast_build_group_coscheduled is set, then we also add an edge if
+ * there is any pair of elements in the two domains that are scheduled
+ * together.
+ * Code is then generated (by generate_component)
+ * for each of the strongly connected components in this graph
+ * in their topological order.
+ *
+ * Since the test is performed on the domain of the inverse schedules of
+ * the different domains, we precompute these domains and store
+ * them in data.domain.
+ */
+static __isl_give isl_ast_graft_list *generate_components(
+	__isl_take isl_union_map *executed, __isl_take isl_ast_build *build)
+{
+	int i;
+	isl_ctx *ctx = isl_ast_build_get_ctx(build);
+	int n = isl_union_map_n_map(executed);
+	struct isl_any_scheduled_after_data data;
+	struct isl_set_map_pair *next;
+	struct isl_tarjan_graph *g = NULL;
+	isl_ast_graft_list *list = NULL;
+	int n_domain = 0;
+
+	data.domain = isl_calloc_array(ctx, struct isl_set_map_pair, n);
+	if (!data.domain)
+		goto error;
+	n_domain = n;
+
+	next = data.domain;
+	if (isl_union_map_foreach_map(executed, &extract_domain, &next) < 0)
+		goto error;
+
+	if (!build)
+		goto error;
+	data.depth = isl_ast_build_get_depth(build);
+	data.group_coscheduled = isl_options_get_ast_build_group_coscheduled(ctx);
+	g = isl_tarjan_graph_init(ctx, n, &any_scheduled_after, &data);
+	if (!g)
+		goto error;
+
+	list = isl_ast_graft_list_alloc(ctx, 0);
+
+	i = 0;
+	while (list && n) {
+		isl_ast_graft_list *list_c;
+		int first = i;
+
+		if (g->order[i] == -1)
+			isl_die(ctx, isl_error_internal, "cannot happen",
+				goto error);
+		++i; --n;
+		while (g->order[i] != -1) {
+			++i; --n;
+		}
+
+		list_c = generate_component(data.domain,
+					    g->order + first, i - first,
+					    isl_ast_build_copy(build));
+		list = isl_ast_graft_list_merge(list, list_c, build);
+
+		++i;
+	}
+
+	if (0)
+error:		list = isl_ast_graft_list_free(list);
+	isl_tarjan_graph_free(g);
+	for (i = 0; i < n_domain; ++i) {
+		isl_map_free(data.domain[i].map);
+		isl_set_free(data.domain[i].set);
+	}
+	free(data.domain);
+	isl_union_map_free(executed);
+	isl_ast_build_free(build);
+
+	return list;
+}
+
+/* Generate code for the next level (and all inner levels).
+ *
+ * If "executed" is empty, i.e., no code needs to be generated,
+ * then we return an empty list.
+ *
+ * If we have already generated code for all loop levels, then we pass
+ * control to generate_inner_level.
+ *
+ * If "executed" lives in a single space, i.e., if code needs to be
+ * generated for a single domain, then there can only be a single
+ * component and we go directly to generate_shifted_component.
+ * Otherwise, we call generate_components to detect the components
+ * and to call generate_component on each of them separately.
+ */
+static __isl_give isl_ast_graft_list *generate_next_level(
+	__isl_take isl_union_map *executed, __isl_take isl_ast_build *build)
+{
+	int depth;
+
+	if (!build || !executed)
+		goto error;
+
+	if (isl_union_map_is_empty(executed)) {
+		isl_ctx *ctx = isl_ast_build_get_ctx(build);
+		isl_union_map_free(executed);
+		isl_ast_build_free(build);
+		return isl_ast_graft_list_alloc(ctx, 0);
+	}
+
+	depth = isl_ast_build_get_depth(build);
+	if (depth >= isl_ast_build_dim(build, isl_dim_set))
+		return generate_inner_level(executed, build);
+
+	if (isl_union_map_n_map(executed) == 1)
+		return generate_shifted_component(executed, build);
+
+	return generate_components(executed, build);
+error:
+	isl_union_map_free(executed);
+	isl_ast_build_free(build);
+	return NULL;
+}
+
+/* Internal data structure used by isl_ast_build_ast_from_schedule.
+ * internal, executed and build are the inputs to generate_code.
+ * list collects the output.
+ */
+struct isl_generate_code_data {
+	int internal;
+	isl_union_map *executed;
+	isl_ast_build *build;
+
+	isl_ast_graft_list *list;
+};
+
+/* Given an inverse schedule in terms of the external build schedule, i.e.,
+ *
+ *	[E -> S] -> D
+ *
+ * with E the external build schedule and S the additional schedule "space",
+ * reformulate the inverse schedule in terms of the internal schedule domain,
+ * i.e., return
+ *
+ *	[I -> S] -> D
+ *
+ * We first obtain a mapping
+ *
+ *	I -> E
+ *
+ * take the inverse and the product with S -> S, resulting in
+ *
+ *	[I -> S] -> [E -> S]
+ *
+ * Applying the map to the input produces the desired result.
+ */
+static __isl_give isl_union_map *internal_executed(
+	__isl_take isl_union_map *executed, __isl_keep isl_space *space,
+	__isl_keep isl_ast_build *build)
+{
+	isl_map *id, *proj;
+
+	proj = isl_ast_build_get_schedule_map(build);
+	proj = isl_map_reverse(proj);
+	space = isl_space_map_from_set(isl_space_copy(space));
+	id = isl_map_identity(space);
+	proj = isl_map_product(proj, id);
+	executed = isl_union_map_apply_domain(executed,
+						isl_union_map_from_map(proj));
+	return executed;
+}
+
+/* Generate an AST that visits the elements in the range of data->executed
+ * in the relative order specified by the corresponding domain element(s)
+ * for those domain elements that belong to "set".
+ * Add the result to data->list.
+ *
+ * The caller ensures that "set" is a universe domain.
+ * "space" is the space of the additional part of the schedule.
+ * It is equal to the space of "set" if build->domain is parametric.
+ * Otherwise, it is equal to the range of the wrapped space of "set".
+ *
+ * If the build space is not parametric and if isl_ast_build_ast_from_schedule
+ * was called from an outside user (data->internal not set), then
+ * the (inverse) schedule refers to the external build domain and needs to
+ * be transformed to refer to the internal build domain.
+ *
+ * If the build space is parametric, then we add some of the parameter
+ * constraints to the executed relation.  Adding these constraints
+ * allows for an earlier detection of conflicts in some cases.
+ * However, we do not want to divide the executed relation into
+ * more disjuncts than necessary.  We therefore approximate
+ * the constraints on the parameters by a single disjunct set.
+ *
+ * The build is extended to include the additional part of the schedule.
+ * If the original build space was not parametric, then the options
+ * in data->build refer only to the additional part of the schedule
+ * and they need to be adjusted to refer to the complete AST build
+ * domain.
+ *
+ * After having adjusted inverse schedule and build, we start generating
+ * code with the outer loop of the current code generation
+ * in generate_next_level.
+ *
+ * If the original build space was not parametric, we undo the embedding
+ * on the resulting isl_ast_node_list so that it can be used within
+ * the outer AST build.
+ */
+static int generate_code_in_space(struct isl_generate_code_data *data,
+	__isl_take isl_set *set, __isl_take isl_space *space)
+{
+	isl_union_map *executed;
+	isl_ast_build *build;
+	isl_ast_graft_list *list;
+	int embed;
+
+	executed = isl_union_map_copy(data->executed);
+	executed = isl_union_map_intersect_domain(executed,
+						 isl_union_set_from_set(set));
+
+	embed = !isl_set_is_params(data->build->domain);
+	if (embed && !data->internal)
+		executed = internal_executed(executed, space, data->build);
+	if (!embed) {
+		isl_set *domain;
+		domain = isl_ast_build_get_domain(data->build);
+		domain = isl_set_from_basic_set(isl_set_simple_hull(domain));
+		executed = isl_union_map_intersect_params(executed, domain);
+	}
+
+	build = isl_ast_build_copy(data->build);
+	build = isl_ast_build_product(build, space);
+
+	list = generate_next_level(executed, build);
+
+	list = isl_ast_graft_list_unembed(list, embed);
+
+	data->list = isl_ast_graft_list_concat(data->list, list);
+
+	return 0;
+}
+
+/* Generate an AST that visits the elements in the range of data->executed
+ * in the relative order specified by the corresponding domain element(s)
+ * for those domain elements that belong to "set".
+ * Add the result to data->list.
+ *
+ * The caller ensures that "set" is a universe domain.
+ *
+ * If the build space S is not parametric, then the space of "set"
+ * need to be a wrapped relation with S as domain.  That is, it needs
+ * to be of the form
+ *
+ *	[S -> T]
+ *
+ * Check this property and pass control to generate_code_in_space
+ * passing along T.
+ * If the build space is not parametric, then T is the space of "set".
+ */
+static int generate_code_set(__isl_take isl_set *set, void *user)
+{
+	struct isl_generate_code_data *data = user;
+	isl_space *space, *build_space;
+	int is_domain;
+
+	space = isl_set_get_space(set);
+
+	if (isl_set_is_params(data->build->domain))
+		return generate_code_in_space(data, set, space);
+
+	build_space = isl_ast_build_get_space(data->build, data->internal);
+	space = isl_space_unwrap(space);
+	is_domain = isl_space_is_domain(build_space, space);
+	isl_space_free(build_space);
+	space = isl_space_range(space);
+
+	if (is_domain < 0)
+		goto error;
+	if (!is_domain)
+		isl_die(isl_set_get_ctx(set), isl_error_invalid,
+			"invalid nested schedule space", goto error);
+
+	return generate_code_in_space(data, set, space);
+error:
+	isl_set_free(set);
+	isl_space_free(space);
+	return -1;
+}
+
+/* Generate an AST that visits the elements in the range of "executed"
+ * in the relative order specified by the corresponding domain element(s).
+ *
+ * "build" is an isl_ast_build that has either been constructed by
+ * isl_ast_build_from_context or passed to a callback set by
+ * isl_ast_build_set_create_leaf.
+ * In the first case, the space of the isl_ast_build is typically
+ * a parametric space, although this is currently not enforced.
+ * In the second case, the space is never a parametric space.
+ * If the space S is not parametric, then the domain space(s) of "executed"
+ * need to be wrapped relations with S as domain.
+ *
+ * If the domain of "executed" consists of several spaces, then an AST
+ * is generated for each of them (in arbitrary order) and the results
+ * are concatenated.
+ *
+ * If "internal" is set, then the domain "S" above refers to the internal
+ * schedule domain representation.  Otherwise, it refers to the external
+ * representation, as returned by isl_ast_build_get_schedule_space.
+ *
+ * We essentially run over all the spaces in the domain of "executed"
+ * and call generate_code_set on each of them.
+ */
+static __isl_give isl_ast_graft_list *generate_code(
+	__isl_take isl_union_map *executed, __isl_take isl_ast_build *build,
+	int internal)
+{
+	isl_ctx *ctx;
+	struct isl_generate_code_data data = { 0 };
+	isl_space *space;
+	isl_union_set *schedule_domain;
+	isl_union_map *universe;
+
+	if (!build)
+		goto error;
+	space = isl_ast_build_get_space(build, 1);
+	space = isl_space_align_params(space,
+				    isl_union_map_get_space(executed));
+	space = isl_space_align_params(space,
+				    isl_union_map_get_space(build->options));
+	build = isl_ast_build_align_params(build, isl_space_copy(space));
+	executed = isl_union_map_align_params(executed, space);
+	if (!executed || !build)
+		goto error;
+
+	ctx = isl_ast_build_get_ctx(build);
+
+	data.internal = internal;
+	data.executed = executed;
+	data.build = build;
+	data.list = isl_ast_graft_list_alloc(ctx, 0);
+
+	universe = isl_union_map_universe(isl_union_map_copy(executed));
+	schedule_domain = isl_union_map_domain(universe);
+	if (isl_union_set_foreach_set(schedule_domain, &generate_code_set,
+					&data) < 0)
+		data.list = isl_ast_graft_list_free(data.list);
+
+	isl_union_set_free(schedule_domain);
+	isl_union_map_free(executed);
+
+	isl_ast_build_free(build);
+	return data.list;
+error:
+	isl_union_map_free(executed);
+	isl_ast_build_free(build);
+	return NULL;
+}
+
+/* Generate an AST that visits the elements in the domain of "schedule"
+ * in the relative order specified by the corresponding image element(s).
+ *
+ * "build" is an isl_ast_build that has either been constructed by
+ * isl_ast_build_from_context or passed to a callback set by
+ * isl_ast_build_set_create_leaf.
+ * In the first case, the space of the isl_ast_build is typically
+ * a parametric space, although this is currently not enforced.
+ * In the second case, the space is never a parametric space.
+ * If the space S is not parametric, then the range space(s) of "schedule"
+ * need to be wrapped relations with S as domain.
+ *
+ * If the range of "schedule" consists of several spaces, then an AST
+ * is generated for each of them (in arbitrary order) and the results
+ * are concatenated.
+ *
+ * We first initialize the local copies of the relevant options.
+ * We do this here rather than when the isl_ast_build is created
+ * because the options may have changed between the construction
+ * of the isl_ast_build and the call to isl_generate_code.
+ *
+ * The main computation is performed on an inverse schedule (with
+ * the schedule domain in the domain and the elements to be executed
+ * in the range) called "executed".
+ */
+__isl_give isl_ast_node *isl_ast_build_ast_from_schedule(
+	__isl_keep isl_ast_build *build, __isl_take isl_union_map *schedule)
+{
+	isl_ast_graft_list *list;
+	isl_ast_node *node;
+	isl_union_map *executed;
+
+	build = isl_ast_build_copy(build);
+	build = isl_ast_build_set_single_valued(build, 0);
+	schedule = isl_union_map_coalesce(schedule);
+	executed = isl_union_map_reverse(schedule);
+	list = generate_code(executed, isl_ast_build_copy(build), 0);
+	node = isl_ast_node_from_graft_list(list, build);
+	isl_ast_build_free(build);
+
+	return node;
+}
diff --git a/polly/lib/External/isl/isl_ast_graft.c b/polly/lib/External/isl/isl_ast_graft.c
new file mode 100644
index 00000000000..611ef5dcf6e
--- /dev/null
+++ b/polly/lib/External/isl/isl_ast_graft.c
@@ -0,0 +1,1242 @@
+/*
+ * Copyright 2012      Ecole Normale Superieure
+ * Copyright 2014      INRIA Rocquencourt
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege,
+ * Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
+ * B.P. 105 - 78153 Le Chesnay, France
+ */
+
+#include <isl_ast_private.h>
+#include <isl_ast_build_expr.h>
+#include <isl_ast_build_private.h>
+#include <isl_ast_graft_private.h>
+
+static __isl_give isl_ast_graft *isl_ast_graft_copy(
+	__isl_keep isl_ast_graft *graft);
+
+#undef BASE
+#define BASE ast_graft
+
+#include <isl_list_templ.c>
+
+#undef BASE
+#define BASE ast_graft
+#include <print_templ.c>
+
+isl_ctx *isl_ast_graft_get_ctx(__isl_keep isl_ast_graft *graft)
+{
+	if (!graft)
+		return NULL;
+	return isl_basic_set_get_ctx(graft->enforced);
+}
+
+__isl_give isl_ast_node *isl_ast_graft_get_node(
+	__isl_keep isl_ast_graft *graft)
+{
+	return graft ? isl_ast_node_copy(graft->node) : NULL;
+}
+
+/* Create a graft for "node" with no guards and no enforced conditions.
+ */
+__isl_give isl_ast_graft *isl_ast_graft_alloc(
+	__isl_take isl_ast_node *node, __isl_keep isl_ast_build *build)
+{
+	isl_ctx *ctx;
+	isl_space *space;
+	isl_ast_graft *graft;
+
+	if (!node)
+		return NULL;
+
+	ctx = isl_ast_node_get_ctx(node);
+	graft = isl_calloc_type(ctx, isl_ast_graft);
+	if (!graft)
+		goto error;
+
+	space = isl_ast_build_get_space(build, 1);
+
+	graft->ref = 1;
+	graft->node = node;
+	graft->guard = isl_set_universe(isl_space_copy(space));
+	graft->enforced = isl_basic_set_universe(space);
+
+	if (!graft->guard || !graft->enforced)
+		return isl_ast_graft_free(graft);
+
+	return graft;
+error:
+	isl_ast_node_free(node);
+	return NULL;
+}
+
+/* Create a graft with no guards and no enforced conditions
+ * encapsulating a call to the domain element specified by "executed".
+ * "executed" is assumed to be single-valued.
+ */
+__isl_give isl_ast_graft *isl_ast_graft_alloc_domain(
+	__isl_take isl_map *executed, __isl_keep isl_ast_build *build)
+{
+	isl_ast_node *node;
+
+	node = isl_ast_build_call_from_executed(build, executed);
+
+	return isl_ast_graft_alloc(node, build);
+}
+
+static __isl_give isl_ast_graft *isl_ast_graft_copy(
+	__isl_keep isl_ast_graft *graft)
+{
+	if (!graft)
+		return NULL;
+
+	graft->ref++;
+	return graft;
+}
+
+/* Do all the grafts in "list" have the same guard and is this guard
+ * independent of the current depth?
+ */
+static int equal_independent_guards(__isl_keep isl_ast_graft_list *list,
+	__isl_keep isl_ast_build *build)
+{
+	int i, n;
+	int depth;
+	isl_ast_graft *graft_0;
+	int equal = 1;
+	int skip;
+
+	graft_0 = isl_ast_graft_list_get_ast_graft(list, 0);
+	if (!graft_0)
+		return -1;
+
+	depth = isl_ast_build_get_depth(build);
+	skip = isl_set_involves_dims(graft_0->guard, isl_dim_set, depth, 1);
+	if (skip < 0 || skip) {
+		isl_ast_graft_free(graft_0);
+		return skip < 0 ? -1 : 0;
+	}
+
+	n = isl_ast_graft_list_n_ast_graft(list);
+	for (i = 1; i < n; ++i) {
+		isl_ast_graft *graft;
+		graft = isl_ast_graft_list_get_ast_graft(list, i);
+		if (!graft)
+			equal = -1;
+		else
+			equal = isl_set_is_equal(graft_0->guard, graft->guard);
+		isl_ast_graft_free(graft);
+		if (equal < 0 || !equal)
+			break;
+	}
+
+	isl_ast_graft_free(graft_0);
+
+	return equal;
+}
+
+/* Hoist "guard" out of the current level (given by "build").
+ *
+ * In particular, eliminate the dimension corresponding to the current depth.
+ */
+static __isl_give isl_set *hoist_guard(__isl_take isl_set *guard,
+	__isl_keep isl_ast_build *build)
+{
+	int depth;
+
+	depth = isl_ast_build_get_depth(build);
+	if (depth < isl_set_dim(guard, isl_dim_set)) {
+		guard = isl_set_remove_divs_involving_dims(guard,
+						isl_dim_set, depth, 1);
+		guard = isl_set_eliminate(guard, isl_dim_set, depth, 1);
+		guard = isl_set_compute_divs(guard);
+	}
+
+	return guard;
+}
+
+/* Extract a common guard from the grafts in "list" that can be hoisted
+ * out of the current level.  If no such guard can be found, then return
+ * a universal set.
+ *
+ * If all the grafts in the list have the same guard and if this guard
+ * is independent of the current level, then it can be hoisted out.
+ * If there is only one graft in the list and if its guard
+ * depends on the current level, then we eliminate this level and
+ * return the result.
+ *
+ * Otherwise, we return the unshifted simple hull of the guards.
+ * In order to be able to hoist as many constraints as possible,
+ * but at the same time avoid hoisting constraints that did not
+ * appear in the guards in the first place, we intersect the guards
+ * with all the information that is available (i.e., the domain
+ * from the build and the enforced constraints of the graft) and
+ * compute the unshifted hull of the result using only constraints
+ * from the original guards.
+ * In particular, intersecting the guards with other known information
+ * allows us to hoist guards that are only explicit is some of
+ * the grafts and implicit in the others.
+ *
+ * The special case for equal guards is needed in case those guards
+ * are non-convex.  Taking the simple hull would remove information
+ * and would not allow for these guards to be hoisted completely.
+ */
+__isl_give isl_set *isl_ast_graft_list_extract_hoistable_guard(
+	__isl_keep isl_ast_graft_list *list, __isl_keep isl_ast_build *build)
+{
+	int i, n;
+	int equal;
+	isl_ctx *ctx;
+	isl_set *guard;
+	isl_set_list *set_list;
+	isl_basic_set *hull;
+
+	if (!list || !build)
+		return NULL;
+
+	n = isl_ast_graft_list_n_ast_graft(list);
+	if (n == 0)
+		return isl_set_universe(isl_ast_build_get_space(build, 1));
+
+	equal = equal_independent_guards(list, build);
+	if (equal < 0)
+		return NULL;
+
+	if (equal || n == 1) {
+		isl_ast_graft *graft_0;
+
+		graft_0 = isl_ast_graft_list_get_ast_graft(list, 0);
+		if (!graft_0)
+			return NULL;
+		guard = isl_set_copy(graft_0->guard);
+		if (!equal)
+			guard = hoist_guard(guard, build);
+		isl_ast_graft_free(graft_0);
+		return guard;
+	}
+
+	ctx = isl_ast_build_get_ctx(build);
+	set_list = isl_set_list_alloc(ctx, n);
+	guard = isl_set_empty(isl_ast_build_get_space(build, 1));
+	for (i = 0; i < n; ++i) {
+		isl_ast_graft *graft;
+		isl_basic_set *enforced;
+		isl_set *guard_i;
+
+		graft = isl_ast_graft_list_get_ast_graft(list, i);
+		enforced = isl_ast_graft_get_enforced(graft);
+		guard_i = isl_set_copy(graft->guard);
+		isl_ast_graft_free(graft);
+		set_list = isl_set_list_add(set_list, isl_set_copy(guard_i));
+		guard_i = isl_set_intersect(guard_i,
+					    isl_set_from_basic_set(enforced));
+		guard_i = isl_set_intersect(guard_i,
+					    isl_ast_build_get_domain(build));
+		guard = isl_set_union(guard, guard_i);
+	}
+	hull = isl_set_unshifted_simple_hull_from_set_list(guard, set_list);
+	guard = isl_set_from_basic_set(hull);
+	return hoist_guard(guard, build);
+}
+
+/* Internal data structure used inside insert_if.
+ *
+ * list is the list of guarded nodes created by each call to insert_if.
+ * node is the original node that is guarded by insert_if.
+ * build is the build in which the AST is constructed.
+ */
+struct isl_insert_if_data {
+	isl_ast_node_list *list;
+	isl_ast_node *node;
+	isl_ast_build *build;
+};
+
+static int insert_if(__isl_take isl_basic_set *bset, void *user);
+
+/* Insert an if node around "node" testing the condition encoded
+ * in guard "guard".
+ *
+ * If the user does not want any disjunctions in the if conditions
+ * and if "guard" does involve a disjunction, then we make the different
+ * disjuncts disjoint and insert an if node corresponding to each disjunct
+ * around a copy of "node".  The result is then a block node containing
+ * this sequence of guarded copies of "node".
+ */
+static __isl_give isl_ast_node *ast_node_insert_if(
+	__isl_take isl_ast_node *node, __isl_take isl_set *guard,
+	__isl_keep isl_ast_build *build)
+{
+	struct isl_insert_if_data data;
+	isl_ctx *ctx;
+
+	ctx = isl_ast_build_get_ctx(build);
+	if (isl_options_get_ast_build_allow_or(ctx) ||
+	    isl_set_n_basic_set(guard) <= 1) {
+		isl_ast_node *if_node;
+		isl_ast_expr *expr;
+
+		expr = isl_ast_build_expr_from_set(build, guard);
+
+		if_node = isl_ast_node_alloc_if(expr);
+		return isl_ast_node_if_set_then(if_node, node);
+	}
+
+	guard = isl_set_make_disjoint(guard);
+
+	data.list = isl_ast_node_list_alloc(ctx, 0);
+	data.node = node;
+	data.build = build;
+	if (isl_set_foreach_basic_set(guard, &insert_if, &data) < 0)
+		data.list = isl_ast_node_list_free(data.list);
+
+	isl_set_free(guard);
+	isl_ast_node_free(data.node);
+	return isl_ast_node_alloc_block(data.list);
+}
+
+/* Insert an if node around a copy of "data->node" testing the condition
+ * encoded in guard "bset" and add the result to data->list.
+ */
+static int insert_if(__isl_take isl_basic_set *bset, void *user)
+{
+	struct isl_insert_if_data *data = user;
+	isl_ast_node *node;
+	isl_set *set;
+
+	set = isl_set_from_basic_set(bset);
+	node = isl_ast_node_copy(data->node);
+	node = ast_node_insert_if(node, set, data->build);
+	data->list = isl_ast_node_list_add(data->list, node);
+
+	return 0;
+}
+
+/* Insert an if node around graft->node testing the condition encoded
+ * in guard "guard", assuming guard involves any conditions.
+ */
+static __isl_give isl_ast_graft *insert_if_node(
+	__isl_take isl_ast_graft *graft, __isl_take isl_set *guard,
+	__isl_keep isl_ast_build *build)
+{
+	int univ;
+
+	if (!graft)
+		goto error;
+
+	univ = isl_set_plain_is_universe(guard);
+	if (univ < 0)
+		goto error;
+	if (univ) {
+		isl_set_free(guard);
+		return graft;
+	}
+
+	build = isl_ast_build_copy(build);
+	graft->node = ast_node_insert_if(graft->node, guard, build);
+	isl_ast_build_free(build);
+
+	if (!graft->node)
+		return isl_ast_graft_free(graft);
+
+	return graft;
+error:
+	isl_set_free(guard);
+	return isl_ast_graft_free(graft);
+}
+
+/* Insert an if node around graft->node testing the condition encoded
+ * in graft->guard, assuming graft->guard involves any conditions.
+ */
+static __isl_give isl_ast_graft *insert_pending_guard_node(
+	__isl_take isl_ast_graft *graft, __isl_keep isl_ast_build *build)
+{
+	if (!graft)
+		return NULL;
+
+	return insert_if_node(graft, isl_set_copy(graft->guard), build);
+}
+
+/* Replace graft->enforced by "enforced".
+ */
+__isl_give isl_ast_graft *isl_ast_graft_set_enforced(
+	__isl_take isl_ast_graft *graft, __isl_take isl_basic_set *enforced)
+{
+	if (!graft || !enforced)
+		goto error;
+
+	isl_basic_set_free(graft->enforced);
+	graft->enforced = enforced;
+
+	return graft;
+error:
+	isl_basic_set_free(enforced);
+	return isl_ast_graft_free(graft);
+}
+
+/* Update "enforced" such that it only involves constraints that are
+ * also enforced by "graft".
+ */
+static __isl_give isl_basic_set *update_enforced(
+	__isl_take isl_basic_set *enforced, __isl_keep isl_ast_graft *graft,
+	int depth)
+{
+	isl_basic_set *enforced_g;
+
+	enforced_g = isl_ast_graft_get_enforced(graft);
+	if (depth < isl_basic_set_dim(enforced_g, isl_dim_set))
+		enforced_g = isl_basic_set_eliminate(enforced_g,
+							isl_dim_set, depth, 1);
+	enforced_g = isl_basic_set_remove_unknown_divs(enforced_g);
+	enforced_g = isl_basic_set_align_params(enforced_g,
+				isl_basic_set_get_space(enforced));
+	enforced = isl_basic_set_align_params(enforced,
+				isl_basic_set_get_space(enforced_g));
+	enforced = isl_set_simple_hull(isl_basic_set_union(enforced,
+						enforced_g));
+
+	return enforced;
+}
+
+/* Extend the node at *body with node.
+ *
+ * If body points to the else branch, then *body may still be NULL.
+ * If so, we simply attach node to this else branch.
+ * Otherwise, we attach a list containing the statements already
+ * attached at *body followed by node.
+ */
+static void extend_body(__isl_keep isl_ast_node **body,
+	__isl_take isl_ast_node *node)
+{
+	isl_ast_node_list *list;
+
+	if  (!*body) {
+		*body = node;
+		return;
+	}
+
+	if ((*body)->type == isl_ast_node_block) {
+		list = isl_ast_node_block_get_children(*body);
+		isl_ast_node_free(*body);
+	} else
+		list = isl_ast_node_list_from_ast_node(*body);
+	list = isl_ast_node_list_add(list, node);
+	*body = isl_ast_node_alloc_block(list);
+}
+
+/* Merge "graft" into the last graft of "list".
+ * body points to the then or else branch of an if node in that last graft.
+ *
+ * We attach graft->node to this branch and update the enforced
+ * set of the last graft of "list" to take into account the enforced
+ * set of "graft".
+ */
+static __isl_give isl_ast_graft_list *graft_extend_body(
+	__isl_take isl_ast_graft_list *list,
+	__isl_keep isl_ast_node **body, __isl_take isl_ast_graft *graft,
+	__isl_keep isl_ast_build *build)
+{
+	int n;
+	int depth;
+	isl_ast_graft *last;
+	isl_space *space;
+	isl_basic_set *enforced;
+
+	if (!list || !graft)
+		goto error;
+	extend_body(body, isl_ast_node_copy(graft->node));
+	if (!*body)
+		goto error;
+
+	n = isl_ast_graft_list_n_ast_graft(list);
+	last = isl_ast_graft_list_get_ast_graft(list, n - 1);
+
+	depth = isl_ast_build_get_depth(build);
+	space = isl_ast_build_get_space(build, 1);
+	enforced = isl_basic_set_empty(space);
+	enforced = update_enforced(enforced, last, depth);
+	enforced = update_enforced(enforced, graft, depth);
+	last = isl_ast_graft_set_enforced(last, enforced);
+
+	list = isl_ast_graft_list_set_ast_graft(list, n - 1, last);
+	isl_ast_graft_free(graft);
+	return list;
+error:
+	isl_ast_graft_free(graft);
+	return isl_ast_graft_list_free(list);
+}
+
+/* Merge "graft" into the last graft of "list", attaching graft->node
+ * to the then branch of "last_if".
+ */
+static __isl_give isl_ast_graft_list *extend_then(
+	__isl_take isl_ast_graft_list *list,
+	__isl_keep isl_ast_node *last_if, __isl_take isl_ast_graft *graft,
+	__isl_keep isl_ast_build *build)
+{
+	return graft_extend_body(list, &last_if->u.i.then, graft, build);
+}
+
+/* Merge "graft" into the last graft of "list", attaching graft->node
+ * to the else branch of "last_if".
+ */
+static __isl_give isl_ast_graft_list *extend_else(
+	__isl_take isl_ast_graft_list *list,
+	__isl_keep isl_ast_node *last_if, __isl_take isl_ast_graft *graft,
+	__isl_keep isl_ast_build *build)
+{
+	return graft_extend_body(list, &last_if->u.i.else_node, graft, build);
+}
+
+/* This data structure keeps track of an if node.
+ *
+ * "node" is the actual if-node
+ * "guard" is the original, non-simplified guard of the node
+ * "complement" is the complement of "guard" in the context of outer if nodes
+ */
+struct isl_if_node {
+	isl_ast_node *node;
+	isl_set *guard;
+	isl_set *complement;
+};
+
+/* Given a list of "n" if nodes, clear those starting at "first"
+ * and return "first" (i.e., the updated size of the array).
+ */
+static int clear_if_nodes(struct isl_if_node *if_node, int first, int n)
+{
+	int i;
+
+	for (i = first; i < n; ++i) {
+		isl_set_free(if_node[i].guard);
+		isl_set_free(if_node[i].complement);
+	}
+
+	return first;
+}
+
+/* For each graft in "list",
+ * insert an if node around graft->node testing the condition encoded
+ * in graft->guard, assuming graft->guard involves any conditions.
+ *
+ * We keep track of a list of generated if nodes that can be extended
+ * without changing the order of the elements in "list".
+ * If the guard of a graft is a subset of either the guard or its complement
+ * of one of those if nodes, then the node
+ * of the new graft is inserted into the then or else branch of the last graft
+ * and the current graft is discarded.
+ * The guard of the node is then simplified based on the conditions
+ * enforced at that then or else branch.
+ * Otherwise, the current graft is appended to the list.
+ *
+ * We only construct else branches if allowed by the user.
+ */
+static __isl_give isl_ast_graft_list *insert_pending_guard_nodes(
+	__isl_take isl_ast_graft_list *list,
+	__isl_keep isl_ast_build *build)
+{
+	int i, j, n, n_if;
+	int allow_else;
+	isl_ctx *ctx;
+	isl_ast_graft_list *res;
+	struct isl_if_node *if_node = NULL;
+
+	if (!build || !list)
+		return isl_ast_graft_list_free(list);
+
+	ctx = isl_ast_build_get_ctx(build);
+	n = isl_ast_graft_list_n_ast_graft(list);
+
+	allow_else = isl_options_get_ast_build_allow_else(ctx);
+
+	n_if = 0;
+	if (n > 1) {
+		if_node = isl_alloc_array(ctx, struct isl_if_node, n - 1);
+		if (!if_node)
+			return isl_ast_graft_list_free(list);
+	}
+
+	res = isl_ast_graft_list_alloc(ctx, n);
+
+	for (i = 0; i < n; ++i) {
+		isl_set *guard;
+		isl_ast_graft *graft;
+		int subset, found_then, found_else;
+		isl_ast_node *node;
+
+		graft = isl_ast_graft_list_get_ast_graft(list, i);
+		if (!graft)
+			break;
+		subset = 0;
+		found_then = found_else = -1;
+		if (n_if > 0) {
+			isl_set *test;
+			test = isl_set_copy(graft->guard);
+			test = isl_set_intersect(test,
+						isl_set_copy(build->domain));
+			for (j = n_if - 1; j >= 0; --j) {
+				subset = isl_set_is_subset(test,
+							if_node[j].guard);
+				if (subset < 0 || subset) {
+					found_then = j;
+					break;
+				}
+				if (!allow_else)
+					continue;
+				subset = isl_set_is_subset(test,
+							if_node[j].complement);
+				if (subset < 0 || subset) {
+					found_else = j;
+					break;
+				}
+			}
+			n_if = clear_if_nodes(if_node, j + 1, n_if);
+			isl_set_free(test);
+		}
+		if (subset < 0) {
+			graft = isl_ast_graft_free(graft);
+			break;
+		}
+
+		guard = isl_set_copy(graft->guard);
+		if (found_then >= 0)
+			graft->guard = isl_set_gist(graft->guard,
+				isl_set_copy(if_node[found_then].guard));
+		else if (found_else >= 0)
+			graft->guard = isl_set_gist(graft->guard,
+				isl_set_copy(if_node[found_else].complement));
+
+		node = graft->node;
+		if (!graft->guard)
+			graft = isl_ast_graft_free(graft);
+		graft = insert_pending_guard_node(graft, build);
+		if (graft && graft->node != node && i != n - 1) {
+			isl_set *set;
+			if_node[n_if].node = graft->node;
+			if_node[n_if].guard = guard;
+			if (found_then >= 0)
+				set = if_node[found_then].guard;
+			else if (found_else >= 0)
+				set = if_node[found_else].complement;
+			else
+				set = build->domain;
+			set = isl_set_copy(set);
+			set = isl_set_subtract(set, isl_set_copy(guard));
+			if_node[n_if].complement = set;
+			n_if++;
+		} else
+			isl_set_free(guard);
+		if (!graft)
+			break;
+
+		if (found_then >= 0)
+			res = extend_then(res, if_node[found_then].node,
+						graft, build);
+		else if (found_else >= 0)
+			res = extend_else(res, if_node[found_else].node,
+						graft, build);
+		else
+			res = isl_ast_graft_list_add(res, graft);
+	}
+	if (i < n)
+		res = isl_ast_graft_list_free(res);
+
+	isl_ast_graft_list_free(list);
+	clear_if_nodes(if_node, 0, n_if);
+	free(if_node);
+	return res;
+}
+
+/* Collect the nodes contained in the grafts in "list" in a node list.
+ */
+static __isl_give isl_ast_node_list *extract_node_list(
+	__isl_keep isl_ast_graft_list *list)
+{
+	int i, n;
+	isl_ctx *ctx;
+	isl_ast_node_list *node_list;
+
+	if (!list)
+		return NULL;
+	ctx = isl_ast_graft_list_get_ctx(list);
+	n = isl_ast_graft_list_n_ast_graft(list);
+	node_list = isl_ast_node_list_alloc(ctx, n);
+	for (i = 0; i < n; ++i) {
+		isl_ast_node *node;
+		isl_ast_graft *graft;
+
+		graft = isl_ast_graft_list_get_ast_graft(list, i);
+		node = isl_ast_graft_get_node(graft);
+		node_list = isl_ast_node_list_add(node_list, node);
+		isl_ast_graft_free(graft);
+	}
+
+	return node_list;
+}
+
+/* Look for shared enforced constraints by all the elements in "list"
+ * on outer loops (with respect to the current depth) and return the result.
+ *
+ * We assume that the number of children is at least one.
+ */
+__isl_give isl_basic_set *isl_ast_graft_list_extract_shared_enforced(
+	__isl_keep isl_ast_graft_list *list,
+	__isl_keep isl_ast_build *build)
+{
+	int i, n;
+	int depth;
+	isl_space *space;
+	isl_basic_set *enforced;
+
+	if (!list)
+		return NULL;
+
+	n = isl_ast_graft_list_n_ast_graft(list);
+	if (n == 0)
+		isl_die(isl_ast_graft_list_get_ctx(list), isl_error_invalid,
+			"for node should have at least one child",
+			return NULL);
+
+	space = isl_ast_build_get_space(build, 1);
+	enforced = isl_basic_set_empty(space);
+
+	depth = isl_ast_build_get_depth(build);
+	for (i = 0; i < n; ++i) {
+		isl_ast_graft *graft;
+
+		graft = isl_ast_graft_list_get_ast_graft(list, i);
+		enforced = update_enforced(enforced, graft, depth);
+		isl_ast_graft_free(graft);
+	}
+
+	return enforced;
+}
+
+/* Record "guard" in "graft" so that it will be enforced somewhere
+ * up the tree.  If the graft already has a guard, then it may be partially
+ * redundant in combination with the new guard and in the context
+ * the generated constraints of "build".  In fact, the new guard
+ * may in itself have some redundant constraints.
+ * We therefore (re)compute the gist of the intersection
+ * and coalesce the result.
+ */
+static __isl_give isl_ast_graft *store_guard(__isl_take isl_ast_graft *graft,
+	__isl_take isl_set *guard, __isl_keep isl_ast_build *build)
+{
+	int is_universe;
+
+	if (!graft)
+		goto error;
+
+	is_universe = isl_set_plain_is_universe(guard);
+	if (is_universe < 0)
+		goto error;
+	if (is_universe) {
+		isl_set_free(guard);
+		return graft;
+	}
+
+	graft->guard = isl_set_intersect(graft->guard, guard);
+	graft->guard = isl_set_gist(graft->guard,
+				    isl_ast_build_get_generated(build));
+	graft->guard = isl_set_coalesce(graft->guard);
+	if (!graft->guard)
+		return isl_ast_graft_free(graft);
+
+	return graft;
+error:
+	isl_set_free(guard);
+	return isl_ast_graft_free(graft);
+}
+
+/* For each graft in "list", replace its guard with the gist with
+ * respect to "context".
+ */
+static __isl_give isl_ast_graft_list *gist_guards(
+	__isl_take isl_ast_graft_list *list, __isl_keep isl_set *context)
+{
+	int i, n;
+
+	if (!list)
+		return NULL;
+
+	n = isl_ast_graft_list_n_ast_graft(list);
+	for (i = 0; i < n; ++i) {
+		isl_ast_graft *graft;
+
+		graft = isl_ast_graft_list_get_ast_graft(list, i);
+		if (!graft)
+			break;
+		graft->guard = isl_set_gist(graft->guard,
+						isl_set_copy(context));
+		if (!graft->guard)
+			graft = isl_ast_graft_free(graft);
+		list = isl_ast_graft_list_set_ast_graft(list, i, graft);
+	}
+	if (i < n)
+		return isl_ast_graft_list_free(list);
+
+	return list;
+}
+
+/* For each graft in "list", replace its guard with the gist with
+ * respect to "context".
+ */
+__isl_give isl_ast_graft_list *isl_ast_graft_list_gist_guards(
+	__isl_take isl_ast_graft_list *list, __isl_take isl_set *context)
+{
+	list = gist_guards(list, context);
+	isl_set_free(context);
+
+	return list;
+}
+
+/* Allocate a graft in "build" based on the list of grafts in "sub_build".
+ * "guard" and "enforced" are the guard and enforced constraints
+ * of the allocated graft.  The guard is used to simplify the guards
+ * of the elements in "list".
+ *
+ * The node is initialized to either a block containing the nodes of "children"
+ * or, if there is only a single child, the node of that child.
+ * If the current level requires a for node, it should be inserted by
+ * a subsequent call to isl_ast_graft_insert_for.
+ */
+__isl_give isl_ast_graft *isl_ast_graft_alloc_from_children(
+	__isl_take isl_ast_graft_list *list, __isl_take isl_set *guard,
+	__isl_take isl_basic_set *enforced, __isl_keep isl_ast_build *build,
+	__isl_keep isl_ast_build *sub_build)
+{
+	isl_ast_build *guard_build;
+	isl_ast_node *node;
+	isl_ast_node_list *node_list;
+	isl_ast_graft *graft;
+
+	guard_build = isl_ast_build_copy(sub_build);
+	guard_build = isl_ast_build_replace_pending_by_guard(guard_build,
+						isl_set_copy(guard));
+	list = gist_guards(list, guard);
+	list = insert_pending_guard_nodes(list, guard_build);
+	isl_ast_build_free(guard_build);
+
+	node_list = extract_node_list(list);
+	node = isl_ast_node_from_ast_node_list(node_list);
+	isl_ast_graft_list_free(list);
+
+	graft = isl_ast_graft_alloc(node, build);
+	graft = store_guard(graft, guard, build);
+	graft = isl_ast_graft_enforce(graft, enforced);
+
+	return graft;
+}
+
+/* Combine the grafts in the list into a single graft.
+ *
+ * The guard is initialized to the shared guard of the list elements (if any),
+ * provided it does not depend on the current dimension.
+ * The guards in the elements are then simplified with respect to the
+ * hoisted guard and materialized as if nodes around the contained AST nodes
+ * in the context of "sub_build".
+ *
+ * The enforced set is initialized to the simple hull of the enforced sets
+ * of the elements, provided the ast_build_exploit_nested_bounds option is set
+ * or the new graft will be used at the same level.
+ *
+ * The node is initialized to either a block containing the nodes of "list"
+ * or, if there is only a single element, the node of that element.
+ */
+static __isl_give isl_ast_graft *ast_graft_list_fuse(
+	__isl_take isl_ast_graft_list *list, __isl_keep isl_ast_build *build)
+{
+	isl_ast_graft *graft;
+	isl_basic_set *enforced;
+	isl_set *guard;
+
+	if (!list)
+		return NULL;
+
+	enforced = isl_ast_graft_list_extract_shared_enforced(list, build);
+	guard = isl_ast_graft_list_extract_hoistable_guard(list, build);
+	graft = isl_ast_graft_alloc_from_children(list, guard, enforced,
+						    build, build);
+
+	return graft;
+}
+
+/* Combine the grafts in the list into a single graft.
+ * Return a list containing this single graft.
+ * If the original list is empty, then return an empty list.
+ */
+__isl_give isl_ast_graft_list *isl_ast_graft_list_fuse(
+	__isl_take isl_ast_graft_list *list,
+	__isl_keep isl_ast_build *build)
+{
+	isl_ast_graft *graft;
+
+	if (!list)
+		return NULL;
+	if (isl_ast_graft_list_n_ast_graft(list) <= 1)
+		return list;
+	graft = ast_graft_list_fuse(list, build);
+	return isl_ast_graft_list_from_ast_graft(graft);
+}
+
+/* Combine the two grafts into a single graft.
+ * Return a list containing this single graft.
+ */
+static __isl_give isl_ast_graft *isl_ast_graft_fuse(
+	__isl_take isl_ast_graft *graft1, __isl_take isl_ast_graft *graft2,
+	__isl_keep isl_ast_build *build)
+{
+	isl_ctx *ctx;
+	isl_ast_graft_list *list;
+
+	ctx = isl_ast_build_get_ctx(build);
+
+	list = isl_ast_graft_list_alloc(ctx, 2);
+	list = isl_ast_graft_list_add(list, graft1);
+	list = isl_ast_graft_list_add(list, graft2);
+
+	return ast_graft_list_fuse(list, build);
+}
+
+/* Insert a for node enclosing the current graft->node.
+ */
+__isl_give isl_ast_graft *isl_ast_graft_insert_for(
+	__isl_take isl_ast_graft *graft, __isl_take isl_ast_node *node)
+{
+	if (!graft)
+		goto error;
+
+	graft->node = isl_ast_node_for_set_body(node, graft->node);
+	if (!graft->node)
+		return isl_ast_graft_free(graft);
+
+	return graft;
+error:
+	isl_ast_node_free(node);
+	isl_ast_graft_free(graft);
+	return NULL;
+}
+
+/* Represent the graft list as an AST node.
+ * This operation drops the information about guards in the grafts, so
+ * if there are any pending guards, then they are materialized as if nodes.
+ */
+__isl_give isl_ast_node *isl_ast_node_from_graft_list(
+	__isl_take isl_ast_graft_list *list,
+	__isl_keep isl_ast_build *build)
+{
+	isl_ast_node_list *node_list;
+
+	list = insert_pending_guard_nodes(list, build);
+	node_list = extract_node_list(list);
+	isl_ast_graft_list_free(list);
+
+	return isl_ast_node_from_ast_node_list(node_list);
+}
+
+void *isl_ast_graft_free(__isl_take isl_ast_graft *graft)
+{
+	if (!graft)
+		return NULL;
+
+	if (--graft->ref > 0)
+		return NULL;
+
+	isl_ast_node_free(graft->node);
+	isl_set_free(graft->guard);
+	isl_basic_set_free(graft->enforced);
+	free(graft);
+
+	return NULL;
+}
+
+/* Record that the grafted tree enforces
+ * "enforced" by intersecting graft->enforced with "enforced".
+ */
+__isl_give isl_ast_graft *isl_ast_graft_enforce(
+	__isl_take isl_ast_graft *graft, __isl_take isl_basic_set *enforced)
+{
+	if (!graft || !enforced)
+		goto error;
+
+	enforced = isl_basic_set_align_params(enforced,
+				isl_basic_set_get_space(graft->enforced));
+	graft->enforced = isl_basic_set_align_params(graft->enforced,
+				isl_basic_set_get_space(enforced));
+	graft->enforced = isl_basic_set_intersect(graft->enforced, enforced);
+	if (!graft->enforced)
+		return isl_ast_graft_free(graft);
+
+	return graft;
+error:
+	isl_basic_set_free(enforced);
+	return isl_ast_graft_free(graft);
+}
+
+__isl_give isl_basic_set *isl_ast_graft_get_enforced(
+	__isl_keep isl_ast_graft *graft)
+{
+	return graft ? isl_basic_set_copy(graft->enforced) : NULL;
+}
+
+__isl_give isl_set *isl_ast_graft_get_guard(__isl_keep isl_ast_graft *graft)
+{
+	return graft ? isl_set_copy(graft->guard) : NULL;
+}
+
+/* Record that "guard" needs to be inserted in "graft".
+ */
+__isl_give isl_ast_graft *isl_ast_graft_add_guard(
+	__isl_take isl_ast_graft *graft,
+	__isl_take isl_set *guard, __isl_keep isl_ast_build *build)
+{
+	return store_guard(graft, guard, build);
+}
+
+/* Reformulate the "graft", which was generated in the context
+ * of an inner code generation, in terms of the outer code generation
+ * AST build.
+ *
+ * If "product" is set, then the domain of the inner code generation build is
+ *
+ *	[O -> S]
+ *
+ * with O the domain of the outer code generation build.
+ * We essentially need to project out S.
+ *
+ * If "product" is not set, then we need to project the domains onto
+ * their parameter spaces.
+ */
+__isl_give isl_ast_graft *isl_ast_graft_unembed(__isl_take isl_ast_graft *graft,
+	int product)
+{
+	isl_basic_set *enforced;
+
+	if (!graft)
+		return NULL;
+
+	if (product) {
+		enforced = graft->enforced;
+		enforced = isl_basic_map_domain(isl_basic_set_unwrap(enforced));
+		graft->enforced = enforced;
+		graft->guard = isl_map_domain(isl_set_unwrap(graft->guard));
+	} else {
+		graft->enforced = isl_basic_set_params(graft->enforced);
+		graft->guard = isl_set_params(graft->guard);
+	}
+	graft->guard = isl_set_compute_divs(graft->guard);
+
+	if (!graft->enforced || !graft->guard)
+		return isl_ast_graft_free(graft);
+
+	return graft;
+}
+
+/* Reformulate the grafts in "list", which were generated in the context
+ * of an inner code generation, in terms of the outer code generation
+ * AST build.
+ */
+__isl_give isl_ast_graft_list *isl_ast_graft_list_unembed(
+	__isl_take isl_ast_graft_list *list, int product)
+{
+	int i, n;
+
+	n = isl_ast_graft_list_n_ast_graft(list);
+	for (i = 0; i < n; ++i) {
+		isl_ast_graft *graft;
+
+		graft = isl_ast_graft_list_get_ast_graft(list, i);
+		graft = isl_ast_graft_unembed(graft, product);
+		list = isl_ast_graft_list_set_ast_graft(list, i, graft);
+	}
+
+	return list;
+}
+
+/* Compute the preimage of "graft" under the function represented by "ma".
+ * In other words, plug in "ma" in "enforced" and "guard" fields of "graft".
+ */
+__isl_give isl_ast_graft *isl_ast_graft_preimage_multi_aff(
+	__isl_take isl_ast_graft *graft, __isl_take isl_multi_aff *ma)
+{
+	isl_basic_set *enforced;
+
+	if (!graft)
+		return NULL;
+
+	enforced = graft->enforced;
+	graft->enforced = isl_basic_set_preimage_multi_aff(enforced,
+						isl_multi_aff_copy(ma));
+	graft->guard = isl_set_preimage_multi_aff(graft->guard, ma);
+
+	if (!graft->enforced || !graft->guard)
+		return isl_ast_graft_free(graft);
+
+	return graft;
+}
+
+/* Compute the preimage of all the grafts in "list" under
+ * the function represented by "ma".
+ */
+__isl_give isl_ast_graft_list *isl_ast_graft_list_preimage_multi_aff(
+	__isl_take isl_ast_graft_list *list, __isl_take isl_multi_aff *ma)
+{
+	int i, n;
+
+	n = isl_ast_graft_list_n_ast_graft(list);
+	for (i = 0; i < n; ++i) {
+		isl_ast_graft *graft;
+
+		graft = isl_ast_graft_list_get_ast_graft(list, i);
+		graft = isl_ast_graft_preimage_multi_aff(graft,
+						    isl_multi_aff_copy(ma));
+		list = isl_ast_graft_list_set_ast_graft(list, i, graft);
+	}
+
+	isl_multi_aff_free(ma);
+	return list;
+}
+
+/* Compare two grafts based on their guards.
+ */
+static int cmp_graft(__isl_keep isl_ast_graft *a, __isl_keep isl_ast_graft *b,
+	void *user)
+{
+	return isl_set_plain_cmp(a->guard, b->guard);
+}
+
+/* Order the elements in "list" based on their guards.
+ */
+__isl_give isl_ast_graft_list *isl_ast_graft_list_sort_guard(
+	__isl_take isl_ast_graft_list *list)
+{
+	return isl_ast_graft_list_sort(list, &cmp_graft, NULL);
+}
+
+/* Merge the given two lists into a single list of grafts,
+ * merging grafts with the same guard into a single graft.
+ *
+ * "list2" has been sorted using isl_ast_graft_list_sort.
+ * "list1" may be the result of a previous call to isl_ast_graft_list_merge
+ * and may therefore not be completely sorted.
+ *
+ * The elements in "list2" need to be executed after those in "list1",
+ * but if the guard of a graft in "list2" is disjoint from the guards
+ * of some final elements in "list1", then it can be moved up to before
+ * those final elements.
+ *
+ * In particular, we look at each element g of "list2" in turn
+ * and move it up beyond elements of "list1" that would be sorted
+ * after g as long as each of these elements has a guard that is disjoint
+ * from that of g.
+ *
+ * We do not allow the second or any later element of "list2" to be moved
+ * before a previous elements of "list2" even if the reason that
+ * that element didn't move up further was that its guard was not disjoint
+ * from that of the previous element in "list1".
+ */
+__isl_give isl_ast_graft_list *isl_ast_graft_list_merge(
+	__isl_take isl_ast_graft_list *list1,
+	__isl_take isl_ast_graft_list *list2,
+	__isl_keep isl_ast_build *build)
+{
+	int i, j, first;
+
+	if (!list1 || !list2 || !build)
+		goto error;
+	if (list2->n == 0) {
+		isl_ast_graft_list_free(list2);
+		return list1;
+	}
+	if (list1->n == 0) {
+		isl_ast_graft_list_free(list1);
+		return list2;
+	}
+
+	first = 0;
+	for (i = 0; i < list2->n; ++i) {
+		isl_ast_graft *graft;
+		graft = isl_ast_graft_list_get_ast_graft(list2, i);
+		if (!graft)
+			break;
+
+		for (j = list1->n; j >= 0; --j) {
+			int cmp, disjoint;
+			isl_ast_graft *graft_j;
+
+			if (j == first)
+				cmp = -1;
+			else
+				cmp = isl_set_plain_cmp(list1->p[j - 1]->guard,
+							graft->guard);
+			if (cmp > 0) {
+				disjoint = isl_set_is_disjoint(graft->guard,
+							list1->p[j - 1]->guard);
+				if (disjoint < 0) {
+					list1 = isl_ast_graft_list_free(list1);
+					break;
+				}
+				if (!disjoint)
+					cmp = -1;
+			}
+			if (cmp > 0)
+				continue;
+			if (cmp < 0) {
+				list1 = isl_ast_graft_list_insert(list1, j,
+								graft);
+				break;
+			}
+
+			--j;
+
+			graft_j = isl_ast_graft_list_get_ast_graft(list1, j);
+			graft_j = isl_ast_graft_fuse(graft_j, graft, build);
+			list1 = isl_ast_graft_list_set_ast_graft(list1, j,
+								graft_j);
+			break;
+		}
+
+		if (j < 0)
+			isl_die(isl_ast_build_get_ctx(build),
+				isl_error_internal,
+				"element failed to get inserted", break);
+
+		first = j + 1;
+		if (!list1)
+			break;
+	}
+	if (i < list2->n)
+		list1 = isl_ast_graft_list_free(list1);
+	isl_ast_graft_list_free(list2);
+
+	return list1;
+error:
+	isl_ast_graft_list_free(list1);
+	isl_ast_graft_list_free(list2);
+	return NULL;
+}
+
+__isl_give isl_printer *isl_printer_print_ast_graft(__isl_take isl_printer *p,
+	__isl_keep isl_ast_graft *graft)
+{
+	if (!p)
+		return NULL;
+	if (!graft)
+		return isl_printer_free(p);
+
+	p = isl_printer_print_str(p, "(");
+	p = isl_printer_print_str(p, "guard: ");
+	p = isl_printer_print_set(p, graft->guard);
+	p = isl_printer_print_str(p, ", ");
+	p = isl_printer_print_str(p, "enforced: ");
+	p = isl_printer_print_basic_set(p, graft->enforced);
+	p = isl_printer_print_str(p, ", ");
+	p = isl_printer_print_str(p, "node: ");
+	p = isl_printer_print_ast_node(p, graft->node);
+	p = isl_printer_print_str(p, ")");
+
+	return p;
+}
diff --git a/polly/lib/External/isl/isl_ast_graft_private.h b/polly/lib/External/isl/isl_ast_graft_private.h
new file mode 100644
index 00000000000..82b1c1ccaa0
--- /dev/null
+++ b/polly/lib/External/isl/isl_ast_graft_private.h
@@ -0,0 +1,97 @@
+#ifndef ISL_AST_GRAFT_PRIVATE_H
+#define ISL_AST_GRAFT_PRIVATE_H
+
+#include <isl/ast.h>
+#include <isl/set.h>
+#include <isl/list.h>
+#include <isl/printer.h>
+
+struct isl_ast_graft;
+typedef struct isl_ast_graft isl_ast_graft;
+
+/* Representation of part of an AST ("node") with some additional polyhedral
+ * information about the tree.
+ *
+ * "guard" contains conditions that should still be enforced by
+ * some ancestor of the current tree.  In particular, the already
+ * generated tree assumes that these conditions hold, but may not
+ * have enforced them itself.
+ * The guard should not contain any unknown divs as it will be used
+ * to generate an if condition.
+ *
+ * "enforced" expresses constraints that are already enforced by the for
+ * nodes in the current tree and that therefore do not need to be enforced
+ * by any ancestor.
+ * The constraints only involve outer loop iterators.
+ */
+struct isl_ast_graft {
+	int ref;
+
+	isl_ast_node *node;
+
+	isl_set *guard;
+	isl_basic_set *enforced;
+};
+
+ISL_DECLARE_LIST(ast_graft)
+
+#undef EL
+#define EL isl_ast_graft
+
+#include <isl_list_templ.h>
+
+isl_ctx *isl_ast_graft_get_ctx(__isl_keep isl_ast_graft *graft);
+
+__isl_give isl_ast_graft *isl_ast_graft_alloc(
+	__isl_take isl_ast_node *node, __isl_keep isl_ast_build *build);
+__isl_give isl_ast_graft *isl_ast_graft_alloc_from_children(
+	__isl_take isl_ast_graft_list *list, __isl_take isl_set *guard,
+	__isl_take isl_basic_set *enforced, __isl_keep isl_ast_build *build,
+	__isl_keep isl_ast_build *sub_build);
+__isl_give isl_ast_graft_list *isl_ast_graft_list_fuse(
+	__isl_take isl_ast_graft_list *children,
+	__isl_keep isl_ast_build *build);
+__isl_give isl_ast_graft *isl_ast_graft_alloc_domain(
+	__isl_take isl_map *schedule, __isl_keep isl_ast_build *build);
+void *isl_ast_graft_free(__isl_take isl_ast_graft *graft);
+__isl_give isl_ast_graft_list *isl_ast_graft_list_sort_guard(
+	__isl_take isl_ast_graft_list *list);
+
+__isl_give isl_ast_graft_list *isl_ast_graft_list_merge(
+	__isl_take isl_ast_graft_list *list1,
+	__isl_take isl_ast_graft_list *list2,
+	__isl_keep isl_ast_build *build);
+
+__isl_give isl_ast_node *isl_ast_graft_get_node(
+	__isl_keep isl_ast_graft *graft);
+__isl_give isl_basic_set *isl_ast_graft_get_enforced(
+	__isl_keep isl_ast_graft *graft);
+__isl_give isl_set *isl_ast_graft_get_guard(__isl_keep isl_ast_graft *graft);
+
+__isl_give isl_ast_graft *isl_ast_graft_insert_for(
+	__isl_take isl_ast_graft *graft, __isl_take isl_ast_node *node);
+__isl_give isl_ast_graft *isl_ast_graft_add_guard(
+	__isl_take isl_ast_graft *graft,
+	__isl_take isl_set *guard, __isl_keep isl_ast_build *build);
+__isl_give isl_ast_graft *isl_ast_graft_enforce(
+	__isl_take isl_ast_graft *graft, __isl_take isl_basic_set *enforced);
+
+__isl_give isl_ast_graft_list *isl_ast_graft_list_unembed(
+	__isl_take isl_ast_graft_list *list, int product);
+__isl_give isl_ast_graft_list *isl_ast_graft_list_preimage_multi_aff(
+	__isl_take isl_ast_graft_list *list, __isl_take isl_multi_aff *ma);
+
+__isl_give isl_ast_node *isl_ast_node_from_graft_list(
+	__isl_take isl_ast_graft_list *list, __isl_keep isl_ast_build *build);
+
+__isl_give isl_basic_set *isl_ast_graft_list_extract_shared_enforced(
+	__isl_keep isl_ast_graft_list *list, __isl_keep isl_ast_build *build);
+__isl_give isl_set *isl_ast_graft_list_extract_hoistable_guard(
+	__isl_keep isl_ast_graft_list *list, __isl_keep isl_ast_build *build);
+__isl_give isl_ast_graft_list *isl_ast_graft_list_gist_guards(
+	__isl_take isl_ast_graft_list *list, __isl_take isl_set *context);
+
+__isl_give isl_printer *isl_printer_print_ast_graft(__isl_take isl_printer *p,
+	__isl_keep isl_ast_graft *graft);
+
+#endif
diff --git a/polly/lib/External/isl/isl_ast_int.c b/polly/lib/External/isl/isl_ast_int.c
new file mode 100644
index 00000000000..178c38cf32a
--- /dev/null
+++ b/polly/lib/External/isl/isl_ast_int.c
@@ -0,0 +1,13 @@
+#include <isl/deprecated/ast_int.h>
+#include <isl/deprecated/val_int.h>
+#include <isl_ast_private.h>
+
+int isl_ast_expr_get_int(__isl_keep isl_ast_expr *expr, isl_int *v)
+{
+	if (!expr)
+		return -1;
+	if (expr->type != isl_ast_expr_int)
+		isl_die(isl_ast_expr_get_ctx(expr), isl_error_invalid,
+			"expression not an int", return -1);
+	return isl_val_get_num_isl_int(expr->u.v, v);
+}
diff --git a/polly/lib/External/isl/isl_ast_private.h b/polly/lib/External/isl/isl_ast_private.h
new file mode 100644
index 00000000000..c1bf334403b
--- /dev/null
+++ b/polly/lib/External/isl/isl_ast_private.h
@@ -0,0 +1,114 @@
+#ifndef ISL_AST_PRIVATE_H
+#define ISL_AST_PRIVATE_H
+
+#include <isl/aff.h>
+#include <isl/ast.h>
+#include <isl/set.h>
+#include <isl/map.h>
+#include <isl/vec.h>
+#include <isl/list.h>
+
+/* An expression is either an integer, an identifier or an operation
+ * with zero or more arguments.
+ */
+struct isl_ast_expr {
+	int ref;
+
+	isl_ctx *ctx;
+
+	enum isl_ast_expr_type type;
+
+	union {
+		isl_val *v;
+		isl_id *id;
+		struct {
+			enum isl_ast_op_type op;
+			unsigned n_arg;
+			isl_ast_expr **args;
+		} op;
+	} u;
+};
+
+#undef EL
+#define EL isl_ast_expr
+
+#include <isl_list_templ.h>
+
+__isl_give isl_ast_expr *isl_ast_expr_alloc_int_si(isl_ctx *ctx, int i);
+__isl_give isl_ast_expr *isl_ast_expr_alloc_op(isl_ctx *ctx,
+	enum isl_ast_op_type op, int n_arg);
+__isl_give isl_ast_expr *isl_ast_expr_alloc_binary(enum isl_ast_op_type type,
+	__isl_take isl_ast_expr *expr1, __isl_take isl_ast_expr *expr2);
+
+#undef EL
+#define EL isl_ast_node
+
+#include <isl_list_templ.h>
+
+/* A node is either a block, an if, a for or a user node.
+ * "else_node" is NULL if the if node does not have an else branch.
+ * "cond" and "inc" are NULL for degenerate for nodes.
+ */
+struct isl_ast_node {
+	int ref;
+
+	isl_ctx *ctx;
+	enum isl_ast_node_type type;
+
+	union {
+		struct {
+			isl_ast_node_list *children;
+		} b;
+		struct {
+			isl_ast_expr *guard;
+			isl_ast_node *then;
+			isl_ast_node *else_node;
+		} i;
+		struct {
+			unsigned degenerate : 1;
+			isl_ast_expr *iterator;
+			isl_ast_expr *init;
+			isl_ast_expr *cond;
+			isl_ast_expr *inc;
+			isl_ast_node *body;
+		} f;
+		struct {
+			isl_ast_expr *expr;
+		} e;
+	} u;
+
+	isl_id *annotation;
+};
+
+__isl_give isl_ast_node *isl_ast_node_alloc_for(__isl_take isl_id *id);
+__isl_give isl_ast_node *isl_ast_node_for_mark_degenerate(
+	__isl_take isl_ast_node *node);
+__isl_give isl_ast_node *isl_ast_node_alloc_if(__isl_take isl_ast_expr *guard);
+__isl_give isl_ast_node *isl_ast_node_alloc_block(
+	__isl_take isl_ast_node_list *list);
+__isl_give isl_ast_node *isl_ast_node_from_ast_node_list(
+	__isl_take isl_ast_node_list *list);
+__isl_give isl_ast_node *isl_ast_node_for_set_body(
+	__isl_take isl_ast_node *node, __isl_take isl_ast_node *body);
+__isl_give isl_ast_node *isl_ast_node_if_set_then(
+	__isl_take isl_ast_node *node, __isl_take isl_ast_node *child);
+
+struct isl_ast_print_options {
+	int ref;
+	isl_ctx *ctx;
+
+	__isl_give isl_printer *(*print_for)(__isl_take isl_printer *p,
+		__isl_take isl_ast_print_options *options,
+		__isl_keep isl_ast_node *node, void *user);
+	void *print_for_user;
+	__isl_give isl_printer *(*print_user)(__isl_take isl_printer *p,
+		__isl_take isl_ast_print_options *options,
+		__isl_keep isl_ast_node *node, void *user);
+	void *print_user_user;
+};
+
+__isl_give isl_printer *isl_ast_node_list_print(
+	__isl_keep isl_ast_node_list *list, __isl_take isl_printer *p,
+	__isl_keep isl_ast_print_options *options);
+
+#endif
diff --git a/polly/lib/External/isl/isl_band.c b/polly/lib/External/isl/isl_band.c
new file mode 100644
index 00000000000..aaee1f1c951
--- /dev/null
+++ b/polly/lib/External/isl/isl_band.c
@@ -0,0 +1,726 @@
+/*
+ * Copyright 2011      INRIA Saclay
+ * Copyright 2012-2013 Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France
+ * and Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France
+ */
+
+#include <isl_band_private.h>
+#include <isl_schedule_private.h>
+
+#undef BASE
+#define BASE band
+
+#include <isl_list_templ.c>
+
+isl_ctx *isl_band_get_ctx(__isl_keep isl_band *band)
+{
+	return band ? isl_union_pw_multi_aff_get_ctx(band->pma) : NULL;
+}
+
+__isl_give isl_band *isl_band_alloc(isl_ctx *ctx)
+{
+	isl_band *band;
+
+	band = isl_calloc_type(ctx, isl_band);
+	if (!band)
+		return NULL;
+
+	band->ref = 1;
+
+	return band;
+}
+
+/* Create a duplicate of the given band.  The duplicate refers
+ * to the same schedule and parent as the input, but does not
+ * increment their reference counts.
+ */
+__isl_give isl_band *isl_band_dup(__isl_keep isl_band *band)
+{
+	int i;
+	isl_ctx *ctx;
+	isl_band *dup;
+
+	if (!band)
+		return NULL;
+
+	ctx = isl_band_get_ctx(band);
+	dup = isl_band_alloc(ctx);
+	if (!dup)
+		return NULL;
+
+	dup->n = band->n;
+	dup->coincident = isl_alloc_array(ctx, int, band->n);
+	if (band->n && !dup->coincident)
+		goto error;
+
+	for (i = 0; i < band->n; ++i)
+		dup->coincident[i] = band->coincident[i];
+
+	dup->pma = isl_union_pw_multi_aff_copy(band->pma);
+	dup->schedule = band->schedule;
+	dup->parent = band->parent;
+
+	if (!dup->pma)
+		goto error;
+
+	return dup;
+error:
+	isl_band_free(dup);
+	return NULL;
+}
+
+/* We not only increment the reference count of the band,
+ * but also that of the schedule that contains this band.
+ * This ensures that the schedule won't disappear while there
+ * is still a reference to the band outside of the schedule.
+ * There is no need to increment the reference count of the parent
+ * band as the parent band is part of the same schedule.
+ */
+__isl_give isl_band *isl_band_copy(__isl_keep isl_band *band)
+{
+	if (!band)
+		return NULL;
+
+	band->ref++;
+	band->schedule->ref++;
+	return band;
+}
+
+/* If this is not the last reference to the band (the one from within the
+ * schedule), then we also need to decrement the reference count of the
+ * containing schedule as it was incremented in isl_band_copy.
+ */
+__isl_null isl_band *isl_band_free(__isl_take isl_band *band)
+{
+	if (!band)
+		return NULL;
+
+	if (--band->ref > 0) {
+		isl_schedule_free(band->schedule);
+		return NULL;
+	}
+
+	isl_union_pw_multi_aff_free(band->pma);
+	isl_band_list_free(band->children);
+	free(band->coincident);
+	free(band);
+
+	return NULL;
+}
+
+int isl_band_has_children(__isl_keep isl_band *band)
+{
+	if (!band)
+		return -1;
+
+	return band->children != NULL;
+}
+
+__isl_give isl_band_list *isl_band_get_children(
+	__isl_keep isl_band *band)
+{
+	if (!band)
+		return NULL;
+	if (!band->children)
+		isl_die(isl_band_get_ctx(band), isl_error_invalid,
+			"band has no children", return NULL);
+	return isl_band_list_dup(band->children);
+}
+
+int isl_band_n_member(__isl_keep isl_band *band)
+{
+	return band ? band->n : 0;
+}
+
+/* Is the given scheduling dimension coincident within the band and
+ * with respect to the coincidence constraints.
+ */
+int isl_band_member_is_coincident(__isl_keep isl_band *band, int pos)
+{
+	if (!band)
+		return -1;
+
+	if (pos < 0 || pos >= band->n)
+		isl_die(isl_band_get_ctx(band), isl_error_invalid,
+			"invalid member position", return -1);
+
+	return band->coincident[pos];
+}
+
+/* Return the schedule that leads up to this band.
+ */
+__isl_give isl_union_map *isl_band_get_prefix_schedule(
+	__isl_keep isl_band *band)
+{
+	isl_union_set *domain;
+	isl_union_pw_multi_aff *prefix;
+	isl_band *a;
+
+	if (!band)
+		return NULL;
+
+	prefix = isl_union_pw_multi_aff_copy(band->pma);
+	domain = isl_union_pw_multi_aff_domain(prefix);
+	prefix = isl_union_pw_multi_aff_from_domain(domain);
+
+	for (a = band->parent; a; a = a->parent) {
+		isl_union_pw_multi_aff *partial;
+
+		partial = isl_union_pw_multi_aff_copy(a->pma);
+		prefix = isl_union_pw_multi_aff_flat_range_product(partial,
+								   prefix);
+	}
+
+	return isl_union_map_from_union_pw_multi_aff(prefix);
+}
+
+/* Return the schedule of the band in isolation.
+ */
+__isl_give isl_union_pw_multi_aff *
+isl_band_get_partial_schedule_union_pw_multi_aff(__isl_keep isl_band *band)
+{
+	return band ? isl_union_pw_multi_aff_copy(band->pma) : NULL;
+}
+
+/* Return the schedule of the band in isolation.
+ */
+__isl_give isl_union_map *isl_band_get_partial_schedule(
+	__isl_keep isl_band *band)
+{
+	isl_union_pw_multi_aff *sched;
+
+	sched = isl_band_get_partial_schedule_union_pw_multi_aff(band);
+	return isl_union_map_from_union_pw_multi_aff(sched);
+}
+
+__isl_give isl_union_pw_multi_aff *
+isl_band_get_suffix_schedule_union_pw_multi_aff(__isl_keep isl_band *band);
+
+/* Return the schedule for the given band list.
+ * For each band in the list, the schedule is composed of the partial
+ * and suffix schedules of that band.
+ */
+__isl_give isl_union_pw_multi_aff *
+isl_band_list_get_suffix_schedule_union_pw_multi_aff(
+	__isl_keep isl_band_list *list)
+{
+	isl_ctx *ctx;
+	int i, n;
+	isl_space *space;
+	isl_union_pw_multi_aff *suffix;
+
+	if (!list)
+		return NULL;
+
+	ctx = isl_band_list_get_ctx(list);
+	space = isl_space_alloc(ctx, 0, 0, 0);
+	suffix = isl_union_pw_multi_aff_empty(space);
+	n = isl_band_list_n_band(list);
+	for (i = 0; i < n; ++i) {
+		isl_band *el;
+		isl_union_pw_multi_aff *partial;
+		isl_union_pw_multi_aff *suffix_i;
+
+		el = isl_band_list_get_band(list, i);
+		partial = isl_band_get_partial_schedule_union_pw_multi_aff(el);
+		suffix_i = isl_band_get_suffix_schedule_union_pw_multi_aff(el);
+		suffix_i = isl_union_pw_multi_aff_flat_range_product(
+				partial, suffix_i);
+		suffix = isl_union_pw_multi_aff_union_add(suffix, suffix_i);
+
+		isl_band_free(el);
+	}
+
+	return suffix;
+}
+
+/* Return the schedule for the given band list.
+ * For each band in the list, the schedule is composed of the partial
+ * and suffix schedules of that band.
+ */
+__isl_give isl_union_map *isl_band_list_get_suffix_schedule(
+	__isl_keep isl_band_list *list)
+{
+	isl_union_pw_multi_aff *suffix;
+
+	suffix = isl_band_list_get_suffix_schedule_union_pw_multi_aff(list);
+	return isl_union_map_from_union_pw_multi_aff(suffix);
+}
+
+/* Return the schedule for the forest underneath the given band.
+ */
+__isl_give isl_union_pw_multi_aff *
+isl_band_get_suffix_schedule_union_pw_multi_aff(__isl_keep isl_band *band)
+{
+	isl_union_pw_multi_aff *suffix;
+
+	if (!band)
+		return NULL;
+
+	if (!isl_band_has_children(band)) {
+		isl_union_set *domain;
+
+		suffix = isl_union_pw_multi_aff_copy(band->pma);
+		domain = isl_union_pw_multi_aff_domain(suffix);
+		suffix = isl_union_pw_multi_aff_from_domain(domain);
+	} else {
+		isl_band_list *list;
+
+		list = isl_band_get_children(band);
+		suffix =
+		    isl_band_list_get_suffix_schedule_union_pw_multi_aff(list);
+		isl_band_list_free(list);
+	}
+
+	return suffix;
+}
+
+/* Return the schedule for the forest underneath the given band.
+ */
+__isl_give isl_union_map *isl_band_get_suffix_schedule(
+	__isl_keep isl_band *band)
+{
+	isl_union_pw_multi_aff *suffix;
+
+	suffix = isl_band_get_suffix_schedule_union_pw_multi_aff(band);
+	return isl_union_map_from_union_pw_multi_aff(suffix);
+}
+
+/* Call "fn" on each band (recursively) in the list
+ * in depth-first post-order.
+ */
+int isl_band_list_foreach_band(__isl_keep isl_band_list *list,
+	int (*fn)(__isl_keep isl_band *band, void *user), void *user)
+{
+	int i, n;
+
+	if (!list)
+		return -1;
+
+	n = isl_band_list_n_band(list);
+	for (i = 0; i < n; ++i) {
+		isl_band *band;
+		int r = 0;
+
+		band = isl_band_list_get_band(list, i);
+		if (isl_band_has_children(band)) {
+			isl_band_list *children;
+
+			children = isl_band_get_children(band);
+			r = isl_band_list_foreach_band(children, fn, user);
+			isl_band_list_free(children);
+		}
+
+		if (!band)
+			r = -1;
+		if (r == 0)
+			r = fn(band, user);
+
+		isl_band_free(band);
+		if (r)
+			return r;
+	}
+
+	return 0;
+}
+
+/* Internal data used during the construction of the schedule
+ * for the tile loops.
+ *
+ * sizes contains the tile sizes
+ * scale is set if the tile loops should be scaled
+ * tiled collects the result for a single statement
+ * res collects the result for all statements
+ */
+struct isl_band_tile_data {
+	isl_multi_val *sizes;
+	isl_union_pw_multi_aff *res;
+	isl_pw_multi_aff *tiled;
+	int scale;
+};
+
+/* Given part of the schedule of a band, construct the corresponding
+ * schedule for the tile loops based on the tile sizes in data->sizes
+ * and add the result to data->tiled.
+ *
+ * If data->scale is set, then dimension i of the schedule will be
+ * of the form
+ *
+ *	m_i * floor(s_i(x) / m_i)
+ *
+ * where s_i(x) refers to the original schedule and m_i is the tile size.
+ * If data->scale is not set, then dimension i of the schedule will be
+ * of the form
+ *
+ *	floor(s_i(x) / m_i)
+ *
+ */
+static int multi_aff_tile(__isl_take isl_set *set,
+	__isl_take isl_multi_aff *ma, void *user)
+{
+	struct isl_band_tile_data *data = user;
+	isl_pw_multi_aff *pma;
+	int i, n;
+	isl_val *v;
+
+	n = isl_multi_aff_dim(ma, isl_dim_out);
+
+	for (i = 0; i < n; ++i) {
+		isl_aff *aff;
+
+		aff = isl_multi_aff_get_aff(ma, i);
+		v = isl_multi_val_get_val(data->sizes, i);
+
+		aff = isl_aff_scale_down_val(aff, isl_val_copy(v));
+		aff = isl_aff_floor(aff);
+		if (data->scale)
+			aff = isl_aff_scale_val(aff, isl_val_copy(v));
+		isl_val_free(v);
+
+		ma = isl_multi_aff_set_aff(ma, i, aff);
+	}
+
+	pma = isl_pw_multi_aff_alloc(set, ma);
+	data->tiled = isl_pw_multi_aff_union_add(data->tiled, pma);
+
+	return 0;
+}
+
+/* Given part of the schedule of a band, construct the corresponding
+ * schedule for the tile loops based on the tile sizes in data->sizes
+ * and add the result to data->res.
+ */
+static int pw_multi_aff_tile(__isl_take isl_pw_multi_aff *pma, void *user)
+{
+	struct isl_band_tile_data *data = user;
+
+	data->tiled = isl_pw_multi_aff_empty(isl_pw_multi_aff_get_space(pma));
+
+	if (isl_pw_multi_aff_foreach_piece(pma, &multi_aff_tile, data) < 0)
+		goto error;
+
+	isl_pw_multi_aff_free(pma);
+	data->res = isl_union_pw_multi_aff_add_pw_multi_aff(data->res,
+								data->tiled);
+
+	return 0;
+error:
+	isl_pw_multi_aff_free(pma);
+	isl_pw_multi_aff_free(data->tiled);
+	return -1;
+}
+
+/* Given the schedule of a band, construct the corresponding
+ * schedule for the tile loops based on the given tile sizes
+ * and return the result.
+ */
+static isl_union_pw_multi_aff *isl_union_pw_multi_aff_tile(
+	__isl_take isl_union_pw_multi_aff *sched,
+	__isl_keep isl_multi_val *sizes)
+{
+	isl_ctx *ctx;
+	isl_space *space;
+	struct isl_band_tile_data data = { sizes };
+
+	ctx = isl_multi_val_get_ctx(sizes);
+
+	space = isl_union_pw_multi_aff_get_space(sched);
+	data.res = isl_union_pw_multi_aff_empty(space);
+	data.scale = isl_options_get_tile_scale_tile_loops(ctx);
+
+	if (isl_union_pw_multi_aff_foreach_pw_multi_aff(sched,
+						&pw_multi_aff_tile, &data) < 0)
+		goto error;
+
+	isl_union_pw_multi_aff_free(sched);
+	return data.res;
+error:
+	isl_union_pw_multi_aff_free(sched);
+	isl_union_pw_multi_aff_free(data.res);
+	return NULL;
+}
+
+/* Extract the range space from "pma" and store it in *user.
+ * All entries are expected to have the same range space, so we can
+ * stop after extracting the range space from the first entry.
+ */
+static int extract_range_space(__isl_take isl_pw_multi_aff *pma, void *user)
+{
+	isl_space **space = user;
+
+	*space = isl_space_range(isl_pw_multi_aff_get_space(pma));
+	isl_pw_multi_aff_free(pma);
+
+	return -1;
+}
+
+/* Extract the range space of "band".  All entries in band->pma should
+ * have the same range space.  Furthermore, band->pma should have at least
+ * one entry.
+ */
+static __isl_give isl_space *band_get_range_space(__isl_keep isl_band *band)
+{
+	isl_space *space;
+
+	if (!band)
+		return NULL;
+
+	space = NULL;
+	isl_union_pw_multi_aff_foreach_pw_multi_aff(band->pma,
+						&extract_range_space, &space);
+
+	return space;
+}
+
+/* Construct and return an isl_multi_val in the given space, with as entries
+ * the first elements of "v", padded with ones if the size of "v" is smaller
+ * than the dimension of "space".
+ */
+static __isl_give isl_multi_val *multi_val_from_vec(__isl_take isl_space *space,
+	__isl_take isl_vec *v)
+{
+	isl_ctx *ctx;
+	isl_multi_val *mv;
+	int i, n, size;
+
+	if (!space || !v)
+		goto error;
+
+	ctx = isl_space_get_ctx(space);
+	mv = isl_multi_val_zero(space);
+	n = isl_multi_val_dim(mv, isl_dim_set);
+	size = isl_vec_size(v);
+	if (n < size)
+		size = n;
+
+	for (i = 0; i < size; ++i) {
+		isl_val *val = isl_vec_get_element_val(v, i);
+		mv = isl_multi_val_set_val(mv, i, val);
+	}
+	for (i = size; i < n; ++i)
+		mv = isl_multi_val_set_val(mv, i, isl_val_one(ctx));
+
+	isl_vec_free(v);
+	return mv;
+error:
+	isl_space_free(space);
+	isl_vec_free(v);
+	return NULL;
+}
+
+/* Tile the given band using the specified tile sizes.
+ * The given band is modified to refer to the tile loops and
+ * a child band is created to refer to the point loops.
+ * The children of this point loop band are the children
+ * of the original band.
+ *
+ * If the scale tile loops option is set, then the tile loops
+ * are scaled by the tile sizes.  If the shift point loops option is set,
+ * then the point loops are shifted to start at zero.
+ * In particular, these options affect the tile and point loop schedules
+ * as follows
+ *
+ *	scale	shift	original	tile		point
+ *
+ *	0	0	i		floor(i/s)	i
+ *	1	0	i		s * floor(i/s)	i
+ *	0	1	i		floor(i/s)	i - s * floor(i/s)
+ *	1	1	i		s * floor(i/s)	i - s * floor(i/s)
+ */
+int isl_band_tile(__isl_keep isl_band *band, __isl_take isl_vec *sizes)
+{
+	isl_ctx *ctx;
+	isl_band *child;
+	isl_band_list *list = NULL;
+	isl_union_pw_multi_aff *sched = NULL, *child_sched = NULL;
+	isl_space *space;
+	isl_multi_val *mv_sizes;
+
+	if (!band || !sizes)
+		goto error;
+
+	ctx = isl_vec_get_ctx(sizes);
+	child = isl_band_dup(band);
+	list = isl_band_list_alloc(ctx, 1);
+	list = isl_band_list_add(list, child);
+	if (!list)
+		goto error;
+
+	space = band_get_range_space(band);
+	mv_sizes = multi_val_from_vec(space, isl_vec_copy(sizes));
+	sched = isl_union_pw_multi_aff_copy(band->pma);
+	sched = isl_union_pw_multi_aff_tile(sched, mv_sizes);
+
+	child_sched = isl_union_pw_multi_aff_copy(child->pma);
+	if (isl_options_get_tile_shift_point_loops(ctx)) {
+		isl_union_pw_multi_aff *scaled;
+		scaled = isl_union_pw_multi_aff_copy(sched);
+		if (!isl_options_get_tile_scale_tile_loops(ctx))
+			scaled = isl_union_pw_multi_aff_scale_multi_val(scaled,
+						isl_multi_val_copy(mv_sizes));
+		child_sched = isl_union_pw_multi_aff_sub(child_sched, scaled);
+	}
+	isl_multi_val_free(mv_sizes);
+	if (!sched || !child_sched)
+		goto error;
+
+	child->children = band->children;
+	band->children = list;
+	child->parent = band;
+	isl_union_pw_multi_aff_free(band->pma);
+	band->pma = sched;
+	isl_union_pw_multi_aff_free(child->pma);
+	child->pma = child_sched;
+
+	isl_vec_free(sizes);
+	return 0;
+error:
+	isl_union_pw_multi_aff_free(sched);
+	isl_union_pw_multi_aff_free(child_sched);
+	isl_band_list_free(list);
+	isl_vec_free(sizes);
+	return -1;
+}
+
+/* Internal data structure used inside isl_union_pw_multi_aff_drop.
+ *
+ * "pos" is the position of the first dimension to drop.
+ * "n" is the number of dimensions to drop.
+ * "res" accumulates the result.
+ */
+struct isl_union_pw_multi_aff_drop_data {
+	int pos;
+	int n;
+	isl_union_pw_multi_aff *res;
+};
+
+/* Drop the data->n output dimensions starting at data->pos from "pma"
+ * and add the result to data->res.
+ */
+static int pw_multi_aff_drop(__isl_take isl_pw_multi_aff *pma, void *user)
+{
+	struct isl_union_pw_multi_aff_drop_data *data = user;
+
+	pma = isl_pw_multi_aff_drop_dims(pma, isl_dim_out, data->pos, data->n);
+
+	data->res = isl_union_pw_multi_aff_add_pw_multi_aff(data->res, pma);
+	if (!data->res)
+		return -1;
+
+	return 0;
+}
+
+/* Drop the "n" output dimensions starting at "pos" from "sched".
+ */
+static isl_union_pw_multi_aff *isl_union_pw_multi_aff_drop(
+	__isl_take isl_union_pw_multi_aff *sched, int pos, int n)
+{
+	isl_space *space;
+	struct isl_union_pw_multi_aff_drop_data data = { pos, n };
+
+	space = isl_union_pw_multi_aff_get_space(sched);
+	data.res = isl_union_pw_multi_aff_empty(space);
+
+	if (isl_union_pw_multi_aff_foreach_pw_multi_aff(sched,
+						&pw_multi_aff_drop, &data) < 0)
+		data.res = isl_union_pw_multi_aff_free(data.res);
+
+	isl_union_pw_multi_aff_free(sched);
+	return data.res;
+}
+
+/* Drop the "n" dimensions starting at "pos" from "band".
+ */
+static int isl_band_drop(__isl_keep isl_band *band, int pos, int n)
+{
+	int i;
+	isl_union_pw_multi_aff *sched;
+
+	if (!band)
+		return -1;
+	if (n == 0)
+		return 0;
+
+	sched = isl_union_pw_multi_aff_copy(band->pma);
+	sched = isl_union_pw_multi_aff_drop(sched, pos, n);
+	if (!sched)
+		return -1;
+
+	isl_union_pw_multi_aff_free(band->pma);
+	band->pma = sched;
+
+	for (i = pos + n; i < band->n; ++i)
+		band->coincident[i - n] = band->coincident[i];
+
+	band->n -= n;
+
+	return 0;
+}
+
+/* Split the given band into two nested bands, one with the first "pos"
+ * dimensions of "band" and one with the remaining band->n - pos dimensions.
+ */
+int isl_band_split(__isl_keep isl_band *band, int pos)
+{
+	isl_ctx *ctx;
+	isl_band *child;
+	isl_band_list *list;
+
+	if (!band)
+		return -1;
+
+	ctx = isl_band_get_ctx(band);
+
+	if (pos < 0 || pos > band->n)
+		isl_die(ctx, isl_error_invalid, "position out of bounds",
+			return -1);
+
+	child = isl_band_dup(band);
+	if (isl_band_drop(child, 0, pos) < 0)
+		child = isl_band_free(child);
+	list = isl_band_list_alloc(ctx, 1);
+	list = isl_band_list_add(list, child);
+	if (!list)
+		return -1;
+
+	if (isl_band_drop(band, pos, band->n - pos) < 0) {
+		isl_band_list_free(list);
+		return -1;
+	}
+
+	child->children = band->children;
+	band->children = list;
+	child->parent = band;
+
+	return 0;
+}
+
+__isl_give isl_printer *isl_printer_print_band(__isl_take isl_printer *p,
+	__isl_keep isl_band *band)
+{
+	isl_union_map *prefix, *partial, *suffix;
+
+	prefix = isl_band_get_prefix_schedule(band);
+	partial = isl_band_get_partial_schedule(band);
+	suffix = isl_band_get_suffix_schedule(band);
+
+	p = isl_printer_print_str(p, "(");
+	p = isl_printer_print_union_map(p, prefix);
+	p = isl_printer_print_str(p, ",");
+	p = isl_printer_print_union_map(p, partial);
+	p = isl_printer_print_str(p, ",");
+	p = isl_printer_print_union_map(p, suffix);
+	p = isl_printer_print_str(p, ")");
+
+	isl_union_map_free(prefix);
+	isl_union_map_free(partial);
+	isl_union_map_free(suffix);
+
+	return p;
+}
diff --git a/polly/lib/External/isl/isl_band_private.h b/polly/lib/External/isl/isl_band_private.h
new file mode 100644
index 00000000000..8b9c0b77da7
--- /dev/null
+++ b/polly/lib/External/isl/isl_band_private.h
@@ -0,0 +1,47 @@
+#ifndef ISL_BAND_PRIVATE_H
+#define ISL_BAND_PRIVATE_H
+
+#include <isl/aff.h>
+#include <isl/band.h>
+#include <isl/list.h>
+#include <isl/schedule.h>
+
+/* Information about a band within a schedule.
+ *
+ * n is the number of scheduling dimensions within the band.
+ * coincident is an array of length n, indicating whether a scheduling dimension
+ *	satisfies the coincidence constraints in the sense that
+ *	the corresponding dependence distances are zero.
+ * pma is the partial schedule corresponding to this band.
+ * schedule is the schedule that contains this band.
+ * parent is the parent of this band (or NULL if the band is a root).
+ * children are the children of this band (or NULL if the band is a leaf).
+ *
+ * To avoid circular dependences in the reference counting,
+ * the schedule and parent pointers are not reference counted.
+ * isl_band_copy increments the reference count of schedule to ensure
+ * that outside references to the band keep the schedule alive.
+ */
+struct isl_band {
+	int ref;
+
+	int n;
+	int *coincident;
+
+	isl_union_pw_multi_aff *pma;
+	isl_schedule *schedule;
+	isl_band *parent;
+	isl_band_list *children;
+};
+
+#undef EL
+#define EL isl_band
+
+#include <isl_list_templ.h>
+
+__isl_give isl_band *isl_band_alloc(isl_ctx *ctx);
+
+__isl_give isl_union_map *isl_band_list_get_suffix_schedule(
+	__isl_keep isl_band_list *list);
+
+#endif
diff --git a/polly/lib/External/isl/isl_basis_reduction.h b/polly/lib/External/isl/isl_basis_reduction.h
new file mode 100644
index 00000000000..2517c2f1605
--- /dev/null
+++ b/polly/lib/External/isl/isl_basis_reduction.h
@@ -0,0 +1,27 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_BASIS_REDUCTION_H
+#define ISL_BASIS_REDUCTION_H
+
+#include <isl/set.h>
+#include <isl_mat_private.h>
+#include "isl_tab.h"
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_tab *isl_tab_compute_reduced_basis(struct isl_tab *tab);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/isl_bernstein.c b/polly/lib/External/isl/isl_bernstein.c
new file mode 100644
index 00000000000..a34a0d5f947
--- /dev/null
+++ b/polly/lib/External/isl/isl_bernstein.c
@@ -0,0 +1,555 @@
+/*
+ * Copyright 2006-2007 Universiteit Leiden
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2010      INRIA Saclay
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, Leiden Institute of Advanced Computer Science,
+ * Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
+ * and K.U.Leuven, Departement Computerwetenschappen, Celestijnenlaan 200A,
+ * B-3001 Leuven, Belgium
+ * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl/set.h>
+#include <isl_seq.h>
+#include <isl_morph.h>
+#include <isl_factorization.h>
+#include <isl_vertices_private.h>
+#include <isl_polynomial_private.h>
+#include <isl_options_private.h>
+#include <isl_vec_private.h>
+#include <isl_bernstein.h>
+
+struct bernstein_data {
+	enum isl_fold type;
+	isl_qpolynomial *poly;
+	int check_tight;
+
+	isl_cell *cell;
+
+	isl_qpolynomial_fold *fold;
+	isl_qpolynomial_fold *fold_tight;
+	isl_pw_qpolynomial_fold *pwf;
+	isl_pw_qpolynomial_fold *pwf_tight;
+};
+
+static int vertex_is_integral(__isl_keep isl_basic_set *vertex)
+{
+	unsigned nvar;
+	unsigned nparam;
+	int i;
+
+	nvar = isl_basic_set_dim(vertex, isl_dim_set);
+	nparam = isl_basic_set_dim(vertex, isl_dim_param);
+	for (i = 0; i < nvar; ++i) {
+		int r = nvar - 1 - i;
+		if (!isl_int_is_one(vertex->eq[r][1 + nparam + i]) &&
+		    !isl_int_is_negone(vertex->eq[r][1 + nparam + i]))
+			return 0;
+	}
+
+	return 1;
+}
+
+static __isl_give isl_qpolynomial *vertex_coordinate(
+	__isl_keep isl_basic_set *vertex, int i, __isl_take isl_space *dim)
+{
+	unsigned nvar;
+	unsigned nparam;
+	int r;
+	isl_int denom;
+	isl_qpolynomial *v;
+
+	nvar = isl_basic_set_dim(vertex, isl_dim_set);
+	nparam = isl_basic_set_dim(vertex, isl_dim_param);
+	r = nvar - 1 - i;
+
+	isl_int_init(denom);
+	isl_int_set(denom, vertex->eq[r][1 + nparam + i]);
+	isl_assert(vertex->ctx, !isl_int_is_zero(denom), goto error);
+
+	if (isl_int_is_pos(denom))
+		isl_seq_neg(vertex->eq[r], vertex->eq[r],
+				1 + isl_basic_set_total_dim(vertex));
+	else
+		isl_int_neg(denom, denom);
+
+	v = isl_qpolynomial_from_affine(dim, vertex->eq[r], denom);
+	isl_int_clear(denom);
+
+	return v;
+error:
+	isl_space_free(dim);
+	isl_int_clear(denom);
+	return NULL;
+}
+
+/* Check whether the bound associated to the selection "k" is tight,
+ * which is the case if we select exactly one vertex and if that vertex
+ * is integral for all values of the parameters.
+ */
+static int is_tight(int *k, int n, int d, isl_cell *cell)
+{
+	int i;
+
+	for (i = 0; i < n; ++i) {
+		int v;
+		if (k[i] != d) {
+			if (k[i])
+				return 0;
+			continue;
+		}
+		v = cell->ids[n - 1 - i];
+		return vertex_is_integral(cell->vertices->v[v].vertex);
+	}
+
+	return 0;
+}
+
+static void add_fold(__isl_take isl_qpolynomial *b, __isl_keep isl_set *dom,
+	int *k, int n, int d, struct bernstein_data *data)
+{
+	isl_qpolynomial_fold *fold;
+
+	fold = isl_qpolynomial_fold_alloc(data->type, b);
+
+	if (data->check_tight && is_tight(k, n, d, data->cell))
+		data->fold_tight = isl_qpolynomial_fold_fold_on_domain(dom,
+							data->fold_tight, fold);
+	else
+		data->fold = isl_qpolynomial_fold_fold_on_domain(dom,
+							data->fold, fold);
+}
+
+/* Extract the coefficients of the Bernstein base polynomials and store
+ * them in data->fold and data->fold_tight.
+ *
+ * In particular, the coefficient of each monomial
+ * of multi-degree (k[0], k[1], ..., k[n-1]) is divided by the corresponding
+ * multinomial coefficient d!/k[0]! k[1]! ... k[n-1]!
+ *
+ * c[i] contains the coefficient of the selected powers of the first i+1 vars.
+ * multinom[i] contains the partial multinomial coefficient.
+ */
+static void extract_coefficients(isl_qpolynomial *poly,
+	__isl_keep isl_set *dom, struct bernstein_data *data)
+{
+	int i;
+	int d;
+	int n;
+	isl_ctx *ctx;
+	isl_qpolynomial **c = NULL;
+	int *k = NULL;
+	int *left = NULL;
+	isl_vec *multinom = NULL;
+
+	if (!poly)
+		return;
+
+	ctx = isl_qpolynomial_get_ctx(poly);
+	n = isl_qpolynomial_dim(poly, isl_dim_in);
+	d = isl_qpolynomial_degree(poly);
+	isl_assert(ctx, n >= 2, return);
+
+	c = isl_calloc_array(ctx, isl_qpolynomial *, n);
+	k = isl_alloc_array(ctx, int, n);
+	left = isl_alloc_array(ctx, int, n);
+	multinom = isl_vec_alloc(ctx, n);
+	if (!c || !k || !left || !multinom)
+		goto error;
+
+	isl_int_set_si(multinom->el[0], 1);
+	for (k[0] = d; k[0] >= 0; --k[0]) {
+		int i = 1;
+		isl_qpolynomial_free(c[0]);
+		c[0] = isl_qpolynomial_coeff(poly, isl_dim_in, n - 1, k[0]);
+		left[0] = d - k[0];
+		k[1] = -1;
+		isl_int_set(multinom->el[1], multinom->el[0]);
+		while (i > 0) {
+			if (i == n - 1) {
+				int j;
+				isl_space *dim;
+				isl_qpolynomial *b;
+				isl_qpolynomial *f;
+				for (j = 2; j <= left[i - 1]; ++j)
+					isl_int_divexact_ui(multinom->el[i],
+						multinom->el[i], j);
+				b = isl_qpolynomial_coeff(c[i - 1], isl_dim_in,
+					n - 1 - i, left[i - 1]);
+				b = isl_qpolynomial_project_domain_on_params(b);
+				dim = isl_qpolynomial_get_domain_space(b);
+				f = isl_qpolynomial_rat_cst_on_domain(dim, ctx->one,
+					multinom->el[i]);
+				b = isl_qpolynomial_mul(b, f);
+				k[n - 1] = left[n - 2];
+				add_fold(b, dom, k, n, d, data);
+				--i;
+				continue;
+			}
+			if (k[i] >= left[i - 1]) {
+				--i;
+				continue;
+			}
+			++k[i];
+			if (k[i])
+				isl_int_divexact_ui(multinom->el[i],
+					multinom->el[i], k[i]);
+			isl_qpolynomial_free(c[i]);
+			c[i] = isl_qpolynomial_coeff(c[i - 1], isl_dim_in,
+					n - 1 - i, k[i]);
+			left[i] = left[i - 1] - k[i];
+			k[i + 1] = -1;
+			isl_int_set(multinom->el[i + 1], multinom->el[i]);
+			++i;
+		}
+		isl_int_mul_ui(multinom->el[0], multinom->el[0], k[0]);
+	}
+
+	for (i = 0; i < n; ++i)
+		isl_qpolynomial_free(c[i]);
+
+	isl_vec_free(multinom);
+	free(left);
+	free(k);
+	free(c);
+	return;
+error:
+	isl_vec_free(multinom);
+	free(left);
+	free(k);
+	if (c)
+		for (i = 0; i < n; ++i)
+			isl_qpolynomial_free(c[i]);
+	free(c);
+	return;
+}
+
+/* Perform bernstein expansion on the parametric vertices that are active
+ * on "cell".
+ *
+ * data->poly has been homogenized in the calling function.
+ *
+ * We plug in the barycentric coordinates for the set variables
+ *
+ *		\vec x = \sum_i \alpha_i v_i(\vec p)
+ *
+ * and the constant "1 = \sum_i \alpha_i" for the homogeneous dimension.
+ * Next, we extract the coefficients of the Bernstein base polynomials.
+ */
+static int bernstein_coefficients_cell(__isl_take isl_cell *cell, void *user)
+{
+	int i, j;
+	struct bernstein_data *data = (struct bernstein_data *)user;
+	isl_space *dim_param;
+	isl_space *dim_dst;
+	isl_qpolynomial *poly = data->poly;
+	unsigned nvar;
+	int n_vertices;
+	isl_qpolynomial **subs;
+	isl_pw_qpolynomial_fold *pwf;
+	isl_set *dom;
+	isl_ctx *ctx;
+
+	if (!poly)
+		goto error;
+
+	nvar = isl_qpolynomial_dim(poly, isl_dim_in) - 1;
+	n_vertices = cell->n_vertices;
+
+	ctx = isl_qpolynomial_get_ctx(poly);
+	if (n_vertices > nvar + 1 && ctx->opt->bernstein_triangulate)
+		return isl_cell_foreach_simplex(cell,
+					    &bernstein_coefficients_cell, user);
+
+	subs = isl_alloc_array(ctx, isl_qpolynomial *, 1 + nvar);
+	if (!subs)
+		goto error;
+
+	dim_param = isl_basic_set_get_space(cell->dom);
+	dim_dst = isl_qpolynomial_get_domain_space(poly);
+	dim_dst = isl_space_add_dims(dim_dst, isl_dim_set, n_vertices);
+
+	for (i = 0; i < 1 + nvar; ++i)
+		subs[i] = isl_qpolynomial_zero_on_domain(isl_space_copy(dim_dst));
+
+	for (i = 0; i < n_vertices; ++i) {
+		isl_qpolynomial *c;
+		c = isl_qpolynomial_var_on_domain(isl_space_copy(dim_dst), isl_dim_set,
+					1 + nvar + i);
+		for (j = 0; j < nvar; ++j) {
+			int k = cell->ids[i];
+			isl_qpolynomial *v;
+			v = vertex_coordinate(cell->vertices->v[k].vertex, j,
+						isl_space_copy(dim_param));
+			v = isl_qpolynomial_add_dims(v, isl_dim_in,
+							1 + nvar + n_vertices);
+			v = isl_qpolynomial_mul(v, isl_qpolynomial_copy(c));
+			subs[1 + j] = isl_qpolynomial_add(subs[1 + j], v);
+		}
+		subs[0] = isl_qpolynomial_add(subs[0], c);
+	}
+	isl_space_free(dim_dst);
+
+	poly = isl_qpolynomial_copy(poly);
+
+	poly = isl_qpolynomial_add_dims(poly, isl_dim_in, n_vertices);
+	poly = isl_qpolynomial_substitute(poly, isl_dim_in, 0, 1 + nvar, subs);
+	poly = isl_qpolynomial_drop_dims(poly, isl_dim_in, 0, 1 + nvar);
+
+	data->cell = cell;
+	dom = isl_set_from_basic_set(isl_basic_set_copy(cell->dom));
+	data->fold = isl_qpolynomial_fold_empty(data->type, isl_space_copy(dim_param));
+	data->fold_tight = isl_qpolynomial_fold_empty(data->type, dim_param);
+	extract_coefficients(poly, dom, data);
+
+	pwf = isl_pw_qpolynomial_fold_alloc(data->type, isl_set_copy(dom),
+					    data->fold);
+	data->pwf = isl_pw_qpolynomial_fold_fold(data->pwf, pwf);
+	pwf = isl_pw_qpolynomial_fold_alloc(data->type, dom, data->fold_tight);
+	data->pwf_tight = isl_pw_qpolynomial_fold_fold(data->pwf_tight, pwf);
+
+	isl_qpolynomial_free(poly);
+	isl_cell_free(cell);
+	for (i = 0; i < 1 + nvar; ++i)
+		isl_qpolynomial_free(subs[i]);
+	free(subs);
+	return 0;
+error:
+	isl_cell_free(cell);
+	return -1;
+}
+
+/* Base case of applying bernstein expansion.
+ *
+ * We compute the chamber decomposition of the parametric polytope "bset"
+ * and then perform bernstein expansion on the parametric vertices
+ * that are active on each chamber.
+ */
+static __isl_give isl_pw_qpolynomial_fold *bernstein_coefficients_base(
+	__isl_take isl_basic_set *bset,
+	__isl_take isl_qpolynomial *poly, struct bernstein_data *data, int *tight)
+{
+	unsigned nvar;
+	isl_space *dim;
+	isl_pw_qpolynomial_fold *pwf;
+	isl_vertices *vertices;
+	int covers;
+
+	nvar = isl_basic_set_dim(bset, isl_dim_set);
+	if (nvar == 0) {
+		isl_set *dom;
+		isl_qpolynomial_fold *fold;
+
+		fold = isl_qpolynomial_fold_alloc(data->type, poly);
+		dom = isl_set_from_basic_set(bset);
+		if (tight)
+			*tight = 1;
+		pwf = isl_pw_qpolynomial_fold_alloc(data->type, dom, fold);
+		return isl_pw_qpolynomial_fold_project_domain_on_params(pwf);
+	}
+
+	if (isl_qpolynomial_is_zero(poly)) {
+		isl_set *dom;
+		isl_qpolynomial_fold *fold;
+		fold = isl_qpolynomial_fold_alloc(data->type, poly);
+		dom = isl_set_from_basic_set(bset);
+		pwf = isl_pw_qpolynomial_fold_alloc(data->type, dom, fold);
+		if (tight)
+			*tight = 1;
+		return isl_pw_qpolynomial_fold_project_domain_on_params(pwf);
+	}
+
+	dim = isl_basic_set_get_space(bset);
+	dim = isl_space_params(dim);
+	dim = isl_space_from_domain(dim);
+	dim = isl_space_add_dims(dim, isl_dim_set, 1);
+	data->pwf = isl_pw_qpolynomial_fold_zero(isl_space_copy(dim), data->type);
+	data->pwf_tight = isl_pw_qpolynomial_fold_zero(dim, data->type);
+	data->poly = isl_qpolynomial_homogenize(isl_qpolynomial_copy(poly));
+	vertices = isl_basic_set_compute_vertices(bset);
+	isl_vertices_foreach_disjoint_cell(vertices,
+		&bernstein_coefficients_cell, data);
+	isl_vertices_free(vertices);
+	isl_qpolynomial_free(data->poly);
+
+	isl_basic_set_free(bset);
+	isl_qpolynomial_free(poly);
+
+	covers = isl_pw_qpolynomial_fold_covers(data->pwf_tight, data->pwf);
+	if (covers < 0)
+		goto error;
+
+	if (tight)
+		*tight = covers;
+
+	if (covers) {
+		isl_pw_qpolynomial_fold_free(data->pwf);
+		return data->pwf_tight;
+	}
+
+	data->pwf = isl_pw_qpolynomial_fold_fold(data->pwf, data->pwf_tight);
+
+	return data->pwf;
+error:
+	isl_pw_qpolynomial_fold_free(data->pwf_tight);
+	isl_pw_qpolynomial_fold_free(data->pwf);
+	return NULL;
+}
+
+/* Apply bernstein expansion recursively by working in on len[i]
+ * set variables at a time, with i ranging from n_group - 1 to 0.
+ */
+static __isl_give isl_pw_qpolynomial_fold *bernstein_coefficients_recursive(
+	__isl_take isl_pw_qpolynomial *pwqp,
+	int n_group, int *len, struct bernstein_data *data, int *tight)
+{
+	int i;
+	unsigned nparam;
+	unsigned nvar;
+	isl_pw_qpolynomial_fold *pwf;
+
+	if (!pwqp)
+		return NULL;
+
+	nparam = isl_pw_qpolynomial_dim(pwqp, isl_dim_param);
+	nvar = isl_pw_qpolynomial_dim(pwqp, isl_dim_in);
+
+	pwqp = isl_pw_qpolynomial_move_dims(pwqp, isl_dim_param, nparam,
+					isl_dim_in, 0, nvar - len[n_group - 1]);
+	pwf = isl_pw_qpolynomial_bound(pwqp, data->type, tight);
+
+	for (i = n_group - 2; i >= 0; --i) {
+		nparam = isl_pw_qpolynomial_fold_dim(pwf, isl_dim_param);
+		pwf = isl_pw_qpolynomial_fold_move_dims(pwf, isl_dim_in, 0,
+				isl_dim_param, nparam - len[i], len[i]);
+		if (tight && !*tight)
+			tight = NULL;
+		pwf = isl_pw_qpolynomial_fold_bound(pwf, tight);
+	}
+
+	return pwf;
+}
+
+static __isl_give isl_pw_qpolynomial_fold *bernstein_coefficients_factors(
+	__isl_take isl_basic_set *bset,
+	__isl_take isl_qpolynomial *poly, struct bernstein_data *data, int *tight)
+{
+	isl_factorizer *f;
+	isl_set *set;
+	isl_pw_qpolynomial *pwqp;
+	isl_pw_qpolynomial_fold *pwf;
+
+	f = isl_basic_set_factorizer(bset);
+	if (!f)
+		goto error;
+	if (f->n_group == 0) {
+		isl_factorizer_free(f);
+		return  bernstein_coefficients_base(bset, poly, data, tight);
+	}
+
+	set = isl_set_from_basic_set(bset);
+	pwqp = isl_pw_qpolynomial_alloc(set, poly);
+	pwqp = isl_pw_qpolynomial_morph_domain(pwqp, isl_morph_copy(f->morph));
+
+	pwf = bernstein_coefficients_recursive(pwqp, f->n_group, f->len, data,
+						tight);
+
+	isl_factorizer_free(f);
+
+	return pwf;
+error:
+	isl_basic_set_free(bset);
+	isl_qpolynomial_free(poly);
+	return NULL;
+}
+
+static __isl_give isl_pw_qpolynomial_fold *bernstein_coefficients_full_recursive(
+	__isl_take isl_basic_set *bset,
+	__isl_take isl_qpolynomial *poly, struct bernstein_data *data, int *tight)
+{
+	int i;
+	int *len;
+	unsigned nvar;
+	isl_pw_qpolynomial_fold *pwf;
+	isl_set *set;
+	isl_pw_qpolynomial *pwqp;
+
+	if (!bset || !poly)
+		goto error;
+
+	nvar = isl_basic_set_dim(bset, isl_dim_set);
+	
+	len = isl_alloc_array(bset->ctx, int, nvar);
+	if (nvar && !len)
+		goto error;
+
+	for (i = 0; i < nvar; ++i)
+		len[i] = 1;
+
+	set = isl_set_from_basic_set(bset);
+	pwqp = isl_pw_qpolynomial_alloc(set, poly);
+
+	pwf = bernstein_coefficients_recursive(pwqp, nvar, len, data, tight);
+
+	free(len);
+
+	return pwf;
+error:
+	isl_basic_set_free(bset);
+	isl_qpolynomial_free(poly);
+	return NULL;
+}
+
+/* Compute a bound on the polynomial defined over the parametric polytope
+ * using bernstein expansion and store the result
+ * in bound->pwf and bound->pwf_tight.
+ *
+ * If bernstein_recurse is set to ISL_BERNSTEIN_FACTORS, we check if
+ * the polytope can be factorized and apply bernstein expansion recursively
+ * on the factors.
+ * If bernstein_recurse is set to ISL_BERNSTEIN_INTERVALS, we apply
+ * bernstein expansion recursively on each dimension.
+ * Otherwise, we apply bernstein expansion on the entire polytope.
+ */
+int isl_qpolynomial_bound_on_domain_bernstein(__isl_take isl_basic_set *bset,
+	__isl_take isl_qpolynomial *poly, struct isl_bound *bound)
+{
+	struct bernstein_data data;
+	isl_pw_qpolynomial_fold *pwf;
+	unsigned nvar;
+	int tight = 0;
+	int *tp = bound->check_tight ? &tight : NULL;
+
+	if (!bset || !poly)
+		goto error;
+
+	data.type = bound->type;
+	data.check_tight = bound->check_tight;
+
+	nvar = isl_basic_set_dim(bset, isl_dim_set);
+
+	if (bset->ctx->opt->bernstein_recurse & ISL_BERNSTEIN_FACTORS)
+		pwf = bernstein_coefficients_factors(bset, poly, &data, tp);
+	else if (nvar > 1 &&
+	    (bset->ctx->opt->bernstein_recurse & ISL_BERNSTEIN_INTERVALS))
+		pwf = bernstein_coefficients_full_recursive(bset, poly, &data, tp);
+	else
+		pwf = bernstein_coefficients_base(bset, poly, &data, tp);
+
+	if (tight)
+		bound->pwf_tight = isl_pw_qpolynomial_fold_fold(bound->pwf_tight, pwf);
+	else
+		bound->pwf = isl_pw_qpolynomial_fold_fold(bound->pwf, pwf);
+
+	return 0;
+error:
+	isl_basic_set_free(bset);
+	isl_qpolynomial_free(poly);
+	return -1;
+}
diff --git a/polly/lib/External/isl/isl_bernstein.h b/polly/lib/External/isl/isl_bernstein.h
new file mode 100644
index 00000000000..7694b04e566
--- /dev/null
+++ b/polly/lib/External/isl/isl_bernstein.h
@@ -0,0 +1,4 @@
+#include <isl_bound.h>
+
+int isl_qpolynomial_bound_on_domain_bernstein(__isl_take isl_basic_set *bset,
+	__isl_take isl_qpolynomial *poly, struct isl_bound *bound);
diff --git a/polly/lib/External/isl/isl_blk.c b/polly/lib/External/isl/isl_blk.c
new file mode 100644
index 00000000000..d9f80dbb032
--- /dev/null
+++ b/polly/lib/External/isl/isl_blk.c
@@ -0,0 +1,134 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <isl_blk.h>
+#include <isl_ctx_private.h>
+
+/* The maximal number of cache misses before first element is evicted */
+#define ISL_BLK_MAX_MISS	100
+
+struct isl_blk isl_blk_empty()
+{
+	struct isl_blk block;
+	block.size = 0;
+	block.data = NULL;
+	return block;
+}
+
+static int isl_blk_is_empty(struct isl_blk block)
+{
+	return block.size == 0 && block.data == NULL;
+}
+
+static struct isl_blk isl_blk_error()
+{
+	struct isl_blk block;
+	block.size = -1;
+	block.data = NULL;
+	return block;
+}
+
+int isl_blk_is_error(struct isl_blk block)
+{
+	return block.size == -1 && block.data == NULL;
+}
+
+static struct isl_blk extend(struct isl_ctx *ctx, struct isl_blk block,
+				size_t new_n)
+{
+	int i;
+	isl_int *p;
+
+	if (block.size >= new_n)
+		return block;
+
+	p = block.data;
+	block.data = isl_realloc_array(ctx, block.data, isl_int, new_n);
+	if (!block.data) {
+		free(p);
+		return isl_blk_error();
+	}
+
+	for (i = block.size; i < new_n; ++i)
+		isl_int_init(block.data[i]);
+	block.size = new_n;
+
+	return block;
+}
+
+static void isl_blk_free_force(struct isl_ctx *ctx, struct isl_blk block)
+{
+	int i;
+
+	for (i = 0; i < block.size; ++i)
+		isl_int_clear(block.data[i]);
+	free(block.data);
+}
+
+struct isl_blk isl_blk_alloc(struct isl_ctx *ctx, size_t n)
+{
+	int i;
+	struct isl_blk block;
+
+	block = isl_blk_empty();
+	if (n && ctx->n_cached) {
+		int best = 0;
+		for (i = 1; ctx->cache[best].size != n && i < ctx->n_cached; ++i) {
+			if (ctx->cache[best].size < n) {
+				if (ctx->cache[i].size > ctx->cache[best].size)
+					best = i;
+			} else if (ctx->cache[i].size >= n &&
+				   ctx->cache[i].size < ctx->cache[best].size)
+					best = i;
+		}
+		if (ctx->cache[best].size < 2 * n + 100) {
+			block = ctx->cache[best];
+			if (--ctx->n_cached != best)
+				ctx->cache[best] = ctx->cache[ctx->n_cached];
+			if (best == 0)
+				ctx->n_miss = 0;
+		} else if (ctx->n_miss++ >= ISL_BLK_MAX_MISS) {
+			isl_blk_free_force(ctx, ctx->cache[0]);
+			if (--ctx->n_cached != 0)
+				ctx->cache[0] = ctx->cache[ctx->n_cached];
+			ctx->n_miss = 0;
+		}
+	}
+
+	return extend(ctx, block, n);
+}
+
+struct isl_blk isl_blk_extend(struct isl_ctx *ctx, struct isl_blk block,
+				size_t new_n)
+{
+	if (isl_blk_is_empty(block))
+		return isl_blk_alloc(ctx, new_n);
+
+	return extend(ctx, block, new_n);
+}
+
+void isl_blk_free(struct isl_ctx *ctx, struct isl_blk block)
+{
+	if (isl_blk_is_empty(block) || isl_blk_is_error(block))
+		return;
+
+	if (ctx->n_cached < ISL_BLK_CACHE_SIZE)
+		ctx->cache[ctx->n_cached++] = block;
+	else
+		isl_blk_free_force(ctx, block);
+}
+
+void isl_blk_clear_cache(struct isl_ctx *ctx)
+{
+	int i;
+
+	for (i = 0; i < ctx->n_cached; ++i)
+		isl_blk_free_force(ctx, ctx->cache[i]);
+	ctx->n_cached = 0;
+}
diff --git a/polly/lib/External/isl/isl_blk.h b/polly/lib/External/isl/isl_blk.h
new file mode 100644
index 00000000000..7756e010ec2
--- /dev/null
+++ b/polly/lib/External/isl/isl_blk.h
@@ -0,0 +1,40 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_BLK_H
+#define ISL_BLK_H
+
+#include <isl_int.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_blk {
+	size_t size;
+	isl_int *data;
+};
+
+#define ISL_BLK_CACHE_SIZE	20
+
+struct isl_ctx;
+
+struct isl_blk isl_blk_alloc(struct isl_ctx *ctx, size_t n);
+struct isl_blk isl_blk_empty(void);
+int isl_blk_is_error(struct isl_blk block);
+struct isl_blk isl_blk_extend(struct isl_ctx *ctx, struct isl_blk block,
+				size_t new_n);
+void isl_blk_free(struct isl_ctx *ctx, struct isl_blk block);
+void isl_blk_clear_cache(struct isl_ctx *ctx);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/isl_bound.c b/polly/lib/External/isl/isl_bound.c
new file mode 100644
index 00000000000..04c76a0ffb2
--- /dev/null
+++ b/polly/lib/External/isl/isl_bound.c
@@ -0,0 +1,331 @@
+/*
+ * Copyright 2010      INRIA Saclay
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France 
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl_bound.h>
+#include <isl_bernstein.h>
+#include <isl_range.h>
+#include <isl_polynomial_private.h>
+#include <isl_options_private.h>
+
+/* Compute a bound on the polynomial defined over the parametric polytope
+ * using either range propagation or bernstein expansion and
+ * store the result in bound->pwf and bound->pwf_tight.
+ * Since bernstein expansion requires bounded domains, we apply
+ * range propagation on unbounded domains.  Otherwise, we respect the choice
+ * of the user.
+ */
+static int compressed_guarded_poly_bound(__isl_take isl_basic_set *bset,
+	__isl_take isl_qpolynomial *poly, void *user)
+{
+	struct isl_bound *bound = (struct isl_bound *)user;
+	int bounded;
+
+	if (!bset || !poly)
+		goto error;
+
+	if (bset->ctx->opt->bound == ISL_BOUND_RANGE)
+		return isl_qpolynomial_bound_on_domain_range(bset, poly, bound);
+
+	bounded = isl_basic_set_is_bounded(bset);
+	if (bounded < 0)
+		goto error;
+	if (bounded)
+		return isl_qpolynomial_bound_on_domain_bernstein(bset, poly, bound);
+	else
+		return isl_qpolynomial_bound_on_domain_range(bset, poly, bound);
+error:
+	isl_basic_set_free(bset);
+	isl_qpolynomial_free(poly);
+	return -1;
+}
+
+static int unwrapped_guarded_poly_bound(__isl_take isl_basic_set *bset,
+	__isl_take isl_qpolynomial *poly, void *user)
+{
+	struct isl_bound *bound = (struct isl_bound *)user;
+	isl_pw_qpolynomial_fold *top_pwf;
+	isl_pw_qpolynomial_fold *top_pwf_tight;
+	isl_space *dim;
+	isl_morph *morph;
+	int r;
+
+	bset = isl_basic_set_detect_equalities(bset);
+
+	if (!bset)
+		goto error;
+
+	if (bset->n_eq == 0)
+		return compressed_guarded_poly_bound(bset, poly, user);
+
+	morph = isl_basic_set_full_compression(bset);
+
+	bset = isl_morph_basic_set(isl_morph_copy(morph), bset);
+	poly = isl_qpolynomial_morph_domain(poly, isl_morph_copy(morph));
+
+	dim = isl_morph_get_ran_space(morph);
+	dim = isl_space_params(dim);
+
+	top_pwf = bound->pwf;
+	top_pwf_tight = bound->pwf_tight;
+
+	dim = isl_space_from_domain(dim);
+	dim = isl_space_add_dims(dim, isl_dim_out, 1);
+	bound->pwf = isl_pw_qpolynomial_fold_zero(isl_space_copy(dim),
+						  bound->type);
+	bound->pwf_tight = isl_pw_qpolynomial_fold_zero(dim, bound->type);
+
+	r = compressed_guarded_poly_bound(bset, poly, user);
+
+	morph = isl_morph_dom_params(morph);
+	morph = isl_morph_ran_params(morph);
+	morph = isl_morph_inverse(morph);
+
+	bound->pwf = isl_pw_qpolynomial_fold_morph_domain(bound->pwf,
+							isl_morph_copy(morph));
+	bound->pwf_tight = isl_pw_qpolynomial_fold_morph_domain(
+						bound->pwf_tight, morph);
+
+	bound->pwf = isl_pw_qpolynomial_fold_fold(top_pwf, bound->pwf);
+	bound->pwf_tight = isl_pw_qpolynomial_fold_fold(top_pwf_tight,
+							bound->pwf_tight);
+
+	return r;
+error:
+	isl_basic_set_free(bset);
+	isl_qpolynomial_free(poly);
+	return -1;
+}
+
+static int guarded_poly_bound(__isl_take isl_basic_set *bset,
+	__isl_take isl_qpolynomial *poly, void *user)
+{
+	struct isl_bound *bound = (struct isl_bound *)user;
+	isl_space *dim;
+	isl_pw_qpolynomial_fold *top_pwf;
+	isl_pw_qpolynomial_fold *top_pwf_tight;
+	int nparam;
+	int n_in;
+	int r;
+
+	if (!bound->wrapping)
+		return unwrapped_guarded_poly_bound(bset, poly, user);
+
+	nparam = isl_space_dim(bound->dim, isl_dim_param);
+	n_in = isl_space_dim(bound->dim, isl_dim_in);
+
+	bset = isl_basic_set_move_dims(bset, isl_dim_param, nparam,
+					isl_dim_set, 0, n_in);
+	poly = isl_qpolynomial_move_dims(poly, isl_dim_param, nparam,
+					isl_dim_in, 0, n_in);
+
+	dim = isl_basic_set_get_space(bset);
+	dim = isl_space_params(dim);
+
+	top_pwf = bound->pwf;
+	top_pwf_tight = bound->pwf_tight;
+
+	dim = isl_space_from_domain(dim);
+	dim = isl_space_add_dims(dim, isl_dim_out, 1);
+	bound->pwf = isl_pw_qpolynomial_fold_zero(isl_space_copy(dim),
+						  bound->type);
+	bound->pwf_tight = isl_pw_qpolynomial_fold_zero(dim, bound->type);
+
+	r = unwrapped_guarded_poly_bound(bset, poly, user);
+
+	bound->pwf = isl_pw_qpolynomial_fold_reset_space(bound->pwf,
+						    isl_space_copy(bound->dim));
+	bound->pwf_tight = isl_pw_qpolynomial_fold_reset_space(bound->pwf_tight,
+						    isl_space_copy(bound->dim));
+
+	bound->pwf = isl_pw_qpolynomial_fold_fold(top_pwf, bound->pwf);
+	bound->pwf_tight = isl_pw_qpolynomial_fold_fold(top_pwf_tight,
+							bound->pwf_tight);
+
+	return r;
+}
+
+static int guarded_qp(__isl_take isl_qpolynomial *qp, void *user)
+{
+	struct isl_bound *bound = (struct isl_bound *)user;
+	int r;
+
+	r = isl_qpolynomial_as_polynomial_on_domain(qp, bound->bset,
+						    &guarded_poly_bound, user);
+	isl_qpolynomial_free(qp);
+	return r;
+}
+
+static int basic_guarded_fold(__isl_take isl_basic_set *bset, void *user)
+{
+	struct isl_bound *bound = (struct isl_bound *)user;
+	int r;
+
+	bound->bset = bset;
+	r = isl_qpolynomial_fold_foreach_qpolynomial(bound->fold,
+							&guarded_qp, user);
+	isl_basic_set_free(bset);
+	return r;
+}
+
+static int guarded_fold(__isl_take isl_set *set,
+	__isl_take isl_qpolynomial_fold *fold, void *user)
+{
+	struct isl_bound *bound = (struct isl_bound *)user;
+
+	if (!set || !fold)
+		goto error;
+
+	set = isl_set_make_disjoint(set);
+
+	bound->fold = fold;
+	bound->type = isl_qpolynomial_fold_get_type(fold);
+
+	if (isl_set_foreach_basic_set(set, &basic_guarded_fold, bound) < 0)
+		goto error;
+
+	isl_set_free(set);
+	isl_qpolynomial_fold_free(fold);
+
+	return 0;
+error:
+	isl_set_free(set);
+	isl_qpolynomial_fold_free(fold);
+	return -1;
+}
+
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_bound(
+	__isl_take isl_pw_qpolynomial_fold *pwf, int *tight)
+{
+	unsigned nvar;
+	struct isl_bound bound;
+	int covers;
+
+	if (!pwf)
+		return NULL;
+
+	bound.dim = isl_pw_qpolynomial_fold_get_domain_space(pwf);
+
+	bound.wrapping = isl_space_is_wrapping(bound.dim);
+	if (bound.wrapping)
+		bound.dim = isl_space_unwrap(bound.dim);
+	nvar = isl_space_dim(bound.dim, isl_dim_out);
+	bound.dim = isl_space_domain(bound.dim);
+	bound.dim = isl_space_from_domain(bound.dim);
+	bound.dim = isl_space_add_dims(bound.dim, isl_dim_out, 1);
+
+	if (nvar == 0) {
+		if (tight)
+			*tight = 1;
+		return isl_pw_qpolynomial_fold_reset_space(pwf, bound.dim);
+	}
+
+	if (isl_pw_qpolynomial_fold_is_zero(pwf)) {
+		enum isl_fold type = pwf->type;
+		isl_pw_qpolynomial_fold_free(pwf);
+		if (tight)
+			*tight = 1;
+		return isl_pw_qpolynomial_fold_zero(bound.dim, type);
+	}
+
+	bound.pwf = isl_pw_qpolynomial_fold_zero(isl_space_copy(bound.dim),
+							pwf->type);
+	bound.pwf_tight = isl_pw_qpolynomial_fold_zero(isl_space_copy(bound.dim),
+							pwf->type);
+	bound.check_tight = !!tight;
+
+	if (isl_pw_qpolynomial_fold_foreach_lifted_piece(pwf,
+							guarded_fold, &bound) < 0)
+		goto error;
+
+	covers = isl_pw_qpolynomial_fold_covers(bound.pwf_tight, bound.pwf);
+	if (covers < 0)
+		goto error;
+
+	if (tight)
+		*tight = covers;
+
+	isl_space_free(bound.dim);
+	isl_pw_qpolynomial_fold_free(pwf);
+
+	if (covers) {
+		isl_pw_qpolynomial_fold_free(bound.pwf);
+		return bound.pwf_tight;
+	}
+
+	bound.pwf = isl_pw_qpolynomial_fold_fold(bound.pwf, bound.pwf_tight);
+
+	return bound.pwf;
+error:
+	isl_pw_qpolynomial_fold_free(bound.pwf_tight);
+	isl_pw_qpolynomial_fold_free(bound.pwf);
+	isl_pw_qpolynomial_fold_free(pwf);
+	isl_space_free(bound.dim);
+	return NULL;
+}
+
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_bound(
+	__isl_take isl_pw_qpolynomial *pwqp, enum isl_fold type, int *tight)
+{
+	isl_pw_qpolynomial_fold *pwf;
+
+	pwf = isl_pw_qpolynomial_fold_from_pw_qpolynomial(type, pwqp);
+	return isl_pw_qpolynomial_fold_bound(pwf, tight);
+}
+
+struct isl_union_bound_data {
+	enum isl_fold type;
+	int tight;
+	isl_union_pw_qpolynomial_fold *res;
+};
+
+static int bound_pw(__isl_take isl_pw_qpolynomial *pwqp, void *user)
+{
+	struct isl_union_bound_data *data = user;
+	isl_pw_qpolynomial_fold *pwf;
+
+	pwf = isl_pw_qpolynomial_bound(pwqp, data->type,
+					data->tight ? &data->tight : NULL);
+	data->res = isl_union_pw_qpolynomial_fold_fold_pw_qpolynomial_fold(
+								data->res, pwf);
+
+	return 0;
+}
+
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_bound(
+	__isl_take isl_union_pw_qpolynomial *upwqp,
+	enum isl_fold type, int *tight)
+{
+	isl_space *dim;
+	struct isl_union_bound_data data = { type, 1, NULL };
+
+	if (!upwqp)
+		return NULL;
+
+	if (!tight)
+		data.tight = 0;
+
+	dim = isl_union_pw_qpolynomial_get_space(upwqp);
+	data.res = isl_union_pw_qpolynomial_fold_zero(dim, type);
+	if (isl_union_pw_qpolynomial_foreach_pw_qpolynomial(upwqp,
+						    &bound_pw, &data) < 0)
+		goto error;
+
+	isl_union_pw_qpolynomial_free(upwqp);
+	if (tight)
+		*tight = data.tight;
+
+	return data.res;
+error:
+	isl_union_pw_qpolynomial_free(upwqp);
+	isl_union_pw_qpolynomial_fold_free(data.res);
+	return NULL;
+}
diff --git a/polly/lib/External/isl/isl_bound.h b/polly/lib/External/isl/isl_bound.h
new file mode 100644
index 00000000000..1a9d390eab9
--- /dev/null
+++ b/polly/lib/External/isl/isl_bound.h
@@ -0,0 +1,20 @@
+#ifndef ISL_BOUND_H
+#define ISL_BOUND_H
+
+#include <isl/polynomial.h>
+
+struct isl_bound {
+	/* input */
+	int check_tight;
+	int wrapping;
+	enum isl_fold type;
+	isl_space *dim;
+	isl_basic_set *bset;
+	isl_qpolynomial_fold *fold;
+
+	/* output */
+	isl_pw_qpolynomial_fold *pwf;
+	isl_pw_qpolynomial_fold *pwf_tight;
+};
+
+#endif
diff --git a/polly/lib/External/isl/isl_coalesce.c b/polly/lib/External/isl/isl_coalesce.c
new file mode 100644
index 00000000000..2899e03b2d3
--- /dev/null
+++ b/polly/lib/External/isl/isl_coalesce.c
@@ -0,0 +1,1745 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2010      INRIA Saclay
+ * Copyright 2012-2013 Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France 
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+#include "isl_map_private.h"
+#include <isl_seq.h>
+#include <isl/options.h>
+#include "isl_tab.h"
+#include <isl_mat_private.h>
+#include <isl_local_space_private.h>
+#include <isl_vec_private.h>
+
+#define STATUS_ERROR		-1
+#define STATUS_REDUNDANT	 1
+#define STATUS_VALID	 	 2
+#define STATUS_SEPARATE	 	 3
+#define STATUS_CUT	 	 4
+#define STATUS_ADJ_EQ	 	 5
+#define STATUS_ADJ_INEQ	 	 6
+
+static int status_in(isl_int *ineq, struct isl_tab *tab)
+{
+	enum isl_ineq_type type = isl_tab_ineq_type(tab, ineq);
+	switch (type) {
+	default:
+	case isl_ineq_error:		return STATUS_ERROR;
+	case isl_ineq_redundant:	return STATUS_VALID;
+	case isl_ineq_separate:		return STATUS_SEPARATE;
+	case isl_ineq_cut:		return STATUS_CUT;
+	case isl_ineq_adj_eq:		return STATUS_ADJ_EQ;
+	case isl_ineq_adj_ineq:		return STATUS_ADJ_INEQ;
+	}
+}
+
+/* Compute the position of the equalities of basic map "bmap_i"
+ * with respect to the basic map represented by "tab_j".
+ * The resulting array has twice as many entries as the number
+ * of equalities corresponding to the two inequalties to which
+ * each equality corresponds.
+ */
+static int *eq_status_in(__isl_keep isl_basic_map *bmap_i,
+	struct isl_tab *tab_j)
+{
+	int k, l;
+	int *eq = isl_calloc_array(bmap_i->ctx, int, 2 * bmap_i->n_eq);
+	unsigned dim;
+
+	if (!eq)
+		return NULL;
+
+	dim = isl_basic_map_total_dim(bmap_i);
+	for (k = 0; k < bmap_i->n_eq; ++k) {
+		for (l = 0; l < 2; ++l) {
+			isl_seq_neg(bmap_i->eq[k], bmap_i->eq[k], 1+dim);
+			eq[2 * k + l] = status_in(bmap_i->eq[k], tab_j);
+			if (eq[2 * k + l] == STATUS_ERROR)
+				goto error;
+		}
+		if (eq[2 * k] == STATUS_SEPARATE ||
+		    eq[2 * k + 1] == STATUS_SEPARATE)
+			break;
+	}
+
+	return eq;
+error:
+	free(eq);
+	return NULL;
+}
+
+/* Compute the position of the inequalities of basic map "bmap_i"
+ * (also represented by "tab_i", if not NULL) with respect to the basic map
+ * represented by "tab_j".
+ */
+static int *ineq_status_in(__isl_keep isl_basic_map *bmap_i,
+	struct isl_tab *tab_i, struct isl_tab *tab_j)
+{
+	int k;
+	unsigned n_eq = bmap_i->n_eq;
+	int *ineq = isl_calloc_array(bmap_i->ctx, int, bmap_i->n_ineq);
+
+	if (!ineq)
+		return NULL;
+
+	for (k = 0; k < bmap_i->n_ineq; ++k) {
+		if (tab_i && isl_tab_is_redundant(tab_i, n_eq + k)) {
+			ineq[k] = STATUS_REDUNDANT;
+			continue;
+		}
+		ineq[k] = status_in(bmap_i->ineq[k], tab_j);
+		if (ineq[k] == STATUS_ERROR)
+			goto error;
+		if (ineq[k] == STATUS_SEPARATE)
+			break;
+	}
+
+	return ineq;
+error:
+	free(ineq);
+	return NULL;
+}
+
+static int any(int *con, unsigned len, int status)
+{
+	int i;
+
+	for (i = 0; i < len ; ++i)
+		if (con[i] == status)
+			return 1;
+	return 0;
+}
+
+static int count(int *con, unsigned len, int status)
+{
+	int i;
+	int c = 0;
+
+	for (i = 0; i < len ; ++i)
+		if (con[i] == status)
+			c++;
+	return c;
+}
+
+static int all(int *con, unsigned len, int status)
+{
+	int i;
+
+	for (i = 0; i < len ; ++i) {
+		if (con[i] == STATUS_REDUNDANT)
+			continue;
+		if (con[i] != status)
+			return 0;
+	}
+	return 1;
+}
+
+static void drop(struct isl_map *map, int i, struct isl_tab **tabs)
+{
+	isl_basic_map_free(map->p[i]);
+	isl_tab_free(tabs[i]);
+
+	if (i != map->n - 1) {
+		map->p[i] = map->p[map->n - 1];
+		tabs[i] = tabs[map->n - 1];
+	}
+	tabs[map->n - 1] = NULL;
+	map->n--;
+}
+
+/* Replace the pair of basic maps i and j by the basic map bounded
+ * by the valid constraints in both basic maps and the constraint
+ * in extra (if not NULL).
+ */
+static int fuse(struct isl_map *map, int i, int j,
+	struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j,
+	__isl_keep isl_mat *extra)
+{
+	int k, l;
+	struct isl_basic_map *fused = NULL;
+	struct isl_tab *fused_tab = NULL;
+	unsigned total = isl_basic_map_total_dim(map->p[i]);
+	unsigned extra_rows = extra ? extra->n_row : 0;
+
+	fused = isl_basic_map_alloc_space(isl_space_copy(map->p[i]->dim),
+			map->p[i]->n_div,
+			map->p[i]->n_eq + map->p[j]->n_eq,
+			map->p[i]->n_ineq + map->p[j]->n_ineq + extra_rows);
+	if (!fused)
+		goto error;
+
+	for (k = 0; k < map->p[i]->n_eq; ++k) {
+		if (eq_i && (eq_i[2 * k] != STATUS_VALID ||
+			     eq_i[2 * k + 1] != STATUS_VALID))
+			continue;
+		l = isl_basic_map_alloc_equality(fused);
+		if (l < 0)
+			goto error;
+		isl_seq_cpy(fused->eq[l], map->p[i]->eq[k], 1 + total);
+	}
+
+	for (k = 0; k < map->p[j]->n_eq; ++k) {
+		if (eq_j && (eq_j[2 * k] != STATUS_VALID ||
+			     eq_j[2 * k + 1] != STATUS_VALID))
+			continue;
+		l = isl_basic_map_alloc_equality(fused);
+		if (l < 0)
+			goto error;
+		isl_seq_cpy(fused->eq[l], map->p[j]->eq[k], 1 + total);
+	}
+
+	for (k = 0; k < map->p[i]->n_ineq; ++k) {
+		if (ineq_i[k] != STATUS_VALID)
+			continue;
+		l = isl_basic_map_alloc_inequality(fused);
+		if (l < 0)
+			goto error;
+		isl_seq_cpy(fused->ineq[l], map->p[i]->ineq[k], 1 + total);
+	}
+
+	for (k = 0; k < map->p[j]->n_ineq; ++k) {
+		if (ineq_j[k] != STATUS_VALID)
+			continue;
+		l = isl_basic_map_alloc_inequality(fused);
+		if (l < 0)
+			goto error;
+		isl_seq_cpy(fused->ineq[l], map->p[j]->ineq[k], 1 + total);
+	}
+
+	for (k = 0; k < map->p[i]->n_div; ++k) {
+		int l = isl_basic_map_alloc_div(fused);
+		if (l < 0)
+			goto error;
+		isl_seq_cpy(fused->div[l], map->p[i]->div[k], 1 + 1 + total);
+	}
+
+	for (k = 0; k < extra_rows; ++k) {
+		l = isl_basic_map_alloc_inequality(fused);
+		if (l < 0)
+			goto error;
+		isl_seq_cpy(fused->ineq[l], extra->row[k], 1 + total);
+	}
+
+	fused = isl_basic_map_gauss(fused, NULL);
+	ISL_F_SET(fused, ISL_BASIC_MAP_FINAL);
+	if (ISL_F_ISSET(map->p[i], ISL_BASIC_MAP_RATIONAL) &&
+	    ISL_F_ISSET(map->p[j], ISL_BASIC_MAP_RATIONAL))
+		ISL_F_SET(fused, ISL_BASIC_MAP_RATIONAL);
+
+	fused_tab = isl_tab_from_basic_map(fused, 0);
+	if (isl_tab_detect_redundant(fused_tab) < 0)
+		goto error;
+
+	isl_basic_map_free(map->p[i]);
+	map->p[i] = fused;
+	isl_tab_free(tabs[i]);
+	tabs[i] = fused_tab;
+	drop(map, j, tabs);
+
+	return 1;
+error:
+	isl_tab_free(fused_tab);
+	isl_basic_map_free(fused);
+	return -1;
+}
+
+/* Given a pair of basic maps i and j such that all constraints are either
+ * "valid" or "cut", check if the facets corresponding to the "cut"
+ * constraints of i lie entirely within basic map j.
+ * If so, replace the pair by the basic map consisting of the valid
+ * constraints in both basic maps.
+ *
+ * To see that we are not introducing any extra points, call the
+ * two basic maps A and B and the resulting map U and let x
+ * be an element of U \setminus ( A \cup B ).
+ * Then there is a pair of cut constraints c_1 and c_2 in A and B such that x
+ * violates them.  Let X be the intersection of U with the opposites
+ * of these constraints.  Then x \in X.
+ * The facet corresponding to c_1 contains the corresponding facet of A.
+ * This facet is entirely contained in B, so c_2 is valid on the facet.
+ * However, since it is also (part of) a facet of X, -c_2 is also valid
+ * on the facet.  This means c_2 is saturated on the facet, so c_1 and
+ * c_2 must be opposites of each other, but then x could not violate
+ * both of them.
+ */
+static int check_facets(struct isl_map *map, int i, int j,
+	struct isl_tab **tabs, int *ineq_i, int *ineq_j)
+{
+	int k, l;
+	struct isl_tab_undo *snap;
+	unsigned n_eq = map->p[i]->n_eq;
+
+	snap = isl_tab_snap(tabs[i]);
+
+	for (k = 0; k < map->p[i]->n_ineq; ++k) {
+		if (ineq_i[k] != STATUS_CUT)
+			continue;
+		if (isl_tab_select_facet(tabs[i], n_eq + k) < 0)
+			return -1;
+		for (l = 0; l < map->p[j]->n_ineq; ++l) {
+			int stat;
+			if (ineq_j[l] != STATUS_CUT)
+				continue;
+			stat = status_in(map->p[j]->ineq[l], tabs[i]);
+			if (stat != STATUS_VALID)
+				break;
+		}
+		if (isl_tab_rollback(tabs[i], snap) < 0)
+			return -1;
+		if (l < map->p[j]->n_ineq)
+			break;
+	}
+
+	if (k < map->p[i]->n_ineq)
+		/* BAD CUT PAIR */
+		return 0;
+	return fuse(map, i, j, tabs, NULL, ineq_i, NULL, ineq_j, NULL);
+}
+
+/* Check if basic map "i" contains the basic map represented
+ * by the tableau "tab".
+ */
+static int contains(struct isl_map *map, int i, int *ineq_i,
+	struct isl_tab *tab)
+{
+	int k, l;
+	unsigned dim;
+
+	dim = isl_basic_map_total_dim(map->p[i]);
+	for (k = 0; k < map->p[i]->n_eq; ++k) {
+		for (l = 0; l < 2; ++l) {
+			int stat;
+			isl_seq_neg(map->p[i]->eq[k], map->p[i]->eq[k], 1+dim);
+			stat = status_in(map->p[i]->eq[k], tab);
+			if (stat != STATUS_VALID)
+				return 0;
+		}
+	}
+
+	for (k = 0; k < map->p[i]->n_ineq; ++k) {
+		int stat;
+		if (ineq_i[k] == STATUS_REDUNDANT)
+			continue;
+		stat = status_in(map->p[i]->ineq[k], tab);
+		if (stat != STATUS_VALID)
+			return 0;
+	}
+	return 1;
+}
+
+/* Basic map "i" has an inequality (say "k") that is adjacent
+ * to some inequality of basic map "j".  All the other inequalities
+ * are valid for "j".
+ * Check if basic map "j" forms an extension of basic map "i".
+ *
+ * Note that this function is only called if some of the equalities or
+ * inequalities of basic map "j" do cut basic map "i".  The function is
+ * correct even if there are no such cut constraints, but in that case
+ * the additional checks performed by this function are overkill.
+ *
+ * In particular, we replace constraint k, say f >= 0, by constraint
+ * f <= -1, add the inequalities of "j" that are valid for "i"
+ * and check if the result is a subset of basic map "j".
+ * If so, then we know that this result is exactly equal to basic map "j"
+ * since all its constraints are valid for basic map "j".
+ * By combining the valid constraints of "i" (all equalities and all
+ * inequalities except "k") and the valid constraints of "j" we therefore
+ * obtain a basic map that is equal to their union.
+ * In this case, there is no need to perform a rollback of the tableau
+ * since it is going to be destroyed in fuse().
+ *
+ *
+ *	|\__			|\__
+ *	|   \__			|   \__
+ *	|      \_	=>	|      \__
+ *	|_______| _		|_________\
+ *
+ *
+ *	|\			|\
+ *	| \			| \
+ *	|  \			|  \
+ *	|  |			|   \
+ *	|  ||\		=>      |    \
+ *	|  || \			|     \
+ *	|  ||  |		|      |
+ *	|__||_/			|_____/
+ */
+static int is_adj_ineq_extension(__isl_keep isl_map *map, int i, int j,
+	struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
+{
+	int k;
+	struct isl_tab_undo *snap;
+	unsigned n_eq = map->p[i]->n_eq;
+	unsigned total = isl_basic_map_total_dim(map->p[i]);
+	int r;
+
+	if (isl_tab_extend_cons(tabs[i], 1 + map->p[j]->n_ineq) < 0)
+		return -1;
+
+	for (k = 0; k < map->p[i]->n_ineq; ++k)
+		if (ineq_i[k] == STATUS_ADJ_INEQ)
+			break;
+	if (k >= map->p[i]->n_ineq)
+		isl_die(isl_map_get_ctx(map), isl_error_internal,
+			"ineq_i should have exactly one STATUS_ADJ_INEQ",
+			return -1);
+
+	snap = isl_tab_snap(tabs[i]);
+
+	if (isl_tab_unrestrict(tabs[i], n_eq + k) < 0)
+		return -1;
+
+	isl_seq_neg(map->p[i]->ineq[k], map->p[i]->ineq[k], 1 + total);
+	isl_int_sub_ui(map->p[i]->ineq[k][0], map->p[i]->ineq[k][0], 1);
+	r = isl_tab_add_ineq(tabs[i], map->p[i]->ineq[k]);
+	isl_seq_neg(map->p[i]->ineq[k], map->p[i]->ineq[k], 1 + total);
+	isl_int_sub_ui(map->p[i]->ineq[k][0], map->p[i]->ineq[k][0], 1);
+	if (r < 0)
+		return -1;
+
+	for (k = 0; k < map->p[j]->n_ineq; ++k) {
+		if (ineq_j[k] != STATUS_VALID)
+			continue;
+		if (isl_tab_add_ineq(tabs[i], map->p[j]->ineq[k]) < 0)
+			return -1;
+	}
+
+	if (contains(map, j, ineq_j, tabs[i]))
+		return fuse(map, i, j, tabs, eq_i, ineq_i, eq_j, ineq_j, NULL);
+
+	if (isl_tab_rollback(tabs[i], snap) < 0)
+		return -1;
+
+	return 0;
+}
+
+
+/* Both basic maps have at least one inequality with and adjacent
+ * (but opposite) inequality in the other basic map.
+ * Check that there are no cut constraints and that there is only
+ * a single pair of adjacent inequalities.
+ * If so, we can replace the pair by a single basic map described
+ * by all but the pair of adjacent inequalities.
+ * Any additional points introduced lie strictly between the two
+ * adjacent hyperplanes and can therefore be integral.
+ *
+ *        ____			  _____
+ *       /    ||\		 /     \
+ *      /     || \		/       \
+ *      \     ||  \	=>	\        \
+ *       \    ||  /		 \       /
+ *        \___||_/		  \_____/
+ *
+ * The test for a single pair of adjancent inequalities is important
+ * for avoiding the combination of two basic maps like the following
+ *
+ *       /|
+ *      / |
+ *     /__|
+ *         _____
+ *         |   |
+ *         |   |
+ *         |___|
+ *
+ * If there are some cut constraints on one side, then we may
+ * still be able to fuse the two basic maps, but we need to perform
+ * some additional checks in is_adj_ineq_extension.
+ */
+static int check_adj_ineq(struct isl_map *map, int i, int j,
+	struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
+{
+	int count_i, count_j;
+	int cut_i, cut_j;
+
+	count_i = count(ineq_i, map->p[i]->n_ineq, STATUS_ADJ_INEQ);
+	count_j = count(ineq_j, map->p[j]->n_ineq, STATUS_ADJ_INEQ);
+
+	if (count_i != 1 && count_j != 1)
+		return 0;
+
+	cut_i = any(eq_i, 2 * map->p[i]->n_eq, STATUS_CUT) ||
+		any(ineq_i, map->p[i]->n_ineq, STATUS_CUT);
+	cut_j = any(eq_j, 2 * map->p[j]->n_eq, STATUS_CUT) ||
+		any(ineq_j, map->p[j]->n_ineq, STATUS_CUT);
+
+	if (!cut_i && !cut_j && count_i == 1 && count_j == 1)
+		return fuse(map, i, j, tabs, NULL, ineq_i, NULL, ineq_j, NULL);
+
+	if (count_i == 1 && !cut_i)
+		return is_adj_ineq_extension(map, i, j, tabs,
+						eq_i, ineq_i, eq_j, ineq_j);
+
+	if (count_j == 1 && !cut_j)
+		return is_adj_ineq_extension(map, j, i, tabs,
+						eq_j, ineq_j, eq_i, ineq_i);
+
+	return 0;
+}
+
+/* Basic map "i" has an inequality "k" that is adjacent to some equality
+ * of basic map "j".  All the other inequalities are valid for "j".
+ * Check if basic map "j" forms an extension of basic map "i".
+ *
+ * In particular, we relax constraint "k", compute the corresponding
+ * facet and check whether it is included in the other basic map.
+ * If so, we know that relaxing the constraint extends the basic
+ * map with exactly the other basic map (we already know that this
+ * other basic map is included in the extension, because there
+ * were no "cut" inequalities in "i") and we can replace the
+ * two basic maps by this extension.
+ *        ____			  _____
+ *       /    || 		 /     |
+ *      /     ||  		/      |
+ *      \     ||   	=>	\      |
+ *       \    ||		 \     |
+ *        \___||		  \____|
+ */
+static int is_adj_eq_extension(struct isl_map *map, int i, int j, int k,
+	struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
+{
+	int changed = 0;
+	int super;
+	struct isl_tab_undo *snap, *snap2;
+	unsigned n_eq = map->p[i]->n_eq;
+
+	if (isl_tab_is_equality(tabs[i], n_eq + k))
+		return 0;
+
+	snap = isl_tab_snap(tabs[i]);
+	tabs[i] = isl_tab_relax(tabs[i], n_eq + k);
+	snap2 = isl_tab_snap(tabs[i]);
+	if (isl_tab_select_facet(tabs[i], n_eq + k) < 0)
+		return -1;
+	super = contains(map, j, ineq_j, tabs[i]);
+	if (super) {
+		if (isl_tab_rollback(tabs[i], snap2) < 0)
+			return -1;
+		map->p[i] = isl_basic_map_cow(map->p[i]);
+		if (!map->p[i])
+			return -1;
+		isl_int_add_ui(map->p[i]->ineq[k][0], map->p[i]->ineq[k][0], 1);
+		ISL_F_SET(map->p[i], ISL_BASIC_MAP_FINAL);
+		drop(map, j, tabs);
+		changed = 1;
+	} else
+		if (isl_tab_rollback(tabs[i], snap) < 0)
+			return -1;
+
+	return changed;
+}
+
+/* Data structure that keeps track of the wrapping constraints
+ * and of information to bound the coefficients of those constraints.
+ *
+ * bound is set if we want to apply a bound on the coefficients
+ * mat contains the wrapping constraints
+ * max is the bound on the coefficients (if bound is set)
+ */
+struct isl_wraps {
+	int bound;
+	isl_mat *mat;
+	isl_int max;
+};
+
+/* Update wraps->max to be greater than or equal to the coefficients
+ * in the equalities and inequalities of bmap that can be removed if we end up
+ * applying wrapping.
+ */
+static void wraps_update_max(struct isl_wraps *wraps,
+	__isl_keep isl_basic_map *bmap, int *eq, int *ineq)
+{
+	int k;
+	isl_int max_k;
+	unsigned total = isl_basic_map_total_dim(bmap);
+
+	isl_int_init(max_k);
+
+	for (k = 0; k < bmap->n_eq; ++k) {
+		if (eq[2 * k] == STATUS_VALID &&
+		    eq[2 * k + 1] == STATUS_VALID)
+			continue;
+		isl_seq_abs_max(bmap->eq[k] + 1, total, &max_k);
+		if (isl_int_abs_gt(max_k, wraps->max))
+			isl_int_set(wraps->max, max_k);
+	}
+
+	for (k = 0; k < bmap->n_ineq; ++k) {
+		if (ineq[k] == STATUS_VALID || ineq[k] == STATUS_REDUNDANT)
+			continue;
+		isl_seq_abs_max(bmap->ineq[k] + 1, total, &max_k);
+		if (isl_int_abs_gt(max_k, wraps->max))
+			isl_int_set(wraps->max, max_k);
+	}
+
+	isl_int_clear(max_k);
+}
+
+/* Initialize the isl_wraps data structure.
+ * If we want to bound the coefficients of the wrapping constraints,
+ * we set wraps->max to the largest coefficient
+ * in the equalities and inequalities that can be removed if we end up
+ * applying wrapping.
+ */
+static void wraps_init(struct isl_wraps *wraps, __isl_take isl_mat *mat,
+	__isl_keep isl_map *map, int i, int j,
+	int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
+{
+	isl_ctx *ctx;
+
+	wraps->bound = 0;
+	wraps->mat = mat;
+	if (!mat)
+		return;
+	ctx = isl_mat_get_ctx(mat);
+	wraps->bound = isl_options_get_coalesce_bounded_wrapping(ctx);
+	if (!wraps->bound)
+		return;
+	isl_int_init(wraps->max);
+	isl_int_set_si(wraps->max, 0);
+	wraps_update_max(wraps, map->p[i], eq_i, ineq_i);
+	wraps_update_max(wraps, map->p[j], eq_j, ineq_j);
+}
+
+/* Free the contents of the isl_wraps data structure.
+ */
+static void wraps_free(struct isl_wraps *wraps)
+{
+	isl_mat_free(wraps->mat);
+	if (wraps->bound)
+		isl_int_clear(wraps->max);
+}
+
+/* Is the wrapping constraint in row "row" allowed?
+ *
+ * If wraps->bound is set, we check that none of the coefficients
+ * is greater than wraps->max.
+ */
+static int allow_wrap(struct isl_wraps *wraps, int row)
+{
+	int i;
+
+	if (!wraps->bound)
+		return 1;
+
+	for (i = 1; i < wraps->mat->n_col; ++i)
+		if (isl_int_abs_gt(wraps->mat->row[row][i], wraps->max))
+			return 0;
+
+	return 1;
+}
+
+/* For each non-redundant constraint in "bmap" (as determined by "tab"),
+ * wrap the constraint around "bound" such that it includes the whole
+ * set "set" and append the resulting constraint to "wraps".
+ * "wraps" is assumed to have been pre-allocated to the appropriate size.
+ * wraps->n_row is the number of actual wrapped constraints that have
+ * been added.
+ * If any of the wrapping problems results in a constraint that is
+ * identical to "bound", then this means that "set" is unbounded in such
+ * way that no wrapping is possible.  If this happens then wraps->n_row
+ * is reset to zero.
+ * Similarly, if we want to bound the coefficients of the wrapping
+ * constraints and a newly added wrapping constraint does not
+ * satisfy the bound, then wraps->n_row is also reset to zero.
+ */
+static int add_wraps(struct isl_wraps *wraps, __isl_keep isl_basic_map *bmap,
+	struct isl_tab *tab, isl_int *bound, __isl_keep isl_set *set)
+{
+	int l;
+	int w;
+	unsigned total = isl_basic_map_total_dim(bmap);
+
+	w = wraps->mat->n_row;
+
+	for (l = 0; l < bmap->n_ineq; ++l) {
+		if (isl_seq_is_neg(bound, bmap->ineq[l], 1 + total))
+			continue;
+		if (isl_seq_eq(bound, bmap->ineq[l], 1 + total))
+			continue;
+		if (isl_tab_is_redundant(tab, bmap->n_eq + l))
+			continue;
+
+		isl_seq_cpy(wraps->mat->row[w], bound, 1 + total);
+		if (!isl_set_wrap_facet(set, wraps->mat->row[w], bmap->ineq[l]))
+			return -1;
+		if (isl_seq_eq(wraps->mat->row[w], bound, 1 + total))
+			goto unbounded;
+		if (!allow_wrap(wraps, w))
+			goto unbounded;
+		++w;
+	}
+	for (l = 0; l < bmap->n_eq; ++l) {
+		if (isl_seq_is_neg(bound, bmap->eq[l], 1 + total))
+			continue;
+		if (isl_seq_eq(bound, bmap->eq[l], 1 + total))
+			continue;
+
+		isl_seq_cpy(wraps->mat->row[w], bound, 1 + total);
+		isl_seq_neg(wraps->mat->row[w + 1], bmap->eq[l], 1 + total);
+		if (!isl_set_wrap_facet(set, wraps->mat->row[w],
+					wraps->mat->row[w + 1]))
+			return -1;
+		if (isl_seq_eq(wraps->mat->row[w], bound, 1 + total))
+			goto unbounded;
+		if (!allow_wrap(wraps, w))
+			goto unbounded;
+		++w;
+
+		isl_seq_cpy(wraps->mat->row[w], bound, 1 + total);
+		if (!isl_set_wrap_facet(set, wraps->mat->row[w], bmap->eq[l]))
+			return -1;
+		if (isl_seq_eq(wraps->mat->row[w], bound, 1 + total))
+			goto unbounded;
+		if (!allow_wrap(wraps, w))
+			goto unbounded;
+		++w;
+	}
+
+	wraps->mat->n_row = w;
+	return 0;
+unbounded:
+	wraps->mat->n_row = 0;
+	return 0;
+}
+
+/* Check if the constraints in "wraps" from "first" until the last
+ * are all valid for the basic set represented by "tab".
+ * If not, wraps->n_row is set to zero.
+ */
+static int check_wraps(__isl_keep isl_mat *wraps, int first,
+	struct isl_tab *tab)
+{
+	int i;
+
+	for (i = first; i < wraps->n_row; ++i) {
+		enum isl_ineq_type type;
+		type = isl_tab_ineq_type(tab, wraps->row[i]);
+		if (type == isl_ineq_error)
+			return -1;
+		if (type == isl_ineq_redundant)
+			continue;
+		wraps->n_row = 0;
+		return 0;
+	}
+
+	return 0;
+}
+
+/* Return a set that corresponds to the non-redudant constraints
+ * (as recorded in tab) of bmap.
+ *
+ * It's important to remove the redundant constraints as some
+ * of the other constraints may have been modified after the
+ * constraints were marked redundant.
+ * In particular, a constraint may have been relaxed.
+ * Redundant constraints are ignored when a constraint is relaxed
+ * and should therefore continue to be ignored ever after.
+ * Otherwise, the relaxation might be thwarted by some of
+ * these constraints.
+ */
+static __isl_give isl_set *set_from_updated_bmap(__isl_keep isl_basic_map *bmap,
+	struct isl_tab *tab)
+{
+	bmap = isl_basic_map_copy(bmap);
+	bmap = isl_basic_map_cow(bmap);
+	bmap = isl_basic_map_update_from_tab(bmap, tab);
+	return isl_set_from_basic_set(isl_basic_map_underlying_set(bmap));
+}
+
+/* Given a basic set i with a constraint k that is adjacent to either the
+ * whole of basic set j or a facet of basic set j, check if we can wrap
+ * both the facet corresponding to k and the facet of j (or the whole of j)
+ * around their ridges to include the other set.
+ * If so, replace the pair of basic sets by their union.
+ *
+ * All constraints of i (except k) are assumed to be valid for j.
+ *
+ * However, the constraints of j may not be valid for i and so
+ * we have to check that the wrapping constraints for j are valid for i.
+ *
+ * In the case where j has a facet adjacent to i, tab[j] is assumed
+ * to have been restricted to this facet, so that the non-redundant
+ * constraints in tab[j] are the ridges of the facet.
+ * Note that for the purpose of wrapping, it does not matter whether
+ * we wrap the ridges of i around the whole of j or just around
+ * the facet since all the other constraints are assumed to be valid for j.
+ * In practice, we wrap to include the whole of j.
+ *        ____			  _____
+ *       /    | 		 /     \
+ *      /     ||  		/      |
+ *      \     ||   	=>	\      |
+ *       \    ||		 \     |
+ *        \___||		  \____|
+ *
+ */
+static int can_wrap_in_facet(struct isl_map *map, int i, int j, int k,
+	struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
+{
+	int changed = 0;
+	struct isl_wraps wraps;
+	isl_mat *mat;
+	struct isl_set *set_i = NULL;
+	struct isl_set *set_j = NULL;
+	struct isl_vec *bound = NULL;
+	unsigned total = isl_basic_map_total_dim(map->p[i]);
+	struct isl_tab_undo *snap;
+	int n;
+
+	set_i = set_from_updated_bmap(map->p[i], tabs[i]);
+	set_j = set_from_updated_bmap(map->p[j], tabs[j]);
+	mat = isl_mat_alloc(map->ctx, 2 * (map->p[i]->n_eq + map->p[j]->n_eq) +
+					map->p[i]->n_ineq + map->p[j]->n_ineq,
+					1 + total);
+	wraps_init(&wraps, mat, map, i, j, eq_i, ineq_i, eq_j, ineq_j);
+	bound = isl_vec_alloc(map->ctx, 1 + total);
+	if (!set_i || !set_j || !wraps.mat || !bound)
+		goto error;
+
+	isl_seq_cpy(bound->el, map->p[i]->ineq[k], 1 + total);
+	isl_int_add_ui(bound->el[0], bound->el[0], 1);
+
+	isl_seq_cpy(wraps.mat->row[0], bound->el, 1 + total);
+	wraps.mat->n_row = 1;
+
+	if (add_wraps(&wraps, map->p[j], tabs[j], bound->el, set_i) < 0)
+		goto error;
+	if (!wraps.mat->n_row)
+		goto unbounded;
+
+	snap = isl_tab_snap(tabs[i]);
+
+	if (isl_tab_select_facet(tabs[i], map->p[i]->n_eq + k) < 0)
+		goto error;
+	if (isl_tab_detect_redundant(tabs[i]) < 0)
+		goto error;
+
+	isl_seq_neg(bound->el, map->p[i]->ineq[k], 1 + total);
+
+	n = wraps.mat->n_row;
+	if (add_wraps(&wraps, map->p[i], tabs[i], bound->el, set_j) < 0)
+		goto error;
+
+	if (isl_tab_rollback(tabs[i], snap) < 0)
+		goto error;
+	if (check_wraps(wraps.mat, n, tabs[i]) < 0)
+		goto error;
+	if (!wraps.mat->n_row)
+		goto unbounded;
+
+	changed = fuse(map, i, j, tabs, eq_i, ineq_i, eq_j, ineq_j, wraps.mat);
+
+unbounded:
+	wraps_free(&wraps);
+
+	isl_set_free(set_i);
+	isl_set_free(set_j);
+
+	isl_vec_free(bound);
+
+	return changed;
+error:
+	wraps_free(&wraps);
+	isl_vec_free(bound);
+	isl_set_free(set_i);
+	isl_set_free(set_j);
+	return -1;
+}
+
+/* Set the is_redundant property of the "n" constraints in "cuts",
+ * except "k" to "v".
+ * This is a fairly tricky operation as it bypasses isl_tab.c.
+ * The reason we want to temporarily mark some constraints redundant
+ * is that we want to ignore them in add_wraps.
+ *
+ * Initially all cut constraints are non-redundant, but the
+ * selection of a facet right before the call to this function
+ * may have made some of them redundant.
+ * Likewise, the same constraints are marked non-redundant
+ * in the second call to this function, before they are officially
+ * made non-redundant again in the subsequent rollback.
+ */
+static void set_is_redundant(struct isl_tab *tab, unsigned n_eq,
+	int *cuts, int n, int k, int v)
+{
+	int l;
+
+	for (l = 0; l < n; ++l) {
+		if (l == k)
+			continue;
+		tab->con[n_eq + cuts[l]].is_redundant = v;
+	}
+}
+
+/* Given a pair of basic maps i and j such that j sticks out
+ * of i at n cut constraints, each time by at most one,
+ * try to compute wrapping constraints and replace the two
+ * basic maps by a single basic map.
+ * The other constraints of i are assumed to be valid for j.
+ *
+ * The facets of i corresponding to the cut constraints are
+ * wrapped around their ridges, except those ridges determined
+ * by any of the other cut constraints.
+ * The intersections of cut constraints need to be ignored
+ * as the result of wrapping one cut constraint around another
+ * would result in a constraint cutting the union.
+ * In each case, the facets are wrapped to include the union
+ * of the two basic maps.
+ *
+ * The pieces of j that lie at an offset of exactly one from
+ * one of the cut constraints of i are wrapped around their edges.
+ * Here, there is no need to ignore intersections because we
+ * are wrapping around the union of the two basic maps.
+ *
+ * If any wrapping fails, i.e., if we cannot wrap to touch
+ * the union, then we give up.
+ * Otherwise, the pair of basic maps is replaced by their union.
+ */
+static int wrap_in_facets(struct isl_map *map, int i, int j,
+	int *cuts, int n, struct isl_tab **tabs,
+	int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
+{
+	int changed = 0;
+	struct isl_wraps wraps;
+	isl_mat *mat;
+	isl_set *set = NULL;
+	isl_vec *bound = NULL;
+	unsigned total = isl_basic_map_total_dim(map->p[i]);
+	int max_wrap;
+	int k;
+	struct isl_tab_undo *snap_i, *snap_j;
+
+	if (isl_tab_extend_cons(tabs[j], 1) < 0)
+		goto error;
+
+	max_wrap = 2 * (map->p[i]->n_eq + map->p[j]->n_eq) +
+		    map->p[i]->n_ineq + map->p[j]->n_ineq;
+	max_wrap *= n;
+
+	set = isl_set_union(set_from_updated_bmap(map->p[i], tabs[i]),
+			    set_from_updated_bmap(map->p[j], tabs[j]));
+	mat = isl_mat_alloc(map->ctx, max_wrap, 1 + total);
+	wraps_init(&wraps, mat, map, i, j, eq_i, ineq_i, eq_j, ineq_j);
+	bound = isl_vec_alloc(map->ctx, 1 + total);
+	if (!set || !wraps.mat || !bound)
+		goto error;
+
+	snap_i = isl_tab_snap(tabs[i]);
+	snap_j = isl_tab_snap(tabs[j]);
+
+	wraps.mat->n_row = 0;
+
+	for (k = 0; k < n; ++k) {
+		if (isl_tab_select_facet(tabs[i], map->p[i]->n_eq + cuts[k]) < 0)
+			goto error;
+		if (isl_tab_detect_redundant(tabs[i]) < 0)
+			goto error;
+		set_is_redundant(tabs[i], map->p[i]->n_eq, cuts, n, k, 1);
+
+		isl_seq_neg(bound->el, map->p[i]->ineq[cuts[k]], 1 + total);
+		if (!tabs[i]->empty &&
+		    add_wraps(&wraps, map->p[i], tabs[i], bound->el, set) < 0)
+			goto error;
+
+		set_is_redundant(tabs[i], map->p[i]->n_eq, cuts, n, k, 0);
+		if (isl_tab_rollback(tabs[i], snap_i) < 0)
+			goto error;
+
+		if (tabs[i]->empty)
+			break;
+		if (!wraps.mat->n_row)
+			break;
+
+		isl_seq_cpy(bound->el, map->p[i]->ineq[cuts[k]], 1 + total);
+		isl_int_add_ui(bound->el[0], bound->el[0], 1);
+		if (isl_tab_add_eq(tabs[j], bound->el) < 0)
+			goto error;
+		if (isl_tab_detect_redundant(tabs[j]) < 0)
+			goto error;
+
+		if (!tabs[j]->empty &&
+		    add_wraps(&wraps, map->p[j], tabs[j], bound->el, set) < 0)
+			goto error;
+
+		if (isl_tab_rollback(tabs[j], snap_j) < 0)
+			goto error;
+
+		if (!wraps.mat->n_row)
+			break;
+	}
+
+	if (k == n)
+		changed = fuse(map, i, j, tabs,
+				eq_i, ineq_i, eq_j, ineq_j, wraps.mat);
+
+	isl_vec_free(bound);
+	wraps_free(&wraps);
+	isl_set_free(set);
+
+	return changed;
+error:
+	isl_vec_free(bound);
+	wraps_free(&wraps);
+	isl_set_free(set);
+	return -1;
+}
+
+/* Given two basic sets i and j such that i has no cut equalities,
+ * check if relaxing all the cut inequalities of i by one turns
+ * them into valid constraint for j and check if we can wrap in
+ * the bits that are sticking out.
+ * If so, replace the pair by their union.
+ *
+ * We first check if all relaxed cut inequalities of i are valid for j
+ * and then try to wrap in the intersections of the relaxed cut inequalities
+ * with j.
+ *
+ * During this wrapping, we consider the points of j that lie at a distance
+ * of exactly 1 from i.  In particular, we ignore the points that lie in
+ * between this lower-dimensional space and the basic map i.
+ * We can therefore only apply this to integer maps.
+ *        ____			  _____
+ *       / ___|_		 /     \
+ *      / |    |  		/      |
+ *      \ |    |   	=>	\      |
+ *       \|____|		 \     |
+ *        \___| 		  \____/
+ *
+ *	 _____			 ______
+ *	| ____|_		|      \
+ *	| |     |		|       |
+ *	| |	|	=>	|       |
+ *	|_|     |		|       |
+ *	  |_____|		 \______|
+ *
+ *	 _______
+ *	|       |
+ *	|  |\   |
+ *	|  | \  |
+ *	|  |  \ |
+ *	|  |   \|
+ *	|  |    \
+ *	|  |_____\
+ *	|       |
+ *	|_______|
+ *
+ * Wrapping can fail if the result of wrapping one of the facets
+ * around its edges does not produce any new facet constraint.
+ * In particular, this happens when we try to wrap in unbounded sets.
+ *
+ *	 _______________________________________________________________________
+ *	|
+ *	|  ___
+ *	| |   |
+ *	|_|   |_________________________________________________________________
+ *	  |___|
+ *
+ * The following is not an acceptable result of coalescing the above two
+ * sets as it includes extra integer points.
+ *	 _______________________________________________________________________
+ *	|
+ *	|     
+ *	|      
+ *	|
+ *	 \______________________________________________________________________
+ */
+static int can_wrap_in_set(struct isl_map *map, int i, int j,
+	struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
+{
+	int changed = 0;
+	int k, m;
+	int n;
+	int *cuts = NULL;
+
+	if (ISL_F_ISSET(map->p[i], ISL_BASIC_MAP_RATIONAL) ||
+	    ISL_F_ISSET(map->p[j], ISL_BASIC_MAP_RATIONAL))
+		return 0;
+
+	n = count(ineq_i, map->p[i]->n_ineq, STATUS_CUT);
+	if (n == 0)
+		return 0;
+
+	cuts = isl_alloc_array(map->ctx, int, n);
+	if (!cuts)
+		return -1;
+
+	for (k = 0, m = 0; m < n; ++k) {
+		enum isl_ineq_type type;
+
+		if (ineq_i[k] != STATUS_CUT)
+			continue;
+
+		isl_int_add_ui(map->p[i]->ineq[k][0], map->p[i]->ineq[k][0], 1);
+		type = isl_tab_ineq_type(tabs[j], map->p[i]->ineq[k]);
+		isl_int_sub_ui(map->p[i]->ineq[k][0], map->p[i]->ineq[k][0], 1);
+		if (type == isl_ineq_error)
+			goto error;
+		if (type != isl_ineq_redundant)
+			break;
+		cuts[m] = k;
+		++m;
+	}
+
+	if (m == n)
+		changed = wrap_in_facets(map, i, j, cuts, n, tabs,
+					 eq_i, ineq_i, eq_j, ineq_j);
+
+	free(cuts);
+
+	return changed;
+error:
+	free(cuts);
+	return -1;
+}
+
+/* Check if either i or j has a single cut constraint that can
+ * be used to wrap in (a facet of) the other basic set.
+ * if so, replace the pair by their union.
+ */
+static int check_wrap(struct isl_map *map, int i, int j,
+	struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
+{
+	int changed = 0;
+
+	if (!any(eq_i, 2 * map->p[i]->n_eq, STATUS_CUT))
+		changed = can_wrap_in_set(map, i, j, tabs,
+					    eq_i, ineq_i, eq_j, ineq_j);
+	if (changed)
+		return changed;
+
+	if (!any(eq_j, 2 * map->p[j]->n_eq, STATUS_CUT))
+		changed = can_wrap_in_set(map, j, i, tabs,
+					    eq_j, ineq_j, eq_i, ineq_i);
+	return changed;
+}
+
+/* At least one of the basic maps has an equality that is adjacent
+ * to inequality.  Make sure that only one of the basic maps has
+ * such an equality and that the other basic map has exactly one
+ * inequality adjacent to an equality.
+ * We call the basic map that has the inequality "i" and the basic
+ * map that has the equality "j".
+ * If "i" has any "cut" (in)equality, then relaxing the inequality
+ * by one would not result in a basic map that contains the other
+ * basic map.
+ */
+static int check_adj_eq(struct isl_map *map, int i, int j,
+	struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
+{
+	int changed = 0;
+	int k;
+
+	if (any(eq_i, 2 * map->p[i]->n_eq, STATUS_ADJ_INEQ) &&
+	    any(eq_j, 2 * map->p[j]->n_eq, STATUS_ADJ_INEQ))
+		/* ADJ EQ TOO MANY */
+		return 0;
+
+	if (any(eq_i, 2 * map->p[i]->n_eq, STATUS_ADJ_INEQ))
+		return check_adj_eq(map, j, i, tabs,
+					eq_j, ineq_j, eq_i, ineq_i);
+
+	/* j has an equality adjacent to an inequality in i */
+
+	if (any(eq_i, 2 * map->p[i]->n_eq, STATUS_CUT))
+		return 0;
+	if (any(ineq_i, map->p[i]->n_ineq, STATUS_CUT))
+		/* ADJ EQ CUT */
+		return 0;
+	if (count(ineq_i, map->p[i]->n_ineq, STATUS_ADJ_EQ) != 1 ||
+	    any(ineq_j, map->p[j]->n_ineq, STATUS_ADJ_EQ) ||
+	    any(ineq_i, map->p[i]->n_ineq, STATUS_ADJ_INEQ) ||
+	    any(ineq_j, map->p[j]->n_ineq, STATUS_ADJ_INEQ))
+		/* ADJ EQ TOO MANY */
+		return 0;
+
+	for (k = 0; k < map->p[i]->n_ineq; ++k)
+		if (ineq_i[k] == STATUS_ADJ_EQ)
+			break;
+
+	changed = is_adj_eq_extension(map, i, j, k, tabs,
+					eq_i, ineq_i, eq_j, ineq_j);
+	if (changed)
+		return changed;
+
+	if (count(eq_j, 2 * map->p[j]->n_eq, STATUS_ADJ_INEQ) != 1)
+		return 0;
+
+	changed = can_wrap_in_facet(map, i, j, k, tabs, eq_i, ineq_i, eq_j, ineq_j);
+
+	return changed;
+}
+
+/* The two basic maps lie on adjacent hyperplanes.  In particular,
+ * basic map "i" has an equality that lies parallel to basic map "j".
+ * Check if we can wrap the facets around the parallel hyperplanes
+ * to include the other set.
+ *
+ * We perform basically the same operations as can_wrap_in_facet,
+ * except that we don't need to select a facet of one of the sets.
+ *				_
+ *	\\			\\
+ *	 \\		=>	 \\
+ *	  \			  \|
+ *
+ * We only allow one equality of "i" to be adjacent to an equality of "j"
+ * to avoid coalescing
+ *
+ *	[m, n] -> { [x, y] -> [x, 1 + y] : x >= 1 and y >= 1 and
+ *					    x <= 10 and y <= 10;
+ *		    [x, y] -> [1 + x, y] : x >= 1 and x <= 20 and
+ *					    y >= 5 and y <= 15 }
+ *
+ * to
+ *
+ *	[m, n] -> { [x, y] -> [x2, y2] : x >= 1 and 10y2 <= 20 - x + 10y and
+ *					4y2 >= 5 + 3y and 5y2 <= 15 + 4y and
+ *					y2 <= 1 + x + y - x2 and y2 >= y and
+ *					y2 >= 1 + x + y - x2 }
+ */
+static int check_eq_adj_eq(struct isl_map *map, int i, int j,
+	struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
+{
+	int k;
+	int changed = 0;
+	struct isl_wraps wraps;
+	isl_mat *mat;
+	struct isl_set *set_i = NULL;
+	struct isl_set *set_j = NULL;
+	struct isl_vec *bound = NULL;
+	unsigned total = isl_basic_map_total_dim(map->p[i]);
+
+	if (count(eq_i, 2 * map->p[i]->n_eq, STATUS_ADJ_EQ) != 1)
+		return 0;
+
+	for (k = 0; k < 2 * map->p[i]->n_eq ; ++k)
+		if (eq_i[k] == STATUS_ADJ_EQ)
+			break;
+
+	set_i = set_from_updated_bmap(map->p[i], tabs[i]);
+	set_j = set_from_updated_bmap(map->p[j], tabs[j]);
+	mat = isl_mat_alloc(map->ctx, 2 * (map->p[i]->n_eq + map->p[j]->n_eq) +
+					map->p[i]->n_ineq + map->p[j]->n_ineq,
+					1 + total);
+	wraps_init(&wraps, mat, map, i, j, eq_i, ineq_i, eq_j, ineq_j);
+	bound = isl_vec_alloc(map->ctx, 1 + total);
+	if (!set_i || !set_j || !wraps.mat || !bound)
+		goto error;
+
+	if (k % 2 == 0)
+		isl_seq_neg(bound->el, map->p[i]->eq[k / 2], 1 + total);
+	else
+		isl_seq_cpy(bound->el, map->p[i]->eq[k / 2], 1 + total);
+	isl_int_add_ui(bound->el[0], bound->el[0], 1);
+
+	isl_seq_cpy(wraps.mat->row[0], bound->el, 1 + total);
+	wraps.mat->n_row = 1;
+
+	if (add_wraps(&wraps, map->p[j], tabs[j], bound->el, set_i) < 0)
+		goto error;
+	if (!wraps.mat->n_row)
+		goto unbounded;
+
+	isl_int_sub_ui(bound->el[0], bound->el[0], 1);
+	isl_seq_neg(bound->el, bound->el, 1 + total);
+
+	isl_seq_cpy(wraps.mat->row[wraps.mat->n_row], bound->el, 1 + total);
+	wraps.mat->n_row++;
+
+	if (add_wraps(&wraps, map->p[i], tabs[i], bound->el, set_j) < 0)
+		goto error;
+	if (!wraps.mat->n_row)
+		goto unbounded;
+
+	changed = fuse(map, i, j, tabs, eq_i, ineq_i, eq_j, ineq_j, wraps.mat);
+
+	if (0) {
+error:		changed = -1;
+	}
+unbounded:
+
+	wraps_free(&wraps);
+	isl_set_free(set_i);
+	isl_set_free(set_j);
+	isl_vec_free(bound);
+
+	return changed;
+}
+
+/* Check if the union of the given pair of basic maps
+ * can be represented by a single basic map.
+ * If so, replace the pair by the single basic map and return 1.
+ * Otherwise, return 0;
+ * The two basic maps are assumed to live in the same local space.
+ *
+ * We first check the effect of each constraint of one basic map
+ * on the other basic map.
+ * The constraint may be
+ *	redundant	the constraint is redundant in its own
+ *			basic map and should be ignore and removed
+ *			in the end
+ *	valid		all (integer) points of the other basic map
+ *			satisfy the constraint
+ *	separate	no (integer) point of the other basic map
+ *			satisfies the constraint
+ *	cut		some but not all points of the other basic map
+ *			satisfy the constraint
+ *	adj_eq		the given constraint is adjacent (on the outside)
+ *			to an equality of the other basic map
+ *	adj_ineq	the given constraint is adjacent (on the outside)
+ *			to an inequality of the other basic map
+ *
+ * We consider seven cases in which we can replace the pair by a single
+ * basic map.  We ignore all "redundant" constraints.
+ *
+ *	1. all constraints of one basic map are valid
+ *		=> the other basic map is a subset and can be removed
+ *
+ *	2. all constraints of both basic maps are either "valid" or "cut"
+ *	   and the facets corresponding to the "cut" constraints
+ *	   of one of the basic maps lies entirely inside the other basic map
+ *		=> the pair can be replaced by a basic map consisting
+ *		   of the valid constraints in both basic maps
+ *
+ *	3. there is a single pair of adjacent inequalities
+ *	   (all other constraints are "valid")
+ *		=> the pair can be replaced by a basic map consisting
+ *		   of the valid constraints in both basic maps
+ *
+ *	4. one basic map has a single adjacent inequality, while the other
+ *	   constraints are "valid".  The other basic map has some
+ *	   "cut" constraints, but replacing the adjacent inequality by
+ *	   its opposite and adding the valid constraints of the other
+ *	   basic map results in a subset of the other basic map
+ *		=> the pair can be replaced by a basic map consisting
+ *		   of the valid constraints in both basic maps
+ *
+ *	5. there is a single adjacent pair of an inequality and an equality,
+ *	   the other constraints of the basic map containing the inequality are
+ *	   "valid".  Moreover, if the inequality the basic map is relaxed
+ *	   and then turned into an equality, then resulting facet lies
+ *	   entirely inside the other basic map
+ *		=> the pair can be replaced by the basic map containing
+ *		   the inequality, with the inequality relaxed.
+ *
+ *	6. there is a single adjacent pair of an inequality and an equality,
+ *	   the other constraints of the basic map containing the inequality are
+ *	   "valid".  Moreover, the facets corresponding to both
+ *	   the inequality and the equality can be wrapped around their
+ *	   ridges to include the other basic map
+ *		=> the pair can be replaced by a basic map consisting
+ *		   of the valid constraints in both basic maps together
+ *		   with all wrapping constraints
+ *
+ *	7. one of the basic maps extends beyond the other by at most one.
+ *	   Moreover, the facets corresponding to the cut constraints and
+ *	   the pieces of the other basic map at offset one from these cut
+ *	   constraints can be wrapped around their ridges to include
+ *	   the union of the two basic maps
+ *		=> the pair can be replaced by a basic map consisting
+ *		   of the valid constraints in both basic maps together
+ *		   with all wrapping constraints
+ *
+ *	8. the two basic maps live in adjacent hyperplanes.  In principle
+ *	   such sets can always be combined through wrapping, but we impose
+ *	   that there is only one such pair, to avoid overeager coalescing.
+ *
+ * Throughout the computation, we maintain a collection of tableaus
+ * corresponding to the basic maps.  When the basic maps are dropped
+ * or combined, the tableaus are modified accordingly.
+ */
+static int coalesce_local_pair(__isl_keep isl_map *map, int i, int j,
+	struct isl_tab **tabs)
+{
+	int changed = 0;
+	int *eq_i = NULL;
+	int *eq_j = NULL;
+	int *ineq_i = NULL;
+	int *ineq_j = NULL;
+
+	eq_i = eq_status_in(map->p[i], tabs[j]);
+	if (map->p[i]->n_eq && !eq_i)
+		goto error;
+	if (any(eq_i, 2 * map->p[i]->n_eq, STATUS_ERROR))
+		goto error;
+	if (any(eq_i, 2 * map->p[i]->n_eq, STATUS_SEPARATE))
+		goto done;
+
+	eq_j = eq_status_in(map->p[j], tabs[i]);
+	if (map->p[j]->n_eq && !eq_j)
+		goto error;
+	if (any(eq_j, 2 * map->p[j]->n_eq, STATUS_ERROR))
+		goto error;
+	if (any(eq_j, 2 * map->p[j]->n_eq, STATUS_SEPARATE))
+		goto done;
+
+	ineq_i = ineq_status_in(map->p[i], tabs[i], tabs[j]);
+	if (map->p[i]->n_ineq && !ineq_i)
+		goto error;
+	if (any(ineq_i, map->p[i]->n_ineq, STATUS_ERROR))
+		goto error;
+	if (any(ineq_i, map->p[i]->n_ineq, STATUS_SEPARATE))
+		goto done;
+
+	ineq_j = ineq_status_in(map->p[j], tabs[j], tabs[i]);
+	if (map->p[j]->n_ineq && !ineq_j)
+		goto error;
+	if (any(ineq_j, map->p[j]->n_ineq, STATUS_ERROR))
+		goto error;
+	if (any(ineq_j, map->p[j]->n_ineq, STATUS_SEPARATE))
+		goto done;
+
+	if (all(eq_i, 2 * map->p[i]->n_eq, STATUS_VALID) &&
+	    all(ineq_i, map->p[i]->n_ineq, STATUS_VALID)) {
+		drop(map, j, tabs);
+		changed = 1;
+	} else if (all(eq_j, 2 * map->p[j]->n_eq, STATUS_VALID) &&
+		   all(ineq_j, map->p[j]->n_ineq, STATUS_VALID)) {
+		drop(map, i, tabs);
+		changed = 1;
+	} else if (any(eq_i, 2 * map->p[i]->n_eq, STATUS_ADJ_EQ)) {
+		changed = check_eq_adj_eq(map, i, j, tabs,
+					eq_i, ineq_i, eq_j, ineq_j);
+	} else if (any(eq_j, 2 * map->p[j]->n_eq, STATUS_ADJ_EQ)) {
+		changed = check_eq_adj_eq(map, j, i, tabs,
+					eq_j, ineq_j, eq_i, ineq_i);
+	} else if (any(eq_i, 2 * map->p[i]->n_eq, STATUS_ADJ_INEQ) ||
+		   any(eq_j, 2 * map->p[j]->n_eq, STATUS_ADJ_INEQ)) {
+		changed = check_adj_eq(map, i, j, tabs,
+					eq_i, ineq_i, eq_j, ineq_j);
+	} else if (any(ineq_i, map->p[i]->n_ineq, STATUS_ADJ_EQ) ||
+		   any(ineq_j, map->p[j]->n_ineq, STATUS_ADJ_EQ)) {
+		/* Can't happen */
+		/* BAD ADJ INEQ */
+	} else if (any(ineq_i, map->p[i]->n_ineq, STATUS_ADJ_INEQ) ||
+		   any(ineq_j, map->p[j]->n_ineq, STATUS_ADJ_INEQ)) {
+		changed = check_adj_ineq(map, i, j, tabs,
+					eq_i, ineq_i, eq_j, ineq_j);
+	} else {
+		if (!any(eq_i, 2 * map->p[i]->n_eq, STATUS_CUT) &&
+		    !any(eq_j, 2 * map->p[j]->n_eq, STATUS_CUT))
+			changed = check_facets(map, i, j, tabs, ineq_i, ineq_j);
+		if (!changed)
+			changed = check_wrap(map, i, j, tabs,
+						eq_i, ineq_i, eq_j, ineq_j);
+	}
+
+done:
+	free(eq_i);
+	free(eq_j);
+	free(ineq_i);
+	free(ineq_j);
+	return changed;
+error:
+	free(eq_i);
+	free(eq_j);
+	free(ineq_i);
+	free(ineq_j);
+	return -1;
+}
+
+/* Do the two basic maps live in the same local space, i.e.,
+ * do they have the same (known) divs?
+ * If either basic map has any unknown divs, then we can only assume
+ * that they do not live in the same local space.
+ */
+static int same_divs(__isl_keep isl_basic_map *bmap1,
+	__isl_keep isl_basic_map *bmap2)
+{
+	int i;
+	int known;
+	int total;
+
+	if (!bmap1 || !bmap2)
+		return -1;
+	if (bmap1->n_div != bmap2->n_div)
+		return 0;
+
+	if (bmap1->n_div == 0)
+		return 1;
+
+	known = isl_basic_map_divs_known(bmap1);
+	if (known < 0 || !known)
+		return known;
+	known = isl_basic_map_divs_known(bmap2);
+	if (known < 0 || !known)
+		return known;
+
+	total = isl_basic_map_total_dim(bmap1);
+	for (i = 0; i < bmap1->n_div; ++i)
+		if (!isl_seq_eq(bmap1->div[i], bmap2->div[i], 2 + total))
+			return 0;
+
+	return 1;
+}
+
+/* Given two basic maps "i" and "j", where the divs of "i" form a subset
+ * of those of "j", check if basic map "j" is a subset of basic map "i"
+ * and, if so, drop basic map "j".
+ *
+ * We first expand the divs of basic map "i" to match those of basic map "j",
+ * using the divs and expansion computed by the caller.
+ * Then we check if all constraints of the expanded "i" are valid for "j".
+ */
+static int coalesce_subset(__isl_keep isl_map *map, int i, int j,
+	struct isl_tab **tabs, __isl_keep isl_mat *div, int *exp)
+{
+	isl_basic_map *bmap;
+	int changed = 0;
+	int *eq_i = NULL;
+	int *ineq_i = NULL;
+
+	bmap = isl_basic_map_copy(map->p[i]);
+	bmap = isl_basic_set_expand_divs(bmap, isl_mat_copy(div), exp);
+
+	if (!bmap)
+		goto error;
+
+	eq_i = eq_status_in(bmap, tabs[j]);
+	if (bmap->n_eq && !eq_i)
+		goto error;
+	if (any(eq_i, 2 * bmap->n_eq, STATUS_ERROR))
+		goto error;
+	if (any(eq_i, 2 * bmap->n_eq, STATUS_SEPARATE))
+		goto done;
+
+	ineq_i = ineq_status_in(bmap, NULL, tabs[j]);
+	if (bmap->n_ineq && !ineq_i)
+		goto error;
+	if (any(ineq_i, bmap->n_ineq, STATUS_ERROR))
+		goto error;
+	if (any(ineq_i, bmap->n_ineq, STATUS_SEPARATE))
+		goto done;
+
+	if (all(eq_i, 2 * map->p[i]->n_eq, STATUS_VALID) &&
+	    all(ineq_i, map->p[i]->n_ineq, STATUS_VALID)) {
+		drop(map, j, tabs);
+		changed = 1;
+	}
+
+done:
+	isl_basic_map_free(bmap);
+	free(eq_i);
+	free(ineq_i);
+	return 0;
+error:
+	isl_basic_map_free(bmap);
+	free(eq_i);
+	free(ineq_i);
+	return -1;
+}
+
+/* Check if the basic map "j" is a subset of basic map "i",
+ * assuming that "i" has fewer divs that "j".
+ * If not, then we change the order.
+ *
+ * If the two basic maps have the same number of divs, then
+ * they must necessarily be different.  Otherwise, we would have
+ * called coalesce_local_pair.  We therefore don't try anything
+ * in this case.
+ *
+ * We first check if the divs of "i" are all known and form a subset
+ * of those of "j".  If so, we pass control over to coalesce_subset.
+ */
+static int check_coalesce_subset(__isl_keep isl_map *map, int i, int j,
+	struct isl_tab **tabs)
+{
+	int known;
+	isl_mat *div_i, *div_j, *div;
+	int *exp1 = NULL;
+	int *exp2 = NULL;
+	isl_ctx *ctx;
+	int subset;
+
+	if (map->p[i]->n_div == map->p[j]->n_div)
+		return 0;
+	if (map->p[j]->n_div < map->p[i]->n_div)
+		return check_coalesce_subset(map, j, i, tabs);
+
+	known = isl_basic_map_divs_known(map->p[i]);
+	if (known < 0 || !known)
+		return known;
+
+	ctx = isl_map_get_ctx(map);
+
+	div_i = isl_basic_map_get_divs(map->p[i]);
+	div_j = isl_basic_map_get_divs(map->p[j]);
+
+	if (!div_i || !div_j)
+		goto error;
+
+	exp1 = isl_alloc_array(ctx, int, div_i->n_row);
+	exp2 = isl_alloc_array(ctx, int, div_j->n_row);
+	if ((div_i->n_row && !exp1) || (div_j->n_row && !exp2))
+		goto error;
+
+	div = isl_merge_divs(div_i, div_j, exp1, exp2);
+	if (!div)
+		goto error;
+
+	if (div->n_row == div_j->n_row)
+		subset = coalesce_subset(map, i, j, tabs, div, exp1);
+	else
+		subset = 0;
+
+	isl_mat_free(div);
+
+	isl_mat_free(div_i);
+	isl_mat_free(div_j);
+
+	free(exp2);
+	free(exp1);
+
+	return subset;
+error:
+	isl_mat_free(div_i);
+	isl_mat_free(div_j);
+	free(exp1);
+	free(exp2);
+	return -1;
+}
+
+/* Check if the union of the given pair of basic maps
+ * can be represented by a single basic map.
+ * If so, replace the pair by the single basic map and return 1.
+ * Otherwise, return 0;
+ *
+ * We first check if the two basic maps live in the same local space.
+ * If so, we do the complete check.  Otherwise, we check if one is
+ * an obvious subset of the other.
+ */
+static int coalesce_pair(__isl_keep isl_map *map, int i, int j,
+	struct isl_tab **tabs)
+{
+	int same;
+
+	same = same_divs(map->p[i], map->p[j]);
+	if (same < 0)
+		return -1;
+	if (same)
+		return coalesce_local_pair(map, i, j, tabs);
+
+	return check_coalesce_subset(map, i, j, tabs);
+}
+
+static struct isl_map *coalesce(struct isl_map *map, struct isl_tab **tabs)
+{
+	int i, j;
+
+	for (i = map->n - 2; i >= 0; --i)
+restart:
+		for (j = i + 1; j < map->n; ++j) {
+			int changed;
+			changed = coalesce_pair(map, i, j, tabs);
+			if (changed < 0)
+				goto error;
+			if (changed)
+				goto restart;
+		}
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* For each pair of basic maps in the map, check if the union of the two
+ * can be represented by a single basic map.
+ * If so, replace the pair by the single basic map and start over.
+ *
+ * Since we are constructing the tableaus of the basic maps anyway,
+ * we exploit them to detect implicit equalities and redundant constraints.
+ * This also helps the coalescing as it can ignore the redundant constraints.
+ * In order to avoid confusion, we make all implicit equalities explicit
+ * in the basic maps.  We don't call isl_basic_map_gauss, though,
+ * as that may affect the number of constraints.
+ * This means that we have to call isl_basic_map_gauss at the end
+ * of the computation to ensure that the basic maps are not left
+ * in an unexpected state.
+ */
+struct isl_map *isl_map_coalesce(struct isl_map *map)
+{
+	int i;
+	unsigned n;
+	struct isl_tab **tabs = NULL;
+
+	map = isl_map_remove_empty_parts(map);
+	if (!map)
+		return NULL;
+
+	if (map->n <= 1)
+		return map;
+
+	map = isl_map_sort_divs(map);
+	map = isl_map_cow(map);
+
+	if (!map)
+		return NULL;
+
+	tabs = isl_calloc_array(map->ctx, struct isl_tab *, map->n);
+	if (!tabs)
+		goto error;
+
+	n = map->n;
+	for (i = 0; i < map->n; ++i) {
+		tabs[i] = isl_tab_from_basic_map(map->p[i], 0);
+		if (!tabs[i])
+			goto error;
+		if (!ISL_F_ISSET(map->p[i], ISL_BASIC_MAP_NO_IMPLICIT))
+			if (isl_tab_detect_implicit_equalities(tabs[i]) < 0)
+				goto error;
+		map->p[i] = isl_tab_make_equalities_explicit(tabs[i],
+								map->p[i]);
+		if (!map->p[i])
+			goto error;
+		if (!ISL_F_ISSET(map->p[i], ISL_BASIC_MAP_NO_REDUNDANT))
+			if (isl_tab_detect_redundant(tabs[i]) < 0)
+				goto error;
+	}
+	for (i = map->n - 1; i >= 0; --i)
+		if (tabs[i]->empty)
+			drop(map, i, tabs);
+
+	map = coalesce(map, tabs);
+
+	if (map)
+		for (i = 0; i < map->n; ++i) {
+			map->p[i] = isl_basic_map_update_from_tab(map->p[i],
+								    tabs[i]);
+			map->p[i] = isl_basic_map_gauss(map->p[i], NULL);
+			map->p[i] = isl_basic_map_finalize(map->p[i]);
+			if (!map->p[i])
+				goto error;
+			ISL_F_SET(map->p[i], ISL_BASIC_MAP_NO_IMPLICIT);
+			ISL_F_SET(map->p[i], ISL_BASIC_MAP_NO_REDUNDANT);
+		}
+
+	for (i = 0; i < n; ++i)
+		isl_tab_free(tabs[i]);
+
+	free(tabs);
+
+	return map;
+error:
+	if (tabs)
+		for (i = 0; i < n; ++i)
+			isl_tab_free(tabs[i]);
+	free(tabs);
+	isl_map_free(map);
+	return NULL;
+}
+
+/* For each pair of basic sets in the set, check if the union of the two
+ * can be represented by a single basic set.
+ * If so, replace the pair by the single basic set and start over.
+ */
+struct isl_set *isl_set_coalesce(struct isl_set *set)
+{
+	return (struct isl_set *)isl_map_coalesce((struct isl_map *)set);
+}
diff --git a/polly/lib/External/isl/isl_config.h b/polly/lib/External/isl/isl_config.h
new file mode 100644
index 00000000000..906614f6a01
--- /dev/null
+++ b/polly/lib/External/isl/isl_config.h
@@ -0,0 +1,169 @@
+/* isl_config.h.in.  Generated from configure.ac by autoheader.  */
+
+/* Define if HeaderSearchOptions::AddPath takes 4 arguments */
+#undef ADDPATH_TAKES_4_ARGUMENTS
+
+/* Clang installation prefix */
+#undef CLANG_PREFIX
+
+/* Define if CompilerInstance::createDiagnostics takes argc and argv */
+#undef CREATEDIAGNOSTICS_TAKES_ARG
+
+/* Define if CompilerInstance::createPreprocessor takes TranslationUnitKind */
+#undef CREATEPREPROCESSOR_TAKES_TUKIND
+
+/* Define if TargetInfo::CreateTargetInfo takes pointer */
+#undef CREATETARGETINFO_TAKES_POINTER
+
+/* Define if TargetInfo::CreateTargetInfo takes shared_ptr */
+#undef CREATETARGETINFO_TAKES_SHARED_PTR
+
+/* Define if Driver constructor takes default image name */
+#undef DRIVER_CTOR_TAKES_DEFAULTIMAGENAME
+
+/* Define to Diagnostic for older versions of clang */
+#undef DiagnosticsEngine
+
+/* most gcc compilers know a function __attribute__((__warn_unused_result__))
+   */
+#undef GCC_WARN_UNUSED_RESULT
+
+/* Define if llvm/ADT/OwningPtr.h exists */
+#undef HAVE_ADT_OWNINGPTR_H
+
+/* Define if clang/Basic/DiagnosticOptions.h exists */
+#undef HAVE_BASIC_DIAGNOSTICOPTIONS_H
+
+/* Define if Driver constructor takes CXXIsProduction argument */
+#undef HAVE_CXXISPRODUCTION
+
+/* Define to 1 if you have the declaration of `ffs', and to 0 if you don't. */
+#undef HAVE_DECL_FFS
+
+/* Define to 1 if you have the declaration of `mp_get_memory_functions', and
+   to 0 if you don't. */
+#undef HAVE_DECL_MP_GET_MEMORY_FUNCTIONS
+
+/* Define to 1 if you have the declaration of `__builtin_ffs', and to 0 if you
+   don't. */
+#undef HAVE_DECL___BUILTIN_FFS
+
+/* Define to 1 if you have the <dlfcn.h> header file. */
+#undef HAVE_DLFCN_H
+
+/* Define to 1 if you have the <inttypes.h> header file. */
+#undef HAVE_INTTYPES_H
+
+/* Define if Driver constructor takes IsProduction argument */
+#undef HAVE_ISPRODUCTION
+
+/* Define to 1 if you have the `gmp' library (-lgmp). */
+#undef HAVE_LIBGMP
+
+/* Define to 1 if you have the <memory.h> header file. */
+#undef HAVE_MEMORY_H
+
+/* Define if SourceManager has a setMainFileID method */
+#undef HAVE_SETMAINFILEID
+
+/* Define to 1 if you have the <stdint.h> header file. */
+#undef HAVE_STDINT_H
+
+/* Define to 1 if you have the <stdlib.h> header file. */
+#undef HAVE_STDLIB_H
+
+/* Define to 1 if you have the <strings.h> header file. */
+#undef HAVE_STRINGS_H
+
+/* Define to 1 if you have the <string.h> header file. */
+#undef HAVE_STRING_H
+
+/* Define to 1 if you have the <sys/stat.h> header file. */
+#undef HAVE_SYS_STAT_H
+
+/* Define to 1 if you have the <sys/types.h> header file. */
+#undef HAVE_SYS_TYPES_H
+
+/* Define to 1 if you have the <unistd.h> header file. */
+#undef HAVE_UNISTD_H
+
+/* define if your compiler has __attribute__ */
+#undef HAVE___ATTRIBUTE__
+
+/* Return type of HandleTopLevelDeclReturn */
+#undef HandleTopLevelDeclContinue
+
+/* Return type of HandleTopLevelDeclReturn */
+#undef HandleTopLevelDeclReturn
+
+/* Define to the sub-directory in which libtool stores uninstalled libraries.
+   */
+#undef LT_OBJDIR
+
+/* Name of package */
+#undef PACKAGE
+
+/* Define to the address where bug reports for this package should be sent. */
+#undef PACKAGE_BUGREPORT
+
+/* Define to the full name of this package. */
+#undef PACKAGE_NAME
+
+/* Define to the full name and version of this package. */
+#undef PACKAGE_STRING
+
+/* Define to the one symbol short name of this package. */
+#undef PACKAGE_TARNAME
+
+/* Define to the home page for this package. */
+#undef PACKAGE_URL
+
+/* Define to the version of this package. */
+#undef PACKAGE_VERSION
+
+/* The size of `char', as computed by sizeof. */
+#undef SIZEOF_CHAR
+
+/* The size of `int', as computed by sizeof. */
+#undef SIZEOF_INT
+
+/* The size of `long', as computed by sizeof. */
+#undef SIZEOF_LONG
+
+/* The size of `short', as computed by sizeof. */
+#undef SIZEOF_SHORT
+
+/* The size of `void*', as computed by sizeof. */
+#undef SIZEOF_VOIDP
+
+/* Define to 1 if you have the ANSI C header files. */
+#undef STDC_HEADERS
+
+/* Define if Driver::BuildCompilation takes ArrayRef */
+#undef USE_ARRAYREF
+
+/* use gmp to implement isl_int */
+#undef USE_GMP_FOR_MP
+
+/* use imath to implement isl_int */
+#define USE_IMATH_FOR_MP 1
+
+/* Version number of package */
+#undef VERSION
+
+/* Define to getParamType for newer versions of clang */
+#undef getArgType
+
+/* Define to getHostTriple for older versions of clang */
+#undef getDefaultTargetTriple
+
+/* Define to getInstantiationLineNumber for older versions of clang */
+#undef getExpansionLineNumber
+
+/* Define to getNumParams for newer versions of clang */
+#undef getNumArgs
+
+/* Define to getResultType for older versions of clang */
+#undef getReturnType
+
+#include <isl_config_post.h>
diff --git a/polly/lib/External/isl/isl_config_post.h b/polly/lib/External/isl/isl_config_post.h
new file mode 100644
index 00000000000..5d67aae406d
--- /dev/null
+++ b/polly/lib/External/isl/isl_config_post.h
@@ -0,0 +1,13 @@
+#ifndef HAVE___ATTRIBUTE__
+#define __attribute__(x)
+#endif
+
+#if (HAVE_DECL_FFS==0) && (HAVE_DECL___BUILTIN_FFS==1)
+#define ffs __builtin_ffs
+#endif
+
+#ifdef GCC_WARN_UNUSED_RESULT
+#define WARN_UNUSED	GCC_WARN_UNUSED_RESULT
+#else
+#define WARN_UNUSED
+#endif
diff --git a/polly/lib/External/isl/isl_constraint.c b/polly/lib/External/isl/isl_constraint.c
new file mode 100644
index 00000000000..049175ba7f9
--- /dev/null
+++ b/polly/lib/External/isl/isl_constraint.c
@@ -0,0 +1,1437 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2010      INRIA Saclay
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France 
+ */
+
+#include <isl_map_private.h>
+#include <isl_constraint_private.h>
+#include <isl_space_private.h>
+#include <isl_seq.h>
+#include <isl_aff_private.h>
+#include <isl_local_space_private.h>
+#include <isl_val_private.h>
+#include <isl_vec_private.h>
+#include <isl/deprecated/constraint_int.h>
+
+#undef BASE
+#define BASE constraint
+
+#include <isl_list_templ.c>
+
+isl_ctx *isl_constraint_get_ctx(__isl_keep isl_constraint *c)
+{
+	return c ? isl_local_space_get_ctx(c->ls) : NULL;
+}
+
+static unsigned n(struct isl_constraint *c, enum isl_dim_type type)
+{
+	return isl_local_space_dim(c->ls, type);
+}
+
+static unsigned offset(struct isl_constraint *c, enum isl_dim_type type)
+{
+	return isl_local_space_offset(c->ls, type);
+}
+
+static unsigned basic_map_offset(__isl_keep isl_basic_map *bmap,
+							enum isl_dim_type type)
+{
+	return type == isl_dim_div ? 1 + isl_space_dim(bmap->dim, isl_dim_all)
+				   : 1 + isl_space_offset(bmap->dim, type);
+}
+
+static unsigned basic_set_offset(struct isl_basic_set *bset,
+							enum isl_dim_type type)
+{
+	isl_space *dim = bset->dim;
+	switch (type) {
+	case isl_dim_param:	return 1;
+	case isl_dim_in:	return 1 + dim->nparam;
+	case isl_dim_out:	return 1 + dim->nparam + dim->n_in;
+	case isl_dim_div:	return 1 + dim->nparam + dim->n_in + dim->n_out;
+	default:		return 0;
+	}
+}
+
+__isl_give isl_constraint *isl_constraint_alloc_vec(int eq,
+	__isl_take isl_local_space *ls, __isl_take isl_vec *v)
+{
+	isl_constraint *constraint;
+
+	if (!ls || !v)
+		goto error;
+
+	constraint = isl_alloc_type(isl_vec_get_ctx(v), isl_constraint);
+	if (!constraint)
+		goto error;
+
+	constraint->ref = 1;
+	constraint->eq = eq;
+	constraint->ls = ls;
+	constraint->v = v;
+
+	return constraint;
+error:
+	isl_local_space_free(ls);
+	isl_vec_free(v);
+	return NULL;
+}
+
+__isl_give isl_constraint *isl_constraint_alloc(int eq,
+	__isl_take isl_local_space *ls)
+{
+	isl_ctx *ctx;
+	isl_vec *v;
+
+	if (!ls)
+		return NULL;
+
+	ctx = isl_local_space_get_ctx(ls);
+	v = isl_vec_alloc(ctx, 1 + isl_local_space_dim(ls, isl_dim_all));
+	v = isl_vec_clr(v);
+	return isl_constraint_alloc_vec(eq, ls, v);
+}
+
+struct isl_constraint *isl_basic_map_constraint(struct isl_basic_map *bmap,
+	isl_int **line)
+{
+	int eq;
+	isl_ctx *ctx;
+	isl_vec *v;
+	isl_local_space *ls = NULL;
+	isl_constraint *constraint;
+
+	if (!bmap || !line)
+		goto error;
+
+	eq = line >= bmap->eq;
+
+	ctx = isl_basic_map_get_ctx(bmap);
+	ls = isl_basic_map_get_local_space(bmap);
+	v = isl_vec_alloc(ctx, 1 + isl_local_space_dim(ls, isl_dim_all));
+	if (!v)
+		goto error;
+	isl_seq_cpy(v->el, line[0], v->size);
+	constraint = isl_constraint_alloc_vec(eq, ls, v);
+
+	isl_basic_map_free(bmap);
+	return constraint;
+error:
+	isl_local_space_free(ls);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+struct isl_constraint *isl_basic_set_constraint(struct isl_basic_set *bset,
+	isl_int **line)
+{
+	return isl_basic_map_constraint((struct isl_basic_map *)bset, line);
+}
+
+__isl_give isl_constraint *isl_equality_alloc(__isl_take isl_local_space *ls)
+{
+	return isl_constraint_alloc(1, ls);
+}
+
+__isl_give isl_constraint *isl_inequality_alloc(__isl_take isl_local_space *ls)
+{
+	return isl_constraint_alloc(0, ls);
+}
+
+struct isl_constraint *isl_constraint_dup(struct isl_constraint *c)
+{
+	if (!c)
+		return NULL;
+
+	return isl_constraint_alloc_vec(c->eq, isl_local_space_copy(c->ls),
+						isl_vec_copy(c->v));
+}
+
+struct isl_constraint *isl_constraint_cow(struct isl_constraint *c)
+{
+	if (!c)
+		return NULL;
+
+	if (c->ref == 1)
+		return c;
+	c->ref--;
+	return isl_constraint_dup(c);
+}
+
+struct isl_constraint *isl_constraint_copy(struct isl_constraint *constraint)
+{
+	if (!constraint)
+		return NULL;
+
+	constraint->ref++;
+	return constraint;
+}
+
+__isl_null isl_constraint *isl_constraint_free(__isl_take isl_constraint *c)
+{
+	if (!c)
+		return NULL;
+
+	if (--c->ref > 0)
+		return NULL;
+
+	isl_local_space_free(c->ls);
+	isl_vec_free(c->v);
+	free(c);
+
+	return NULL;
+}
+
+/* Return the number of constraints in "bmap", i.e., the
+ * number of times isl_basic_map_foreach_constraint will
+ * call the callback.
+ */
+int isl_basic_map_n_constraint(__isl_keep isl_basic_map *bmap)
+{
+	if (!bmap)
+		return -1;
+
+	return bmap->n_eq + bmap->n_ineq;
+}
+
+/* Return the number of constraints in "bset", i.e., the
+ * number of times isl_basic_set_foreach_constraint will
+ * call the callback.
+ */
+int isl_basic_set_n_constraint(__isl_keep isl_basic_set *bset)
+{
+	return isl_basic_map_n_constraint(bset);
+}
+
+int isl_basic_map_foreach_constraint(__isl_keep isl_basic_map *bmap,
+	int (*fn)(__isl_take isl_constraint *c, void *user), void *user)
+{
+	int i;
+	struct isl_constraint *c;
+
+	if (!bmap)
+		return -1;
+
+	isl_assert(bmap->ctx, ISL_F_ISSET(bmap, ISL_BASIC_MAP_FINAL),
+			return -1);
+
+	for (i = 0; i < bmap->n_eq; ++i) {
+		c = isl_basic_map_constraint(isl_basic_map_copy(bmap),
+						&bmap->eq[i]);
+		if (!c)
+			return -1;
+		if (fn(c, user) < 0)
+			return -1;
+	}
+
+	for (i = 0; i < bmap->n_ineq; ++i) {
+		c = isl_basic_map_constraint(isl_basic_map_copy(bmap),
+						&bmap->ineq[i]);
+		if (!c)
+			return -1;
+		if (fn(c, user) < 0)
+			return -1;
+	}
+
+	return 0;
+}
+
+int isl_basic_set_foreach_constraint(__isl_keep isl_basic_set *bset,
+	int (*fn)(__isl_take isl_constraint *c, void *user), void *user)
+{
+	return isl_basic_map_foreach_constraint((isl_basic_map *)bset, fn, user);
+}
+
+/* Add the constraint to the list that "user" points to, if it is not
+ * a div constraint.
+ */
+static int collect_constraint(__isl_take isl_constraint *constraint,
+	void *user)
+{
+	isl_constraint_list **list = user;
+
+	if (isl_constraint_is_div_constraint(constraint))
+		isl_constraint_free(constraint);
+	else
+		*list = isl_constraint_list_add(*list, constraint);
+
+	return 0;
+}
+
+/* Return a list of constraints that, when combined, are equivalent
+ * to "bmap".  The input is required to have only known divs.
+ *
+ * There is no need to include the div constraints as they are
+ * implied by the div expressions.
+ */
+__isl_give isl_constraint_list *isl_basic_map_get_constraint_list(
+	__isl_keep isl_basic_map *bmap)
+{
+	int n;
+	int known;
+	isl_ctx *ctx;
+	isl_constraint_list *list;
+
+	known = isl_basic_map_divs_known(bmap);
+	if (known < 0)
+		return NULL;
+	ctx = isl_basic_map_get_ctx(bmap);
+	if (!known)
+		isl_die(ctx, isl_error_invalid,
+			"input involves unknown divs", return NULL);
+
+	n = isl_basic_map_n_constraint(bmap);
+	list = isl_constraint_list_alloc(ctx, n);
+	if (isl_basic_map_foreach_constraint(bmap,
+					    &collect_constraint, &list) < 0)
+		list = isl_constraint_list_free(list);
+
+	return list;
+}
+
+/* Return a list of constraints that, when combined, are equivalent
+ * to "bset".  The input is required to have only known divs.
+ */
+__isl_give isl_constraint_list *isl_basic_set_get_constraint_list(
+	__isl_keep isl_basic_set *bset)
+{
+	return isl_basic_map_get_constraint_list(bset);
+}
+
+int isl_constraint_is_equal(struct isl_constraint *constraint1,
+	struct isl_constraint *constraint2)
+{
+	int equal;
+
+	if (!constraint1 || !constraint2)
+		return 0;
+	if (constraint1->eq != constraint2->eq)
+		return 0;
+	equal = isl_local_space_is_equal(constraint1->ls, constraint2->ls);
+	if (equal < 0 || !equal)
+		return equal;
+	return isl_vec_is_equal(constraint1->v, constraint2->v);
+}
+
+struct isl_basic_map *isl_basic_map_add_constraint(
+	struct isl_basic_map *bmap, struct isl_constraint *constraint)
+{
+	isl_ctx *ctx;
+	isl_space *dim;
+	int equal_space;
+
+	if (!bmap || !constraint)
+		goto error;
+
+	ctx = isl_constraint_get_ctx(constraint);
+	dim = isl_constraint_get_space(constraint);
+	equal_space = isl_space_is_equal(bmap->dim, dim);
+	isl_space_free(dim);
+	isl_assert(ctx, equal_space, goto error);
+
+	bmap = isl_basic_map_intersect(bmap,
+				isl_basic_map_from_constraint(constraint));
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	isl_constraint_free(constraint);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_add_constraint(
+	struct isl_basic_set *bset, struct isl_constraint *constraint)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_add_constraint((struct isl_basic_map *)bset,
+						constraint);
+}
+
+__isl_give isl_map *isl_map_add_constraint(__isl_take isl_map *map,
+	__isl_take isl_constraint *constraint)
+{
+	isl_basic_map *bmap;
+
+	bmap = isl_basic_map_from_constraint(constraint);
+	map = isl_map_intersect(map, isl_map_from_basic_map(bmap));
+
+	return map;
+}
+
+__isl_give isl_set *isl_set_add_constraint(__isl_take isl_set *set,
+	__isl_take isl_constraint *constraint)
+{
+	return isl_map_add_constraint(set, constraint);
+}
+
+__isl_give isl_space *isl_constraint_get_space(
+	__isl_keep isl_constraint *constraint)
+{
+	return constraint ? isl_local_space_get_space(constraint->ls) : NULL;
+}
+
+__isl_give isl_local_space *isl_constraint_get_local_space(
+	__isl_keep isl_constraint *constraint)
+{
+	return constraint ? isl_local_space_copy(constraint->ls) : NULL;
+}
+
+int isl_constraint_dim(struct isl_constraint *constraint,
+	enum isl_dim_type type)
+{
+	if (!constraint)
+		return -1;
+	return n(constraint, type);
+}
+
+int isl_constraint_involves_dims(__isl_keep isl_constraint *constraint,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+	isl_ctx *ctx;
+	int *active = NULL;
+	int involves = 0;
+
+	if (!constraint)
+		return -1;
+	if (n == 0)
+		return 0;
+
+	ctx = isl_constraint_get_ctx(constraint);
+	if (first + n > isl_constraint_dim(constraint, type))
+		isl_die(ctx, isl_error_invalid,
+			"range out of bounds", return -1);
+
+	active = isl_local_space_get_active(constraint->ls,
+					    constraint->v->el + 1);
+	if (!active)
+		goto error;
+
+	first += isl_local_space_offset(constraint->ls, type) - 1;
+	for (i = 0; i < n; ++i)
+		if (active[first + i]) {
+			involves = 1;
+			break;
+		}
+
+	free(active);
+
+	return involves;
+error:
+	free(active);
+	return -1;
+}
+
+/* Does the given constraint represent a lower bound on the given
+ * dimension?
+ */
+int isl_constraint_is_lower_bound(__isl_keep isl_constraint *constraint,
+	enum isl_dim_type type, unsigned pos)
+{
+	if (!constraint)
+		return -1;
+
+	if (pos >= isl_local_space_dim(constraint->ls, type))
+		isl_die(isl_constraint_get_ctx(constraint), isl_error_invalid,
+			"position out of bounds", return -1);
+
+	pos += isl_local_space_offset(constraint->ls, type);
+	return isl_int_is_pos(constraint->v->el[pos]);
+}
+
+/* Does the given constraint represent an upper bound on the given
+ * dimension?
+ */
+int isl_constraint_is_upper_bound(__isl_keep isl_constraint *constraint,
+	enum isl_dim_type type, unsigned pos)
+{
+	if (!constraint)
+		return -1;
+
+	if (pos >= isl_local_space_dim(constraint->ls, type))
+		isl_die(isl_constraint_get_ctx(constraint), isl_error_invalid,
+			"position out of bounds", return -1);
+
+	pos += isl_local_space_offset(constraint->ls, type);
+	return isl_int_is_neg(constraint->v->el[pos]);
+}
+
+const char *isl_constraint_get_dim_name(__isl_keep isl_constraint *constraint,
+	enum isl_dim_type type, unsigned pos)
+{
+	return constraint ?
+	    isl_local_space_get_dim_name(constraint->ls, type, pos) : NULL;
+}
+
+void isl_constraint_get_constant(struct isl_constraint *constraint, isl_int *v)
+{
+	if (!constraint)
+		return;
+	isl_int_set(*v, constraint->v->el[0]);
+}
+
+/* Return the constant term of "constraint".
+ */
+__isl_give isl_val *isl_constraint_get_constant_val(
+	__isl_keep isl_constraint *constraint)
+{
+	isl_ctx *ctx;
+
+	if (!constraint)
+		return NULL;
+
+	ctx = isl_constraint_get_ctx(constraint);
+	return isl_val_int_from_isl_int(ctx, constraint->v->el[0]);
+}
+
+void isl_constraint_get_coefficient(struct isl_constraint *constraint,
+	enum isl_dim_type type, int pos, isl_int *v)
+{
+	if (!constraint)
+		return;
+
+	if (pos >= isl_local_space_dim(constraint->ls, type))
+		isl_die(constraint->v->ctx, isl_error_invalid,
+			"position out of bounds", return);
+
+	pos += isl_local_space_offset(constraint->ls, type);
+	isl_int_set(*v, constraint->v->el[pos]);
+}
+
+/* Return the coefficient of the variable of type "type" at position "pos"
+ * of "constraint".
+ */
+__isl_give isl_val *isl_constraint_get_coefficient_val(
+	__isl_keep isl_constraint *constraint, enum isl_dim_type type, int pos)
+{
+	isl_ctx *ctx;
+
+	if (!constraint)
+		return NULL;
+
+	ctx = isl_constraint_get_ctx(constraint);
+	if (pos < 0 || pos >= isl_local_space_dim(constraint->ls, type))
+		isl_die(ctx, isl_error_invalid,
+			"position out of bounds", return NULL);
+
+	pos += isl_local_space_offset(constraint->ls, type);
+	return isl_val_int_from_isl_int(ctx, constraint->v->el[pos]);
+}
+
+__isl_give isl_aff *isl_constraint_get_div(__isl_keep isl_constraint *constraint,
+	int pos)
+{
+	if (!constraint)
+		return NULL;
+
+	return isl_local_space_get_div(constraint->ls, pos);
+}
+
+__isl_give isl_constraint *isl_constraint_set_constant(
+	__isl_take isl_constraint *constraint, isl_int v)
+{
+	constraint = isl_constraint_cow(constraint);
+	if (!constraint)
+		return NULL;
+
+	constraint->v = isl_vec_cow(constraint->v);
+	if (!constraint->v)
+		return isl_constraint_free(constraint);
+
+	isl_int_set(constraint->v->el[0], v);
+	return constraint;
+}
+
+/* Replace the constant term of "constraint" by "v".
+ */
+__isl_give isl_constraint *isl_constraint_set_constant_val(
+	__isl_take isl_constraint *constraint, __isl_take isl_val *v)
+{
+	constraint = isl_constraint_cow(constraint);
+	if (!constraint || !v)
+		goto error;
+	if (!isl_val_is_int(v))
+		isl_die(isl_constraint_get_ctx(constraint), isl_error_invalid,
+			"expecting integer value", goto error);
+	constraint->v = isl_vec_set_element_val(constraint->v, 0, v);
+	if (!constraint->v)
+		constraint = isl_constraint_free(constraint);
+	return constraint;
+error:
+	isl_val_free(v);
+	return isl_constraint_free(constraint);
+}
+
+__isl_give isl_constraint *isl_constraint_set_constant_si(
+	__isl_take isl_constraint *constraint, int v)
+{
+	constraint = isl_constraint_cow(constraint);
+	if (!constraint)
+		return NULL;
+
+	constraint->v = isl_vec_cow(constraint->v);
+	if (!constraint->v)
+		return isl_constraint_free(constraint);
+
+	isl_int_set_si(constraint->v->el[0], v);
+	return constraint;
+}
+
+__isl_give isl_constraint *isl_constraint_set_coefficient(
+	__isl_take isl_constraint *constraint,
+	enum isl_dim_type type, int pos, isl_int v)
+{
+	constraint = isl_constraint_cow(constraint);
+	if (!constraint)
+		return NULL;
+
+	if (pos >= isl_local_space_dim(constraint->ls, type))
+		isl_die(constraint->v->ctx, isl_error_invalid,
+			"position out of bounds",
+			return isl_constraint_free(constraint));
+
+	constraint = isl_constraint_cow(constraint);
+	if (!constraint)
+		return NULL;
+
+	constraint->v = isl_vec_cow(constraint->v);
+	if (!constraint->v)
+		return isl_constraint_free(constraint);
+
+	pos += isl_local_space_offset(constraint->ls, type);
+	isl_int_set(constraint->v->el[pos], v);
+
+	return constraint;
+}
+
+/* Replace the coefficient of the variable of type "type" at position "pos"
+ * of "constraint" by "v".
+ */
+__isl_give isl_constraint *isl_constraint_set_coefficient_val(
+	__isl_take isl_constraint *constraint,
+	enum isl_dim_type type, int pos, __isl_take isl_val *v)
+{
+	constraint = isl_constraint_cow(constraint);
+	if (!constraint || !v)
+		goto error;
+	if (!isl_val_is_int(v))
+		isl_die(isl_constraint_get_ctx(constraint), isl_error_invalid,
+			"expecting integer value", goto error);
+
+	if (pos >= isl_local_space_dim(constraint->ls, type))
+		isl_die(isl_constraint_get_ctx(constraint), isl_error_invalid,
+			"position out of bounds", goto error);
+
+	pos += isl_local_space_offset(constraint->ls, type);
+	constraint->v = isl_vec_set_element_val(constraint->v, pos, v);
+	if (!constraint->v)
+		constraint = isl_constraint_free(constraint);
+	return constraint;
+error:
+	isl_val_free(v);
+	return isl_constraint_free(constraint);
+}
+
+__isl_give isl_constraint *isl_constraint_set_coefficient_si(
+	__isl_take isl_constraint *constraint,
+	enum isl_dim_type type, int pos, int v)
+{
+	constraint = isl_constraint_cow(constraint);
+	if (!constraint)
+		return NULL;
+
+	if (pos >= isl_local_space_dim(constraint->ls, type))
+		isl_die(constraint->v->ctx, isl_error_invalid,
+			"position out of bounds",
+			return isl_constraint_free(constraint));
+
+	constraint = isl_constraint_cow(constraint);
+	if (!constraint)
+		return NULL;
+
+	constraint->v = isl_vec_cow(constraint->v);
+	if (!constraint->v)
+		return isl_constraint_free(constraint);
+
+	pos += isl_local_space_offset(constraint->ls, type);
+	isl_int_set_si(constraint->v->el[pos], v);
+
+	return constraint;
+}
+
+/* Drop any constraint from "bset" that is identical to "constraint".
+ * In particular, this means that the local spaces of "bset" and
+ * "constraint" need to be the same.
+ *
+ * Since the given constraint may actually be a pointer into the bset,
+ * we have to be careful not to reorder the constraints as the user
+ * may be holding on to other constraints from the same bset.
+ * This should be cleaned up when the internal representation of
+ * isl_constraint is changed to use isl_aff.
+ */
+__isl_give isl_basic_set *isl_basic_set_drop_constraint(
+	__isl_take isl_basic_set *bset, __isl_take isl_constraint *constraint)
+{
+	int i;
+	unsigned n;
+	isl_int **row;
+	unsigned total;
+	isl_local_space *ls1;
+	int equal;
+
+	if (!bset || !constraint)
+		goto error;
+
+	ls1 = isl_basic_set_get_local_space(bset);
+	equal = isl_local_space_is_equal(ls1, constraint->ls);
+	isl_local_space_free(ls1);
+	if (equal < 0)
+		goto error;
+	if (!equal) {
+		isl_constraint_free(constraint);
+		return bset;
+	}
+
+	if (isl_constraint_is_equality(constraint)) {
+		n = bset->n_eq;
+		row = bset->eq;
+	} else {
+		n = bset->n_ineq;
+		row = bset->ineq;
+	}
+
+	total = isl_constraint_dim(constraint, isl_dim_all);
+	for (i = 0; i < n; ++i)
+		if (isl_seq_eq(row[i], constraint->v->el, 1 + total))
+			isl_seq_clr(row[i], 1 + total);
+			
+	isl_constraint_free(constraint);
+	return bset;
+error:
+	isl_constraint_free(constraint);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+struct isl_constraint *isl_constraint_negate(struct isl_constraint *constraint)
+{
+	isl_ctx *ctx;
+
+	constraint = isl_constraint_cow(constraint);
+	if (!constraint)
+		return NULL;
+
+	ctx = isl_constraint_get_ctx(constraint);
+	if (isl_constraint_is_equality(constraint))
+		isl_die(ctx, isl_error_invalid, "cannot negate equality",
+			return isl_constraint_free(constraint));
+	constraint->v = isl_vec_neg(constraint->v);
+	constraint->v = isl_vec_cow(constraint->v);
+	if (!constraint->v)
+		return isl_constraint_free(constraint);
+	isl_int_sub_ui(constraint->v->el[0], constraint->v->el[0], 1);
+	return constraint;
+}
+
+int isl_constraint_is_equality(struct isl_constraint *constraint)
+{
+	if (!constraint)
+		return -1;
+	return constraint->eq;
+}
+
+int isl_constraint_is_div_constraint(__isl_keep isl_constraint *constraint)
+{
+	int i;
+	int n_div;
+
+	if (!constraint)
+		return -1;
+	if (isl_constraint_is_equality(constraint))
+		return 0;
+	n_div = isl_constraint_dim(constraint, isl_dim_div);
+	for (i = 0; i < n_div; ++i) {
+		if (isl_local_space_is_div_constraint(constraint->ls,
+							constraint->v->el, i))
+			return 1;
+	}
+
+	return 0;
+}
+
+/* We manually set ISL_BASIC_SET_FINAL instead of calling
+ * isl_basic_map_finalize because we want to keep the position
+ * of the divs and we therefore do not want to throw away redundant divs.
+ * This is arguably a bit fragile.
+ */
+__isl_give isl_basic_map *isl_basic_map_from_constraint(
+	__isl_take isl_constraint *constraint)
+{
+	int k;
+	isl_local_space *ls;
+	struct isl_basic_map *bmap;
+	isl_int *c;
+	unsigned total;
+
+	if (!constraint)
+		return NULL;
+
+	ls = isl_local_space_copy(constraint->ls);
+	bmap = isl_basic_map_from_local_space(ls);
+	bmap = isl_basic_map_extend_constraints(bmap, 1, 1);
+	if (isl_constraint_is_equality(constraint)) {
+		k = isl_basic_map_alloc_equality(bmap);
+		if (k < 0)
+			goto error;
+		c = bmap->eq[k];
+	}
+	else {
+		k = isl_basic_map_alloc_inequality(bmap);
+		if (k < 0)
+			goto error;
+		c = bmap->ineq[k];
+	}
+	total = isl_basic_map_total_dim(bmap);
+	isl_seq_cpy(c, constraint->v->el, 1 + total);
+	isl_constraint_free(constraint);
+	if (bmap)
+		ISL_F_SET(bmap, ISL_BASIC_SET_FINAL);
+	return bmap;
+error:
+	isl_constraint_free(constraint);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_from_constraint(
+	struct isl_constraint *constraint)
+{
+	if (!constraint)
+		return NULL;
+
+	if (isl_constraint_dim(constraint, isl_dim_in) != 0)
+		isl_die(isl_constraint_get_ctx(constraint), isl_error_invalid,
+			"not a set constraint", goto error);
+	return (isl_basic_set *)isl_basic_map_from_constraint(constraint);
+error:
+	isl_constraint_free(constraint);
+	return NULL;
+}
+
+/* Is the variable of "type" at position "pos" of "bmap" defined
+ * in terms of earlier dimensions through an equality?
+ *
+ * If so, and if c is not NULL, then return a copy of this equality in *c.
+ */
+int isl_basic_map_has_defining_equality(
+	__isl_keep isl_basic_map *bmap, enum isl_dim_type type, int pos,
+	__isl_give isl_constraint **c)
+{
+	int i;
+	unsigned offset;
+	unsigned total;
+
+	if (!bmap)
+		return -1;
+	offset = basic_map_offset(bmap, type);
+	total = isl_basic_map_total_dim(bmap);
+	isl_assert(bmap->ctx, pos < isl_basic_map_dim(bmap, type), return -1);
+	for (i = 0; i < bmap->n_eq; ++i) {
+		if (isl_int_is_zero(bmap->eq[i][offset + pos]) ||
+		    isl_seq_first_non_zero(bmap->eq[i]+offset+pos+1,
+					   1+total-offset-pos-1) != -1)
+			continue;
+		if (c)
+			*c = isl_basic_map_constraint(isl_basic_map_copy(bmap),
+								&bmap->eq[i]);
+		return 1;
+	}
+	return 0;
+}
+
+/* Is the variable of "type" at position "pos" of "bset" defined
+ * in terms of earlier dimensions through an equality?
+ *
+ * If so, and if c is not NULL, then return a copy of this equality in *c.
+ */
+int isl_basic_set_has_defining_equality(
+	__isl_keep isl_basic_set *bset, enum isl_dim_type type, int pos,
+	__isl_give isl_constraint **c)
+{
+	return isl_basic_map_has_defining_equality((isl_basic_map *)bset,
+						    type, pos, c);
+}
+
+int isl_basic_set_has_defining_inequalities(
+	struct isl_basic_set *bset, enum isl_dim_type type, int pos,
+	struct isl_constraint **lower,
+	struct isl_constraint **upper)
+{
+	int i, j;
+	unsigned offset;
+	unsigned total;
+	isl_int m;
+	isl_int **lower_line, **upper_line;
+
+	if (!bset)
+		return -1;
+	offset = basic_set_offset(bset, type);
+	total = isl_basic_set_total_dim(bset);
+	isl_assert(bset->ctx, pos < isl_basic_set_dim(bset, type), return -1);
+	isl_int_init(m);
+	for (i = 0; i < bset->n_ineq; ++i) {
+		if (isl_int_is_zero(bset->ineq[i][offset + pos]))
+			continue;
+		if (isl_int_is_one(bset->ineq[i][offset + pos]))
+			continue;
+		if (isl_int_is_negone(bset->ineq[i][offset + pos]))
+			continue;
+		if (isl_seq_first_non_zero(bset->ineq[i]+offset+pos+1,
+						1+total-offset-pos-1) != -1)
+			continue;
+		for (j = i + 1; j < bset->n_ineq; ++j) {
+			if (!isl_seq_is_neg(bset->ineq[i]+1, bset->ineq[j]+1,
+					    total))
+				continue;
+			isl_int_add(m, bset->ineq[i][0], bset->ineq[j][0]);
+			if (isl_int_abs_ge(m, bset->ineq[i][offset+pos]))
+				continue;
+
+			if (isl_int_is_pos(bset->ineq[i][offset+pos])) {
+				lower_line = &bset->ineq[i];
+				upper_line = &bset->ineq[j];
+			} else {
+				lower_line = &bset->ineq[j];
+				upper_line = &bset->ineq[i];
+			}
+			*lower = isl_basic_set_constraint(
+					isl_basic_set_copy(bset), lower_line);
+			*upper = isl_basic_set_constraint(
+					isl_basic_set_copy(bset), upper_line);
+			isl_int_clear(m);
+			return 1;
+		}
+	}
+	*lower = NULL;
+	*upper = NULL;
+	isl_int_clear(m);
+	return 0;
+}
+
+/* Given two constraints "a" and "b" on the variable at position "abs_pos"
+ * (in "a" and "b"), add a constraint to "bset" that ensures that the
+ * bound implied by "a" is (strictly) larger than the bound implied by "b".
+ *
+ * If both constraints imply lower bounds, then this means that "a" is
+ * active in the result.
+ * If both constraints imply upper bounds, then this means that "b" is
+ * active in the result.
+ */
+static __isl_give isl_basic_set *add_larger_bound_constraint(
+	__isl_take isl_basic_set *bset, isl_int *a, isl_int *b,
+	unsigned abs_pos, int strict)
+{
+	int k;
+	isl_int t;
+	unsigned total;
+
+	k = isl_basic_set_alloc_inequality(bset);
+	if (k < 0)
+		goto error;
+
+	total = isl_basic_set_dim(bset, isl_dim_all);
+
+	isl_int_init(t);
+	isl_int_neg(t, b[1 + abs_pos]);
+
+	isl_seq_combine(bset->ineq[k], t, a, a[1 + abs_pos], b, 1 + abs_pos);
+	isl_seq_combine(bset->ineq[k] + 1 + abs_pos,
+		t, a + 1 + abs_pos + 1, a[1 + abs_pos], b + 1 + abs_pos + 1,
+		total - abs_pos);
+
+	if (strict)
+		isl_int_sub_ui(bset->ineq[k][0], bset->ineq[k][0], 1);
+
+	isl_int_clear(t);
+
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Add constraints to "context" that ensure that "u" is the smallest
+ * (and therefore active) upper bound on "abs_pos" in "bset" and return
+ * the resulting basic set.
+ */
+static __isl_give isl_basic_set *set_smallest_upper_bound(
+	__isl_keep isl_basic_set *context,
+	__isl_keep isl_basic_set *bset, unsigned abs_pos, int n_upper, int u)
+{
+	int j;
+
+	context = isl_basic_set_copy(context);
+	context = isl_basic_set_cow(context);
+
+	context = isl_basic_set_extend_constraints(context, 0, n_upper - 1);
+
+	for (j = 0; j < bset->n_ineq; ++j) {
+		if (j == u)
+			continue;
+		if (!isl_int_is_neg(bset->ineq[j][1 + abs_pos]))
+			continue;
+		context = add_larger_bound_constraint(context,
+			bset->ineq[j], bset->ineq[u], abs_pos, j > u);
+	}
+
+	context = isl_basic_set_simplify(context);
+	context = isl_basic_set_finalize(context);
+
+	return context;
+}
+
+/* Add constraints to "context" that ensure that "u" is the largest
+ * (and therefore active) upper bound on "abs_pos" in "bset" and return
+ * the resulting basic set.
+ */
+static __isl_give isl_basic_set *set_largest_lower_bound(
+	__isl_keep isl_basic_set *context,
+	__isl_keep isl_basic_set *bset, unsigned abs_pos, int n_lower, int l)
+{
+	int j;
+
+	context = isl_basic_set_copy(context);
+	context = isl_basic_set_cow(context);
+
+	context = isl_basic_set_extend_constraints(context, 0, n_lower - 1);
+
+	for (j = 0; j < bset->n_ineq; ++j) {
+		if (j == l)
+			continue;
+		if (!isl_int_is_pos(bset->ineq[j][1 + abs_pos]))
+			continue;
+		context = add_larger_bound_constraint(context,
+			bset->ineq[l], bset->ineq[j], abs_pos, j > l);
+	}
+
+	context = isl_basic_set_simplify(context);
+	context = isl_basic_set_finalize(context);
+
+	return context;
+}
+
+static int foreach_upper_bound(__isl_keep isl_basic_set *bset,
+	enum isl_dim_type type, unsigned abs_pos,
+	__isl_take isl_basic_set *context, int n_upper,
+	int (*fn)(__isl_take isl_constraint *lower,
+		  __isl_take isl_constraint *upper,
+		  __isl_take isl_basic_set *bset, void *user), void *user)
+{
+	isl_basic_set *context_i;
+	isl_constraint *upper = NULL;
+	int i;
+
+	for (i = 0; i < bset->n_ineq; ++i) {
+		if (isl_int_is_zero(bset->ineq[i][1 + abs_pos]))
+			continue;
+
+		context_i = set_smallest_upper_bound(context, bset,
+							abs_pos, n_upper, i);
+		if (isl_basic_set_is_empty(context_i)) {
+			isl_basic_set_free(context_i);
+			continue;
+		}
+		upper = isl_basic_set_constraint(isl_basic_set_copy(bset),
+						&bset->ineq[i]);
+		if (!upper || !context_i)
+			goto error;
+		if (fn(NULL, upper, context_i, user) < 0)
+			break;
+	}
+
+	isl_basic_set_free(context);
+
+	if (i < bset->n_ineq)
+		return -1;
+
+	return 0;
+error:
+	isl_constraint_free(upper);
+	isl_basic_set_free(context_i);
+	isl_basic_set_free(context);
+	return -1;
+}
+
+static int foreach_lower_bound(__isl_keep isl_basic_set *bset,
+	enum isl_dim_type type, unsigned abs_pos,
+	__isl_take isl_basic_set *context, int n_lower,
+	int (*fn)(__isl_take isl_constraint *lower,
+		  __isl_take isl_constraint *upper,
+		  __isl_take isl_basic_set *bset, void *user), void *user)
+{
+	isl_basic_set *context_i;
+	isl_constraint *lower = NULL;
+	int i;
+
+	for (i = 0; i < bset->n_ineq; ++i) {
+		if (isl_int_is_zero(bset->ineq[i][1 + abs_pos]))
+			continue;
+
+		context_i = set_largest_lower_bound(context, bset,
+							abs_pos, n_lower, i);
+		if (isl_basic_set_is_empty(context_i)) {
+			isl_basic_set_free(context_i);
+			continue;
+		}
+		lower = isl_basic_set_constraint(isl_basic_set_copy(bset),
+						&bset->ineq[i]);
+		if (!lower || !context_i)
+			goto error;
+		if (fn(lower, NULL, context_i, user) < 0)
+			break;
+	}
+
+	isl_basic_set_free(context);
+
+	if (i < bset->n_ineq)
+		return -1;
+
+	return 0;
+error:
+	isl_constraint_free(lower);
+	isl_basic_set_free(context_i);
+	isl_basic_set_free(context);
+	return -1;
+}
+
+static int foreach_bound_pair(__isl_keep isl_basic_set *bset,
+	enum isl_dim_type type, unsigned abs_pos,
+	__isl_take isl_basic_set *context, int n_lower, int n_upper,
+	int (*fn)(__isl_take isl_constraint *lower,
+		  __isl_take isl_constraint *upper,
+		  __isl_take isl_basic_set *bset, void *user), void *user)
+{
+	isl_basic_set *context_i, *context_j;
+	isl_constraint *lower = NULL;
+	isl_constraint *upper = NULL;
+	int i, j;
+
+	for (i = 0; i < bset->n_ineq; ++i) {
+		if (!isl_int_is_pos(bset->ineq[i][1 + abs_pos]))
+			continue;
+
+		context_i = set_largest_lower_bound(context, bset,
+							abs_pos, n_lower, i);
+		if (isl_basic_set_is_empty(context_i)) {
+			isl_basic_set_free(context_i);
+			continue;
+		}
+
+		for (j = 0; j < bset->n_ineq; ++j) {
+			if (!isl_int_is_neg(bset->ineq[j][1 + abs_pos]))
+				continue;
+
+			context_j = set_smallest_upper_bound(context_i, bset,
+							    abs_pos, n_upper, j);
+			context_j = isl_basic_set_extend_constraints(context_j,
+									0, 1);
+			context_j = add_larger_bound_constraint(context_j,
+				bset->ineq[i], bset->ineq[j], abs_pos, 0);
+			context_j = isl_basic_set_simplify(context_j);
+			context_j = isl_basic_set_finalize(context_j);
+			if (isl_basic_set_is_empty(context_j)) {
+				isl_basic_set_free(context_j);
+				continue;
+			}
+			lower = isl_basic_set_constraint(isl_basic_set_copy(bset),
+							&bset->ineq[i]);
+			upper = isl_basic_set_constraint(isl_basic_set_copy(bset),
+							&bset->ineq[j]);
+			if (!lower || !upper || !context_j)
+				goto error;
+			if (fn(lower, upper, context_j, user) < 0)
+				break;
+		}
+
+		isl_basic_set_free(context_i);
+
+		if (j < bset->n_ineq)
+			break;
+	}
+
+	isl_basic_set_free(context);
+
+	if (i < bset->n_ineq)
+		return -1;
+
+	return 0;
+error:
+	isl_constraint_free(lower);
+	isl_constraint_free(upper);
+	isl_basic_set_free(context_i);
+	isl_basic_set_free(context_j);
+	isl_basic_set_free(context);
+	return -1;
+}
+
+/* For each pair of lower and upper bounds on the variable "pos"
+ * of type "type", call "fn" with these lower and upper bounds and the
+ * set of constraints on the remaining variables where these bounds
+ * are active, i.e., (stricly) larger/smaller than the other lower/upper bounds.
+ *
+ * If the designated variable is equal to an affine combination of the
+ * other variables then fn is called with both lower and upper
+ * set to the corresponding equality.
+ *
+ * If there is no lower (or upper) bound, then NULL is passed
+ * as the corresponding bound.
+ *
+ * We first check if the variable is involved in any equality.
+ * If not, we count the number of lower and upper bounds and
+ * act accordingly.
+ */
+int isl_basic_set_foreach_bound_pair(__isl_keep isl_basic_set *bset,
+	enum isl_dim_type type, unsigned pos,
+	int (*fn)(__isl_take isl_constraint *lower,
+		  __isl_take isl_constraint *upper,
+		  __isl_take isl_basic_set *bset, void *user), void *user)
+{
+	int i;
+	isl_constraint *lower = NULL;
+	isl_constraint *upper = NULL;
+	isl_basic_set *context = NULL;
+	unsigned abs_pos;
+	int n_lower, n_upper;
+
+	if (!bset)
+		return -1;
+	isl_assert(bset->ctx, pos < isl_basic_set_dim(bset, type), return -1);
+	isl_assert(bset->ctx, type == isl_dim_param || type == isl_dim_set,
+		return -1);
+
+	abs_pos = pos;
+	if (type == isl_dim_set)
+		abs_pos += isl_basic_set_dim(bset, isl_dim_param);
+
+	for (i = 0; i < bset->n_eq; ++i) {
+		if (isl_int_is_zero(bset->eq[i][1 + abs_pos]))
+			continue;
+
+		lower = isl_basic_set_constraint(isl_basic_set_copy(bset),
+						&bset->eq[i]);
+		upper = isl_constraint_copy(lower);
+		context = isl_basic_set_remove_dims(isl_basic_set_copy(bset),
+					type, pos, 1);
+		if (!lower || !upper || !context)
+			goto error;
+		return fn(lower, upper, context, user);
+	}
+
+	n_lower = 0;
+	n_upper = 0;
+	for (i = 0; i < bset->n_ineq; ++i) {
+		if (isl_int_is_pos(bset->ineq[i][1 + abs_pos]))
+			n_lower++;
+		else if (isl_int_is_neg(bset->ineq[i][1 + abs_pos]))
+			n_upper++;
+	}
+
+	context = isl_basic_set_copy(bset);
+	context = isl_basic_set_cow(context);
+	if (!context)
+		goto error;
+	for (i = context->n_ineq - 1; i >= 0; --i)
+		if (!isl_int_is_zero(context->ineq[i][1 + abs_pos]))
+			isl_basic_set_drop_inequality(context, i);
+
+	context = isl_basic_set_drop(context, type, pos, 1);
+	if (!n_lower && !n_upper)
+		return fn(NULL, NULL, context, user);
+	if (!n_lower)
+		return foreach_upper_bound(bset, type, abs_pos, context, n_upper,
+						fn, user);
+	if (!n_upper)
+		return foreach_lower_bound(bset, type, abs_pos, context, n_lower,
+						fn, user);
+	return foreach_bound_pair(bset, type, abs_pos, context, n_lower, n_upper,
+					fn, user);
+error:
+	isl_constraint_free(lower);
+	isl_constraint_free(upper);
+	isl_basic_set_free(context);
+	return -1;
+}
+
+__isl_give isl_aff *isl_constraint_get_bound(
+	__isl_keep isl_constraint *constraint, enum isl_dim_type type, int pos)
+{
+	isl_aff *aff;
+	isl_ctx *ctx;
+
+	if (!constraint)
+		return NULL;
+	ctx = isl_constraint_get_ctx(constraint);
+	if (pos >= isl_constraint_dim(constraint, type))
+		isl_die(ctx, isl_error_invalid,
+			"index out of bounds", return NULL);
+	if (isl_constraint_dim(constraint, isl_dim_in) != 0)
+		isl_die(ctx, isl_error_invalid,
+			"not a set constraint", return NULL);
+
+	pos += offset(constraint, type);
+	if (isl_int_is_zero(constraint->v->el[pos]))
+		isl_die(ctx, isl_error_invalid,
+			"constraint does not define a bound on given dimension",
+			return NULL);
+
+	aff = isl_aff_alloc(isl_local_space_copy(constraint->ls));
+	if (!aff)
+		return NULL;
+
+	if (isl_int_is_neg(constraint->v->el[pos]))
+		isl_seq_cpy(aff->v->el + 1, constraint->v->el, aff->v->size - 1);
+	else
+		isl_seq_neg(aff->v->el + 1, constraint->v->el, aff->v->size - 1);
+	isl_int_set_si(aff->v->el[1 + pos], 0);
+	isl_int_abs(aff->v->el[0], constraint->v->el[pos]);
+
+	return aff;
+}
+
+/* For an inequality constraint
+ *
+ *	f >= 0
+ *
+ * or an equality constraint
+ *
+ *	f = 0
+ *
+ * return the affine expression f.
+ */
+__isl_give isl_aff *isl_constraint_get_aff(
+	__isl_keep isl_constraint *constraint)
+{
+	isl_aff *aff;
+
+	if (!constraint)
+		return NULL;
+
+	aff = isl_aff_alloc(isl_local_space_copy(constraint->ls));
+	if (!aff)
+		return NULL;
+
+	isl_seq_cpy(aff->v->el + 1, constraint->v->el, aff->v->size - 1);
+	isl_int_set_si(aff->v->el[0], 1);
+
+	return aff;
+}
+
+/* Construct an inequality (eq = 0) or equality (eq = 1) constraint from "aff".
+ * In particular, construct aff >= 0 or aff = 0.
+ *
+ * The denominator of "aff" can be ignored.
+ */
+static __isl_give isl_constraint *isl_constraint_alloc_aff(int eq,
+	__isl_take isl_aff *aff)
+{
+	isl_local_space *ls;
+	isl_vec *v;
+
+	if (!aff)
+		return NULL;
+	ls = isl_aff_get_domain_local_space(aff);
+	v = isl_vec_drop_els(isl_vec_copy(aff->v), 0, 1);
+	isl_aff_free(aff);
+
+	return isl_constraint_alloc_vec(eq, ls, v);
+}
+
+/* Construct an equality constraint equating the given affine expression
+ * to zero.
+ */
+__isl_give isl_constraint *isl_equality_from_aff(__isl_take isl_aff *aff)
+{
+	return isl_constraint_alloc_aff(1, aff);
+}
+
+/* Construct an inequality constraint enforcing the given affine expression
+ * to be non-negative.
+ */
+__isl_give isl_constraint *isl_inequality_from_aff(__isl_take isl_aff *aff)
+{
+	return isl_constraint_alloc_aff(0, aff);
+}
+
+/* Compare two isl_constraints.
+ *
+ * Return -1 if "c1" is "smaller" than "c2", 1 if "c1" is "greater"
+ * than "c2" and 0 if they are equal.
+ *
+ * The order is fairly arbitrary.  We do consider constraints that only involve
+ * earlier dimensions as "smaller".
+ */
+int isl_constraint_plain_cmp(__isl_keep isl_constraint *c1,
+	__isl_keep isl_constraint *c2)
+{
+	int cmp;
+	int last1, last2;
+
+	if (c1 == c2)
+		return 0;
+	if (!c1)
+		return -1;
+	if (!c2)
+		return 1;
+	cmp = isl_local_space_cmp(c1->ls, c2->ls);
+	if (cmp != 0)
+		return cmp;
+
+	last1 = isl_seq_last_non_zero(c1->v->el + 1, c1->v->size - 1);
+	last2 = isl_seq_last_non_zero(c2->v->el + 1, c1->v->size - 1);
+	if (last1 != last2)
+		return last1 - last2;
+
+	return isl_seq_cmp(c1->v->el, c2->v->el, c1->v->size);
+}
+
+/* Compare two constraints based on their final (non-zero) coefficients.
+ * In particular, the constraint that involves later variables or
+ * that has a larger coefficient for a shared latest variable
+ * is considered "greater" than the other constraint.
+ *
+ * Return -1 if "c1" is "smaller" than "c2", 1 if "c1" is "greater"
+ * than "c2" and 0 if they are equal.
+ *
+ * If the constraints live in different local spaces, then we cannot
+ * really compare the constraints so we compare the local spaces instead.
+ */
+int isl_constraint_cmp_last_non_zero(__isl_keep isl_constraint *c1,
+	__isl_keep isl_constraint *c2)
+{
+	int cmp;
+	int last1, last2;
+
+	if (c1 == c2)
+		return 0;
+	if (!c1)
+		return -1;
+	if (!c2)
+		return 1;
+	cmp = isl_local_space_cmp(c1->ls, c2->ls);
+	if (cmp != 0)
+		return cmp;
+
+	last1 = isl_seq_last_non_zero(c1->v->el + 1, c1->v->size - 1);
+	last2 = isl_seq_last_non_zero(c2->v->el + 1, c1->v->size - 1);
+	if (last1 != last2)
+		return last1 - last2;
+	if (last1 == -1)
+		return 0;
+	return isl_int_abs_cmp(c1->v->el[1 + last1], c2->v->el[1 + last2]);
+}
diff --git a/polly/lib/External/isl/isl_constraint_private.h b/polly/lib/External/isl/isl_constraint_private.h
new file mode 100644
index 00000000000..5cef0170e80
--- /dev/null
+++ b/polly/lib/External/isl/isl_constraint_private.h
@@ -0,0 +1,32 @@
+#ifndef ISL_CONSTRAINT_PRIVATE_H
+#define ISL_CONSTRAINT_PRIVATE_H
+
+#include <isl/constraint.h>
+#include <isl/local_space.h>
+#include <isl/vec.h>
+
+struct isl_constraint {
+	int ref;
+
+	int eq;
+	isl_local_space	*ls;
+	isl_vec		*v;
+};
+
+#undef EL
+#define EL isl_constraint
+
+#include <isl_list_templ.h>
+
+struct isl_constraint *isl_basic_set_constraint(struct isl_basic_set *bset,
+	isl_int **line);
+
+void isl_constraint_get_coefficient(__isl_keep isl_constraint *constraint,
+	enum isl_dim_type type, int pos, isl_int *v);
+__isl_give isl_constraint *isl_constraint_set_constant(
+	__isl_take isl_constraint *constraint, isl_int v);
+__isl_give isl_constraint *isl_constraint_set_coefficient(
+	__isl_take isl_constraint *constraint,
+	enum isl_dim_type type, int pos, isl_int v);
+
+#endif
diff --git a/polly/lib/External/isl/isl_convex_hull.c b/polly/lib/External/isl/isl_convex_hull.c
new file mode 100644
index 00000000000..b1254be1783
--- /dev/null
+++ b/polly/lib/External/isl/isl_convex_hull.c
@@ -0,0 +1,2829 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2014      INRIA Rocquencourt
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
+ * B.P. 105 - 78153 Le Chesnay, France
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl_lp_private.h>
+#include <isl/map.h>
+#include <isl_mat_private.h>
+#include <isl_vec_private.h>
+#include <isl/set.h>
+#include <isl_seq.h>
+#include <isl_options_private.h>
+#include "isl_equalities.h"
+#include "isl_tab.h"
+#include <isl_sort.h>
+
+static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set);
+
+/* Return 1 if constraint c is redundant with respect to the constraints
+ * in bmap.  If c is a lower [upper] bound in some variable and bmap
+ * does not have a lower [upper] bound in that variable, then c cannot
+ * be redundant and we do not need solve any lp.
+ */
+int isl_basic_map_constraint_is_redundant(struct isl_basic_map **bmap,
+	isl_int *c, isl_int *opt_n, isl_int *opt_d)
+{
+	enum isl_lp_result res;
+	unsigned total;
+	int i, j;
+
+	if (!bmap)
+		return -1;
+
+	total = isl_basic_map_total_dim(*bmap);
+	for (i = 0; i < total; ++i) {
+		int sign;
+		if (isl_int_is_zero(c[1+i]))
+			continue;
+		sign = isl_int_sgn(c[1+i]);
+		for (j = 0; j < (*bmap)->n_ineq; ++j)
+			if (sign == isl_int_sgn((*bmap)->ineq[j][1+i]))
+				break;
+		if (j == (*bmap)->n_ineq)
+			break;
+	}
+	if (i < total)
+		return 0;
+
+	res = isl_basic_map_solve_lp(*bmap, 0, c, (*bmap)->ctx->one,
+					opt_n, opt_d, NULL);
+	if (res == isl_lp_unbounded)
+		return 0;
+	if (res == isl_lp_error)
+		return -1;
+	if (res == isl_lp_empty) {
+		*bmap = isl_basic_map_set_to_empty(*bmap);
+		return 0;
+	}
+	return !isl_int_is_neg(*opt_n);
+}
+
+int isl_basic_set_constraint_is_redundant(struct isl_basic_set **bset,
+	isl_int *c, isl_int *opt_n, isl_int *opt_d)
+{
+	return isl_basic_map_constraint_is_redundant(
+			(struct isl_basic_map **)bset, c, opt_n, opt_d);
+}
+
+/* Remove redundant
+ * constraints.  If the minimal value along the normal of a constraint
+ * is the same if the constraint is removed, then the constraint is redundant.
+ *
+ * Alternatively, we could have intersected the basic map with the
+ * corresponding equality and the checked if the dimension was that
+ * of a facet.
+ */
+__isl_give isl_basic_map *isl_basic_map_remove_redundancies(
+	__isl_take isl_basic_map *bmap)
+{
+	struct isl_tab *tab;
+
+	if (!bmap)
+		return NULL;
+
+	bmap = isl_basic_map_gauss(bmap, NULL);
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
+		return bmap;
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_REDUNDANT))
+		return bmap;
+	if (bmap->n_ineq <= 1)
+		return bmap;
+
+	tab = isl_tab_from_basic_map(bmap, 0);
+	if (isl_tab_detect_implicit_equalities(tab) < 0)
+		goto error;
+	if (isl_tab_detect_redundant(tab) < 0)
+		goto error;
+	bmap = isl_basic_map_update_from_tab(bmap, tab);
+	isl_tab_free(tab);
+	ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
+	ISL_F_SET(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
+	return bmap;
+error:
+	isl_tab_free(tab);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_remove_redundancies(
+	__isl_take isl_basic_set *bset)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_remove_redundancies((struct isl_basic_map *)bset);
+}
+
+/* Remove redundant constraints in each of the basic maps.
+ */
+__isl_give isl_map *isl_map_remove_redundancies(__isl_take isl_map *map)
+{
+	return isl_map_inline_foreach_basic_map(map,
+					    &isl_basic_map_remove_redundancies);
+}
+
+__isl_give isl_set *isl_set_remove_redundancies(__isl_take isl_set *set)
+{
+	return isl_map_remove_redundancies(set);
+}
+
+/* Check if the set set is bound in the direction of the affine
+ * constraint c and if so, set the constant term such that the
+ * resulting constraint is a bounding constraint for the set.
+ */
+static int uset_is_bound(struct isl_set *set, isl_int *c, unsigned len)
+{
+	int first;
+	int j;
+	isl_int opt;
+	isl_int opt_denom;
+
+	isl_int_init(opt);
+	isl_int_init(opt_denom);
+	first = 1;
+	for (j = 0; j < set->n; ++j) {
+		enum isl_lp_result res;
+
+		if (ISL_F_ISSET(set->p[j], ISL_BASIC_SET_EMPTY))
+			continue;
+
+		res = isl_basic_set_solve_lp(set->p[j],
+				0, c, set->ctx->one, &opt, &opt_denom, NULL);
+		if (res == isl_lp_unbounded)
+			break;
+		if (res == isl_lp_error)
+			goto error;
+		if (res == isl_lp_empty) {
+			set->p[j] = isl_basic_set_set_to_empty(set->p[j]);
+			if (!set->p[j])
+				goto error;
+			continue;
+		}
+		if (first || isl_int_is_neg(opt)) {
+			if (!isl_int_is_one(opt_denom))
+				isl_seq_scale(c, c, opt_denom, len);
+			isl_int_sub(c[0], c[0], opt);
+		}
+		first = 0;
+	}
+	isl_int_clear(opt);
+	isl_int_clear(opt_denom);
+	return j >= set->n;
+error:
+	isl_int_clear(opt);
+	isl_int_clear(opt_denom);
+	return -1;
+}
+
+__isl_give isl_basic_map *isl_basic_map_set_rational(
+	__isl_take isl_basic_set *bmap)
+{
+	if (!bmap)
+		return NULL;
+
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
+		return bmap;
+
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+
+	ISL_F_SET(bmap, ISL_BASIC_MAP_RATIONAL);
+
+	return isl_basic_map_finalize(bmap);
+}
+
+__isl_give isl_basic_set *isl_basic_set_set_rational(
+	__isl_take isl_basic_set *bset)
+{
+	return isl_basic_map_set_rational(bset);
+}
+
+__isl_give isl_map *isl_map_set_rational(__isl_take isl_map *map)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_set_rational(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_set_rational(__isl_take isl_set *set)
+{
+	return isl_map_set_rational(set);
+}
+
+static struct isl_basic_set *isl_basic_set_add_equality(
+	struct isl_basic_set *bset, isl_int *c)
+{
+	int i;
+	unsigned dim;
+
+	if (!bset)
+		return NULL;
+
+	if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
+		return bset;
+
+	isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
+	isl_assert(bset->ctx, bset->n_div == 0, goto error);
+	dim = isl_basic_set_n_dim(bset);
+	bset = isl_basic_set_cow(bset);
+	bset = isl_basic_set_extend(bset, 0, dim, 0, 1, 0);
+	i = isl_basic_set_alloc_equality(bset);
+	if (i < 0)
+		goto error;
+	isl_seq_cpy(bset->eq[i], c, 1 + dim);
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+static struct isl_set *isl_set_add_basic_set_equality(struct isl_set *set, isl_int *c)
+{
+	int i;
+
+	set = isl_set_cow(set);
+	if (!set)
+		return NULL;
+	for (i = 0; i < set->n; ++i) {
+		set->p[i] = isl_basic_set_add_equality(set->p[i], c);
+		if (!set->p[i])
+			goto error;
+	}
+	return set;
+error:
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Given a union of basic sets, construct the constraints for wrapping
+ * a facet around one of its ridges.
+ * In particular, if each of n the d-dimensional basic sets i in "set"
+ * contains the origin, satisfies the constraints x_1 >= 0 and x_2 >= 0
+ * and is defined by the constraints
+ *				    [ 1 ]
+ *				A_i [ x ]  >= 0
+ *
+ * then the resulting set is of dimension n*(1+d) and has as constraints
+ *
+ *				    [ a_i ]
+ *				A_i [ x_i ] >= 0
+ *
+ *				      a_i   >= 0
+ *
+ *			\sum_i x_{i,1} = 1
+ */
+static struct isl_basic_set *wrap_constraints(struct isl_set *set)
+{
+	struct isl_basic_set *lp;
+	unsigned n_eq;
+	unsigned n_ineq;
+	int i, j, k;
+	unsigned dim, lp_dim;
+
+	if (!set)
+		return NULL;
+
+	dim = 1 + isl_set_n_dim(set);
+	n_eq = 1;
+	n_ineq = set->n;
+	for (i = 0; i < set->n; ++i) {
+		n_eq += set->p[i]->n_eq;
+		n_ineq += set->p[i]->n_ineq;
+	}
+	lp = isl_basic_set_alloc(set->ctx, 0, dim * set->n, 0, n_eq, n_ineq);
+	lp = isl_basic_set_set_rational(lp);
+	if (!lp)
+		return NULL;
+	lp_dim = isl_basic_set_n_dim(lp);
+	k = isl_basic_set_alloc_equality(lp);
+	isl_int_set_si(lp->eq[k][0], -1);
+	for (i = 0; i < set->n; ++i) {
+		isl_int_set_si(lp->eq[k][1+dim*i], 0);
+		isl_int_set_si(lp->eq[k][1+dim*i+1], 1);
+		isl_seq_clr(lp->eq[k]+1+dim*i+2, dim-2);
+	}
+	for (i = 0; i < set->n; ++i) {
+		k = isl_basic_set_alloc_inequality(lp);
+		isl_seq_clr(lp->ineq[k], 1+lp_dim);
+		isl_int_set_si(lp->ineq[k][1+dim*i], 1);
+
+		for (j = 0; j < set->p[i]->n_eq; ++j) {
+			k = isl_basic_set_alloc_equality(lp);
+			isl_seq_clr(lp->eq[k], 1+dim*i);
+			isl_seq_cpy(lp->eq[k]+1+dim*i, set->p[i]->eq[j], dim);
+			isl_seq_clr(lp->eq[k]+1+dim*(i+1), dim*(set->n-i-1));
+		}
+
+		for (j = 0; j < set->p[i]->n_ineq; ++j) {
+			k = isl_basic_set_alloc_inequality(lp);
+			isl_seq_clr(lp->ineq[k], 1+dim*i);
+			isl_seq_cpy(lp->ineq[k]+1+dim*i, set->p[i]->ineq[j], dim);
+			isl_seq_clr(lp->ineq[k]+1+dim*(i+1), dim*(set->n-i-1));
+		}
+	}
+	return lp;
+}
+
+/* Given a facet "facet" of the convex hull of "set" and a facet "ridge"
+ * of that facet, compute the other facet of the convex hull that contains
+ * the ridge.
+ *
+ * We first transform the set such that the facet constraint becomes
+ *
+ *			x_1 >= 0
+ *
+ * I.e., the facet lies in
+ *
+ *			x_1 = 0
+ *
+ * and on that facet, the constraint that defines the ridge is
+ *
+ *			x_2 >= 0
+ *
+ * (This transformation is not strictly needed, all that is needed is
+ * that the ridge contains the origin.)
+ *
+ * Since the ridge contains the origin, the cone of the convex hull
+ * will be of the form
+ *
+ *			x_1 >= 0
+ *			x_2 >= a x_1
+ *
+ * with this second constraint defining the new facet.
+ * The constant a is obtained by settting x_1 in the cone of the
+ * convex hull to 1 and minimizing x_2.
+ * Now, each element in the cone of the convex hull is the sum
+ * of elements in the cones of the basic sets.
+ * If a_i is the dilation factor of basic set i, then the problem
+ * we need to solve is
+ *
+ *			min \sum_i x_{i,2}
+ *			st
+ *				\sum_i x_{i,1} = 1
+ *				    a_i   >= 0
+ *				  [ a_i ]
+ *				A [ x_i ] >= 0
+ *
+ * with
+ *				    [  1  ]
+ *				A_i [ x_i ] >= 0
+ *
+ * the constraints of each (transformed) basic set.
+ * If a = n/d, then the constraint defining the new facet (in the transformed
+ * space) is
+ *
+ *			-n x_1 + d x_2 >= 0
+ *
+ * In the original space, we need to take the same combination of the
+ * corresponding constraints "facet" and "ridge".
+ *
+ * If a = -infty = "-1/0", then we just return the original facet constraint.
+ * This means that the facet is unbounded, but has a bounded intersection
+ * with the union of sets.
+ */
+isl_int *isl_set_wrap_facet(__isl_keep isl_set *set,
+	isl_int *facet, isl_int *ridge)
+{
+	int i;
+	isl_ctx *ctx;
+	struct isl_mat *T = NULL;
+	struct isl_basic_set *lp = NULL;
+	struct isl_vec *obj;
+	enum isl_lp_result res;
+	isl_int num, den;
+	unsigned dim;
+
+	if (!set)
+		return NULL;
+	ctx = set->ctx;
+	set = isl_set_copy(set);
+	set = isl_set_set_rational(set);
+
+	dim = 1 + isl_set_n_dim(set);
+	T = isl_mat_alloc(ctx, 3, dim);
+	if (!T)
+		goto error;
+	isl_int_set_si(T->row[0][0], 1);
+	isl_seq_clr(T->row[0]+1, dim - 1);
+	isl_seq_cpy(T->row[1], facet, dim);
+	isl_seq_cpy(T->row[2], ridge, dim);
+	T = isl_mat_right_inverse(T);
+	set = isl_set_preimage(set, T);
+	T = NULL;
+	if (!set)
+		goto error;
+	lp = wrap_constraints(set);
+	obj = isl_vec_alloc(ctx, 1 + dim*set->n);
+	if (!obj)
+		goto error;
+	isl_int_set_si(obj->block.data[0], 0);
+	for (i = 0; i < set->n; ++i) {
+		isl_seq_clr(obj->block.data + 1 + dim*i, 2);
+		isl_int_set_si(obj->block.data[1 + dim*i+2], 1);
+		isl_seq_clr(obj->block.data + 1 + dim*i+3, dim-3);
+	}
+	isl_int_init(num);
+	isl_int_init(den);
+	res = isl_basic_set_solve_lp(lp, 0,
+			    obj->block.data, ctx->one, &num, &den, NULL);
+	if (res == isl_lp_ok) {
+		isl_int_neg(num, num);
+		isl_seq_combine(facet, num, facet, den, ridge, dim);
+		isl_seq_normalize(ctx, facet, dim);
+	}
+	isl_int_clear(num);
+	isl_int_clear(den);
+	isl_vec_free(obj);
+	isl_basic_set_free(lp);
+	isl_set_free(set);
+	if (res == isl_lp_error)
+		return NULL;
+	isl_assert(ctx, res == isl_lp_ok || res == isl_lp_unbounded, 
+		   return NULL);
+	return facet;
+error:
+	isl_basic_set_free(lp);
+	isl_mat_free(T);
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Compute the constraint of a facet of "set".
+ *
+ * We first compute the intersection with a bounding constraint
+ * that is orthogonal to one of the coordinate axes.
+ * If the affine hull of this intersection has only one equality,
+ * we have found a facet.
+ * Otherwise, we wrap the current bounding constraint around
+ * one of the equalities of the face (one that is not equal to
+ * the current bounding constraint).
+ * This process continues until we have found a facet.
+ * The dimension of the intersection increases by at least
+ * one on each iteration, so termination is guaranteed.
+ */
+static __isl_give isl_mat *initial_facet_constraint(__isl_keep isl_set *set)
+{
+	struct isl_set *slice = NULL;
+	struct isl_basic_set *face = NULL;
+	int i;
+	unsigned dim = isl_set_n_dim(set);
+	int is_bound;
+	isl_mat *bounds = NULL;
+
+	isl_assert(set->ctx, set->n > 0, goto error);
+	bounds = isl_mat_alloc(set->ctx, 1, 1 + dim);
+	if (!bounds)
+		return NULL;
+
+	isl_seq_clr(bounds->row[0], dim);
+	isl_int_set_si(bounds->row[0][1 + dim - 1], 1);
+	is_bound = uset_is_bound(set, bounds->row[0], 1 + dim);
+	if (is_bound < 0)
+		goto error;
+	isl_assert(set->ctx, is_bound, goto error);
+	isl_seq_normalize(set->ctx, bounds->row[0], 1 + dim);
+	bounds->n_row = 1;
+
+	for (;;) {
+		slice = isl_set_copy(set);
+		slice = isl_set_add_basic_set_equality(slice, bounds->row[0]);
+		face = isl_set_affine_hull(slice);
+		if (!face)
+			goto error;
+		if (face->n_eq == 1) {
+			isl_basic_set_free(face);
+			break;
+		}
+		for (i = 0; i < face->n_eq; ++i)
+			if (!isl_seq_eq(bounds->row[0], face->eq[i], 1 + dim) &&
+			    !isl_seq_is_neg(bounds->row[0],
+						face->eq[i], 1 + dim))
+				break;
+		isl_assert(set->ctx, i < face->n_eq, goto error);
+		if (!isl_set_wrap_facet(set, bounds->row[0], face->eq[i]))
+			goto error;
+		isl_seq_normalize(set->ctx, bounds->row[0], bounds->n_col);
+		isl_basic_set_free(face);
+	}
+
+	return bounds;
+error:
+	isl_basic_set_free(face);
+	isl_mat_free(bounds);
+	return NULL;
+}
+
+/* Given the bounding constraint "c" of a facet of the convex hull of "set",
+ * compute a hyperplane description of the facet, i.e., compute the facets
+ * of the facet.
+ *
+ * We compute an affine transformation that transforms the constraint
+ *
+ *			  [ 1 ]
+ *			c [ x ] = 0
+ *
+ * to the constraint
+ *
+ *			   z_1  = 0
+ *
+ * by computing the right inverse U of a matrix that starts with the rows
+ *
+ *			[ 1 0 ]
+ *			[  c  ]
+ *
+ * Then
+ *			[ 1 ]     [ 1 ]
+ *			[ x ] = U [ z ]
+ * and
+ *			[ 1 ]     [ 1 ]
+ *			[ z ] = Q [ x ]
+ *
+ * with Q = U^{-1}
+ * Since z_1 is zero, we can drop this variable as well as the corresponding
+ * column of U to obtain
+ *
+ *			[ 1 ]      [ 1  ]
+ *			[ x ] = U' [ z' ]
+ * and
+ *			[ 1  ]      [ 1 ]
+ *			[ z' ] = Q' [ x ]
+ *
+ * with Q' equal to Q, but without the corresponding row.
+ * After computing the facets of the facet in the z' space,
+ * we convert them back to the x space through Q.
+ */
+static struct isl_basic_set *compute_facet(struct isl_set *set, isl_int *c)
+{
+	struct isl_mat *m, *U, *Q;
+	struct isl_basic_set *facet = NULL;
+	struct isl_ctx *ctx;
+	unsigned dim;
+
+	ctx = set->ctx;
+	set = isl_set_copy(set);
+	dim = isl_set_n_dim(set);
+	m = isl_mat_alloc(set->ctx, 2, 1 + dim);
+	if (!m)
+		goto error;
+	isl_int_set_si(m->row[0][0], 1);
+	isl_seq_clr(m->row[0]+1, dim);
+	isl_seq_cpy(m->row[1], c, 1+dim);
+	U = isl_mat_right_inverse(m);
+	Q = isl_mat_right_inverse(isl_mat_copy(U));
+	U = isl_mat_drop_cols(U, 1, 1);
+	Q = isl_mat_drop_rows(Q, 1, 1);
+	set = isl_set_preimage(set, U);
+	facet = uset_convex_hull_wrap_bounded(set);
+	facet = isl_basic_set_preimage(facet, Q);
+	if (facet)
+		isl_assert(ctx, facet->n_eq == 0, goto error);
+	return facet;
+error:
+	isl_basic_set_free(facet);
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Given an initial facet constraint, compute the remaining facets.
+ * We do this by running through all facets found so far and computing
+ * the adjacent facets through wrapping, adding those facets that we
+ * hadn't already found before.
+ *
+ * For each facet we have found so far, we first compute its facets
+ * in the resulting convex hull.  That is, we compute the ridges
+ * of the resulting convex hull contained in the facet.
+ * We also compute the corresponding facet in the current approximation
+ * of the convex hull.  There is no need to wrap around the ridges
+ * in this facet since that would result in a facet that is already
+ * present in the current approximation.
+ *
+ * This function can still be significantly optimized by checking which of
+ * the facets of the basic sets are also facets of the convex hull and
+ * using all the facets so far to help in constructing the facets of the
+ * facets
+ * and/or
+ * using the technique in section "3.1 Ridge Generation" of
+ * "Extended Convex Hull" by Fukuda et al.
+ */
+static struct isl_basic_set *extend(struct isl_basic_set *hull,
+	struct isl_set *set)
+{
+	int i, j, f;
+	int k;
+	struct isl_basic_set *facet = NULL;
+	struct isl_basic_set *hull_facet = NULL;
+	unsigned dim;
+
+	if (!hull)
+		return NULL;
+
+	isl_assert(set->ctx, set->n > 0, goto error);
+
+	dim = isl_set_n_dim(set);
+
+	for (i = 0; i < hull->n_ineq; ++i) {
+		facet = compute_facet(set, hull->ineq[i]);
+		facet = isl_basic_set_add_equality(facet, hull->ineq[i]);
+		facet = isl_basic_set_gauss(facet, NULL);
+		facet = isl_basic_set_normalize_constraints(facet);
+		hull_facet = isl_basic_set_copy(hull);
+		hull_facet = isl_basic_set_add_equality(hull_facet, hull->ineq[i]);
+		hull_facet = isl_basic_set_gauss(hull_facet, NULL);
+		hull_facet = isl_basic_set_normalize_constraints(hull_facet);
+		if (!facet || !hull_facet)
+			goto error;
+		hull = isl_basic_set_cow(hull);
+		hull = isl_basic_set_extend_space(hull,
+			isl_space_copy(hull->dim), 0, 0, facet->n_ineq);
+		if (!hull)
+			goto error;
+		for (j = 0; j < facet->n_ineq; ++j) {
+			for (f = 0; f < hull_facet->n_ineq; ++f)
+				if (isl_seq_eq(facet->ineq[j],
+						hull_facet->ineq[f], 1 + dim))
+					break;
+			if (f < hull_facet->n_ineq)
+				continue;
+			k = isl_basic_set_alloc_inequality(hull);
+			if (k < 0)
+				goto error;
+			isl_seq_cpy(hull->ineq[k], hull->ineq[i], 1+dim);
+			if (!isl_set_wrap_facet(set, hull->ineq[k], facet->ineq[j]))
+				goto error;
+		}
+		isl_basic_set_free(hull_facet);
+		isl_basic_set_free(facet);
+	}
+	hull = isl_basic_set_simplify(hull);
+	hull = isl_basic_set_finalize(hull);
+	return hull;
+error:
+	isl_basic_set_free(hull_facet);
+	isl_basic_set_free(facet);
+	isl_basic_set_free(hull);
+	return NULL;
+}
+
+/* Special case for computing the convex hull of a one dimensional set.
+ * We simply collect the lower and upper bounds of each basic set
+ * and the biggest of those.
+ */
+static struct isl_basic_set *convex_hull_1d(struct isl_set *set)
+{
+	struct isl_mat *c = NULL;
+	isl_int *lower = NULL;
+	isl_int *upper = NULL;
+	int i, j, k;
+	isl_int a, b;
+	struct isl_basic_set *hull;
+
+	for (i = 0; i < set->n; ++i) {
+		set->p[i] = isl_basic_set_simplify(set->p[i]);
+		if (!set->p[i])
+			goto error;
+	}
+	set = isl_set_remove_empty_parts(set);
+	if (!set)
+		goto error;
+	isl_assert(set->ctx, set->n > 0, goto error);
+	c = isl_mat_alloc(set->ctx, 2, 2);
+	if (!c)
+		goto error;
+
+	if (set->p[0]->n_eq > 0) {
+		isl_assert(set->ctx, set->p[0]->n_eq == 1, goto error);
+		lower = c->row[0];
+		upper = c->row[1];
+		if (isl_int_is_pos(set->p[0]->eq[0][1])) {
+			isl_seq_cpy(lower, set->p[0]->eq[0], 2);
+			isl_seq_neg(upper, set->p[0]->eq[0], 2);
+		} else {
+			isl_seq_neg(lower, set->p[0]->eq[0], 2);
+			isl_seq_cpy(upper, set->p[0]->eq[0], 2);
+		}
+	} else {
+		for (j = 0; j < set->p[0]->n_ineq; ++j) {
+			if (isl_int_is_pos(set->p[0]->ineq[j][1])) {
+				lower = c->row[0];
+				isl_seq_cpy(lower, set->p[0]->ineq[j], 2);
+			} else {
+				upper = c->row[1];
+				isl_seq_cpy(upper, set->p[0]->ineq[j], 2);
+			}
+		}
+	}
+
+	isl_int_init(a);
+	isl_int_init(b);
+	for (i = 0; i < set->n; ++i) {
+		struct isl_basic_set *bset = set->p[i];
+		int has_lower = 0;
+		int has_upper = 0;
+
+		for (j = 0; j < bset->n_eq; ++j) {
+			has_lower = 1;
+			has_upper = 1;
+			if (lower) {
+				isl_int_mul(a, lower[0], bset->eq[j][1]);
+				isl_int_mul(b, lower[1], bset->eq[j][0]);
+				if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
+					isl_seq_cpy(lower, bset->eq[j], 2);
+				if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
+					isl_seq_neg(lower, bset->eq[j], 2);
+			}
+			if (upper) {
+				isl_int_mul(a, upper[0], bset->eq[j][1]);
+				isl_int_mul(b, upper[1], bset->eq[j][0]);
+				if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
+					isl_seq_neg(upper, bset->eq[j], 2);
+				if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
+					isl_seq_cpy(upper, bset->eq[j], 2);
+			}
+		}
+		for (j = 0; j < bset->n_ineq; ++j) {
+			if (isl_int_is_pos(bset->ineq[j][1]))
+				has_lower = 1;
+			if (isl_int_is_neg(bset->ineq[j][1]))
+				has_upper = 1;
+			if (lower && isl_int_is_pos(bset->ineq[j][1])) {
+				isl_int_mul(a, lower[0], bset->ineq[j][1]);
+				isl_int_mul(b, lower[1], bset->ineq[j][0]);
+				if (isl_int_lt(a, b))
+					isl_seq_cpy(lower, bset->ineq[j], 2);
+			}
+			if (upper && isl_int_is_neg(bset->ineq[j][1])) {
+				isl_int_mul(a, upper[0], bset->ineq[j][1]);
+				isl_int_mul(b, upper[1], bset->ineq[j][0]);
+				if (isl_int_gt(a, b))
+					isl_seq_cpy(upper, bset->ineq[j], 2);
+			}
+		}
+		if (!has_lower)
+			lower = NULL;
+		if (!has_upper)
+			upper = NULL;
+	}
+	isl_int_clear(a);
+	isl_int_clear(b);
+
+	hull = isl_basic_set_alloc(set->ctx, 0, 1, 0, 0, 2);
+	hull = isl_basic_set_set_rational(hull);
+	if (!hull)
+		goto error;
+	if (lower) {
+		k = isl_basic_set_alloc_inequality(hull);
+		isl_seq_cpy(hull->ineq[k], lower, 2);
+	}
+	if (upper) {
+		k = isl_basic_set_alloc_inequality(hull);
+		isl_seq_cpy(hull->ineq[k], upper, 2);
+	}
+	hull = isl_basic_set_finalize(hull);
+	isl_set_free(set);
+	isl_mat_free(c);
+	return hull;
+error:
+	isl_set_free(set);
+	isl_mat_free(c);
+	return NULL;
+}
+
+static struct isl_basic_set *convex_hull_0d(struct isl_set *set)
+{
+	struct isl_basic_set *convex_hull;
+
+	if (!set)
+		return NULL;
+
+	if (isl_set_is_empty(set))
+		convex_hull = isl_basic_set_empty(isl_space_copy(set->dim));
+	else
+		convex_hull = isl_basic_set_universe(isl_space_copy(set->dim));
+	isl_set_free(set);
+	return convex_hull;
+}
+
+/* Compute the convex hull of a pair of basic sets without any parameters or
+ * integer divisions using Fourier-Motzkin elimination.
+ * The convex hull is the set of all points that can be written as
+ * the sum of points from both basic sets (in homogeneous coordinates).
+ * We set up the constraints in a space with dimensions for each of
+ * the three sets and then project out the dimensions corresponding
+ * to the two original basic sets, retaining only those corresponding
+ * to the convex hull.
+ */
+static struct isl_basic_set *convex_hull_pair_elim(struct isl_basic_set *bset1,
+	struct isl_basic_set *bset2)
+{
+	int i, j, k;
+	struct isl_basic_set *bset[2];
+	struct isl_basic_set *hull = NULL;
+	unsigned dim;
+
+	if (!bset1 || !bset2)
+		goto error;
+
+	dim = isl_basic_set_n_dim(bset1);
+	hull = isl_basic_set_alloc(bset1->ctx, 0, 2 + 3 * dim, 0,
+				1 + dim + bset1->n_eq + bset2->n_eq,
+				2 + bset1->n_ineq + bset2->n_ineq);
+	bset[0] = bset1;
+	bset[1] = bset2;
+	for (i = 0; i < 2; ++i) {
+		for (j = 0; j < bset[i]->n_eq; ++j) {
+			k = isl_basic_set_alloc_equality(hull);
+			if (k < 0)
+				goto error;
+			isl_seq_clr(hull->eq[k], (i+1) * (1+dim));
+			isl_seq_clr(hull->eq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
+			isl_seq_cpy(hull->eq[k]+(i+1)*(1+dim), bset[i]->eq[j],
+					1+dim);
+		}
+		for (j = 0; j < bset[i]->n_ineq; ++j) {
+			k = isl_basic_set_alloc_inequality(hull);
+			if (k < 0)
+				goto error;
+			isl_seq_clr(hull->ineq[k], (i+1) * (1+dim));
+			isl_seq_clr(hull->ineq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
+			isl_seq_cpy(hull->ineq[k]+(i+1)*(1+dim),
+					bset[i]->ineq[j], 1+dim);
+		}
+		k = isl_basic_set_alloc_inequality(hull);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(hull->ineq[k], 1+2+3*dim);
+		isl_int_set_si(hull->ineq[k][(i+1)*(1+dim)], 1);
+	}
+	for (j = 0; j < 1+dim; ++j) {
+		k = isl_basic_set_alloc_equality(hull);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(hull->eq[k], 1+2+3*dim);
+		isl_int_set_si(hull->eq[k][j], -1);
+		isl_int_set_si(hull->eq[k][1+dim+j], 1);
+		isl_int_set_si(hull->eq[k][2*(1+dim)+j], 1);
+	}
+	hull = isl_basic_set_set_rational(hull);
+	hull = isl_basic_set_remove_dims(hull, isl_dim_set, dim, 2*(1+dim));
+	hull = isl_basic_set_remove_redundancies(hull);
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return hull;
+error:
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	isl_basic_set_free(hull);
+	return NULL;
+}
+
+/* Is the set bounded for each value of the parameters?
+ */
+int isl_basic_set_is_bounded(__isl_keep isl_basic_set *bset)
+{
+	struct isl_tab *tab;
+	int bounded;
+
+	if (!bset)
+		return -1;
+	if (isl_basic_set_plain_is_empty(bset))
+		return 1;
+
+	tab = isl_tab_from_recession_cone(bset, 1);
+	bounded = isl_tab_cone_is_bounded(tab);
+	isl_tab_free(tab);
+	return bounded;
+}
+
+/* Is the image bounded for each value of the parameters and
+ * the domain variables?
+ */
+int isl_basic_map_image_is_bounded(__isl_keep isl_basic_map *bmap)
+{
+	unsigned nparam = isl_basic_map_dim(bmap, isl_dim_param);
+	unsigned n_in = isl_basic_map_dim(bmap, isl_dim_in);
+	int bounded;
+
+	bmap = isl_basic_map_copy(bmap);
+	bmap = isl_basic_map_cow(bmap);
+	bmap = isl_basic_map_move_dims(bmap, isl_dim_param, nparam,
+					isl_dim_in, 0, n_in);
+	bounded = isl_basic_set_is_bounded((isl_basic_set *)bmap);
+	isl_basic_map_free(bmap);
+
+	return bounded;
+}
+
+/* Is the set bounded for each value of the parameters?
+ */
+int isl_set_is_bounded(__isl_keep isl_set *set)
+{
+	int i;
+
+	if (!set)
+		return -1;
+
+	for (i = 0; i < set->n; ++i) {
+		int bounded = isl_basic_set_is_bounded(set->p[i]);
+		if (!bounded || bounded < 0)
+			return bounded;
+	}
+	return 1;
+}
+
+/* Compute the lineality space of the convex hull of bset1 and bset2.
+ *
+ * We first compute the intersection of the recession cone of bset1
+ * with the negative of the recession cone of bset2 and then compute
+ * the linear hull of the resulting cone.
+ */
+static struct isl_basic_set *induced_lineality_space(
+	struct isl_basic_set *bset1, struct isl_basic_set *bset2)
+{
+	int i, k;
+	struct isl_basic_set *lin = NULL;
+	unsigned dim;
+
+	if (!bset1 || !bset2)
+		goto error;
+
+	dim = isl_basic_set_total_dim(bset1);
+	lin = isl_basic_set_alloc_space(isl_basic_set_get_space(bset1), 0,
+					bset1->n_eq + bset2->n_eq,
+					bset1->n_ineq + bset2->n_ineq);
+	lin = isl_basic_set_set_rational(lin);
+	if (!lin)
+		goto error;
+	for (i = 0; i < bset1->n_eq; ++i) {
+		k = isl_basic_set_alloc_equality(lin);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(lin->eq[k][0], 0);
+		isl_seq_cpy(lin->eq[k] + 1, bset1->eq[i] + 1, dim);
+	}
+	for (i = 0; i < bset1->n_ineq; ++i) {
+		k = isl_basic_set_alloc_inequality(lin);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(lin->ineq[k][0], 0);
+		isl_seq_cpy(lin->ineq[k] + 1, bset1->ineq[i] + 1, dim);
+	}
+	for (i = 0; i < bset2->n_eq; ++i) {
+		k = isl_basic_set_alloc_equality(lin);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(lin->eq[k][0], 0);
+		isl_seq_neg(lin->eq[k] + 1, bset2->eq[i] + 1, dim);
+	}
+	for (i = 0; i < bset2->n_ineq; ++i) {
+		k = isl_basic_set_alloc_inequality(lin);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(lin->ineq[k][0], 0);
+		isl_seq_neg(lin->ineq[k] + 1, bset2->ineq[i] + 1, dim);
+	}
+
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return isl_basic_set_affine_hull(lin);
+error:
+	isl_basic_set_free(lin);
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return NULL;
+}
+
+static struct isl_basic_set *uset_convex_hull(struct isl_set *set);
+
+/* Given a set and a linear space "lin" of dimension n > 0,
+ * project the linear space from the set, compute the convex hull
+ * and then map the set back to the original space.
+ *
+ * Let
+ *
+ *	M x = 0
+ *
+ * describe the linear space.  We first compute the Hermite normal
+ * form H = M U of M = H Q, to obtain
+ *
+ *	H Q x = 0
+ *
+ * The last n rows of H will be zero, so the last n variables of x' = Q x
+ * are the one we want to project out.  We do this by transforming each
+ * basic set A x >= b to A U x' >= b and then removing the last n dimensions.
+ * After computing the convex hull in x'_1, i.e., A' x'_1 >= b',
+ * we transform the hull back to the original space as A' Q_1 x >= b',
+ * with Q_1 all but the last n rows of Q.
+ */
+static struct isl_basic_set *modulo_lineality(struct isl_set *set,
+	struct isl_basic_set *lin)
+{
+	unsigned total = isl_basic_set_total_dim(lin);
+	unsigned lin_dim;
+	struct isl_basic_set *hull;
+	struct isl_mat *M, *U, *Q;
+
+	if (!set || !lin)
+		goto error;
+	lin_dim = total - lin->n_eq;
+	M = isl_mat_sub_alloc6(set->ctx, lin->eq, 0, lin->n_eq, 1, total);
+	M = isl_mat_left_hermite(M, 0, &U, &Q);
+	if (!M)
+		goto error;
+	isl_mat_free(M);
+	isl_basic_set_free(lin);
+
+	Q = isl_mat_drop_rows(Q, Q->n_row - lin_dim, lin_dim);
+
+	U = isl_mat_lin_to_aff(U);
+	Q = isl_mat_lin_to_aff(Q);
+
+	set = isl_set_preimage(set, U);
+	set = isl_set_remove_dims(set, isl_dim_set, total - lin_dim, lin_dim);
+	hull = uset_convex_hull(set);
+	hull = isl_basic_set_preimage(hull, Q);
+
+	return hull;
+error:
+	isl_basic_set_free(lin);
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Given two polyhedra with as constraints h_{ij} x >= 0 in homegeneous space,
+ * set up an LP for solving
+ *
+ *	\sum_j \alpha_{1j} h_{1j} = \sum_j \alpha_{2j} h_{2j}
+ *
+ * \alpha{i0} corresponds to the (implicit) positivity constraint 1 >= 0
+ * The next \alpha{ij} correspond to the equalities and come in pairs.
+ * The final \alpha{ij} correspond to the inequalities.
+ */
+static struct isl_basic_set *valid_direction_lp(
+	struct isl_basic_set *bset1, struct isl_basic_set *bset2)
+{
+	isl_space *dim;
+	struct isl_basic_set *lp;
+	unsigned d;
+	int n;
+	int i, j, k;
+
+	if (!bset1 || !bset2)
+		goto error;
+	d = 1 + isl_basic_set_total_dim(bset1);
+	n = 2 +
+	    2 * bset1->n_eq + bset1->n_ineq + 2 * bset2->n_eq + bset2->n_ineq;
+	dim = isl_space_set_alloc(bset1->ctx, 0, n);
+	lp = isl_basic_set_alloc_space(dim, 0, d, n);
+	if (!lp)
+		goto error;
+	for (i = 0; i < n; ++i) {
+		k = isl_basic_set_alloc_inequality(lp);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(lp->ineq[k] + 1, n);
+		isl_int_set_si(lp->ineq[k][0], -1);
+		isl_int_set_si(lp->ineq[k][1 + i], 1);
+	}
+	for (i = 0; i < d; ++i) {
+		k = isl_basic_set_alloc_equality(lp);
+		if (k < 0)
+			goto error;
+		n = 0;
+		isl_int_set_si(lp->eq[k][n], 0); n++;
+		/* positivity constraint 1 >= 0 */
+		isl_int_set_si(lp->eq[k][n], i == 0); n++;
+		for (j = 0; j < bset1->n_eq; ++j) {
+			isl_int_set(lp->eq[k][n], bset1->eq[j][i]); n++;
+			isl_int_neg(lp->eq[k][n], bset1->eq[j][i]); n++;
+		}
+		for (j = 0; j < bset1->n_ineq; ++j) {
+			isl_int_set(lp->eq[k][n], bset1->ineq[j][i]); n++;
+		}
+		/* positivity constraint 1 >= 0 */
+		isl_int_set_si(lp->eq[k][n], -(i == 0)); n++;
+		for (j = 0; j < bset2->n_eq; ++j) {
+			isl_int_neg(lp->eq[k][n], bset2->eq[j][i]); n++;
+			isl_int_set(lp->eq[k][n], bset2->eq[j][i]); n++;
+		}
+		for (j = 0; j < bset2->n_ineq; ++j) {
+			isl_int_neg(lp->eq[k][n], bset2->ineq[j][i]); n++;
+		}
+	}
+	lp = isl_basic_set_gauss(lp, NULL);
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return lp;
+error:
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return NULL;
+}
+
+/* Compute a vector s in the homogeneous space such that <s, r> > 0
+ * for all rays in the homogeneous space of the two cones that correspond
+ * to the input polyhedra bset1 and bset2.
+ *
+ * We compute s as a vector that satisfies
+ *
+ *	s = \sum_j \alpha_{ij} h_{ij}	for i = 1,2			(*)
+ *
+ * with h_{ij} the normals of the facets of polyhedron i
+ * (including the "positivity constraint" 1 >= 0) and \alpha_{ij}
+ * strictly positive numbers.  For simplicity we impose \alpha_{ij} >= 1.
+ * We first set up an LP with as variables the \alpha{ij}.
+ * In this formulation, for each polyhedron i,
+ * the first constraint is the positivity constraint, followed by pairs
+ * of variables for the equalities, followed by variables for the inequalities.
+ * We then simply pick a feasible solution and compute s using (*).
+ *
+ * Note that we simply pick any valid direction and make no attempt
+ * to pick a "good" or even the "best" valid direction.
+ */
+static struct isl_vec *valid_direction(
+	struct isl_basic_set *bset1, struct isl_basic_set *bset2)
+{
+	struct isl_basic_set *lp;
+	struct isl_tab *tab;
+	struct isl_vec *sample = NULL;
+	struct isl_vec *dir;
+	unsigned d;
+	int i;
+	int n;
+
+	if (!bset1 || !bset2)
+		goto error;
+	lp = valid_direction_lp(isl_basic_set_copy(bset1),
+				isl_basic_set_copy(bset2));
+	tab = isl_tab_from_basic_set(lp, 0);
+	sample = isl_tab_get_sample_value(tab);
+	isl_tab_free(tab);
+	isl_basic_set_free(lp);
+	if (!sample)
+		goto error;
+	d = isl_basic_set_total_dim(bset1);
+	dir = isl_vec_alloc(bset1->ctx, 1 + d);
+	if (!dir)
+		goto error;
+	isl_seq_clr(dir->block.data + 1, dir->size - 1);
+	n = 1;
+	/* positivity constraint 1 >= 0 */
+	isl_int_set(dir->block.data[0], sample->block.data[n]); n++;
+	for (i = 0; i < bset1->n_eq; ++i) {
+		isl_int_sub(sample->block.data[n],
+			    sample->block.data[n], sample->block.data[n+1]);
+		isl_seq_combine(dir->block.data,
+				bset1->ctx->one, dir->block.data,
+				sample->block.data[n], bset1->eq[i], 1 + d);
+
+		n += 2;
+	}
+	for (i = 0; i < bset1->n_ineq; ++i)
+		isl_seq_combine(dir->block.data,
+				bset1->ctx->one, dir->block.data,
+				sample->block.data[n++], bset1->ineq[i], 1 + d);
+	isl_vec_free(sample);
+	isl_seq_normalize(bset1->ctx, dir->el, dir->size);
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return dir;
+error:
+	isl_vec_free(sample);
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return NULL;
+}
+
+/* Given a polyhedron b_i + A_i x >= 0 and a map T = S^{-1},
+ * compute b_i' + A_i' x' >= 0, with
+ *
+ *	[ b_i A_i ]        [ y' ]		              [ y' ]
+ *	[  1   0  ] S^{-1} [ x' ] >= 0	or	[ b_i' A_i' ] [ x' ] >= 0
+ *
+ * In particular, add the "positivity constraint" and then perform
+ * the mapping.
+ */
+static struct isl_basic_set *homogeneous_map(struct isl_basic_set *bset,
+	struct isl_mat *T)
+{
+	int k;
+
+	if (!bset)
+		goto error;
+	bset = isl_basic_set_extend_constraints(bset, 0, 1);
+	k = isl_basic_set_alloc_inequality(bset);
+	if (k < 0)
+		goto error;
+	isl_seq_clr(bset->ineq[k] + 1, isl_basic_set_total_dim(bset));
+	isl_int_set_si(bset->ineq[k][0], 1);
+	bset = isl_basic_set_preimage(bset, T);
+	return bset;
+error:
+	isl_mat_free(T);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Compute the convex hull of a pair of basic sets without any parameters or
+ * integer divisions, where the convex hull is known to be pointed,
+ * but the basic sets may be unbounded.
+ *
+ * We turn this problem into the computation of a convex hull of a pair
+ * _bounded_ polyhedra by "changing the direction of the homogeneous
+ * dimension".  This idea is due to Matthias Koeppe.
+ *
+ * Consider the cones in homogeneous space that correspond to the
+ * input polyhedra.  The rays of these cones are also rays of the
+ * polyhedra if the coordinate that corresponds to the homogeneous
+ * dimension is zero.  That is, if the inner product of the rays
+ * with the homogeneous direction is zero.
+ * The cones in the homogeneous space can also be considered to
+ * correspond to other pairs of polyhedra by chosing a different
+ * homogeneous direction.  To ensure that both of these polyhedra
+ * are bounded, we need to make sure that all rays of the cones
+ * correspond to vertices and not to rays.
+ * Let s be a direction such that <s, r> > 0 for all rays r of both cones.
+ * Then using s as a homogeneous direction, we obtain a pair of polytopes.
+ * The vector s is computed in valid_direction.
+ *
+ * Note that we need to consider _all_ rays of the cones and not just
+ * the rays that correspond to rays in the polyhedra.  If we were to
+ * only consider those rays and turn them into vertices, then we
+ * may inadvertently turn some vertices into rays.
+ *
+ * The standard homogeneous direction is the unit vector in the 0th coordinate.
+ * We therefore transform the two polyhedra such that the selected
+ * direction is mapped onto this standard direction and then proceed
+ * with the normal computation.
+ * Let S be a non-singular square matrix with s as its first row,
+ * then we want to map the polyhedra to the space
+ *
+ *	[ y' ]     [ y ]		[ y ]          [ y' ]
+ *	[ x' ] = S [ x ]	i.e.,	[ x ] = S^{-1} [ x' ]
+ *
+ * We take S to be the unimodular completion of s to limit the growth
+ * of the coefficients in the following computations.
+ *
+ * Let b_i + A_i x >= 0 be the constraints of polyhedron i.
+ * We first move to the homogeneous dimension
+ *
+ *	b_i y + A_i x >= 0		[ b_i A_i ] [ y ]    [ 0 ]
+ *	    y         >= 0	or	[  1   0  ] [ x ] >= [ 0 ]
+ *
+ * Then we change directoin
+ *
+ *	[ b_i A_i ]        [ y' ]		              [ y' ]
+ *	[  1   0  ] S^{-1} [ x' ] >= 0	or	[ b_i' A_i' ] [ x' ] >= 0
+ *
+ * Then we compute the convex hull of the polytopes b_i' + A_i' x' >= 0
+ * resulting in b' + A' x' >= 0, which we then convert back
+ *
+ *	            [ y ]		        [ y ]
+ *	[ b' A' ] S [ x ] >= 0	or	[ b A ] [ x ] >= 0
+ *
+ * The polyhedron b + A x >= 0 is then the convex hull of the input polyhedra.
+ */
+static struct isl_basic_set *convex_hull_pair_pointed(
+	struct isl_basic_set *bset1, struct isl_basic_set *bset2)
+{
+	struct isl_ctx *ctx = NULL;
+	struct isl_vec *dir = NULL;
+	struct isl_mat *T = NULL;
+	struct isl_mat *T2 = NULL;
+	struct isl_basic_set *hull;
+	struct isl_set *set;
+
+	if (!bset1 || !bset2)
+		goto error;
+	ctx = bset1->ctx;
+	dir = valid_direction(isl_basic_set_copy(bset1),
+				isl_basic_set_copy(bset2));
+	if (!dir)
+		goto error;
+	T = isl_mat_alloc(bset1->ctx, dir->size, dir->size);
+	if (!T)
+		goto error;
+	isl_seq_cpy(T->row[0], dir->block.data, dir->size);
+	T = isl_mat_unimodular_complete(T, 1);
+	T2 = isl_mat_right_inverse(isl_mat_copy(T));
+
+	bset1 = homogeneous_map(bset1, isl_mat_copy(T2));
+	bset2 = homogeneous_map(bset2, T2);
+	set = isl_set_alloc_space(isl_basic_set_get_space(bset1), 2, 0);
+	set = isl_set_add_basic_set(set, bset1);
+	set = isl_set_add_basic_set(set, bset2);
+	hull = uset_convex_hull(set);
+	hull = isl_basic_set_preimage(hull, T);
+	 
+	isl_vec_free(dir);
+
+	return hull;
+error:
+	isl_vec_free(dir);
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return NULL;
+}
+
+static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set);
+static struct isl_basic_set *modulo_affine_hull(
+	struct isl_set *set, struct isl_basic_set *affine_hull);
+
+/* Compute the convex hull of a pair of basic sets without any parameters or
+ * integer divisions.
+ *
+ * This function is called from uset_convex_hull_unbounded, which
+ * means that the complete convex hull is unbounded.  Some pairs
+ * of basic sets may still be bounded, though.
+ * They may even lie inside a lower dimensional space, in which
+ * case they need to be handled inside their affine hull since
+ * the main algorithm assumes that the result is full-dimensional.
+ *
+ * If the convex hull of the two basic sets would have a non-trivial
+ * lineality space, we first project out this lineality space.
+ */
+static struct isl_basic_set *convex_hull_pair(struct isl_basic_set *bset1,
+	struct isl_basic_set *bset2)
+{
+	isl_basic_set *lin, *aff;
+	int bounded1, bounded2;
+
+	if (bset1->ctx->opt->convex == ISL_CONVEX_HULL_FM)
+		return convex_hull_pair_elim(bset1, bset2);
+
+	aff = isl_set_affine_hull(isl_basic_set_union(isl_basic_set_copy(bset1),
+						    isl_basic_set_copy(bset2)));
+	if (!aff)
+		goto error;
+	if (aff->n_eq != 0) 
+		return modulo_affine_hull(isl_basic_set_union(bset1, bset2), aff);
+	isl_basic_set_free(aff);
+
+	bounded1 = isl_basic_set_is_bounded(bset1);
+	bounded2 = isl_basic_set_is_bounded(bset2);
+
+	if (bounded1 < 0 || bounded2 < 0)
+		goto error;
+
+	if (bounded1 && bounded2)
+		return uset_convex_hull_wrap(isl_basic_set_union(bset1, bset2));
+
+	if (bounded1 || bounded2)
+		return convex_hull_pair_pointed(bset1, bset2);
+
+	lin = induced_lineality_space(isl_basic_set_copy(bset1),
+				      isl_basic_set_copy(bset2));
+	if (!lin)
+		goto error;
+	if (isl_basic_set_is_universe(lin)) {
+		isl_basic_set_free(bset1);
+		isl_basic_set_free(bset2);
+		return lin;
+	}
+	if (lin->n_eq < isl_basic_set_total_dim(lin)) {
+		struct isl_set *set;
+		set = isl_set_alloc_space(isl_basic_set_get_space(bset1), 2, 0);
+		set = isl_set_add_basic_set(set, bset1);
+		set = isl_set_add_basic_set(set, bset2);
+		return modulo_lineality(set, lin);
+	}
+	isl_basic_set_free(lin);
+
+	return convex_hull_pair_pointed(bset1, bset2);
+error:
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return NULL;
+}
+
+/* Compute the lineality space of a basic set.
+ * We currently do not allow the basic set to have any divs.
+ * We basically just drop the constants and turn every inequality
+ * into an equality.
+ */
+struct isl_basic_set *isl_basic_set_lineality_space(struct isl_basic_set *bset)
+{
+	int i, k;
+	struct isl_basic_set *lin = NULL;
+	unsigned dim;
+
+	if (!bset)
+		goto error;
+	isl_assert(bset->ctx, bset->n_div == 0, goto error);
+	dim = isl_basic_set_total_dim(bset);
+
+	lin = isl_basic_set_alloc_space(isl_basic_set_get_space(bset), 0, dim, 0);
+	if (!lin)
+		goto error;
+	for (i = 0; i < bset->n_eq; ++i) {
+		k = isl_basic_set_alloc_equality(lin);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(lin->eq[k][0], 0);
+		isl_seq_cpy(lin->eq[k] + 1, bset->eq[i] + 1, dim);
+	}
+	lin = isl_basic_set_gauss(lin, NULL);
+	if (!lin)
+		goto error;
+	for (i = 0; i < bset->n_ineq && lin->n_eq < dim; ++i) {
+		k = isl_basic_set_alloc_equality(lin);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(lin->eq[k][0], 0);
+		isl_seq_cpy(lin->eq[k] + 1, bset->ineq[i] + 1, dim);
+		lin = isl_basic_set_gauss(lin, NULL);
+		if (!lin)
+			goto error;
+	}
+	isl_basic_set_free(bset);
+	return lin;
+error:
+	isl_basic_set_free(lin);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Compute the (linear) hull of the lineality spaces of the basic sets in the
+ * "underlying" set "set".
+ */
+static struct isl_basic_set *uset_combined_lineality_space(struct isl_set *set)
+{
+	int i;
+	struct isl_set *lin = NULL;
+
+	if (!set)
+		return NULL;
+	if (set->n == 0) {
+		isl_space *dim = isl_set_get_space(set);
+		isl_set_free(set);
+		return isl_basic_set_empty(dim);
+	}
+
+	lin = isl_set_alloc_space(isl_set_get_space(set), set->n, 0);
+	for (i = 0; i < set->n; ++i)
+		lin = isl_set_add_basic_set(lin,
+		    isl_basic_set_lineality_space(isl_basic_set_copy(set->p[i])));
+	isl_set_free(set);
+	return isl_set_affine_hull(lin);
+}
+
+/* Compute the convex hull of a set without any parameters or
+ * integer divisions.
+ * In each step, we combined two basic sets until only one
+ * basic set is left.
+ * The input basic sets are assumed not to have a non-trivial
+ * lineality space.  If any of the intermediate results has
+ * a non-trivial lineality space, it is projected out.
+ */
+static struct isl_basic_set *uset_convex_hull_unbounded(struct isl_set *set)
+{
+	struct isl_basic_set *convex_hull = NULL;
+
+	convex_hull = isl_set_copy_basic_set(set);
+	set = isl_set_drop_basic_set(set, convex_hull);
+	if (!set)
+		goto error;
+	while (set->n > 0) {
+		struct isl_basic_set *t;
+		t = isl_set_copy_basic_set(set);
+		if (!t)
+			goto error;
+		set = isl_set_drop_basic_set(set, t);
+		if (!set)
+			goto error;
+		convex_hull = convex_hull_pair(convex_hull, t);
+		if (set->n == 0)
+			break;
+		t = isl_basic_set_lineality_space(isl_basic_set_copy(convex_hull));
+		if (!t)
+			goto error;
+		if (isl_basic_set_is_universe(t)) {
+			isl_basic_set_free(convex_hull);
+			convex_hull = t;
+			break;
+		}
+		if (t->n_eq < isl_basic_set_total_dim(t)) {
+			set = isl_set_add_basic_set(set, convex_hull);
+			return modulo_lineality(set, t);
+		}
+		isl_basic_set_free(t);
+	}
+	isl_set_free(set);
+	return convex_hull;
+error:
+	isl_set_free(set);
+	isl_basic_set_free(convex_hull);
+	return NULL;
+}
+
+/* Compute an initial hull for wrapping containing a single initial
+ * facet.
+ * This function assumes that the given set is bounded.
+ */
+static struct isl_basic_set *initial_hull(struct isl_basic_set *hull,
+	struct isl_set *set)
+{
+	struct isl_mat *bounds = NULL;
+	unsigned dim;
+	int k;
+
+	if (!hull)
+		goto error;
+	bounds = initial_facet_constraint(set);
+	if (!bounds)
+		goto error;
+	k = isl_basic_set_alloc_inequality(hull);
+	if (k < 0)
+		goto error;
+	dim = isl_set_n_dim(set);
+	isl_assert(set->ctx, 1 + dim == bounds->n_col, goto error);
+	isl_seq_cpy(hull->ineq[k], bounds->row[0], bounds->n_col);
+	isl_mat_free(bounds);
+
+	return hull;
+error:
+	isl_basic_set_free(hull);
+	isl_mat_free(bounds);
+	return NULL;
+}
+
+struct max_constraint {
+	struct isl_mat *c;
+	int	 	count;
+	int		ineq;
+};
+
+static int max_constraint_equal(const void *entry, const void *val)
+{
+	struct max_constraint *a = (struct max_constraint *)entry;
+	isl_int *b = (isl_int *)val;
+
+	return isl_seq_eq(a->c->row[0] + 1, b, a->c->n_col - 1);
+}
+
+static void update_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
+	isl_int *con, unsigned len, int n, int ineq)
+{
+	struct isl_hash_table_entry *entry;
+	struct max_constraint *c;
+	uint32_t c_hash;
+
+	c_hash = isl_seq_get_hash(con + 1, len);
+	entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
+			con + 1, 0);
+	if (!entry)
+		return;
+	c = entry->data;
+	if (c->count < n) {
+		isl_hash_table_remove(ctx, table, entry);
+		return;
+	}
+	c->count++;
+	if (isl_int_gt(c->c->row[0][0], con[0]))
+		return;
+	if (isl_int_eq(c->c->row[0][0], con[0])) {
+		if (ineq)
+			c->ineq = ineq;
+		return;
+	}
+	c->c = isl_mat_cow(c->c);
+	isl_int_set(c->c->row[0][0], con[0]);
+	c->ineq = ineq;
+}
+
+/* Check whether the constraint hash table "table" constains the constraint
+ * "con".
+ */
+static int has_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
+	isl_int *con, unsigned len, int n)
+{
+	struct isl_hash_table_entry *entry;
+	struct max_constraint *c;
+	uint32_t c_hash;
+
+	c_hash = isl_seq_get_hash(con + 1, len);
+	entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
+			con + 1, 0);
+	if (!entry)
+		return 0;
+	c = entry->data;
+	if (c->count < n)
+		return 0;
+	return isl_int_eq(c->c->row[0][0], con[0]);
+}
+
+/* Check for inequality constraints of a basic set without equalities
+ * such that the same or more stringent copies of the constraint appear
+ * in all of the basic sets.  Such constraints are necessarily facet
+ * constraints of the convex hull.
+ *
+ * If the resulting basic set is by chance identical to one of
+ * the basic sets in "set", then we know that this basic set contains
+ * all other basic sets and is therefore the convex hull of set.
+ * In this case we set *is_hull to 1.
+ */
+static struct isl_basic_set *common_constraints(struct isl_basic_set *hull,
+	struct isl_set *set, int *is_hull)
+{
+	int i, j, s, n;
+	int min_constraints;
+	int best;
+	struct max_constraint *constraints = NULL;
+	struct isl_hash_table *table = NULL;
+	unsigned total;
+
+	*is_hull = 0;
+
+	for (i = 0; i < set->n; ++i)
+		if (set->p[i]->n_eq == 0)
+			break;
+	if (i >= set->n)
+		return hull;
+	min_constraints = set->p[i]->n_ineq;
+	best = i;
+	for (i = best + 1; i < set->n; ++i) {
+		if (set->p[i]->n_eq != 0)
+			continue;
+		if (set->p[i]->n_ineq >= min_constraints)
+			continue;
+		min_constraints = set->p[i]->n_ineq;
+		best = i;
+	}
+	constraints = isl_calloc_array(hull->ctx, struct max_constraint,
+					min_constraints);
+	if (!constraints)
+		return hull;
+	table = isl_alloc_type(hull->ctx, struct isl_hash_table);
+	if (isl_hash_table_init(hull->ctx, table, min_constraints))
+		goto error;
+
+	total = isl_space_dim(set->dim, isl_dim_all);
+	for (i = 0; i < set->p[best]->n_ineq; ++i) {
+		constraints[i].c = isl_mat_sub_alloc6(hull->ctx,
+			set->p[best]->ineq + i, 0, 1, 0, 1 + total);
+		if (!constraints[i].c)
+			goto error;
+		constraints[i].ineq = 1;
+	}
+	for (i = 0; i < min_constraints; ++i) {
+		struct isl_hash_table_entry *entry;
+		uint32_t c_hash;
+		c_hash = isl_seq_get_hash(constraints[i].c->row[0] + 1, total);
+		entry = isl_hash_table_find(hull->ctx, table, c_hash,
+			max_constraint_equal, constraints[i].c->row[0] + 1, 1);
+		if (!entry)
+			goto error;
+		isl_assert(hull->ctx, !entry->data, goto error);
+		entry->data = &constraints[i];
+	}
+
+	n = 0;
+	for (s = 0; s < set->n; ++s) {
+		if (s == best)
+			continue;
+
+		for (i = 0; i < set->p[s]->n_eq; ++i) {
+			isl_int *eq = set->p[s]->eq[i];
+			for (j = 0; j < 2; ++j) {
+				isl_seq_neg(eq, eq, 1 + total);
+				update_constraint(hull->ctx, table,
+							    eq, total, n, 0);
+			}
+		}
+		for (i = 0; i < set->p[s]->n_ineq; ++i) {
+			isl_int *ineq = set->p[s]->ineq[i];
+			update_constraint(hull->ctx, table, ineq, total, n,
+				set->p[s]->n_eq == 0);
+		}
+		++n;
+	}
+
+	for (i = 0; i < min_constraints; ++i) {
+		if (constraints[i].count < n)
+			continue;
+		if (!constraints[i].ineq)
+			continue;
+		j = isl_basic_set_alloc_inequality(hull);
+		if (j < 0)
+			goto error;
+		isl_seq_cpy(hull->ineq[j], constraints[i].c->row[0], 1 + total);
+	}
+
+	for (s = 0; s < set->n; ++s) {
+		if (set->p[s]->n_eq)
+			continue;
+		if (set->p[s]->n_ineq != hull->n_ineq)
+			continue;
+		for (i = 0; i < set->p[s]->n_ineq; ++i) {
+			isl_int *ineq = set->p[s]->ineq[i];
+			if (!has_constraint(hull->ctx, table, ineq, total, n))
+				break;
+		}
+		if (i == set->p[s]->n_ineq)
+			*is_hull = 1;
+	}
+
+	isl_hash_table_clear(table);
+	for (i = 0; i < min_constraints; ++i)
+		isl_mat_free(constraints[i].c);
+	free(constraints);
+	free(table);
+	return hull;
+error:
+	isl_hash_table_clear(table);
+	free(table);
+	if (constraints)
+		for (i = 0; i < min_constraints; ++i)
+			isl_mat_free(constraints[i].c);
+	free(constraints);
+	return hull;
+}
+
+/* Create a template for the convex hull of "set" and fill it up
+ * obvious facet constraints, if any.  If the result happens to
+ * be the convex hull of "set" then *is_hull is set to 1.
+ */
+static struct isl_basic_set *proto_hull(struct isl_set *set, int *is_hull)
+{
+	struct isl_basic_set *hull;
+	unsigned n_ineq;
+	int i;
+
+	n_ineq = 1;
+	for (i = 0; i < set->n; ++i) {
+		n_ineq += set->p[i]->n_eq;
+		n_ineq += set->p[i]->n_ineq;
+	}
+	hull = isl_basic_set_alloc_space(isl_space_copy(set->dim), 0, 0, n_ineq);
+	hull = isl_basic_set_set_rational(hull);
+	if (!hull)
+		return NULL;
+	return common_constraints(hull, set, is_hull);
+}
+
+static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set)
+{
+	struct isl_basic_set *hull;
+	int is_hull;
+
+	hull = proto_hull(set, &is_hull);
+	if (hull && !is_hull) {
+		if (hull->n_ineq == 0)
+			hull = initial_hull(hull, set);
+		hull = extend(hull, set);
+	}
+	isl_set_free(set);
+
+	return hull;
+}
+
+/* Compute the convex hull of a set without any parameters or
+ * integer divisions.  Depending on whether the set is bounded,
+ * we pass control to the wrapping based convex hull or
+ * the Fourier-Motzkin elimination based convex hull.
+ * We also handle a few special cases before checking the boundedness.
+ */
+static struct isl_basic_set *uset_convex_hull(struct isl_set *set)
+{
+	struct isl_basic_set *convex_hull = NULL;
+	struct isl_basic_set *lin;
+
+	if (isl_set_n_dim(set) == 0)
+		return convex_hull_0d(set);
+
+	set = isl_set_coalesce(set);
+	set = isl_set_set_rational(set);
+
+	if (!set)
+		goto error;
+	if (!set)
+		return NULL;
+	if (set->n == 1) {
+		convex_hull = isl_basic_set_copy(set->p[0]);
+		isl_set_free(set);
+		return convex_hull;
+	}
+	if (isl_set_n_dim(set) == 1)
+		return convex_hull_1d(set);
+
+	if (isl_set_is_bounded(set) &&
+	    set->ctx->opt->convex == ISL_CONVEX_HULL_WRAP)
+		return uset_convex_hull_wrap(set);
+
+	lin = uset_combined_lineality_space(isl_set_copy(set));
+	if (!lin)
+		goto error;
+	if (isl_basic_set_is_universe(lin)) {
+		isl_set_free(set);
+		return lin;
+	}
+	if (lin->n_eq < isl_basic_set_total_dim(lin))
+		return modulo_lineality(set, lin);
+	isl_basic_set_free(lin);
+
+	return uset_convex_hull_unbounded(set);
+error:
+	isl_set_free(set);
+	isl_basic_set_free(convex_hull);
+	return NULL;
+}
+
+/* This is the core procedure, where "set" is a "pure" set, i.e.,
+ * without parameters or divs and where the convex hull of set is
+ * known to be full-dimensional.
+ */
+static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set)
+{
+	struct isl_basic_set *convex_hull = NULL;
+
+	if (!set)
+		goto error;
+
+	if (isl_set_n_dim(set) == 0) {
+		convex_hull = isl_basic_set_universe(isl_space_copy(set->dim));
+		isl_set_free(set);
+		convex_hull = isl_basic_set_set_rational(convex_hull);
+		return convex_hull;
+	}
+
+	set = isl_set_set_rational(set);
+	set = isl_set_coalesce(set);
+	if (!set)
+		goto error;
+	if (set->n == 1) {
+		convex_hull = isl_basic_set_copy(set->p[0]);
+		isl_set_free(set);
+		convex_hull = isl_basic_map_remove_redundancies(convex_hull);
+		return convex_hull;
+	}
+	if (isl_set_n_dim(set) == 1)
+		return convex_hull_1d(set);
+
+	return uset_convex_hull_wrap(set);
+error:
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Compute the convex hull of set "set" with affine hull "affine_hull",
+ * We first remove the equalities (transforming the set), compute the
+ * convex hull of the transformed set and then add the equalities back
+ * (after performing the inverse transformation.
+ */
+static struct isl_basic_set *modulo_affine_hull(
+	struct isl_set *set, struct isl_basic_set *affine_hull)
+{
+	struct isl_mat *T;
+	struct isl_mat *T2;
+	struct isl_basic_set *dummy;
+	struct isl_basic_set *convex_hull;
+
+	dummy = isl_basic_set_remove_equalities(
+			isl_basic_set_copy(affine_hull), &T, &T2);
+	if (!dummy)
+		goto error;
+	isl_basic_set_free(dummy);
+	set = isl_set_preimage(set, T);
+	convex_hull = uset_convex_hull(set);
+	convex_hull = isl_basic_set_preimage(convex_hull, T2);
+	convex_hull = isl_basic_set_intersect(convex_hull, affine_hull);
+	return convex_hull;
+error:
+	isl_basic_set_free(affine_hull);
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Compute the convex hull of a map.
+ *
+ * The implementation was inspired by "Extended Convex Hull" by Fukuda et al.,
+ * specifically, the wrapping of facets to obtain new facets.
+ */
+struct isl_basic_map *isl_map_convex_hull(struct isl_map *map)
+{
+	struct isl_basic_set *bset;
+	struct isl_basic_map *model = NULL;
+	struct isl_basic_set *affine_hull = NULL;
+	struct isl_basic_map *convex_hull = NULL;
+	struct isl_set *set = NULL;
+	struct isl_ctx *ctx;
+
+	map = isl_map_detect_equalities(map);
+	map = isl_map_align_divs(map);
+	if (!map)
+		goto error;
+
+	ctx = map->ctx;
+	if (map->n == 0) {
+		convex_hull = isl_basic_map_empty_like_map(map);
+		isl_map_free(map);
+		return convex_hull;
+	}
+
+	model = isl_basic_map_copy(map->p[0]);
+	set = isl_map_underlying_set(map);
+	if (!set)
+		goto error;
+
+	affine_hull = isl_set_affine_hull(isl_set_copy(set));
+	if (!affine_hull)
+		goto error;
+	if (affine_hull->n_eq != 0)
+		bset = modulo_affine_hull(set, affine_hull);
+	else {
+		isl_basic_set_free(affine_hull);
+		bset = uset_convex_hull(set);
+	}
+
+	convex_hull = isl_basic_map_overlying_set(bset, model);
+	if (!convex_hull)
+		return NULL;
+
+	ISL_F_SET(convex_hull, ISL_BASIC_MAP_NO_IMPLICIT);
+	ISL_F_SET(convex_hull, ISL_BASIC_MAP_ALL_EQUALITIES);
+	ISL_F_CLR(convex_hull, ISL_BASIC_MAP_RATIONAL);
+	return convex_hull;
+error:
+	isl_set_free(set);
+	isl_basic_map_free(model);
+	return NULL;
+}
+
+struct isl_basic_set *isl_set_convex_hull(struct isl_set *set)
+{
+	return (struct isl_basic_set *)
+		isl_map_convex_hull((struct isl_map *)set);
+}
+
+__isl_give isl_basic_map *isl_map_polyhedral_hull(__isl_take isl_map *map)
+{
+	isl_basic_map *hull;
+
+	hull = isl_map_convex_hull(map);
+	return isl_basic_map_remove_divs(hull);
+}
+
+__isl_give isl_basic_set *isl_set_polyhedral_hull(__isl_take isl_set *set)
+{
+	return (isl_basic_set *)isl_map_polyhedral_hull((isl_map *)set);
+}
+
+struct sh_data_entry {
+	struct isl_hash_table	*table;
+	struct isl_tab		*tab;
+};
+
+/* Holds the data needed during the simple hull computation.
+ * In particular,
+ *	n		the number of basic sets in the original set
+ *	hull_table	a hash table of already computed constraints
+ *			in the simple hull
+ *	p		for each basic set,
+ *		table		a hash table of the constraints
+ *		tab		the tableau corresponding to the basic set
+ */
+struct sh_data {
+	struct isl_ctx		*ctx;
+	unsigned		n;
+	struct isl_hash_table	*hull_table;
+	struct sh_data_entry	p[1];
+};
+
+static void sh_data_free(struct sh_data *data)
+{
+	int i;
+
+	if (!data)
+		return;
+	isl_hash_table_free(data->ctx, data->hull_table);
+	for (i = 0; i < data->n; ++i) {
+		isl_hash_table_free(data->ctx, data->p[i].table);
+		isl_tab_free(data->p[i].tab);
+	}
+	free(data);
+}
+
+struct ineq_cmp_data {
+	unsigned	len;
+	isl_int		*p;
+};
+
+static int has_ineq(const void *entry, const void *val)
+{
+	isl_int *row = (isl_int *)entry;
+	struct ineq_cmp_data *v = (struct ineq_cmp_data *)val;
+
+	return isl_seq_eq(row + 1, v->p + 1, v->len) ||
+	       isl_seq_is_neg(row + 1, v->p + 1, v->len);
+}
+
+static int hash_ineq(struct isl_ctx *ctx, struct isl_hash_table *table,
+			isl_int *ineq, unsigned len)
+{
+	uint32_t c_hash;
+	struct ineq_cmp_data v;
+	struct isl_hash_table_entry *entry;
+
+	v.len = len;
+	v.p = ineq;
+	c_hash = isl_seq_get_hash(ineq + 1, len);
+	entry = isl_hash_table_find(ctx, table, c_hash, has_ineq, &v, 1);
+	if (!entry)
+		return - 1;
+	entry->data = ineq;
+	return 0;
+}
+
+/* Fill hash table "table" with the constraints of "bset".
+ * Equalities are added as two inequalities.
+ * The value in the hash table is a pointer to the (in)equality of "bset".
+ */
+static int hash_basic_set(struct isl_hash_table *table,
+				struct isl_basic_set *bset)
+{
+	int i, j;
+	unsigned dim = isl_basic_set_total_dim(bset);
+
+	for (i = 0; i < bset->n_eq; ++i) {
+		for (j = 0; j < 2; ++j) {
+			isl_seq_neg(bset->eq[i], bset->eq[i], 1 + dim);
+			if (hash_ineq(bset->ctx, table, bset->eq[i], dim) < 0)
+				return -1;
+		}
+	}
+	for (i = 0; i < bset->n_ineq; ++i) {
+		if (hash_ineq(bset->ctx, table, bset->ineq[i], dim) < 0)
+			return -1;
+	}
+	return 0;
+}
+
+static struct sh_data *sh_data_alloc(struct isl_set *set, unsigned n_ineq)
+{
+	struct sh_data *data;
+	int i;
+
+	data = isl_calloc(set->ctx, struct sh_data,
+		sizeof(struct sh_data) +
+		(set->n - 1) * sizeof(struct sh_data_entry));
+	if (!data)
+		return NULL;
+	data->ctx = set->ctx;
+	data->n = set->n;
+	data->hull_table = isl_hash_table_alloc(set->ctx, n_ineq);
+	if (!data->hull_table)
+		goto error;
+	for (i = 0; i < set->n; ++i) {
+		data->p[i].table = isl_hash_table_alloc(set->ctx,
+				    2 * set->p[i]->n_eq + set->p[i]->n_ineq);
+		if (!data->p[i].table)
+			goto error;
+		if (hash_basic_set(data->p[i].table, set->p[i]) < 0)
+			goto error;
+	}
+	return data;
+error:
+	sh_data_free(data);
+	return NULL;
+}
+
+/* Check if inequality "ineq" is a bound for basic set "j" or if
+ * it can be relaxed (by increasing the constant term) to become
+ * a bound for that basic set.  In the latter case, the constant
+ * term is updated.
+ * Relaxation of the constant term is only allowed if "shift" is set.
+ *
+ * Return 1 if "ineq" is a bound
+ *	  0 if "ineq" may attain arbitrarily small values on basic set "j"
+ *	 -1 if some error occurred
+ */
+static int is_bound(struct sh_data *data, struct isl_set *set, int j,
+	isl_int *ineq, int shift)
+{
+	enum isl_lp_result res;
+	isl_int opt;
+
+	if (!data->p[j].tab) {
+		data->p[j].tab = isl_tab_from_basic_set(set->p[j], 0);
+		if (!data->p[j].tab)
+			return -1;
+	}
+
+	isl_int_init(opt);
+
+	res = isl_tab_min(data->p[j].tab, ineq, data->ctx->one,
+				&opt, NULL, 0);
+	if (res == isl_lp_ok && isl_int_is_neg(opt)) {
+		if (shift)
+			isl_int_sub(ineq[0], ineq[0], opt);
+		else
+			res = isl_lp_unbounded;
+	}
+
+	isl_int_clear(opt);
+
+	return (res == isl_lp_ok || res == isl_lp_empty) ? 1 :
+	       res == isl_lp_unbounded ? 0 : -1;
+}
+
+/* Check if inequality "ineq" from basic set "i" is or can be relaxed to
+ * become a bound on the whole set.  If so, add the (relaxed) inequality
+ * to "hull".  Relaxation is only allowed if "shift" is set.
+ *
+ * We first check if "hull" already contains a translate of the inequality.
+ * If so, we are done.
+ * Then, we check if any of the previous basic sets contains a translate
+ * of the inequality.  If so, then we have already considered this
+ * inequality and we are done.
+ * Otherwise, for each basic set other than "i", we check if the inequality
+ * is a bound on the basic set.
+ * For previous basic sets, we know that they do not contain a translate
+ * of the inequality, so we directly call is_bound.
+ * For following basic sets, we first check if a translate of the
+ * inequality appears in its description and if so directly update
+ * the inequality accordingly.
+ */
+static struct isl_basic_set *add_bound(struct isl_basic_set *hull,
+	struct sh_data *data, struct isl_set *set, int i, isl_int *ineq,
+	int shift)
+{
+	uint32_t c_hash;
+	struct ineq_cmp_data v;
+	struct isl_hash_table_entry *entry;
+	int j, k;
+
+	if (!hull)
+		return NULL;
+
+	v.len = isl_basic_set_total_dim(hull);
+	v.p = ineq;
+	c_hash = isl_seq_get_hash(ineq + 1, v.len);
+
+	entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
+					has_ineq, &v, 0);
+	if (entry)
+		return hull;
+
+	for (j = 0; j < i; ++j) {
+		entry = isl_hash_table_find(hull->ctx, data->p[j].table,
+						c_hash, has_ineq, &v, 0);
+		if (entry)
+			break;
+	}
+	if (j < i)
+		return hull;
+
+	k = isl_basic_set_alloc_inequality(hull);
+	if (k < 0)
+		goto error;
+	isl_seq_cpy(hull->ineq[k], ineq, 1 + v.len);
+
+	for (j = 0; j < i; ++j) {
+		int bound;
+		bound = is_bound(data, set, j, hull->ineq[k], shift);
+		if (bound < 0)
+			goto error;
+		if (!bound)
+			break;
+	}
+	if (j < i) {
+		isl_basic_set_free_inequality(hull, 1);
+		return hull;
+	}
+
+	for (j = i + 1; j < set->n; ++j) {
+		int bound, neg;
+		isl_int *ineq_j;
+		entry = isl_hash_table_find(hull->ctx, data->p[j].table,
+						c_hash, has_ineq, &v, 0);
+		if (entry) {
+			ineq_j = entry->data;
+			neg = isl_seq_is_neg(ineq_j + 1,
+					     hull->ineq[k] + 1, v.len);
+			if (neg)
+				isl_int_neg(ineq_j[0], ineq_j[0]);
+			if (isl_int_gt(ineq_j[0], hull->ineq[k][0]))
+				isl_int_set(hull->ineq[k][0], ineq_j[0]);
+			if (neg)
+				isl_int_neg(ineq_j[0], ineq_j[0]);
+			continue;
+		}
+		bound = is_bound(data, set, j, hull->ineq[k], shift);
+		if (bound < 0)
+			goto error;
+		if (!bound)
+			break;
+	}
+	if (j < set->n) {
+		isl_basic_set_free_inequality(hull, 1);
+		return hull;
+	}
+
+	entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
+					has_ineq, &v, 1);
+	if (!entry)
+		goto error;
+	entry->data = hull->ineq[k];
+
+	return hull;
+error:
+	isl_basic_set_free(hull);
+	return NULL;
+}
+
+/* Check if any inequality from basic set "i" is or can be relaxed to
+ * become a bound on the whole set.  If so, add the (relaxed) inequality
+ * to "hull".  Relaxation is only allowed if "shift" is set.
+ */
+static struct isl_basic_set *add_bounds(struct isl_basic_set *bset,
+	struct sh_data *data, struct isl_set *set, int i, int shift)
+{
+	int j, k;
+	unsigned dim = isl_basic_set_total_dim(bset);
+
+	for (j = 0; j < set->p[i]->n_eq; ++j) {
+		for (k = 0; k < 2; ++k) {
+			isl_seq_neg(set->p[i]->eq[j], set->p[i]->eq[j], 1+dim);
+			bset = add_bound(bset, data, set, i, set->p[i]->eq[j],
+					    shift);
+		}
+	}
+	for (j = 0; j < set->p[i]->n_ineq; ++j)
+		bset = add_bound(bset, data, set, i, set->p[i]->ineq[j], shift);
+	return bset;
+}
+
+/* Compute a superset of the convex hull of set that is described
+ * by only (translates of) the constraints in the constituents of set.
+ * Translation is only allowed if "shift" is set.
+ */
+static __isl_give isl_basic_set *uset_simple_hull(__isl_take isl_set *set,
+	int shift)
+{
+	struct sh_data *data = NULL;
+	struct isl_basic_set *hull = NULL;
+	unsigned n_ineq;
+	int i;
+
+	if (!set)
+		return NULL;
+
+	n_ineq = 0;
+	for (i = 0; i < set->n; ++i) {
+		if (!set->p[i])
+			goto error;
+		n_ineq += 2 * set->p[i]->n_eq + set->p[i]->n_ineq;
+	}
+
+	hull = isl_basic_set_alloc_space(isl_space_copy(set->dim), 0, 0, n_ineq);
+	if (!hull)
+		goto error;
+
+	data = sh_data_alloc(set, n_ineq);
+	if (!data)
+		goto error;
+
+	for (i = 0; i < set->n; ++i)
+		hull = add_bounds(hull, data, set, i, shift);
+
+	sh_data_free(data);
+	isl_set_free(set);
+
+	return hull;
+error:
+	sh_data_free(data);
+	isl_basic_set_free(hull);
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Compute a superset of the convex hull of map that is described
+ * by only (translates of) the constraints in the constituents of map.
+ * Translation is only allowed if "shift" is set.
+ */
+static __isl_give isl_basic_map *map_simple_hull(__isl_take isl_map *map,
+	int shift)
+{
+	struct isl_set *set = NULL;
+	struct isl_basic_map *model = NULL;
+	struct isl_basic_map *hull;
+	struct isl_basic_map *affine_hull;
+	struct isl_basic_set *bset = NULL;
+
+	if (!map)
+		return NULL;
+	if (map->n == 0) {
+		hull = isl_basic_map_empty_like_map(map);
+		isl_map_free(map);
+		return hull;
+	}
+	if (map->n == 1) {
+		hull = isl_basic_map_copy(map->p[0]);
+		isl_map_free(map);
+		return hull;
+	}
+
+	map = isl_map_detect_equalities(map);
+	affine_hull = isl_map_affine_hull(isl_map_copy(map));
+	map = isl_map_align_divs(map);
+	model = map ? isl_basic_map_copy(map->p[0]) : NULL;
+
+	set = isl_map_underlying_set(map);
+
+	bset = uset_simple_hull(set, shift);
+
+	hull = isl_basic_map_overlying_set(bset, model);
+
+	hull = isl_basic_map_intersect(hull, affine_hull);
+	hull = isl_basic_map_remove_redundancies(hull);
+
+	if (!hull)
+		return NULL;
+	ISL_F_SET(hull, ISL_BASIC_MAP_NO_IMPLICIT);
+	ISL_F_SET(hull, ISL_BASIC_MAP_ALL_EQUALITIES);
+
+	hull = isl_basic_map_finalize(hull);
+
+	return hull;
+}
+
+/* Compute a superset of the convex hull of map that is described
+ * by only translates of the constraints in the constituents of map.
+ */
+__isl_give isl_basic_map *isl_map_simple_hull(__isl_take isl_map *map)
+{
+	return map_simple_hull(map, 1);
+}
+
+struct isl_basic_set *isl_set_simple_hull(struct isl_set *set)
+{
+	return (struct isl_basic_set *)
+		isl_map_simple_hull((struct isl_map *)set);
+}
+
+/* Compute a superset of the convex hull of map that is described
+ * by only the constraints in the constituents of map.
+ */
+__isl_give isl_basic_map *isl_map_unshifted_simple_hull(
+	__isl_take isl_map *map)
+{
+	return map_simple_hull(map, 0);
+}
+
+__isl_give isl_basic_set *isl_set_unshifted_simple_hull(
+	__isl_take isl_set *set)
+{
+	return isl_map_unshifted_simple_hull(set);
+}
+
+/* Check if "ineq" is a bound on "set" and, if so, add it to "hull".
+ *
+ * For each basic set in "set", we first check if the basic set
+ * contains a translate of "ineq".  If this translate is more relaxed,
+ * then we assume that "ineq" is not a bound on this basic set.
+ * Otherwise, we know that it is a bound.
+ * If the basic set does not contain a translate of "ineq", then
+ * we call is_bound to perform the test.
+ */
+static __isl_give isl_basic_set *add_bound_from_constraint(
+	__isl_take isl_basic_set *hull, struct sh_data *data,
+	__isl_keep isl_set *set, isl_int *ineq)
+{
+	int i, k;
+	isl_ctx *ctx;
+	uint32_t c_hash;
+	struct ineq_cmp_data v;
+
+	if (!hull || !set)
+		return isl_basic_set_free(hull);
+
+	v.len = isl_basic_set_total_dim(hull);
+	v.p = ineq;
+	c_hash = isl_seq_get_hash(ineq + 1, v.len);
+
+	ctx = isl_basic_set_get_ctx(hull);
+	for (i = 0; i < set->n; ++i) {
+		int bound;
+		struct isl_hash_table_entry *entry;
+
+		entry = isl_hash_table_find(ctx, data->p[i].table,
+						c_hash, &has_ineq, &v, 0);
+		if (entry) {
+			isl_int *ineq_i = entry->data;
+			int neg, more_relaxed;
+
+			neg = isl_seq_is_neg(ineq_i + 1, ineq + 1, v.len);
+			if (neg)
+				isl_int_neg(ineq_i[0], ineq_i[0]);
+			more_relaxed = isl_int_gt(ineq_i[0], ineq[0]);
+			if (neg)
+				isl_int_neg(ineq_i[0], ineq_i[0]);
+			if (more_relaxed)
+				break;
+			else
+				continue;
+		}
+		bound = is_bound(data, set, i, ineq, 0);
+		if (bound < 0)
+			return isl_basic_set_free(hull);
+		if (!bound)
+			break;
+	}
+	if (i < set->n)
+		return hull;
+
+	k = isl_basic_set_alloc_inequality(hull);
+	if (k < 0)
+		return isl_basic_set_free(hull);
+	isl_seq_cpy(hull->ineq[k], ineq, 1 + v.len);
+
+	return hull;
+}
+
+/* Compute a superset of the convex hull of "set" that is described
+ * by only some of the "n_ineq" constraints in the list "ineq", where "set"
+ * has no parameters or integer divisions.
+ *
+ * The inequalities in "ineq" are assumed to have been sorted such
+ * that constraints with the same linear part appear together and
+ * that among constraints with the same linear part, those with
+ * smaller constant term appear first.
+ *
+ * We reuse the same data structure that is used by uset_simple_hull,
+ * but we do not need the hull table since we will not consider the
+ * same constraint more than once.  We therefore allocate it with zero size.
+ *
+ * We run through the constraints and try to add them one by one,
+ * skipping identical constraints.  If we have added a constraint and
+ * the next constraint is a more relaxed translate, then we skip this
+ * next constraint as well.
+ */
+static __isl_give isl_basic_set *uset_unshifted_simple_hull_from_constraints(
+	__isl_take isl_set *set, int n_ineq, isl_int **ineq)
+{
+	int i;
+	int last_added = 0;
+	struct sh_data *data = NULL;
+	isl_basic_set *hull = NULL;
+	unsigned dim;
+
+	hull = isl_basic_set_alloc_space(isl_set_get_space(set), 0, 0, n_ineq);
+	if (!hull)
+		goto error;
+
+	data = sh_data_alloc(set, 0);
+	if (!data)
+		goto error;
+
+	dim = isl_set_dim(set, isl_dim_set);
+	for (i = 0; i < n_ineq; ++i) {
+		int hull_n_ineq = hull->n_ineq;
+		int parallel;
+
+		parallel = i > 0 && isl_seq_eq(ineq[i - 1] + 1, ineq[i] + 1,
+						dim);
+		if (parallel &&
+		    (last_added || isl_int_eq(ineq[i - 1][0], ineq[i][0])))
+			continue;
+		hull = add_bound_from_constraint(hull, data, set, ineq[i]);
+		if (!hull)
+			goto error;
+		last_added = hull->n_ineq > hull_n_ineq;
+	}
+
+	sh_data_free(data);
+	isl_set_free(set);
+	return hull;
+error:
+	sh_data_free(data);
+	isl_set_free(set);
+	isl_basic_set_free(hull);
+	return NULL;
+}
+
+/* Collect pointers to all the inequalities in the elements of "list"
+ * in "ineq".  For equalities, store both a pointer to the equality and
+ * a pointer to its opposite, which is first copied to "mat".
+ * "ineq" and "mat" are assumed to have been preallocated to the right size
+ * (the number of inequalities + 2 times the number of equalites and
+ * the number of equalities, respectively).
+ */
+static __isl_give isl_mat *collect_inequalities(__isl_take isl_mat *mat,
+	__isl_keep isl_basic_set_list *list, isl_int **ineq)
+{
+	int i, j, n, n_eq, n_ineq;
+
+	if (!mat)
+		return NULL;
+
+	n_eq = 0;
+	n_ineq = 0;
+	n = isl_basic_set_list_n_basic_set(list);
+	for (i = 0; i < n; ++i) {
+		isl_basic_set *bset;
+		bset = isl_basic_set_list_get_basic_set(list, i);
+		if (!bset)
+			return isl_mat_free(mat);
+		for (j = 0; j < bset->n_eq; ++j) {
+			ineq[n_ineq++] = mat->row[n_eq];
+			ineq[n_ineq++] = bset->eq[j];
+			isl_seq_neg(mat->row[n_eq++], bset->eq[j], mat->n_col);
+		}
+		for (j = 0; j < bset->n_ineq; ++j)
+			ineq[n_ineq++] = bset->ineq[j];
+		isl_basic_set_free(bset);
+	}
+
+	return mat;
+}
+
+/* Comparison routine for use as an isl_sort callback.
+ *
+ * Constraints with the same linear part are sorted together and
+ * among constraints with the same linear part, those with smaller
+ * constant term are sorted first.
+ */
+static int cmp_ineq(const void *a, const void *b, void *arg)
+{
+	unsigned dim = *(unsigned *) arg;
+	isl_int * const *ineq1 = a;
+	isl_int * const *ineq2 = b;
+	int cmp;
+
+	cmp = isl_seq_cmp((*ineq1) + 1, (*ineq2) + 1, dim);
+	if (cmp != 0)
+		return cmp;
+	return isl_int_cmp((*ineq1)[0], (*ineq2)[0]);
+}
+
+/* Compute a superset of the convex hull of "set" that is described
+ * by only constraints in the elements of "list", where "set" has
+ * no parameters or integer divisions.
+ *
+ * We collect all the constraints in those elements and then
+ * sort the constraints such that constraints with the same linear part
+ * are sorted together and that those with smaller constant term are
+ * sorted first.
+ */
+static __isl_give isl_basic_set *uset_unshifted_simple_hull_from_basic_set_list(
+	__isl_take isl_set *set, __isl_take isl_basic_set_list *list)
+{
+	int i, n, n_eq, n_ineq;
+	unsigned dim;
+	isl_ctx *ctx;
+	isl_mat *mat = NULL;
+	isl_int **ineq = NULL;
+	isl_basic_set *hull;
+
+	if (!set)
+		goto error;
+	ctx = isl_set_get_ctx(set);
+
+	n_eq = 0;
+	n_ineq = 0;
+	n = isl_basic_set_list_n_basic_set(list);
+	for (i = 0; i < n; ++i) {
+		isl_basic_set *bset;
+		bset = isl_basic_set_list_get_basic_set(list, i);
+		if (!bset)
+			goto error;
+		n_eq += bset->n_eq;
+		n_ineq += 2 * bset->n_eq + bset->n_ineq;
+		isl_basic_set_free(bset);
+	}
+
+	ineq = isl_alloc_array(ctx, isl_int *, n_ineq);
+	if (n_ineq > 0 && !ineq)
+		goto error;
+
+	dim = isl_set_dim(set, isl_dim_set);
+	mat = isl_mat_alloc(ctx, n_eq, 1 + dim);
+	mat = collect_inequalities(mat, list, ineq);
+	if (!mat)
+		goto error;
+
+	if (isl_sort(ineq, n_ineq, sizeof(ineq[0]), &cmp_ineq, &dim) < 0)
+		goto error;
+
+	hull = uset_unshifted_simple_hull_from_constraints(set, n_ineq, ineq);
+
+	isl_mat_free(mat);
+	free(ineq);
+	isl_basic_set_list_free(list);
+	return hull;
+error:
+	isl_mat_free(mat);
+	free(ineq);
+	isl_set_free(set);
+	isl_basic_set_list_free(list);
+	return NULL;
+}
+
+/* Compute a superset of the convex hull of "set" that is described
+ * by only constraints in the elements of "list".
+ *
+ * If the list is empty, then we can only describe the universe set.
+ * If the input set is empty, then all constraints are valid, so
+ * we return the intersection of the elements in "list".
+ *
+ * Otherwise, we align all divs and temporarily treat them
+ * as regular variables, computing the unshifted simple hull in
+ * uset_unshifted_simple_hull_from_basic_set_list.
+ */
+static __isl_give isl_basic_set *set_unshifted_simple_hull_from_basic_set_list(
+	__isl_take isl_set *set, __isl_take isl_basic_set_list *list)
+{
+	isl_basic_set *model;
+	isl_basic_set *hull;
+
+	if (!set || !list)
+		goto error;
+
+	if (isl_basic_set_list_n_basic_set(list) == 0) {
+		isl_space *space;
+
+		space = isl_set_get_space(set);
+		isl_set_free(set);
+		isl_basic_set_list_free(list);
+		return isl_basic_set_universe(space);
+	}
+	if (isl_set_plain_is_empty(set)) {
+		isl_set_free(set);
+		return isl_basic_set_list_intersect(list);
+	}
+
+	set = isl_set_align_divs_to_basic_set_list(set, list);
+	if (!set)
+		goto error;
+	list = isl_basic_set_list_align_divs_to_basic_set(list, set->p[0]);
+
+	model = isl_basic_set_list_get_basic_set(list, 0);
+
+	set = isl_set_to_underlying_set(set);
+	list = isl_basic_set_list_underlying_set(list);
+
+	hull = uset_unshifted_simple_hull_from_basic_set_list(set, list);
+	hull = isl_basic_map_overlying_set(hull, model);
+
+	return hull;
+error:
+	isl_set_free(set);
+	isl_basic_set_list_free(list);
+	return NULL;
+}
+
+/* Return a sequence of the basic sets that make up the sets in "list".
+ */
+static __isl_give isl_basic_set_list *collect_basic_sets(
+	__isl_take isl_set_list *list)
+{
+	int i, n;
+	isl_ctx *ctx;
+	isl_basic_set_list *bset_list;
+
+	if (!list)
+		return NULL;
+	n = isl_set_list_n_set(list);
+	ctx = isl_set_list_get_ctx(list);
+	bset_list = isl_basic_set_list_alloc(ctx, 0);
+
+	for (i = 0; i < n; ++i) {
+		isl_set *set;
+		isl_basic_set_list *list_i;
+
+		set = isl_set_list_get_set(list, i);
+		set = isl_set_compute_divs(set);
+		list_i = isl_set_get_basic_set_list(set);
+		isl_set_free(set);
+		bset_list = isl_basic_set_list_concat(bset_list, list_i);
+	}
+
+	isl_set_list_free(list);
+	return bset_list;
+}
+
+/* Compute a superset of the convex hull of "set" that is described
+ * by only constraints in the elements of "list".
+ *
+ * If "set" is the universe, then the convex hull (and therefore
+ * any superset of the convexhull) is the universe as well.
+ *
+ * Otherwise, we collect all the basic sets in the set list and
+ * continue with set_unshifted_simple_hull_from_basic_set_list.
+ */
+__isl_give isl_basic_set *isl_set_unshifted_simple_hull_from_set_list(
+	__isl_take isl_set *set, __isl_take isl_set_list *list)
+{
+	isl_basic_set_list *bset_list;
+	int is_universe;
+
+	is_universe = isl_set_plain_is_universe(set);
+	if (is_universe < 0)
+		set = isl_set_free(set);
+	if (is_universe < 0 || is_universe) {
+		isl_set_list_free(list);
+		return isl_set_unshifted_simple_hull(set);
+	}
+
+	bset_list = collect_basic_sets(list);
+	return set_unshifted_simple_hull_from_basic_set_list(set, bset_list);
+}
+
+/* Given a set "set", return parametric bounds on the dimension "dim".
+ */
+static struct isl_basic_set *set_bounds(struct isl_set *set, int dim)
+{
+	unsigned set_dim = isl_set_dim(set, isl_dim_set);
+	set = isl_set_copy(set);
+	set = isl_set_eliminate_dims(set, dim + 1, set_dim - (dim + 1));
+	set = isl_set_eliminate_dims(set, 0, dim);
+	return isl_set_convex_hull(set);
+}
+
+/* Computes a "simple hull" and then check if each dimension in the
+ * resulting hull is bounded by a symbolic constant.  If not, the
+ * hull is intersected with the corresponding bounds on the whole set.
+ */
+struct isl_basic_set *isl_set_bounded_simple_hull(struct isl_set *set)
+{
+	int i, j;
+	struct isl_basic_set *hull;
+	unsigned nparam, left;
+	int removed_divs = 0;
+
+	hull = isl_set_simple_hull(isl_set_copy(set));
+	if (!hull)
+		goto error;
+
+	nparam = isl_basic_set_dim(hull, isl_dim_param);
+	for (i = 0; i < isl_basic_set_dim(hull, isl_dim_set); ++i) {
+		int lower = 0, upper = 0;
+		struct isl_basic_set *bounds;
+
+		left = isl_basic_set_total_dim(hull) - nparam - i - 1;
+		for (j = 0; j < hull->n_eq; ++j) {
+			if (isl_int_is_zero(hull->eq[j][1 + nparam + i]))
+				continue;
+			if (isl_seq_first_non_zero(hull->eq[j]+1+nparam+i+1,
+						    left) == -1)
+				break;
+		}
+		if (j < hull->n_eq)
+			continue;
+
+		for (j = 0; j < hull->n_ineq; ++j) {
+			if (isl_int_is_zero(hull->ineq[j][1 + nparam + i]))
+				continue;
+			if (isl_seq_first_non_zero(hull->ineq[j]+1+nparam+i+1,
+						    left) != -1 ||
+			    isl_seq_first_non_zero(hull->ineq[j]+1+nparam,
+						    i) != -1)
+				continue;
+			if (isl_int_is_pos(hull->ineq[j][1 + nparam + i]))
+				lower = 1;
+			else
+				upper = 1;
+			if (lower && upper)
+				break;
+		}
+
+		if (lower && upper)
+			continue;
+
+		if (!removed_divs) {
+			set = isl_set_remove_divs(set);
+			if (!set)
+				goto error;
+			removed_divs = 1;
+		}
+		bounds = set_bounds(set, i);
+		hull = isl_basic_set_intersect(hull, bounds);
+		if (!hull)
+			goto error;
+	}
+
+	isl_set_free(set);
+	return hull;
+error:
+	isl_set_free(set);
+	return NULL;
+}
diff --git a/polly/lib/External/isl/isl_ctx.c b/polly/lib/External/isl/isl_ctx.c
new file mode 100644
index 00000000000..95880345ab8
--- /dev/null
+++ b/polly/lib/External/isl/isl_ctx.c
@@ -0,0 +1,334 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <isl_ctx_private.h>
+#include <isl/vec.h>
+#include <isl_options_private.h>
+
+#define __isl_calloc(type,size)		((type *)calloc(1, size))
+#define __isl_calloc_type(type)		__isl_calloc(type,sizeof(type))
+
+/* Check that the result of an allocation ("p") is not NULL and
+ * complain if it is.
+ * The only exception is when allocation size ("size") is equal to zero.
+ */
+static void *check_non_null(isl_ctx *ctx, void *p, size_t size)
+{
+	if (p || size == 0)
+		return p;
+	isl_die(ctx, isl_error_alloc, "allocation failure", return NULL);
+}
+
+/* Prepare for performing the next "operation" in the context.
+ * Return 0 if we are allowed to perform this operation and
+ * return -1 if we should abort the computation.
+ *
+ * In particular, we should stop if the user has explicitly aborted
+ * the computation or if the maximal number of operations has been exceeded.
+ */
+int isl_ctx_next_operation(isl_ctx *ctx)
+{
+	if (!ctx)
+		return -1;
+	if (ctx->abort) {
+		isl_ctx_set_error(ctx, isl_error_abort);
+		return -1;
+	}
+	if (ctx->max_operations && ctx->operations >= ctx->max_operations)
+		isl_die(ctx, isl_error_quota,
+			"maximal number of operations exceeded", return -1);
+	ctx->operations++;
+	return 0;
+}
+
+/* Call malloc and complain if it fails.
+ * If ctx is NULL, then return NULL.
+ */
+void *isl_malloc_or_die(isl_ctx *ctx, size_t size)
+{
+	if (isl_ctx_next_operation(ctx) < 0)
+		return NULL;
+	return ctx ? check_non_null(ctx, malloc(size), size) : NULL;
+}
+
+/* Call calloc and complain if it fails.
+ * If ctx is NULL, then return NULL.
+ */
+void *isl_calloc_or_die(isl_ctx *ctx, size_t nmemb, size_t size)
+{
+	if (isl_ctx_next_operation(ctx) < 0)
+		return NULL;
+	return ctx ? check_non_null(ctx, calloc(nmemb, size), nmemb) : NULL;
+}
+
+/* Call realloc and complain if it fails.
+ * If ctx is NULL, then return NULL.
+ */
+void *isl_realloc_or_die(isl_ctx *ctx, void *ptr, size_t size)
+{
+	if (isl_ctx_next_operation(ctx) < 0)
+		return NULL;
+	return ctx ? check_non_null(ctx, realloc(ptr, size), size) : NULL;
+}
+
+void isl_handle_error(isl_ctx *ctx, enum isl_error error, const char *msg,
+	const char *file, int line)
+{
+	if (!ctx)
+		return;
+
+	isl_ctx_set_error(ctx, error);
+
+	switch (ctx->opt->on_error) {
+	case ISL_ON_ERROR_WARN:
+		fprintf(stderr, "%s:%d: %s\n", file, line, msg);
+		return;
+	case ISL_ON_ERROR_CONTINUE:
+		return;
+	case ISL_ON_ERROR_ABORT:
+		fprintf(stderr, "%s:%d: %s\n", file, line, msg);
+		abort();
+		return;
+	}
+}
+
+static struct isl_options *find_nested_options(struct isl_args *args,
+	void *opt, struct isl_args *wanted)
+{
+	int i;
+	struct isl_options *options;
+
+	if (args == wanted)
+		return opt;
+
+	for (i = 0; args->args[i].type != isl_arg_end; ++i) {
+		struct isl_arg *arg = &args->args[i];
+		void *child;
+
+		if (arg->type != isl_arg_child)
+			continue;
+
+		if (arg->offset == (size_t) -1)
+			child = opt;
+		else
+			child = *(void **)(((char *)opt) + arg->offset);
+
+		options = find_nested_options(arg->u.child.child,
+						child, wanted);
+		if (options)
+			return options;
+	}
+
+	return NULL;
+}
+
+static struct isl_options *find_nested_isl_options(struct isl_args *args,
+	void *opt)
+{
+	return find_nested_options(args, opt, &isl_options_args);
+}
+
+void *isl_ctx_peek_options(isl_ctx *ctx, struct isl_args *args)
+{
+	if (!ctx)
+		return NULL;
+	if (args == &isl_options_args)
+		return ctx->opt;
+	return find_nested_options(ctx->user_args, ctx->user_opt, args);
+}
+
+isl_ctx *isl_ctx_alloc_with_options(struct isl_args *args, void *user_opt)
+{
+	struct isl_ctx *ctx = NULL;
+	struct isl_options *opt = NULL;
+	int opt_allocated = 0;
+
+	if (!user_opt)
+		return NULL;
+
+	opt = find_nested_isl_options(args, user_opt);
+	if (!opt) {
+		opt = isl_options_new_with_defaults();
+		if (!opt)
+			goto error;
+		opt_allocated = 1;
+	}
+
+	ctx = __isl_calloc_type(struct isl_ctx);
+	if (!ctx)
+		goto error;
+
+	if (isl_hash_table_init(ctx, &ctx->id_table, 0))
+		goto error;
+
+	ctx->stats = isl_calloc_type(ctx, struct isl_stats);
+	if (!ctx->stats)
+		goto error;
+
+	ctx->user_args = args;
+	ctx->user_opt = user_opt;
+	ctx->opt_allocated = opt_allocated;
+	ctx->opt = opt;
+	ctx->ref = 0;
+
+	isl_int_init(ctx->zero);
+	isl_int_set_si(ctx->zero, 0);
+
+	isl_int_init(ctx->one);
+	isl_int_set_si(ctx->one, 1);
+
+	isl_int_init(ctx->two);
+	isl_int_set_si(ctx->two, 2);
+
+	isl_int_init(ctx->negone);
+	isl_int_set_si(ctx->negone, -1);
+
+	isl_int_init(ctx->normalize_gcd);
+
+	ctx->n_cached = 0;
+	ctx->n_miss = 0;
+
+	ctx->error = isl_error_none;
+
+	ctx->operations = 0;
+	isl_ctx_set_max_operations(ctx, ctx->opt->max_operations);
+
+	return ctx;
+error:
+	isl_args_free(args, user_opt);
+	if (opt_allocated)
+		isl_options_free(opt);
+	free(ctx);
+	return NULL;
+}
+
+struct isl_ctx *isl_ctx_alloc()
+{
+	struct isl_options *opt;
+
+	opt = isl_options_new_with_defaults();
+
+	return isl_ctx_alloc_with_options(&isl_options_args, opt);
+}
+
+void isl_ctx_ref(struct isl_ctx *ctx)
+{
+	ctx->ref++;
+}
+
+void isl_ctx_deref(struct isl_ctx *ctx)
+{
+	isl_assert(ctx, ctx->ref > 0, return);
+	ctx->ref--;
+}
+
+/* Print statistics on usage.
+ */
+static void print_stats(isl_ctx *ctx)
+{
+	fprintf(stderr, "operations: %lu\n", ctx->operations);
+}
+
+void isl_ctx_free(struct isl_ctx *ctx)
+{
+	if (!ctx)
+		return;
+	if (ctx->ref != 0)
+		isl_die(ctx, isl_error_invalid,
+			"isl_ctx freed, but some objects still reference it",
+			return);
+
+	if (ctx->opt->print_stats)
+		print_stats(ctx);
+
+	isl_hash_table_clear(&ctx->id_table);
+	isl_blk_clear_cache(ctx);
+	isl_int_clear(ctx->zero);
+	isl_int_clear(ctx->one);
+	isl_int_clear(ctx->two);
+	isl_int_clear(ctx->negone);
+	isl_int_clear(ctx->normalize_gcd);
+	isl_args_free(ctx->user_args, ctx->user_opt);
+	if (ctx->opt_allocated)
+		isl_options_free(ctx->opt);
+	free(ctx->stats);
+	free(ctx);
+}
+
+struct isl_options *isl_ctx_options(isl_ctx *ctx)
+{
+	if (!ctx)
+		return NULL;
+	return ctx->opt;
+}
+
+enum isl_error isl_ctx_last_error(isl_ctx *ctx)
+{
+	return ctx->error;
+}
+
+void isl_ctx_reset_error(isl_ctx *ctx)
+{
+	ctx->error = isl_error_none;
+}
+
+void isl_ctx_set_error(isl_ctx *ctx, enum isl_error error)
+{
+	if (ctx)
+		ctx->error = error;
+}
+
+void isl_ctx_abort(isl_ctx *ctx)
+{
+	if (ctx)
+		ctx->abort = 1;
+}
+
+void isl_ctx_resume(isl_ctx *ctx)
+{
+	if (ctx)
+		ctx->abort = 0;
+}
+
+int isl_ctx_aborted(isl_ctx *ctx)
+{
+	return ctx ? ctx->abort : -1;
+}
+
+int isl_ctx_parse_options(isl_ctx *ctx, int argc, char **argv, unsigned flags)
+{
+	if (!ctx)
+		return -1;
+	return isl_args_parse(ctx->user_args, argc, argv, ctx->user_opt, flags);
+}
+
+/* Set the maximal number of iterations of "ctx" to "max_operations".
+ */
+void isl_ctx_set_max_operations(isl_ctx *ctx, unsigned long max_operations)
+{
+	if (!ctx)
+		return;
+	ctx->max_operations = max_operations;
+}
+
+/* Return the maximal number of iterations of "ctx".
+ */
+unsigned long isl_ctx_get_max_operations(isl_ctx *ctx)
+{
+	return ctx ? ctx->max_operations : 0;
+}
+
+/* Reset the number of operations performed by "ctx".
+ */
+void isl_ctx_reset_operations(isl_ctx *ctx)
+{
+	if (!ctx)
+		return;
+	ctx->operations = 0;
+}
diff --git a/polly/lib/External/isl/isl_ctx_private.h b/polly/lib/External/isl/isl_ctx_private.h
new file mode 100644
index 00000000000..b4bb2a5e46c
--- /dev/null
+++ b/polly/lib/External/isl/isl_ctx_private.h
@@ -0,0 +1,34 @@
+#include <isl/ctx.h>
+#include <isl_blk.h>
+
+struct isl_ctx {
+	int			ref;
+
+	struct isl_stats	*stats;
+
+	int			 opt_allocated;
+	struct isl_options	*opt;
+	void			*user_opt;
+	struct isl_args		*user_args;
+
+	isl_int			zero;
+	isl_int			one;
+	isl_int			two;
+	isl_int			negone;
+
+	isl_int			normalize_gcd;
+
+	int			n_cached;
+	int			n_miss;
+	struct isl_blk		cache[ISL_BLK_CACHE_SIZE];
+	struct isl_hash_table	id_table;
+
+	enum isl_error		error;
+
+	int			abort;
+
+	unsigned long		operations;
+	unsigned long		max_operations;
+};
+
+int isl_ctx_next_operation(isl_ctx *ctx);
diff --git a/polly/lib/External/isl/isl_deprecated.c b/polly/lib/External/isl/isl_deprecated.c
new file mode 100644
index 00000000000..babe0def461
--- /dev/null
+++ b/polly/lib/External/isl/isl_deprecated.c
@@ -0,0 +1,10 @@
+#include <isl/set.h>
+
+/* This function was never documented and has been replaced by
+ * isl_basic_set_add_dims.
+ */
+__isl_give isl_basic_set *isl_basic_set_add(__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned n)
+{
+	return isl_basic_set_add_dims(bset, type, n);
+}
diff --git a/polly/lib/External/isl/isl_dim.c b/polly/lib/External/isl/isl_dim.c
new file mode 100644
index 00000000000..914d666b38e
--- /dev/null
+++ b/polly/lib/External/isl/isl_dim.c
@@ -0,0 +1,258 @@
+#include <isl/dim.h>
+#include <isl/aff.h>
+#include <isl/map.h>
+#include <isl/set.h>
+#include <isl/polynomial.h>
+
+isl_ctx *isl_dim_get_ctx(__isl_keep isl_space *dim)
+{
+	return isl_space_get_ctx(dim);
+}
+
+__isl_give isl_space *isl_dim_alloc(isl_ctx *ctx,
+	unsigned nparam, unsigned n_in, unsigned n_out)
+{
+	return isl_space_alloc(ctx, nparam, n_in, n_out);
+}
+__isl_give isl_space *isl_dim_set_alloc(isl_ctx *ctx,
+	unsigned nparam, unsigned dim)
+{
+	return isl_space_set_alloc(ctx, nparam, dim);
+}
+__isl_give isl_space *isl_dim_copy(__isl_keep isl_space *dim)
+{
+	return isl_space_copy(dim);
+}
+void isl_dim_free(__isl_take isl_space *dim)
+{
+	isl_space_free(dim);
+}
+
+unsigned isl_dim_size(__isl_keep isl_space *dim, enum isl_dim_type type)
+{
+	return isl_space_dim(dim, type);
+}
+
+__isl_give isl_space *isl_dim_set_dim_id(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id)
+{
+	return isl_space_set_dim_id(dim, type, pos, id);
+}
+int isl_dim_has_dim_id(__isl_keep isl_space *dim,
+	enum isl_dim_type type, unsigned pos)
+{
+	return isl_space_has_dim_id(dim, type, pos);
+}
+__isl_give isl_id *isl_dim_get_dim_id(__isl_keep isl_space *dim,
+	enum isl_dim_type type, unsigned pos)
+{
+	return isl_space_get_dim_id(dim, type, pos);
+}
+
+int isl_dim_find_dim_by_id(__isl_keep isl_space *dim,
+	enum isl_dim_type type, __isl_keep isl_id *id)
+{
+	return isl_space_find_dim_by_id(dim, type, id);
+}
+
+__isl_give isl_space *isl_dim_set_tuple_id(__isl_take isl_space *dim,
+	enum isl_dim_type type, __isl_take isl_id *id)
+{
+	return isl_space_set_tuple_id(dim, type, id);
+}
+__isl_give isl_space *isl_dim_reset_tuple_id(__isl_take isl_space *dim,
+	enum isl_dim_type type)
+{
+	return isl_space_reset_tuple_id(dim, type);
+}
+int isl_dim_has_tuple_id(__isl_keep isl_space *dim, enum isl_dim_type type)
+{
+	return isl_space_has_tuple_id(dim, type);
+}
+__isl_give isl_id *isl_dim_get_tuple_id(__isl_keep isl_space *dim,
+	enum isl_dim_type type)
+{
+	return isl_space_get_tuple_id(dim, type);
+}
+
+__isl_give isl_space *isl_dim_set_name(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned pos, __isl_keep const char *name)
+{
+	return isl_space_set_dim_name(dim, type, pos, name);
+}
+__isl_keep const char *isl_dim_get_name(__isl_keep isl_space *dim,
+	enum isl_dim_type type, unsigned pos)
+{
+	return isl_space_get_dim_name(dim, type, pos);
+}
+
+__isl_give isl_space *isl_dim_set_tuple_name(__isl_take isl_space *dim,
+	enum isl_dim_type type, const char *s)
+{
+	return isl_space_set_tuple_name(dim, type, s);
+}
+const char *isl_dim_get_tuple_name(__isl_keep isl_space *dim,
+	enum isl_dim_type type)
+{
+	return isl_space_get_tuple_name(dim, type);
+}
+
+int isl_dim_is_wrapping(__isl_keep isl_space *dim)
+{
+	return isl_space_is_wrapping(dim);
+}
+__isl_give isl_space *isl_dim_wrap(__isl_take isl_space *dim)
+{
+	return isl_space_wrap(dim);
+}
+__isl_give isl_space *isl_dim_unwrap(__isl_take isl_space *dim)
+{
+	return isl_space_unwrap(dim);
+}
+
+__isl_give isl_space *isl_dim_domain(__isl_take isl_space *dim)
+{
+	return isl_space_domain(dim);
+}
+__isl_give isl_space *isl_dim_from_domain(__isl_take isl_space *dim)
+{
+	return isl_space_from_domain(dim);
+}
+__isl_give isl_space *isl_dim_range(__isl_take isl_space *dim)
+{
+	return isl_space_range(dim);
+}
+__isl_give isl_space *isl_dim_from_range(__isl_take isl_space *dim)
+{
+	return isl_space_from_range(dim);
+}
+__isl_give isl_space *isl_dim_reverse(__isl_take isl_space *dim)
+{
+	return isl_space_reverse(dim);
+}
+__isl_give isl_space *isl_dim_join(__isl_take isl_space *left,
+	__isl_take isl_space *right)
+{
+	return isl_space_join(left, right);
+}
+__isl_give isl_space *isl_dim_align_params(__isl_take isl_space *dim1,
+	__isl_take isl_space *dim2)
+{
+	return isl_space_align_params(dim1, dim2);
+}
+__isl_give isl_space *isl_dim_insert(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned pos, unsigned n)
+{
+	return isl_space_insert_dims(dim, type, pos, n);
+}
+__isl_give isl_space *isl_dim_add(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned n)
+{
+	return isl_space_add_dims(dim, type, n);
+}
+__isl_give isl_space *isl_dim_drop(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return isl_space_drop_dims(dim, type, first, n);
+}
+__isl_give isl_space *isl_dim_move(__isl_take isl_space *dim,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n)
+{
+	return isl_space_move_dims(dim, dst_type, dst_pos, src_type, src_pos, n);
+}
+__isl_give isl_space *isl_dim_map_from_set(__isl_take isl_space *dim)
+{
+	return isl_space_map_from_set(dim);
+}
+__isl_give isl_space *isl_dim_zip(__isl_take isl_space *dim)
+{
+	return isl_space_zip(dim);
+}
+
+__isl_give isl_local_space *isl_local_space_from_dim(
+	__isl_take isl_space *dim)
+{
+	return isl_local_space_from_space(dim);
+}
+__isl_give isl_space *isl_local_space_get_dim(
+	__isl_keep isl_local_space *ls)
+{
+	return isl_local_space_get_space(ls);
+}
+
+__isl_give isl_space *isl_aff_get_dim(__isl_keep isl_aff *aff)
+{
+	return isl_aff_get_space(aff);
+}
+__isl_give isl_space *isl_pw_aff_get_dim(__isl_keep isl_pw_aff *pwaff)
+{
+	return isl_pw_aff_get_space(pwaff);
+}
+
+__isl_give isl_space *isl_constraint_get_dim(
+	__isl_keep isl_constraint *constraint)
+{
+	return isl_constraint_get_space(constraint);
+}
+
+__isl_give isl_space *isl_basic_map_get_dim(__isl_keep isl_basic_map *bmap)
+{
+	return isl_basic_map_get_space(bmap);
+}
+__isl_give isl_space *isl_map_get_dim(__isl_keep isl_map *map)
+{
+	return isl_map_get_space(map);
+}
+__isl_give isl_space *isl_union_map_get_dim(__isl_keep isl_union_map *umap)
+{
+	return isl_union_map_get_space(umap);
+}
+
+__isl_give isl_space *isl_basic_set_get_dim(__isl_keep isl_basic_set *bset)
+{
+	return isl_basic_set_get_space(bset);
+}
+__isl_give isl_space *isl_set_get_dim(__isl_keep isl_set *set)
+{
+	return isl_set_get_space(set);
+}
+__isl_give isl_space *isl_union_set_get_dim(__isl_keep isl_union_set *uset)
+{
+	return isl_union_set_get_space(uset);
+}
+
+__isl_give isl_space *isl_point_get_dim(__isl_keep isl_point *pnt)
+{
+	return isl_point_get_space(pnt);
+}
+
+__isl_give isl_space *isl_qpolynomial_get_dim(__isl_keep isl_qpolynomial *qp)
+{
+	return isl_qpolynomial_get_space(qp);
+}
+__isl_give isl_space *isl_pw_qpolynomial_get_dim(
+	__isl_keep isl_pw_qpolynomial *pwqp)
+{
+	return isl_pw_qpolynomial_get_space(pwqp);
+}
+__isl_give isl_space *isl_qpolynomial_fold_get_dim(
+	__isl_keep isl_qpolynomial_fold *fold)
+{
+	return isl_qpolynomial_fold_get_space(fold);
+}
+__isl_give isl_space *isl_pw_qpolynomial_fold_get_dim(
+	__isl_keep isl_pw_qpolynomial_fold *pwf)
+{
+	return isl_pw_qpolynomial_fold_get_space(pwf);
+}
+__isl_give isl_space *isl_union_pw_qpolynomial_get_dim(
+	__isl_keep isl_union_pw_qpolynomial *upwqp)
+{
+	return isl_union_pw_qpolynomial_get_space(upwqp);
+}
+__isl_give isl_space *isl_union_pw_qpolynomial_fold_get_dim(
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf)
+{
+	return isl_union_pw_qpolynomial_fold_get_space(upwf);
+}
diff --git a/polly/lib/External/isl/isl_dim_map.c b/polly/lib/External/isl/isl_dim_map.c
new file mode 100644
index 00000000000..102b8e8a6a4
--- /dev/null
+++ b/polly/lib/External/isl/isl_dim_map.c
@@ -0,0 +1,233 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2010-2011 INRIA Saclay
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France 
+ */
+
+#include <isl_map_private.h>
+#include <isl_space_private.h>
+#include <isl_dim_map.h>
+#include <isl_reordering.h>
+
+struct isl_dim_map_entry {
+	int pos;
+	int sgn;
+};
+
+/* Maps dst positions to src positions */
+struct isl_dim_map {
+	unsigned len;
+	struct isl_dim_map_entry m[1];
+};
+
+__isl_give isl_dim_map *isl_dim_map_alloc(isl_ctx *ctx, unsigned len)
+{
+	int i;
+	struct isl_dim_map *dim_map;
+	dim_map = isl_alloc(ctx, struct isl_dim_map,
+	    sizeof(struct isl_dim_map) + len * sizeof(struct isl_dim_map_entry));
+	if (!dim_map)
+		return NULL;
+	dim_map->len = 1 + len;
+	dim_map->m[0].pos = 0;
+	dim_map->m[0].sgn = 1;
+	for (i = 0; i < len; ++i)
+		dim_map->m[1 + i].sgn = 0;
+	return dim_map;
+}
+
+void isl_dim_map_range(__isl_keep isl_dim_map *dim_map,
+	unsigned dst_pos, unsigned dst_stride,
+	unsigned src_pos, unsigned src_stride,
+	unsigned n, int sign)
+{
+	int i;
+
+	if (!dim_map)
+		return;
+
+	for (i = 0; i < n; ++i) {
+		unsigned d = 1 + dst_pos + dst_stride * i;
+		unsigned s = 1 + src_pos + src_stride * i;
+		dim_map->m[d].pos = s;
+		dim_map->m[d].sgn = sign;
+	}
+}
+
+void isl_dim_map_dim_range(__isl_keep isl_dim_map *dim_map,
+	__isl_keep isl_space *dim, enum isl_dim_type type,
+	unsigned first, unsigned n, unsigned dst_pos)
+{
+	int i;
+	unsigned src_pos;
+
+	if (!dim_map || !dim)
+		return;
+	
+	src_pos = 1 + isl_space_offset(dim, type);
+	for (i = 0; i < n; ++i) {
+		dim_map->m[1 + dst_pos + i].pos = src_pos + first + i;
+		dim_map->m[1 + dst_pos + i].sgn = 1;
+	}
+}
+
+void isl_dim_map_dim(__isl_keep isl_dim_map *dim_map, __isl_keep isl_space *dim,
+	enum isl_dim_type type, unsigned dst_pos)
+{
+	isl_dim_map_dim_range(dim_map, dim, type,
+			      0, isl_space_dim(dim, type), dst_pos);
+}
+
+void isl_dim_map_div(__isl_keep isl_dim_map *dim_map,
+	__isl_keep isl_basic_map *bmap, unsigned dst_pos)
+{
+	int i;
+	unsigned src_pos;
+
+	if (!dim_map || !bmap)
+		return;
+	
+	src_pos = 1 + isl_space_dim(bmap->dim, isl_dim_all);
+	for (i = 0; i < bmap->n_div; ++i) {
+		dim_map->m[1 + dst_pos + i].pos = src_pos + i;
+		dim_map->m[1 + dst_pos + i].sgn = 1;
+	}
+}
+
+void isl_dim_map_dump(struct isl_dim_map *dim_map)
+{
+	int i;
+
+	for (i = 0; i < dim_map->len; ++i)
+		fprintf(stderr, "%d -> %d * %d; ", i,
+			dim_map->m[i].sgn, dim_map->m[i].pos);
+	fprintf(stderr, "\n");
+}
+
+static void copy_constraint_dim_map(isl_int *dst, isl_int *src,
+					struct isl_dim_map *dim_map)
+{
+	int i;
+
+	for (i = 0; i < dim_map->len; ++i) {
+		if (dim_map->m[i].sgn == 0)
+			isl_int_set_si(dst[i], 0);
+		else if (dim_map->m[i].sgn > 0)
+			isl_int_set(dst[i], src[dim_map->m[i].pos]);
+		else
+			isl_int_neg(dst[i], src[dim_map->m[i].pos]);
+	}
+}
+
+static void copy_div_dim_map(isl_int *dst, isl_int *src,
+					struct isl_dim_map *dim_map)
+{
+	isl_int_set(dst[0], src[0]);
+	copy_constraint_dim_map(dst+1, src+1, dim_map);
+}
+
+__isl_give isl_basic_map *isl_basic_map_add_constraints_dim_map(
+	__isl_take isl_basic_map *dst, __isl_take isl_basic_map *src,
+	__isl_take isl_dim_map *dim_map)
+{
+	int i;
+
+	if (!src || !dst || !dim_map)
+		goto error;
+
+	for (i = 0; i < src->n_eq; ++i) {
+		int i1 = isl_basic_map_alloc_equality(dst);
+		if (i1 < 0)
+			goto error;
+		copy_constraint_dim_map(dst->eq[i1], src->eq[i], dim_map);
+	}
+
+	for (i = 0; i < src->n_ineq; ++i) {
+		int i1 = isl_basic_map_alloc_inequality(dst);
+		if (i1 < 0)
+			goto error;
+		copy_constraint_dim_map(dst->ineq[i1], src->ineq[i], dim_map);
+	}
+
+	for (i = 0; i < src->n_div; ++i) {
+		int i1 = isl_basic_map_alloc_div(dst);
+		if (i1 < 0)
+			goto error;
+		copy_div_dim_map(dst->div[i1], src->div[i], dim_map);
+	}
+
+	free(dim_map);
+	isl_basic_map_free(src);
+
+	return dst;
+error:
+	free(dim_map);
+	isl_basic_map_free(src);
+	isl_basic_map_free(dst);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_add_constraints_dim_map(
+	__isl_take isl_basic_set *dst, __isl_take isl_basic_set *src,
+	__isl_take isl_dim_map *dim_map)
+{
+	return isl_basic_map_add_constraints_dim_map(dst, src, dim_map);
+}
+
+/* Extend the given dim_map with mappings for the divs in bmap.
+ */
+__isl_give isl_dim_map *isl_dim_map_extend(__isl_keep isl_dim_map *dim_map,
+	__isl_keep isl_basic_map *bmap)
+{
+	int i;
+	struct isl_dim_map *res;
+	int offset;
+
+	offset = isl_basic_map_offset(bmap, isl_dim_div);
+
+	res = isl_dim_map_alloc(bmap->ctx, dim_map->len - 1 + bmap->n_div);
+	if (!res)
+		return NULL;
+
+	for (i = 0; i < dim_map->len; ++i)
+		res->m[i] = dim_map->m[i];
+	for (i = 0; i < bmap->n_div; ++i) {
+		res->m[dim_map->len + i].pos = offset + i;
+		res->m[dim_map->len + i].sgn = 1;
+	}
+
+	return res;
+}
+
+/* Extract a dim_map from a reordering.
+ * We essentially need to reverse the mapping, and add an offset
+ * of 1 for the constant term.
+ */
+__isl_give isl_dim_map *isl_dim_map_from_reordering(
+	__isl_keep isl_reordering *exp)
+{
+	int i;
+	isl_ctx *ctx;
+	struct isl_dim_map *dim_map;
+
+	if (!exp)
+		return NULL;
+
+	ctx = isl_space_get_ctx(exp->dim);
+	dim_map = isl_dim_map_alloc(ctx, isl_space_dim(exp->dim, isl_dim_all));
+	if (!dim_map)
+		return NULL;
+
+	for (i = 0; i < exp->len; ++i) {
+		dim_map->m[1 + exp->pos[i]].pos = 1 + i;
+		dim_map->m[1 + exp->pos[i]].sgn = 1;
+	}
+
+	return dim_map;
+}
diff --git a/polly/lib/External/isl/isl_dim_map.h b/polly/lib/External/isl/isl_dim_map.h
new file mode 100644
index 00000000000..0f988c3166d
--- /dev/null
+++ b/polly/lib/External/isl/isl_dim_map.h
@@ -0,0 +1,36 @@
+#ifndef ISL_DIM_MAP_H
+#define ISL_DIM_MAP_H
+
+#include <isl/ctx.h>
+#include <isl/space.h>
+#include <isl/map.h>
+
+struct isl_dim_map;
+typedef struct isl_dim_map isl_dim_map;
+
+__isl_give isl_dim_map *isl_dim_map_alloc(isl_ctx *ctx, unsigned len);
+void isl_dim_map_range(__isl_keep isl_dim_map *dim_map,
+	unsigned dst_pos, unsigned dst_stride,
+	unsigned src_pos, unsigned src_stride,
+	unsigned n, int sign);
+void isl_dim_map_dim_range(__isl_keep isl_dim_map *dim_map,
+	isl_space *dim, enum isl_dim_type type,
+	unsigned first, unsigned n, unsigned dst_pos);
+void isl_dim_map_dim(__isl_keep isl_dim_map *dim_map, __isl_keep isl_space *dim,
+	enum isl_dim_type type, unsigned dst_pos);
+void isl_dim_map_div(__isl_keep isl_dim_map *dim_map,
+	__isl_keep isl_basic_map *bmap, unsigned dst_pos);
+__isl_give isl_basic_set *isl_basic_set_add_constraints_dim_map(
+	__isl_take isl_basic_set *dst, __isl_take isl_basic_set *src,
+	__isl_take isl_dim_map *dim_map);
+__isl_give isl_basic_map *isl_basic_map_add_constraints_dim_map(
+	__isl_take isl_basic_map *dst, __isl_take isl_basic_map *src,
+	__isl_take isl_dim_map *dim_map);
+
+__isl_give isl_dim_map *isl_dim_map_extend(__isl_keep isl_dim_map *dim_map,
+	__isl_keep isl_basic_map *bmap);
+
+__isl_give isl_dim_map *isl_dim_map_from_reordering(
+	__isl_keep isl_reordering *exp);
+
+#endif
diff --git a/polly/lib/External/isl/isl_equalities.c b/polly/lib/External/isl/isl_equalities.c
new file mode 100644
index 00000000000..e257f705f53
--- /dev/null
+++ b/polly/lib/External/isl/isl_equalities.c
@@ -0,0 +1,782 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2010      INRIA Saclay
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France
+ */
+
+#include <isl_mat_private.h>
+#include <isl_vec_private.h>
+#include <isl_seq.h>
+#include "isl_map_private.h"
+#include "isl_equalities.h"
+#include <isl_val_private.h>
+
+/* Given a set of modulo constraints
+ *
+ *		c + A y = 0 mod d
+ *
+ * this function computes a particular solution y_0
+ *
+ * The input is given as a matrix B = [ c A ] and a vector d.
+ *
+ * The output is matrix containing the solution y_0 or
+ * a zero-column matrix if the constraints admit no integer solution.
+ *
+ * The given set of constrains is equivalent to
+ *
+ *		c + A y = -D x
+ *
+ * with D = diag d and x a fresh set of variables.
+ * Reducing both c and A modulo d does not change the
+ * value of y in the solution and may lead to smaller coefficients.
+ * Let M = [ D A ] and [ H 0 ] = M U, the Hermite normal form of M.
+ * Then
+ *		  [ x ]
+ *		M [ y ] = - c
+ * and so
+ *		               [ x ]
+ *		[ H 0 ] U^{-1} [ y ] = - c
+ * Let
+ *		[ A ]          [ x ]
+ *		[ B ] = U^{-1} [ y ]
+ * then
+ *		H A + 0 B = -c
+ *
+ * so B may be chosen arbitrarily, e.g., B = 0, and then
+ *
+ *		       [ x ] = [ -c ]
+ *		U^{-1} [ y ] = [  0 ]
+ * or
+ *		[ x ]     [ -c ]
+ *		[ y ] = U [  0 ]
+ * specifically,
+ *
+ *		y = U_{2,1} (-c)
+ *
+ * If any of the coordinates of this y are non-integer
+ * then the constraints admit no integer solution and
+ * a zero-column matrix is returned.
+ */
+static struct isl_mat *particular_solution(struct isl_mat *B, struct isl_vec *d)
+{
+	int i, j;
+	struct isl_mat *M = NULL;
+	struct isl_mat *C = NULL;
+	struct isl_mat *U = NULL;
+	struct isl_mat *H = NULL;
+	struct isl_mat *cst = NULL;
+	struct isl_mat *T = NULL;
+
+	M = isl_mat_alloc(B->ctx, B->n_row, B->n_row + B->n_col - 1);
+	C = isl_mat_alloc(B->ctx, 1 + B->n_row, 1);
+	if (!M || !C)
+		goto error;
+	isl_int_set_si(C->row[0][0], 1);
+	for (i = 0; i < B->n_row; ++i) {
+		isl_seq_clr(M->row[i], B->n_row);
+		isl_int_set(M->row[i][i], d->block.data[i]);
+		isl_int_neg(C->row[1 + i][0], B->row[i][0]);
+		isl_int_fdiv_r(C->row[1+i][0], C->row[1+i][0], M->row[i][i]);
+		for (j = 0; j < B->n_col - 1; ++j)
+			isl_int_fdiv_r(M->row[i][B->n_row + j],
+					B->row[i][1 + j], M->row[i][i]);
+	}
+	M = isl_mat_left_hermite(M, 0, &U, NULL);
+	if (!M || !U)
+		goto error;
+	H = isl_mat_sub_alloc(M, 0, B->n_row, 0, B->n_row);
+	H = isl_mat_lin_to_aff(H);
+	C = isl_mat_inverse_product(H, C);
+	if (!C)
+		goto error;
+	for (i = 0; i < B->n_row; ++i) {
+		if (!isl_int_is_divisible_by(C->row[1+i][0], C->row[0][0]))
+			break;
+		isl_int_divexact(C->row[1+i][0], C->row[1+i][0], C->row[0][0]);
+	}
+	if (i < B->n_row)
+		cst = isl_mat_alloc(B->ctx, B->n_row, 0);
+	else
+		cst = isl_mat_sub_alloc(C, 1, B->n_row, 0, 1);
+	T = isl_mat_sub_alloc(U, B->n_row, B->n_col - 1, 0, B->n_row);
+	cst = isl_mat_product(T, cst);
+	isl_mat_free(M);
+	isl_mat_free(C);
+	isl_mat_free(U);
+	return cst;
+error:
+	isl_mat_free(M);
+	isl_mat_free(C);
+	isl_mat_free(U);
+	return NULL;
+}
+
+/* Compute and return the matrix
+ *
+ *		U_1^{-1} diag(d_1, 1, ..., 1)
+ *
+ * with U_1 the unimodular completion of the first (and only) row of B.
+ * The columns of this matrix generate the lattice that satisfies
+ * the single (linear) modulo constraint.
+ */
+static struct isl_mat *parameter_compression_1(
+			struct isl_mat *B, struct isl_vec *d)
+{
+	struct isl_mat *U;
+
+	U = isl_mat_alloc(B->ctx, B->n_col - 1, B->n_col - 1);
+	if (!U)
+		return NULL;
+	isl_seq_cpy(U->row[0], B->row[0] + 1, B->n_col - 1);
+	U = isl_mat_unimodular_complete(U, 1);
+	U = isl_mat_right_inverse(U);
+	if (!U)
+		return NULL;
+	isl_mat_col_mul(U, 0, d->block.data[0], 0);
+	U = isl_mat_lin_to_aff(U);
+	return U;
+}
+
+/* Compute a common lattice of solutions to the linear modulo
+ * constraints specified by B and d.
+ * See also the documentation of isl_mat_parameter_compression.
+ * We put the matrix
+ * 
+ *		A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
+ *
+ * on a common denominator.  This denominator D is the lcm of modulos d.
+ * Since L_i = U_i^{-1} diag(d_i, 1, ... 1), we have
+ * L_i^{-T} = U_i^T diag(d_i, 1, ... 1)^{-T} = U_i^T diag(1/d_i, 1, ..., 1).
+ * Putting this on the common denominator, we have
+ * D * L_i^{-T} = U_i^T diag(D/d_i, D, ..., D).
+ */
+static struct isl_mat *parameter_compression_multi(
+			struct isl_mat *B, struct isl_vec *d)
+{
+	int i, j, k;
+	isl_int D;
+	struct isl_mat *A = NULL, *U = NULL;
+	struct isl_mat *T;
+	unsigned size;
+
+	isl_int_init(D);
+
+	isl_vec_lcm(d, &D);
+
+	size = B->n_col - 1;
+	A = isl_mat_alloc(B->ctx, size, B->n_row * size);
+	U = isl_mat_alloc(B->ctx, size, size);
+	if (!U || !A)
+		goto error;
+	for (i = 0; i < B->n_row; ++i) {
+		isl_seq_cpy(U->row[0], B->row[i] + 1, size);
+		U = isl_mat_unimodular_complete(U, 1);
+		if (!U)
+			goto error;
+		isl_int_divexact(D, D, d->block.data[i]);
+		for (k = 0; k < U->n_col; ++k)
+			isl_int_mul(A->row[k][i*size+0], D, U->row[0][k]);
+		isl_int_mul(D, D, d->block.data[i]);
+		for (j = 1; j < U->n_row; ++j)
+			for (k = 0; k < U->n_col; ++k)
+				isl_int_mul(A->row[k][i*size+j],
+						D, U->row[j][k]);
+	}
+	A = isl_mat_left_hermite(A, 0, NULL, NULL);
+	T = isl_mat_sub_alloc(A, 0, A->n_row, 0, A->n_row);
+	T = isl_mat_lin_to_aff(T);
+	if (!T)
+		goto error;
+	isl_int_set(T->row[0][0], D);
+	T = isl_mat_right_inverse(T);
+	if (!T)
+		goto error;
+	isl_assert(T->ctx, isl_int_is_one(T->row[0][0]), goto error);
+	T = isl_mat_transpose(T);
+	isl_mat_free(A);
+	isl_mat_free(U);
+
+	isl_int_clear(D);
+	return T;
+error:
+	isl_mat_free(A);
+	isl_mat_free(U);
+	isl_int_clear(D);
+	return NULL;
+}
+
+/* Given a set of modulo constraints
+ *
+ *		c + A y = 0 mod d
+ *
+ * this function returns an affine transformation T,
+ *
+ *		y = T y'
+ *
+ * that bijectively maps the integer vectors y' to integer
+ * vectors y that satisfy the modulo constraints.
+ *
+ * This function is inspired by Section 2.5.3
+ * of B. Meister, "Stating and Manipulating Periodicity in the Polytope
+ * Model.  Applications to Program Analysis and Optimization".
+ * However, the implementation only follows the algorithm of that
+ * section for computing a particular solution and not for computing
+ * a general homogeneous solution.  The latter is incomplete and
+ * may remove some valid solutions.
+ * Instead, we use an adaptation of the algorithm in Section 7 of
+ * B. Meister, S. Verdoolaege, "Polynomial Approximations in the Polytope
+ * Model: Bringing the Power of Quasi-Polynomials to the Masses".
+ *
+ * The input is given as a matrix B = [ c A ] and a vector d.
+ * Each element of the vector d corresponds to a row in B.
+ * The output is a lower triangular matrix.
+ * If no integer vector y satisfies the given constraints then
+ * a matrix with zero columns is returned.
+ *
+ * We first compute a particular solution y_0 to the given set of
+ * modulo constraints in particular_solution.  If no such solution
+ * exists, then we return a zero-columned transformation matrix.
+ * Otherwise, we compute the generic solution to
+ *
+ *		A y = 0 mod d
+ *
+ * That is we want to compute G such that
+ *
+ *		y = G y''
+ *
+ * with y'' integer, describes the set of solutions.
+ *
+ * We first remove the common factors of each row.
+ * In particular if gcd(A_i,d_i) != 1, then we divide the whole
+ * row i (including d_i) by this common factor.  If afterwards gcd(A_i) != 1,
+ * then we divide this row of A by the common factor, unless gcd(A_i) = 0.
+ * In the later case, we simply drop the row (in both A and d).
+ *
+ * If there are no rows left in A, then G is the identity matrix. Otherwise,
+ * for each row i, we now determine the lattice of integer vectors
+ * that satisfies this row.  Let U_i be the unimodular extension of the
+ * row A_i.  This unimodular extension exists because gcd(A_i) = 1.
+ * The first component of
+ *
+ *		y' = U_i y
+ *
+ * needs to be a multiple of d_i.  Let y' = diag(d_i, 1, ..., 1) y''.
+ * Then,
+ *
+ *		y = U_i^{-1} diag(d_i, 1, ..., 1) y''
+ *
+ * for arbitrary integer vectors y''.  That is, y belongs to the lattice
+ * generated by the columns of L_i = U_i^{-1} diag(d_i, 1, ..., 1).
+ * If there is only one row, then G = L_1.
+ *
+ * If there is more than one row left, we need to compute the intersection
+ * of the lattices.  That is, we need to compute an L such that
+ *
+ *		L = L_i L_i'	for all i
+ *
+ * with L_i' some integer matrices.  Let A be constructed as follows
+ *
+ *		A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
+ *
+ * and computed the Hermite Normal Form of A = [ H 0 ] U
+ * Then,
+ *
+ *		L_i^{-T} = H U_{1,i}
+ *
+ * or
+ *
+ *		H^{-T} = L_i U_{1,i}^T
+ *
+ * In other words G = L = H^{-T}.
+ * To ensure that G is lower triangular, we compute and use its Hermite
+ * normal form.
+ *
+ * The affine transformation matrix returned is then
+ *
+ *		[  1   0  ]
+ *		[ y_0  G  ]
+ *
+ * as any y = y_0 + G y' with y' integer is a solution to the original
+ * modulo constraints.
+ */
+struct isl_mat *isl_mat_parameter_compression(
+			struct isl_mat *B, struct isl_vec *d)
+{
+	int i;
+	struct isl_mat *cst = NULL;
+	struct isl_mat *T = NULL;
+	isl_int D;
+
+	if (!B || !d)
+		goto error;
+	isl_assert(B->ctx, B->n_row == d->size, goto error);
+	cst = particular_solution(B, d);
+	if (!cst)
+		goto error;
+	if (cst->n_col == 0) {
+		T = isl_mat_alloc(B->ctx, B->n_col, 0);
+		isl_mat_free(cst);
+		isl_mat_free(B);
+		isl_vec_free(d);
+		return T;
+	}
+	isl_int_init(D);
+	/* Replace a*g*row = 0 mod g*m by row = 0 mod m */
+	for (i = 0; i < B->n_row; ++i) {
+		isl_seq_gcd(B->row[i] + 1, B->n_col - 1, &D);
+		if (isl_int_is_one(D))
+			continue;
+		if (isl_int_is_zero(D)) {
+			B = isl_mat_drop_rows(B, i, 1);
+			d = isl_vec_cow(d);
+			if (!B || !d)
+				goto error2;
+			isl_seq_cpy(d->block.data+i, d->block.data+i+1,
+							d->size - (i+1));
+			d->size--;
+			i--;
+			continue;
+		}
+		B = isl_mat_cow(B);
+		if (!B)
+			goto error2;
+		isl_seq_scale_down(B->row[i] + 1, B->row[i] + 1, D, B->n_col-1);
+		isl_int_gcd(D, D, d->block.data[i]);
+		d = isl_vec_cow(d);
+		if (!d)
+			goto error2;
+		isl_int_divexact(d->block.data[i], d->block.data[i], D);
+	}
+	isl_int_clear(D);
+	if (B->n_row == 0)
+		T = isl_mat_identity(B->ctx, B->n_col);
+	else if (B->n_row == 1)
+		T = parameter_compression_1(B, d);
+	else
+		T = parameter_compression_multi(B, d);
+	T = isl_mat_left_hermite(T, 0, NULL, NULL);
+	if (!T)
+		goto error;
+	isl_mat_sub_copy(T->ctx, T->row + 1, cst->row, cst->n_row, 0, 0, 1);
+	isl_mat_free(cst);
+	isl_mat_free(B);
+	isl_vec_free(d);
+	return T;
+error2:
+	isl_int_clear(D);
+error:
+	isl_mat_free(cst);
+	isl_mat_free(B);
+	isl_vec_free(d);
+	return NULL;
+}
+
+/* Given a set of equalities
+ *
+ *		B(y) + A x = 0						(*)
+ *
+ * compute and return an affine transformation T,
+ *
+ *		y = T y'
+ *
+ * that bijectively maps the integer vectors y' to integer
+ * vectors y that satisfy the modulo constraints for some value of x.
+ *
+ * Let [H 0] be the Hermite Normal Form of A, i.e.,
+ *
+ *		A = [H 0] Q
+ *
+ * Then y is a solution of (*) iff
+ *
+ *		H^-1 B(y) (= - [I 0] Q x)
+ *
+ * is an integer vector.  Let d be the common denominator of H^-1.
+ * We impose
+ *
+ *		d H^-1 B(y) = 0 mod d
+ *
+ * and compute the solution using isl_mat_parameter_compression.
+ */
+__isl_give isl_mat *isl_mat_parameter_compression_ext(__isl_take isl_mat *B,
+	__isl_take isl_mat *A)
+{
+	isl_ctx *ctx;
+	isl_vec *d;
+	int n_row, n_col;
+
+	if (!A)
+		return isl_mat_free(B);
+
+	ctx = isl_mat_get_ctx(A);
+	n_row = A->n_row;
+	n_col = A->n_col;
+	A = isl_mat_left_hermite(A, 0, NULL, NULL);
+	A = isl_mat_drop_cols(A, n_row, n_col - n_row);
+	A = isl_mat_lin_to_aff(A);
+	A = isl_mat_right_inverse(A);
+	d = isl_vec_alloc(ctx, n_row);
+	if (A)
+		d = isl_vec_set(d, A->row[0][0]);
+	A = isl_mat_drop_rows(A, 0, 1);
+	A = isl_mat_drop_cols(A, 0, 1);
+	B = isl_mat_product(A, B);
+
+	return isl_mat_parameter_compression(B, d);
+}
+
+/* Given a set of equalities
+ *
+ *		M x - c = 0
+ *
+ * this function computes a unimodular transformation from a lower-dimensional
+ * space to the original space that bijectively maps the integer points x'
+ * in the lower-dimensional space to the integer points x in the original
+ * space that satisfy the equalities.
+ *
+ * The input is given as a matrix B = [ -c M ] and the output is a
+ * matrix that maps [1 x'] to [1 x].
+ * If T2 is not NULL, then *T2 is set to a matrix mapping [1 x] to [1 x'].
+ *
+ * First compute the (left) Hermite normal form of M,
+ *
+ *		M [U1 U2] = M U = H = [H1 0]
+ * or
+ *		              M = H Q = [H1 0] [Q1]
+ *                                             [Q2]
+ *
+ * with U, Q unimodular, Q = U^{-1} (and H lower triangular).
+ * Define the transformed variables as
+ *
+ *		x = [U1 U2] [ x1' ] = [U1 U2] [Q1] x
+ *		            [ x2' ]           [Q2]
+ *
+ * The equalities then become
+ *
+ *		H1 x1' - c = 0   or   x1' = H1^{-1} c = c'
+ *
+ * If any of the c' is non-integer, then the original set has no
+ * integer solutions (since the x' are a unimodular transformation
+ * of the x) and a zero-column matrix is returned.
+ * Otherwise, the transformation is given by
+ *
+ *		x = U1 H1^{-1} c + U2 x2'
+ *
+ * The inverse transformation is simply
+ *
+ *		x2' = Q2 x
+ */
+__isl_give isl_mat *isl_mat_variable_compression(__isl_take isl_mat *B,
+	__isl_give isl_mat **T2)
+{
+	int i;
+	struct isl_mat *H = NULL, *C = NULL, *H1, *U = NULL, *U1, *U2, *TC;
+	unsigned dim;
+
+	if (T2)
+		*T2 = NULL;
+	if (!B)
+		goto error;
+
+	dim = B->n_col - 1;
+	H = isl_mat_sub_alloc(B, 0, B->n_row, 1, dim);
+	H = isl_mat_left_hermite(H, 0, &U, T2);
+	if (!H || !U || (T2 && !*T2))
+		goto error;
+	if (T2) {
+		*T2 = isl_mat_drop_rows(*T2, 0, B->n_row);
+		*T2 = isl_mat_lin_to_aff(*T2);
+		if (!*T2)
+			goto error;
+	}
+	C = isl_mat_alloc(B->ctx, 1+B->n_row, 1);
+	if (!C)
+		goto error;
+	isl_int_set_si(C->row[0][0], 1);
+	isl_mat_sub_neg(C->ctx, C->row+1, B->row, B->n_row, 0, 0, 1);
+	H1 = isl_mat_sub_alloc(H, 0, H->n_row, 0, H->n_row);
+	H1 = isl_mat_lin_to_aff(H1);
+	TC = isl_mat_inverse_product(H1, C);
+	if (!TC)
+		goto error;
+	isl_mat_free(H);
+	if (!isl_int_is_one(TC->row[0][0])) {
+		for (i = 0; i < B->n_row; ++i) {
+			if (!isl_int_is_divisible_by(TC->row[1+i][0], TC->row[0][0])) {
+				struct isl_ctx *ctx = B->ctx;
+				isl_mat_free(B);
+				isl_mat_free(TC);
+				isl_mat_free(U);
+				if (T2) {
+					isl_mat_free(*T2);
+					*T2 = NULL;
+				}
+				return isl_mat_alloc(ctx, 1 + dim, 0);
+			}
+			isl_seq_scale_down(TC->row[1+i], TC->row[1+i], TC->row[0][0], 1);
+		}
+		isl_int_set_si(TC->row[0][0], 1);
+	}
+	U1 = isl_mat_sub_alloc(U, 0, U->n_row, 0, B->n_row);
+	U1 = isl_mat_lin_to_aff(U1);
+	U2 = isl_mat_sub_alloc(U, 0, U->n_row, B->n_row, U->n_row - B->n_row);
+	U2 = isl_mat_lin_to_aff(U2);
+	isl_mat_free(U);
+	TC = isl_mat_product(U1, TC);
+	TC = isl_mat_aff_direct_sum(TC, U2);
+
+	isl_mat_free(B);
+
+	return TC;
+error:
+	isl_mat_free(B);
+	isl_mat_free(H);
+	isl_mat_free(U);
+	if (T2) {
+		isl_mat_free(*T2);
+		*T2 = NULL;
+	}
+	return NULL;
+}
+
+/* Use the n equalities of bset to unimodularly transform the
+ * variables x such that n transformed variables x1' have a constant value
+ * and rewrite the constraints of bset in terms of the remaining
+ * transformed variables x2'.  The matrix pointed to by T maps
+ * the new variables x2' back to the original variables x, while T2
+ * maps the original variables to the new variables.
+ */
+static struct isl_basic_set *compress_variables(
+	struct isl_basic_set *bset, struct isl_mat **T, struct isl_mat **T2)
+{
+	struct isl_mat *B, *TC;
+	unsigned dim;
+
+	if (T)
+		*T = NULL;
+	if (T2)
+		*T2 = NULL;
+	if (!bset)
+		goto error;
+	isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
+	isl_assert(bset->ctx, bset->n_div == 0, goto error);
+	dim = isl_basic_set_n_dim(bset);
+	isl_assert(bset->ctx, bset->n_eq <= dim, goto error);
+	if (bset->n_eq == 0)
+		return bset;
+
+	B = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, bset->n_eq, 0, 1 + dim);
+	TC = isl_mat_variable_compression(B, T2);
+	if (!TC)
+		goto error;
+	if (TC->n_col == 0) {
+		isl_mat_free(TC);
+		if (T2) {
+			isl_mat_free(*T2);
+			*T2 = NULL;
+		}
+		return isl_basic_set_set_to_empty(bset);
+	}
+
+	bset = isl_basic_set_preimage(bset, T ? isl_mat_copy(TC) : TC);
+	if (T)
+		*T = TC;
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_remove_equalities(
+	struct isl_basic_set *bset, struct isl_mat **T, struct isl_mat **T2)
+{
+	if (T)
+		*T = NULL;
+	if (T2)
+		*T2 = NULL;
+	if (!bset)
+		return NULL;
+	isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
+	bset = isl_basic_set_gauss(bset, NULL);
+	if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
+		return bset;
+	bset = compress_variables(bset, T, T2);
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	*T = NULL;
+	return NULL;
+}
+
+/* Check if dimension dim belongs to a residue class
+ *		i_dim \equiv r mod m
+ * with m != 1 and if so return m in *modulo and r in *residue.
+ * As a special case, when i_dim has a fixed value v, then
+ * *modulo is set to 0 and *residue to v.
+ *
+ * If i_dim does not belong to such a residue class, then *modulo
+ * is set to 1 and *residue is set to 0.
+ */
+int isl_basic_set_dim_residue_class(struct isl_basic_set *bset,
+	int pos, isl_int *modulo, isl_int *residue)
+{
+	struct isl_ctx *ctx;
+	struct isl_mat *H = NULL, *U = NULL, *C, *H1, *U1;
+	unsigned total;
+	unsigned nparam;
+
+	if (!bset || !modulo || !residue)
+		return -1;
+
+	if (isl_basic_set_plain_dim_is_fixed(bset, pos, residue)) {
+		isl_int_set_si(*modulo, 0);
+		return 0;
+	}
+
+	ctx = bset->ctx;
+	total = isl_basic_set_total_dim(bset);
+	nparam = isl_basic_set_n_param(bset);
+	H = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, bset->n_eq, 1, total);
+	H = isl_mat_left_hermite(H, 0, &U, NULL);
+	if (!H)
+		return -1;
+
+	isl_seq_gcd(U->row[nparam + pos]+bset->n_eq,
+			total-bset->n_eq, modulo);
+	if (isl_int_is_zero(*modulo))
+		isl_int_set_si(*modulo, 1);
+	if (isl_int_is_one(*modulo)) {
+		isl_int_set_si(*residue, 0);
+		isl_mat_free(H);
+		isl_mat_free(U);
+		return 0;
+	}
+
+	C = isl_mat_alloc(bset->ctx, 1+bset->n_eq, 1);
+	if (!C)
+		goto error;
+	isl_int_set_si(C->row[0][0], 1);
+	isl_mat_sub_neg(C->ctx, C->row+1, bset->eq, bset->n_eq, 0, 0, 1);
+	H1 = isl_mat_sub_alloc(H, 0, H->n_row, 0, H->n_row);
+	H1 = isl_mat_lin_to_aff(H1);
+	C = isl_mat_inverse_product(H1, C);
+	isl_mat_free(H);
+	U1 = isl_mat_sub_alloc(U, nparam+pos, 1, 0, bset->n_eq);
+	U1 = isl_mat_lin_to_aff(U1);
+	isl_mat_free(U);
+	C = isl_mat_product(U1, C);
+	if (!C)
+		return -1;
+	if (!isl_int_is_divisible_by(C->row[1][0], C->row[0][0])) {
+		bset = isl_basic_set_copy(bset);
+		bset = isl_basic_set_set_to_empty(bset);
+		isl_basic_set_free(bset);
+		isl_int_set_si(*modulo, 1);
+		isl_int_set_si(*residue, 0);
+		return 0;
+	}
+	isl_int_divexact(*residue, C->row[1][0], C->row[0][0]);
+	isl_int_fdiv_r(*residue, *residue, *modulo);
+	isl_mat_free(C);
+	return 0;
+error:
+	isl_mat_free(H);
+	isl_mat_free(U);
+	return -1;
+}
+
+/* Check if dimension dim belongs to a residue class
+ *		i_dim \equiv r mod m
+ * with m != 1 and if so return m in *modulo and r in *residue.
+ * As a special case, when i_dim has a fixed value v, then
+ * *modulo is set to 0 and *residue to v.
+ *
+ * If i_dim does not belong to such a residue class, then *modulo
+ * is set to 1 and *residue is set to 0.
+ */
+int isl_set_dim_residue_class(struct isl_set *set,
+	int pos, isl_int *modulo, isl_int *residue)
+{
+	isl_int m;
+	isl_int r;
+	int i;
+
+	if (!set || !modulo || !residue)
+		return -1;
+
+	if (set->n == 0) {
+		isl_int_set_si(*modulo, 0);
+		isl_int_set_si(*residue, 0);
+		return 0;
+	}
+
+	if (isl_basic_set_dim_residue_class(set->p[0], pos, modulo, residue)<0)
+		return -1;
+
+	if (set->n == 1)
+		return 0;
+
+	if (isl_int_is_one(*modulo))
+		return 0;
+
+	isl_int_init(m);
+	isl_int_init(r);
+
+	for (i = 1; i < set->n; ++i) {
+		if (isl_basic_set_dim_residue_class(set->p[i], pos, &m, &r) < 0)
+			goto error;
+		isl_int_gcd(*modulo, *modulo, m);
+		isl_int_sub(m, *residue, r);
+		isl_int_gcd(*modulo, *modulo, m);
+		if (!isl_int_is_zero(*modulo))
+			isl_int_fdiv_r(*residue, *residue, *modulo);
+		if (isl_int_is_one(*modulo))
+			break;
+	}
+
+	isl_int_clear(m);
+	isl_int_clear(r);
+
+	return 0;
+error:
+	isl_int_clear(m);
+	isl_int_clear(r);
+	return -1;
+}
+
+/* Check if dimension "dim" belongs to a residue class
+ *		i_dim \equiv r mod m
+ * with m != 1 and if so return m in *modulo and r in *residue.
+ * As a special case, when i_dim has a fixed value v, then
+ * *modulo is set to 0 and *residue to v.
+ *
+ * If i_dim does not belong to such a residue class, then *modulo
+ * is set to 1 and *residue is set to 0.
+ */
+int isl_set_dim_residue_class_val(__isl_keep isl_set *set,
+	int pos, __isl_give isl_val **modulo, __isl_give isl_val **residue)
+{
+	*modulo = NULL;
+	*residue = NULL;
+	if (!set)
+		return -1;
+	*modulo = isl_val_alloc(isl_set_get_ctx(set));
+	*residue = isl_val_alloc(isl_set_get_ctx(set));
+	if (!*modulo || !*residue)
+		goto error;
+	if (isl_set_dim_residue_class(set, pos,
+					&(*modulo)->n, &(*residue)->n) < 0)
+		goto error;
+	isl_int_set_si((*modulo)->d, 1);
+	isl_int_set_si((*residue)->d, 1);
+	return 0;
+error:
+	isl_val_free(*modulo);
+	isl_val_free(*residue);
+	return -1;
+}
diff --git a/polly/lib/External/isl/isl_equalities.h b/polly/lib/External/isl/isl_equalities.h
new file mode 100644
index 00000000000..0206cd1c222
--- /dev/null
+++ b/polly/lib/External/isl/isl_equalities.h
@@ -0,0 +1,33 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_EQUALITIES_H
+#define ISL_EQUALITIES_H
+
+#include <isl/set.h>
+#include <isl/mat.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+__isl_give isl_mat *isl_mat_variable_compression(__isl_take isl_mat *B,
+	__isl_give isl_mat **T2);
+struct isl_mat *isl_mat_parameter_compression(
+			struct isl_mat *B, struct isl_vec *d);
+__isl_give isl_mat *isl_mat_parameter_compression_ext(__isl_take isl_mat *B,
+	__isl_take isl_mat *A);
+struct isl_basic_set *isl_basic_set_remove_equalities(
+	struct isl_basic_set *bset, struct isl_mat **T, struct isl_mat **T2);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/isl_factorization.c b/polly/lib/External/isl/isl_factorization.c
new file mode 100644
index 00000000000..f692cdbcd0b
--- /dev/null
+++ b/polly/lib/External/isl/isl_factorization.c
@@ -0,0 +1,331 @@
+/*
+ * Copyright 2005-2007 Universiteit Leiden
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2010      INRIA Saclay
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, Leiden Institute of Advanced Computer Science,
+ * Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
+ * and K.U.Leuven, Departement Computerwetenschappen, Celestijnenlaan 200A,
+ * B-3001 Leuven, Belgium
+ * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France 
+ */
+
+#include <isl_map_private.h>
+#include <isl_factorization.h>
+#include <isl_space_private.h>
+#include <isl_mat_private.h>
+
+static __isl_give isl_factorizer *isl_factorizer_alloc(
+	__isl_take isl_morph *morph, int n_group)
+{
+	isl_factorizer *f = NULL;
+	int *len = NULL;
+
+	if (!morph)
+		return NULL;
+
+	if (n_group > 0) {
+		len = isl_alloc_array(morph->dom->ctx, int, n_group);
+		if (!len)
+			goto error;
+	}
+
+	f = isl_alloc_type(morph->dom->ctx, struct isl_factorizer);
+	if (!f)
+		goto error;
+
+	f->morph = morph;
+	f->n_group = n_group;
+	f->len = len;
+
+	return f;
+error:
+	free(len);
+	isl_morph_free(morph);
+	return NULL;
+}
+
+void isl_factorizer_free(__isl_take isl_factorizer *f)
+{
+	if (!f)
+		return;
+
+	isl_morph_free(f->morph);
+	free(f->len);
+	free(f);
+}
+
+void isl_factorizer_dump(__isl_take isl_factorizer *f)
+{
+	int i;
+
+	if (!f)
+		return;
+
+	isl_morph_print_internal(f->morph, stderr);
+	fprintf(stderr, "[");
+	for (i = 0; i < f->n_group; ++i) {
+		if (i)
+			fprintf(stderr, ", ");
+		fprintf(stderr, "%d", f->len[i]);
+	}
+	fprintf(stderr, "]\n");
+}
+
+__isl_give isl_factorizer *isl_factorizer_identity(__isl_keep isl_basic_set *bset)
+{
+	return isl_factorizer_alloc(isl_morph_identity(bset), 0);
+}
+
+__isl_give isl_factorizer *isl_factorizer_groups(__isl_keep isl_basic_set *bset,
+	__isl_take isl_mat *Q, __isl_take isl_mat *U, int n, int *len)
+{
+	int i;
+	unsigned nvar;
+	unsigned ovar;
+	isl_space *dim;
+	isl_basic_set *dom;
+	isl_basic_set *ran;
+	isl_morph *morph;
+	isl_factorizer *f;
+	isl_mat *id;
+
+	if (!bset || !Q || !U)
+		goto error;
+
+	ovar = 1 + isl_space_offset(bset->dim, isl_dim_set);
+	id = isl_mat_identity(bset->ctx, ovar);
+	Q = isl_mat_diagonal(isl_mat_copy(id), Q);
+	U = isl_mat_diagonal(id, U);
+
+	nvar = isl_basic_set_dim(bset, isl_dim_set);
+	dim = isl_basic_set_get_space(bset);
+	dom = isl_basic_set_universe(isl_space_copy(dim));
+	dim = isl_space_drop_dims(dim, isl_dim_set, 0, nvar);
+	dim = isl_space_add_dims(dim, isl_dim_set, nvar);
+	ran = isl_basic_set_universe(dim);
+	morph = isl_morph_alloc(dom, ran, Q, U);
+	f = isl_factorizer_alloc(morph, n);
+	if (!f)
+		return NULL;
+	for (i = 0; i < n; ++i)
+		f->len[i] = len[i];
+	return f;
+error:
+	isl_mat_free(Q);
+	isl_mat_free(U);
+	return NULL;
+}
+
+struct isl_factor_groups {
+	int *pos;		/* for each column: row position of pivot */
+	int *group;		/* group to which a column belongs */
+	int *cnt;		/* number of columns in the group */
+	int *rowgroup;		/* group to which a constraint belongs */
+};
+
+/* Initialize isl_factor_groups structure: find pivot row positions,
+ * each column initially belongs to its own group and the groups
+ * of the constraints are still unknown.
+ */
+static int init_groups(struct isl_factor_groups *g, __isl_keep isl_mat *H)
+{
+	int i, j;
+
+	if (!H)
+		return -1;
+
+	g->pos = isl_alloc_array(H->ctx, int, H->n_col);
+	g->group = isl_alloc_array(H->ctx, int, H->n_col);
+	g->cnt = isl_alloc_array(H->ctx, int, H->n_col);
+	g->rowgroup = isl_alloc_array(H->ctx, int, H->n_row);
+
+	if (!g->pos || !g->group || !g->cnt || !g->rowgroup)
+		return -1;
+
+	for (i = 0; i < H->n_row; ++i)
+		g->rowgroup[i] = -1;
+	for (i = 0, j = 0; i < H->n_col; ++i) {
+		for ( ; j < H->n_row; ++j)
+			if (!isl_int_is_zero(H->row[j][i]))
+				break;
+		g->pos[i] = j;
+	}
+	for (i = 0; i < H->n_col; ++i) {
+		g->group[i] = i;
+		g->cnt[i] = 1;
+	}
+
+	return 0;
+}
+
+/* Update group[k] to the group column k belongs to.
+ * When merging two groups, only the group of the current
+ * group leader is changed.  Here we change the group of
+ * the other members to also point to the group that the
+ * old group leader now points to.
+ */
+static void update_group(struct isl_factor_groups *g, int k)
+{
+	int p = g->group[k];
+	while (g->cnt[p] == 0)
+		p = g->group[p];
+	g->group[k] = p;
+}
+
+/* Merge group i with all groups of the subsequent columns
+ * with non-zero coefficients in row j of H.
+ * (The previous columns are all zero; otherwise we would have handled
+ * the row before.)
+ */
+static int update_group_i_with_row_j(struct isl_factor_groups *g, int i, int j,
+	__isl_keep isl_mat *H)
+{
+	int k;
+
+	g->rowgroup[j] = g->group[i];
+	for (k = i + 1; k < H->n_col && j >= g->pos[k]; ++k) {
+		update_group(g, k);
+		update_group(g, i);
+		if (g->group[k] != g->group[i] &&
+		    !isl_int_is_zero(H->row[j][k])) {
+			isl_assert(H->ctx, g->cnt[g->group[k]] != 0, return -1);
+			isl_assert(H->ctx, g->cnt[g->group[i]] != 0, return -1);
+			if (g->group[i] < g->group[k]) {
+				g->cnt[g->group[i]] += g->cnt[g->group[k]];
+				g->cnt[g->group[k]] = 0;
+				g->group[g->group[k]] = g->group[i];
+			} else {
+				g->cnt[g->group[k]] += g->cnt[g->group[i]];
+				g->cnt[g->group[i]] = 0;
+				g->group[g->group[i]] = g->group[k];
+			}
+		}
+	}
+
+	return 0;
+}
+
+/* Update the group information based on the constraint matrix.
+ */
+static int update_groups(struct isl_factor_groups *g, __isl_keep isl_mat *H)
+{
+	int i, j;
+
+	for (i = 0; i < H->n_col && g->cnt[0] < H->n_col; ++i) {
+		if (g->pos[i] == H->n_row)
+			continue; /* A line direction */
+		if (g->rowgroup[g->pos[i]] == -1)
+			g->rowgroup[g->pos[i]] = i;
+		for (j = g->pos[i] + 1; j < H->n_row; ++j) {
+			if (isl_int_is_zero(H->row[j][i]))
+				continue;
+			if (g->rowgroup[j] != -1)
+				continue;
+			if (update_group_i_with_row_j(g, i, j, H) < 0)
+				return -1;
+		}
+	}
+	for (i = 1; i < H->n_col; ++i)
+		update_group(g, i);
+
+	return 0;
+}
+
+static void clear_groups(struct isl_factor_groups *g)
+{
+	if (!g)
+		return;
+	free(g->pos);
+	free(g->group);
+	free(g->cnt);
+	free(g->rowgroup);
+}
+
+/* Determine if the set variables of the basic set can be factorized and
+ * return the results in an isl_factorizer.
+ *
+ * The algorithm works by first computing the Hermite normal form
+ * and then grouping columns linked by one or more constraints together,
+ * where a constraints "links" two or more columns if the constraint
+ * has nonzero coefficients in the columns.
+ */
+__isl_give isl_factorizer *isl_basic_set_factorizer(
+	__isl_keep isl_basic_set *bset)
+{
+	int i, j, n, done;
+	isl_mat *H, *U, *Q;
+	unsigned nvar;
+	struct isl_factor_groups g = { 0 };
+	isl_factorizer *f;
+
+	if (!bset)
+		return NULL;
+
+	isl_assert(bset->ctx, isl_basic_set_dim(bset, isl_dim_div) == 0,
+		return NULL);
+
+	nvar = isl_basic_set_dim(bset, isl_dim_set);
+	if (nvar <= 1)
+		return isl_factorizer_identity(bset);
+
+	H = isl_mat_alloc(bset->ctx, bset->n_eq + bset->n_ineq, nvar);
+	if (!H)
+		return NULL;
+	isl_mat_sub_copy(bset->ctx, H->row, bset->eq, bset->n_eq,
+		0, 1 + isl_space_offset(bset->dim, isl_dim_set), nvar);
+	isl_mat_sub_copy(bset->ctx, H->row + bset->n_eq, bset->ineq, bset->n_ineq,
+		0, 1 + isl_space_offset(bset->dim, isl_dim_set), nvar);
+	H = isl_mat_left_hermite(H, 0, &U, &Q);
+
+	if (init_groups(&g, H) < 0)
+		goto error;
+	if (update_groups(&g, H) < 0)
+		goto error;
+
+	if (g.cnt[0] == nvar) {
+		isl_mat_free(H);
+		isl_mat_free(U);
+		isl_mat_free(Q);
+		clear_groups(&g);
+
+		return isl_factorizer_identity(bset);
+	}
+
+	done = 0;
+	n = 0;
+	while (done != nvar) {
+		int group = g.group[done];
+		for (i = 1; i < g.cnt[group]; ++i) {
+			if (g.group[done + i] == group)
+				continue;
+			for (j = done + g.cnt[group]; j < nvar; ++j)
+				if (g.group[j] == group)
+					break;
+			if (j == nvar)
+				isl_die(bset->ctx, isl_error_internal,
+					"internal error", goto error);
+			g.group[j] = g.group[done + i];
+			Q = isl_mat_swap_rows(Q, done + i, j);
+			U = isl_mat_swap_cols(U, done + i, j);
+		}
+		done += g.cnt[group];
+		g.pos[n++] = g.cnt[group];
+	}
+
+	f = isl_factorizer_groups(bset, Q, U, n, g.pos);
+
+	isl_mat_free(H);
+	clear_groups(&g);
+
+	return f;
+error:
+	isl_mat_free(H);
+	isl_mat_free(U);
+	isl_mat_free(Q);
+	clear_groups(&g);
+	return NULL;
+}
diff --git a/polly/lib/External/isl/isl_factorization.h b/polly/lib/External/isl/isl_factorization.h
new file mode 100644
index 00000000000..4687b9cfd97
--- /dev/null
+++ b/polly/lib/External/isl/isl_factorization.h
@@ -0,0 +1,29 @@
+#include <isl/set.h>
+#include <isl_morph.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+/* Data for factorizing a particular basic set.
+ * After applying "morph" to the basic set, there are "n_group"
+ * groups of consecutive set variables, each of length "len[i]",
+ * with 0 <= i < n_group.
+ * If no factorization is possible, then "n_group" is set to 0.
+ */
+struct isl_factorizer {
+	isl_morph	*morph;
+	int		n_group;
+	int		*len;
+};
+typedef struct isl_factorizer isl_factorizer;
+
+__isl_give isl_factorizer *isl_basic_set_factorizer(
+	__isl_keep isl_basic_set *bset);
+
+void isl_factorizer_free(__isl_take isl_factorizer *f);
+void isl_factorizer_dump(__isl_take isl_factorizer *f);
+
+#if defined(__cplusplus)
+}
+#endif
diff --git a/polly/lib/External/isl/isl_farkas.c b/polly/lib/External/isl/isl_farkas.c
new file mode 100644
index 00000000000..dddb49a76c2
--- /dev/null
+++ b/polly/lib/External/isl/isl_farkas.c
@@ -0,0 +1,403 @@
+/*
+ * Copyright 2010      INRIA Saclay
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France 
+ */
+
+#include <isl_map_private.h>
+#include <isl/set.h>
+#include <isl_space_private.h>
+#include <isl_seq.h>
+
+/*
+ * Let C be a cone and define
+ *
+ *	C' := { y | forall x in C : y x >= 0 }
+ *
+ * C' contains the coefficients of all linear constraints
+ * that are valid for C.
+ * Furthermore, C'' = C.
+ *
+ * If C is defined as { x | A x >= 0 }
+ * then any element in C' must be a non-negative combination
+ * of the rows of A, i.e., y = t A with t >= 0.  That is,
+ *
+ *	C' = { y | exists t >= 0 : y = t A }
+ *
+ * If any of the rows in A actually represents an equality, then
+ * also negative combinations of this row are allowed and so the
+ * non-negativity constraint on the corresponding element of t
+ * can be dropped.
+ *
+ * A polyhedron P = { x | b + A x >= 0 } can be represented
+ * in homogeneous coordinates by the cone
+ * C = { [z,x] | b z + A x >= and z >= 0 }
+ * The valid linear constraints on C correspond to the valid affine
+ * constraints on P.
+ * This is essentially Farkas' lemma.
+ *
+ * Since
+ *				  [ 1 0 ]
+ *		[ w y ] = [t_0 t] [ b A ]
+ *
+ * we have
+ *
+ *	C' = { w, y | exists t_0, t >= 0 : y = t A and w = t_0 + t b }
+ * or
+ *
+ *	C' = { w, y | exists t >= 0 : y = t A and w - t b >= 0 }
+ *
+ * In practice, we introduce an extra variable (w), shifting all
+ * other variables to the right, and an extra inequality
+ * (w - t b >= 0) corresponding to the positivity constraint on
+ * the homogeneous coordinate.
+ *
+ * When going back from coefficients to solutions, we immediately
+ * plug in 1 for z, which corresponds to shifting all variables
+ * to the left, with the leftmost ending up in the constant position.
+ */
+
+/* Add the given prefix to all named isl_dim_set dimensions in "dim".
+ */
+static __isl_give isl_space *isl_space_prefix(__isl_take isl_space *dim,
+	const char *prefix)
+{
+	int i;
+	isl_ctx *ctx;
+	unsigned nvar;
+	size_t prefix_len = strlen(prefix);
+
+	if (!dim)
+		return NULL;
+
+	ctx = isl_space_get_ctx(dim);
+	nvar = isl_space_dim(dim, isl_dim_set);
+
+	for (i = 0; i < nvar; ++i) {
+		const char *name;
+		char *prefix_name;
+
+		name = isl_space_get_dim_name(dim, isl_dim_set, i);
+		if (!name)
+			continue;
+
+		prefix_name = isl_alloc_array(ctx, char,
+					      prefix_len + strlen(name) + 1);
+		if (!prefix_name)
+			goto error;
+		memcpy(prefix_name, prefix, prefix_len);
+		strcpy(prefix_name + prefix_len, name);
+
+		dim = isl_space_set_dim_name(dim, isl_dim_set, i, prefix_name);
+		free(prefix_name);
+	}
+
+	return dim;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+/* Given a dimension specification of the solutions space, construct
+ * a dimension specification for the space of coefficients.
+ *
+ * In particular transform
+ *
+ *	[params] -> { S }
+ *
+ * to
+ *
+ *	{ coefficients[[cst, params] -> S] }
+ *
+ * and prefix each dimension name with "c_".
+ */
+static __isl_give isl_space *isl_space_coefficients(__isl_take isl_space *dim)
+{
+	isl_space *dim_param;
+	unsigned nvar;
+	unsigned nparam;
+
+	nvar = isl_space_dim(dim, isl_dim_set);
+	nparam = isl_space_dim(dim, isl_dim_param);
+	dim_param = isl_space_copy(dim);
+	dim_param = isl_space_drop_dims(dim_param, isl_dim_set, 0, nvar);
+	dim_param = isl_space_move_dims(dim_param, isl_dim_set, 0,
+				 isl_dim_param, 0, nparam);
+	dim_param = isl_space_prefix(dim_param, "c_");
+	dim_param = isl_space_insert_dims(dim_param, isl_dim_set, 0, 1);
+	dim_param = isl_space_set_dim_name(dim_param, isl_dim_set, 0, "c_cst");
+	dim = isl_space_drop_dims(dim, isl_dim_param, 0, nparam);
+	dim = isl_space_prefix(dim, "c_");
+	dim = isl_space_join(isl_space_from_domain(dim_param),
+			   isl_space_from_range(dim));
+	dim = isl_space_wrap(dim);
+	dim = isl_space_set_tuple_name(dim, isl_dim_set, "coefficients");
+
+	return dim;
+}
+
+/* Drop the given prefix from all named dimensions of type "type" in "dim".
+ */
+static __isl_give isl_space *isl_space_unprefix(__isl_take isl_space *dim,
+	enum isl_dim_type type, const char *prefix)
+{
+	int i;
+	unsigned n;
+	size_t prefix_len = strlen(prefix);
+
+	n = isl_space_dim(dim, type);
+
+	for (i = 0; i < n; ++i) {
+		const char *name;
+
+		name = isl_space_get_dim_name(dim, type, i);
+		if (!name)
+			continue;
+		if (strncmp(name, prefix, prefix_len))
+			continue;
+
+		dim = isl_space_set_dim_name(dim, type, i, name + prefix_len);
+	}
+
+	return dim;
+}
+
+/* Given a dimension specification of the space of coefficients, construct
+ * a dimension specification for the space of solutions.
+ *
+ * In particular transform
+ *
+ *	{ coefficients[[cst, params] -> S] }
+ *
+ * to
+ *
+ *	[params] -> { S }
+ *
+ * and drop the "c_" prefix from the dimension names.
+ */
+static __isl_give isl_space *isl_space_solutions(__isl_take isl_space *dim)
+{
+	unsigned nparam;
+
+	dim = isl_space_unwrap(dim);
+	dim = isl_space_drop_dims(dim, isl_dim_in, 0, 1);
+	dim = isl_space_unprefix(dim, isl_dim_in, "c_");
+	dim = isl_space_unprefix(dim, isl_dim_out, "c_");
+	nparam = isl_space_dim(dim, isl_dim_in);
+	dim = isl_space_move_dims(dim, isl_dim_param, 0, isl_dim_in, 0, nparam);
+	dim = isl_space_range(dim);
+
+	return dim;
+}
+
+/* Return the rational universe basic set in the given space.
+ */
+static __isl_give isl_basic_set *rational_universe(__isl_take isl_space *space)
+{
+	isl_basic_set *bset;
+
+	bset = isl_basic_set_universe(space);
+	bset = isl_basic_set_set_rational(bset);
+
+	return bset;
+}
+
+/* Compute the dual of "bset" by applying Farkas' lemma.
+ * As explained above, we add an extra dimension to represent
+ * the coefficient of the constant term when going from solutions
+ * to coefficients (shift == 1) and we drop the extra dimension when going
+ * in the opposite direction (shift == -1).  "dim" is the space in which
+ * the dual should be created.
+ *
+ * If "bset" is (obviously) empty, then the way this emptiness
+ * is represented by the constraints does not allow for the application
+ * of the standard farkas algorithm.  We therefore handle this case
+ * specifically and return the universe basic set.
+ */
+static __isl_give isl_basic_set *farkas(__isl_take isl_space *space,
+	__isl_take isl_basic_set *bset, int shift)
+{
+	int i, j, k;
+	isl_basic_set *dual = NULL;
+	unsigned total;
+
+	if (isl_basic_set_plain_is_empty(bset)) {
+		isl_basic_set_free(bset);
+		return rational_universe(space);
+	}
+
+	total = isl_basic_set_total_dim(bset);
+
+	dual = isl_basic_set_alloc_space(space, bset->n_eq + bset->n_ineq,
+					total, bset->n_ineq + (shift > 0));
+	dual = isl_basic_set_set_rational(dual);
+
+	for (i = 0; i < bset->n_eq + bset->n_ineq; ++i) {
+		k = isl_basic_set_alloc_div(dual);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(dual->div[k][0], 0);
+	}
+
+	for (i = 0; i < total; ++i) {
+		k = isl_basic_set_alloc_equality(dual);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(dual->eq[k], 1 + shift + total);
+		isl_int_set_si(dual->eq[k][1 + shift + i], -1);
+		for (j = 0; j < bset->n_eq; ++j)
+			isl_int_set(dual->eq[k][1 + shift + total + j],
+				    bset->eq[j][1 + i]);
+		for (j = 0; j < bset->n_ineq; ++j)
+			isl_int_set(dual->eq[k][1 + shift + total + bset->n_eq + j],
+				    bset->ineq[j][1 + i]);
+	}
+
+	for (i = 0; i < bset->n_ineq; ++i) {
+		k = isl_basic_set_alloc_inequality(dual);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(dual->ineq[k],
+			    1 + shift + total + bset->n_eq + bset->n_ineq);
+		isl_int_set_si(dual->ineq[k][1 + shift + total + bset->n_eq + i], 1);
+	}
+
+	if (shift > 0) {
+		k = isl_basic_set_alloc_inequality(dual);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(dual->ineq[k], 2 + total);
+		isl_int_set_si(dual->ineq[k][1], 1);
+		for (j = 0; j < bset->n_eq; ++j)
+			isl_int_neg(dual->ineq[k][2 + total + j],
+				    bset->eq[j][0]);
+		for (j = 0; j < bset->n_ineq; ++j)
+			isl_int_neg(dual->ineq[k][2 + total + bset->n_eq + j],
+				    bset->ineq[j][0]);
+	}
+
+	dual = isl_basic_set_remove_divs(dual);
+	isl_basic_set_simplify(dual);
+	isl_basic_set_finalize(dual);
+
+	isl_basic_set_free(bset);
+	return dual;
+error:
+	isl_basic_set_free(bset);
+	isl_basic_set_free(dual);
+	return NULL;
+}
+
+/* Construct a basic set containing the tuples of coefficients of all
+ * valid affine constraints on the given basic set.
+ */
+__isl_give isl_basic_set *isl_basic_set_coefficients(
+	__isl_take isl_basic_set *bset)
+{
+	isl_space *dim;
+
+	if (!bset)
+		return NULL;
+	if (bset->n_div)
+		isl_die(bset->ctx, isl_error_invalid,
+			"input set not allowed to have local variables",
+			goto error);
+
+	dim = isl_basic_set_get_space(bset);
+	dim = isl_space_coefficients(dim);
+
+	return farkas(dim, bset, 1);
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Construct a basic set containing the elements that satisfy all
+ * affine constraints whose coefficient tuples are
+ * contained in the given basic set.
+ */
+__isl_give isl_basic_set *isl_basic_set_solutions(
+	__isl_take isl_basic_set *bset)
+{
+	isl_space *dim;
+
+	if (!bset)
+		return NULL;
+	if (bset->n_div)
+		isl_die(bset->ctx, isl_error_invalid,
+			"input set not allowed to have local variables",
+			goto error);
+
+	dim = isl_basic_set_get_space(bset);
+	dim = isl_space_solutions(dim);
+
+	return farkas(dim, bset, -1);
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Construct a basic set containing the tuples of coefficients of all
+ * valid affine constraints on the given set.
+ */
+__isl_give isl_basic_set *isl_set_coefficients(__isl_take isl_set *set)
+{
+	int i;
+	isl_basic_set *coeff;
+
+	if (!set)
+		return NULL;
+	if (set->n == 0) {
+		isl_space *space = isl_set_get_space(set);
+		space = isl_space_coefficients(space);
+		isl_set_free(set);
+		return rational_universe(space);
+	}
+
+	coeff = isl_basic_set_coefficients(isl_basic_set_copy(set->p[0]));
+
+	for (i = 1; i < set->n; ++i) {
+		isl_basic_set *bset, *coeff_i;
+		bset = isl_basic_set_copy(set->p[i]);
+		coeff_i = isl_basic_set_coefficients(bset);
+		coeff = isl_basic_set_intersect(coeff, coeff_i);
+	}
+
+	isl_set_free(set);
+	return coeff;
+}
+
+/* Construct a basic set containing the elements that satisfy all
+ * affine constraints whose coefficient tuples are
+ * contained in the given set.
+ */
+__isl_give isl_basic_set *isl_set_solutions(__isl_take isl_set *set)
+{
+	int i;
+	isl_basic_set *sol;
+
+	if (!set)
+		return NULL;
+	if (set->n == 0) {
+		isl_space *space = isl_set_get_space(set);
+		space = isl_space_solutions(space);
+		isl_set_free(set);
+		return rational_universe(space);
+	}
+
+	sol = isl_basic_set_solutions(isl_basic_set_copy(set->p[0]));
+
+	for (i = 1; i < set->n; ++i) {
+		isl_basic_set *bset, *sol_i;
+		bset = isl_basic_set_copy(set->p[i]);
+		sol_i = isl_basic_set_solutions(bset);
+		sol = isl_basic_set_intersect(sol, sol_i);
+	}
+
+	isl_set_free(set);
+	return sol;
+}
diff --git a/polly/lib/External/isl/isl_flow.c b/polly/lib/External/isl/isl_flow.c
new file mode 100644
index 00000000000..36ba0ed51f6
--- /dev/null
+++ b/polly/lib/External/isl/isl_flow.c
@@ -0,0 +1,1496 @@
+/*
+ * Copyright 2005-2007 Universiteit Leiden
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2010      INRIA Saclay
+ * Copyright 2012      Universiteit Leiden
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, Leiden Institute of Advanced Computer Science,
+ * Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
+ * and K.U.Leuven, Departement Computerwetenschappen, Celestijnenlaan 200A,
+ * B-3001 Leuven, Belgium
+ * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France 
+ */
+
+#include <isl/set.h>
+#include <isl/map.h>
+#include <isl/flow.h>
+#include <isl_sort.h>
+
+enum isl_restriction_type {
+	isl_restriction_type_empty,
+	isl_restriction_type_none,
+	isl_restriction_type_input,
+	isl_restriction_type_output
+};
+
+struct isl_restriction {
+	enum isl_restriction_type type;
+
+	isl_set *source;
+	isl_set *sink;
+};
+
+/* Create a restriction of the given type.
+ */
+static __isl_give isl_restriction *isl_restriction_alloc(
+	__isl_take isl_map *source_map, enum isl_restriction_type type)
+{
+	isl_ctx *ctx;
+	isl_restriction *restr;
+
+	if (!source_map)
+		return NULL;
+
+	ctx = isl_map_get_ctx(source_map);
+	restr = isl_calloc_type(ctx, struct isl_restriction);
+	if (!restr)
+		goto error;
+
+	restr->type = type;
+
+	isl_map_free(source_map);
+	return restr;
+error:
+	isl_map_free(source_map);
+	return NULL;
+}
+
+/* Create a restriction that doesn't restrict anything.
+ */
+__isl_give isl_restriction *isl_restriction_none(__isl_take isl_map *source_map)
+{
+	return isl_restriction_alloc(source_map, isl_restriction_type_none);
+}
+
+/* Create a restriction that removes everything.
+ */
+__isl_give isl_restriction *isl_restriction_empty(
+	__isl_take isl_map *source_map)
+{
+	return isl_restriction_alloc(source_map, isl_restriction_type_empty);
+}
+
+/* Create a restriction on the input of the maximization problem
+ * based on the given source and sink restrictions.
+ */
+__isl_give isl_restriction *isl_restriction_input(
+	__isl_take isl_set *source_restr, __isl_take isl_set *sink_restr)
+{
+	isl_ctx *ctx;
+	isl_restriction *restr;
+
+	if (!source_restr || !sink_restr)
+		goto error;
+
+	ctx = isl_set_get_ctx(source_restr);
+	restr = isl_calloc_type(ctx, struct isl_restriction);
+	if (!restr)
+		goto error;
+
+	restr->type = isl_restriction_type_input;
+	restr->source = source_restr;
+	restr->sink = sink_restr;
+
+	return restr;
+error:
+	isl_set_free(source_restr);
+	isl_set_free(sink_restr);
+	return NULL;
+}
+
+/* Create a restriction on the output of the maximization problem
+ * based on the given source restriction.
+ */
+__isl_give isl_restriction *isl_restriction_output(
+	__isl_take isl_set *source_restr)
+{
+	isl_ctx *ctx;
+	isl_restriction *restr;
+
+	if (!source_restr)
+		return NULL;
+
+	ctx = isl_set_get_ctx(source_restr);
+	restr = isl_calloc_type(ctx, struct isl_restriction);
+	if (!restr)
+		goto error;
+
+	restr->type = isl_restriction_type_output;
+	restr->source = source_restr;
+
+	return restr;
+error:
+	isl_set_free(source_restr);
+	return NULL;
+}
+
+__isl_null isl_restriction *isl_restriction_free(
+	__isl_take isl_restriction *restr)
+{
+	if (!restr)
+		return NULL;
+
+	isl_set_free(restr->source);
+	isl_set_free(restr->sink);
+	free(restr);
+	return NULL;
+}
+
+isl_ctx *isl_restriction_get_ctx(__isl_keep isl_restriction *restr)
+{
+	return restr ? isl_set_get_ctx(restr->source) : NULL;
+}
+
+/* A private structure to keep track of a mapping together with
+ * a user-specified identifier and a boolean indicating whether
+ * the map represents a must or may access/dependence.
+ */
+struct isl_labeled_map {
+	struct isl_map	*map;
+	void		*data;
+	int		must;
+};
+
+/* A structure containing the input for dependence analysis:
+ * - a sink
+ * - n_must + n_may (<= max_source) sources
+ * - a function for determining the relative order of sources and sink
+ * The must sources are placed before the may sources.
+ *
+ * domain_map is an auxiliary map that maps the sink access relation
+ * to the domain of this access relation.
+ *
+ * restrict_fn is a callback that (if not NULL) will be called
+ * right before any lexicographical maximization.
+ */
+struct isl_access_info {
+	isl_map				*domain_map;
+	struct isl_labeled_map		sink;
+	isl_access_level_before		level_before;
+
+	isl_access_restrict		restrict_fn;
+	void				*restrict_user;
+
+	int		    		max_source;
+	int		    		n_must;
+	int		    		n_may;
+	struct isl_labeled_map		source[1];
+};
+
+/* A structure containing the output of dependence analysis:
+ * - n_source dependences
+ * - a wrapped subset of the sink for which definitely no source could be found
+ * - a wrapped subset of the sink for which possibly no source could be found
+ */
+struct isl_flow {
+	isl_set			*must_no_source;
+	isl_set			*may_no_source;
+	int			n_source;
+	struct isl_labeled_map	*dep;
+};
+
+/* Construct an isl_access_info structure and fill it up with
+ * the given data.  The number of sources is set to 0.
+ */
+__isl_give isl_access_info *isl_access_info_alloc(__isl_take isl_map *sink,
+	void *sink_user, isl_access_level_before fn, int max_source)
+{
+	isl_ctx *ctx;
+	struct isl_access_info *acc;
+
+	if (!sink)
+		return NULL;
+
+	ctx = isl_map_get_ctx(sink);
+	isl_assert(ctx, max_source >= 0, goto error);
+
+	acc = isl_calloc(ctx, struct isl_access_info,
+			sizeof(struct isl_access_info) +
+			(max_source - 1) * sizeof(struct isl_labeled_map));
+	if (!acc)
+		goto error;
+
+	acc->sink.map = sink;
+	acc->sink.data = sink_user;
+	acc->level_before = fn;
+	acc->max_source = max_source;
+	acc->n_must = 0;
+	acc->n_may = 0;
+
+	return acc;
+error:
+	isl_map_free(sink);
+	return NULL;
+}
+
+/* Free the given isl_access_info structure.
+ */
+__isl_null isl_access_info *isl_access_info_free(
+	__isl_take isl_access_info *acc)
+{
+	int i;
+
+	if (!acc)
+		return NULL;
+	isl_map_free(acc->domain_map);
+	isl_map_free(acc->sink.map);
+	for (i = 0; i < acc->n_must + acc->n_may; ++i)
+		isl_map_free(acc->source[i].map);
+	free(acc);
+	return NULL;
+}
+
+isl_ctx *isl_access_info_get_ctx(__isl_keep isl_access_info *acc)
+{
+	return acc ? isl_map_get_ctx(acc->sink.map) : NULL;
+}
+
+__isl_give isl_access_info *isl_access_info_set_restrict(
+	__isl_take isl_access_info *acc, isl_access_restrict fn, void *user)
+{
+	if (!acc)
+		return NULL;
+	acc->restrict_fn = fn;
+	acc->restrict_user = user;
+	return acc;
+}
+
+/* Add another source to an isl_access_info structure, making
+ * sure the "must" sources are placed before the "may" sources.
+ * This function may be called at most max_source times on a
+ * given isl_access_info structure, with max_source as specified
+ * in the call to isl_access_info_alloc that constructed the structure.
+ */
+__isl_give isl_access_info *isl_access_info_add_source(
+	__isl_take isl_access_info *acc, __isl_take isl_map *source,
+	int must, void *source_user)
+{
+	isl_ctx *ctx;
+
+	if (!acc)
+		goto error;
+	ctx = isl_map_get_ctx(acc->sink.map);
+	isl_assert(ctx, acc->n_must + acc->n_may < acc->max_source, goto error);
+	
+	if (must) {
+		if (acc->n_may)
+			acc->source[acc->n_must + acc->n_may] =
+				acc->source[acc->n_must];
+		acc->source[acc->n_must].map = source;
+		acc->source[acc->n_must].data = source_user;
+		acc->source[acc->n_must].must = 1;
+		acc->n_must++;
+	} else {
+		acc->source[acc->n_must + acc->n_may].map = source;
+		acc->source[acc->n_must + acc->n_may].data = source_user;
+		acc->source[acc->n_must + acc->n_may].must = 0;
+		acc->n_may++;
+	}
+
+	return acc;
+error:
+	isl_map_free(source);
+	isl_access_info_free(acc);
+	return NULL;
+}
+
+/* Return -n, 0 or n (with n a positive value), depending on whether
+ * the source access identified by p1 should be sorted before, together
+ * or after that identified by p2.
+ *
+ * If p1 appears before p2, then it should be sorted first.
+ * For more generic initial schedules, it is possible that neither
+ * p1 nor p2 appears before the other, or at least not in any obvious way.
+ * We therefore also check if p2 appears before p1, in which case p2
+ * should be sorted first.
+ * If not, we try to order the two statements based on the description
+ * of the iteration domains.  This results in an arbitrary, but fairly
+ * stable ordering.
+ */
+static int access_sort_cmp(const void *p1, const void *p2, void *user)
+{
+	isl_access_info *acc = user;
+	const struct isl_labeled_map *i1, *i2;
+	int level1, level2;
+	uint32_t h1, h2;
+	i1 = (const struct isl_labeled_map *) p1;
+	i2 = (const struct isl_labeled_map *) p2;
+
+	level1 = acc->level_before(i1->data, i2->data);
+	if (level1 % 2)
+		return -1;
+
+	level2 = acc->level_before(i2->data, i1->data);
+	if (level2 % 2)
+		return 1;
+
+	h1 = isl_map_get_hash(i1->map);
+	h2 = isl_map_get_hash(i2->map);
+	return h1 > h2 ? 1 : h1 < h2 ? -1 : 0;
+}
+
+/* Sort the must source accesses in their textual order.
+ */
+static __isl_give isl_access_info *isl_access_info_sort_sources(
+	__isl_take isl_access_info *acc)
+{
+	if (!acc)
+		return NULL;
+	if (acc->n_must <= 1)
+		return acc;
+
+	if (isl_sort(acc->source, acc->n_must, sizeof(struct isl_labeled_map),
+		    access_sort_cmp, acc) < 0)
+		return isl_access_info_free(acc);
+
+	return acc;
+}
+
+/* Align the parameters of the two spaces if needed and then call
+ * isl_space_join.
+ */
+static __isl_give isl_space *space_align_and_join(__isl_take isl_space *left,
+	__isl_take isl_space *right)
+{
+	if (isl_space_match(left, isl_dim_param, right, isl_dim_param))
+		return isl_space_join(left, right);
+
+	left = isl_space_align_params(left, isl_space_copy(right));
+	right = isl_space_align_params(right, isl_space_copy(left));
+	return isl_space_join(left, right);
+}
+
+/* Initialize an empty isl_flow structure corresponding to a given
+ * isl_access_info structure.
+ * For each must access, two dependences are created (initialized
+ * to the empty relation), one for the resulting must dependences
+ * and one for the resulting may dependences.  May accesses can
+ * only lead to may dependences, so only one dependence is created
+ * for each of them.
+ * This function is private as isl_flow structures are only supposed
+ * to be created by isl_access_info_compute_flow.
+ */
+static __isl_give isl_flow *isl_flow_alloc(__isl_keep isl_access_info *acc)
+{
+	int i, n;
+	struct isl_ctx *ctx;
+	struct isl_flow *dep;
+
+	if (!acc)
+		return NULL;
+
+	ctx = isl_map_get_ctx(acc->sink.map);
+	dep = isl_calloc_type(ctx, struct isl_flow);
+	if (!dep)
+		return NULL;
+
+	n = 2 * acc->n_must + acc->n_may;
+	dep->dep = isl_calloc_array(ctx, struct isl_labeled_map, n);
+	if (n && !dep->dep)
+		goto error;
+
+	dep->n_source = n;
+	for (i = 0; i < acc->n_must; ++i) {
+		isl_space *dim;
+		dim = space_align_and_join(
+			isl_map_get_space(acc->source[i].map),
+			isl_space_reverse(isl_map_get_space(acc->sink.map)));
+		dep->dep[2 * i].map = isl_map_empty(dim);
+		dep->dep[2 * i + 1].map = isl_map_copy(dep->dep[2 * i].map);
+		dep->dep[2 * i].data = acc->source[i].data;
+		dep->dep[2 * i + 1].data = acc->source[i].data;
+		dep->dep[2 * i].must = 1;
+		dep->dep[2 * i + 1].must = 0;
+		if (!dep->dep[2 * i].map || !dep->dep[2 * i + 1].map)
+			goto error;
+	}
+	for (i = acc->n_must; i < acc->n_must + acc->n_may; ++i) {
+		isl_space *dim;
+		dim = space_align_and_join(
+			isl_map_get_space(acc->source[i].map),
+			isl_space_reverse(isl_map_get_space(acc->sink.map)));
+		dep->dep[acc->n_must + i].map = isl_map_empty(dim);
+		dep->dep[acc->n_must + i].data = acc->source[i].data;
+		dep->dep[acc->n_must + i].must = 0;
+		if (!dep->dep[acc->n_must + i].map)
+			goto error;
+	}
+
+	return dep;
+error:
+	isl_flow_free(dep);
+	return NULL;
+}
+
+/* Iterate over all sources and for each resulting flow dependence
+ * that is not empty, call the user specfied function.
+ * The second argument in this function call identifies the source,
+ * while the third argument correspond to the final argument of
+ * the isl_flow_foreach call.
+ */
+int isl_flow_foreach(__isl_keep isl_flow *deps,
+	int (*fn)(__isl_take isl_map *dep, int must, void *dep_user, void *user),
+	void *user)
+{
+	int i;
+
+	if (!deps)
+		return -1;
+
+	for (i = 0; i < deps->n_source; ++i) {
+		if (isl_map_plain_is_empty(deps->dep[i].map))
+			continue;
+		if (fn(isl_map_copy(deps->dep[i].map), deps->dep[i].must,
+				deps->dep[i].data, user) < 0)
+			return -1;
+	}
+
+	return 0;
+}
+
+/* Return a copy of the subset of the sink for which no source could be found.
+ */
+__isl_give isl_map *isl_flow_get_no_source(__isl_keep isl_flow *deps, int must)
+{
+	if (!deps)
+		return NULL;
+	
+	if (must)
+		return isl_set_unwrap(isl_set_copy(deps->must_no_source));
+	else
+		return isl_set_unwrap(isl_set_copy(deps->may_no_source));
+}
+
+void isl_flow_free(__isl_take isl_flow *deps)
+{
+	int i;
+
+	if (!deps)
+		return;
+	isl_set_free(deps->must_no_source);
+	isl_set_free(deps->may_no_source);
+	if (deps->dep) {
+		for (i = 0; i < deps->n_source; ++i)
+			isl_map_free(deps->dep[i].map);
+		free(deps->dep);
+	}
+	free(deps);
+}
+
+isl_ctx *isl_flow_get_ctx(__isl_keep isl_flow *deps)
+{
+	return deps ? isl_set_get_ctx(deps->must_no_source) : NULL;
+}
+
+/* Return a map that enforces that the domain iteration occurs after
+ * the range iteration at the given level.
+ * If level is odd, then the domain iteration should occur after
+ * the target iteration in their shared level/2 outermost loops.
+ * In this case we simply need to enforce that these outermost
+ * loop iterations are the same.
+ * If level is even, then the loop iterator of the domain should
+ * be greater than the loop iterator of the range at the last
+ * of the level/2 shared loops, i.e., loop level/2 - 1.
+ */
+static __isl_give isl_map *after_at_level(__isl_take isl_space *dim, int level)
+{
+	struct isl_basic_map *bmap;
+
+	if (level % 2)
+		bmap = isl_basic_map_equal(dim, level/2);
+	else
+		bmap = isl_basic_map_more_at(dim, level/2 - 1);
+
+	return isl_map_from_basic_map(bmap);
+}
+
+/* Compute the partial lexicographic maximum of "dep" on domain "sink",
+ * but first check if the user has set acc->restrict_fn and if so
+ * update either the input or the output of the maximization problem
+ * with respect to the resulting restriction.
+ *
+ * Since the user expects a mapping from sink iterations to source iterations,
+ * whereas the domain of "dep" is a wrapped map, mapping sink iterations
+ * to accessed array elements, we first need to project out the accessed
+ * sink array elements by applying acc->domain_map.
+ * Similarly, the sink restriction specified by the user needs to be
+ * converted back to the wrapped map.
+ */
+static __isl_give isl_map *restricted_partial_lexmax(
+	__isl_keep isl_access_info *acc, __isl_take isl_map *dep,
+	int source, __isl_take isl_set *sink, __isl_give isl_set **empty)
+{
+	isl_map *source_map;
+	isl_restriction *restr;
+	isl_set *sink_domain;
+	isl_set *sink_restr;
+	isl_map *res;
+
+	if (!acc->restrict_fn)
+		return isl_map_partial_lexmax(dep, sink, empty);
+
+	source_map = isl_map_copy(dep);
+	source_map = isl_map_apply_domain(source_map,
+					    isl_map_copy(acc->domain_map));
+	sink_domain = isl_set_copy(sink);
+	sink_domain = isl_set_apply(sink_domain, isl_map_copy(acc->domain_map));
+	restr = acc->restrict_fn(source_map, sink_domain,
+				acc->source[source].data, acc->restrict_user);
+	isl_set_free(sink_domain);
+	isl_map_free(source_map);
+
+	if (!restr)
+		goto error;
+	if (restr->type == isl_restriction_type_input) {
+		dep = isl_map_intersect_range(dep, isl_set_copy(restr->source));
+		sink_restr = isl_set_copy(restr->sink);
+		sink_restr = isl_set_apply(sink_restr,
+				isl_map_reverse(isl_map_copy(acc->domain_map)));
+		sink = isl_set_intersect(sink, sink_restr);
+	} else if (restr->type == isl_restriction_type_empty) {
+		isl_space *space = isl_map_get_space(dep);
+		isl_map_free(dep);
+		dep = isl_map_empty(space);
+	}
+
+	res = isl_map_partial_lexmax(dep, sink, empty);
+
+	if (restr->type == isl_restriction_type_output)
+		res = isl_map_intersect_range(res, isl_set_copy(restr->source));
+
+	isl_restriction_free(restr);
+	return res;
+error:
+	isl_map_free(dep);
+	isl_set_free(sink);
+	*empty = NULL;
+	return NULL;
+}
+
+/* Compute the last iteration of must source j that precedes the sink
+ * at the given level for sink iterations in set_C.
+ * The subset of set_C for which no such iteration can be found is returned
+ * in *empty.
+ */
+static struct isl_map *last_source(struct isl_access_info *acc, 
+				    struct isl_set *set_C,
+				    int j, int level, struct isl_set **empty)
+{
+	struct isl_map *read_map;
+	struct isl_map *write_map;
+	struct isl_map *dep_map;
+	struct isl_map *after;
+	struct isl_map *result;
+
+	read_map = isl_map_copy(acc->sink.map);
+	write_map = isl_map_copy(acc->source[j].map);
+	write_map = isl_map_reverse(write_map);
+	dep_map = isl_map_apply_range(read_map, write_map);
+	after = after_at_level(isl_map_get_space(dep_map), level);
+	dep_map = isl_map_intersect(dep_map, after);
+	result = restricted_partial_lexmax(acc, dep_map, j, set_C, empty);
+	result = isl_map_reverse(result);
+
+	return result;
+}
+
+/* For a given mapping between iterations of must source j and iterations
+ * of the sink, compute the last iteration of must source k preceding
+ * the sink at level before_level for any of the sink iterations,
+ * but following the corresponding iteration of must source j at level
+ * after_level.
+ */
+static struct isl_map *last_later_source(struct isl_access_info *acc,
+					 struct isl_map *old_map,
+					 int j, int before_level,
+					 int k, int after_level,
+					 struct isl_set **empty)
+{
+	isl_space *dim;
+	struct isl_set *set_C;
+	struct isl_map *read_map;
+	struct isl_map *write_map;
+	struct isl_map *dep_map;
+	struct isl_map *after_write;
+	struct isl_map *before_read;
+	struct isl_map *result;
+
+	set_C = isl_map_range(isl_map_copy(old_map));
+	read_map = isl_map_copy(acc->sink.map);
+	write_map = isl_map_copy(acc->source[k].map);
+
+	write_map = isl_map_reverse(write_map);
+	dep_map = isl_map_apply_range(read_map, write_map);
+	dim = space_align_and_join(isl_map_get_space(acc->source[k].map),
+		    isl_space_reverse(isl_map_get_space(acc->source[j].map)));
+	after_write = after_at_level(dim, after_level);
+	after_write = isl_map_apply_range(after_write, old_map);
+	after_write = isl_map_reverse(after_write);
+	dep_map = isl_map_intersect(dep_map, after_write);
+	before_read = after_at_level(isl_map_get_space(dep_map), before_level);
+	dep_map = isl_map_intersect(dep_map, before_read);
+	result = restricted_partial_lexmax(acc, dep_map, k, set_C, empty);
+	result = isl_map_reverse(result);
+
+	return result;
+}
+
+/* Given a shared_level between two accesses, return 1 if the
+ * the first can precede the second at the requested target_level.
+ * If the target level is odd, i.e., refers to a statement level
+ * dimension, then first needs to precede second at the requested
+ * level, i.e., shared_level must be equal to target_level.
+ * If the target level is odd, then the two loops should share
+ * at least the requested number of outer loops.
+ */
+static int can_precede_at_level(int shared_level, int target_level)
+{
+	if (shared_level < target_level)
+		return 0;
+	if ((target_level % 2) && shared_level > target_level)
+		return 0;
+	return 1;
+}
+
+/* Given a possible flow dependence temp_rel[j] between source j and the sink
+ * at level sink_level, remove those elements for which
+ * there is an iteration of another source k < j that is closer to the sink.
+ * The flow dependences temp_rel[k] are updated with the improved sources.
+ * Any improved source needs to precede the sink at the same level
+ * and needs to follow source j at the same or a deeper level.
+ * The lower this level, the later the execution date of source k.
+ * We therefore consider lower levels first.
+ *
+ * If temp_rel[j] is empty, then there can be no improvement and
+ * we return immediately.
+ */
+static int intermediate_sources(__isl_keep isl_access_info *acc,
+	struct isl_map **temp_rel, int j, int sink_level)
+{
+	int k, level;
+	int depth = 2 * isl_map_dim(acc->source[j].map, isl_dim_in) + 1;
+
+	if (isl_map_plain_is_empty(temp_rel[j]))
+		return 0;
+
+	for (k = j - 1; k >= 0; --k) {
+		int plevel, plevel2;
+		plevel = acc->level_before(acc->source[k].data, acc->sink.data);
+		if (!can_precede_at_level(plevel, sink_level))
+			continue;
+
+		plevel2 = acc->level_before(acc->source[j].data,
+						acc->source[k].data);
+
+		for (level = sink_level; level <= depth; ++level) {
+			struct isl_map *T;
+			struct isl_set *trest;
+			struct isl_map *copy;
+
+			if (!can_precede_at_level(plevel2, level))
+				continue;
+
+			copy = isl_map_copy(temp_rel[j]);
+			T = last_later_source(acc, copy, j, sink_level, k,
+					      level, &trest);
+			if (isl_map_plain_is_empty(T)) {
+				isl_set_free(trest);
+				isl_map_free(T);
+				continue;
+			}
+			temp_rel[j] = isl_map_intersect_range(temp_rel[j], trest);
+			temp_rel[k] = isl_map_union_disjoint(temp_rel[k], T);
+		}
+	}
+
+	return 0;
+}
+
+/* Compute all iterations of may source j that precedes the sink at the given
+ * level for sink iterations in set_C.
+ */
+static __isl_give isl_map *all_sources(__isl_keep isl_access_info *acc,
+				    __isl_take isl_set *set_C, int j, int level)
+{
+	isl_map *read_map;
+	isl_map *write_map;
+	isl_map *dep_map;
+	isl_map *after;
+
+	read_map = isl_map_copy(acc->sink.map);
+	read_map = isl_map_intersect_domain(read_map, set_C);
+	write_map = isl_map_copy(acc->source[acc->n_must + j].map);
+	write_map = isl_map_reverse(write_map);
+	dep_map = isl_map_apply_range(read_map, write_map);
+	after = after_at_level(isl_map_get_space(dep_map), level);
+	dep_map = isl_map_intersect(dep_map, after);
+
+	return isl_map_reverse(dep_map);
+}
+
+/* For a given mapping between iterations of must source k and iterations
+ * of the sink, compute the all iteration of may source j preceding
+ * the sink at level before_level for any of the sink iterations,
+ * but following the corresponding iteration of must source k at level
+ * after_level.
+ */
+static __isl_give isl_map *all_later_sources(__isl_keep isl_access_info *acc,
+	__isl_take isl_map *old_map,
+	int j, int before_level, int k, int after_level)
+{
+	isl_space *dim;
+	isl_set *set_C;
+	isl_map *read_map;
+	isl_map *write_map;
+	isl_map *dep_map;
+	isl_map *after_write;
+	isl_map *before_read;
+
+	set_C = isl_map_range(isl_map_copy(old_map));
+	read_map = isl_map_copy(acc->sink.map);
+	read_map = isl_map_intersect_domain(read_map, set_C);
+	write_map = isl_map_copy(acc->source[acc->n_must + j].map);
+
+	write_map = isl_map_reverse(write_map);
+	dep_map = isl_map_apply_range(read_map, write_map);
+	dim = isl_space_join(isl_map_get_space(acc->source[acc->n_must + j].map),
+		    isl_space_reverse(isl_map_get_space(acc->source[k].map)));
+	after_write = after_at_level(dim, after_level);
+	after_write = isl_map_apply_range(after_write, old_map);
+	after_write = isl_map_reverse(after_write);
+	dep_map = isl_map_intersect(dep_map, after_write);
+	before_read = after_at_level(isl_map_get_space(dep_map), before_level);
+	dep_map = isl_map_intersect(dep_map, before_read);
+	return isl_map_reverse(dep_map);
+}
+
+/* Given the must and may dependence relations for the must accesses
+ * for level sink_level, check if there are any accesses of may access j
+ * that occur in between and return their union.
+ * If some of these accesses are intermediate with respect to
+ * (previously thought to be) must dependences, then these
+ * must dependences are turned into may dependences.
+ */
+static __isl_give isl_map *all_intermediate_sources(
+	__isl_keep isl_access_info *acc, __isl_take isl_map *map,
+	struct isl_map **must_rel, struct isl_map **may_rel,
+	int j, int sink_level)
+{
+	int k, level;
+	int depth = 2 * isl_map_dim(acc->source[acc->n_must + j].map,
+					isl_dim_in) + 1;
+
+	for (k = 0; k < acc->n_must; ++k) {
+		int plevel;
+
+		if (isl_map_plain_is_empty(may_rel[k]) &&
+		    isl_map_plain_is_empty(must_rel[k]))
+			continue;
+
+		plevel = acc->level_before(acc->source[k].data,
+					acc->source[acc->n_must + j].data);
+
+		for (level = sink_level; level <= depth; ++level) {
+			isl_map *T;
+			isl_map *copy;
+			isl_set *ran;
+
+			if (!can_precede_at_level(plevel, level))
+				continue;
+
+			copy = isl_map_copy(may_rel[k]);
+			T = all_later_sources(acc, copy, j, sink_level, k, level);
+			map = isl_map_union(map, T);
+
+			copy = isl_map_copy(must_rel[k]);
+			T = all_later_sources(acc, copy, j, sink_level, k, level);
+			ran = isl_map_range(isl_map_copy(T));
+			map = isl_map_union(map, T);
+			may_rel[k] = isl_map_union_disjoint(may_rel[k],
+			    isl_map_intersect_range(isl_map_copy(must_rel[k]),
+						    isl_set_copy(ran)));
+			T = isl_map_from_domain_and_range(
+			    isl_set_universe(
+				isl_space_domain(isl_map_get_space(must_rel[k]))),
+			    ran);
+			must_rel[k] = isl_map_subtract(must_rel[k], T);
+		}
+	}
+
+	return map;
+}
+
+/* Compute dependences for the case where all accesses are "may"
+ * accesses, which boils down to computing memory based dependences.
+ * The generic algorithm would also work in this case, but it would
+ * be overkill to use it.
+ */
+static __isl_give isl_flow *compute_mem_based_dependences(
+	__isl_keep isl_access_info *acc)
+{
+	int i;
+	isl_set *mustdo;
+	isl_set *maydo;
+	isl_flow *res;
+
+	res = isl_flow_alloc(acc);
+	if (!res)
+		return NULL;
+
+	mustdo = isl_map_domain(isl_map_copy(acc->sink.map));
+	maydo = isl_set_copy(mustdo);
+
+	for (i = 0; i < acc->n_may; ++i) {
+		int plevel;
+		int is_before;
+		isl_space *dim;
+		isl_map *before;
+		isl_map *dep;
+
+		plevel = acc->level_before(acc->source[i].data, acc->sink.data);
+		is_before = plevel & 1;
+		plevel >>= 1;
+
+		dim = isl_map_get_space(res->dep[i].map);
+		if (is_before)
+			before = isl_map_lex_le_first(dim, plevel);
+		else
+			before = isl_map_lex_lt_first(dim, plevel);
+		dep = isl_map_apply_range(isl_map_copy(acc->source[i].map),
+			isl_map_reverse(isl_map_copy(acc->sink.map)));
+		dep = isl_map_intersect(dep, before);
+		mustdo = isl_set_subtract(mustdo,
+					    isl_map_range(isl_map_copy(dep)));
+		res->dep[i].map = isl_map_union(res->dep[i].map, dep);
+	}
+
+	res->may_no_source = isl_set_subtract(maydo, isl_set_copy(mustdo));
+	res->must_no_source = mustdo;
+
+	return res;
+}
+
+/* Compute dependences for the case where there is at least one
+ * "must" access.
+ *
+ * The core algorithm considers all levels in which a source may precede
+ * the sink, where a level may either be a statement level or a loop level.
+ * The outermost statement level is 1, the first loop level is 2, etc...
+ * The algorithm basically does the following:
+ * for all levels l of the read access from innermost to outermost
+ *	for all sources w that may precede the sink access at that level
+ *	    compute the last iteration of the source that precedes the sink access
+ *					    at that level
+ *	    add result to possible last accesses at level l of source w
+ *	    for all sources w2 that we haven't considered yet at this level that may
+ *					    also precede the sink access
+ *		for all levels l2 of w from l to innermost
+ *		    for all possible last accesses dep of w at l
+ *			compute last iteration of w2 between the source and sink
+ *								of dep
+ *			add result to possible last accesses at level l of write w2
+ *			and replace possible last accesses dep by the remainder
+ *
+ *
+ * The above algorithm is applied to the must access.  During the course
+ * of the algorithm, we keep track of sink iterations that still
+ * need to be considered.  These iterations are split into those that
+ * haven't been matched to any source access (mustdo) and those that have only
+ * been matched to may accesses (maydo).
+ * At the end of each level, we also consider the may accesses.
+ * In particular, we consider may accesses that precede the remaining
+ * sink iterations, moving elements from mustdo to maydo when appropriate,
+ * and may accesses that occur between a must source and a sink of any 
+ * dependences found at the current level, turning must dependences into
+ * may dependences when appropriate.
+ * 
+ */
+static __isl_give isl_flow *compute_val_based_dependences(
+	__isl_keep isl_access_info *acc)
+{
+	isl_ctx *ctx;
+	isl_flow *res;
+	isl_set *mustdo = NULL;
+	isl_set *maydo = NULL;
+	int level, j;
+	int depth;
+	isl_map **must_rel = NULL;
+	isl_map **may_rel = NULL;
+
+	if (!acc)
+		return NULL;
+
+	res = isl_flow_alloc(acc);
+	if (!res)
+		goto error;
+	ctx = isl_map_get_ctx(acc->sink.map);
+
+	depth = 2 * isl_map_dim(acc->sink.map, isl_dim_in) + 1;
+	mustdo = isl_map_domain(isl_map_copy(acc->sink.map));
+	maydo = isl_set_empty_like(mustdo);
+	if (!mustdo || !maydo)
+		goto error;
+	if (isl_set_plain_is_empty(mustdo))
+		goto done;
+
+	must_rel = isl_alloc_array(ctx, struct isl_map *, acc->n_must);
+	may_rel = isl_alloc_array(ctx, struct isl_map *, acc->n_must);
+	if (!must_rel || !may_rel)
+		goto error;
+
+	for (level = depth; level >= 1; --level) {
+		for (j = acc->n_must-1; j >=0; --j) {
+			must_rel[j] = isl_map_empty_like(res->dep[2 * j].map);
+			may_rel[j] = isl_map_copy(must_rel[j]);
+		}
+
+		for (j = acc->n_must - 1; j >= 0; --j) {
+			struct isl_map *T;
+			struct isl_set *rest;
+			int plevel;
+
+			plevel = acc->level_before(acc->source[j].data,
+						     acc->sink.data);
+			if (!can_precede_at_level(plevel, level))
+				continue;
+
+			T = last_source(acc, mustdo, j, level, &rest);
+			must_rel[j] = isl_map_union_disjoint(must_rel[j], T);
+			mustdo = rest;
+
+			intermediate_sources(acc, must_rel, j, level);
+
+			T = last_source(acc, maydo, j, level, &rest);
+			may_rel[j] = isl_map_union_disjoint(may_rel[j], T);
+			maydo = rest;
+
+			intermediate_sources(acc, may_rel, j, level);
+
+			if (isl_set_plain_is_empty(mustdo) &&
+			    isl_set_plain_is_empty(maydo))
+				break;
+		}
+		for (j = j - 1; j >= 0; --j) {
+			int plevel;
+
+			plevel = acc->level_before(acc->source[j].data,
+						     acc->sink.data);
+			if (!can_precede_at_level(plevel, level))
+				continue;
+
+			intermediate_sources(acc, must_rel, j, level);
+			intermediate_sources(acc, may_rel, j, level);
+		}
+
+		for (j = 0; j < acc->n_may; ++j) {
+			int plevel;
+			isl_map *T;
+			isl_set *ran;
+
+			plevel = acc->level_before(acc->source[acc->n_must + j].data,
+						     acc->sink.data);
+			if (!can_precede_at_level(plevel, level))
+				continue;
+
+			T = all_sources(acc, isl_set_copy(maydo), j, level);
+			res->dep[2 * acc->n_must + j].map =
+			    isl_map_union(res->dep[2 * acc->n_must + j].map, T);
+			T = all_sources(acc, isl_set_copy(mustdo), j, level);
+			ran = isl_map_range(isl_map_copy(T));
+			res->dep[2 * acc->n_must + j].map =
+			    isl_map_union(res->dep[2 * acc->n_must + j].map, T);
+			mustdo = isl_set_subtract(mustdo, isl_set_copy(ran));
+			maydo = isl_set_union_disjoint(maydo, ran);
+
+			T = res->dep[2 * acc->n_must + j].map;
+			T = all_intermediate_sources(acc, T, must_rel, may_rel,
+							j, level);
+			res->dep[2 * acc->n_must + j].map = T;
+		}
+
+		for (j = acc->n_must - 1; j >= 0; --j) {
+			res->dep[2 * j].map =
+				isl_map_union_disjoint(res->dep[2 * j].map,
+							     must_rel[j]);
+			res->dep[2 * j + 1].map =
+				isl_map_union_disjoint(res->dep[2 * j + 1].map,
+							     may_rel[j]);
+		}
+
+		if (isl_set_plain_is_empty(mustdo) &&
+		    isl_set_plain_is_empty(maydo))
+			break;
+	}
+
+	free(must_rel);
+	free(may_rel);
+done:
+	res->must_no_source = mustdo;
+	res->may_no_source = maydo;
+	return res;
+error:
+	isl_flow_free(res);
+	isl_set_free(mustdo);
+	isl_set_free(maydo);
+	free(must_rel);
+	free(may_rel);
+	return NULL;
+}
+
+/* Given a "sink" access, a list of n "source" accesses,
+ * compute for each iteration of the sink access
+ * and for each element accessed by that iteration,
+ * the source access in the list that last accessed the
+ * element accessed by the sink access before this sink access.
+ * Each access is given as a map from the loop iterators
+ * to the array indices.
+ * The result is a list of n relations between source and sink
+ * iterations and a subset of the domain of the sink access,
+ * corresponding to those iterations that access an element
+ * not previously accessed.
+ *
+ * To deal with multi-valued sink access relations, the sink iteration
+ * domain is first extended with dimensions that correspond to the data
+ * space.  After the computation is finished, these extra dimensions are
+ * projected out again.
+ */
+__isl_give isl_flow *isl_access_info_compute_flow(__isl_take isl_access_info *acc)
+{
+	int j;
+	struct isl_flow *res = NULL;
+
+	if (!acc)
+		return NULL;
+
+	acc->domain_map = isl_map_domain_map(isl_map_copy(acc->sink.map));
+	acc->sink.map = isl_map_range_map(acc->sink.map);
+	if (!acc->sink.map)
+		goto error;
+
+	if (acc->n_must == 0)
+		res = compute_mem_based_dependences(acc);
+	else {
+		acc = isl_access_info_sort_sources(acc);
+		res = compute_val_based_dependences(acc);
+	}
+	if (!res)
+		goto error;
+
+	for (j = 0; j < res->n_source; ++j) {
+		res->dep[j].map = isl_map_apply_range(res->dep[j].map,
+					isl_map_copy(acc->domain_map));
+		if (!res->dep[j].map)
+			goto error;
+	}
+	if (!res->must_no_source || !res->may_no_source)
+		goto error;
+
+	isl_access_info_free(acc);
+	return res;
+error:
+	isl_access_info_free(acc);
+	isl_flow_free(res);
+	return NULL;
+}
+
+
+/* Keep track of some information about a schedule for a given
+ * access.  In particular, keep track of which dimensions
+ * have a constant value and of the actual constant values.
+ */
+struct isl_sched_info {
+	int *is_cst;
+	isl_vec *cst;
+};
+
+static void sched_info_free(__isl_take struct isl_sched_info *info)
+{
+	if (!info)
+		return;
+	isl_vec_free(info->cst);
+	free(info->is_cst);
+	free(info);
+}
+
+/* Extract information on the constant dimensions of the schedule
+ * for a given access.  The "map" is of the form
+ *
+ *	[S -> D] -> A
+ *
+ * with S the schedule domain, D the iteration domain and A the data domain.
+ */
+static __isl_give struct isl_sched_info *sched_info_alloc(
+	__isl_keep isl_map *map)
+{
+	isl_ctx *ctx;
+	isl_space *dim;
+	struct isl_sched_info *info;
+	int i, n;
+
+	if (!map)
+		return NULL;
+
+	dim = isl_space_unwrap(isl_space_domain(isl_map_get_space(map)));
+	if (!dim)
+		return NULL;
+	n = isl_space_dim(dim, isl_dim_in);
+	isl_space_free(dim);
+
+	ctx = isl_map_get_ctx(map);
+	info = isl_alloc_type(ctx, struct isl_sched_info);
+	if (!info)
+		return NULL;
+	info->is_cst = isl_alloc_array(ctx, int, n);
+	info->cst = isl_vec_alloc(ctx, n);
+	if (n && (!info->is_cst || !info->cst))
+		goto error;
+
+	for (i = 0; i < n; ++i) {
+		isl_val *v;
+
+		v = isl_map_plain_get_val_if_fixed(map, isl_dim_in, i);
+		if (!v)
+			goto error;
+		info->is_cst[i] = !isl_val_is_nan(v);
+		if (info->is_cst[i])
+			info->cst = isl_vec_set_element_val(info->cst, i, v);
+		else
+			isl_val_free(v);
+	}
+
+	return info;
+error:
+	sched_info_free(info);
+	return NULL;
+}
+
+struct isl_compute_flow_data {
+	isl_union_map *must_source;
+	isl_union_map *may_source;
+	isl_union_map *must_dep;
+	isl_union_map *may_dep;
+	isl_union_map *must_no_source;
+	isl_union_map *may_no_source;
+
+	int count;
+	int must;
+	isl_space *dim;
+	struct isl_sched_info *sink_info;
+	struct isl_sched_info **source_info;
+	isl_access_info *accesses;
+};
+
+static int count_matching_array(__isl_take isl_map *map, void *user)
+{
+	int eq;
+	isl_space *dim;
+	struct isl_compute_flow_data *data;
+
+	data = (struct isl_compute_flow_data *)user;
+
+	dim = isl_space_range(isl_map_get_space(map));
+
+	eq = isl_space_is_equal(dim, data->dim);
+
+	isl_space_free(dim);
+	isl_map_free(map);
+
+	if (eq < 0)
+		return -1;
+	if (eq)
+		data->count++;
+
+	return 0;
+}
+
+static int collect_matching_array(__isl_take isl_map *map, void *user)
+{
+	int eq;
+	isl_space *dim;
+	struct isl_sched_info *info;
+	struct isl_compute_flow_data *data;
+
+	data = (struct isl_compute_flow_data *)user;
+
+	dim = isl_space_range(isl_map_get_space(map));
+
+	eq = isl_space_is_equal(dim, data->dim);
+
+	isl_space_free(dim);
+
+	if (eq < 0)
+		goto error;
+	if (!eq) {
+		isl_map_free(map);
+		return 0;
+	}
+
+	info = sched_info_alloc(map);
+	data->source_info[data->count] = info;
+
+	data->accesses = isl_access_info_add_source(data->accesses,
+						    map, data->must, info);
+
+	data->count++;
+
+	return 0;
+error:
+	isl_map_free(map);
+	return -1;
+}
+
+/* Determine the shared nesting level and the "textual order" of
+ * the given accesses.
+ *
+ * We first determine the minimal schedule dimension for both accesses.
+ *
+ * If among those dimensions, we can find one where both have a fixed
+ * value and if moreover those values are different, then the previous
+ * dimension is the last shared nesting level and the textual order
+ * is determined based on the order of the fixed values.
+ * If no such fixed values can be found, then we set the shared
+ * nesting level to the minimal schedule dimension, with no textual ordering.
+ */
+static int before(void *first, void *second)
+{
+	struct isl_sched_info *info1 = first;
+	struct isl_sched_info *info2 = second;
+	int n1, n2;
+	int i;
+
+	n1 = isl_vec_size(info1->cst);
+	n2 = isl_vec_size(info2->cst);
+
+	if (n2 < n1)
+		n1 = n2;
+
+	for (i = 0; i < n1; ++i) {
+		int r;
+		int cmp;
+
+		if (!info1->is_cst[i])
+			continue;
+		if (!info2->is_cst[i])
+			continue;
+		cmp = isl_vec_cmp_element(info1->cst, info2->cst, i);
+		if (cmp == 0)
+			continue;
+
+		r = 2 * i + (cmp < 0);
+
+		return r;
+	}
+
+	return 2 * n1;
+}
+
+/* Given a sink access, look for all the source accesses that access
+ * the same array and perform dataflow analysis on them using
+ * isl_access_info_compute_flow.
+ */
+static int compute_flow(__isl_take isl_map *map, void *user)
+{
+	int i;
+	isl_ctx *ctx;
+	struct isl_compute_flow_data *data;
+	isl_flow *flow;
+
+	data = (struct isl_compute_flow_data *)user;
+
+	ctx = isl_map_get_ctx(map);
+
+	data->accesses = NULL;
+	data->sink_info = NULL;
+	data->source_info = NULL;
+	data->count = 0;
+	data->dim = isl_space_range(isl_map_get_space(map));
+
+	if (isl_union_map_foreach_map(data->must_source,
+					&count_matching_array, data) < 0)
+		goto error;
+	if (isl_union_map_foreach_map(data->may_source,
+					&count_matching_array, data) < 0)
+		goto error;
+
+	data->sink_info = sched_info_alloc(map);
+	data->source_info = isl_calloc_array(ctx, struct isl_sched_info *,
+					     data->count);
+
+	data->accesses = isl_access_info_alloc(isl_map_copy(map),
+				data->sink_info, &before, data->count);
+	if (!data->sink_info || (data->count && !data->source_info) ||
+	    !data->accesses)
+		goto error;
+	data->count = 0;
+	data->must = 1;
+	if (isl_union_map_foreach_map(data->must_source,
+					&collect_matching_array, data) < 0)
+		goto error;
+	data->must = 0;
+	if (isl_union_map_foreach_map(data->may_source,
+					&collect_matching_array, data) < 0)
+		goto error;
+
+	flow = isl_access_info_compute_flow(data->accesses);
+	data->accesses = NULL;
+
+	if (!flow)
+		goto error;
+
+	data->must_no_source = isl_union_map_union(data->must_no_source,
+		    isl_union_map_from_map(isl_flow_get_no_source(flow, 1)));
+	data->may_no_source = isl_union_map_union(data->may_no_source,
+		    isl_union_map_from_map(isl_flow_get_no_source(flow, 0)));
+
+	for (i = 0; i < flow->n_source; ++i) {
+		isl_union_map *dep;
+		dep = isl_union_map_from_map(isl_map_copy(flow->dep[i].map));
+		if (flow->dep[i].must)
+			data->must_dep = isl_union_map_union(data->must_dep, dep);
+		else
+			data->may_dep = isl_union_map_union(data->may_dep, dep);
+	}
+
+	isl_flow_free(flow);
+
+	sched_info_free(data->sink_info);
+	if (data->source_info) {
+		for (i = 0; i < data->count; ++i)
+			sched_info_free(data->source_info[i]);
+		free(data->source_info);
+	}
+	isl_space_free(data->dim);
+	isl_map_free(map);
+
+	return 0;
+error:
+	isl_access_info_free(data->accesses);
+	sched_info_free(data->sink_info);
+	if (data->source_info) {
+		for (i = 0; i < data->count; ++i)
+			sched_info_free(data->source_info[i]);
+		free(data->source_info);
+	}
+	isl_space_free(data->dim);
+	isl_map_free(map);
+
+	return -1;
+}
+
+/* Given a collection of "sink" and "source" accesses,
+ * compute for each iteration of a sink access
+ * and for each element accessed by that iteration,
+ * the source access in the list that last accessed the
+ * element accessed by the sink access before this sink access.
+ * Each access is given as a map from the loop iterators
+ * to the array indices.
+ * The result is a relations between source and sink
+ * iterations and a subset of the domain of the sink accesses,
+ * corresponding to those iterations that access an element
+ * not previously accessed.
+ *
+ * We first prepend the schedule dimensions to the domain
+ * of the accesses so that we can easily compare their relative order.
+ * Then we consider each sink access individually in compute_flow.
+ */
+int isl_union_map_compute_flow(__isl_take isl_union_map *sink,
+	__isl_take isl_union_map *must_source,
+	__isl_take isl_union_map *may_source,
+	__isl_take isl_union_map *schedule,
+	__isl_give isl_union_map **must_dep, __isl_give isl_union_map **may_dep,
+	__isl_give isl_union_map **must_no_source,
+	__isl_give isl_union_map **may_no_source)
+{
+	isl_space *dim;
+	isl_union_map *range_map = NULL;
+	struct isl_compute_flow_data data;
+
+	sink = isl_union_map_align_params(sink,
+					    isl_union_map_get_space(must_source));
+	sink = isl_union_map_align_params(sink,
+					    isl_union_map_get_space(may_source));
+	sink = isl_union_map_align_params(sink,
+					    isl_union_map_get_space(schedule));
+	dim = isl_union_map_get_space(sink);
+	must_source = isl_union_map_align_params(must_source, isl_space_copy(dim));
+	may_source = isl_union_map_align_params(may_source, isl_space_copy(dim));
+	schedule = isl_union_map_align_params(schedule, isl_space_copy(dim));
+
+	schedule = isl_union_map_reverse(schedule);
+	range_map = isl_union_map_range_map(schedule);
+	schedule = isl_union_map_reverse(isl_union_map_copy(range_map));
+	sink = isl_union_map_apply_domain(sink, isl_union_map_copy(schedule));
+	must_source = isl_union_map_apply_domain(must_source,
+						isl_union_map_copy(schedule));
+	may_source = isl_union_map_apply_domain(may_source, schedule);
+
+	data.must_source = must_source;
+	data.may_source = may_source;
+	data.must_dep = must_dep ?
+		isl_union_map_empty(isl_space_copy(dim)) : NULL;
+	data.may_dep = may_dep ? isl_union_map_empty(isl_space_copy(dim)) : NULL;
+	data.must_no_source = must_no_source ?
+		isl_union_map_empty(isl_space_copy(dim)) : NULL;
+	data.may_no_source = may_no_source ?
+		isl_union_map_empty(isl_space_copy(dim)) : NULL;
+
+	isl_space_free(dim);
+
+	if (isl_union_map_foreach_map(sink, &compute_flow, &data) < 0)
+		goto error;
+
+	isl_union_map_free(sink);
+	isl_union_map_free(must_source);
+	isl_union_map_free(may_source);
+
+	if (must_dep) {
+		data.must_dep = isl_union_map_apply_domain(data.must_dep,
+					isl_union_map_copy(range_map));
+		data.must_dep = isl_union_map_apply_range(data.must_dep,
+					isl_union_map_copy(range_map));
+		*must_dep = data.must_dep;
+	}
+	if (may_dep) {
+		data.may_dep = isl_union_map_apply_domain(data.may_dep,
+					isl_union_map_copy(range_map));
+		data.may_dep = isl_union_map_apply_range(data.may_dep,
+					isl_union_map_copy(range_map));
+		*may_dep = data.may_dep;
+	}
+	if (must_no_source) {
+		data.must_no_source = isl_union_map_apply_domain(
+			data.must_no_source, isl_union_map_copy(range_map));
+		*must_no_source = data.must_no_source;
+	}
+	if (may_no_source) {
+		data.may_no_source = isl_union_map_apply_domain(
+			data.may_no_source, isl_union_map_copy(range_map));
+		*may_no_source = data.may_no_source;
+	}
+
+	isl_union_map_free(range_map);
+
+	return 0;
+error:
+	isl_union_map_free(range_map);
+	isl_union_map_free(sink);
+	isl_union_map_free(must_source);
+	isl_union_map_free(may_source);
+	isl_union_map_free(data.must_dep);
+	isl_union_map_free(data.may_dep);
+	isl_union_map_free(data.must_no_source);
+	isl_union_map_free(data.may_no_source);
+
+	if (must_dep)
+		*must_dep = NULL;
+	if (may_dep)
+		*may_dep = NULL;
+	if (must_no_source)
+		*must_no_source = NULL;
+	if (may_no_source)
+		*may_no_source = NULL;
+	return -1;
+}
diff --git a/polly/lib/External/isl/isl_fold.c b/polly/lib/External/isl/isl_fold.c
new file mode 100644
index 00000000000..dee86c31f47
--- /dev/null
+++ b/polly/lib/External/isl/isl_fold.c
@@ -0,0 +1,1711 @@
+/*
+ * Copyright 2010      INRIA Saclay
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France 
+ */
+
+#define ISL_DIM_H
+#include <isl_map_private.h>
+#include <isl_union_map_private.h>
+#include <isl_polynomial_private.h>
+#include <isl_point_private.h>
+#include <isl_space_private.h>
+#include <isl_lp_private.h>
+#include <isl_seq.h>
+#include <isl_mat_private.h>
+#include <isl_val_private.h>
+#include <isl_vec_private.h>
+#include <isl_config.h>
+#include <isl/deprecated/polynomial_int.h>
+
+enum isl_fold isl_fold_type_negate(enum isl_fold type)
+{
+	switch (type) {
+	case isl_fold_min:
+		return isl_fold_max;
+	case isl_fold_max:
+		return isl_fold_min;
+	case isl_fold_list:
+		return isl_fold_list;
+	}
+
+	isl_die(NULL, isl_error_internal, "unhandled isl_fold type", abort());
+}
+
+static __isl_give isl_qpolynomial_fold *qpolynomial_fold_alloc(
+	enum isl_fold type, __isl_take isl_space *dim, int n)
+{
+	isl_qpolynomial_fold *fold;
+
+	if (!dim)
+		goto error;
+
+	isl_assert(dim->ctx, n >= 0, goto error);
+	fold = isl_calloc(dim->ctx, struct isl_qpolynomial_fold,
+			sizeof(struct isl_qpolynomial_fold) +
+			(n - 1) * sizeof(struct isl_qpolynomial *));
+	if (!fold)
+		goto error;
+
+	fold->ref = 1;
+	fold->size = n;
+	fold->n = 0;
+	fold->type = type;
+	fold->dim = dim;
+
+	return fold;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+isl_ctx *isl_qpolynomial_fold_get_ctx(__isl_keep isl_qpolynomial_fold *fold)
+{
+	return fold ? fold->dim->ctx : NULL;
+}
+
+__isl_give isl_space *isl_qpolynomial_fold_get_domain_space(
+	__isl_keep isl_qpolynomial_fold *fold)
+{
+	return fold ? isl_space_copy(fold->dim) : NULL;
+}
+
+__isl_give isl_space *isl_qpolynomial_fold_get_space(
+	__isl_keep isl_qpolynomial_fold *fold)
+{
+	isl_space *space;
+	if (!fold)
+		return NULL;
+	space = isl_space_copy(fold->dim);
+	space = isl_space_from_domain(space);
+	space = isl_space_add_dims(space, isl_dim_out, 1);
+	return space;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_reset_domain_space(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_space *dim)
+{
+	int i;
+
+	fold = isl_qpolynomial_fold_cow(fold);
+	if (!fold || !dim)
+		goto error;
+
+	for (i = 0; i < fold->n; ++i) {
+		fold->qp[i] = isl_qpolynomial_reset_domain_space(fold->qp[i],
+							isl_space_copy(dim));
+		if (!fold->qp[i])
+			goto error;
+	}
+
+	isl_space_free(fold->dim);
+	fold->dim = dim;
+
+	return fold;
+error:
+	isl_qpolynomial_fold_free(fold);
+	isl_space_free(dim);
+	return NULL;
+}
+
+/* Reset the space of "fold".  This function is called from isl_pw_templ.c
+ * and doesn't know if the space of an element object is represented
+ * directly or through its domain.  It therefore passes along both.
+ */
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_reset_space_and_domain(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_space *space,
+	__isl_take isl_space *domain)
+{
+	isl_space_free(space);
+	return isl_qpolynomial_fold_reset_domain_space(fold, domain);
+}
+
+int isl_qpolynomial_fold_involves_dims(__isl_keep isl_qpolynomial_fold *fold,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+
+	if (!fold)
+		return -1;
+	if (fold->n == 0 || n == 0)
+		return 0;
+
+	for (i = 0; i < fold->n; ++i) {
+		int involves = isl_qpolynomial_involves_dims(fold->qp[i],
+							    type, first, n);
+		if (involves < 0 || involves)
+			return involves;
+	}
+	return 0;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_set_dim_name(
+	__isl_take isl_qpolynomial_fold *fold,
+	enum isl_dim_type type, unsigned pos, const char *s)
+{
+	int i;
+
+	fold = isl_qpolynomial_fold_cow(fold);
+	if (!fold)
+		return NULL;
+	fold->dim = isl_space_set_dim_name(fold->dim, type, pos, s);
+	if (!fold->dim)
+		goto error;
+
+	for (i = 0; i < fold->n; ++i) {
+		fold->qp[i] = isl_qpolynomial_set_dim_name(fold->qp[i],
+							    type, pos, s);
+		if (!fold->qp[i])
+			goto error;
+	}
+
+	return fold;
+error:
+	isl_qpolynomial_fold_free(fold);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_drop_dims(
+	__isl_take isl_qpolynomial_fold *fold,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+	enum isl_dim_type set_type;
+
+	if (!fold)
+		return NULL;
+	if (n == 0)
+		return fold;
+
+	set_type = type == isl_dim_in ? isl_dim_set : type;
+
+	fold = isl_qpolynomial_fold_cow(fold);
+	if (!fold)
+		return NULL;
+	fold->dim = isl_space_drop_dims(fold->dim, set_type, first, n);
+	if (!fold->dim)
+		goto error;
+
+	for (i = 0; i < fold->n; ++i) {
+		fold->qp[i] = isl_qpolynomial_drop_dims(fold->qp[i],
+							    type, first, n);
+		if (!fold->qp[i])
+			goto error;
+	}
+
+	return fold;
+error:
+	isl_qpolynomial_fold_free(fold);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_insert_dims(
+	__isl_take isl_qpolynomial_fold *fold,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+
+	if (!fold)
+		return NULL;
+	if (n == 0 && !isl_space_is_named_or_nested(fold->dim, type))
+		return fold;
+
+	fold = isl_qpolynomial_fold_cow(fold);
+	if (!fold)
+		return NULL;
+	fold->dim = isl_space_insert_dims(fold->dim, type, first, n);
+	if (!fold->dim)
+		goto error;
+
+	for (i = 0; i < fold->n; ++i) {
+		fold->qp[i] = isl_qpolynomial_insert_dims(fold->qp[i],
+							    type, first, n);
+		if (!fold->qp[i])
+			goto error;
+	}
+
+	return fold;
+error:
+	isl_qpolynomial_fold_free(fold);
+	return NULL;
+}
+
+static int isl_qpolynomial_cst_sign(__isl_keep isl_qpolynomial *qp)
+{
+	struct isl_upoly_cst *cst;
+
+	cst = isl_upoly_as_cst(qp->upoly);
+	if (!cst)
+		return 0;
+
+	return isl_int_sgn(cst->n) < 0 ? -1 : 1;
+}
+
+static int isl_qpolynomial_aff_sign(__isl_keep isl_set *set,
+	__isl_keep isl_qpolynomial *qp)
+{
+	enum isl_lp_result res;
+	isl_vec *aff;
+	isl_int opt;
+	int sgn = 0;
+
+	aff = isl_qpolynomial_extract_affine(qp);
+	if (!aff)
+		return 0;
+
+	isl_int_init(opt);
+
+	res = isl_set_solve_lp(set, 0, aff->el + 1, aff->el[0],
+				&opt, NULL, NULL);
+	if (res == isl_lp_error)
+		goto done;
+	if (res == isl_lp_empty ||
+	    (res == isl_lp_ok && !isl_int_is_neg(opt))) {
+		sgn = 1;
+		goto done;
+	}
+
+	res = isl_set_solve_lp(set, 1, aff->el + 1, aff->el[0],
+				&opt, NULL, NULL);
+	if (res == isl_lp_ok && !isl_int_is_pos(opt))
+		sgn = -1;
+
+done:
+	isl_int_clear(opt);
+	isl_vec_free(aff);
+	return sgn;
+}
+
+/* Determine, if possible, the sign of the quasipolynomial "qp" on
+ * the domain "set".
+ *
+ * If qp is a constant, then the problem is trivial.
+ * If qp is linear, then we check if the minimum of the corresponding
+ * affine constraint is non-negative or if the maximum is non-positive.
+ *
+ * Otherwise, we check if the outermost variable "v" has a lower bound "l"
+ * in "set".  If so, we write qp(v,v') as
+ *
+ *	q(v,v') * (v - l) + r(v')
+ *
+ * if q(v,v') and r(v') have the same known sign, then the original
+ * quasipolynomial has the same sign as well.
+ *
+ * Return
+ *	-1 if qp <= 0
+ *	 1 if qp >= 0
+ *	 0 if unknown
+ */
+static int isl_qpolynomial_sign(__isl_keep isl_set *set,
+	__isl_keep isl_qpolynomial *qp)
+{
+	int d;
+	int i;
+	int is;
+	struct isl_upoly_rec *rec;
+	isl_vec *v;
+	isl_int l;
+	enum isl_lp_result res;
+	int sgn = 0;
+
+	is = isl_qpolynomial_is_cst(qp, NULL, NULL);
+	if (is < 0)
+		return 0;
+	if (is)
+		return isl_qpolynomial_cst_sign(qp);
+
+	is = isl_qpolynomial_is_affine(qp);
+	if (is < 0)
+		return 0;
+	if (is)
+		return isl_qpolynomial_aff_sign(set, qp);
+
+	if (qp->div->n_row > 0)
+		return 0;
+
+	rec = isl_upoly_as_rec(qp->upoly);
+	if (!rec)
+		return 0;
+
+	d = isl_space_dim(qp->dim, isl_dim_all);
+	v = isl_vec_alloc(set->ctx, 2 + d);
+	if (!v)
+		return 0;
+
+	isl_seq_clr(v->el + 1, 1 + d);
+	isl_int_set_si(v->el[0], 1);
+	isl_int_set_si(v->el[2 + qp->upoly->var], 1);
+
+	isl_int_init(l);
+
+	res = isl_set_solve_lp(set, 0, v->el + 1, v->el[0], &l, NULL, NULL);
+	if (res == isl_lp_ok) {
+		isl_qpolynomial *min;
+		isl_qpolynomial *base;
+		isl_qpolynomial *r, *q;
+		isl_qpolynomial *t;
+
+		min = isl_qpolynomial_cst_on_domain(isl_space_copy(qp->dim), l);
+		base = isl_qpolynomial_var_pow_on_domain(isl_space_copy(qp->dim),
+						qp->upoly->var, 1);
+
+		r = isl_qpolynomial_alloc(isl_space_copy(qp->dim), 0,
+					  isl_upoly_copy(rec->p[rec->n - 1]));
+		q = isl_qpolynomial_copy(r);
+
+		for (i = rec->n - 2; i >= 0; --i) {
+			r = isl_qpolynomial_mul(r, isl_qpolynomial_copy(min));
+			t = isl_qpolynomial_alloc(isl_space_copy(qp->dim), 0,
+						  isl_upoly_copy(rec->p[i]));
+			r = isl_qpolynomial_add(r, t);
+			if (i == 0)
+				break;
+			q = isl_qpolynomial_mul(q, isl_qpolynomial_copy(base));
+			q = isl_qpolynomial_add(q, isl_qpolynomial_copy(r));
+		}
+
+		if (isl_qpolynomial_is_zero(q))
+			sgn = isl_qpolynomial_sign(set, r);
+		else if (isl_qpolynomial_is_zero(r))
+			sgn = isl_qpolynomial_sign(set, q);
+		else {
+			int sgn_q, sgn_r;
+			sgn_r = isl_qpolynomial_sign(set, r);
+			sgn_q = isl_qpolynomial_sign(set, q);
+			if (sgn_r == sgn_q)
+				sgn = sgn_r;
+		}
+
+		isl_qpolynomial_free(min);
+		isl_qpolynomial_free(base);
+		isl_qpolynomial_free(q);
+		isl_qpolynomial_free(r);
+	}
+
+	isl_int_clear(l);
+
+	isl_vec_free(v);
+
+	return sgn;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_fold_on_domain(
+	__isl_keep isl_set *set,
+	__isl_take isl_qpolynomial_fold *fold1,
+	__isl_take isl_qpolynomial_fold *fold2)
+{
+	int i, j;
+	int n1;
+	struct isl_qpolynomial_fold *res = NULL;
+	int better;
+
+	if (!fold1 || !fold2)
+		goto error;
+
+	isl_assert(fold1->dim->ctx, fold1->type == fold2->type, goto error);
+	isl_assert(fold1->dim->ctx, isl_space_is_equal(fold1->dim, fold2->dim),
+			goto error);
+
+	better = fold1->type == isl_fold_max ? -1 : 1;
+
+	if (isl_qpolynomial_fold_is_empty(fold1)) {
+		isl_qpolynomial_fold_free(fold1);
+		return fold2;
+	}
+
+	if (isl_qpolynomial_fold_is_empty(fold2)) {
+		isl_qpolynomial_fold_free(fold2);
+		return fold1;
+	}
+
+	res = qpolynomial_fold_alloc(fold1->type, isl_space_copy(fold1->dim),
+					fold1->n + fold2->n);
+	if (!res)
+		goto error;
+
+	for (i = 0; i < fold1->n; ++i) {
+		res->qp[res->n] = isl_qpolynomial_copy(fold1->qp[i]);
+		if (!res->qp[res->n])
+			goto error;
+		res->n++;
+	}
+	n1 = res->n;
+
+	for (i = 0; i < fold2->n; ++i) {
+		for (j = n1 - 1; j >= 0; --j) {
+			isl_qpolynomial *d;
+			int sgn;
+			d = isl_qpolynomial_sub(
+				isl_qpolynomial_copy(res->qp[j]),
+				isl_qpolynomial_copy(fold2->qp[i]));
+			sgn = isl_qpolynomial_sign(set, d);
+			isl_qpolynomial_free(d);
+			if (sgn == 0)
+				continue;
+			if (sgn != better)
+				break;
+			isl_qpolynomial_free(res->qp[j]);
+			if (j != n1 - 1)
+				res->qp[j] = res->qp[n1 - 1];
+			n1--;
+			if (n1 != res->n - 1)
+				res->qp[n1] = res->qp[res->n - 1];
+			res->n--;
+		}
+		if (j >= 0)
+			continue;
+		res->qp[res->n] = isl_qpolynomial_copy(fold2->qp[i]);
+		if (!res->qp[res->n])
+			goto error;
+		res->n++;
+	}
+
+	isl_qpolynomial_fold_free(fold1);
+	isl_qpolynomial_fold_free(fold2);
+
+	return res;
+error:
+	isl_qpolynomial_fold_free(res);
+	isl_qpolynomial_fold_free(fold1);
+	isl_qpolynomial_fold_free(fold2);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_add_qpolynomial(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_qpolynomial *qp)
+{
+	int i;
+
+	if (!fold || !qp)
+		goto error;
+
+	if (isl_qpolynomial_is_zero(qp)) {
+		isl_qpolynomial_free(qp);
+		return fold;
+	}
+
+	fold = isl_qpolynomial_fold_cow(fold);
+	if (!fold)
+		goto error;
+
+	for (i = 0; i < fold->n; ++i) {
+		fold->qp[i] = isl_qpolynomial_add(fold->qp[i],
+						isl_qpolynomial_copy(qp));
+		if (!fold->qp[i])
+			goto error;
+	}
+
+	isl_qpolynomial_free(qp);
+	return fold;
+error:
+	isl_qpolynomial_fold_free(fold);
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_add_on_domain(
+	__isl_keep isl_set *dom,
+	__isl_take isl_qpolynomial_fold *fold1,
+	__isl_take isl_qpolynomial_fold *fold2)
+{
+	int i;
+	isl_qpolynomial_fold *res = NULL;
+
+	if (!fold1 || !fold2)
+		goto error;
+
+	if (isl_qpolynomial_fold_is_empty(fold1)) {
+		isl_qpolynomial_fold_free(fold1);
+		return fold2;
+	}
+
+	if (isl_qpolynomial_fold_is_empty(fold2)) {
+		isl_qpolynomial_fold_free(fold2);
+		return fold1;
+	}
+
+	if (fold1->n == 1 && fold2->n != 1)
+		return isl_qpolynomial_fold_add_on_domain(dom, fold2, fold1);
+
+	if (fold2->n == 1) {
+		res = isl_qpolynomial_fold_add_qpolynomial(fold1,
+					isl_qpolynomial_copy(fold2->qp[0]));
+		isl_qpolynomial_fold_free(fold2);
+		return res;
+	}
+
+	res = isl_qpolynomial_fold_add_qpolynomial(
+				isl_qpolynomial_fold_copy(fold1),
+				isl_qpolynomial_copy(fold2->qp[0]));
+
+	for (i = 1; i < fold2->n; ++i) {
+		isl_qpolynomial_fold *res_i;
+		res_i = isl_qpolynomial_fold_add_qpolynomial(
+					isl_qpolynomial_fold_copy(fold1),
+					isl_qpolynomial_copy(fold2->qp[i]));
+		res = isl_qpolynomial_fold_fold_on_domain(dom, res, res_i);
+	}
+
+	isl_qpolynomial_fold_free(fold1);
+	isl_qpolynomial_fold_free(fold2);
+	return res;
+error:
+	isl_qpolynomial_fold_free(res);
+	isl_qpolynomial_fold_free(fold1);
+	isl_qpolynomial_fold_free(fold2);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_substitute_equalities(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_basic_set *eq)
+{
+	int i;
+
+	if (!fold || !eq)
+		goto error;
+
+	fold = isl_qpolynomial_fold_cow(fold);
+	if (!fold)
+		return NULL;
+
+	for (i = 0; i < fold->n; ++i) {
+		fold->qp[i] = isl_qpolynomial_substitute_equalities(fold->qp[i],
+							isl_basic_set_copy(eq));
+		if (!fold->qp[i])
+			goto error;
+	}
+
+	isl_basic_set_free(eq);
+	return fold;
+error:
+	isl_basic_set_free(eq);
+	isl_qpolynomial_fold_free(fold);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_gist(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_set *context)
+{
+	int i;
+
+	if (!fold || !context)
+		goto error;
+
+	fold = isl_qpolynomial_fold_cow(fold);
+	if (!fold)
+		return NULL;
+
+	for (i = 0; i < fold->n; ++i) {
+		fold->qp[i] = isl_qpolynomial_gist(fold->qp[i],
+							isl_set_copy(context));
+		if (!fold->qp[i])
+			goto error;
+	}
+
+	isl_set_free(context);
+	return fold;
+error:
+	isl_set_free(context);
+	isl_qpolynomial_fold_free(fold);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_gist_params(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_set *context)
+{
+	isl_space *space = isl_qpolynomial_fold_get_domain_space(fold);
+	isl_set *dom_context = isl_set_universe(space);
+	dom_context = isl_set_intersect_params(dom_context, context);
+	return isl_qpolynomial_fold_gist(fold, dom_context);
+}
+
+#define HAS_TYPE
+
+#undef PW
+#define PW isl_pw_qpolynomial_fold
+#undef EL
+#define EL isl_qpolynomial_fold
+#undef EL_IS_ZERO
+#define EL_IS_ZERO is_empty
+#undef ZERO
+#define ZERO zero
+#undef IS_ZERO
+#define IS_ZERO is_zero
+#undef FIELD
+#define FIELD fold
+#undef DEFAULT_IS_ZERO
+#define DEFAULT_IS_ZERO 1
+
+#define NO_NEG
+#define NO_PULLBACK
+
+#include <isl_pw_templ.c>
+
+#undef UNION
+#define UNION isl_union_pw_qpolynomial_fold
+#undef PART
+#define PART isl_pw_qpolynomial_fold
+#undef PARTS
+#define PARTS pw_qpolynomial_fold
+#define ALIGN_DOMAIN
+
+#define NO_SUB
+
+#include <isl_union_templ.c>
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_empty(enum isl_fold type,
+	__isl_take isl_space *dim)
+{
+	return qpolynomial_fold_alloc(type, dim, 0);
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_alloc(
+	enum isl_fold type, __isl_take isl_qpolynomial *qp)
+{
+	isl_qpolynomial_fold *fold;
+
+	if (!qp)
+		return NULL;
+
+	fold = qpolynomial_fold_alloc(type, isl_space_copy(qp->dim), 1);
+	if (!fold)
+		goto error;
+
+	fold->qp[0] = qp;
+	fold->n++;
+
+	return fold;
+error:
+	isl_qpolynomial_fold_free(fold);
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_copy(
+	__isl_keep isl_qpolynomial_fold *fold)
+{
+	if (!fold)
+		return NULL;
+
+	fold->ref++;
+	return fold;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_dup(
+	__isl_keep isl_qpolynomial_fold *fold)
+{
+	int i;
+	isl_qpolynomial_fold *dup;
+
+	if (!fold)
+		return NULL;
+	dup = qpolynomial_fold_alloc(fold->type,
+					isl_space_copy(fold->dim), fold->n);
+	if (!dup)
+		return NULL;
+	
+	dup->n = fold->n;
+	for (i = 0; i < fold->n; ++i) {
+		dup->qp[i] = isl_qpolynomial_copy(fold->qp[i]);
+		if (!dup->qp[i])
+			goto error;
+	}
+
+	return dup;
+error:
+	isl_qpolynomial_fold_free(dup);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_cow(
+	__isl_take isl_qpolynomial_fold *fold)
+{
+	if (!fold)
+		return NULL;
+
+	if (fold->ref == 1)
+		return fold;
+	fold->ref--;
+	return isl_qpolynomial_fold_dup(fold);
+}
+
+void isl_qpolynomial_fold_free(__isl_take isl_qpolynomial_fold *fold)
+{
+	int i;
+
+	if (!fold)
+		return;
+	if (--fold->ref > 0)
+		return;
+
+	for (i = 0; i < fold->n; ++i)
+		isl_qpolynomial_free(fold->qp[i]);
+	isl_space_free(fold->dim);
+	free(fold);
+}
+
+int isl_qpolynomial_fold_is_empty(__isl_keep isl_qpolynomial_fold *fold)
+{
+	if (!fold)
+		return -1;
+
+	return fold->n == 0;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_fold(
+	__isl_take isl_qpolynomial_fold *fold1,
+	__isl_take isl_qpolynomial_fold *fold2)
+{
+	int i;
+	struct isl_qpolynomial_fold *res = NULL;
+
+	if (!fold1 || !fold2)
+		goto error;
+
+	isl_assert(fold1->dim->ctx, fold1->type == fold2->type, goto error);
+	isl_assert(fold1->dim->ctx, isl_space_is_equal(fold1->dim, fold2->dim),
+			goto error);
+
+	if (isl_qpolynomial_fold_is_empty(fold1)) {
+		isl_qpolynomial_fold_free(fold1);
+		return fold2;
+	}
+
+	if (isl_qpolynomial_fold_is_empty(fold2)) {
+		isl_qpolynomial_fold_free(fold2);
+		return fold1;
+	}
+
+	res = qpolynomial_fold_alloc(fold1->type, isl_space_copy(fold1->dim),
+					fold1->n + fold2->n);
+	if (!res)
+		goto error;
+
+	for (i = 0; i < fold1->n; ++i) {
+		res->qp[res->n] = isl_qpolynomial_copy(fold1->qp[i]);
+		if (!res->qp[res->n])
+			goto error;
+		res->n++;
+	}
+
+	for (i = 0; i < fold2->n; ++i) {
+		res->qp[res->n] = isl_qpolynomial_copy(fold2->qp[i]);
+		if (!res->qp[res->n])
+			goto error;
+		res->n++;
+	}
+
+	isl_qpolynomial_fold_free(fold1);
+	isl_qpolynomial_fold_free(fold2);
+
+	return res;
+error:
+	isl_qpolynomial_fold_free(res);
+	isl_qpolynomial_fold_free(fold1);
+	isl_qpolynomial_fold_free(fold2);
+	return NULL;
+}
+
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_fold(
+	__isl_take isl_pw_qpolynomial_fold *pw1,
+	__isl_take isl_pw_qpolynomial_fold *pw2)
+{
+	int i, j, n;
+	struct isl_pw_qpolynomial_fold *res;
+	isl_set *set;
+
+	if (!pw1 || !pw2)
+		goto error;
+
+	isl_assert(pw1->dim->ctx, isl_space_is_equal(pw1->dim, pw2->dim), goto error);
+
+	if (isl_pw_qpolynomial_fold_is_zero(pw1)) {
+		isl_pw_qpolynomial_fold_free(pw1);
+		return pw2;
+	}
+
+	if (isl_pw_qpolynomial_fold_is_zero(pw2)) {
+		isl_pw_qpolynomial_fold_free(pw2);
+		return pw1;
+	}
+
+	if (pw1->type != pw2->type)
+		isl_die(pw1->dim->ctx, isl_error_invalid,
+			"fold types don't match", goto error);
+
+	n = (pw1->n + 1) * (pw2->n + 1);
+	res = isl_pw_qpolynomial_fold_alloc_size(isl_space_copy(pw1->dim),
+						pw1->type, n);
+
+	for (i = 0; i < pw1->n; ++i) {
+		set = isl_set_copy(pw1->p[i].set);
+		for (j = 0; j < pw2->n; ++j) {
+			struct isl_set *common;
+			isl_qpolynomial_fold *sum;
+			set = isl_set_subtract(set,
+					isl_set_copy(pw2->p[j].set));
+			common = isl_set_intersect(isl_set_copy(pw1->p[i].set),
+						isl_set_copy(pw2->p[j].set));
+			if (isl_set_plain_is_empty(common)) {
+				isl_set_free(common);
+				continue;
+			}
+
+			sum = isl_qpolynomial_fold_fold_on_domain(common,
+			       isl_qpolynomial_fold_copy(pw1->p[i].fold),
+			       isl_qpolynomial_fold_copy(pw2->p[j].fold));
+
+			res = isl_pw_qpolynomial_fold_add_piece(res, common, sum);
+		}
+		res = isl_pw_qpolynomial_fold_add_piece(res, set,
+			isl_qpolynomial_fold_copy(pw1->p[i].fold));
+	}
+
+	for (j = 0; j < pw2->n; ++j) {
+		set = isl_set_copy(pw2->p[j].set);
+		for (i = 0; i < pw1->n; ++i)
+			set = isl_set_subtract(set, isl_set_copy(pw1->p[i].set));
+		res = isl_pw_qpolynomial_fold_add_piece(res, set,
+				    isl_qpolynomial_fold_copy(pw2->p[j].fold));
+	}
+
+	isl_pw_qpolynomial_fold_free(pw1);
+	isl_pw_qpolynomial_fold_free(pw2);
+
+	return res;
+error:
+	isl_pw_qpolynomial_fold_free(pw1);
+	isl_pw_qpolynomial_fold_free(pw2);
+	return NULL;
+}
+
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_fold_pw_qpolynomial_fold(
+	__isl_take isl_union_pw_qpolynomial_fold *u,
+	__isl_take isl_pw_qpolynomial_fold *part)
+{
+	uint32_t hash;
+	struct isl_hash_table_entry *entry;
+
+	u = isl_union_pw_qpolynomial_fold_cow(u);
+
+	if (!part || !u)
+		goto error;
+
+	isl_assert(u->space->ctx,
+	    isl_space_match(part->dim, isl_dim_param, u->space, isl_dim_param),
+	    goto error);
+
+	hash = isl_space_get_hash(part->dim);
+	entry = isl_hash_table_find(u->space->ctx, &u->table, hash,
+				    &has_same_domain_space, part->dim, 1);
+	if (!entry)
+		goto error;
+
+	if (!entry->data)
+		entry->data = part;
+	else {
+		entry->data = isl_pw_qpolynomial_fold_fold(entry->data,
+					    isl_pw_qpolynomial_fold_copy(part));
+		if (!entry->data)
+			goto error;
+		isl_pw_qpolynomial_fold_free(part);
+	}
+
+	return u;
+error:
+	isl_pw_qpolynomial_fold_free(part);
+	isl_union_pw_qpolynomial_fold_free(u);
+	return NULL;
+}
+
+static int fold_part(__isl_take isl_pw_qpolynomial_fold *part, void *user)
+{
+	isl_union_pw_qpolynomial_fold **u;
+	u = (isl_union_pw_qpolynomial_fold **)user;
+
+	*u = isl_union_pw_qpolynomial_fold_fold_pw_qpolynomial_fold(*u, part);
+
+	return 0;
+}
+
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_fold(
+	__isl_take isl_union_pw_qpolynomial_fold *u1,
+	__isl_take isl_union_pw_qpolynomial_fold *u2)
+{
+	u1 = isl_union_pw_qpolynomial_fold_cow(u1);
+
+	if (!u1 || !u2)
+		goto error;
+
+	if (isl_union_pw_qpolynomial_fold_foreach_pw_qpolynomial_fold(u2,
+							&fold_part, &u1) < 0)
+		goto error;
+
+	isl_union_pw_qpolynomial_fold_free(u2);
+
+	return u1;
+error:
+	isl_union_pw_qpolynomial_fold_free(u1);
+	isl_union_pw_qpolynomial_fold_free(u2);
+	return NULL;
+}
+
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_from_pw_qpolynomial(
+	enum isl_fold type, __isl_take isl_pw_qpolynomial *pwqp)
+{
+	int i;
+	isl_pw_qpolynomial_fold *pwf;
+
+	if (!pwqp)
+		return NULL;
+	
+	pwf = isl_pw_qpolynomial_fold_alloc_size(isl_space_copy(pwqp->dim),
+						type, pwqp->n);
+
+	for (i = 0; i < pwqp->n; ++i)
+		pwf = isl_pw_qpolynomial_fold_add_piece(pwf,
+			isl_set_copy(pwqp->p[i].set),
+			isl_qpolynomial_fold_alloc(type,
+				isl_qpolynomial_copy(pwqp->p[i].qp)));
+
+	isl_pw_qpolynomial_free(pwqp);
+
+	return pwf;
+}
+
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_add(
+	__isl_take isl_pw_qpolynomial_fold *pwf1,
+	__isl_take isl_pw_qpolynomial_fold *pwf2)
+{
+	return isl_pw_qpolynomial_fold_union_add_(pwf1, pwf2);
+}
+
+int isl_qpolynomial_fold_plain_is_equal(__isl_keep isl_qpolynomial_fold *fold1,
+	__isl_keep isl_qpolynomial_fold *fold2)
+{
+	int i;
+
+	if (!fold1 || !fold2)
+		return -1;
+
+	if (fold1->n != fold2->n)
+		return 0;
+
+	/* We probably want to sort the qps first... */
+	for (i = 0; i < fold1->n; ++i) {
+		int eq = isl_qpolynomial_plain_is_equal(fold1->qp[i], fold2->qp[i]);
+		if (eq < 0 || !eq)
+			return eq;
+	}
+
+	return 1;
+}
+
+__isl_give isl_val *isl_qpolynomial_fold_eval(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_point *pnt)
+{
+	isl_ctx *ctx;
+	isl_val *v;
+
+	if (!fold || !pnt)
+		goto error;
+	ctx = isl_point_get_ctx(pnt);
+	isl_assert(pnt->dim->ctx, isl_space_is_equal(pnt->dim, fold->dim), goto error);
+	isl_assert(pnt->dim->ctx,
+		fold->type == isl_fold_max || fold->type == isl_fold_min,
+		goto error);
+
+	if (fold->n == 0)
+		v = isl_val_zero(ctx);
+	else {
+		int i;
+		v = isl_qpolynomial_eval(isl_qpolynomial_copy(fold->qp[0]),
+						isl_point_copy(pnt));
+		for (i = 1; i < fold->n; ++i) {
+			isl_val *v_i;
+			v_i = isl_qpolynomial_eval(
+					    isl_qpolynomial_copy(fold->qp[i]),
+					    isl_point_copy(pnt));
+			if (fold->type == isl_fold_max)
+				v = isl_val_max(v, v_i);
+			else
+				v = isl_val_min(v, v_i);
+		}
+	}
+	isl_qpolynomial_fold_free(fold);
+	isl_point_free(pnt);
+
+	return v;
+error:
+	isl_qpolynomial_fold_free(fold);
+	isl_point_free(pnt);
+	return NULL;
+}
+
+size_t isl_pw_qpolynomial_fold_size(__isl_keep isl_pw_qpolynomial_fold *pwf)
+{
+	int i;
+	size_t n = 0;
+
+	for (i = 0; i < pwf->n; ++i)
+		n += pwf->p[i].fold->n;
+
+	return n;
+}
+
+__isl_give isl_val *isl_qpolynomial_fold_opt_on_domain(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_set *set, int max)
+{
+	int i;
+	isl_val *opt;
+
+	if (!set || !fold)
+		goto error;
+
+	if (fold->n == 0) {
+		opt = isl_val_zero(isl_set_get_ctx(set));
+		isl_set_free(set);
+		isl_qpolynomial_fold_free(fold);
+		return opt;
+	}
+
+	opt = isl_qpolynomial_opt_on_domain(isl_qpolynomial_copy(fold->qp[0]),
+						isl_set_copy(set), max);
+	for (i = 1; i < fold->n; ++i) {
+		isl_val *opt_i;
+		opt_i = isl_qpolynomial_opt_on_domain(
+				isl_qpolynomial_copy(fold->qp[i]),
+				isl_set_copy(set), max);
+		if (max)
+			opt = isl_val_max(opt, opt_i);
+		else
+			opt = isl_val_min(opt, opt_i);
+	}
+
+	isl_set_free(set);
+	isl_qpolynomial_fold_free(fold);
+
+	return opt;
+error:
+	isl_set_free(set);
+	isl_qpolynomial_fold_free(fold);
+	return NULL;
+}
+
+/* Check whether for each quasi-polynomial in "fold2" there is
+ * a quasi-polynomial in "fold1" that dominates it on "set".
+ */
+static int qpolynomial_fold_covers_on_domain(__isl_keep isl_set *set,
+	__isl_keep isl_qpolynomial_fold *fold1,
+	__isl_keep isl_qpolynomial_fold *fold2)
+{
+	int i, j;
+	int covers;
+
+	if (!set || !fold1 || !fold2)
+		return -1;
+
+	covers = fold1->type == isl_fold_max ? 1 : -1;
+
+	for (i = 0; i < fold2->n; ++i) {
+		for (j = 0; j < fold1->n; ++j) {
+			isl_qpolynomial *d;
+			int sgn;
+
+			d = isl_qpolynomial_sub(
+				isl_qpolynomial_copy(fold1->qp[j]),
+				isl_qpolynomial_copy(fold2->qp[i]));
+			sgn = isl_qpolynomial_sign(set, d);
+			isl_qpolynomial_free(d);
+			if (sgn == covers)
+				break;
+		}
+		if (j >= fold1->n)
+			return 0;
+	}
+
+	return 1;
+}
+
+/* Check whether "pwf1" dominated "pwf2", i.e., the domain of "pwf1" contains
+ * that of "pwf2" and on each cell, the corresponding fold from pwf1 dominates
+ * that of pwf2.
+ */
+int isl_pw_qpolynomial_fold_covers(__isl_keep isl_pw_qpolynomial_fold *pwf1,
+	__isl_keep isl_pw_qpolynomial_fold *pwf2)
+{
+	int i, j;
+	isl_set *dom1, *dom2;
+	int is_subset;
+
+	if (!pwf1 || !pwf2)
+		return -1;
+
+	if (pwf2->n == 0)
+		return 1;
+	if (pwf1->n == 0)
+		return 0;
+
+	dom1 = isl_pw_qpolynomial_fold_domain(isl_pw_qpolynomial_fold_copy(pwf1));
+	dom2 = isl_pw_qpolynomial_fold_domain(isl_pw_qpolynomial_fold_copy(pwf2));
+	is_subset = isl_set_is_subset(dom2, dom1);
+	isl_set_free(dom1);
+	isl_set_free(dom2);
+
+	if (is_subset < 0 || !is_subset)
+		return is_subset;
+
+	for (i = 0; i < pwf2->n; ++i) {
+		for (j = 0; j < pwf1->n; ++j) {
+			int is_empty;
+			isl_set *common;
+			int covers;
+
+			common = isl_set_intersect(isl_set_copy(pwf1->p[j].set),
+						   isl_set_copy(pwf2->p[i].set));
+			is_empty = isl_set_is_empty(common);
+			if (is_empty < 0 || is_empty) {
+				isl_set_free(common);
+				if (is_empty < 0)
+					return -1;
+				continue;
+			}
+			covers = qpolynomial_fold_covers_on_domain(common,
+					pwf1->p[j].fold, pwf2->p[i].fold);
+			isl_set_free(common);
+			if (covers < 0 || !covers)
+				return covers;
+		}
+	}
+
+	return 1;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_morph_domain(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_morph *morph)
+{
+	int i;
+	isl_ctx *ctx;
+
+	if (!fold || !morph)
+		goto error;
+
+	ctx = fold->dim->ctx;
+	isl_assert(ctx, isl_space_is_equal(fold->dim, morph->dom->dim), goto error);
+
+	fold = isl_qpolynomial_fold_cow(fold);
+	if (!fold)
+		goto error;
+
+	isl_space_free(fold->dim);
+	fold->dim = isl_space_copy(morph->ran->dim);
+	if (!fold->dim)
+		goto error;
+
+	for (i = 0; i < fold->n; ++i) {
+		fold->qp[i] = isl_qpolynomial_morph_domain(fold->qp[i],
+						isl_morph_copy(morph));
+		if (!fold->qp[i])
+			goto error;
+	}
+
+	isl_morph_free(morph);
+
+	return fold;
+error:
+	isl_qpolynomial_fold_free(fold);
+	isl_morph_free(morph);
+	return NULL;
+}
+
+enum isl_fold isl_qpolynomial_fold_get_type(__isl_keep isl_qpolynomial_fold *fold)
+{
+	if (!fold)
+		return isl_fold_list;
+	return fold->type;
+}
+
+enum isl_fold isl_union_pw_qpolynomial_fold_get_type(
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf)
+{
+	if (!upwf)
+		return isl_fold_list;
+	return upwf->type;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_lift(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_space *dim)
+{
+	int i;
+
+	if (!fold || !dim)
+		goto error;
+
+	if (isl_space_is_equal(fold->dim, dim)) {
+		isl_space_free(dim);
+		return fold;
+	}
+
+	fold = isl_qpolynomial_fold_cow(fold);
+	if (!fold)
+		goto error;
+
+	isl_space_free(fold->dim);
+	fold->dim = isl_space_copy(dim);
+	if (!fold->dim)
+		goto error;
+
+	for (i = 0; i < fold->n; ++i) {
+		fold->qp[i] = isl_qpolynomial_lift(fold->qp[i],
+						isl_space_copy(dim));
+		if (!fold->qp[i])
+			goto error;
+	}
+
+	isl_space_free(dim);
+
+	return fold;
+error:
+	isl_qpolynomial_fold_free(fold);
+	isl_space_free(dim);
+	return NULL;
+}
+
+int isl_qpolynomial_fold_foreach_qpolynomial(
+	__isl_keep isl_qpolynomial_fold *fold,
+	int (*fn)(__isl_take isl_qpolynomial *qp, void *user), void *user)
+{
+	int i;
+
+	if (!fold)
+		return -1;
+
+	for (i = 0; i < fold->n; ++i)
+		if (fn(isl_qpolynomial_copy(fold->qp[i]), user) < 0)
+			return -1;
+
+	return 0;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_move_dims(
+	__isl_take isl_qpolynomial_fold *fold,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n)
+{
+	int i;
+
+	if (n == 0)
+		return fold;
+
+	fold = isl_qpolynomial_fold_cow(fold);
+	if (!fold)
+		return NULL;
+
+	fold->dim = isl_space_move_dims(fold->dim, dst_type, dst_pos,
+						src_type, src_pos, n);
+	if (!fold->dim)
+		goto error;
+
+	for (i = 0; i < fold->n; ++i) {
+		fold->qp[i] = isl_qpolynomial_move_dims(fold->qp[i],
+				dst_type, dst_pos, src_type, src_pos, n);
+		if (!fold->qp[i])
+			goto error;
+	}
+
+	return fold;
+error:
+	isl_qpolynomial_fold_free(fold);
+	return NULL;
+}
+
+/* For each 0 <= i < "n", replace variable "first" + i of type "type"
+ * in fold->qp[k] by subs[i].
+ */
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_substitute(
+	__isl_take isl_qpolynomial_fold *fold,
+	enum isl_dim_type type, unsigned first, unsigned n,
+	__isl_keep isl_qpolynomial **subs)
+{
+	int i;
+
+	if (n == 0)
+		return fold;
+
+	fold = isl_qpolynomial_fold_cow(fold);
+	if (!fold)
+		return NULL;
+
+	for (i = 0; i < fold->n; ++i) {
+		fold->qp[i] = isl_qpolynomial_substitute(fold->qp[i],
+				type, first, n, subs);
+		if (!fold->qp[i])
+			goto error;
+	}
+
+	return fold;
+error:
+	isl_qpolynomial_fold_free(fold);
+	return NULL;
+}
+
+static int add_pwqp(__isl_take isl_pw_qpolynomial *pwqp, void *user)
+{
+	isl_ctx *ctx;
+	isl_pw_qpolynomial_fold *pwf;
+	isl_union_pw_qpolynomial_fold **upwf;
+	uint32_t hash;
+	struct isl_hash_table_entry *entry;
+
+	upwf = (isl_union_pw_qpolynomial_fold **)user;
+
+	ctx = pwqp->dim->ctx;
+	hash = isl_space_get_hash(pwqp->dim);
+	entry = isl_hash_table_find(ctx, &(*upwf)->table, hash,
+				     &has_same_domain_space, pwqp->dim, 1);
+	if (!entry)
+		goto error;
+
+	pwf = isl_pw_qpolynomial_fold_from_pw_qpolynomial((*upwf)->type, pwqp);
+	if (!entry->data)
+		entry->data = pwf;
+	else {
+		entry->data = isl_pw_qpolynomial_fold_add(entry->data, pwf);
+		if (!entry->data)
+			return -1;
+		if (isl_pw_qpolynomial_fold_is_zero(entry->data)) {
+			isl_pw_qpolynomial_fold_free(entry->data);
+			isl_hash_table_remove(ctx, &(*upwf)->table, entry);
+		}
+	}
+
+	return 0;
+error:
+	isl_pw_qpolynomial_free(pwqp);
+	return -1;
+}
+
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_add_union_pw_qpolynomial(
+	__isl_take isl_union_pw_qpolynomial_fold *upwf,
+	__isl_take isl_union_pw_qpolynomial *upwqp)
+{
+	upwf = isl_union_pw_qpolynomial_fold_align_params(upwf,
+				isl_union_pw_qpolynomial_get_space(upwqp));
+	upwqp = isl_union_pw_qpolynomial_align_params(upwqp,
+				isl_union_pw_qpolynomial_fold_get_space(upwf));
+
+	upwf = isl_union_pw_qpolynomial_fold_cow(upwf);
+	if (!upwf || !upwqp)
+		goto error;
+
+	if (isl_union_pw_qpolynomial_foreach_pw_qpolynomial(upwqp, &add_pwqp,
+							 &upwf) < 0)
+		goto error;
+
+	isl_union_pw_qpolynomial_free(upwqp);
+
+	return upwf;
+error:
+	isl_union_pw_qpolynomial_fold_free(upwf);
+	isl_union_pw_qpolynomial_free(upwqp);
+	return NULL;
+}
+
+static int join_compatible(__isl_keep isl_space *dim1, __isl_keep isl_space *dim2)
+{
+	int m;
+	m = isl_space_match(dim1, isl_dim_param, dim2, isl_dim_param);
+	if (m < 0 || !m)
+		return m;
+	return isl_space_tuple_is_equal(dim1, isl_dim_out, dim2, isl_dim_in);
+}
+
+/* Compute the intersection of the range of the map and the domain
+ * of the piecewise quasipolynomial reduction and then compute a bound
+ * on the associated quasipolynomial reduction over all elements
+ * in this intersection.
+ *
+ * We first introduce some unconstrained dimensions in the
+ * piecewise quasipolynomial, intersect the resulting domain
+ * with the wrapped map and the compute the sum.
+ */
+__isl_give isl_pw_qpolynomial_fold *isl_map_apply_pw_qpolynomial_fold(
+	__isl_take isl_map *map, __isl_take isl_pw_qpolynomial_fold *pwf,
+	int *tight)
+{
+	isl_ctx *ctx;
+	isl_set *dom;
+	isl_space *map_dim;
+	isl_space *pwf_dim;
+	unsigned n_in;
+	int ok;
+
+	ctx = isl_map_get_ctx(map);
+	if (!ctx)
+		goto error;
+
+	map_dim = isl_map_get_space(map);
+	pwf_dim = isl_pw_qpolynomial_fold_get_space(pwf);
+	ok = join_compatible(map_dim, pwf_dim);
+	isl_space_free(map_dim);
+	isl_space_free(pwf_dim);
+	if (!ok)
+		isl_die(ctx, isl_error_invalid, "incompatible dimensions",
+			goto error);
+
+	n_in = isl_map_dim(map, isl_dim_in);
+	pwf = isl_pw_qpolynomial_fold_insert_dims(pwf, isl_dim_in, 0, n_in);
+
+	dom = isl_map_wrap(map);
+	pwf = isl_pw_qpolynomial_fold_reset_domain_space(pwf,
+						isl_set_get_space(dom));
+
+	pwf = isl_pw_qpolynomial_fold_intersect_domain(pwf, dom);
+	pwf = isl_pw_qpolynomial_fold_bound(pwf, tight);
+	
+	return pwf;
+error:
+	isl_map_free(map);
+	isl_pw_qpolynomial_fold_free(pwf);
+	return NULL;
+}
+
+__isl_give isl_pw_qpolynomial_fold *isl_set_apply_pw_qpolynomial_fold(
+	__isl_take isl_set *set, __isl_take isl_pw_qpolynomial_fold *pwf,
+	int *tight)
+{
+	return isl_map_apply_pw_qpolynomial_fold(set, pwf, tight);
+}
+
+struct isl_apply_fold_data {
+	isl_union_pw_qpolynomial_fold *upwf;
+	isl_union_pw_qpolynomial_fold *res;
+	isl_map *map;
+	int tight;
+};
+
+static int pw_qpolynomial_fold_apply(__isl_take isl_pw_qpolynomial_fold *pwf,
+	void *user)
+{
+	isl_space *map_dim;
+	isl_space *pwf_dim;
+	struct isl_apply_fold_data *data = user;
+	int ok;
+
+	map_dim = isl_map_get_space(data->map);
+	pwf_dim = isl_pw_qpolynomial_fold_get_space(pwf);
+	ok = join_compatible(map_dim, pwf_dim);
+	isl_space_free(map_dim);
+	isl_space_free(pwf_dim);
+
+	if (ok) {
+		pwf = isl_map_apply_pw_qpolynomial_fold(isl_map_copy(data->map),
+				    pwf, data->tight ? &data->tight : NULL);
+		data->res = isl_union_pw_qpolynomial_fold_fold_pw_qpolynomial_fold(
+							data->res, pwf);
+	} else
+		isl_pw_qpolynomial_fold_free(pwf);
+
+	return 0;
+}
+
+static int map_apply(__isl_take isl_map *map, void *user)
+{
+	struct isl_apply_fold_data *data = user;
+	int r;
+
+	data->map = map;
+	r = isl_union_pw_qpolynomial_fold_foreach_pw_qpolynomial_fold(
+				data->upwf, &pw_qpolynomial_fold_apply, data);
+
+	isl_map_free(map);
+	return r;
+}
+
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_map_apply_union_pw_qpolynomial_fold(
+	__isl_take isl_union_map *umap,
+	__isl_take isl_union_pw_qpolynomial_fold *upwf, int *tight)
+{
+	isl_space *dim;
+	enum isl_fold type;
+	struct isl_apply_fold_data data;
+
+	upwf = isl_union_pw_qpolynomial_fold_align_params(upwf,
+				isl_union_map_get_space(umap));
+	umap = isl_union_map_align_params(umap,
+				isl_union_pw_qpolynomial_fold_get_space(upwf));
+
+	data.upwf = upwf;
+	data.tight = tight ? 1 : 0;
+	dim = isl_union_pw_qpolynomial_fold_get_space(upwf);
+	type = isl_union_pw_qpolynomial_fold_get_type(upwf);
+	data.res = isl_union_pw_qpolynomial_fold_zero(dim, type);
+	if (isl_union_map_foreach_map(umap, &map_apply, &data) < 0)
+		goto error;
+
+	isl_union_map_free(umap);
+	isl_union_pw_qpolynomial_fold_free(upwf);
+
+	if (tight)
+		*tight = data.tight;
+
+	return data.res;
+error:
+	isl_union_map_free(umap);
+	isl_union_pw_qpolynomial_fold_free(upwf);
+	isl_union_pw_qpolynomial_fold_free(data.res);
+	return NULL;
+}
+
+__isl_give isl_union_pw_qpolynomial_fold *isl_union_set_apply_union_pw_qpolynomial_fold(
+	__isl_take isl_union_set *uset,
+	__isl_take isl_union_pw_qpolynomial_fold *upwf, int *tight)
+{
+	return isl_union_map_apply_union_pw_qpolynomial_fold(uset, upwf, tight);
+}
+
+/* Reorder the dimension of "fold" according to the given reordering.
+ */
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_realign_domain(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_reordering *r)
+{
+	int i;
+
+	fold = isl_qpolynomial_fold_cow(fold);
+	if (!fold || !r)
+		goto error;
+
+	for (i = 0; i < fold->n; ++i) {
+		fold->qp[i] = isl_qpolynomial_realign_domain(fold->qp[i],
+						    isl_reordering_copy(r));
+		if (!fold->qp[i])
+			goto error;
+	}
+
+	fold = isl_qpolynomial_fold_reset_domain_space(fold,
+						    isl_space_copy(r->dim));
+
+	isl_reordering_free(r);
+
+	return fold;
+error:
+	isl_qpolynomial_fold_free(fold);
+	isl_reordering_free(r);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_mul_isl_int(
+	__isl_take isl_qpolynomial_fold *fold, isl_int v)
+{
+	int i;
+
+	if (isl_int_is_one(v))
+		return fold;
+	if (fold && isl_int_is_zero(v)) {
+		isl_qpolynomial_fold *zero;
+		isl_space *dim = isl_space_copy(fold->dim);
+		zero = isl_qpolynomial_fold_empty(fold->type, dim);
+		isl_qpolynomial_fold_free(fold);
+		return zero;
+	}
+
+	fold = isl_qpolynomial_fold_cow(fold);
+	if (!fold)
+		return NULL;
+
+	if (isl_int_is_neg(v))
+		fold->type = isl_fold_type_negate(fold->type);
+	for (i = 0; i < fold->n; ++i) {
+		fold->qp[i] = isl_qpolynomial_mul_isl_int(fold->qp[i], v);
+		if (!fold->qp[i])
+			goto error;
+	}
+
+	return fold;
+error:
+	isl_qpolynomial_fold_free(fold);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_scale(
+	__isl_take isl_qpolynomial_fold *fold, isl_int v)
+{
+	return isl_qpolynomial_fold_mul_isl_int(fold, v);
+}
+
+/* Multiply "fold" by "v".
+ */
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_scale_val(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_val *v)
+{
+	int i;
+
+	if (!fold || !v)
+		goto error;
+
+	if (isl_val_is_one(v)) {
+		isl_val_free(v);
+		return fold;
+	}
+	if (isl_val_is_zero(v)) {
+		isl_qpolynomial_fold *zero;
+		isl_space *space = isl_qpolynomial_fold_get_domain_space(fold);
+		zero = isl_qpolynomial_fold_empty(fold->type, space);
+		isl_qpolynomial_fold_free(fold);
+		isl_val_free(v);
+		return zero;
+	}
+	if (!isl_val_is_rat(v))
+		isl_die(isl_qpolynomial_fold_get_ctx(fold), isl_error_invalid,
+			"expecting rational factor", goto error);
+
+	fold = isl_qpolynomial_fold_cow(fold);
+	if (!fold)
+		goto error;
+
+	if (isl_val_is_neg(v))
+		fold->type = isl_fold_type_negate(fold->type);
+	for (i = 0; i < fold->n; ++i) {
+		fold->qp[i] = isl_qpolynomial_scale_val(fold->qp[i],
+							isl_val_copy(v));
+		if (!fold->qp[i])
+			goto error;
+	}
+
+	isl_val_free(v);
+	return fold;
+error:
+	isl_val_free(v);
+	isl_qpolynomial_fold_free(fold);
+	return NULL;
+}
+
+/* Divide "fold" by "v".
+ */
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_scale_down_val(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_val *v)
+{
+	if (!fold || !v)
+		goto error;
+
+	if (isl_val_is_one(v)) {
+		isl_val_free(v);
+		return fold;
+	}
+	if (!isl_val_is_rat(v))
+		isl_die(isl_qpolynomial_fold_get_ctx(fold), isl_error_invalid,
+			"expecting rational factor", goto error);
+	if (isl_val_is_zero(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"cannot scale down by zero", goto error);
+
+	return isl_qpolynomial_fold_scale_val(fold, isl_val_inv(v));
+error:
+	isl_val_free(v);
+	isl_qpolynomial_fold_free(fold);
+	return NULL;
+}
diff --git a/polly/lib/External/isl/isl_gmp.c b/polly/lib/External/isl/isl_gmp.c
new file mode 100644
index 00000000000..d488fd2eb30
--- /dev/null
+++ b/polly/lib/External/isl/isl_gmp.c
@@ -0,0 +1,24 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <isl_int.h>
+
+uint32_t isl_gmp_hash(mpz_t v, uint32_t hash)
+{
+	int sa = v[0]._mp_size;
+	int abs_sa = sa < 0 ? -sa : sa;
+	unsigned char *data = (unsigned char *)v[0]._mp_d;
+	unsigned char *end = data + abs_sa * sizeof(v[0]._mp_d[0]);
+
+	if (sa < 0)
+		isl_hash_byte(hash, 0xFF);
+	for (; data < end; ++data)
+		isl_hash_byte(hash, *data);
+	return hash;
+}
diff --git a/polly/lib/External/isl/isl_hash.c b/polly/lib/External/isl/isl_hash.c
new file mode 100644
index 00000000000..36817d7fb42
--- /dev/null
+++ b/polly/lib/External/isl/isl_hash.c
@@ -0,0 +1,230 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <stdlib.h>
+#include <strings.h>
+#include <isl/hash.h>
+#include <isl/ctx.h>
+#include "isl_config.h"
+
+uint32_t isl_hash_string(uint32_t hash, const char *s)
+{
+	for (; *s; s++)
+		isl_hash_byte(hash, *s);
+	return hash;
+}
+
+uint32_t isl_hash_mem(uint32_t hash, const void *p, size_t len)
+{
+	int i;
+	const char *s = p;
+	for (i = 0; i < len; ++i)
+		isl_hash_byte(hash, s[i]);
+	return hash;
+}
+
+static unsigned int round_up(unsigned int v)
+{
+	int old_v = v;
+
+	while (v) {
+		old_v = v;
+		v ^= v & -v;
+	}
+	return old_v << 1;
+}
+
+int isl_hash_table_init(struct isl_ctx *ctx, struct isl_hash_table *table,
+			int min_size)
+{
+	size_t size;
+
+	if (!table)
+		return -1;
+
+	if (min_size < 2)
+		min_size = 2;
+	table->bits = ffs(round_up(4 * (min_size + 1) / 3 - 1)) - 1;
+	table->n = 0;
+
+	size = 1 << table->bits;
+	table->entries = isl_calloc_array(ctx, struct isl_hash_table_entry,
+					  size);
+	if (!table->entries)
+		return -1;
+
+	return 0;
+}
+
+/* Dummy comparison function that always returns false.
+ */
+static int no(const void *entry, const void *val)
+{
+	return 0;
+}
+
+/* Extend "table" to twice its size.
+ * Return 0 on success and -1 on error.
+ *
+ * We reuse isl_hash_table_find to create entries in the extended table.
+ * Since all entries in the original table are assumed to be different,
+ * there is no need to compare them against each other.
+ */
+static int grow_table(struct isl_ctx *ctx, struct isl_hash_table *table)
+{
+	int n;
+	size_t old_size, size;
+	struct isl_hash_table_entry *entries;
+	uint32_t h;
+
+	entries = table->entries;
+	old_size = 1 << table->bits;
+	size = 2 * old_size;
+	table->entries = isl_calloc_array(ctx, struct isl_hash_table_entry,
+					  size);
+	if (!table->entries) {
+		table->entries = entries;
+		return -1;
+	}
+
+	n = table->n;
+	table->n = 0;
+	table->bits++;
+
+	for (h = 0; h < old_size; ++h) {
+		struct isl_hash_table_entry *entry;
+
+		if (!entries[h].data)
+			continue;
+
+		entry = isl_hash_table_find(ctx, table, entries[h].hash,
+					    &no, NULL, 1);
+		if (!entry) {
+			table->bits--;
+			free(table->entries);
+			table->entries = entries;
+			table->n = n;
+			return -1;
+		}
+
+		*entry = entries[h];
+	}
+
+	free(entries);
+
+	return 0;
+}
+
+struct isl_hash_table *isl_hash_table_alloc(struct isl_ctx *ctx, int min_size)
+{
+	struct isl_hash_table *table = NULL;
+
+	table = isl_alloc_type(ctx, struct isl_hash_table);
+	if (isl_hash_table_init(ctx, table, min_size))
+		goto error;
+	return table;
+error:
+	isl_hash_table_free(ctx, table);
+	return NULL;
+}
+
+void isl_hash_table_clear(struct isl_hash_table *table)
+{
+	if (!table)
+		return;
+	free(table->entries);
+}
+
+void isl_hash_table_free(struct isl_ctx *ctx, struct isl_hash_table *table)
+{
+	if (!table)
+		return;
+	isl_hash_table_clear(table);
+	free(table);
+}
+
+struct isl_hash_table_entry *isl_hash_table_find(struct isl_ctx *ctx,
+				struct isl_hash_table *table,
+				uint32_t key_hash,
+				int (*eq)(const void *entry, const void *val),
+				const void *val, int reserve)
+{
+	size_t size;
+	uint32_t h, key_bits;
+
+	key_bits = isl_hash_bits(key_hash, table->bits);
+	size = 1 << table->bits;
+	for (h = key_bits; table->entries[h].data; h = (h+1) % size)
+		if (table->entries[h].hash == key_hash &&
+		    eq(table->entries[h].data, val))
+			return &table->entries[h];
+
+	if (!reserve)
+		return NULL;
+
+	if (4 * table->n >= 3 * size) {
+		if (grow_table(ctx, table) < 0)
+			return NULL;
+		return isl_hash_table_find(ctx, table, key_hash, eq, val, 1);
+	}
+
+	table->n++;
+	table->entries[h].hash = key_hash;
+
+	return &table->entries[h];
+}
+
+int isl_hash_table_foreach(struct isl_ctx *ctx,
+				struct isl_hash_table *table,
+				int (*fn)(void **entry, void *user), void *user)
+{
+	size_t size;
+	uint32_t h;
+
+	if (!table->entries)
+		return -1;
+
+	size = 1 << table->bits;
+	for (h = 0; h < size; ++ h)
+		if (table->entries[h].data &&
+		    fn(&table->entries[h].data, user) < 0)
+			return -1;
+	
+	return 0;
+}
+
+void isl_hash_table_remove(struct isl_ctx *ctx,
+				struct isl_hash_table *table,
+				struct isl_hash_table_entry *entry)
+{
+	int h, h2;
+	size_t size;
+
+	if (!table || !entry)
+		return;
+
+	size = 1 << table->bits;
+	h = entry - table->entries;
+	isl_assert(ctx, h >= 0 && h < size, return);
+
+	for (h2 = h+1; table->entries[h2 % size].data; h2++) {
+		uint32_t bits = isl_hash_bits(table->entries[h2 % size].hash,
+						table->bits);
+		uint32_t offset = (size + bits - (h+1)) % size;
+		if (offset <= h2 - (h+1))
+			continue;
+		*entry = table->entries[h2 % size];
+		h = h2;
+		entry = &table->entries[h % size];
+	}
+
+	entry->hash = 0;
+	entry->data = NULL;
+	table->n--;
+}
diff --git a/polly/lib/External/isl/isl_hide_deprecated.h b/polly/lib/External/isl/isl_hide_deprecated.h
new file mode 100644
index 00000000000..006e796f88f
--- /dev/null
+++ b/polly/lib/External/isl/isl_hide_deprecated.h
@@ -0,0 +1,53 @@
+#define isl_aff_get_constant	isl_gmp_aff_get_constant
+#define isl_aff_get_coefficient	isl_gmp_aff_get_coefficient
+#define isl_aff_get_denominator	isl_gmp_aff_get_denominator
+#define isl_aff_set_constant	isl_gmp_aff_set_constant
+#define isl_aff_set_coefficient	isl_gmp_aff_set_coefficient
+#define isl_aff_set_denominator	isl_gmp_aff_set_denominator
+#define isl_aff_add_constant	isl_gmp_aff_add_constant
+#define isl_aff_add_constant_num	isl_gmp_aff_add_constant_num
+#define isl_aff_add_coefficient	isl_gmp_aff_add_coefficient
+#define isl_aff_mod	isl_gmp_aff_mod
+#define isl_aff_scale	isl_gmp_aff_scale
+#define isl_aff_scale_down	isl_gmp_aff_scale_down
+#define isl_pw_aff_mod	isl_gmp_pw_aff_mod
+#define isl_pw_aff_scale	isl_gmp_pw_aff_scale
+#define isl_pw_aff_scale_down	isl_gmp_pw_aff_scale_down
+#define isl_multi_aff_scale	isl_gmp_multi_aff_scale
+#define isl_ast_expr_get_int	isl_gmp_ast_expr_get_int
+#define isl_constraint_get_constant	isl_gmp_constraint_get_constant
+#define isl_constraint_get_coefficient	isl_gmp_constraint_get_coefficient
+#define isl_constraint_set_constant	isl_gmp_constraint_set_constant
+#define isl_constraint_set_coefficient	isl_gmp_constraint_set_coefficient
+#define isl_basic_set_max	isl_gmp_basic_set_max
+#define isl_set_min	isl_gmp_set_min
+#define isl_set_max	isl_gmp_set_max
+#define isl_gmp_hash	isl_gmp_gmp_hash
+#define isl_basic_map_plain_is_fixed	isl_gmp_basic_map_plain_is_fixed
+#define isl_map_fix	isl_gmp_map_fix
+#define isl_map_plain_is_fixed	isl_gmp_map_plain_is_fixed
+#define isl_map_fast_is_fixed	isl_gmp_map_fast_is_fixed
+#define isl_map_fixed_power	isl_gmp_map_fixed_power
+#define isl_mat_get_element	isl_gmp_mat_get_element
+#define isl_mat_set_element	isl_gmp_mat_set_element
+#define isl_point_get_coordinate	isl_gmp_point_get_coordinate
+#define isl_point_set_coordinate	isl_gmp_point_set_coordinate
+#define isl_qpolynomial_rat_cst_on_domain	isl_gmp_qpolynomial_rat_cst_on_domain
+#define isl_qpolynomial_is_cst	isl_gmp_qpolynomial_is_cst
+#define isl_qpolynomial_scale	isl_gmp_qpolynomial_scale
+#define isl_term_get_num	isl_gmp_term_get_num
+#define isl_term_get_den	isl_gmp_term_get_den
+#define isl_qpolynomial_fold_scale	isl_gmp_qpolynomial_fold_scale
+#define isl_pw_qpolynomial_fold_fix_dim	isl_gmp_pw_qpolynomial_fold_fix_dim
+#define isl_basic_set_fix	isl_gmp_basic_set_fix
+#define isl_set_lower_bound	isl_gmp_set_lower_bound
+#define isl_set_upper_bound	isl_gmp_set_upper_bound
+#define isl_set_fix	isl_gmp_set_fix
+#define isl_set_plain_is_fixed	isl_gmp_set_plain_is_fixed
+#define isl_union_map_fixed_power	isl_gmp_union_map_fixed_power
+#define isl_val_int_from_isl_int	isl_gmp_val_int_from_isl_int
+#define isl_val_get_num_isl_int	isl_gmp_val_get_num_isl_int
+#define isl_vec_get_element	isl_gmp_vec_get_element
+#define isl_vec_set_element	isl_gmp_vec_set_element
+#define isl_vec_set	isl_gmp_vec_set
+#define isl_vec_fdiv_r	isl_gmp_vec_fdiv_r
diff --git a/polly/lib/External/isl/isl_hmap_templ.c b/polly/lib/External/isl/isl_hmap_templ.c
new file mode 100644
index 00000000000..983dfbddcad
--- /dev/null
+++ b/polly/lib/External/isl/isl_hmap_templ.c
@@ -0,0 +1,388 @@
+/*
+ * Copyright 2011      INRIA Saclay
+ * Copyright 2013      Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+#include <isl/ctx.h>
+#include <isl/hash.h>
+
+#define xCAT(A,B) A ## B
+#define CAT(A,B) xCAT(A,B)
+#define KEY CAT(isl_,KEY_BASE)
+#define VAL CAT(isl_,VAL_BASE)
+#define xFN(TYPE,NAME) TYPE ## _ ## NAME
+#define FN(TYPE,NAME) xFN(TYPE,NAME)
+#define xHMAP(KEY,VAL_BASE) KEY ## _to_ ## VAL_BASE
+#define yHMAP(KEY,VAL_BASE) xHMAP(KEY,VAL_BASE)
+#define HMAP yHMAP(KEY,VAL_BASE)
+#define HMAP_BASE yHMAP(KEY_BASE,VAL_BASE)
+#define xS(TYPE1,TYPE2,NAME) struct isl_ ## TYPE1 ## _ ## TYPE2 ## _ ## NAME
+#define yS(TYPE1,TYPE2,NAME) xS(TYPE1,TYPE2,NAME)
+#define S(NAME) yS(KEY_BASE,VAL_BASE,NAME)
+
+struct HMAP {
+	int ref;
+	isl_ctx *ctx;
+	struct isl_hash_table table;
+};
+
+S(pair) {
+	KEY *key;
+	VAL *val;
+};
+
+__isl_give HMAP *FN(HMAP,alloc)(isl_ctx *ctx, int min_size)
+{
+	HMAP *hmap;
+
+	hmap = isl_calloc_type(ctx, HMAP);
+	if (!hmap)
+		return NULL;
+
+	hmap->ctx = ctx;
+	isl_ctx_ref(ctx);
+	hmap->ref = 1;
+
+	if (isl_hash_table_init(ctx, &hmap->table, min_size) < 0)
+		return FN(HMAP,free)(hmap);
+
+	return hmap;
+}
+
+static int free_pair(void **entry, void *user)
+{
+	S(pair) *pair = *entry;
+	FN(KEY,free)(pair->key);
+	FN(VAL,free)(pair->val);
+	free(pair);
+	*entry = NULL;
+	return 0;
+}
+
+__isl_null HMAP *FN(HMAP,free)(__isl_take HMAP *hmap)
+{
+	if (!hmap)
+		return NULL;
+	if (--hmap->ref > 0)
+		return NULL;
+	isl_hash_table_foreach(hmap->ctx, &hmap->table, &free_pair, NULL);
+	isl_hash_table_clear(&hmap->table);
+	isl_ctx_deref(hmap->ctx);
+	free(hmap);
+	return NULL;
+}
+
+isl_ctx *FN(HMAP,get_ctx)(__isl_keep HMAP *hmap)
+{
+	return hmap ? hmap->ctx : NULL;
+}
+
+/* Add a mapping from "key" to "val" to the associative array
+ * pointed to by user.
+ */
+static int add_key_val(__isl_take KEY *key, __isl_take VAL *val, void *user)
+{
+	HMAP **hmap = (HMAP **) user;
+
+	*hmap = FN(HMAP,set)(*hmap, key, val);
+
+	if (!*hmap)
+		return -1;
+
+	return 0;
+}
+
+__isl_give HMAP *FN(HMAP,dup)(__isl_keep HMAP *hmap)
+{
+	HMAP *dup;
+
+	if (!hmap)
+		return NULL;
+
+	dup = FN(HMAP,alloc)(hmap->ctx, hmap->table.n);
+	if (FN(HMAP,foreach)(hmap, &add_key_val, &dup) < 0)
+		return FN(HMAP,free)(dup);
+
+	return dup;
+}
+
+__isl_give HMAP *FN(HMAP,cow)(__isl_take HMAP *hmap)
+{
+	if (!hmap)
+		return NULL;
+
+	if (hmap->ref == 1)
+		return hmap;
+	hmap->ref--;
+	return FN(HMAP,dup)(hmap);
+}
+
+__isl_give HMAP *FN(HMAP,copy)(__isl_keep HMAP *hmap)
+{
+	if (!hmap)
+		return NULL;
+
+	hmap->ref++;
+	return hmap;
+}
+
+static int has_key(const void *entry, const void *c_key)
+{
+	const S(pair) *pair = entry;
+	KEY *key = (KEY *) c_key;
+
+	return KEY_EQUAL(pair->key, key);
+}
+
+int FN(HMAP,has)(__isl_keep HMAP *hmap, __isl_keep KEY *key)
+{
+	uint32_t hash;
+
+	if (!hmap)
+		return -1;
+
+	hash = FN(KEY,get_hash)(key);
+	return !!isl_hash_table_find(hmap->ctx, &hmap->table, hash,
+					&has_key, key, 0);
+}
+
+__isl_give VAL *FN(HMAP,get)(__isl_keep HMAP *hmap, __isl_take KEY *key)
+{
+	struct isl_hash_table_entry *entry;
+	S(pair) *pair;
+	uint32_t hash;
+
+	if (!hmap || !key)
+		goto error;
+
+	hash = FN(KEY,get_hash)(key);
+	entry = isl_hash_table_find(hmap->ctx, &hmap->table, hash,
+					&has_key, key, 0);
+	FN(KEY,free)(key);
+
+	if (!entry)
+		return NULL;
+
+	pair = entry->data;
+
+	return FN(VAL,copy)(pair->val);
+error:
+	FN(KEY,free)(key);
+	return NULL;
+}
+
+/* Remove the mapping between "key" and its associated value (if any)
+ * from "hmap".
+ *
+ * If "key" is not mapped to anything, then we leave "hmap" untouched"
+ */
+__isl_give HMAP *FN(HMAP,drop)(__isl_take HMAP *hmap, __isl_take KEY *key)
+{
+	struct isl_hash_table_entry *entry;
+	S(pair) *pair;
+	uint32_t hash;
+
+	if (!hmap || !key)
+		goto error;
+
+	hash = FN(KEY,get_hash)(key);
+	entry = isl_hash_table_find(hmap->ctx, &hmap->table, hash,
+					&has_key, key, 0);
+	if (!entry) {
+		FN(KEY,free)(key);
+		return hmap;
+	}
+
+	hmap = FN(HMAP,cow)(hmap);
+	if (!hmap)
+		goto error;
+	entry = isl_hash_table_find(hmap->ctx, &hmap->table, hash,
+					&has_key, key, 0);
+	FN(KEY,free)(key);
+
+	if (!entry)
+		isl_die(hmap->ctx, isl_error_internal,
+			"missing entry" , goto error);
+
+	pair = entry->data;
+	isl_hash_table_remove(hmap->ctx, &hmap->table, entry);
+	FN(KEY,free)(pair->key);
+	FN(VAL,free)(pair->val);
+	free(pair);
+
+	return hmap;
+error:
+	FN(KEY,free)(key);
+	FN(HMAP,free)(hmap);
+	return NULL;
+}
+
+/* Add a mapping from "key" to "val" to "hmap".
+ * If "key" was already mapped to something else, then that mapping
+ * is replaced.
+ * If key happened to be mapped to "val" already, then we leave
+ * "hmap" untouched.
+ */
+__isl_give HMAP *FN(HMAP,set)(__isl_take HMAP *hmap,
+	__isl_take KEY *key, __isl_take VAL *val)
+{
+	struct isl_hash_table_entry *entry;
+	S(pair) *pair;
+	uint32_t hash;
+
+	if (!hmap || !key || !val)
+		goto error;
+
+	hash = FN(KEY,get_hash)(key);
+	entry = isl_hash_table_find(hmap->ctx, &hmap->table, hash,
+					&has_key, key, 0);
+	if (entry) {
+		int equal;
+		pair = entry->data;
+		equal = VAL_EQUAL(pair->val, val);
+		if (equal < 0)
+			goto error;
+		if (equal) {
+			FN(KEY,free)(key);
+			FN(VAL,free)(val);
+			return hmap;
+		}
+	}
+
+	hmap = FN(HMAP,cow)(hmap);
+	if (!hmap)
+		goto error;
+
+	entry = isl_hash_table_find(hmap->ctx, &hmap->table, hash,
+					&has_key, key, 1);
+
+	if (!entry)
+		goto error;
+
+	if (entry->data) {
+		pair = entry->data;
+		FN(VAL,free)(pair->val);
+		pair->val = val;
+		FN(KEY,free)(key);
+		return hmap;
+	}
+
+	pair = isl_alloc_type(hmap->ctx, S(pair));
+	if (!pair)
+		goto error;
+
+	entry->data = pair;
+	pair->key = key;
+	pair->val = val;
+	return hmap;
+error:
+	FN(KEY,free)(key);
+	FN(VAL,free)(val);
+	return FN(HMAP,free)(hmap);
+}
+
+/* Internal data structure for isl_map_to_basic_set_foreach.
+ *
+ * fn is the function that should be called on each entry.
+ * user is the user-specified final argument to fn.
+ */
+S(foreach_data) {
+	int (*fn)(__isl_take KEY *key, __isl_take VAL *val, void *user);
+	void *user;
+};
+
+/* Call data->fn on a copy of the key and value in *entry.
+ */
+static int call_on_copy(void **entry, void *user)
+{
+	S(pair) *pair = *entry;
+	S(foreach_data) *data = (S(foreach_data) *) user;
+
+	return data->fn(FN(KEY,copy)(pair->key), FN(VAL,copy)(pair->val),
+			data->user);
+}
+
+/* Call "fn" on each pair of key and value in "hmap".
+ */
+int FN(HMAP,foreach)(__isl_keep HMAP *hmap,
+	int (*fn)(__isl_take KEY *key, __isl_take VAL *val, void *user),
+	void *user)
+{
+	S(foreach_data) data = { fn, user };
+
+	if (!hmap)
+		return -1;
+
+	return isl_hash_table_foreach(hmap->ctx, &hmap->table,
+				      &call_on_copy, &data);
+}
+
+/* Internal data structure for print_pair.
+ *
+ * p is the printer on which the associative array is being printed.
+ * first is set if the current key-value pair is the first to be printed.
+ */
+S(print_data) {
+	isl_printer *p;
+	int first;
+};
+
+/* Print the given key-value pair to data->p.
+ */
+static int print_pair(__isl_take KEY *key, __isl_take VAL *val, void *user)
+{
+	S(print_data) *data = user;
+
+	if (!data->first)
+		data->p = isl_printer_print_str(data->p, ", ");
+	data->p = FN(isl_printer_print,KEY_BASE)(data->p, key);
+	data->p = isl_printer_print_str(data->p, ": ");
+	data->p = FN(isl_printer_print,VAL_BASE)(data->p, val);
+	data->first = 0;
+
+	FN(KEY,free)(key);
+	FN(VAL,free)(val);
+	return 0;
+}
+
+/* Print the associative array to "p".
+ */
+__isl_give isl_printer *FN(isl_printer_print,HMAP_BASE)(
+	__isl_take isl_printer *p, __isl_keep HMAP *hmap)
+{
+	S(print_data) data;
+
+	if (!p || !hmap)
+		return isl_printer_free(p);
+
+	p = isl_printer_print_str(p, "{");
+	data.p = p;
+	data.first = 1;
+	if (FN(HMAP,foreach)(hmap, &print_pair, &data) < 0)
+		data.p = isl_printer_free(data.p);
+	p = data.p;
+	p = isl_printer_print_str(p, "}");
+
+	return p;
+}
+
+void FN(HMAP,dump)(__isl_keep HMAP *hmap)
+{
+	isl_printer *printer;
+
+	if (!hmap)
+		return;
+
+	printer = isl_printer_to_file(FN(HMAP,get_ctx)(hmap), stderr);
+	printer = FN(isl_printer_print,HMAP_BASE)(printer, hmap);
+	printer = isl_printer_end_line(printer);
+
+	isl_printer_free(printer);
+}
diff --git a/polly/lib/External/isl/isl_id.c b/polly/lib/External/isl/isl_id.c
new file mode 100644
index 00000000000..6895570e113
--- /dev/null
+++ b/polly/lib/External/isl/isl_id.c
@@ -0,0 +1,242 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <string.h>
+#include <isl_ctx_private.h>
+#include <isl_id_private.h>
+
+#undef BASE
+#define BASE id
+
+#include <isl_list_templ.c>
+
+/* A special, static isl_id to use as domains (and ranges)
+ * of sets and parameters domains.
+ * The user should never get a hold on this isl_id.
+ */
+isl_id isl_id_none = {
+	.ref = -1,
+	.ctx = NULL,
+	.name = "#none",
+	.user = NULL
+};
+
+isl_ctx *isl_id_get_ctx(__isl_keep isl_id *id)
+{
+	return id ? id->ctx : NULL;
+}
+
+void *isl_id_get_user(__isl_keep isl_id *id)
+{
+	return id ? id->user : NULL;
+}
+
+const char *isl_id_get_name(__isl_keep isl_id *id)
+{
+	return id ? id->name : NULL;
+}
+
+static __isl_give isl_id *id_alloc(isl_ctx *ctx, const char *name, void *user)
+{
+	const char *copy = name ? strdup(name) : NULL;
+	isl_id *id;
+
+	if (name && !copy)
+		return NULL;
+	id = isl_calloc_type(ctx, struct isl_id);
+	if (!id)
+		goto error;
+
+	id->ctx = ctx;
+	isl_ctx_ref(id->ctx);
+	id->ref = 1;
+	id->name = copy;
+	id->user = user;
+
+	id->hash = isl_hash_init();
+	if (name)
+		id->hash = isl_hash_string(id->hash, name);
+	else
+		id->hash = isl_hash_builtin(id->hash, user);
+
+	return id;
+error:
+	free((char *)copy);
+	return NULL;
+}
+
+uint32_t isl_id_get_hash(__isl_keep isl_id *id)
+{
+	return id ? id->hash : 0;
+}
+
+struct isl_name_and_user {
+	const char *name;
+	void *user;
+};
+
+static int isl_id_has_name_and_user(const void *entry, const void *val)
+{
+	isl_id *id = (isl_id *)entry;
+	struct isl_name_and_user *nu = (struct isl_name_and_user *) val;
+
+	if (id->user != nu->user)
+		return 0;
+	if (!id->name && !nu->name)
+		return 1;
+
+	return !strcmp(id->name, nu->name);
+}
+
+__isl_give isl_id *isl_id_alloc(isl_ctx *ctx, const char *name, void *user)
+{
+	struct isl_hash_table_entry *entry;
+	uint32_t id_hash;
+	struct isl_name_and_user nu = { name, user };
+
+	if (!ctx)
+		return NULL;
+
+	id_hash = isl_hash_init();
+	if (name)
+		id_hash = isl_hash_string(id_hash, name);
+	else
+		id_hash = isl_hash_builtin(id_hash, user);
+	entry = isl_hash_table_find(ctx, &ctx->id_table, id_hash,
+					isl_id_has_name_and_user, &nu, 1);
+	if (!entry)
+		return NULL;
+	if (entry->data)
+		return isl_id_copy(entry->data);
+	entry->data = id_alloc(ctx, name, user);
+	if (!entry->data)
+		ctx->id_table.n--;
+	return entry->data;
+}
+
+/* If the id has a negative refcount, then it is a static isl_id
+ * which should not be changed.
+ */
+__isl_give isl_id *isl_id_copy(isl_id *id)
+{
+	if (!id)
+		return NULL;
+
+	if (id->ref < 0)
+		return id;
+
+	id->ref++;
+	return id;
+}
+
+/* Compare two isl_ids.
+ *
+ * The order is fairly arbitrary.  We do keep the comparison of
+ * the user pointers as a last resort since these pointer values
+ * may not be stable across different systems or even different runs.
+ */
+int isl_id_cmp(__isl_keep isl_id *id1, __isl_keep isl_id *id2)
+{
+	if (id1 == id2)
+		return 0;
+	if (!id1)
+		return -1;
+	if (!id2)
+		return 1;
+	if (!id1->name != !id2->name)
+		return !id1->name - !id2->name;
+	if (id1->name) {
+		int cmp = strcmp(id1->name, id2->name);
+		if (cmp != 0)
+			return cmp;
+	}
+	if (id1->user < id2->user)
+		return -1;
+	else
+		return 1;
+}
+
+static int isl_id_eq(const void *entry, const void *name)
+{
+	return entry == name;
+}
+
+uint32_t isl_hash_id(uint32_t hash, __isl_keep isl_id *id)
+{
+	if (id)
+		isl_hash_hash(hash, id->hash);
+
+	return hash;
+}
+
+/* Replace the free_user callback by "free_user".
+ */
+__isl_give isl_id *isl_id_set_free_user(__isl_take isl_id *id,
+	__isl_give void (*free_user)(void *user))
+{
+	if (!id)
+		return NULL;
+
+	id->free_user = free_user;
+
+	return id;
+}
+
+/* If the id has a negative refcount, then it is a static isl_id
+ * and should not be freed.
+ */
+__isl_null isl_id *isl_id_free(__isl_take isl_id *id)
+{
+	struct isl_hash_table_entry *entry;
+
+	if (!id)
+		return NULL;
+
+	if (id->ref < 0)
+		return NULL;
+
+	if (--id->ref > 0)
+		return NULL;
+
+	entry = isl_hash_table_find(id->ctx, &id->ctx->id_table, id->hash,
+					isl_id_eq, id, 0);
+	if (!entry)
+		isl_die(id->ctx, isl_error_unknown,
+			"unable to find id", (void)0);
+	else
+		isl_hash_table_remove(id->ctx, &id->ctx->id_table, entry);
+
+	if (id->free_user)
+		id->free_user(id->user);
+
+	free((char *)id->name);
+	isl_ctx_deref(id->ctx);
+	free(id);
+
+	return NULL;
+}
+
+__isl_give isl_printer *isl_printer_print_id(__isl_take isl_printer *p,
+	__isl_keep isl_id *id)
+{
+	if (!id)
+		goto error;
+
+	if (id->name)
+		p = isl_printer_print_str(p, id->name);
+	if (id->user) {
+		char buffer[50];
+		snprintf(buffer, sizeof(buffer), "@%p", id->user);
+		p = isl_printer_print_str(p, buffer);
+	}
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
diff --git a/polly/lib/External/isl/isl_id_private.h b/polly/lib/External/isl/isl_id_private.h
new file mode 100644
index 00000000000..1d903c8e59e
--- /dev/null
+++ b/polly/lib/External/isl/isl_id_private.h
@@ -0,0 +1,41 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_ID_PRIVATE_H
+#define ISL_ID_PRIVATE_H
+
+#include <isl/id.h>
+
+/* Represent a name and/or user pointer.
+ *
+ * If "free_user" is set, then it will be called on "user" when
+ * the last instance of the isl_id is freed.
+ */
+struct isl_id {
+	int ref;
+	isl_ctx *ctx;
+
+	const char *name;
+	void *user;
+	uint32_t hash;
+
+	__isl_give void (*free_user)(void *user);
+};
+
+#undef EL
+#define EL isl_id
+
+#include <isl_list_templ.h>
+
+uint32_t isl_hash_id(uint32_t hash, __isl_keep isl_id *id);
+int isl_id_cmp(__isl_keep isl_id *id1, __isl_keep isl_id *id2);
+
+extern isl_id isl_id_none;
+
+#endif
diff --git a/polly/lib/External/isl/isl_id_to_ast_expr.c b/polly/lib/External/isl/isl_id_to_ast_expr.c
new file mode 100644
index 00000000000..f8ac7f7c68e
--- /dev/null
+++ b/polly/lib/External/isl/isl_id_to_ast_expr.c
@@ -0,0 +1,11 @@
+#include <isl/id_to_ast_expr.h>
+#include <isl/ast.h>
+
+#define isl_id_is_equal(id1,id2)	id1 == id2
+
+#define KEY_BASE	id
+#define KEY_EQUAL	isl_id_is_equal
+#define VAL_BASE	ast_expr
+#define VAL_EQUAL	isl_ast_expr_is_equal
+
+#include <isl_hmap_templ.c>
diff --git a/polly/lib/External/isl/isl_id_to_pw_aff.c b/polly/lib/External/isl/isl_id_to_pw_aff.c
new file mode 100644
index 00000000000..2fce240dd30
--- /dev/null
+++ b/polly/lib/External/isl/isl_id_to_pw_aff.c
@@ -0,0 +1,11 @@
+#include <isl/id_to_pw_aff.h>
+#include <isl/aff.h>
+
+#define isl_id_is_equal(id1,id2)	id1 == id2
+
+#define KEY_BASE	id
+#define KEY_EQUAL	isl_id_is_equal
+#define VAL_BASE	pw_aff
+#define VAL_EQUAL	isl_pw_aff_plain_is_equal
+
+#include <isl_hmap_templ.c>
diff --git a/polly/lib/External/isl/isl_ilp.c b/polly/lib/External/isl/isl_ilp.c
new file mode 100644
index 00000000000..5aec90639fd
--- /dev/null
+++ b/polly/lib/External/isl/isl_ilp.c
@@ -0,0 +1,642 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl/ilp.h>
+#include "isl_sample.h"
+#include <isl_seq.h>
+#include "isl_equalities.h"
+#include <isl_aff_private.h>
+#include <isl_local_space_private.h>
+#include <isl_mat_private.h>
+#include <isl_val_private.h>
+#include <isl_vec_private.h>
+#include <isl_lp_private.h>
+#include <isl_ilp_private.h>
+#include <isl/deprecated/ilp_int.h>
+
+/* Given a basic set "bset", construct a basic set U such that for
+ * each element x in U, the whole unit box positioned at x is inside
+ * the given basic set.
+ * Note that U may not contain all points that satisfy this property.
+ *
+ * We simply add the sum of all negative coefficients to the constant
+ * term.  This ensures that if x satisfies the resulting constraints,
+ * then x plus any sum of unit vectors satisfies the original constraints.
+ */
+static struct isl_basic_set *unit_box_base_points(struct isl_basic_set *bset)
+{
+	int i, j, k;
+	struct isl_basic_set *unit_box = NULL;
+	unsigned total;
+
+	if (!bset)
+		goto error;
+
+	if (bset->n_eq != 0) {
+		unit_box = isl_basic_set_empty_like(bset);
+		isl_basic_set_free(bset);
+		return unit_box;
+	}
+
+	total = isl_basic_set_total_dim(bset);
+	unit_box = isl_basic_set_alloc_space(isl_basic_set_get_space(bset),
+					0, 0, bset->n_ineq);
+
+	for (i = 0; i < bset->n_ineq; ++i) {
+		k = isl_basic_set_alloc_inequality(unit_box);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(unit_box->ineq[k], bset->ineq[i], 1 + total);
+		for (j = 0; j < total; ++j) {
+			if (isl_int_is_nonneg(unit_box->ineq[k][1 + j]))
+				continue;
+			isl_int_add(unit_box->ineq[k][0],
+				unit_box->ineq[k][0], unit_box->ineq[k][1 + j]);
+		}
+	}
+
+	isl_basic_set_free(bset);
+	return unit_box;
+error:
+	isl_basic_set_free(bset);
+	isl_basic_set_free(unit_box);
+	return NULL;
+}
+
+/* Find an integer point in "bset", preferably one that is
+ * close to minimizing "f".
+ *
+ * We first check if we can easily put unit boxes inside bset.
+ * If so, we take the best base point of any of the unit boxes we can find
+ * and round it up to the nearest integer.
+ * If not, we simply pick any integer point in "bset".
+ */
+static struct isl_vec *initial_solution(struct isl_basic_set *bset, isl_int *f)
+{
+	enum isl_lp_result res;
+	struct isl_basic_set *unit_box;
+	struct isl_vec *sol;
+
+	unit_box = unit_box_base_points(isl_basic_set_copy(bset));
+
+	res = isl_basic_set_solve_lp(unit_box, 0, f, bset->ctx->one,
+					NULL, NULL, &sol);
+	if (res == isl_lp_ok) {
+		isl_basic_set_free(unit_box);
+		return isl_vec_ceil(sol);
+	}
+
+	isl_basic_set_free(unit_box);
+
+	return isl_basic_set_sample_vec(isl_basic_set_copy(bset));
+}
+
+/* Restrict "bset" to those points with values for f in the interval [l, u].
+ */
+static struct isl_basic_set *add_bounds(struct isl_basic_set *bset,
+	isl_int *f, isl_int l, isl_int u)
+{
+	int k;
+	unsigned total;
+
+	total = isl_basic_set_total_dim(bset);
+	bset = isl_basic_set_extend_constraints(bset, 0, 2);
+
+	k = isl_basic_set_alloc_inequality(bset);
+	if (k < 0)
+		goto error;
+	isl_seq_cpy(bset->ineq[k], f, 1 + total);
+	isl_int_sub(bset->ineq[k][0], bset->ineq[k][0], l);
+
+	k = isl_basic_set_alloc_inequality(bset);
+	if (k < 0)
+		goto error;
+	isl_seq_neg(bset->ineq[k], f, 1 + total);
+	isl_int_add(bset->ineq[k][0], bset->ineq[k][0], u);
+
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Find an integer point in "bset" that minimizes f (in any) such that
+ * the value of f lies inside the interval [l, u].
+ * Return this integer point if it can be found.
+ * Otherwise, return sol.
+ *
+ * We perform a number of steps until l > u.
+ * In each step, we look for an integer point with value in either
+ * the whole interval [l, u] or half of the interval [l, l+floor(u-l-1/2)].
+ * The choice depends on whether we have found an integer point in the
+ * previous step.  If so, we look for the next point in half of the remaining
+ * interval.
+ * If we find a point, the current solution is updated and u is set
+ * to its value minus 1.
+ * If no point can be found, we update l to the upper bound of the interval
+ * we checked (u or l+floor(u-l-1/2)) plus 1.
+ */
+static struct isl_vec *solve_ilp_search(struct isl_basic_set *bset,
+	isl_int *f, isl_int *opt, struct isl_vec *sol, isl_int l, isl_int u)
+{
+	isl_int tmp;
+	int divide = 1;
+
+	isl_int_init(tmp);
+
+	while (isl_int_le(l, u)) {
+		struct isl_basic_set *slice;
+		struct isl_vec *sample;
+
+		if (!divide)
+			isl_int_set(tmp, u);
+		else {
+			isl_int_sub(tmp, u, l);
+			isl_int_fdiv_q_ui(tmp, tmp, 2);
+			isl_int_add(tmp, tmp, l);
+		}
+		slice = add_bounds(isl_basic_set_copy(bset), f, l, tmp);
+		sample = isl_basic_set_sample_vec(slice);
+		if (!sample) {
+			isl_vec_free(sol);
+			sol = NULL;
+			break;
+		}
+		if (sample->size > 0) {
+			isl_vec_free(sol);
+			sol = sample;
+			isl_seq_inner_product(f, sol->el, sol->size, opt);
+			isl_int_sub_ui(u, *opt, 1);
+			divide = 1;
+		} else {
+			isl_vec_free(sample);
+			if (!divide)
+				break;
+			isl_int_add_ui(l, tmp, 1);
+			divide = 0;
+		}
+	}
+
+	isl_int_clear(tmp);
+
+	return sol;
+}
+
+/* Find an integer point in "bset" that minimizes f (if any).
+ * If sol_p is not NULL then the integer point is returned in *sol_p.
+ * The optimal value of f is returned in *opt.
+ *
+ * The algorithm maintains a currently best solution and an interval [l, u]
+ * of values of f for which integer solutions could potentially still be found.
+ * The initial value of the best solution so far is any solution.
+ * The initial value of l is minimal value of f over the rationals
+ * (rounded up to the nearest integer).
+ * The initial value of u is the value of f at the initial solution minus 1.
+ *
+ * We then call solve_ilp_search to perform a binary search on the interval.
+ */
+static enum isl_lp_result solve_ilp(struct isl_basic_set *bset,
+				      isl_int *f, isl_int *opt,
+				      struct isl_vec **sol_p)
+{
+	enum isl_lp_result res;
+	isl_int l, u;
+	struct isl_vec *sol;
+
+	res = isl_basic_set_solve_lp(bset, 0, f, bset->ctx->one,
+					opt, NULL, &sol);
+	if (res == isl_lp_ok && isl_int_is_one(sol->el[0])) {
+		if (sol_p)
+			*sol_p = sol;
+		else
+			isl_vec_free(sol);
+		return isl_lp_ok;
+	}
+	isl_vec_free(sol);
+	if (res == isl_lp_error || res == isl_lp_empty)
+		return res;
+
+	sol = initial_solution(bset, f);
+	if (!sol)
+		return isl_lp_error;
+	if (sol->size == 0) {
+		isl_vec_free(sol);
+		return isl_lp_empty;
+	}
+	if (res == isl_lp_unbounded) {
+		isl_vec_free(sol);
+		return isl_lp_unbounded;
+	}
+
+	isl_int_init(l);
+	isl_int_init(u);
+
+	isl_int_set(l, *opt);
+
+	isl_seq_inner_product(f, sol->el, sol->size, opt);
+	isl_int_sub_ui(u, *opt, 1);
+
+	sol = solve_ilp_search(bset, f, opt, sol, l, u);
+	if (!sol)
+		res = isl_lp_error;
+
+	isl_int_clear(l);
+	isl_int_clear(u);
+
+	if (sol_p)
+		*sol_p = sol;
+	else
+		isl_vec_free(sol);
+
+	return res;
+}
+
+static enum isl_lp_result solve_ilp_with_eq(struct isl_basic_set *bset, int max,
+				      isl_int *f, isl_int *opt,
+				      struct isl_vec **sol_p)
+{
+	unsigned dim;
+	enum isl_lp_result res;
+	struct isl_mat *T = NULL;
+	struct isl_vec *v;
+
+	bset = isl_basic_set_copy(bset);
+	dim = isl_basic_set_total_dim(bset);
+	v = isl_vec_alloc(bset->ctx, 1 + dim);
+	if (!v)
+		goto error;
+	isl_seq_cpy(v->el, f, 1 + dim);
+	bset = isl_basic_set_remove_equalities(bset, &T, NULL);
+	v = isl_vec_mat_product(v, isl_mat_copy(T));
+	if (!v)
+		goto error;
+	res = isl_basic_set_solve_ilp(bset, max, v->el, opt, sol_p);
+	isl_vec_free(v);
+	if (res == isl_lp_ok && sol_p) {
+		*sol_p = isl_mat_vec_product(T, *sol_p);
+		if (!*sol_p)
+			res = isl_lp_error;
+	} else
+		isl_mat_free(T);
+	isl_basic_set_free(bset);
+	return res;
+error:
+	isl_mat_free(T);
+	isl_basic_set_free(bset);
+	return isl_lp_error;
+}
+
+/* Find an integer point in "bset" that minimizes (or maximizes if max is set)
+ * f (if any).
+ * If sol_p is not NULL then the integer point is returned in *sol_p.
+ * The optimal value of f is returned in *opt.
+ *
+ * If there is any equality among the points in "bset", then we first
+ * project it out.  Otherwise, we continue with solve_ilp above.
+ */
+enum isl_lp_result isl_basic_set_solve_ilp(struct isl_basic_set *bset, int max,
+				      isl_int *f, isl_int *opt,
+				      struct isl_vec **sol_p)
+{
+	unsigned dim;
+	enum isl_lp_result res;
+
+	if (!bset)
+		return isl_lp_error;
+	if (sol_p)
+		*sol_p = NULL;
+
+	isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
+
+	if (isl_basic_set_plain_is_empty(bset))
+		return isl_lp_empty;
+
+	if (bset->n_eq)
+		return solve_ilp_with_eq(bset, max, f, opt, sol_p);
+
+	dim = isl_basic_set_total_dim(bset);
+
+	if (max)
+		isl_seq_neg(f, f, 1 + dim);
+
+	res = solve_ilp(bset, f, opt, sol_p);
+
+	if (max) {
+		isl_seq_neg(f, f, 1 + dim);
+		isl_int_neg(*opt, *opt);
+	}
+
+	return res;
+error:
+	isl_basic_set_free(bset);
+	return isl_lp_error;
+}
+
+static enum isl_lp_result basic_set_opt(__isl_keep isl_basic_set *bset, int max,
+	__isl_keep isl_aff *obj, isl_int *opt)
+{
+	enum isl_lp_result res;
+
+	if (!obj)
+		return isl_lp_error;
+	bset = isl_basic_set_copy(bset);
+	bset = isl_basic_set_underlying_set(bset);
+	res = isl_basic_set_solve_ilp(bset, max, obj->v->el + 1, opt, NULL);
+	isl_basic_set_free(bset);
+	return res;
+}
+
+static __isl_give isl_mat *extract_divs(__isl_keep isl_basic_set *bset)
+{
+	int i;
+	isl_ctx *ctx = isl_basic_set_get_ctx(bset);
+	isl_mat *div;
+
+	div = isl_mat_alloc(ctx, bset->n_div,
+			    1 + 1 + isl_basic_set_total_dim(bset));
+	if (!div)
+		return NULL;
+
+	for (i = 0; i < bset->n_div; ++i)
+		isl_seq_cpy(div->row[i], bset->div[i], div->n_col);
+
+	return div;
+}
+
+enum isl_lp_result isl_basic_set_opt(__isl_keep isl_basic_set *bset, int max,
+	__isl_keep isl_aff *obj, isl_int *opt)
+{
+	int *exp1 = NULL;
+	int *exp2 = NULL;
+	isl_ctx *ctx;
+	isl_mat *bset_div = NULL;
+	isl_mat *div = NULL;
+	enum isl_lp_result res;
+	int bset_n_div, obj_n_div;
+
+	if (!bset || !obj)
+		return isl_lp_error;
+
+	ctx = isl_aff_get_ctx(obj);
+	if (!isl_space_is_equal(bset->dim, obj->ls->dim))
+		isl_die(ctx, isl_error_invalid,
+			"spaces don't match", return isl_lp_error);
+	if (!isl_int_is_one(obj->v->el[0]))
+		isl_die(ctx, isl_error_unsupported,
+			"expecting integer affine expression",
+			return isl_lp_error);
+
+	bset_n_div = isl_basic_set_dim(bset, isl_dim_div);
+	obj_n_div = isl_aff_dim(obj, isl_dim_div);
+	if (bset_n_div == 0 && obj_n_div == 0)
+		return basic_set_opt(bset, max, obj, opt);
+
+	bset = isl_basic_set_copy(bset);
+	obj = isl_aff_copy(obj);
+
+	bset_div = extract_divs(bset);
+	exp1 = isl_alloc_array(ctx, int, bset_n_div);
+	exp2 = isl_alloc_array(ctx, int, obj_n_div);
+	if (!bset_div || (bset_n_div && !exp1) || (obj_n_div && !exp2))
+		goto error;
+
+	div = isl_merge_divs(bset_div, obj->ls->div, exp1, exp2);
+
+	bset = isl_basic_set_expand_divs(bset, isl_mat_copy(div), exp1);
+	obj = isl_aff_expand_divs(obj, isl_mat_copy(div), exp2);
+
+	res = basic_set_opt(bset, max, obj, opt);
+
+	isl_mat_free(bset_div);
+	isl_mat_free(div);
+	free(exp1);
+	free(exp2);
+	isl_basic_set_free(bset);
+	isl_aff_free(obj);
+
+	return res;
+error:
+	isl_mat_free(div);
+	isl_mat_free(bset_div);
+	free(exp1);
+	free(exp2);
+	isl_basic_set_free(bset);
+	isl_aff_free(obj);
+	return isl_lp_error;
+}
+
+/* Compute the minimum (maximum if max is set) of the integer affine
+ * expression obj over the points in set and put the result in *opt.
+ *
+ * The parameters are assumed to have been aligned.
+ */
+static enum isl_lp_result isl_set_opt_aligned(__isl_keep isl_set *set, int max,
+	__isl_keep isl_aff *obj, isl_int *opt)
+{
+	int i;
+	enum isl_lp_result res;
+	int empty = 1;
+	isl_int opt_i;
+
+	if (!set || !obj)
+		return isl_lp_error;
+	if (set->n == 0)
+		return isl_lp_empty;
+
+	res = isl_basic_set_opt(set->p[0], max, obj, opt);
+	if (res == isl_lp_error || res == isl_lp_unbounded)
+		return res;
+	if (set->n == 1)
+		return res;
+	if (res == isl_lp_ok)
+		empty = 0;
+
+	isl_int_init(opt_i);
+	for (i = 1; i < set->n; ++i) {
+		res = isl_basic_set_opt(set->p[i], max, obj, &opt_i);
+		if (res == isl_lp_error || res == isl_lp_unbounded) {
+			isl_int_clear(opt_i);
+			return res;
+		}
+		if (res == isl_lp_ok)
+			empty = 0;
+		if (max ? isl_int_gt(opt_i, *opt) : isl_int_lt(opt_i, *opt))
+			isl_int_set(*opt, opt_i);
+	}
+	isl_int_clear(opt_i);
+
+	return empty ? isl_lp_empty : isl_lp_ok;
+}
+
+/* Compute the minimum (maximum if max is set) of the integer affine
+ * expression obj over the points in set and put the result in *opt.
+ */
+enum isl_lp_result isl_set_opt(__isl_keep isl_set *set, int max,
+	__isl_keep isl_aff *obj, isl_int *opt)
+{
+	enum isl_lp_result res;
+
+	if (!set || !obj)
+		return isl_lp_error;
+
+	if (isl_space_match(set->dim, isl_dim_param,
+			    obj->ls->dim, isl_dim_param))
+		return isl_set_opt_aligned(set, max, obj, opt);
+
+	set = isl_set_copy(set);
+	obj = isl_aff_copy(obj);
+	set = isl_set_align_params(set, isl_aff_get_domain_space(obj));
+	obj = isl_aff_align_params(obj, isl_set_get_space(set));
+
+	res = isl_set_opt_aligned(set, max, obj, opt);
+
+	isl_set_free(set);
+	isl_aff_free(obj);
+
+	return res;
+}
+
+enum isl_lp_result isl_basic_set_max(__isl_keep isl_basic_set *bset,
+	__isl_keep isl_aff *obj, isl_int *opt)
+{
+	return isl_basic_set_opt(bset, 1, obj, opt);
+}
+
+enum isl_lp_result isl_set_max(__isl_keep isl_set *set,
+	__isl_keep isl_aff *obj, isl_int *opt)
+{
+	return isl_set_opt(set, 1, obj, opt);
+}
+
+enum isl_lp_result isl_set_min(__isl_keep isl_set *set,
+	__isl_keep isl_aff *obj, isl_int *opt)
+{
+	return isl_set_opt(set, 0, obj, opt);
+}
+
+/* Convert the result of a function that returns an isl_lp_result
+ * to an isl_val.  The numerator of "v" is set to the optimal value
+ * if lp_res is isl_lp_ok.  "max" is set if a maximum was computed.
+ *
+ * Return "v" with denominator set to 1 if lp_res is isl_lp_ok.
+ * Return NULL on error.
+ * Return a NaN if lp_res is isl_lp_empty.
+ * Return infinity or negative infinity if lp_res is isl_lp_unbounded,
+ * depending on "max".
+ */
+static __isl_give isl_val *convert_lp_result(enum isl_lp_result lp_res,
+	__isl_take isl_val *v, int max)
+{
+	isl_ctx *ctx;
+
+	if (lp_res == isl_lp_ok) {
+		isl_int_set_si(v->d, 1);
+		return isl_val_normalize(v);
+	}
+	ctx = isl_val_get_ctx(v);
+	isl_val_free(v);
+	if (lp_res == isl_lp_error)
+		return NULL;
+	if (lp_res == isl_lp_empty)
+		return isl_val_nan(ctx);
+	if (max)
+		return isl_val_infty(ctx);
+	else
+		return isl_val_neginfty(ctx);
+}
+
+/* Return the minimum (maximum if max is set) of the integer affine
+ * expression "obj" over the points in "bset".
+ *
+ * Return infinity or negative infinity if the optimal value is unbounded and
+ * NaN if "bset" is empty.
+ *
+ * Call isl_basic_set_opt and translate the results.
+ */
+__isl_give isl_val *isl_basic_set_opt_val(__isl_keep isl_basic_set *bset,
+	int max, __isl_keep isl_aff *obj)
+{
+	isl_ctx *ctx;
+	isl_val *res;
+	enum isl_lp_result lp_res;
+
+	if (!bset || !obj)
+		return NULL;
+
+	ctx = isl_aff_get_ctx(obj);
+	res = isl_val_alloc(ctx);
+	if (!res)
+		return NULL;
+	lp_res = isl_basic_set_opt(bset, max, obj, &res->n);
+	return convert_lp_result(lp_res, res, max);
+}
+
+/* Return the maximum of the integer affine
+ * expression "obj" over the points in "bset".
+ *
+ * Return infinity or negative infinity if the optimal value is unbounded and
+ * NaN if "bset" is empty.
+ */
+__isl_give isl_val *isl_basic_set_max_val(__isl_keep isl_basic_set *bset,
+	__isl_keep isl_aff *obj)
+{
+	return isl_basic_set_opt_val(bset, 1, obj);
+}
+
+/* Return the minimum (maximum if max is set) of the integer affine
+ * expression "obj" over the points in "set".
+ *
+ * Return infinity or negative infinity if the optimal value is unbounded and
+ * NaN if "bset" is empty.
+ *
+ * Call isl_set_opt and translate the results.
+ */
+__isl_give isl_val *isl_set_opt_val(__isl_keep isl_set *set, int max,
+	__isl_keep isl_aff *obj)
+{
+	isl_ctx *ctx;
+	isl_val *res;
+	enum isl_lp_result lp_res;
+
+	if (!set || !obj)
+		return NULL;
+
+	ctx = isl_aff_get_ctx(obj);
+	res = isl_val_alloc(ctx);
+	if (!res)
+		return NULL;
+	lp_res = isl_set_opt(set, max, obj, &res->n);
+	return convert_lp_result(lp_res, res, max);
+}
+
+/* Return the minimum of the integer affine
+ * expression "obj" over the points in "set".
+ *
+ * Return infinity or negative infinity if the optimal value is unbounded and
+ * NaN if "bset" is empty.
+ */
+__isl_give isl_val *isl_set_min_val(__isl_keep isl_set *set,
+	__isl_keep isl_aff *obj)
+{
+	return isl_set_opt_val(set, 0, obj);
+}
+
+/* Return the maximum of the integer affine
+ * expression "obj" over the points in "set".
+ *
+ * Return infinity or negative infinity if the optimal value is unbounded and
+ * NaN if "bset" is empty.
+ */
+__isl_give isl_val *isl_set_max_val(__isl_keep isl_set *set,
+	__isl_keep isl_aff *obj)
+{
+	return isl_set_opt_val(set, 1, obj);
+}
diff --git a/polly/lib/External/isl/isl_ilp_private.h b/polly/lib/External/isl/isl_ilp_private.h
new file mode 100644
index 00000000000..932b2c3d4f2
--- /dev/null
+++ b/polly/lib/External/isl/isl_ilp_private.h
@@ -0,0 +1,11 @@
+#ifndef ISL_ILP_PRIVATE_H
+#define ISL_ILP_PRIVATE_H
+
+#include <isl_int.h>
+#include <isl/lp.h>
+#include <isl/set.h>
+
+enum isl_lp_result isl_basic_set_solve_ilp(__isl_keep isl_basic_set *bset,
+	int max, isl_int *f, isl_int *opt, __isl_give isl_vec **sol_p);
+
+#endif
diff --git a/polly/lib/External/isl/isl_imath.c b/polly/lib/External/isl/isl_imath.c
new file mode 100644
index 00000000000..959b5e78272
--- /dev/null
+++ b/polly/lib/External/isl/isl_imath.c
@@ -0,0 +1,53 @@
+#include <isl_int.h>
+
+uint32_t isl_imath_hash(mp_int v, uint32_t hash)
+{
+	unsigned const char *data = (unsigned char *)v->digits;
+	unsigned const char *end = data + v->used * sizeof(v->digits[0]);
+
+	if (v->sign == 1)
+		isl_hash_byte(hash, 0xFF);
+	for (; data < end; ++data)
+		isl_hash_byte(hash, *data);
+	return hash;
+}
+
+/* Try a standard conversion that fits into a long.
+ */
+int isl_imath_fits_slong_p(mp_int op)
+{
+	unsigned long out;
+	mp_result res = mp_int_to_int(op, &out);
+	return res == MP_OK;
+}
+
+/* Try a standard conversion that fits into an unsigned long.
+ */
+int isl_imath_fits_ulong_p(mp_int op)
+{
+	unsigned long out;
+	mp_result res = mp_int_to_uint(op, &out);
+	return res == MP_OK;
+}
+
+void isl_imath_addmul_ui(mp_int rop, mp_int op1, unsigned long op2)
+{
+	isl_int temp;
+	isl_int_init(temp);
+
+	isl_int_set_ui(temp, op2);
+	isl_int_addmul(rop, op1, temp);
+
+	isl_int_clear(temp);
+}
+
+void isl_imath_submul_ui(mp_int rop, mp_int op1, unsigned long op2)
+{
+	isl_int temp;
+	isl_int_init(temp);
+
+	isl_int_set_ui(temp, op2);
+	isl_int_submul(rop, op1, temp);
+
+	isl_int_clear(temp);
+}
diff --git a/polly/lib/External/isl/isl_imath.h b/polly/lib/External/isl/isl_imath.h
new file mode 100644
index 00000000000..d51e03cb1fd
--- /dev/null
+++ b/polly/lib/External/isl/isl_imath.h
@@ -0,0 +1,8 @@
+#include <imath.h>
+#include <gmp_compat.h>
+
+uint32_t isl_imath_hash(mp_int v, uint32_t hash);
+int isl_imath_fits_ulong_p(mp_int op);
+int isl_imath_fits_slong_p(mp_int op);
+void isl_imath_addmul_ui(mp_int rop, mp_int op1, unsigned long op2);
+void isl_imath_submul_ui(mp_int rop, mp_int op1, unsigned long op2);
diff --git a/polly/lib/External/isl/isl_input.c b/polly/lib/External/isl/isl_input.c
new file mode 100644
index 00000000000..520e7f45c19
--- /dev/null
+++ b/polly/lib/External/isl/isl_input.c
@@ -0,0 +1,3370 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2010      INRIA Saclay
+ * Copyright 2012-2013 Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France 
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+#include <ctype.h>
+#include <stdio.h>
+#include <string.h>
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl/set.h>
+#include <isl_seq.h>
+#include <isl_stream_private.h>
+#include <isl/obj.h>
+#include "isl_polynomial_private.h"
+#include <isl/union_map.h>
+#include <isl_mat_private.h>
+#include <isl_aff_private.h>
+#include <isl_vec_private.h>
+#include <isl/list.h>
+#include <isl_val_private.h>
+
+struct variable {
+	char    	    	*name;
+	int	     		 pos;
+	struct variable		*next;
+};
+
+struct vars {
+	struct isl_ctx	*ctx;
+	int		 n;
+	struct variable	*v;
+};
+
+static struct vars *vars_new(struct isl_ctx *ctx)
+{
+	struct vars *v;
+	v = isl_alloc_type(ctx, struct vars);
+	if (!v)
+		return NULL;
+	v->ctx = ctx;
+	v->n = 0;
+	v->v = NULL;
+	return v;
+}
+
+static void variable_free(struct variable *var)
+{
+	while (var) {
+		struct variable *next = var->next;
+		free(var->name);
+		free(var);
+		var = next;
+	}
+}
+
+static void vars_free(struct vars *v)
+{
+	if (!v)
+		return;
+	variable_free(v->v);
+	free(v);
+}
+
+static void vars_drop(struct vars *v, int n)
+{
+	struct variable *var;
+
+	if (!v || !v->v)
+		return;
+
+	v->n -= n;
+
+	var = v->v;
+	while (--n >= 0) {
+		struct variable *next = var->next;
+		free(var->name);
+		free(var);
+		var = next;
+	}
+	v->v = var;
+}
+
+static struct variable *variable_new(struct vars *v, const char *name, int len,
+				int pos)
+{
+	struct variable *var;
+	var = isl_calloc_type(v->ctx, struct variable);
+	if (!var)
+		goto error;
+	var->name = strdup(name);
+	var->name[len] = '\0';
+	var->pos = pos;
+	var->next = v->v;
+	return var;
+error:
+	variable_free(v->v);
+	return NULL;
+}
+
+static int vars_pos(struct vars *v, const char *s, int len)
+{
+	int pos;
+	struct variable *q;
+
+	if (len == -1)
+		len = strlen(s);
+	for (q = v->v; q; q = q->next) {
+		if (strncmp(q->name, s, len) == 0 && q->name[len] == '\0')
+			break;
+	}
+	if (q)
+		pos = q->pos;
+	else {
+		pos = v->n;
+		v->v = variable_new(v, s, len, v->n);
+		if (!v->v)
+			return -1;
+		v->n++;
+	}
+	return pos;
+}
+
+static int vars_add_anon(struct vars *v)
+{
+	v->v = variable_new(v, "", 0, v->n);
+
+	if (!v->v)
+		return -1;
+	v->n++;
+
+	return 0;
+}
+
+/* Obtain next token, with some preprocessing.
+ * In particular, evaluate expressions of the form x^y,
+ * with x and y values.
+ */
+static struct isl_token *next_token(struct isl_stream *s)
+{
+	struct isl_token *tok, *tok2;
+
+	tok = isl_stream_next_token(s);
+	if (!tok || tok->type != ISL_TOKEN_VALUE)
+		return tok;
+	if (!isl_stream_eat_if_available(s, '^'))
+		return tok;
+	tok2 = isl_stream_next_token(s);
+	if (!tok2 || tok2->type != ISL_TOKEN_VALUE) {
+		isl_stream_error(s, tok2, "expecting constant value");
+		goto error;
+	}
+
+	isl_int_pow_ui(tok->u.v, tok->u.v, isl_int_get_ui(tok2->u.v));
+
+	isl_token_free(tok2);
+	return tok;
+error:
+	isl_token_free(tok);
+	isl_token_free(tok2);
+	return NULL;
+}
+
+/* Read an isl_val from "s".
+ *
+ * The following token sequences are recognized
+ *
+ *	"infty"		->	infty
+ *	"-" "infty"	->	-infty
+ *	"NaN"		->	NaN
+ *	n "/" d		->	n/d
+ *	v		->	v
+ *
+ * where n, d and v are integer constants.
+ */
+__isl_give isl_val *isl_stream_read_val(struct isl_stream *s)
+{
+	struct isl_token *tok = NULL;
+	struct isl_token *tok2 = NULL;
+	isl_val *val;
+
+	tok = next_token(s);
+	if (!tok) {
+		isl_stream_error(s, NULL, "unexpected EOF");
+		goto error;
+	}
+	if (tok->type == ISL_TOKEN_INFTY) {
+		isl_token_free(tok);
+		return isl_val_infty(s->ctx);
+	}
+	if (tok->type == '-' &&
+	    isl_stream_eat_if_available(s, ISL_TOKEN_INFTY)) {
+		isl_token_free(tok);
+		return isl_val_neginfty(s->ctx);
+	}
+	if (tok->type == ISL_TOKEN_NAN) {
+		isl_token_free(tok);
+		return isl_val_nan(s->ctx);
+	}
+	if (tok->type != ISL_TOKEN_VALUE) {
+		isl_stream_error(s, tok, "expecting value");
+		goto error;
+	}
+
+	if (isl_stream_eat_if_available(s, '/')) {
+		tok2 = next_token(s);
+		if (!tok2) {
+			isl_stream_error(s, NULL, "unexpected EOF");
+			goto error;
+		}
+		if (tok2->type != ISL_TOKEN_VALUE) {
+			isl_stream_error(s, tok2, "expecting value");
+			goto error;
+		}
+		val = isl_val_rat_from_isl_int(s->ctx, tok->u.v, tok2->u.v);
+		val = isl_val_normalize(val);
+	} else {
+		val = isl_val_int_from_isl_int(s->ctx, tok->u.v);
+	}
+
+	isl_token_free(tok);
+	isl_token_free(tok2);
+	return val;
+error:
+	isl_token_free(tok);
+	isl_token_free(tok2);
+	return NULL;
+}
+
+/* Read an isl_val from "str".
+ */
+struct isl_val *isl_val_read_from_str(struct isl_ctx *ctx,
+	const char *str)
+{
+	isl_val *val;
+	struct isl_stream *s = isl_stream_new_str(ctx, str);
+	if (!s)
+		return NULL;
+	val = isl_stream_read_val(s);
+	isl_stream_free(s);
+	return val;
+}
+
+static int accept_cst_factor(struct isl_stream *s, isl_int *f)
+{
+	struct isl_token *tok;
+
+	tok = next_token(s);
+	if (!tok || tok->type != ISL_TOKEN_VALUE) {
+		isl_stream_error(s, tok, "expecting constant value");
+		goto error;
+	}
+
+	isl_int_mul(*f, *f, tok->u.v);
+
+	isl_token_free(tok);
+
+	if (isl_stream_eat_if_available(s, '*'))
+		return accept_cst_factor(s, f);
+
+	return 0;
+error:
+	isl_token_free(tok);
+	return -1;
+}
+
+/* Given an affine expression aff, return an affine expression
+ * for aff % d, with d the next token on the stream, which is
+ * assumed to be a constant.
+ *
+ * We introduce an integer division q = [aff/d] and the result
+ * is set to aff - d q.
+ */
+static __isl_give isl_pw_aff *affine_mod(struct isl_stream *s,
+	struct vars *v, __isl_take isl_pw_aff *aff)
+{
+	struct isl_token *tok;
+	isl_pw_aff *q;
+
+	tok = next_token(s);
+	if (!tok || tok->type != ISL_TOKEN_VALUE) {
+		isl_stream_error(s, tok, "expecting constant value");
+		goto error;
+	}
+
+	q = isl_pw_aff_copy(aff);
+	q = isl_pw_aff_scale_down(q, tok->u.v);
+	q = isl_pw_aff_floor(q);
+	q = isl_pw_aff_scale(q, tok->u.v);
+
+	aff = isl_pw_aff_sub(aff, q);
+
+	isl_token_free(tok);
+	return aff;
+error:
+	isl_pw_aff_free(aff);
+	isl_token_free(tok);
+	return NULL;
+}
+
+static __isl_give isl_pw_aff *accept_affine(struct isl_stream *s,
+	__isl_take isl_space *space, struct vars *v);
+static __isl_give isl_pw_aff_list *accept_affine_list(struct isl_stream *s,
+	__isl_take isl_space *dim, struct vars *v);
+
+static __isl_give isl_pw_aff *accept_minmax(struct isl_stream *s,
+	__isl_take isl_space *dim, struct vars *v)
+{
+	struct isl_token *tok;
+	isl_pw_aff_list *list = NULL;
+	int min;
+
+	tok = isl_stream_next_token(s);
+	if (!tok)
+		goto error;
+	min = tok->type == ISL_TOKEN_MIN;
+	isl_token_free(tok);
+
+	if (isl_stream_eat(s, '('))
+		goto error;
+
+	list = accept_affine_list(s, isl_space_copy(dim), v);
+	if (!list)
+		goto error;
+
+	if (isl_stream_eat(s, ')'))
+		goto error;
+
+	isl_space_free(dim);
+	return min ? isl_pw_aff_list_min(list) : isl_pw_aff_list_max(list);
+error:
+	isl_space_free(dim);
+	isl_pw_aff_list_free(list);
+	return NULL;
+}
+
+/* Is "tok" the start of an integer division?
+ */
+static int is_start_of_div(struct isl_token *tok)
+{
+	if (!tok)
+		return 0;
+	if (tok->type == '[')
+		return 1;
+	if (tok->type == ISL_TOKEN_FLOOR)
+		return 1;
+	if (tok->type == ISL_TOKEN_CEIL)
+		return 1;
+	if (tok->type == ISL_TOKEN_FLOORD)
+		return 1;
+	if (tok->type == ISL_TOKEN_CEILD)
+		return 1;
+	return 0;
+}
+
+/* Read an integer division from "s" and return it as an isl_pw_aff.
+ *
+ * The integer division can be of the form
+ *
+ *	[<affine expression>]
+ *	floor(<affine expression>)
+ *	ceil(<affine expression>)
+ *	floord(<affine expression>,<denominator>)
+ *	ceild(<affine expression>,<denominator>)
+ */
+static __isl_give isl_pw_aff *accept_div(struct isl_stream *s,
+	__isl_take isl_space *dim, struct vars *v)
+{
+	struct isl_token *tok;
+	int f = 0;
+	int c = 0;
+	int extra = 0;
+	isl_pw_aff *pwaff = NULL;
+
+	if (isl_stream_eat_if_available(s, ISL_TOKEN_FLOORD))
+		extra = f = 1;
+	else if (isl_stream_eat_if_available(s, ISL_TOKEN_CEILD))
+		extra = c = 1;
+	else if (isl_stream_eat_if_available(s, ISL_TOKEN_FLOOR))
+		f = 1;
+	else if (isl_stream_eat_if_available(s, ISL_TOKEN_CEIL))
+		c = 1;
+	if (f || c) {
+		if (isl_stream_eat(s, '('))
+			goto error;
+	} else {
+		if (isl_stream_eat(s, '['))
+			goto error;
+	}
+
+	pwaff = accept_affine(s, isl_space_copy(dim), v);
+
+	if (extra) {
+		if (isl_stream_eat(s, ','))
+			goto error;
+
+		tok = next_token(s);
+		if (!tok)
+			goto error;
+		if (tok->type != ISL_TOKEN_VALUE) {
+			isl_stream_error(s, tok, "expected denominator");
+			isl_stream_push_token(s, tok);
+			goto error;
+		}
+		isl_pw_aff_scale_down(pwaff,  tok->u.v);
+		isl_token_free(tok);
+	}
+
+	if (c)
+		pwaff = isl_pw_aff_ceil(pwaff);
+	else
+		pwaff = isl_pw_aff_floor(pwaff);
+
+	if (f || c) {
+		if (isl_stream_eat(s, ')'))
+			goto error;
+	} else {
+		if (isl_stream_eat(s, ']'))
+			goto error;
+	}
+
+	isl_space_free(dim);
+	return pwaff;
+error:
+	isl_space_free(dim);
+	isl_pw_aff_free(pwaff);
+	return NULL;
+}
+
+static __isl_give isl_pw_aff *accept_affine_factor(struct isl_stream *s,
+	__isl_take isl_space *dim, struct vars *v)
+{
+	struct isl_token *tok = NULL;
+	isl_pw_aff *res = NULL;
+
+	tok = next_token(s);
+	if (!tok) {
+		isl_stream_error(s, NULL, "unexpected EOF");
+		goto error;
+	}
+
+	if (tok->type == ISL_TOKEN_AFF) {
+		res = isl_pw_aff_copy(tok->u.pwaff);
+		isl_token_free(tok);
+	} else if (tok->type == ISL_TOKEN_IDENT) {
+		int n = v->n;
+		int pos = vars_pos(v, tok->u.s, -1);
+		isl_aff *aff;
+
+		if (pos < 0)
+			goto error;
+		if (pos >= n) {
+			vars_drop(v, v->n - n);
+			isl_stream_error(s, tok, "unknown identifier");
+			goto error;
+		}
+
+		aff = isl_aff_zero_on_domain(isl_local_space_from_space(isl_space_copy(dim)));
+		if (!aff)
+			goto error;
+		isl_int_set_si(aff->v->el[2 + pos], 1);
+		res = isl_pw_aff_from_aff(aff);
+		isl_token_free(tok);
+	} else if (tok->type == ISL_TOKEN_VALUE) {
+		if (isl_stream_eat_if_available(s, '*')) {
+			res = accept_affine_factor(s, isl_space_copy(dim), v);
+			res = isl_pw_aff_scale(res, tok->u.v);
+		} else {
+			isl_local_space *ls;
+			isl_aff *aff;
+			ls = isl_local_space_from_space(isl_space_copy(dim));
+			aff = isl_aff_zero_on_domain(ls);
+			aff = isl_aff_add_constant(aff, tok->u.v);
+			res = isl_pw_aff_from_aff(aff);
+		}
+		isl_token_free(tok);
+	} else if (tok->type == '(') {
+		isl_token_free(tok);
+		tok = NULL;
+		res = accept_affine(s, isl_space_copy(dim), v);
+		if (!res)
+			goto error;
+		if (isl_stream_eat(s, ')'))
+			goto error;
+	} else if (is_start_of_div(tok)) {
+		isl_stream_push_token(s, tok);
+		tok = NULL;
+		res = accept_div(s, isl_space_copy(dim), v);
+	} else if (tok->type == ISL_TOKEN_MIN || tok->type == ISL_TOKEN_MAX) {
+		isl_stream_push_token(s, tok);
+		tok = NULL;
+		res = accept_minmax(s, isl_space_copy(dim), v);
+	} else {
+		isl_stream_error(s, tok, "expecting factor");
+		goto error;
+	}
+	if (isl_stream_eat_if_available(s, '%') ||
+	    isl_stream_eat_if_available(s, ISL_TOKEN_MOD)) {
+		isl_space_free(dim);
+		return affine_mod(s, v, res);
+	}
+	if (isl_stream_eat_if_available(s, '*')) {
+		isl_int f;
+		isl_int_init(f);
+		isl_int_set_si(f, 1);
+		if (accept_cst_factor(s, &f) < 0) {
+			isl_int_clear(f);
+			goto error2;
+		}
+		res = isl_pw_aff_scale(res, f);
+		isl_int_clear(f);
+	}
+	if (isl_stream_eat_if_available(s, '/')) {
+		isl_int f;
+		isl_int_init(f);
+		isl_int_set_si(f, 1);
+		if (accept_cst_factor(s, &f) < 0) {
+			isl_int_clear(f);
+			goto error2;
+		}
+		res = isl_pw_aff_scale_down(res, f);
+		isl_int_clear(f);
+	}
+
+	isl_space_free(dim);
+	return res;
+error:
+	isl_token_free(tok);
+error2:
+	isl_pw_aff_free(res);
+	isl_space_free(dim);
+	return NULL;
+}
+
+static __isl_give isl_pw_aff *add_cst(__isl_take isl_pw_aff *pwaff, isl_int v)
+{
+	isl_aff *aff;
+	isl_space *space;
+
+	space = isl_pw_aff_get_domain_space(pwaff);
+	aff = isl_aff_zero_on_domain(isl_local_space_from_space(space));
+	aff = isl_aff_add_constant(aff, v);
+
+	return isl_pw_aff_add(pwaff, isl_pw_aff_from_aff(aff));
+}
+
+/* Return a piecewise affine expression defined on the specified domain
+ * that represents NaN.
+ */
+static __isl_give isl_pw_aff *nan_on_domain(__isl_keep isl_space *space)
+{
+	isl_local_space *ls;
+
+	ls = isl_local_space_from_space(isl_space_copy(space));
+	return isl_pw_aff_nan_on_domain(ls);
+}
+
+static __isl_give isl_pw_aff *accept_affine(struct isl_stream *s,
+	__isl_take isl_space *space, struct vars *v)
+{
+	struct isl_token *tok = NULL;
+	isl_local_space *ls;
+	isl_pw_aff *res;
+	int sign = 1;
+
+	ls = isl_local_space_from_space(isl_space_copy(space));
+	res = isl_pw_aff_from_aff(isl_aff_zero_on_domain(ls));
+	if (!res)
+		goto error;
+
+	for (;;) {
+		tok = next_token(s);
+		if (!tok) {
+			isl_stream_error(s, NULL, "unexpected EOF");
+			goto error;
+		}
+		if (tok->type == '-') {
+			sign = -sign;
+			isl_token_free(tok);
+			continue;
+		}
+		if (tok->type == '(' || is_start_of_div(tok) ||
+		    tok->type == ISL_TOKEN_MIN || tok->type == ISL_TOKEN_MAX ||
+		    tok->type == ISL_TOKEN_IDENT ||
+		    tok->type == ISL_TOKEN_AFF) {
+			isl_pw_aff *term;
+			isl_stream_push_token(s, tok);
+			tok = NULL;
+			term = accept_affine_factor(s,
+						    isl_space_copy(space), v);
+			if (sign < 0)
+				res = isl_pw_aff_sub(res, term);
+			else
+				res = isl_pw_aff_add(res, term);
+			if (!res)
+				goto error;
+			sign = 1;
+		} else if (tok->type == ISL_TOKEN_VALUE) {
+			if (sign < 0)
+				isl_int_neg(tok->u.v, tok->u.v);
+			if (isl_stream_eat_if_available(s, '*') ||
+			    isl_stream_next_token_is(s, ISL_TOKEN_IDENT)) {
+				isl_pw_aff *term;
+				term = accept_affine_factor(s,
+						    isl_space_copy(space), v);
+				term = isl_pw_aff_scale(term, tok->u.v);
+				res = isl_pw_aff_add(res, term);
+				if (!res)
+					goto error;
+			} else {
+				res = add_cst(res, tok->u.v);
+			}
+			sign = 1;
+		} else if (tok->type == ISL_TOKEN_NAN) {
+			res = isl_pw_aff_add(res, nan_on_domain(space));
+		} else {
+			isl_stream_error(s, tok, "unexpected isl_token");
+			isl_stream_push_token(s, tok);
+			isl_pw_aff_free(res);
+			isl_space_free(space);
+			return NULL;
+		}
+		isl_token_free(tok);
+
+		tok = next_token(s);
+		if (tok && tok->type == '-') {
+			sign = -sign;
+			isl_token_free(tok);
+		} else if (tok && tok->type == '+') {
+			/* nothing */
+			isl_token_free(tok);
+		} else if (tok && tok->type == ISL_TOKEN_VALUE &&
+			   isl_int_is_neg(tok->u.v)) {
+			isl_stream_push_token(s, tok);
+		} else {
+			if (tok)
+				isl_stream_push_token(s, tok);
+			break;
+		}
+	}
+
+	isl_space_free(space);
+	return res;
+error:
+	isl_space_free(space);
+	isl_token_free(tok);
+	isl_pw_aff_free(res);
+	return NULL;
+}
+
+static int is_comparator(struct isl_token *tok)
+{
+	if (!tok)
+		return 0;
+
+	switch (tok->type) {
+	case ISL_TOKEN_LT:
+	case ISL_TOKEN_GT:
+	case ISL_TOKEN_LE:
+	case ISL_TOKEN_GE:
+	case ISL_TOKEN_NE:
+	case '=':
+		return 1;
+	default:
+		return 0;
+	}
+}
+
+static __isl_give isl_map *read_formula(struct isl_stream *s,
+	struct vars *v, __isl_take isl_map *map, int rational);
+static __isl_give isl_pw_aff *accept_extended_affine(struct isl_stream *s,
+	__isl_take isl_space *dim, struct vars *v, int rational);
+
+/* Accept a ternary operator, given the first argument.
+ */
+static __isl_give isl_pw_aff *accept_ternary(struct isl_stream *s,
+	__isl_take isl_map *cond, struct vars *v, int rational)
+{
+	isl_space *dim;
+	isl_pw_aff *pwaff1 = NULL, *pwaff2 = NULL, *pa_cond;
+
+	if (!cond)
+		return NULL;
+
+	if (isl_stream_eat(s, '?'))
+		goto error;
+
+	dim = isl_space_wrap(isl_map_get_space(cond));
+	pwaff1 = accept_extended_affine(s, dim, v, rational);
+	if (!pwaff1)
+		goto error;
+
+	if (isl_stream_eat(s, ':'))
+		goto error;
+
+	dim = isl_pw_aff_get_domain_space(pwaff1);
+	pwaff2 = accept_extended_affine(s, dim, v, rational);
+	if (!pwaff1)
+		goto error;
+
+	pa_cond = isl_set_indicator_function(isl_map_wrap(cond));
+	return isl_pw_aff_cond(pa_cond, pwaff1, pwaff2);
+error:
+	isl_map_free(cond);
+	isl_pw_aff_free(pwaff1);
+	isl_pw_aff_free(pwaff2);
+	return NULL;
+}
+
+/* Set *line and *col to those of the next token, if any.
+ */
+static void set_current_line_col(struct isl_stream *s, int *line, int *col)
+{
+	struct isl_token *tok;
+
+	tok = isl_stream_next_token(s);
+	if (!tok)
+		return;
+
+	*line = tok->line;
+	*col = tok->col;
+	isl_stream_push_token(s, tok);
+}
+
+/* Push a token encapsulating "pa" onto "s", with the given
+ * line and column.
+ */
+static int push_aff(struct isl_stream *s, int line, int col,
+	__isl_take isl_pw_aff *pa)
+{
+	struct isl_token *tok;
+
+	tok = isl_token_new(s->ctx, line, col, 0);
+	if (!tok)
+		goto error;
+	tok->type = ISL_TOKEN_AFF;
+	tok->u.pwaff = pa;
+	isl_stream_push_token(s, tok);
+
+	return 0;
+error:
+	isl_pw_aff_free(pa);
+	return -1;
+}
+
+/* Accept an affine expression that may involve ternary operators.
+ * We first read an affine expression.
+ * If it is not followed by a comparison operator, we simply return it.
+ * Otherwise, we assume the affine expression is part of the first
+ * argument of a ternary operator and try to parse that.
+ */
+static __isl_give isl_pw_aff *accept_extended_affine(struct isl_stream *s,
+	__isl_take isl_space *dim, struct vars *v, int rational)
+{
+	isl_space *space;
+	isl_map *cond;
+	isl_pw_aff *pwaff;
+	struct isl_token *tok;
+	int line = -1, col = -1;
+	int is_comp;
+
+	set_current_line_col(s, &line, &col);
+
+	pwaff = accept_affine(s, dim, v);
+	if (rational)
+		pwaff = isl_pw_aff_set_rational(pwaff);
+	if (!pwaff)
+		return NULL;
+
+	tok = isl_stream_next_token(s);
+	if (!tok)
+		return isl_pw_aff_free(pwaff);
+
+	is_comp = is_comparator(tok);
+	isl_stream_push_token(s, tok);
+	if (!is_comp)
+		return pwaff;
+
+	space = isl_pw_aff_get_domain_space(pwaff);
+	cond = isl_map_universe(isl_space_unwrap(space));
+
+	if (push_aff(s, line, col, pwaff) < 0)
+		cond = isl_map_free(cond);
+	if (!cond)
+		return NULL;
+
+	cond = read_formula(s, v, cond, rational);
+
+	return accept_ternary(s, cond, v, rational);
+}
+
+static __isl_give isl_map *read_var_def(struct isl_stream *s,
+	__isl_take isl_map *map, enum isl_dim_type type, struct vars *v,
+	int rational)
+{
+	isl_pw_aff *def;
+	int pos;
+	isl_map *def_map;
+
+	if (type == isl_dim_param)
+		pos = isl_map_dim(map, isl_dim_param);
+	else {
+		pos = isl_map_dim(map, isl_dim_in);
+		if (type == isl_dim_out)
+			pos += isl_map_dim(map, isl_dim_out);
+		type = isl_dim_in;
+	}
+	--pos;
+
+	def = accept_extended_affine(s, isl_space_wrap(isl_map_get_space(map)),
+					v, rational);
+	def_map = isl_map_from_pw_aff(def);
+	def_map = isl_map_equate(def_map, type, pos, isl_dim_out, 0);
+	def_map = isl_set_unwrap(isl_map_domain(def_map));
+
+	map = isl_map_intersect(map, def_map);
+
+	return map;
+}
+
+static __isl_give isl_pw_aff_list *accept_affine_list(struct isl_stream *s,
+	__isl_take isl_space *dim, struct vars *v)
+{
+	isl_pw_aff *pwaff;
+	isl_pw_aff_list *list;
+	struct isl_token *tok = NULL;
+
+	pwaff = accept_affine(s, isl_space_copy(dim), v);
+	list = isl_pw_aff_list_from_pw_aff(pwaff);
+	if (!list)
+		goto error;
+
+	for (;;) {
+		tok = isl_stream_next_token(s);
+		if (!tok) {
+			isl_stream_error(s, NULL, "unexpected EOF");
+			goto error;
+		}
+		if (tok->type != ',') {
+			isl_stream_push_token(s, tok);
+			break;
+		}
+		isl_token_free(tok);
+
+		pwaff = accept_affine(s, isl_space_copy(dim), v);
+		list = isl_pw_aff_list_concat(list,
+				isl_pw_aff_list_from_pw_aff(pwaff));
+		if (!list)
+			goto error;
+	}
+
+	isl_space_free(dim);
+	return list;
+error:
+	isl_space_free(dim);
+	isl_pw_aff_list_free(list);
+	return NULL;
+}
+
+static __isl_give isl_map *read_defined_var_list(struct isl_stream *s,
+	struct vars *v, __isl_take isl_map *map, int rational)
+{
+	struct isl_token *tok;
+
+	while ((tok = isl_stream_next_token(s)) != NULL) {
+		int p;
+		int n = v->n;
+
+		if (tok->type != ISL_TOKEN_IDENT)
+			break;
+
+		p = vars_pos(v, tok->u.s, -1);
+		if (p < 0)
+			goto error;
+		if (p < n) {
+			isl_stream_error(s, tok, "expecting unique identifier");
+			goto error;
+		}
+
+		map = isl_map_add_dims(map, isl_dim_out, 1);
+
+		isl_token_free(tok);
+		tok = isl_stream_next_token(s);
+		if (tok && tok->type == '=') {
+			isl_token_free(tok);
+			map = read_var_def(s, map, isl_dim_out, v, rational);
+			tok = isl_stream_next_token(s);
+		}
+
+		if (!tok || tok->type != ',')
+			break;
+
+		isl_token_free(tok);
+	}
+	if (tok)
+		isl_stream_push_token(s, tok);
+
+	return map;
+error:
+	isl_token_free(tok);
+	isl_map_free(map);
+	return NULL;
+}
+
+static int next_is_tuple(struct isl_stream *s)
+{
+	struct isl_token *tok;
+	int is_tuple;
+
+	tok = isl_stream_next_token(s);
+	if (!tok)
+		return 0;
+	if (tok->type == '[') {
+		isl_stream_push_token(s, tok);
+		return 1;
+	}
+	if (tok->type != ISL_TOKEN_IDENT && !tok->is_keyword) {
+		isl_stream_push_token(s, tok);
+		return 0;
+	}
+
+	is_tuple = isl_stream_next_token_is(s, '[');
+
+	isl_stream_push_token(s, tok);
+
+	return is_tuple;
+}
+
+/* Allocate an initial tuple with zero dimensions and an anonymous,
+ * unstructured space.
+ * A tuple is represented as an isl_multi_pw_aff.
+ * The range space is the space of the tuple.
+ * The domain space is an anonymous space
+ * with a dimension for each variable in the set of variables in "v".
+ * If a given dimension is not defined in terms of earlier dimensions in
+ * the input, then the corresponding isl_pw_aff is set equal to one time
+ * the variable corresponding to the dimension being defined.
+ */
+static __isl_give isl_multi_pw_aff *tuple_alloc(struct vars *v)
+{
+	return isl_multi_pw_aff_alloc(isl_space_alloc(v->ctx, 0, v->n, 0));
+}
+
+/* Is "pa" an expression in term of earlier dimensions?
+ * The alternative is that the dimension is defined to be equal to itself,
+ * meaning that it has a universe domain and an expression that depends
+ * on itself.  "i" is the position of the expression in a sequence
+ * of "n" expressions.  The final dimensions of "pa" correspond to
+ * these "n" expressions.
+ */
+static int pw_aff_is_expr(__isl_keep isl_pw_aff *pa, int i, int n)
+{
+	isl_aff *aff;
+
+	if (!pa)
+		return -1;
+	if (pa->n != 1)
+		return 1;
+	if (!isl_set_plain_is_universe(pa->p[0].set))
+		return 1;
+
+	aff = pa->p[0].aff;
+	if (isl_int_is_zero(aff->v->el[aff->v->size - n + i]))
+		return 1;
+	return 0;
+}
+
+/* Does the tuple contain any dimensions that are defined
+ * in terms of earlier dimensions?
+ */
+static int tuple_has_expr(__isl_keep isl_multi_pw_aff *tuple)
+{
+	int i, n;
+	int has_expr = 0;
+	isl_pw_aff *pa;
+
+	if (!tuple)
+		return -1;
+	n = isl_multi_pw_aff_dim(tuple, isl_dim_out);
+	for (i = 0; i < n; ++i) {
+		pa = isl_multi_pw_aff_get_pw_aff(tuple, i);
+		has_expr = pw_aff_is_expr(pa, i, n);
+		isl_pw_aff_free(pa);
+		if (has_expr < 0 || has_expr)
+			break;
+	}
+
+	return has_expr;
+}
+
+/* Add a dimension to the given tuple.
+ * The dimension is initially undefined, so it is encoded
+ * as one times itself.
+ */
+static __isl_give isl_multi_pw_aff *tuple_add_dim(
+	__isl_take isl_multi_pw_aff *tuple, struct vars *v)
+{
+	isl_space *space;
+	isl_aff *aff;
+	isl_pw_aff *pa;
+
+	tuple = isl_multi_pw_aff_add_dims(tuple, isl_dim_in, 1);
+	space = isl_multi_pw_aff_get_domain_space(tuple);
+	aff = isl_aff_zero_on_domain(isl_local_space_from_space(space));
+	aff = isl_aff_add_coefficient_si(aff, isl_dim_in, v->n, 1);
+	pa = isl_pw_aff_from_aff(aff);
+	tuple = isl_multi_pw_aff_flat_range_product(tuple,
+					    isl_multi_pw_aff_from_pw_aff(pa));
+
+	return tuple;
+}
+
+/* Set the name of dimension "pos" in "tuple" to "name".
+ * During printing, we add primes if the same name appears more than once
+ * to distinguish the occurrences.  Here, we remove those primes from "name"
+ * before setting the name of the dimension.
+ */
+static __isl_give isl_multi_pw_aff *tuple_set_dim_name(
+	__isl_take isl_multi_pw_aff *tuple, int pos, char *name)
+{
+	char *prime;
+
+	if (!name)
+		return tuple;
+
+	prime = strchr(name, '\'');
+	if (prime)
+		*prime = '\0';
+	tuple = isl_multi_pw_aff_set_dim_name(tuple, isl_dim_set, pos, name);
+	if (prime)
+		*prime = '\'';
+
+	return tuple;
+}
+
+/* Accept a piecewise affine expression.
+ *
+ * At the outer level, the piecewise affine expression may be of the form
+ *
+ *	aff1 : condition1; aff2 : conditions2; ...
+ *
+ * or simply
+ *
+ *	aff
+ *
+ * each of the affine expressions may in turn include ternary operators.
+ *
+ * There may be parentheses around some subexpression of "aff1"
+ * around "aff1" itself, around "aff1 : condition1" and/or
+ * around the entire piecewise affine expression.
+ * We therefore remove the opening parenthesis (if any) from the stream
+ * in case the closing parenthesis follows the colon, but if the closing
+ * parenthesis is the first thing in the stream after the parsed affine
+ * expression, we push the parsed expression onto the stream and parse
+ * again in case the parentheses enclose some subexpression of "aff1".
+ */
+static __isl_give isl_pw_aff *accept_piecewise_affine(struct isl_stream *s,
+	__isl_take isl_space *space, struct vars *v, int rational)
+{
+	isl_pw_aff *res;
+	isl_space *res_space;
+
+	res_space = isl_space_from_domain(isl_space_copy(space));
+	res_space = isl_space_add_dims(res_space, isl_dim_out, 1);
+	res = isl_pw_aff_empty(res_space);
+	do {
+		isl_pw_aff *pa;
+		int seen_paren;
+		int line = -1, col = -1;
+
+		set_current_line_col(s, &line, &col);
+		seen_paren = isl_stream_eat_if_available(s, '(');
+		if (seen_paren)
+			pa = accept_piecewise_affine(s, isl_space_copy(space),
+							v, rational);
+		else
+			pa = accept_extended_affine(s, isl_space_copy(space),
+							v, rational);
+		if (seen_paren && isl_stream_eat_if_available(s, ')')) {
+			seen_paren = 0;
+			if (push_aff(s, line, col, pa) < 0)
+				goto error;
+			pa = accept_extended_affine(s, isl_space_copy(space),
+							v, rational);
+		}
+		if (isl_stream_eat_if_available(s, ':')) {
+			isl_space *dom_space;
+			isl_set *dom;
+
+			dom_space = isl_pw_aff_get_domain_space(pa);
+			dom = isl_set_universe(dom_space);
+			dom = read_formula(s, v, dom, rational);
+			pa = isl_pw_aff_intersect_domain(pa, dom);
+		}
+
+		res = isl_pw_aff_union_add(res, pa);
+
+		if (seen_paren && isl_stream_eat(s, ')'))
+			goto error;
+	} while (isl_stream_eat_if_available(s, ';'));
+
+	isl_space_free(space);
+
+	return res;
+error:
+	isl_space_free(space);
+	return isl_pw_aff_free(res);
+}
+
+/* Read an affine expression from "s" and replace the definition
+ * of dimension "pos" in "tuple" by this expression.
+ *
+ * accept_extended_affine requires a wrapped space as input.
+ * The domain space of "tuple", on the other hand is an anonymous space,
+ * so we have to adjust the space of the isl_pw_aff before adding it
+ * to "tuple".
+ */
+static __isl_give isl_multi_pw_aff *read_tuple_var_def(struct isl_stream *s,
+	__isl_take isl_multi_pw_aff *tuple, int pos, struct vars *v,
+	int rational)
+{
+	isl_space *space;
+	isl_pw_aff *def;
+
+	space = isl_space_wrap(isl_space_alloc(s->ctx, 0, v->n, 0));
+
+	def = accept_piecewise_affine(s, space, v, rational);
+
+	space = isl_space_set_alloc(s->ctx, 0, v->n);
+	def = isl_pw_aff_reset_domain_space(def, space);
+	tuple = isl_multi_pw_aff_set_pw_aff(tuple, pos, def);
+
+	return tuple;
+}
+
+/* Read a list of variables and/or affine expressions and return the list
+ * as an isl_multi_pw_aff.
+ * The elements in the list are separated by either "," or "][".
+ * If "comma" is set then only "," is allowed.
+ */
+static __isl_give isl_multi_pw_aff *read_tuple_var_list(struct isl_stream *s,
+	struct vars *v, int rational, int comma)
+{
+	int i = 0;
+	struct isl_token *tok;
+	isl_multi_pw_aff *res;
+
+	res = tuple_alloc(v);
+
+	if (isl_stream_next_token_is(s, ']'))
+		return res;
+
+	while ((tok = next_token(s)) != NULL) {
+		int new_name = 0;
+
+		res = tuple_add_dim(res, v);
+
+		if (tok->type == ISL_TOKEN_IDENT) {
+			int n = v->n;
+			int p = vars_pos(v, tok->u.s, -1);
+			if (p < 0)
+				goto error;
+			new_name = p >= n;
+		}
+
+		if (tok->type == '*') {
+			if (vars_add_anon(v) < 0)
+				goto error;
+			isl_token_free(tok);
+		} else if (new_name) {
+			res = tuple_set_dim_name(res, i, v->v->name);
+			isl_token_free(tok);
+			if (isl_stream_eat_if_available(s, '='))
+				res = read_tuple_var_def(s, res, i, v,
+							rational);
+		} else {
+			isl_stream_push_token(s, tok);
+			tok = NULL;
+			if (vars_add_anon(v) < 0)
+				goto error;
+			res = read_tuple_var_def(s, res, i, v, rational);
+		}
+
+		tok = isl_stream_next_token(s);
+		if (!comma && tok && tok->type == ']' &&
+		    isl_stream_next_token_is(s, '[')) {
+			isl_token_free(tok);
+			tok = isl_stream_next_token(s);
+		} else if (!tok || tok->type != ',')
+			break;
+
+		isl_token_free(tok);
+		i++;
+	}
+	if (tok)
+		isl_stream_push_token(s, tok);
+
+	return res;
+error:
+	isl_token_free(tok);
+	return isl_multi_pw_aff_free(res);
+}
+
+/* Read a tuple and represent it as an isl_multi_pw_aff.  See tuple_alloc.
+ */
+static __isl_give isl_multi_pw_aff *read_tuple(struct isl_stream *s,
+	struct vars *v, int rational, int comma)
+{
+	struct isl_token *tok;
+	char *name = NULL;
+	isl_multi_pw_aff *res = NULL;
+
+	tok = isl_stream_next_token(s);
+	if (!tok)
+		goto error;
+	if (tok->type == ISL_TOKEN_IDENT || tok->is_keyword) {
+		name = strdup(tok->u.s);
+		isl_token_free(tok);
+		if (!name)
+			goto error;
+	} else
+		isl_stream_push_token(s, tok);
+	if (isl_stream_eat(s, '['))
+		goto error;
+	if (next_is_tuple(s)) {
+		isl_multi_pw_aff *out;
+		int n;
+		res = read_tuple(s, v, rational, comma);
+		if (isl_stream_eat(s, ISL_TOKEN_TO))
+			goto error;
+		out = read_tuple(s, v, rational, comma);
+		n = isl_multi_pw_aff_dim(out, isl_dim_out);
+		res = isl_multi_pw_aff_add_dims(res, isl_dim_in, n);
+		res = isl_multi_pw_aff_range_product(res, out);
+	} else
+		res = read_tuple_var_list(s, v, rational, comma);
+	if (isl_stream_eat(s, ']'))
+		goto error;
+
+	if (name) {
+		res = isl_multi_pw_aff_set_tuple_name(res, isl_dim_out, name);
+		free(name);
+	}
+
+	return res;
+error:
+	free(name);
+	return isl_multi_pw_aff_free(res);
+}
+
+/* Add the tuple represented by the isl_multi_pw_aff "tuple" to "map".
+ * We first create the appropriate space in "map" based on the range
+ * space of this isl_multi_pw_aff.  Then, we add equalities based
+ * on the affine expressions.  These live in an anonymous space,
+ * however, so we first need to reset the space to that of "map".
+ */
+static __isl_give isl_map *map_from_tuple(__isl_take isl_multi_pw_aff *tuple,
+	__isl_take isl_map *map, enum isl_dim_type type, struct vars *v,
+	int rational)
+{
+	int i, n;
+	isl_ctx *ctx;
+	isl_space *space = NULL;
+
+	if (!map || !tuple)
+		goto error;
+	ctx = isl_multi_pw_aff_get_ctx(tuple);
+	n = isl_multi_pw_aff_dim(tuple, isl_dim_out);
+	space = isl_space_range(isl_multi_pw_aff_get_space(tuple));
+	if (!space)
+		goto error;
+
+	if (type == isl_dim_param) {
+		if (isl_space_has_tuple_name(space, isl_dim_set) ||
+		    isl_space_is_wrapping(space)) {
+			isl_die(ctx, isl_error_invalid,
+				"parameter tuples cannot be named or nested",
+				goto error);
+		}
+		map = isl_map_add_dims(map, type, n);
+		for (i = 0; i < n; ++i) {
+			isl_id *id;
+			if (!isl_space_has_dim_name(space, isl_dim_set, i))
+				isl_die(ctx, isl_error_invalid,
+					"parameters must be named",
+					goto error);
+			id = isl_space_get_dim_id(space, isl_dim_set, i);
+			map = isl_map_set_dim_id(map, isl_dim_param, i, id);
+		}
+	} else if (type == isl_dim_in) {
+		isl_set *set;
+
+		set = isl_set_universe(isl_space_copy(space));
+		if (rational)
+			set = isl_set_set_rational(set);
+		set = isl_set_intersect_params(set, isl_map_params(map));
+		map = isl_map_from_domain(set);
+	} else {
+		isl_set *set;
+
+		set = isl_set_universe(isl_space_copy(space));
+		if (rational)
+			set = isl_set_set_rational(set);
+		map = isl_map_from_domain_and_range(isl_map_domain(map), set);
+	}
+
+	for (i = 0; i < n; ++i) {
+		isl_pw_aff *pa;
+		isl_space *space;
+		isl_aff *aff;
+		isl_set *set;
+		isl_map *map_i;
+
+		pa = isl_multi_pw_aff_get_pw_aff(tuple, i);
+		space = isl_pw_aff_get_domain_space(pa);
+		aff = isl_aff_zero_on_domain(isl_local_space_from_space(space));
+		aff = isl_aff_add_coefficient_si(aff,
+						isl_dim_in, v->n - n + i, -1);
+		pa = isl_pw_aff_add(pa, isl_pw_aff_from_aff(aff));
+		if (rational)
+			pa = isl_pw_aff_set_rational(pa);
+		set = isl_pw_aff_zero_set(pa);
+		map_i = isl_map_from_range(set);
+		map_i = isl_map_reset_space(map_i, isl_map_get_space(map));
+		map = isl_map_intersect(map, map_i);
+	}
+
+	isl_space_free(space);
+	isl_multi_pw_aff_free(tuple);
+	return map;
+error:
+	isl_space_free(space);
+	isl_multi_pw_aff_free(tuple);
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Read a tuple from "s" and add it to "map".
+ * The tuple is initially represented as an isl_multi_pw_aff and
+ * then added to "map".
+ */
+static __isl_give isl_map *read_map_tuple(struct isl_stream *s,
+	__isl_take isl_map *map, enum isl_dim_type type, struct vars *v,
+	int rational, int comma)
+{
+	isl_multi_pw_aff *tuple;
+
+	tuple = read_tuple(s, v, rational, comma);
+	if (!tuple)
+		return isl_map_free(map);
+
+	return map_from_tuple(tuple, map, type, v, rational);
+}
+
+static __isl_give isl_set *construct_constraints(
+	__isl_take isl_set *set, int type,
+	__isl_keep isl_pw_aff_list *left, __isl_keep isl_pw_aff_list *right,
+	int rational)
+{
+	isl_set *cond;
+
+	left = isl_pw_aff_list_copy(left);
+	right = isl_pw_aff_list_copy(right);
+	if (rational) {
+		left = isl_pw_aff_list_set_rational(left);
+		right = isl_pw_aff_list_set_rational(right);
+	}
+	if (type == ISL_TOKEN_LE)
+		cond = isl_pw_aff_list_le_set(left, right);
+	else if (type == ISL_TOKEN_GE)
+		cond = isl_pw_aff_list_ge_set(left, right);
+	else if (type == ISL_TOKEN_LT)
+		cond = isl_pw_aff_list_lt_set(left, right);
+	else if (type == ISL_TOKEN_GT)
+		cond = isl_pw_aff_list_gt_set(left, right);
+	else if (type == ISL_TOKEN_NE)
+		cond = isl_pw_aff_list_ne_set(left, right);
+	else
+		cond = isl_pw_aff_list_eq_set(left, right);
+
+	return isl_set_intersect(set, cond);
+}
+
+static __isl_give isl_map *add_constraint(struct isl_stream *s,
+	struct vars *v, __isl_take isl_map *map, int rational)
+{
+	struct isl_token *tok = NULL;
+	isl_pw_aff_list *list1 = NULL, *list2 = NULL;
+	isl_set *set;
+
+	set = isl_map_wrap(map);
+	list1 = accept_affine_list(s, isl_set_get_space(set), v);
+	if (!list1)
+		goto error;
+	tok = isl_stream_next_token(s);
+	if (!is_comparator(tok)) {
+		isl_stream_error(s, tok, "missing operator");
+		if (tok)
+			isl_stream_push_token(s, tok);
+		tok = NULL;
+		goto error;
+	}
+	for (;;) {
+		list2 = accept_affine_list(s, isl_set_get_space(set), v);
+		if (!list2)
+			goto error;
+
+		set = construct_constraints(set, tok->type, list1, list2,
+						rational);
+		isl_token_free(tok);
+		isl_pw_aff_list_free(list1);
+		list1 = list2;
+
+		tok = isl_stream_next_token(s);
+		if (!is_comparator(tok)) {
+			if (tok)
+				isl_stream_push_token(s, tok);
+			break;
+		}
+	}
+	isl_pw_aff_list_free(list1);
+
+	return isl_set_unwrap(set);
+error:
+	if (tok)
+		isl_token_free(tok);
+	isl_pw_aff_list_free(list1);
+	isl_pw_aff_list_free(list2);
+	isl_set_free(set);
+	return NULL;
+}
+
+static __isl_give isl_map *read_exists(struct isl_stream *s,
+	struct vars *v, __isl_take isl_map *map, int rational)
+{
+	int n = v->n;
+	int seen_paren = isl_stream_eat_if_available(s, '(');
+
+	map = isl_map_from_domain(isl_map_wrap(map));
+	map = read_defined_var_list(s, v, map, rational);
+
+	if (isl_stream_eat(s, ':'))
+		goto error;
+
+	map = read_formula(s, v, map, rational);
+	map = isl_set_unwrap(isl_map_domain(map));
+
+	vars_drop(v, v->n - n);
+	if (seen_paren && isl_stream_eat(s, ')'))
+		goto error;
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Parse an expression between parentheses and push the result
+ * back on the stream.
+ *
+ * The parsed expression may be either an affine expression
+ * or a condition.  The first type is pushed onto the stream
+ * as an isl_pw_aff, while the second is pushed as an isl_map.
+ *
+ * If the initial token indicates the start of a condition,
+ * we parse it as such.
+ * Otherwise, we first parse an affine expression and push
+ * that onto the stream.  If the affine expression covers the
+ * entire expression between parentheses, we return.
+ * Otherwise, we assume that the affine expression is the
+ * start of a condition and continue parsing.
+ */
+static int resolve_paren_expr(struct isl_stream *s,
+	struct vars *v, __isl_take isl_map *map, int rational)
+{
+	struct isl_token *tok, *tok2;
+	int line, col;
+	isl_pw_aff *pwaff;
+
+	tok = isl_stream_next_token(s);
+	if (!tok || tok->type != '(')
+		goto error;
+
+	if (isl_stream_next_token_is(s, '('))
+		if (resolve_paren_expr(s, v, isl_map_copy(map), rational))
+			goto error;
+
+	if (isl_stream_next_token_is(s, ISL_TOKEN_EXISTS) ||
+	    isl_stream_next_token_is(s, ISL_TOKEN_NOT) ||
+	    isl_stream_next_token_is(s, ISL_TOKEN_TRUE) ||
+	    isl_stream_next_token_is(s, ISL_TOKEN_FALSE) ||
+	    isl_stream_next_token_is(s, ISL_TOKEN_MAP)) {
+		map = read_formula(s, v, map, rational);
+		if (isl_stream_eat(s, ')'))
+			goto error;
+		tok->type = ISL_TOKEN_MAP;
+		tok->u.map = map;
+		isl_stream_push_token(s, tok);
+		return 0;
+	}
+
+	tok2 = isl_stream_next_token(s);
+	if (!tok2)
+		goto error;
+	line = tok2->line;
+	col = tok2->col;
+	isl_stream_push_token(s, tok2);
+
+	pwaff = accept_affine(s, isl_space_wrap(isl_map_get_space(map)), v);
+	if (!pwaff)
+		goto error;
+
+	tok2 = isl_token_new(s->ctx, line, col, 0);
+	if (!tok2)
+		goto error2;
+	tok2->type = ISL_TOKEN_AFF;
+	tok2->u.pwaff = pwaff;
+
+	if (isl_stream_eat_if_available(s, ')')) {
+		isl_stream_push_token(s, tok2);
+		isl_token_free(tok);
+		isl_map_free(map);
+		return 0;
+	}
+
+	isl_stream_push_token(s, tok2);
+
+	map = read_formula(s, v, map, rational);
+	if (isl_stream_eat(s, ')'))
+		goto error;
+
+	tok->type = ISL_TOKEN_MAP;
+	tok->u.map = map;
+	isl_stream_push_token(s, tok);
+
+	return 0;
+error2:
+	isl_pw_aff_free(pwaff);
+error:
+	isl_token_free(tok);
+	isl_map_free(map);
+	return -1;
+}
+
+static __isl_give isl_map *read_conjunct(struct isl_stream *s,
+	struct vars *v, __isl_take isl_map *map, int rational)
+{
+	if (isl_stream_next_token_is(s, '('))
+		if (resolve_paren_expr(s, v, isl_map_copy(map), rational))
+			goto error;
+
+	if (isl_stream_next_token_is(s, ISL_TOKEN_MAP)) {
+		struct isl_token *tok;
+		tok = isl_stream_next_token(s);
+		if (!tok)
+			goto error;
+		isl_map_free(map);
+		map = isl_map_copy(tok->u.map);
+		isl_token_free(tok);
+		return map;
+	}
+
+	if (isl_stream_eat_if_available(s, ISL_TOKEN_EXISTS))
+		return read_exists(s, v, map, rational);
+
+	if (isl_stream_eat_if_available(s, ISL_TOKEN_TRUE))
+		return map;
+
+	if (isl_stream_eat_if_available(s, ISL_TOKEN_FALSE)) {
+		isl_space *dim = isl_map_get_space(map);
+		isl_map_free(map);
+		return isl_map_empty(dim);
+	}
+		
+	return add_constraint(s, v, map, rational);
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+static __isl_give isl_map *read_conjuncts(struct isl_stream *s,
+	struct vars *v, __isl_take isl_map *map, int rational)
+{
+	isl_map *res;
+	int negate;
+
+	negate = isl_stream_eat_if_available(s, ISL_TOKEN_NOT);
+	res = read_conjunct(s, v, isl_map_copy(map), rational);
+	if (negate)
+		res = isl_map_subtract(isl_map_copy(map), res);
+
+	while (res && isl_stream_eat_if_available(s, ISL_TOKEN_AND)) {
+		isl_map *res_i;
+
+		negate = isl_stream_eat_if_available(s, ISL_TOKEN_NOT);
+		res_i = read_conjunct(s, v, isl_map_copy(map), rational);
+		if (negate)
+			res = isl_map_subtract(res, res_i);
+		else
+			res = isl_map_intersect(res, res_i);
+	}
+
+	isl_map_free(map);
+	return res;
+}
+
+static struct isl_map *read_disjuncts(struct isl_stream *s,
+	struct vars *v, __isl_take isl_map *map, int rational)
+{
+	isl_map *res;
+
+	if (isl_stream_next_token_is(s, '}')) {
+		isl_space *dim = isl_map_get_space(map);
+		isl_map_free(map);
+		return isl_map_universe(dim);
+	}
+
+	res = read_conjuncts(s, v, isl_map_copy(map), rational);
+	while (isl_stream_eat_if_available(s, ISL_TOKEN_OR)) {
+		isl_map *res_i;
+
+		res_i = read_conjuncts(s, v, isl_map_copy(map), rational);
+		res = isl_map_union(res, res_i);
+	}
+
+	isl_map_free(map);
+	return res;
+}
+
+/* Read a first order formula from "s", add the corresponding
+ * constraints to "map" and return the result.
+ *
+ * In particular, read a formula of the form
+ *
+ *	a
+ *
+ * or
+ *
+ *	a implies b
+ *
+ * where a and b are disjunctions.
+ *
+ * In the first case, map is replaced by
+ *
+ *	map \cap { [..] : a }
+ *
+ * In the second case, it is replaced by
+ *
+ *	(map \setminus { [..] : a}) \cup (map \cap { [..] : b })
+ */
+static __isl_give isl_map *read_formula(struct isl_stream *s,
+	struct vars *v, __isl_take isl_map *map, int rational)
+{
+	isl_map *res;
+
+	res = read_disjuncts(s, v, isl_map_copy(map), rational);
+
+	if (isl_stream_eat_if_available(s, ISL_TOKEN_IMPLIES)) {
+		isl_map *res2;
+
+		res = isl_map_subtract(isl_map_copy(map), res);
+		res2 = read_disjuncts(s, v, map, rational);
+		res = isl_map_union(res, res2);
+	} else
+		isl_map_free(map);
+
+	return res;
+}
+
+static int polylib_pos_to_isl_pos(__isl_keep isl_basic_map *bmap, int pos)
+{
+	if (pos < isl_basic_map_dim(bmap, isl_dim_out))
+		return 1 + isl_basic_map_dim(bmap, isl_dim_param) +
+			   isl_basic_map_dim(bmap, isl_dim_in) + pos;
+	pos -= isl_basic_map_dim(bmap, isl_dim_out);
+
+	if (pos < isl_basic_map_dim(bmap, isl_dim_in))
+		return 1 + isl_basic_map_dim(bmap, isl_dim_param) + pos;
+	pos -= isl_basic_map_dim(bmap, isl_dim_in);
+
+	if (pos < isl_basic_map_dim(bmap, isl_dim_div))
+		return 1 + isl_basic_map_dim(bmap, isl_dim_param) +
+			   isl_basic_map_dim(bmap, isl_dim_in) +
+			   isl_basic_map_dim(bmap, isl_dim_out) + pos;
+	pos -= isl_basic_map_dim(bmap, isl_dim_div);
+
+	if (pos < isl_basic_map_dim(bmap, isl_dim_param))
+		return 1 + pos;
+
+	return 0;
+}
+
+static __isl_give isl_basic_map *basic_map_read_polylib_constraint(
+	struct isl_stream *s, __isl_take isl_basic_map *bmap)
+{
+	int j;
+	struct isl_token *tok;
+	int type;
+	int k;
+	isl_int *c;
+	unsigned nparam;
+	unsigned dim;
+
+	if (!bmap)
+		return NULL;
+
+	nparam = isl_basic_map_dim(bmap, isl_dim_param);
+	dim = isl_basic_map_dim(bmap, isl_dim_out);
+
+	tok = isl_stream_next_token(s);
+	if (!tok || tok->type != ISL_TOKEN_VALUE) {
+		isl_stream_error(s, tok, "expecting coefficient");
+		if (tok)
+			isl_stream_push_token(s, tok);
+		goto error;
+	}
+	if (!tok->on_new_line) {
+		isl_stream_error(s, tok, "coefficient should appear on new line");
+		isl_stream_push_token(s, tok);
+		goto error;
+	}
+
+	type = isl_int_get_si(tok->u.v);
+	isl_token_free(tok);
+
+	isl_assert(s->ctx, type == 0 || type == 1, goto error);
+	if (type == 0) {
+		k = isl_basic_map_alloc_equality(bmap);
+		c = bmap->eq[k];
+	} else {
+		k = isl_basic_map_alloc_inequality(bmap);
+		c = bmap->ineq[k];
+	}
+	if (k < 0)
+		goto error;
+
+	for (j = 0; j < 1 + isl_basic_map_total_dim(bmap); ++j) {
+		int pos;
+		tok = isl_stream_next_token(s);
+		if (!tok || tok->type != ISL_TOKEN_VALUE) {
+			isl_stream_error(s, tok, "expecting coefficient");
+			if (tok)
+				isl_stream_push_token(s, tok);
+			goto error;
+		}
+		if (tok->on_new_line) {
+			isl_stream_error(s, tok,
+				"coefficient should not appear on new line");
+			isl_stream_push_token(s, tok);
+			goto error;
+		}
+		pos = polylib_pos_to_isl_pos(bmap, j);
+		isl_int_set(c[pos], tok->u.v);
+		isl_token_free(tok);
+	}
+
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+static __isl_give isl_basic_map *basic_map_read_polylib(struct isl_stream *s)
+{
+	int i;
+	struct isl_token *tok;
+	struct isl_token *tok2;
+	int n_row, n_col;
+	int on_new_line;
+	unsigned in = 0, out, local = 0;
+	struct isl_basic_map *bmap = NULL;
+	int nparam = 0;
+
+	tok = isl_stream_next_token(s);
+	if (!tok) {
+		isl_stream_error(s, NULL, "unexpected EOF");
+		return NULL;
+	}
+	tok2 = isl_stream_next_token(s);
+	if (!tok2) {
+		isl_token_free(tok);
+		isl_stream_error(s, NULL, "unexpected EOF");
+		return NULL;
+	}
+	if (tok->type != ISL_TOKEN_VALUE || tok2->type != ISL_TOKEN_VALUE) {
+		isl_stream_push_token(s, tok2);
+		isl_stream_push_token(s, tok);
+		isl_stream_error(s, NULL,
+				 "expecting constraint matrix dimensions");
+		return NULL;
+	}
+	n_row = isl_int_get_si(tok->u.v);
+	n_col = isl_int_get_si(tok2->u.v);
+	on_new_line = tok2->on_new_line;
+	isl_token_free(tok2);
+	isl_token_free(tok);
+	isl_assert(s->ctx, !on_new_line, return NULL);
+	isl_assert(s->ctx, n_row >= 0, return NULL);
+	isl_assert(s->ctx, n_col >= 2 + nparam, return NULL);
+	tok = isl_stream_next_token_on_same_line(s);
+	if (tok) {
+		if (tok->type != ISL_TOKEN_VALUE) {
+			isl_stream_error(s, tok,
+				    "expecting number of output dimensions");
+			isl_stream_push_token(s, tok);
+			goto error;
+		}
+		out = isl_int_get_si(tok->u.v);
+		isl_token_free(tok);
+
+		tok = isl_stream_next_token_on_same_line(s);
+		if (!tok || tok->type != ISL_TOKEN_VALUE) {
+			isl_stream_error(s, tok,
+				    "expecting number of input dimensions");
+			if (tok)
+				isl_stream_push_token(s, tok);
+			goto error;
+		}
+		in = isl_int_get_si(tok->u.v);
+		isl_token_free(tok);
+
+		tok = isl_stream_next_token_on_same_line(s);
+		if (!tok || tok->type != ISL_TOKEN_VALUE) {
+			isl_stream_error(s, tok,
+				    "expecting number of existentials");
+			if (tok)
+				isl_stream_push_token(s, tok);
+			goto error;
+		}
+		local = isl_int_get_si(tok->u.v);
+		isl_token_free(tok);
+
+		tok = isl_stream_next_token_on_same_line(s);
+		if (!tok || tok->type != ISL_TOKEN_VALUE) {
+			isl_stream_error(s, tok,
+				    "expecting number of parameters");
+			if (tok)
+				isl_stream_push_token(s, tok);
+			goto error;
+		}
+		nparam = isl_int_get_si(tok->u.v);
+		isl_token_free(tok);
+		if (n_col != 1 + out + in + local + nparam + 1) {
+			isl_stream_error(s, NULL,
+				    "dimensions don't match");
+			goto error;
+		}
+	} else
+		out = n_col - 2 - nparam;
+	bmap = isl_basic_map_alloc(s->ctx, nparam, in, out, local, n_row, n_row);
+	if (!bmap)
+		return NULL;
+
+	for (i = 0; i < local; ++i) {
+		int k = isl_basic_map_alloc_div(bmap);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(bmap->div[k], 1 + 1 + nparam + in + out + local);
+	}
+
+	for (i = 0; i < n_row; ++i)
+		bmap = basic_map_read_polylib_constraint(s, bmap);
+
+	tok = isl_stream_next_token_on_same_line(s);
+	if (tok) {
+		isl_stream_error(s, tok, "unexpected extra token on line");
+		isl_stream_push_token(s, tok);
+		goto error;
+	}
+
+	bmap = isl_basic_map_simplify(bmap);
+	bmap = isl_basic_map_finalize(bmap);
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+static struct isl_map *map_read_polylib(struct isl_stream *s)
+{
+	struct isl_token *tok;
+	struct isl_token *tok2;
+	int i, n;
+	struct isl_map *map;
+
+	tok = isl_stream_next_token(s);
+	if (!tok) {
+		isl_stream_error(s, NULL, "unexpected EOF");
+		return NULL;
+	}
+	tok2 = isl_stream_next_token_on_same_line(s);
+	if (tok2 && tok2->type == ISL_TOKEN_VALUE) {
+		isl_stream_push_token(s, tok2);
+		isl_stream_push_token(s, tok);
+		return isl_map_from_basic_map(basic_map_read_polylib(s));
+	}
+	if (tok2) {
+		isl_stream_error(s, tok2, "unexpected token");
+		isl_stream_push_token(s, tok2);
+		isl_stream_push_token(s, tok);
+		return NULL;
+	}
+	n = isl_int_get_si(tok->u.v);
+	isl_token_free(tok);
+
+	isl_assert(s->ctx, n >= 1, return NULL);
+
+	map = isl_map_from_basic_map(basic_map_read_polylib(s));
+
+	for (i = 1; map && i < n; ++i)
+		map = isl_map_union(map,
+			isl_map_from_basic_map(basic_map_read_polylib(s)));
+
+	return map;
+}
+
+static int optional_power(struct isl_stream *s)
+{
+	int pow;
+	struct isl_token *tok;
+
+	tok = isl_stream_next_token(s);
+	if (!tok)
+		return 1;
+	if (tok->type != '^') {
+		isl_stream_push_token(s, tok);
+		return 1;
+	}
+	isl_token_free(tok);
+	tok = isl_stream_next_token(s);
+	if (!tok || tok->type != ISL_TOKEN_VALUE) {
+		isl_stream_error(s, tok, "expecting exponent");
+		if (tok)
+			isl_stream_push_token(s, tok);
+		return 1;
+	}
+	pow = isl_int_get_si(tok->u.v);
+	isl_token_free(tok);
+	return pow;
+}
+
+static __isl_give isl_pw_qpolynomial *read_term(struct isl_stream *s,
+	__isl_keep isl_map *map, struct vars *v);
+
+static __isl_give isl_pw_qpolynomial *read_factor(struct isl_stream *s,
+	__isl_keep isl_map *map, struct vars *v)
+{
+	isl_pw_qpolynomial *pwqp;
+	struct isl_token *tok;
+
+	tok = next_token(s);
+	if (!tok) {
+		isl_stream_error(s, NULL, "unexpected EOF");
+		return NULL;
+	}
+	if (tok->type == '(') {
+		int pow;
+
+		isl_token_free(tok);
+		pwqp = read_term(s, map, v);
+		if (!pwqp)
+			return NULL;
+		if (isl_stream_eat(s, ')'))
+			goto error;
+		pow = optional_power(s);
+		pwqp = isl_pw_qpolynomial_pow(pwqp, pow);
+	} else if (tok->type == ISL_TOKEN_VALUE) {
+		struct isl_token *tok2;
+		isl_qpolynomial *qp;
+
+		tok2 = isl_stream_next_token(s);
+		if (tok2 && tok2->type == '/') {
+			isl_token_free(tok2);
+			tok2 = next_token(s);
+			if (!tok2 || tok2->type != ISL_TOKEN_VALUE) {
+				isl_stream_error(s, tok2, "expected denominator");
+				isl_token_free(tok);
+				isl_token_free(tok2);
+				return NULL;
+			}
+			qp = isl_qpolynomial_rat_cst_on_domain(isl_map_get_space(map),
+						    tok->u.v, tok2->u.v);
+			isl_token_free(tok2);
+		} else {
+			isl_stream_push_token(s, tok2);
+			qp = isl_qpolynomial_cst_on_domain(isl_map_get_space(map),
+						tok->u.v);
+		}
+		isl_token_free(tok);
+		pwqp = isl_pw_qpolynomial_from_qpolynomial(qp);
+	} else if (tok->type == ISL_TOKEN_INFTY) {
+		isl_qpolynomial *qp;
+		isl_token_free(tok);
+		qp = isl_qpolynomial_infty_on_domain(isl_map_get_space(map));
+		pwqp = isl_pw_qpolynomial_from_qpolynomial(qp);
+	} else if (tok->type == ISL_TOKEN_NAN) {
+		isl_qpolynomial *qp;
+		isl_token_free(tok);
+		qp = isl_qpolynomial_nan_on_domain(isl_map_get_space(map));
+		pwqp = isl_pw_qpolynomial_from_qpolynomial(qp);
+	} else if (tok->type == ISL_TOKEN_IDENT) {
+		int n = v->n;
+		int pos = vars_pos(v, tok->u.s, -1);
+		int pow;
+		isl_qpolynomial *qp;
+		if (pos < 0) {
+			isl_token_free(tok);
+			return NULL;
+		}
+		if (pos >= n) {
+			vars_drop(v, v->n - n);
+			isl_stream_error(s, tok, "unknown identifier");
+			isl_token_free(tok);
+			return NULL;
+		}
+		isl_token_free(tok);
+		pow = optional_power(s);
+		qp = isl_qpolynomial_var_pow_on_domain(isl_map_get_space(map), pos, pow);
+		pwqp = isl_pw_qpolynomial_from_qpolynomial(qp);
+	} else if (is_start_of_div(tok)) {
+		isl_pw_aff *pwaff;
+		int pow;
+
+		isl_stream_push_token(s, tok);
+		pwaff = accept_div(s, isl_map_get_space(map), v);
+		pow = optional_power(s);
+		pwqp = isl_pw_qpolynomial_from_pw_aff(pwaff);
+		pwqp = isl_pw_qpolynomial_pow(pwqp, pow);
+	} else if (tok->type == '-') {
+		isl_token_free(tok);
+		pwqp = read_factor(s, map, v);
+		pwqp = isl_pw_qpolynomial_neg(pwqp);
+	} else {
+		isl_stream_error(s, tok, "unexpected isl_token");
+		isl_stream_push_token(s, tok);
+		return NULL;
+	}
+
+	if (isl_stream_eat_if_available(s, '*') ||
+	    isl_stream_next_token_is(s, ISL_TOKEN_IDENT)) {
+		isl_pw_qpolynomial *pwqp2;
+
+		pwqp2 = read_factor(s, map, v);
+		pwqp = isl_pw_qpolynomial_mul(pwqp, pwqp2);
+	}
+
+	return pwqp;
+error:
+	isl_pw_qpolynomial_free(pwqp);
+	return NULL;
+}
+
+static __isl_give isl_pw_qpolynomial *read_term(struct isl_stream *s,
+	__isl_keep isl_map *map, struct vars *v)
+{
+	struct isl_token *tok;
+	isl_pw_qpolynomial *pwqp;
+
+	pwqp = read_factor(s, map, v);
+
+	for (;;) {
+		tok = next_token(s);
+		if (!tok)
+			return pwqp;
+
+		if (tok->type == '+') {
+			isl_pw_qpolynomial *pwqp2;
+
+			isl_token_free(tok);
+			pwqp2 = read_factor(s, map, v);
+			pwqp = isl_pw_qpolynomial_add(pwqp, pwqp2);
+		} else if (tok->type == '-') {
+			isl_pw_qpolynomial *pwqp2;
+
+			isl_token_free(tok);
+			pwqp2 = read_factor(s, map, v);
+			pwqp = isl_pw_qpolynomial_sub(pwqp, pwqp2);
+		} else if (tok->type == ISL_TOKEN_VALUE &&
+			    isl_int_is_neg(tok->u.v)) {
+			isl_pw_qpolynomial *pwqp2;
+
+			isl_stream_push_token(s, tok);
+			pwqp2 = read_factor(s, map, v);
+			pwqp = isl_pw_qpolynomial_add(pwqp, pwqp2);
+		} else {
+			isl_stream_push_token(s, tok);
+			break;
+		}
+	}
+
+	return pwqp;
+}
+
+static __isl_give isl_map *read_optional_formula(struct isl_stream *s,
+	__isl_take isl_map *map, struct vars *v, int rational)
+{
+	struct isl_token *tok;
+
+	tok = isl_stream_next_token(s);
+	if (!tok) {
+		isl_stream_error(s, NULL, "unexpected EOF");
+		goto error;
+	}
+	if (tok->type == ':' ||
+	    (tok->type == ISL_TOKEN_OR && !strcmp(tok->u.s, "|"))) {
+		isl_token_free(tok);
+		map = read_formula(s, v, map, rational);
+	} else
+		isl_stream_push_token(s, tok);
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+static struct isl_obj obj_read_poly(struct isl_stream *s,
+	__isl_take isl_map *map, struct vars *v, int n)
+{
+	struct isl_obj obj = { isl_obj_pw_qpolynomial, NULL };
+	isl_pw_qpolynomial *pwqp;
+	struct isl_set *set;
+
+	pwqp = read_term(s, map, v);
+	map = read_optional_formula(s, map, v, 0);
+	set = isl_map_range(map);
+
+	pwqp = isl_pw_qpolynomial_intersect_domain(pwqp, set);
+
+	vars_drop(v, v->n - n);
+
+	obj.v = pwqp;
+	return obj;
+}
+
+static struct isl_obj obj_read_poly_or_fold(struct isl_stream *s,
+	__isl_take isl_set *set, struct vars *v, int n)
+{
+	struct isl_obj obj = { isl_obj_pw_qpolynomial_fold, NULL };
+	isl_pw_qpolynomial *pwqp;
+	isl_pw_qpolynomial_fold *pwf = NULL;
+
+	if (!isl_stream_eat_if_available(s, ISL_TOKEN_MAX))
+		return obj_read_poly(s, set, v, n);
+
+	if (isl_stream_eat(s, '('))
+		goto error;
+
+	pwqp = read_term(s, set, v);
+	pwf = isl_pw_qpolynomial_fold_from_pw_qpolynomial(isl_fold_max, pwqp);
+
+	while (isl_stream_eat_if_available(s, ',')) {
+		isl_pw_qpolynomial_fold *pwf_i;
+		pwqp = read_term(s, set, v);
+		pwf_i = isl_pw_qpolynomial_fold_from_pw_qpolynomial(isl_fold_max,
+									pwqp);
+		pwf = isl_pw_qpolynomial_fold_fold(pwf, pwf_i);
+	}
+
+	if (isl_stream_eat(s, ')'))
+		goto error;
+
+	set = read_optional_formula(s, set, v, 0);
+	pwf = isl_pw_qpolynomial_fold_intersect_domain(pwf, set);
+
+	vars_drop(v, v->n - n);
+
+	obj.v = pwf;
+	return obj;
+error:
+	isl_set_free(set);
+	isl_pw_qpolynomial_fold_free(pwf);
+	obj.type = isl_obj_none;
+	return obj;
+}
+
+static int is_rational(struct isl_stream *s)
+{
+	struct isl_token *tok;
+
+	tok = isl_stream_next_token(s);
+	if (!tok)
+		return 0;
+	if (tok->type == ISL_TOKEN_RAT && isl_stream_next_token_is(s, ':')) {
+		isl_token_free(tok);
+		isl_stream_eat(s, ':');
+		return 1;
+	}
+
+	isl_stream_push_token(s, tok);
+
+	return 0;
+}
+
+static struct isl_obj obj_read_body(struct isl_stream *s,
+	__isl_take isl_map *map, struct vars *v)
+{
+	struct isl_token *tok;
+	struct isl_obj obj = { isl_obj_set, NULL };
+	int n = v->n;
+	int rational;
+
+	rational = is_rational(s);
+	if (rational)
+		map = isl_map_set_rational(map);
+
+	if (isl_stream_next_token_is(s, ':')) {
+		obj.type = isl_obj_set;
+		obj.v = read_optional_formula(s, map, v, rational);
+		return obj;
+	}
+
+	if (!next_is_tuple(s))
+		return obj_read_poly_or_fold(s, map, v, n);
+
+	map = read_map_tuple(s, map, isl_dim_in, v, rational, 0);
+	if (!map)
+		goto error;
+	tok = isl_stream_next_token(s);
+	if (!tok)
+		goto error;
+	if (tok->type == ISL_TOKEN_TO) {
+		obj.type = isl_obj_map;
+		isl_token_free(tok);
+		if (!next_is_tuple(s)) {
+			isl_set *set = isl_map_domain(map);
+			return obj_read_poly_or_fold(s, set, v, n);
+		}
+		map = read_map_tuple(s, map, isl_dim_out, v, rational, 0);
+		if (!map)
+			goto error;
+	} else {
+		map = isl_map_domain(map);
+		isl_stream_push_token(s, tok);
+	}
+
+	map = read_optional_formula(s, map, v, rational);
+
+	vars_drop(v, v->n - n);
+
+	obj.v = map;
+	return obj;
+error:
+	isl_map_free(map);
+	obj.type = isl_obj_none;
+	return obj;
+}
+
+static struct isl_obj to_union(isl_ctx *ctx, struct isl_obj obj)
+{
+	if (obj.type == isl_obj_map) {
+		obj.v = isl_union_map_from_map(obj.v);
+		obj.type = isl_obj_union_map;
+	} else if (obj.type == isl_obj_set) {
+		obj.v = isl_union_set_from_set(obj.v);
+		obj.type = isl_obj_union_set;
+	} else if (obj.type == isl_obj_pw_qpolynomial) {
+		obj.v = isl_union_pw_qpolynomial_from_pw_qpolynomial(obj.v);
+		obj.type = isl_obj_union_pw_qpolynomial;
+	} else if (obj.type == isl_obj_pw_qpolynomial_fold) {
+		obj.v = isl_union_pw_qpolynomial_fold_from_pw_qpolynomial_fold(obj.v);
+		obj.type = isl_obj_union_pw_qpolynomial_fold;
+	} else
+		isl_assert(ctx, 0, goto error);
+	return obj;
+error:
+	obj.type->free(obj.v);
+	obj.type = isl_obj_none;
+	return obj;
+}
+
+static struct isl_obj obj_add(struct isl_ctx *ctx,
+	struct isl_obj obj1, struct isl_obj obj2)
+{
+	if (obj1.type == isl_obj_set && obj2.type == isl_obj_union_set)
+		obj1 = to_union(ctx, obj1);
+	if (obj1.type == isl_obj_union_set && obj2.type == isl_obj_set)
+		obj2 = to_union(ctx, obj2);
+	if (obj1.type == isl_obj_map && obj2.type == isl_obj_union_map)
+		obj1 = to_union(ctx, obj1);
+	if (obj1.type == isl_obj_union_map && obj2.type == isl_obj_map)
+		obj2 = to_union(ctx, obj2);
+	if (obj1.type == isl_obj_pw_qpolynomial &&
+	    obj2.type == isl_obj_union_pw_qpolynomial)
+		obj1 = to_union(ctx, obj1);
+	if (obj1.type == isl_obj_union_pw_qpolynomial &&
+	    obj2.type == isl_obj_pw_qpolynomial)
+		obj2 = to_union(ctx, obj2);
+	if (obj1.type == isl_obj_pw_qpolynomial_fold &&
+	    obj2.type == isl_obj_union_pw_qpolynomial_fold)
+		obj1 = to_union(ctx, obj1);
+	if (obj1.type == isl_obj_union_pw_qpolynomial_fold &&
+	    obj2.type == isl_obj_pw_qpolynomial_fold)
+		obj2 = to_union(ctx, obj2);
+	isl_assert(ctx, obj1.type == obj2.type, goto error);
+	if (obj1.type == isl_obj_map && !isl_map_has_equal_space(obj1.v, obj2.v)) {
+		obj1 = to_union(ctx, obj1);
+		obj2 = to_union(ctx, obj2);
+	}
+	if (obj1.type == isl_obj_set && !isl_set_has_equal_space(obj1.v, obj2.v)) {
+		obj1 = to_union(ctx, obj1);
+		obj2 = to_union(ctx, obj2);
+	}
+	if (obj1.type == isl_obj_pw_qpolynomial &&
+	    !isl_pw_qpolynomial_has_equal_space(obj1.v, obj2.v)) {
+		obj1 = to_union(ctx, obj1);
+		obj2 = to_union(ctx, obj2);
+	}
+	if (obj1.type == isl_obj_pw_qpolynomial_fold &&
+	    !isl_pw_qpolynomial_fold_has_equal_space(obj1.v, obj2.v)) {
+		obj1 = to_union(ctx, obj1);
+		obj2 = to_union(ctx, obj2);
+	}
+	obj1.v = obj1.type->add(obj1.v, obj2.v);
+	return obj1;
+error:
+	obj1.type->free(obj1.v);
+	obj2.type->free(obj2.v);
+	obj1.type = isl_obj_none;
+	obj1.v = NULL;
+	return obj1;
+}
+
+static struct isl_obj obj_read(struct isl_stream *s)
+{
+	isl_map *map = NULL;
+	struct isl_token *tok;
+	struct vars *v = NULL;
+	struct isl_obj obj = { isl_obj_set, NULL };
+
+	tok = next_token(s);
+	if (!tok) {
+		isl_stream_error(s, NULL, "unexpected EOF");
+		goto error;
+	}
+	if (tok->type == ISL_TOKEN_VALUE) {
+		struct isl_token *tok2;
+		struct isl_map *map;
+
+		tok2 = isl_stream_next_token(s);
+		if (!tok2 || tok2->type != ISL_TOKEN_VALUE ||
+		    isl_int_is_neg(tok2->u.v)) {
+			if (tok2)
+				isl_stream_push_token(s, tok2);
+			obj.type = isl_obj_val;
+			obj.v = isl_val_int_from_isl_int(s->ctx, tok->u.v);
+			isl_token_free(tok);
+			return obj;
+		}
+		isl_stream_push_token(s, tok2);
+		isl_stream_push_token(s, tok);
+		map = map_read_polylib(s);
+		if (!map)
+			goto error;
+		if (isl_map_may_be_set(map))
+			obj.v = isl_map_range(map);
+		else {
+			obj.type = isl_obj_map;
+			obj.v = map;
+		}
+		return obj;
+	}
+	v = vars_new(s->ctx);
+	if (!v) {
+		isl_stream_push_token(s, tok);
+		goto error;
+	}
+	map = isl_map_universe(isl_space_params_alloc(s->ctx, 0));
+	if (tok->type == '[') {
+		isl_stream_push_token(s, tok);
+		map = read_map_tuple(s, map, isl_dim_param, v, 0, 0);
+		if (!map)
+			goto error;
+		tok = isl_stream_next_token(s);
+		if (!tok || tok->type != ISL_TOKEN_TO) {
+			isl_stream_error(s, tok, "expecting '->'");
+			if (tok)
+				isl_stream_push_token(s, tok);
+			goto error;
+		}
+		isl_token_free(tok);
+		tok = isl_stream_next_token(s);
+	}
+	if (!tok || tok->type != '{') {
+		isl_stream_error(s, tok, "expecting '{'");
+		if (tok)
+			isl_stream_push_token(s, tok);
+		goto error;
+	}
+	isl_token_free(tok);
+
+	tok = isl_stream_next_token(s);
+	if (!tok)
+		;
+	else if (tok->type == ISL_TOKEN_IDENT && !strcmp(tok->u.s, "Sym")) {
+		isl_token_free(tok);
+		if (isl_stream_eat(s, '='))
+			goto error;
+		map = read_map_tuple(s, map, isl_dim_param, v, 0, 1);
+		if (!map)
+			goto error;
+	} else if (tok->type == '}') {
+		obj.type = isl_obj_union_set;
+		obj.v = isl_union_set_empty(isl_map_get_space(map));
+		isl_token_free(tok);
+		goto done;
+	} else
+		isl_stream_push_token(s, tok);
+
+	for (;;) {
+		struct isl_obj o;
+		tok = NULL;
+		o = obj_read_body(s, isl_map_copy(map), v);
+		if (o.type == isl_obj_none || !o.v)
+			goto error;
+		if (!obj.v)
+			obj = o;
+		else {
+			obj = obj_add(s->ctx, obj, o);
+			if (obj.type == isl_obj_none || !obj.v)
+				goto error;
+		}
+		tok = isl_stream_next_token(s);
+		if (!tok || tok->type != ';')
+			break;
+		isl_token_free(tok);
+		if (isl_stream_next_token_is(s, '}')) {
+			tok = isl_stream_next_token(s);
+			break;
+		}
+	}
+
+	if (tok && tok->type == '}') {
+		isl_token_free(tok);
+	} else {
+		isl_stream_error(s, tok, "unexpected isl_token");
+		if (tok)
+			isl_token_free(tok);
+		goto error;
+	}
+done:
+	vars_free(v);
+	isl_map_free(map);
+
+	return obj;
+error:
+	isl_map_free(map);
+	obj.type->free(obj.v);
+	if (v)
+		vars_free(v);
+	obj.v = NULL;
+	return obj;
+}
+
+struct isl_obj isl_stream_read_obj(struct isl_stream *s)
+{
+	return obj_read(s);
+}
+
+__isl_give isl_map *isl_stream_read_map(struct isl_stream *s)
+{
+	struct isl_obj obj;
+
+	obj = obj_read(s);
+	if (obj.v)
+		isl_assert(s->ctx, obj.type == isl_obj_map ||
+				   obj.type == isl_obj_set, goto error);
+	
+	if (obj.type == isl_obj_set)
+		obj.v = isl_map_from_range(obj.v);
+
+	return obj.v;
+error:
+	obj.type->free(obj.v);
+	return NULL;
+}
+
+__isl_give isl_set *isl_stream_read_set(struct isl_stream *s)
+{
+	struct isl_obj obj;
+
+	obj = obj_read(s);
+	if (obj.v) {
+		if (obj.type == isl_obj_map && isl_map_may_be_set(obj.v)) {
+			obj.v = isl_map_range(obj.v);
+			obj.type = isl_obj_set;
+		}
+		isl_assert(s->ctx, obj.type == isl_obj_set, goto error);
+	}
+
+	return obj.v;
+error:
+	obj.type->free(obj.v);
+	return NULL;
+}
+
+__isl_give isl_union_map *isl_stream_read_union_map(struct isl_stream *s)
+{
+	struct isl_obj obj;
+
+	obj = obj_read(s);
+	if (obj.type == isl_obj_map) {
+		obj.type = isl_obj_union_map;
+		obj.v = isl_union_map_from_map(obj.v);
+	}
+	if (obj.type == isl_obj_set) {
+		obj.type = isl_obj_union_set;
+		obj.v = isl_union_set_from_set(obj.v);
+	}
+	if (obj.v && obj.type == isl_obj_union_set &&
+	    isl_union_set_is_empty(obj.v))
+		obj.type = isl_obj_union_map;
+	if (obj.v && obj.type != isl_obj_union_map)
+		isl_die(s->ctx, isl_error_invalid, "invalid input", goto error);
+
+	return obj.v;
+error:
+	obj.type->free(obj.v);
+	return NULL;
+}
+
+__isl_give isl_union_set *isl_stream_read_union_set(struct isl_stream *s)
+{
+	struct isl_obj obj;
+
+	obj = obj_read(s);
+	if (obj.type == isl_obj_set) {
+		obj.type = isl_obj_union_set;
+		obj.v = isl_union_set_from_set(obj.v);
+	}
+	if (obj.v)
+		isl_assert(s->ctx, obj.type == isl_obj_union_set, goto error);
+
+	return obj.v;
+error:
+	obj.type->free(obj.v);
+	return NULL;
+}
+
+static __isl_give isl_basic_map *basic_map_read(struct isl_stream *s)
+{
+	struct isl_obj obj;
+	struct isl_map *map;
+	struct isl_basic_map *bmap;
+
+	obj = obj_read(s);
+	if (obj.v && (obj.type != isl_obj_map && obj.type != isl_obj_set))
+		isl_die(s->ctx, isl_error_invalid, "not a (basic) set or map",
+			goto error);
+	map = obj.v;
+	if (!map)
+		return NULL;
+
+	if (map->n > 1)
+		isl_die(s->ctx, isl_error_invalid,
+			"set or map description involves "
+			"more than one disjunct", goto error);
+
+	if (map->n == 0)
+		bmap = isl_basic_map_empty_like_map(map);
+	else
+		bmap = isl_basic_map_copy(map->p[0]);
+
+	isl_map_free(map);
+
+	return bmap;
+error:
+	obj.type->free(obj.v);
+	return NULL;
+}
+
+static __isl_give isl_basic_set *basic_set_read(struct isl_stream *s)
+{
+	isl_basic_map *bmap;
+	bmap = basic_map_read(s);
+	if (!bmap)
+		return NULL;
+	if (!isl_basic_map_may_be_set(bmap))
+		isl_die(s->ctx, isl_error_invalid,
+			"input is not a set", goto error);
+	return isl_basic_map_range(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_read_from_file(isl_ctx *ctx,
+	FILE *input)
+{
+	struct isl_basic_map *bmap;
+	struct isl_stream *s = isl_stream_new_file(ctx, input);
+	if (!s)
+		return NULL;
+	bmap = basic_map_read(s);
+	isl_stream_free(s);
+	return bmap;
+}
+
+__isl_give isl_basic_set *isl_basic_set_read_from_file(isl_ctx *ctx,
+	FILE *input)
+{
+	isl_basic_set *bset;
+	struct isl_stream *s = isl_stream_new_file(ctx, input);
+	if (!s)
+		return NULL;
+	bset = basic_set_read(s);
+	isl_stream_free(s);
+	return bset;
+}
+
+struct isl_basic_map *isl_basic_map_read_from_str(struct isl_ctx *ctx,
+	const char *str)
+{
+	struct isl_basic_map *bmap;
+	struct isl_stream *s = isl_stream_new_str(ctx, str);
+	if (!s)
+		return NULL;
+	bmap = basic_map_read(s);
+	isl_stream_free(s);
+	return bmap;
+}
+
+struct isl_basic_set *isl_basic_set_read_from_str(struct isl_ctx *ctx,
+	const char *str)
+{
+	isl_basic_set *bset;
+	struct isl_stream *s = isl_stream_new_str(ctx, str);
+	if (!s)
+		return NULL;
+	bset = basic_set_read(s);
+	isl_stream_free(s);
+	return bset;
+}
+
+__isl_give isl_map *isl_map_read_from_file(struct isl_ctx *ctx,
+	FILE *input)
+{
+	struct isl_map *map;
+	struct isl_stream *s = isl_stream_new_file(ctx, input);
+	if (!s)
+		return NULL;
+	map = isl_stream_read_map(s);
+	isl_stream_free(s);
+	return map;
+}
+
+__isl_give isl_map *isl_map_read_from_str(struct isl_ctx *ctx,
+	const char *str)
+{
+	struct isl_map *map;
+	struct isl_stream *s = isl_stream_new_str(ctx, str);
+	if (!s)
+		return NULL;
+	map = isl_stream_read_map(s);
+	isl_stream_free(s);
+	return map;
+}
+
+__isl_give isl_set *isl_set_read_from_file(struct isl_ctx *ctx,
+	FILE *input)
+{
+	isl_set *set;
+	struct isl_stream *s = isl_stream_new_file(ctx, input);
+	if (!s)
+		return NULL;
+	set = isl_stream_read_set(s);
+	isl_stream_free(s);
+	return set;
+}
+
+struct isl_set *isl_set_read_from_str(struct isl_ctx *ctx,
+	const char *str)
+{
+	isl_set *set;
+	struct isl_stream *s = isl_stream_new_str(ctx, str);
+	if (!s)
+		return NULL;
+	set = isl_stream_read_set(s);
+	isl_stream_free(s);
+	return set;
+}
+
+__isl_give isl_union_map *isl_union_map_read_from_file(isl_ctx *ctx,
+	FILE *input)
+{
+	isl_union_map *umap;
+	struct isl_stream *s = isl_stream_new_file(ctx, input);
+	if (!s)
+		return NULL;
+	umap = isl_stream_read_union_map(s);
+	isl_stream_free(s);
+	return umap;
+}
+
+__isl_give isl_union_map *isl_union_map_read_from_str(struct isl_ctx *ctx,
+		const char *str)
+{
+	isl_union_map *umap;
+	struct isl_stream *s = isl_stream_new_str(ctx, str);
+	if (!s)
+		return NULL;
+	umap = isl_stream_read_union_map(s);
+	isl_stream_free(s);
+	return umap;
+}
+
+__isl_give isl_union_set *isl_union_set_read_from_file(isl_ctx *ctx,
+	FILE *input)
+{
+	isl_union_set *uset;
+	struct isl_stream *s = isl_stream_new_file(ctx, input);
+	if (!s)
+		return NULL;
+	uset = isl_stream_read_union_set(s);
+	isl_stream_free(s);
+	return uset;
+}
+
+__isl_give isl_union_set *isl_union_set_read_from_str(struct isl_ctx *ctx,
+		const char *str)
+{
+	isl_union_set *uset;
+	struct isl_stream *s = isl_stream_new_str(ctx, str);
+	if (!s)
+		return NULL;
+	uset = isl_stream_read_union_set(s);
+	isl_stream_free(s);
+	return uset;
+}
+
+static __isl_give isl_vec *isl_vec_read_polylib(struct isl_stream *s)
+{
+	struct isl_vec *vec = NULL;
+	struct isl_token *tok;
+	unsigned size;
+	int j;
+
+	tok = isl_stream_next_token(s);
+	if (!tok || tok->type != ISL_TOKEN_VALUE) {
+		isl_stream_error(s, tok, "expecting vector length");
+		goto error;
+	}
+
+	size = isl_int_get_si(tok->u.v);
+	isl_token_free(tok);
+
+	vec = isl_vec_alloc(s->ctx, size);
+
+	for (j = 0; j < size; ++j) {
+		tok = isl_stream_next_token(s);
+		if (!tok || tok->type != ISL_TOKEN_VALUE) {
+			isl_stream_error(s, tok, "expecting constant value");
+			goto error;
+		}
+		isl_int_set(vec->el[j], tok->u.v);
+		isl_token_free(tok);
+	}
+
+	return vec;
+error:
+	isl_token_free(tok);
+	isl_vec_free(vec);
+	return NULL;
+}
+
+static __isl_give isl_vec *vec_read(struct isl_stream *s)
+{
+	return isl_vec_read_polylib(s);
+}
+
+__isl_give isl_vec *isl_vec_read_from_file(isl_ctx *ctx, FILE *input)
+{
+	isl_vec *v;
+	struct isl_stream *s = isl_stream_new_file(ctx, input);
+	if (!s)
+		return NULL;
+	v = vec_read(s);
+	isl_stream_free(s);
+	return v;
+}
+
+__isl_give isl_pw_qpolynomial *isl_stream_read_pw_qpolynomial(
+	struct isl_stream *s)
+{
+	struct isl_obj obj;
+
+	obj = obj_read(s);
+	if (obj.v)
+		isl_assert(s->ctx, obj.type == isl_obj_pw_qpolynomial,
+			   goto error);
+
+	return obj.v;
+error:
+	obj.type->free(obj.v);
+	return NULL;
+}
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_read_from_str(isl_ctx *ctx,
+		const char *str)
+{
+	isl_pw_qpolynomial *pwqp;
+	struct isl_stream *s = isl_stream_new_str(ctx, str);
+	if (!s)
+		return NULL;
+	pwqp = isl_stream_read_pw_qpolynomial(s);
+	isl_stream_free(s);
+	return pwqp;
+}
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_read_from_file(isl_ctx *ctx,
+		FILE *input)
+{
+	isl_pw_qpolynomial *pwqp;
+	struct isl_stream *s = isl_stream_new_file(ctx, input);
+	if (!s)
+		return NULL;
+	pwqp = isl_stream_read_pw_qpolynomial(s);
+	isl_stream_free(s);
+	return pwqp;
+}
+
+/* Is the next token an identifer not in "v"?
+ */
+static int next_is_fresh_ident(struct isl_stream *s, struct vars *v)
+{
+	int n = v->n;
+	int fresh;
+	struct isl_token *tok;
+
+	tok = isl_stream_next_token(s);
+	if (!tok)
+		return 0;
+	fresh = tok->type == ISL_TOKEN_IDENT && vars_pos(v, tok->u.s, -1) >= n;
+	isl_stream_push_token(s, tok);
+
+	vars_drop(v, v->n - n);
+
+	return fresh;
+}
+
+/* First read the domain of the affine expression, which may be
+ * a parameter space or a set.
+ * The tricky part is that we don't know if the domain is a set or not,
+ * so when we are trying to read the domain, we may actually be reading
+ * the affine expression itself (defined on a parameter domains)
+ * If the tuple we are reading is named, we assume it's the domain.
+ * Also, if inside the tuple, the first thing we find is a nested tuple
+ * or a new identifier, we again assume it's the domain.
+ * Otherwise, we assume we are reading an affine expression.
+ */
+static __isl_give isl_set *read_aff_domain(struct isl_stream *s,
+	__isl_take isl_set *dom, struct vars *v)
+{
+	struct isl_token *tok;
+
+	tok = isl_stream_next_token(s);
+	if (tok && (tok->type == ISL_TOKEN_IDENT || tok->is_keyword)) {
+		isl_stream_push_token(s, tok);
+		return read_map_tuple(s, dom, isl_dim_set, v, 1, 0);
+	}
+	if (!tok || tok->type != '[') {
+		isl_stream_error(s, tok, "expecting '['");
+		goto error;
+	}
+	if (next_is_tuple(s) || next_is_fresh_ident(s, v)) {
+		isl_stream_push_token(s, tok);
+		dom = read_map_tuple(s, dom, isl_dim_set, v, 1, 0);
+	} else
+		isl_stream_push_token(s, tok);
+
+	return dom;
+error:
+	if (tok)
+		isl_stream_push_token(s, tok);
+	isl_set_free(dom);
+	return NULL;
+}
+
+/* Read an affine expression from "s".
+ */
+__isl_give isl_aff *isl_stream_read_aff(struct isl_stream *s)
+{
+	isl_aff *aff;
+	isl_multi_aff *ma;
+
+	ma = isl_stream_read_multi_aff(s);
+	if (!ma)
+		return NULL;
+	if (isl_multi_aff_dim(ma, isl_dim_out) != 1)
+		isl_die(s->ctx, isl_error_invalid,
+			"expecting single affine expression",
+			goto error);
+
+	aff = isl_multi_aff_get_aff(ma, 0);
+	isl_multi_aff_free(ma);
+	return aff;
+error:
+	isl_multi_aff_free(ma);
+	return NULL;
+}
+
+/* Read a piecewise affine expression from "s" with domain (space) "dom".
+ */
+static __isl_give isl_pw_aff *read_pw_aff_with_dom(struct isl_stream *s,
+	__isl_take isl_set *dom, struct vars *v)
+{
+	isl_pw_aff *pwaff = NULL;
+
+	if (!isl_set_is_params(dom) && isl_stream_eat(s, ISL_TOKEN_TO))
+		goto error;
+
+	if (isl_stream_eat(s, '['))
+		goto error;
+
+	pwaff = accept_affine(s, isl_set_get_space(dom), v);
+
+	if (isl_stream_eat(s, ']'))
+		goto error;
+
+	dom = read_optional_formula(s, dom, v, 0);
+	pwaff = isl_pw_aff_intersect_domain(pwaff, dom);
+
+	return pwaff;
+error:
+	isl_set_free(dom);
+	isl_pw_aff_free(pwaff);
+	return NULL;
+}
+
+__isl_give isl_pw_aff *isl_stream_read_pw_aff(struct isl_stream *s)
+{
+	struct vars *v;
+	isl_set *dom = NULL;
+	isl_set *aff_dom;
+	isl_pw_aff *pa = NULL;
+	int n;
+
+	v = vars_new(s->ctx);
+	if (!v)
+		return NULL;
+
+	dom = isl_set_universe(isl_space_params_alloc(s->ctx, 0));
+	if (next_is_tuple(s)) {
+		dom = read_map_tuple(s, dom, isl_dim_param, v, 1, 0);
+		if (isl_stream_eat(s, ISL_TOKEN_TO))
+			goto error;
+	}
+	if (isl_stream_eat(s, '{'))
+		goto error;
+
+	n = v->n;
+	aff_dom = read_aff_domain(s, isl_set_copy(dom), v);
+	pa = read_pw_aff_with_dom(s, aff_dom, v);
+	vars_drop(v, v->n - n);
+
+	while (isl_stream_eat_if_available(s, ';')) {
+		isl_pw_aff *pa_i;
+
+		n = v->n;
+		aff_dom = read_aff_domain(s, isl_set_copy(dom), v);
+		pa_i = read_pw_aff_with_dom(s, aff_dom, v);
+		vars_drop(v, v->n - n);
+
+		pa = isl_pw_aff_union_add(pa, pa_i);
+	}
+
+	if (isl_stream_eat(s, '}'))
+		goto error;
+
+	vars_free(v);
+	isl_set_free(dom);
+	return pa;
+error:
+	vars_free(v);
+	isl_set_free(dom);
+	isl_pw_aff_free(pa);
+	return NULL;
+}
+
+__isl_give isl_aff *isl_aff_read_from_str(isl_ctx *ctx, const char *str)
+{
+	isl_aff *aff;
+	struct isl_stream *s = isl_stream_new_str(ctx, str);
+	if (!s)
+		return NULL;
+	aff = isl_stream_read_aff(s);
+	isl_stream_free(s);
+	return aff;
+}
+
+__isl_give isl_pw_aff *isl_pw_aff_read_from_str(isl_ctx *ctx, const char *str)
+{
+	isl_pw_aff *pa;
+	struct isl_stream *s = isl_stream_new_str(ctx, str);
+	if (!s)
+		return NULL;
+	pa = isl_stream_read_pw_aff(s);
+	isl_stream_free(s);
+	return pa;
+}
+
+/* Read an isl_pw_multi_aff from "s".
+ * We currently read a generic object and if it turns out to be a set or
+ * a map, we convert that to an isl_pw_multi_aff.
+ * It would be more efficient if we were to construct the isl_pw_multi_aff
+ * directly.
+ */
+__isl_give isl_pw_multi_aff *isl_stream_read_pw_multi_aff(struct isl_stream *s)
+{
+	struct isl_obj obj;
+
+	obj = obj_read(s);
+	if (!obj.v)
+		return NULL;
+
+	if (obj.type == isl_obj_map)
+		return isl_pw_multi_aff_from_map(obj.v);
+	if (obj.type == isl_obj_set)
+		return isl_pw_multi_aff_from_set(obj.v);
+
+	obj.type->free(obj.v);
+	isl_die(s->ctx, isl_error_invalid, "unexpected object type",
+		return NULL);
+}
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_read_from_str(isl_ctx *ctx,
+	const char *str)
+{
+	isl_pw_multi_aff *pma;
+	struct isl_stream *s = isl_stream_new_str(ctx, str);
+	if (!s)
+		return NULL;
+	pma = isl_stream_read_pw_multi_aff(s);
+	isl_stream_free(s);
+	return pma;
+}
+
+/* Read an isl_union_pw_multi_aff from "s".
+ * We currently read a generic object and if it turns out to be a set or
+ * a map, we convert that to an isl_union_pw_multi_aff.
+ * It would be more efficient if we were to construct
+ * the isl_union_pw_multi_aff directly.
+ */
+__isl_give isl_union_pw_multi_aff *isl_stream_read_union_pw_multi_aff(
+	struct isl_stream *s)
+{
+	struct isl_obj obj;
+
+	obj = obj_read(s);
+	if (!obj.v)
+		return NULL;
+
+	if (obj.type == isl_obj_map || obj.type == isl_obj_set)
+		obj = to_union(s->ctx, obj);
+	if (obj.type == isl_obj_union_map)
+		return isl_union_pw_multi_aff_from_union_map(obj.v);
+	if (obj.type == isl_obj_union_set)
+		return isl_union_pw_multi_aff_from_union_set(obj.v);
+
+	obj.type->free(obj.v);
+	isl_die(s->ctx, isl_error_invalid, "unexpected object type",
+		return NULL);
+}
+
+/* Read an isl_union_pw_multi_aff from "str".
+ */
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_read_from_str(
+	isl_ctx *ctx, const char *str)
+{
+	isl_union_pw_multi_aff *upma;
+	struct isl_stream *s = isl_stream_new_str(ctx, str);
+	if (!s)
+		return NULL;
+	upma = isl_stream_read_union_pw_multi_aff(s);
+	isl_stream_free(s);
+	return upma;
+}
+
+/* Assuming "pa" represents a single affine expression defined on a universe
+ * domain, extract this affine expression.
+ */
+static __isl_give isl_aff *aff_from_pw_aff(__isl_take isl_pw_aff *pa)
+{
+	isl_aff *aff;
+
+	if (!pa)
+		return NULL;
+	if (pa->n != 1)
+		isl_die(isl_pw_aff_get_ctx(pa), isl_error_invalid,
+			"expecting single affine expression",
+			goto error);
+	if (!isl_set_plain_is_universe(pa->p[0].set))
+		isl_die(isl_pw_aff_get_ctx(pa), isl_error_invalid,
+			"expecting universe domain",
+			goto error);
+
+	aff = isl_aff_copy(pa->p[0].aff);
+	isl_pw_aff_free(pa);
+	return aff;
+error:
+	isl_pw_aff_free(pa);
+	return NULL;
+}
+
+/* Read a multi-affine expression from "s".
+ * If the multi-affine expression has a domain, then the tuple
+ * representing this domain cannot involve any affine expressions.
+ * The tuple representing the actual expressions needs to consist
+ * of only affine expressions.  Moreover, these expressions can
+ * only depend on parameters and input dimensions and not on other
+ * output dimensions.
+ */
+__isl_give isl_multi_aff *isl_stream_read_multi_aff(struct isl_stream *s)
+{
+	struct vars *v;
+	isl_set *dom = NULL;
+	isl_multi_pw_aff *tuple = NULL;
+	int dim, i, n;
+	isl_space *space, *dom_space;
+	isl_multi_aff *ma = NULL;
+
+	v = vars_new(s->ctx);
+	if (!v)
+		return NULL;
+
+	dom = isl_set_universe(isl_space_params_alloc(s->ctx, 0));
+	if (next_is_tuple(s)) {
+		dom = read_map_tuple(s, dom, isl_dim_param, v, 1, 0);
+		if (isl_stream_eat(s, ISL_TOKEN_TO))
+			goto error;
+	}
+	if (!isl_set_plain_is_universe(dom))
+		isl_die(s->ctx, isl_error_invalid,
+			"expecting universe parameter domain", goto error);
+	if (isl_stream_eat(s, '{'))
+		goto error;
+
+	tuple = read_tuple(s, v, 0, 0);
+	if (!tuple)
+		goto error;
+	if (isl_stream_eat_if_available(s, ISL_TOKEN_TO)) {
+		isl_set *set;
+		isl_space *space;
+		int has_expr;
+
+		has_expr = tuple_has_expr(tuple);
+		if (has_expr < 0)
+			goto error;
+		if (has_expr)
+			isl_die(s->ctx, isl_error_invalid,
+				"expecting universe domain", goto error);
+		space = isl_space_range(isl_multi_pw_aff_get_space(tuple));
+		set = isl_set_universe(space);
+		dom = isl_set_intersect_params(set, dom);
+		isl_multi_pw_aff_free(tuple);
+		tuple = read_tuple(s, v, 0, 0);
+		if (!tuple)
+			goto error;
+	}
+
+	if (isl_stream_eat(s, '}'))
+		goto error;
+
+	n = isl_multi_pw_aff_dim(tuple, isl_dim_out);
+	dim = isl_set_dim(dom, isl_dim_all);
+	dom_space = isl_set_get_space(dom);
+	space = isl_space_range(isl_multi_pw_aff_get_space(tuple));
+	space = isl_space_align_params(space, isl_space_copy(dom_space));
+	if (!isl_space_is_params(dom_space))
+		space = isl_space_map_from_domain_and_range(
+				isl_space_copy(dom_space), space);
+	isl_space_free(dom_space);
+	ma = isl_multi_aff_alloc(space);
+
+	for (i = 0; i < n; ++i) {
+		isl_pw_aff *pa;
+		isl_aff *aff;
+		pa = isl_multi_pw_aff_get_pw_aff(tuple, i);
+		aff = aff_from_pw_aff(pa);
+		if (!aff)
+			goto error;
+		if (isl_aff_involves_dims(aff, isl_dim_in, dim, i + 1)) {
+			isl_aff_free(aff);
+			isl_die(s->ctx, isl_error_invalid,
+				"not an affine expression", goto error);
+		}
+		aff = isl_aff_drop_dims(aff, isl_dim_in, dim, n);
+		space = isl_multi_aff_get_domain_space(ma);
+		aff = isl_aff_reset_domain_space(aff, space);
+		ma = isl_multi_aff_set_aff(ma, i, aff);
+	}
+
+	isl_multi_pw_aff_free(tuple);
+	vars_free(v);
+	isl_set_free(dom);
+	return ma;
+error:
+	isl_multi_pw_aff_free(tuple);
+	vars_free(v);
+	isl_set_free(dom);
+	isl_multi_aff_free(ma);
+	return NULL;
+}
+
+__isl_give isl_multi_aff *isl_multi_aff_read_from_str(isl_ctx *ctx,
+	const char *str)
+{
+	isl_multi_aff *maff;
+	struct isl_stream *s = isl_stream_new_str(ctx, str);
+	if (!s)
+		return NULL;
+	maff = isl_stream_read_multi_aff(s);
+	isl_stream_free(s);
+	return maff;
+}
+
+/* Read an isl_multi_pw_aff from "s".
+ *
+ * The input format is similar to that of map, except that any conditions
+ * on the domains should be specified inside the tuple since each
+ * piecewise affine expression may have a different domain.
+ *
+ * Since we do not know in advance if the isl_multi_pw_aff lives
+ * in a set or a map space, we first read the first tuple and check
+ * if it is followed by a "->".  If so, we convert the tuple into
+ * the domain of the isl_multi_pw_aff and read in the next tuple.
+ * This tuple (or the first tuple if it was not followed by a "->")
+ * is then converted into the isl_multi_pw_aff.
+ *
+ * Note that the function read_tuple accepts tuples where some output or
+ * set dimensions are defined in terms of other output or set dimensions
+ * since this function is also used to read maps.  As a special case,
+ * read_tuple also accept dimensions that are defined in terms of themselves
+ * (i.e., that are not defined).
+ * These cases are not allowed when reading am isl_multi_pw_aff so we check
+ * that the definition of the output/set dimensions does not involve any
+ * output/set dimensions.
+ * We then drop the output dimensions from the domain of the result
+ * of read_tuple (which is of the form [input, output] -> [output],
+ * with anonymous domain) and reset the space.
+ */
+__isl_give isl_multi_pw_aff *isl_stream_read_multi_pw_aff(struct isl_stream *s)
+{
+	struct vars *v;
+	isl_set *dom = NULL;
+	isl_multi_pw_aff *tuple = NULL;
+	int dim, i, n;
+	isl_space *space, *dom_space;
+	isl_multi_pw_aff *mpa = NULL;
+
+	v = vars_new(s->ctx);
+	if (!v)
+		return NULL;
+
+	dom = isl_set_universe(isl_space_params_alloc(s->ctx, 0));
+	if (next_is_tuple(s)) {
+		dom = read_map_tuple(s, dom, isl_dim_param, v, 1, 0);
+		if (isl_stream_eat(s, ISL_TOKEN_TO))
+			goto error;
+	}
+	if (isl_stream_eat(s, '{'))
+		goto error;
+
+	tuple = read_tuple(s, v, 0, 0);
+	if (!tuple)
+		goto error;
+	if (isl_stream_eat_if_available(s, ISL_TOKEN_TO)) {
+		isl_map *map = map_from_tuple(tuple, dom, isl_dim_in, v, 0);
+		dom = isl_map_domain(map);
+		tuple = read_tuple(s, v, 0, 0);
+		if (!tuple)
+			goto error;
+	}
+
+	if (isl_stream_eat(s, '}'))
+		goto error;
+
+	n = isl_multi_pw_aff_dim(tuple, isl_dim_out);
+	dim = isl_set_dim(dom, isl_dim_all);
+	dom_space = isl_set_get_space(dom);
+	space = isl_space_range(isl_multi_pw_aff_get_space(tuple));
+	space = isl_space_align_params(space, isl_space_copy(dom_space));
+	if (!isl_space_is_params(dom_space))
+		space = isl_space_map_from_domain_and_range(
+				isl_space_copy(dom_space), space);
+	isl_space_free(dom_space);
+	mpa = isl_multi_pw_aff_alloc(space);
+
+	for (i = 0; i < n; ++i) {
+		isl_pw_aff *pa;
+		pa = isl_multi_pw_aff_get_pw_aff(tuple, i);
+		if (!pa)
+			goto error;
+		if (isl_pw_aff_involves_dims(pa, isl_dim_in, dim, i + 1)) {
+			isl_pw_aff_free(pa);
+			isl_die(s->ctx, isl_error_invalid,
+				"not an affine expression", goto error);
+		}
+		pa = isl_pw_aff_drop_dims(pa, isl_dim_in, dim, n);
+		space = isl_multi_pw_aff_get_domain_space(mpa);
+		pa = isl_pw_aff_reset_domain_space(pa, space);
+		mpa = isl_multi_pw_aff_set_pw_aff(mpa, i, pa);
+	}
+
+	isl_multi_pw_aff_free(tuple);
+	vars_free(v);
+	mpa = isl_multi_pw_aff_intersect_domain(mpa, dom);
+	return mpa;
+error:
+	isl_multi_pw_aff_free(tuple);
+	vars_free(v);
+	isl_set_free(dom);
+	isl_multi_pw_aff_free(mpa);
+	return NULL;
+}
+
+/* Read an isl_multi_pw_aff from "str".
+ */
+__isl_give isl_multi_pw_aff *isl_multi_pw_aff_read_from_str(isl_ctx *ctx,
+	const char *str)
+{
+	isl_multi_pw_aff *mpa;
+	struct isl_stream *s = isl_stream_new_str(ctx, str);
+	if (!s)
+		return NULL;
+	mpa = isl_stream_read_multi_pw_aff(s);
+	isl_stream_free(s);
+	return mpa;
+}
+
+__isl_give isl_union_pw_qpolynomial *isl_stream_read_union_pw_qpolynomial(
+	struct isl_stream *s)
+{
+	struct isl_obj obj;
+
+	obj = obj_read(s);
+	if (obj.type == isl_obj_pw_qpolynomial) {
+		obj.type = isl_obj_union_pw_qpolynomial;
+		obj.v = isl_union_pw_qpolynomial_from_pw_qpolynomial(obj.v);
+	}
+	if (obj.v)
+		isl_assert(s->ctx, obj.type == isl_obj_union_pw_qpolynomial,
+			   goto error);
+
+	return obj.v;
+error:
+	obj.type->free(obj.v);
+	return NULL;
+}
+
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_read_from_str(
+	isl_ctx *ctx, const char *str)
+{
+	isl_union_pw_qpolynomial *upwqp;
+	struct isl_stream *s = isl_stream_new_str(ctx, str);
+	if (!s)
+		return NULL;
+	upwqp = isl_stream_read_union_pw_qpolynomial(s);
+	isl_stream_free(s);
+	return upwqp;
+}
diff --git a/polly/lib/External/isl/isl_int.h b/polly/lib/External/isl/isl_int.h
new file mode 100644
index 00000000000..3be0661ee2c
--- /dev/null
+++ b/polly/lib/External/isl/isl_int.h
@@ -0,0 +1,46 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_INT_H
+#define ISL_INT_H
+#define ISL_DEPRECATED_INT_H
+
+#include <isl/hash.h>
+#include <isl/printer.h>
+#include <string.h>
+#include <isl_config.h>
+
+#ifdef USE_GMP_FOR_MP
+#include <isl_int_gmp.h>
+#endif
+
+#ifdef USE_IMATH_FOR_MP
+#include <isl_int_imath.h>
+#endif
+
+#define isl_int_is_zero(i)	(isl_int_sgn(i) == 0)
+#define isl_int_is_one(i)	(isl_int_cmp_si(i,1) == 0)
+#define isl_int_is_negone(i)	(isl_int_cmp_si(i,-1) == 0)
+#define isl_int_is_pos(i)	(isl_int_sgn(i) > 0)
+#define isl_int_is_neg(i)	(isl_int_sgn(i) < 0)
+#define isl_int_is_nonpos(i)	(isl_int_sgn(i) <= 0)
+#define isl_int_is_nonneg(i)	(isl_int_sgn(i) >= 0)
+
+#define isl_int_print(out,i,width)					\
+	do {								\
+		char *s;						\
+		s = isl_int_get_str(i);					\
+		fprintf(out, "%*s", width, s);				\
+		isl_int_free_str(s);                                        \
+	} while (0)
+
+__isl_give isl_printer *isl_printer_print_isl_int(__isl_take isl_printer *p,
+	isl_int i);
+
+#endif /* ISL_INT_H */
diff --git a/polly/lib/External/isl/isl_int_gmp.h b/polly/lib/External/isl/isl_int_gmp.h
new file mode 100644
index 00000000000..08f2857b1e6
--- /dev/null
+++ b/polly/lib/External/isl/isl_int_gmp.h
@@ -0,0 +1,88 @@
+#ifndef ISL_INT_GMP_H
+#define ISL_INT_GMP_H
+
+#include <gmp.h>
+
+/* isl_int is the basic integer type, implemented with GMP's mpz_t.  In the
+ * future, different types such as long long or cln::cl_I will be supported.
+ */
+typedef mpz_t	isl_int;
+
+#define isl_int_init(i)		mpz_init(i)
+#define isl_int_clear(i)	mpz_clear(i)
+
+#define isl_int_set(r,i)	mpz_set(r,i)
+#define isl_int_set_si(r,i)	mpz_set_si(r,i)
+#define isl_int_set_ui(r,i)	mpz_set_ui(r,i)
+#define isl_int_fits_slong(r)	mpz_fits_slong_p(r)
+#define isl_int_get_si(r)	mpz_get_si(r)
+#define isl_int_fits_ulong(r)	mpz_fits_ulong_p(r)
+#define isl_int_get_ui(r)	mpz_get_ui(r)
+#define isl_int_get_d(r)	mpz_get_d(r)
+#define isl_int_get_str(r)	mpz_get_str(0, 10, r)
+#define isl_int_abs(r,i)	mpz_abs(r,i)
+#define isl_int_neg(r,i)	mpz_neg(r,i)
+#define isl_int_swap(i,j)	mpz_swap(i,j)
+#define isl_int_swap_or_set(i,j)	mpz_swap(i,j)
+#define isl_int_add_ui(r,i,j)	mpz_add_ui(r,i,j)
+#define isl_int_sub_ui(r,i,j)	mpz_sub_ui(r,i,j)
+
+#define isl_int_add(r,i,j)	mpz_add(r,i,j)
+#define isl_int_sub(r,i,j)	mpz_sub(r,i,j)
+#define isl_int_mul(r,i,j)	mpz_mul(r,i,j)
+#define isl_int_mul_2exp(r,i,j)	mpz_mul_2exp(r,i,j)
+#define isl_int_mul_si(r,i,j)	mpz_mul_si(r,i,j)
+#define isl_int_mul_ui(r,i,j)	mpz_mul_ui(r,i,j)
+#define isl_int_pow_ui(r,i,j)	mpz_pow_ui(r,i,j)
+#define isl_int_addmul(r,i,j)	mpz_addmul(r,i,j)
+#define isl_int_addmul_ui(r,i,j)	mpz_addmul_ui(r,i,j)
+#define isl_int_submul(r,i,j)	mpz_submul(r,i,j)
+#define isl_int_submul_ui(r,i,j)	mpz_submul_ui(r,i,j)
+
+#define isl_int_gcd(r,i,j)	mpz_gcd(r,i,j)
+#define isl_int_lcm(r,i,j)	mpz_lcm(r,i,j)
+#define isl_int_divexact(r,i,j)	mpz_divexact(r,i,j)
+#define isl_int_divexact_ui(r,i,j)	mpz_divexact_ui(r,i,j)
+#define isl_int_tdiv_q(r,i,j)	mpz_tdiv_q(r,i,j)
+#define isl_int_cdiv_q(r,i,j)	mpz_cdiv_q(r,i,j)
+#define isl_int_fdiv_q(r,i,j)	mpz_fdiv_q(r,i,j)
+#define isl_int_fdiv_r(r,i,j)	mpz_fdiv_r(r,i,j)
+#define isl_int_fdiv_q_ui(r,i,j)	mpz_fdiv_q_ui(r,i,j)
+
+#define isl_int_read(r,s)	mpz_set_str(r,s,10)
+#define isl_int_sgn(i)		mpz_sgn(i)
+#define isl_int_cmp(i,j)	mpz_cmp(i,j)
+#define isl_int_cmp_si(i,si)	mpz_cmp_si(i,si)
+#define isl_int_eq(i,j)		(mpz_cmp(i,j) == 0)
+#define isl_int_ne(i,j)		(mpz_cmp(i,j) != 0)
+#define isl_int_lt(i,j)		(mpz_cmp(i,j) < 0)
+#define isl_int_le(i,j)		(mpz_cmp(i,j) <= 0)
+#define isl_int_gt(i,j)		(mpz_cmp(i,j) > 0)
+#define isl_int_ge(i,j)		(mpz_cmp(i,j) >= 0)
+#define isl_int_abs_cmp(i,j)	mpz_cmpabs(i,j)
+#define isl_int_abs_eq(i,j)	(mpz_cmpabs(i,j) == 0)
+#define isl_int_abs_ne(i,j)	(mpz_cmpabs(i,j) != 0)
+#define isl_int_abs_lt(i,j)	(mpz_cmpabs(i,j) < 0)
+#define isl_int_abs_gt(i,j)	(mpz_cmpabs(i,j) > 0)
+#define isl_int_abs_ge(i,j)	(mpz_cmpabs(i,j) >= 0)
+#define isl_int_is_divisible_by(i,j)	mpz_divisible_p(i,j)
+
+uint32_t isl_gmp_hash(mpz_t v, uint32_t hash);
+#define isl_int_hash(v,h)	isl_gmp_hash(v,h)
+
+#ifndef mp_get_memory_functions
+void mp_get_memory_functions(
+		void *(**alloc_func_ptr) (size_t),
+		void *(**realloc_func_ptr) (void *, size_t, size_t),
+		void (**free_func_ptr) (void *, size_t));
+#endif
+
+typedef void (*isl_int_print_mp_free_t)(void *, size_t);
+#define isl_int_free_str(s)					\
+	do {								\
+		isl_int_print_mp_free_t mp_free;			\
+		mp_get_memory_functions(NULL, NULL, &mp_free);		\
+		(*mp_free)(s, strlen(s) + 1);				\
+	} while (0)
+
+#endif /* ISL_INT_GMP_H */
diff --git a/polly/lib/External/isl/isl_int_imath.h b/polly/lib/External/isl/isl_int_imath.h
new file mode 100644
index 00000000000..e5b7186bf16
--- /dev/null
+++ b/polly/lib/External/isl/isl_int_imath.h
@@ -0,0 +1,76 @@
+#ifndef ISL_INT_IMATH_H
+#define ISL_INT_IMATH_H
+
+#include "isl_hide_deprecated.h"
+
+#include <isl_imath.h>
+
+/* isl_int is the basic integer type, implemented with imath's mp_int. */
+typedef mp_int isl_int;
+
+#define isl_int_init(i)		i = mp_int_alloc()
+#define isl_int_clear(i)	mp_int_free(i)
+
+#define isl_int_set(r,i)	impz_set(r,i)
+#define isl_int_set_si(r,i)	impz_set_si(r,i)
+#define isl_int_set_ui(r,i)	impz_set_ui(r,i)
+#define isl_int_fits_slong(r)	isl_imath_fits_slong_p(r)
+#define isl_int_get_si(r)	impz_get_si(r)
+#define isl_int_fits_ulong(r)	isl_imath_fits_ulong_p(r)
+#define isl_int_get_ui(r)	impz_get_ui(r)
+#define isl_int_get_d(r)	impz_get_si(r)
+#define isl_int_get_str(r)	impz_get_str(0, 10, r)
+#define isl_int_abs(r,i)	impz_abs(r,i)
+#define isl_int_neg(r,i)	impz_neg(r,i)
+#define isl_int_swap(i,j)	impz_swap(i,j)
+#define isl_int_swap_or_set(i,j)	impz_swap(i,j)
+#define isl_int_add_ui(r,i,j)	impz_add_ui(r,i,j)
+#define isl_int_sub_ui(r,i,j)	impz_sub_ui(r,i,j)
+
+#define isl_int_add(r,i,j)	impz_add(r,i,j)
+#define isl_int_sub(r,i,j)	impz_sub(r,i,j)
+#define isl_int_mul(r,i,j)	impz_mul(r,i,j)
+#define isl_int_mul_2exp(r,i,j)	impz_mul_2exp(r,i,j)
+#define isl_int_mul_si(r,i,j)	mp_int_mul_value(i,j,r)
+#define isl_int_mul_ui(r,i,j)	impz_mul_ui(r,i,j)
+#define isl_int_pow_ui(r,i,j)	impz_pow_ui(r,i,j)
+#define isl_int_addmul(r,i,j)	impz_addmul(r,i,j)
+#define isl_int_addmul_ui(r,i,j)	isl_imath_addmul_ui(r,i,j)
+#define isl_int_submul(r,i,j)	impz_submul(r,i,j)
+#define isl_int_submul_ui(r,i,j)	isl_imath_submul_ui(r,i,j)
+
+#define isl_int_gcd(r,i,j)	impz_gcd(r,i,j)
+#define isl_int_lcm(r,i,j)	impz_lcm(r,i,j)
+#define isl_int_divexact(r,i,j)	impz_divexact(r,i,j)
+#define isl_int_divexact_ui(r,i,j)	impz_divexact_ui(r,i,j)
+#define isl_int_tdiv_q(r,i,j)	impz_tdiv_q(r,i,j)
+#define isl_int_cdiv_q(r,i,j)	impz_cdiv_q(r,i,j)
+#define isl_int_fdiv_q(r,i,j)	impz_fdiv_q(r,i,j)
+#define isl_int_fdiv_r(r,i,j)	impz_fdiv_r(r,i,j)
+#define isl_int_fdiv_q_ui(r,i,j)	impz_fdiv_q_ui(r,i,j)
+
+#define isl_int_read(r,s)	impz_set_str(r,s,10)
+#define isl_int_sgn(i)		impz_sgn(i)
+#define isl_int_cmp(i,j)	impz_cmp(i,j)
+#define isl_int_cmp_si(i,si)	impz_cmp_si(i,si)
+#define isl_int_eq(i,j)		(impz_cmp(i,j) == 0)
+#define isl_int_ne(i,j)		(impz_cmp(i,j) != 0)
+#define isl_int_lt(i,j)		(impz_cmp(i,j) < 0)
+#define isl_int_le(i,j)		(impz_cmp(i,j) <= 0)
+#define isl_int_gt(i,j)		(impz_cmp(i,j) > 0)
+#define isl_int_ge(i,j)		(impz_cmp(i,j) >= 0)
+#define isl_int_abs_cmp(i,j)	impz_cmpabs(i,j)
+#define isl_int_abs_eq(i,j)	(impz_cmpabs(i,j) == 0)
+#define isl_int_abs_ne(i,j)	(impz_cmpabs(i,j) != 0)
+#define isl_int_abs_lt(i,j)	(impz_cmpabs(i,j) < 0)
+#define isl_int_abs_gt(i,j)	(impz_cmpabs(i,j) > 0)
+#define isl_int_abs_ge(i,j)	(impz_cmpabs(i,j) >= 0)
+#define isl_int_is_divisible_by(i,j)	impz_divisible_p(i,j)
+
+uint32_t isl_imath_hash(mp_int v, uint32_t hash);
+#define isl_int_hash(v,h)	isl_imath_hash(v,h)
+
+typedef void (*isl_int_print_mp_free_t)(void *, size_t);
+#define isl_int_free_str(s)	free(s)
+
+#endif /* ISL_INT_IMATH_H */
diff --git a/polly/lib/External/isl/isl_list_templ.c b/polly/lib/External/isl/isl_list_templ.c
new file mode 100644
index 00000000000..bee624835a2
--- /dev/null
+++ b/polly/lib/External/isl/isl_list_templ.c
@@ -0,0 +1,522 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2011      INRIA Saclay
+ * Copyright 2012-2013 Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+#include <isl_sort.h>
+#include <isl_tarjan.h>
+
+#define xCAT(A,B) A ## B
+#define CAT(A,B) xCAT(A,B)
+#undef EL
+#define EL CAT(isl_,BASE)
+#define xFN(TYPE,NAME) TYPE ## _ ## NAME
+#define FN(TYPE,NAME) xFN(TYPE,NAME)
+#define xLIST(EL) EL ## _list
+#define LIST(EL) xLIST(EL)
+#define xS(TYPE,NAME) struct TYPE ## _ ## NAME
+#define S(TYPE,NAME) xS(TYPE,NAME)
+
+isl_ctx *FN(LIST(EL),get_ctx)(__isl_keep LIST(EL) *list)
+{
+	return list ? list->ctx : NULL;
+}
+
+__isl_give LIST(EL) *FN(LIST(EL),alloc)(isl_ctx *ctx, int n)
+{
+	LIST(EL) *list;
+
+	if (n < 0)
+		isl_die(ctx, isl_error_invalid,
+			"cannot create list of negative length",
+			return NULL);
+	list = isl_alloc(ctx, LIST(EL),
+			 sizeof(LIST(EL)) + (n - 1) * sizeof(struct EL *));
+	if (!list)
+		return NULL;
+
+	list->ctx = ctx;
+	isl_ctx_ref(ctx);
+	list->ref = 1;
+	list->size = n;
+	list->n = 0;
+	return list;
+}
+
+__isl_give LIST(EL) *FN(LIST(EL),copy)(__isl_keep LIST(EL) *list)
+{
+	if (!list)
+		return NULL;
+
+	list->ref++;
+	return list;
+}
+
+__isl_give LIST(EL) *FN(LIST(EL),dup)(__isl_keep LIST(EL) *list)
+{
+	int i;
+	LIST(EL) *dup;
+
+	if (!list)
+		return NULL;
+
+	dup = FN(LIST(EL),alloc)(FN(LIST(EL),get_ctx)(list), list->n);
+	if (!dup)
+		return NULL;
+	for (i = 0; i < list->n; ++i)
+		dup = FN(LIST(EL),add)(dup, FN(EL,copy)(list->p[i]));
+	return dup;
+}
+
+__isl_give LIST(EL) *FN(LIST(EL),cow)(__isl_take LIST(EL) *list)
+{
+	if (!list)
+		return NULL;
+
+	if (list->ref == 1)
+		return list;
+	list->ref--;
+	return FN(LIST(EL),dup)(list);
+}
+
+/* Make sure "list" has room for at least "n" more pieces.
+ * Always return a list with a single reference.
+ *
+ * If there is only one reference to list, we extend it in place.
+ * Otherwise, we create a new LIST(EL) and copy the elements.
+ */
+static __isl_give LIST(EL) *FN(LIST(EL),grow)(__isl_take LIST(EL) *list, int n)
+{
+	isl_ctx *ctx;
+	int i, new_size;
+	LIST(EL) *res;
+
+	if (!list)
+		return NULL;
+	if (list->ref == 1 && list->n + n <= list->size)
+		return list;
+
+	ctx = FN(LIST(EL),get_ctx)(list);
+	new_size = ((list->n + n + 1) * 3) / 2;
+	if (list->ref == 1) {
+		res = isl_realloc(ctx, list, LIST(EL),
+			    sizeof(LIST(EL)) + (new_size - 1) * sizeof(EL *));
+		if (!res)
+			return FN(LIST(EL),free)(list);
+		res->size = new_size;
+		return res;
+	}
+
+	if (list->n + n <= list->size && list->size < new_size)
+		new_size = list->size;
+
+	res = FN(LIST(EL),alloc)(ctx, new_size);
+	if (!res)
+		return FN(LIST(EL),free)(list);
+
+	for (i = 0; i < list->n; ++i)
+		res = FN(LIST(EL),add)(res, FN(EL,copy)(list->p[i]));
+
+	FN(LIST(EL),free)(list);
+	return res;
+}
+
+__isl_give LIST(EL) *FN(LIST(EL),add)(__isl_take LIST(EL) *list,
+	__isl_take struct EL *el)
+{
+	list = FN(LIST(EL),grow)(list, 1);
+	if (!list || !el)
+		goto error;
+	list->p[list->n] = el;
+	list->n++;
+	return list;
+error:
+	FN(EL,free)(el);
+	FN(LIST(EL),free)(list);
+	return NULL;
+}
+
+/* Remove the "n" elements starting at "first" from "list".
+ */
+__isl_give LIST(EL) *FN(LIST(EL),drop)(__isl_take LIST(EL) *list,
+	unsigned first, unsigned n)
+{
+	int i;
+
+	if (!list)
+		return NULL;
+	if (first + n > list->n || first + n < first)
+		isl_die(list->ctx, isl_error_invalid,
+			"index out of bounds", return FN(LIST(EL),free)(list));
+	if (n == 0)
+		return list;
+	list = FN(LIST(EL),cow)(list);
+	if (!list)
+		return NULL;
+	for (i = 0; i < n; ++i)
+		FN(EL,free)(list->p[first + i]);
+	for (i = first; i + n < list->n; ++i)
+		list->p[i] = list->p[i + n];
+	list->n -= n;
+	return list;
+}
+
+/* Insert "el" at position "pos" in "list".
+ *
+ * If there is only one reference to "list" and if it already has space
+ * for one extra element, we insert it directly into "list".
+ * Otherwise, we create a new list consisting of "el" and copied
+ * elements from "list".
+ */
+__isl_give LIST(EL) *FN(LIST(EL),insert)(__isl_take LIST(EL) *list,
+	unsigned pos, __isl_take struct EL *el)
+{
+	int i;
+	isl_ctx *ctx;
+	LIST(EL) *res;
+
+	if (!list || !el)
+		goto error;
+	ctx = FN(LIST(EL),get_ctx)(list);
+	if (pos > list->n)
+		isl_die(ctx, isl_error_invalid,
+			"index out of bounds", goto error);
+
+	if (list->ref == 1 && list->size > list->n) {
+		for (i = list->n - 1; i >= pos; --i)
+			list->p[i + 1] = list->p[i];
+		list->n++;
+		list->p[pos] = el;
+		return list;
+	}
+
+	res = FN(LIST(EL),alloc)(ctx, list->n + 1);
+	for (i = 0; i < pos; ++i)
+		res = FN(LIST(EL),add)(res, FN(EL,copy)(list->p[i]));
+	res = FN(LIST(EL),add)(res, el);
+	for (i = pos; i < list->n; ++i)
+		res = FN(LIST(EL),add)(res, FN(EL,copy)(list->p[i]));
+	FN(LIST(EL),free)(list);
+
+	return res;
+error:
+	FN(EL,free)(el);
+	FN(LIST(EL),free)(list);
+	return NULL;
+}
+
+__isl_null LIST(EL) *FN(LIST(EL),free)(__isl_take LIST(EL) *list)
+{
+	int i;
+
+	if (!list)
+		return NULL;
+
+	if (--list->ref > 0)
+		return NULL;
+
+	isl_ctx_deref(list->ctx);
+	for (i = 0; i < list->n; ++i)
+		FN(EL,free)(list->p[i]);
+	free(list);
+
+	return NULL;
+}
+
+int FN(FN(LIST(EL),n),BASE)(__isl_keep LIST(EL) *list)
+{
+	return list ? list->n : 0;
+}
+
+__isl_give EL *FN(FN(LIST(EL),get),BASE)(__isl_keep LIST(EL) *list, int index)
+{
+	if (!list)
+		return NULL;
+	if (index < 0 || index >= list->n)
+		isl_die(list->ctx, isl_error_invalid,
+			"index out of bounds", return NULL);
+	return FN(EL,copy)(list->p[index]);
+}
+
+/* Replace the element at position "index" in "list" by "el".
+ */
+__isl_give LIST(EL) *FN(FN(LIST(EL),set),BASE)(__isl_take LIST(EL) *list,
+	int index, __isl_take EL *el)
+{
+	if (!list || !el)
+		goto error;
+	if (index < 0 || index >= list->n)
+		isl_die(list->ctx, isl_error_invalid,
+			"index out of bounds", goto error);
+	if (list->p[index] == el) {
+		FN(EL,free)(el);
+		return list;
+	}
+	list = FN(LIST(EL),cow)(list);
+	if (!list)
+		goto error;
+	FN(EL,free)(list->p[index]);
+	list->p[index] = el;
+	return list;
+error:
+	FN(EL,free)(el);
+	FN(LIST(EL),free)(list);
+	return NULL;
+}
+
+int FN(LIST(EL),foreach)(__isl_keep LIST(EL) *list,
+	int (*fn)(__isl_take EL *el, void *user), void *user)
+{
+	int i;
+
+	if (!list)
+		return -1;
+
+	for (i = 0; i < list->n; ++i) {
+		EL *el = FN(EL,copy(list->p[i]));
+		if (!el)
+			return -1;
+		if (fn(el, user) < 0)
+			return -1;
+	}
+
+	return 0;
+}
+
+/* Internal data structure for isl_*_list_sort.
+ *
+ * "cmp" is the original comparison function.
+ * "user" is a user provided pointer that should be passed to "cmp".
+ */
+S(LIST(EL),sort_data) {
+	int (*cmp)(__isl_keep EL *a, __isl_keep EL *b, void *user);
+	void *user;
+};
+
+/* Compare two entries of an isl_*_list based on the user provided
+ * comparison function on pairs of isl_* objects.
+ */
+static int FN(LIST(EL),cmp)(const void *a, const void *b, void *user)
+{
+	S(LIST(EL),sort_data) *data = user;
+	EL * const *el1 = a;
+	EL * const *el2 = b;
+
+	return data->cmp(*el1, *el2, data->user);
+}
+
+/* Sort the elements of "list" in ascending order according to
+ * comparison function "cmp".
+ */
+__isl_give LIST(EL) *FN(LIST(EL),sort)(__isl_take LIST(EL) *list,
+	int (*cmp)(__isl_keep EL *a, __isl_keep EL *b, void *user), void *user)
+{
+	S(LIST(EL),sort_data) data = { cmp, user };
+
+	if (!list)
+		return NULL;
+	if (list->n <= 1)
+		return list;
+	list = FN(LIST(EL),cow)(list);
+	if (!list)
+		return NULL;
+
+	if (isl_sort(list->p, list->n, sizeof(list->p[0]),
+			&FN(LIST(EL),cmp), &data) < 0)
+		return FN(LIST(EL),free)(list);
+
+	return list;
+}
+
+/* Internal data structure for isl_*_list_foreach_scc.
+ *
+ * "list" is the original list.
+ * "follows" is the user provided callback that defines the edges of the graph.
+ */
+S(LIST(EL),foreach_scc_data) {
+	LIST(EL) *list;
+	int (*follows)(__isl_keep EL *a, __isl_keep EL *b, void *user);
+	void *follows_user;
+};
+
+/* Does element i of data->list follow element j?
+ *
+ * Use the user provided callback to find out.
+ */
+static int FN(LIST(EL),follows)(int i, int j, void *user)
+{
+	S(LIST(EL),foreach_scc_data) *data = user;
+
+	return data->follows(data->list->p[i], data->list->p[j],
+				data->follows_user);
+}
+
+/* Call "fn" on the sublist of "list" that consists of the elements
+ * with indices specified by the "n" elements of "pos".
+ */
+static int FN(LIST(EL),call_on_scc)(__isl_keep LIST(EL) *list, int *pos, int n,
+	int (*fn)(__isl_take LIST(EL) *scc, void *user), void *user)
+{
+	int i;
+	isl_ctx *ctx;
+	LIST(EL) *slice;
+
+	ctx = FN(LIST(EL),get_ctx)(list);
+	slice = FN(LIST(EL),alloc)(ctx, n);
+	for (i = 0; i < n; ++i) {
+		EL *el;
+
+		el = FN(EL,copy)(list->p[pos[i]]);
+		slice = FN(LIST(EL),add)(slice, el);
+	}
+
+	return fn(slice, user);
+}
+
+/* Call "fn" on each of the strongly connected components (SCCs) of
+ * the graph with as vertices the elements of "list" and
+ * a directed edge from node b to node a iff follows(a, b)
+ * returns 1.  follows should return -1 on error.
+ *
+ * If SCC a contains a node i that follows a node j in another SCC b
+ * (i.e., follows(i, j, user) returns 1), then fn will be called on SCC a
+ * after being called on SCC b.
+ *
+ * We simply call isl_tarjan_graph_init, extract the SCCs from the result and
+ * call fn on each of them.
+ */
+int FN(LIST(EL),foreach_scc)(__isl_keep LIST(EL) *list,
+	int (*follows)(__isl_keep EL *a, __isl_keep EL *b, void *user),
+	void *follows_user,
+	int (*fn)(__isl_take LIST(EL) *scc, void *user), void *fn_user)
+{
+	S(LIST(EL),foreach_scc_data) data = { list, follows, follows_user };
+	int i, n;
+	isl_ctx *ctx;
+	struct isl_tarjan_graph *g;
+
+	if (!list)
+		return -1;
+	if (list->n == 0)
+		return 0;
+	if (list->n == 1)
+		return fn(FN(LIST(EL),copy)(list), fn_user);
+
+	ctx = FN(LIST(EL),get_ctx)(list);
+	n = list->n;
+	g = isl_tarjan_graph_init(ctx, n, &FN(LIST(EL),follows), &data);
+	if (!g)
+		return -1;
+
+	i = 0;
+	do {
+		int first;
+
+		if (g->order[i] == -1)
+			isl_die(ctx, isl_error_internal, "cannot happen",
+				break);
+		first = i;
+		while (g->order[i] != -1) {
+			++i; --n;
+		}
+		if (first == 0 && n == 0) {
+			isl_tarjan_graph_free(g);
+			return fn(FN(LIST(EL),copy)(list), fn_user);
+		}
+		if (FN(LIST(EL),call_on_scc)(list, g->order + first, i - first,
+					    fn, fn_user) < 0)
+			break;
+		++i;
+	} while (n);
+
+	isl_tarjan_graph_free(g);
+
+	return n > 0 ? -1 : 0;
+}
+
+__isl_give LIST(EL) *FN(FN(LIST(EL),from),BASE)(__isl_take EL *el)
+{
+	isl_ctx *ctx;
+	LIST(EL) *list;
+
+	if (!el)
+		return NULL;
+	ctx = FN(EL,get_ctx)(el);
+	list = FN(LIST(EL),alloc)(ctx, 1);
+	if (!list)
+		goto error;
+	list = FN(LIST(EL),add)(list, el);
+	return list;
+error:
+	FN(EL,free)(el);
+	return NULL;
+}
+
+__isl_give LIST(EL) *FN(LIST(EL),concat)(__isl_take LIST(EL) *list1,
+	__isl_take LIST(EL) *list2)
+{
+	int i;
+	isl_ctx *ctx;
+	LIST(EL) *res;
+
+	if (!list1 || !list2)
+		goto error;
+
+	ctx = FN(LIST(EL),get_ctx)(list1);
+	res = FN(LIST(EL),alloc)(ctx, list1->n + list2->n);
+	for (i = 0; i < list1->n; ++i)
+		res = FN(LIST(EL),add)(res, FN(EL,copy)(list1->p[i]));
+	for (i = 0; i < list2->n; ++i)
+		res = FN(LIST(EL),add)(res, FN(EL,copy)(list2->p[i]));
+
+	FN(LIST(EL),free)(list1);
+	FN(LIST(EL),free)(list2);
+	return res;
+error:
+	FN(LIST(EL),free)(list1);
+	FN(LIST(EL),free)(list2);
+	return NULL;
+}
+
+__isl_give isl_printer *CAT(isl_printer_print_,LIST(BASE))(
+	__isl_take isl_printer *p, __isl_keep LIST(EL) *list)
+{
+	int i;
+
+	if (!p || !list)
+		goto error;
+	p = isl_printer_print_str(p, "(");
+	for (i = 0; i < list->n; ++i) {
+		if (i)
+			p = isl_printer_print_str(p, ",");
+		p = CAT(isl_printer_print_,BASE)(p, list->p[i]);
+	}
+	p = isl_printer_print_str(p, ")");
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+void FN(LIST(EL),dump)(__isl_keep LIST(EL) *list)
+{
+	isl_printer *printer;
+
+	if (!list)
+		return;
+
+	printer = isl_printer_to_file(FN(LIST(EL),get_ctx)(list), stderr);
+	printer = CAT(isl_printer_print_,LIST(BASE))(printer, list);
+	printer = isl_printer_end_line(printer);
+
+	isl_printer_free(printer);
+}
diff --git a/polly/lib/External/isl/isl_list_templ.h b/polly/lib/External/isl/isl_list_templ.h
new file mode 100644
index 00000000000..893f9d91d4c
--- /dev/null
+++ b/polly/lib/External/isl/isl_list_templ.h
@@ -0,0 +1,16 @@
+#define xFN(TYPE,NAME) TYPE ## _ ## NAME
+#define FN(TYPE,NAME) xFN(TYPE,NAME)
+#define xLIST(EL) EL ## _list
+#define LIST(EL) xLIST(EL)
+
+struct LIST(EL) {
+	int ref;
+	isl_ctx *ctx;
+
+	int n;
+
+	size_t size;
+	struct EL *p[1];
+};
+
+__isl_give LIST(EL) *FN(LIST(EL),dup)(__isl_keep LIST(EL) *list);
diff --git a/polly/lib/External/isl/isl_local_space.c b/polly/lib/External/isl/isl_local_space.c
new file mode 100644
index 00000000000..4de3b357de1
--- /dev/null
+++ b/polly/lib/External/isl/isl_local_space.c
@@ -0,0 +1,1363 @@
+/*
+ * Copyright 2011      INRIA Saclay
+ * Copyright 2012-2014 Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl_local_space_private.h>
+#include <isl_space_private.h>
+#include <isl_mat_private.h>
+#include <isl_aff_private.h>
+#include <isl_vec_private.h>
+#include <isl_seq.h>
+
+isl_ctx *isl_local_space_get_ctx(__isl_keep isl_local_space *ls)
+{
+	return ls ? ls->dim->ctx : NULL;
+}
+
+__isl_give isl_local_space *isl_local_space_alloc_div(__isl_take isl_space *dim,
+	__isl_take isl_mat *div)
+{
+	isl_ctx *ctx;
+	isl_local_space *ls = NULL;
+
+	if (!dim || !div)
+		goto error;
+
+	ctx = isl_space_get_ctx(dim);
+	ls = isl_calloc_type(ctx, struct isl_local_space);
+	if (!ls)
+		goto error;
+
+	ls->ref = 1;
+	ls->dim = dim;
+	ls->div = div;
+
+	return ls;
+error:
+	isl_mat_free(div);
+	isl_space_free(dim);
+	isl_local_space_free(ls);
+	return NULL;
+}
+
+__isl_give isl_local_space *isl_local_space_alloc(__isl_take isl_space *dim,
+	unsigned n_div)
+{
+	isl_ctx *ctx;
+	isl_mat *div;
+	unsigned total;
+
+	if (!dim)
+		return NULL;
+
+	total = isl_space_dim(dim, isl_dim_all);
+
+	ctx = isl_space_get_ctx(dim);
+	div = isl_mat_alloc(ctx, n_div, 1 + 1 + total + n_div);
+	return isl_local_space_alloc_div(dim, div);
+}
+
+__isl_give isl_local_space *isl_local_space_from_space(__isl_take isl_space *dim)
+{
+	return isl_local_space_alloc(dim, 0);
+}
+
+__isl_give isl_local_space *isl_local_space_copy(__isl_keep isl_local_space *ls)
+{
+	if (!ls)
+		return NULL;
+
+	ls->ref++;
+	return ls;
+}
+
+__isl_give isl_local_space *isl_local_space_dup(__isl_keep isl_local_space *ls)
+{
+	if (!ls)
+		return NULL;
+
+	return isl_local_space_alloc_div(isl_space_copy(ls->dim),
+					 isl_mat_copy(ls->div));
+
+}
+
+__isl_give isl_local_space *isl_local_space_cow(__isl_take isl_local_space *ls)
+{
+	if (!ls)
+		return NULL;
+
+	if (ls->ref == 1)
+		return ls;
+	ls->ref--;
+	return isl_local_space_dup(ls);
+}
+
+__isl_null isl_local_space *isl_local_space_free(
+	__isl_take isl_local_space *ls)
+{
+	if (!ls)
+		return NULL;
+
+	if (--ls->ref > 0)
+		return NULL;
+
+	isl_space_free(ls->dim);
+	isl_mat_free(ls->div);
+
+	free(ls);
+
+	return NULL;
+}
+
+/* Is the local space that of a parameter domain?
+ */
+int isl_local_space_is_params(__isl_keep isl_local_space *ls)
+{
+	if (!ls)
+		return -1;
+	return isl_space_is_params(ls->dim);
+}
+
+/* Is the local space that of a set?
+ */
+int isl_local_space_is_set(__isl_keep isl_local_space *ls)
+{
+	return ls ? isl_space_is_set(ls->dim) : -1;
+}
+
+/* Return true if the two local spaces are identical, with identical
+ * expressions for the integer divisions.
+ */
+int isl_local_space_is_equal(__isl_keep isl_local_space *ls1,
+	__isl_keep isl_local_space *ls2)
+{
+	int equal;
+
+	if (!ls1 || !ls2)
+		return -1;
+
+	equal = isl_space_is_equal(ls1->dim, ls2->dim);
+	if (equal < 0 || !equal)
+		return equal;
+
+	if (!isl_local_space_divs_known(ls1))
+		return 0;
+	if (!isl_local_space_divs_known(ls2))
+		return 0;
+
+	return isl_mat_is_equal(ls1->div, ls2->div);
+}
+
+/* Compare two isl_local_spaces.
+ *
+ * Return -1 if "ls1" is "smaller" than "ls2", 1 if "ls1" is "greater"
+ * than "ls2" and 0 if they are equal.
+ *
+ * The order is fairly arbitrary.  We do "prefer" divs that only involve
+ * earlier dimensions in the sense that we consider local spaces where
+ * the first differing div involves earlier dimensions to be smaller.
+ */
+int isl_local_space_cmp(__isl_keep isl_local_space *ls1,
+	__isl_keep isl_local_space *ls2)
+{
+	int i;
+	int cmp;
+	int known1, known2;
+	int last1, last2;
+	int n_col;
+
+	if (ls1 == ls2)
+		return 0;
+	if (!ls1)
+		return -1;
+	if (!ls2)
+		return 1;
+
+	cmp = isl_space_cmp(ls1->dim, ls2->dim);
+	if (cmp != 0)
+		return cmp;
+
+	if (ls1->div->n_row != ls2->div->n_row)
+		return ls1->div->n_row - ls2->div->n_row;
+
+	n_col = isl_mat_cols(ls1->div);
+	for (i = 0; i < ls1->div->n_row; ++i) {
+		known1 = isl_local_space_div_is_known(ls1, i);
+		known2 = isl_local_space_div_is_known(ls2, i);
+		if (!known1 && !known2)
+			continue;
+		if (!known1)
+			return 1;
+		if (!known2)
+			return -1;
+		last1 = isl_seq_last_non_zero(ls1->div->row[i] + 1, n_col - 1);
+		last2 = isl_seq_last_non_zero(ls2->div->row[i] + 1, n_col - 1);
+		if (last1 != last2)
+			return last1 - last2;
+		cmp = isl_seq_cmp(ls1->div->row[i], ls2->div->row[i], n_col);
+		if (cmp != 0)
+			return cmp;
+	}
+
+	return 0;
+}
+
+int isl_local_space_dim(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type)
+{
+	if (!ls)
+		return 0;
+	if (type == isl_dim_div)
+		return ls->div->n_row;
+	if (type == isl_dim_all)
+		return isl_space_dim(ls->dim, isl_dim_all) + ls->div->n_row;
+	return isl_space_dim(ls->dim, type);
+}
+
+unsigned isl_local_space_offset(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type)
+{
+	isl_space *dim;
+
+	if (!ls)
+		return 0;
+
+	dim = ls->dim;
+	switch (type) {
+	case isl_dim_cst:	return 0;
+	case isl_dim_param:	return 1;
+	case isl_dim_in:	return 1 + dim->nparam;
+	case isl_dim_out:	return 1 + dim->nparam + dim->n_in;
+	case isl_dim_div:	return 1 + dim->nparam + dim->n_in + dim->n_out;
+	default:		return 0;
+	}
+}
+
+/* Return the position of the dimension of the given type and name
+ * in "ls".
+ * Return -1 if no such dimension can be found.
+ */
+int isl_local_space_find_dim_by_name(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type, const char *name)
+{
+	if (!ls)
+		return -1;
+	if (type == isl_dim_div)
+		return -1;
+	return isl_space_find_dim_by_name(ls->dim, type, name);
+}
+
+/* Does the given dimension have a name?
+ */
+int isl_local_space_has_dim_name(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos)
+{
+	return ls ? isl_space_has_dim_name(ls->dim, type, pos) : -1;
+}
+
+const char *isl_local_space_get_dim_name(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos)
+{
+	return ls ? isl_space_get_dim_name(ls->dim, type, pos) : NULL;
+}
+
+int isl_local_space_has_dim_id(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos)
+{
+	return ls ? isl_space_has_dim_id(ls->dim, type, pos) : -1;
+}
+
+__isl_give isl_id *isl_local_space_get_dim_id(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos)
+{
+	return ls ? isl_space_get_dim_id(ls->dim, type, pos) : NULL;
+}
+
+__isl_give isl_aff *isl_local_space_get_div(__isl_keep isl_local_space *ls,
+	int pos)
+{
+	isl_aff *aff;
+
+	if (!ls)
+		return NULL;
+
+	if (pos < 0 || pos >= ls->div->n_row)
+		isl_die(isl_local_space_get_ctx(ls), isl_error_invalid,
+			"index out of bounds", return NULL);
+
+	if (isl_int_is_zero(ls->div->row[pos][0]))
+		isl_die(isl_local_space_get_ctx(ls), isl_error_invalid,
+			"expression of div unknown", return NULL);
+	if (!isl_local_space_is_set(ls))
+		isl_die(isl_local_space_get_ctx(ls), isl_error_invalid,
+			"cannot represent divs of map spaces", return NULL);
+
+	aff = isl_aff_alloc(isl_local_space_copy(ls));
+	if (!aff)
+		return NULL;
+	isl_seq_cpy(aff->v->el, ls->div->row[pos], aff->v->size);
+	return aff;
+}
+
+__isl_give isl_space *isl_local_space_get_space(__isl_keep isl_local_space *ls)
+{
+	if (!ls)
+		return NULL;
+
+	return isl_space_copy(ls->dim);
+}
+
+/* Replace the identifier of the tuple of type "type" by "id".
+ */
+__isl_give isl_local_space *isl_local_space_set_tuple_id(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type type, __isl_take isl_id *id)
+{
+	ls = isl_local_space_cow(ls);
+	if (!ls)
+		goto error;
+	ls->dim = isl_space_set_tuple_id(ls->dim, type, id);
+	if (!ls->dim)
+		return isl_local_space_free(ls);
+	return ls;
+error:
+	isl_id_free(id);
+	return NULL;
+}
+
+__isl_give isl_local_space *isl_local_space_set_dim_name(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos, const char *s)
+{
+	ls = isl_local_space_cow(ls);
+	if (!ls)
+		return NULL;
+	ls->dim = isl_space_set_dim_name(ls->dim, type, pos, s);
+	if (!ls->dim)
+		return isl_local_space_free(ls);
+
+	return ls;
+}
+
+__isl_give isl_local_space *isl_local_space_set_dim_id(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id)
+{
+	ls = isl_local_space_cow(ls);
+	if (!ls)
+		goto error;
+	ls->dim = isl_space_set_dim_id(ls->dim, type, pos, id);
+	if (!ls->dim)
+		return isl_local_space_free(ls);
+
+	return ls;
+error:
+	isl_id_free(id);
+	return NULL;
+}
+
+__isl_give isl_local_space *isl_local_space_reset_space(
+	__isl_take isl_local_space *ls, __isl_take isl_space *dim)
+{
+	ls = isl_local_space_cow(ls);
+	if (!ls || !dim)
+		goto error;
+
+	isl_space_free(ls->dim);
+	ls->dim = dim;
+
+	return ls;
+error:
+	isl_local_space_free(ls);
+	isl_space_free(dim);
+	return NULL;
+}
+
+/* Reorder the columns of the given div definitions according to the
+ * given reordering.
+ * The order of the divs themselves is assumed not to change.
+ */
+static __isl_give isl_mat *reorder_divs(__isl_take isl_mat *div,
+	__isl_take isl_reordering *r)
+{
+	int i, j;
+	isl_mat *mat;
+	int extra;
+
+	if (!div || !r)
+		goto error;
+
+	extra = isl_space_dim(r->dim, isl_dim_all) + div->n_row - r->len;
+	mat = isl_mat_alloc(div->ctx, div->n_row, div->n_col + extra);
+	if (!mat)
+		goto error;
+
+	for (i = 0; i < div->n_row; ++i) {
+		isl_seq_cpy(mat->row[i], div->row[i], 2);
+		isl_seq_clr(mat->row[i] + 2, mat->n_col - 2);
+		for (j = 0; j < r->len; ++j)
+			isl_int_set(mat->row[i][2 + r->pos[j]],
+				    div->row[i][2 + j]);
+	}
+
+	isl_reordering_free(r);
+	isl_mat_free(div);
+	return mat;
+error:
+	isl_reordering_free(r);
+	isl_mat_free(div);
+	return NULL;
+}
+
+/* Reorder the dimensions of "ls" according to the given reordering.
+ * The reordering r is assumed to have been extended with the local
+ * variables, leaving them in the same order.
+ */
+__isl_give isl_local_space *isl_local_space_realign(
+	__isl_take isl_local_space *ls, __isl_take isl_reordering *r)
+{
+	ls = isl_local_space_cow(ls);
+	if (!ls || !r)
+		goto error;
+
+	ls->div = reorder_divs(ls->div, isl_reordering_copy(r));
+	if (!ls->div)
+		goto error;
+
+	ls = isl_local_space_reset_space(ls, isl_space_copy(r->dim));
+
+	isl_reordering_free(r);
+	return ls;
+error:
+	isl_local_space_free(ls);
+	isl_reordering_free(r);
+	return NULL;
+}
+
+__isl_give isl_local_space *isl_local_space_add_div(
+	__isl_take isl_local_space *ls, __isl_take isl_vec *div)
+{
+	ls = isl_local_space_cow(ls);
+	if (!ls || !div)
+		goto error;
+
+	if (ls->div->n_col != div->size)
+		isl_die(isl_local_space_get_ctx(ls), isl_error_invalid,
+			"incompatible dimensions", goto error);
+
+	ls->div = isl_mat_add_zero_cols(ls->div, 1);
+	ls->div = isl_mat_add_rows(ls->div, 1);
+	if (!ls->div)
+		goto error;
+
+	isl_seq_cpy(ls->div->row[ls->div->n_row - 1], div->el, div->size);
+	isl_int_set_si(ls->div->row[ls->div->n_row - 1][div->size], 0);
+
+	isl_vec_free(div);
+	return ls;
+error:
+	isl_local_space_free(ls);
+	isl_vec_free(div);
+	return NULL;
+}
+
+__isl_give isl_local_space *isl_local_space_replace_divs(
+	__isl_take isl_local_space *ls, __isl_take isl_mat *div)
+{
+	ls = isl_local_space_cow(ls);
+
+	if (!ls || !div)
+		goto error;
+
+	isl_mat_free(ls->div);
+	ls->div = div;
+	return ls;
+error:
+	isl_mat_free(div);
+	isl_local_space_free(ls);
+	return NULL;
+}
+
+/* Copy row "s" of "src" to row "d" of "dst", applying the expansion
+ * defined by "exp".
+ */
+static void expand_row(__isl_keep isl_mat *dst, int d,
+	__isl_keep isl_mat *src, int s, int *exp)
+{
+	int i;
+	unsigned c = src->n_col - src->n_row;
+
+	isl_seq_cpy(dst->row[d], src->row[s], c);
+	isl_seq_clr(dst->row[d] + c, dst->n_col - c);
+
+	for (i = 0; i < s; ++i)
+		isl_int_set(dst->row[d][c + exp[i]], src->row[s][c + i]);
+}
+
+/* Compare (known) divs.
+ * Return non-zero if at least one of the two divs is unknown.
+ * In particular, if both divs are unknown, we respect their
+ * current order.  Otherwise, we sort the known div after the unknown
+ * div only if the known div depends on the unknown div.
+ */
+static int cmp_row(isl_int *row_i, isl_int *row_j, int i, int j,
+	unsigned n_row, unsigned n_col)
+{
+	int li, lj;
+	int unknown_i, unknown_j;
+
+	unknown_i = isl_int_is_zero(row_i[0]);
+	unknown_j = isl_int_is_zero(row_j[0]);
+
+	if (unknown_i && unknown_j)
+		return i - j;
+
+	if (unknown_i)
+		li = n_col - n_row + i;
+	else
+		li = isl_seq_last_non_zero(row_i, n_col);
+	if (unknown_j)
+		lj = n_col - n_row + j;
+	else
+		lj = isl_seq_last_non_zero(row_j, n_col);
+
+	if (li != lj)
+		return li - lj;
+
+	return isl_seq_cmp(row_i, row_j, n_col);
+}
+
+/* Call cmp_row for divs in a matrix.
+ */
+int isl_mat_cmp_div(__isl_keep isl_mat *div, int i, int j)
+{
+	return cmp_row(div->row[i], div->row[j], i, j, div->n_row, div->n_col);
+}
+
+/* Call cmp_row for divs in a basic map.
+ */
+static int bmap_cmp_row(__isl_keep isl_basic_map *bmap, int i, int j,
+	unsigned total)
+{
+	return cmp_row(bmap->div[i], bmap->div[j], i, j, bmap->n_div, total);
+}
+
+/* Sort the divs in "bmap".
+ *
+ * We first make sure divs are placed after divs on which they depend.
+ * Then we perform a simple insertion sort based on the same ordering
+ * that is used in isl_merge_divs.
+ */
+__isl_give isl_basic_map *isl_basic_map_sort_divs(
+	__isl_take isl_basic_map *bmap)
+{
+	int i, j;
+	unsigned total;
+
+	bmap = isl_basic_map_order_divs(bmap);
+	if (!bmap)
+		return NULL;
+	if (bmap->n_div <= 1)
+		return bmap;
+
+	total = 2 + isl_basic_map_total_dim(bmap);
+	for (i = 1; i < bmap->n_div; ++i) {
+		for (j = i - 1; j >= 0; --j) {
+			if (bmap_cmp_row(bmap, j, j + 1, total) <= 0)
+				break;
+			isl_basic_map_swap_div(bmap, j, j + 1);
+		}
+	}
+
+	return bmap;
+}
+
+/* Sort the divs in the basic maps of "map".
+ */
+__isl_give isl_map *isl_map_sort_divs(__isl_take isl_map *map)
+{
+	return isl_map_inline_foreach_basic_map(map, &isl_basic_map_sort_divs);
+}
+
+/* Combine the two lists of divs into a single list.
+ * For each row i in div1, exp1[i] is set to the position of the corresponding
+ * row in the result.  Similarly for div2 and exp2.
+ * This function guarantees
+ *	exp1[i] >= i
+ *	exp1[i+1] > exp1[i]
+ * For optimal merging, the two input list should have been sorted.
+ */
+__isl_give isl_mat *isl_merge_divs(__isl_keep isl_mat *div1,
+	__isl_keep isl_mat *div2, int *exp1, int *exp2)
+{
+	int i, j, k;
+	isl_mat *div = NULL;
+	unsigned d;
+
+	if (!div1 || !div2)
+		return NULL;
+
+	d = div1->n_col - div1->n_row;
+	div = isl_mat_alloc(div1->ctx, 1 + div1->n_row + div2->n_row,
+				d + div1->n_row + div2->n_row);
+	if (!div)
+		return NULL;
+
+	for (i = 0, j = 0, k = 0; i < div1->n_row && j < div2->n_row; ++k) {
+		int cmp;
+
+		expand_row(div, k, div1, i, exp1);
+		expand_row(div, k + 1, div2, j, exp2);
+
+		cmp = isl_mat_cmp_div(div, k, k + 1);
+		if (cmp == 0) {
+			exp1[i++] = k;
+			exp2[j++] = k;
+		} else if (cmp < 0) {
+			exp1[i++] = k;
+		} else {
+			exp2[j++] = k;
+			isl_seq_cpy(div->row[k], div->row[k + 1], div->n_col);
+		}
+	}
+	for (; i < div1->n_row; ++i, ++k) {
+		expand_row(div, k, div1, i, exp1);
+		exp1[i] = k;
+	}
+	for (; j < div2->n_row; ++j, ++k) {
+		expand_row(div, k, div2, j, exp2);
+		exp2[j] = k;
+	}
+
+	div->n_row = k;
+	div->n_col = d + k;
+
+	return div;
+}
+
+/* Swap divs "a" and "b" in "ls".
+ */
+__isl_give isl_local_space *isl_local_space_swap_div(
+	__isl_take isl_local_space *ls, int a, int b)
+{
+	int offset;
+
+	ls = isl_local_space_cow(ls);
+	if (!ls)
+		return NULL;
+	if (a < 0 || a >= ls->div->n_row || b < 0 || b >= ls->div->n_row)
+		isl_die(isl_local_space_get_ctx(ls), isl_error_invalid,
+			"index out of bounds", return isl_local_space_free(ls));
+	offset = ls->div->n_col - ls->div->n_row;
+	ls->div = isl_mat_swap_cols(ls->div, offset + a, offset + b);
+	ls->div = isl_mat_swap_rows(ls->div, a, b);
+	if (!ls->div)
+		return isl_local_space_free(ls);
+	return ls;
+}
+
+/* Construct a local space that contains all the divs in either
+ * "ls1" or "ls2".
+ */
+__isl_give isl_local_space *isl_local_space_intersect(
+	__isl_take isl_local_space *ls1, __isl_take isl_local_space *ls2)
+{
+	isl_ctx *ctx;
+	int *exp1 = NULL;
+	int *exp2 = NULL;
+	isl_mat *div;
+	int equal;
+
+	if (!ls1 || !ls2)
+		goto error;
+
+	ctx = isl_local_space_get_ctx(ls1);
+	if (!isl_space_is_equal(ls1->dim, ls2->dim))
+		isl_die(ctx, isl_error_invalid,
+			"spaces should be identical", goto error);
+
+	if (ls2->div->n_row == 0) {
+		isl_local_space_free(ls2);
+		return ls1;
+	}
+
+	if (ls1->div->n_row == 0) {
+		isl_local_space_free(ls1);
+		return ls2;
+	}
+
+	exp1 = isl_alloc_array(ctx, int, ls1->div->n_row);
+	exp2 = isl_alloc_array(ctx, int, ls2->div->n_row);
+	if (!exp1 || !exp2)
+		goto error;
+
+	div = isl_merge_divs(ls1->div, ls2->div, exp1, exp2);
+	if (!div)
+		goto error;
+
+	equal = isl_mat_is_equal(ls1->div, div);
+	if (equal < 0)
+		goto error;
+	if (!equal)
+		ls1 = isl_local_space_cow(ls1);
+	if (!ls1)
+		goto error;
+
+	free(exp1);
+	free(exp2);
+	isl_local_space_free(ls2);
+	isl_mat_free(ls1->div);
+	ls1->div = div;
+
+	return ls1;
+error:
+	free(exp1);
+	free(exp2);
+	isl_local_space_free(ls1);
+	isl_local_space_free(ls2);
+	return NULL;
+}
+
+/* Does "ls" have an explicit representation for div "div"?
+ */
+int isl_local_space_div_is_known(__isl_keep isl_local_space *ls, int div)
+{
+	if (!ls)
+		return -1;
+	if (div < 0 || div >= ls->div->n_row)
+		isl_die(isl_local_space_get_ctx(ls), isl_error_invalid,
+			"position out of bounds", return -1);
+	return !isl_int_is_zero(ls->div->row[div][0]);
+}
+
+int isl_local_space_divs_known(__isl_keep isl_local_space *ls)
+{
+	int i;
+
+	if (!ls)
+		return -1;
+
+	for (i = 0; i < ls->div->n_row; ++i)
+		if (isl_int_is_zero(ls->div->row[i][0]))
+			return 0;
+
+	return 1;
+}
+
+__isl_give isl_local_space *isl_local_space_domain(
+	__isl_take isl_local_space *ls)
+{
+	ls = isl_local_space_drop_dims(ls, isl_dim_out,
+					0, isl_local_space_dim(ls, isl_dim_out));
+	ls = isl_local_space_cow(ls);
+	if (!ls)
+		return NULL;
+	ls->dim = isl_space_domain(ls->dim);
+	if (!ls->dim)
+		return isl_local_space_free(ls);
+	return ls;
+}
+
+__isl_give isl_local_space *isl_local_space_range(
+	__isl_take isl_local_space *ls)
+{
+	ls = isl_local_space_drop_dims(ls, isl_dim_in,
+					0, isl_local_space_dim(ls, isl_dim_in));
+	ls = isl_local_space_cow(ls);
+	if (!ls)
+		return NULL;
+
+	ls->dim = isl_space_range(ls->dim);
+	if (!ls->dim)
+		return isl_local_space_free(ls);
+	return ls;
+}
+
+/* Construct a local space for a map that has the given local
+ * space as domain and that has a zero-dimensional range.
+ */
+__isl_give isl_local_space *isl_local_space_from_domain(
+	__isl_take isl_local_space *ls)
+{
+	ls = isl_local_space_cow(ls);
+	if (!ls)
+		return NULL;
+	ls->dim = isl_space_from_domain(ls->dim);
+	if (!ls->dim)
+		return isl_local_space_free(ls);
+	return ls;
+}
+
+__isl_give isl_local_space *isl_local_space_add_dims(
+	__isl_take isl_local_space *ls, enum isl_dim_type type, unsigned n)
+{
+	int pos;
+
+	if (!ls)
+		return NULL;
+	pos = isl_local_space_dim(ls, type);
+	return isl_local_space_insert_dims(ls, type, pos, n);
+}
+
+/* Remove common factor of non-constant terms and denominator.
+ */
+static void normalize_div(__isl_keep isl_local_space *ls, int div)
+{
+	isl_ctx *ctx = ls->div->ctx;
+	unsigned total = ls->div->n_col - 2;
+
+	isl_seq_gcd(ls->div->row[div] + 2, total, &ctx->normalize_gcd);
+	isl_int_gcd(ctx->normalize_gcd,
+		    ctx->normalize_gcd, ls->div->row[div][0]);
+	if (isl_int_is_one(ctx->normalize_gcd))
+		return;
+
+	isl_seq_scale_down(ls->div->row[div] + 2, ls->div->row[div] + 2,
+			    ctx->normalize_gcd, total);
+	isl_int_divexact(ls->div->row[div][0], ls->div->row[div][0],
+			    ctx->normalize_gcd);
+	isl_int_fdiv_q(ls->div->row[div][1], ls->div->row[div][1],
+			    ctx->normalize_gcd);
+}
+
+/* Exploit the equalities in "eq" to simplify the expressions of
+ * the integer divisions in "ls".
+ * The integer divisions in "ls" are assumed to appear as regular
+ * dimensions in "eq".
+ */
+__isl_give isl_local_space *isl_local_space_substitute_equalities(
+	__isl_take isl_local_space *ls, __isl_take isl_basic_set *eq)
+{
+	int i, j, k;
+	unsigned total;
+	unsigned n_div;
+
+	if (!ls || !eq)
+		goto error;
+
+	total = isl_space_dim(eq->dim, isl_dim_all);
+	if (isl_local_space_dim(ls, isl_dim_all) != total)
+		isl_die(isl_local_space_get_ctx(ls), isl_error_invalid,
+			"spaces don't match", goto error);
+	total++;
+	n_div = eq->n_div;
+	for (i = 0; i < eq->n_eq; ++i) {
+		j = isl_seq_last_non_zero(eq->eq[i], total + n_div);
+		if (j < 0 || j == 0 || j >= total)
+			continue;
+
+		for (k = 0; k < ls->div->n_row; ++k) {
+			if (isl_int_is_zero(ls->div->row[k][1 + j]))
+				continue;
+			ls = isl_local_space_cow(ls);
+			if (!ls)
+				goto error;
+			ls->div = isl_mat_cow(ls->div);
+			if (!ls->div)
+				goto error;
+			isl_seq_elim(ls->div->row[k] + 1, eq->eq[i], j, total,
+					&ls->div->row[k][0]);
+			normalize_div(ls, k);
+		}
+	}
+
+	isl_basic_set_free(eq);
+	return ls;
+error:
+	isl_basic_set_free(eq);
+	isl_local_space_free(ls);
+	return NULL;
+}
+
+/* Plug in the affine expressions "subs" of length "subs_len" (including
+ * the denominator and the constant term) into the variable at position "pos"
+ * of the "n" div expressions starting at "first".
+ *
+ * Let i be the dimension to replace and let "subs" be of the form
+ *
+ *	f/d
+ *
+ * Any integer division starting at "first" with a non-zero coefficient for i,
+ *
+ *	floor((a i + g)/m)
+ *
+ * is replaced by
+ *
+ *	floor((a f + d g)/(m d))
+ */
+__isl_give isl_local_space *isl_local_space_substitute_seq(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos, isl_int *subs, int subs_len,
+	int first, int n)
+{
+	int i;
+	isl_int v;
+
+	if (n == 0)
+		return ls;
+	ls = isl_local_space_cow(ls);
+	if (!ls)
+		return NULL;
+	ls->div = isl_mat_cow(ls->div);
+	if (!ls->div)
+		return isl_local_space_free(ls);
+
+	if (first + n > ls->div->n_row)
+		isl_die(isl_local_space_get_ctx(ls), isl_error_invalid,
+			"index out of bounds", return isl_local_space_free(ls));
+
+	pos += isl_local_space_offset(ls, type);
+
+	isl_int_init(v);
+	for (i = first; i < ls->div->n_row; ++i) {
+		if (isl_int_is_zero(ls->div->row[i][1 + pos]))
+			continue;
+		isl_seq_substitute(ls->div->row[i], pos, subs,
+			ls->div->n_col, subs_len, v);
+		normalize_div(ls, i);
+	}
+	isl_int_clear(v);
+
+	return ls;
+}
+
+/* Plug in "subs" for dimension "type", "pos" in the integer divisions
+ * of "ls".
+ *
+ * Let i be the dimension to replace and let "subs" be of the form
+ *
+ *	f/d
+ *
+ * Any integer division with a non-zero coefficient for i,
+ *
+ *	floor((a i + g)/m)
+ *
+ * is replaced by
+ *
+ *	floor((a f + d g)/(m d))
+ */
+__isl_give isl_local_space *isl_local_space_substitute(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos, __isl_keep isl_aff *subs)
+{
+	ls = isl_local_space_cow(ls);
+	if (!ls || !subs)
+		return isl_local_space_free(ls);
+
+	if (!isl_space_is_equal(ls->dim, subs->ls->dim))
+		isl_die(isl_local_space_get_ctx(ls), isl_error_invalid,
+			"spaces don't match", return isl_local_space_free(ls));
+	if (isl_local_space_dim(subs->ls, isl_dim_div) != 0)
+		isl_die(isl_local_space_get_ctx(ls), isl_error_unsupported,
+			"cannot handle divs yet",
+			return isl_local_space_free(ls));
+
+	return isl_local_space_substitute_seq(ls, type, pos, subs->v->el,
+					    subs->v->size, 0, ls->div->n_row);
+}
+
+int isl_local_space_is_named_or_nested(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type)
+{
+	if (!ls)
+		return -1;
+	return isl_space_is_named_or_nested(ls->dim, type);
+}
+
+__isl_give isl_local_space *isl_local_space_drop_dims(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	isl_ctx *ctx;
+
+	if (!ls)
+		return NULL;
+	if (n == 0 && !isl_local_space_is_named_or_nested(ls, type))
+		return ls;
+
+	ctx = isl_local_space_get_ctx(ls);
+	if (first + n > isl_local_space_dim(ls, type))
+		isl_die(ctx, isl_error_invalid, "range out of bounds",
+			return isl_local_space_free(ls));
+
+	ls = isl_local_space_cow(ls);
+	if (!ls)
+		return NULL;
+
+	if (type == isl_dim_div) {
+		ls->div = isl_mat_drop_rows(ls->div, first, n);
+	} else {
+		ls->dim = isl_space_drop_dims(ls->dim, type, first, n);
+		if (!ls->dim)
+			return isl_local_space_free(ls);
+	}
+
+	first += 1 + isl_local_space_offset(ls, type);
+	ls->div = isl_mat_drop_cols(ls->div, first, n);
+	if (!ls->div)
+		return isl_local_space_free(ls);
+
+	return ls;
+}
+
+__isl_give isl_local_space *isl_local_space_insert_dims(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	isl_ctx *ctx;
+
+	if (!ls)
+		return NULL;
+	if (n == 0 && !isl_local_space_is_named_or_nested(ls, type))
+		return ls;
+
+	ctx = isl_local_space_get_ctx(ls);
+	if (first > isl_local_space_dim(ls, type))
+		isl_die(ctx, isl_error_invalid, "position out of bounds",
+			return isl_local_space_free(ls));
+
+	ls = isl_local_space_cow(ls);
+	if (!ls)
+		return NULL;
+
+	if (type == isl_dim_div) {
+		ls->div = isl_mat_insert_zero_rows(ls->div, first, n);
+	} else {
+		ls->dim = isl_space_insert_dims(ls->dim, type, first, n);
+		if (!ls->dim)
+			return isl_local_space_free(ls);
+	}
+
+	first += 1 + isl_local_space_offset(ls, type);
+	ls->div = isl_mat_insert_zero_cols(ls->div, first, n);
+	if (!ls->div)
+		return isl_local_space_free(ls);
+
+	return ls;
+}
+
+/* Check if the constraints pointed to by "constraint" is a div
+ * constraint corresponding to div "div" in "ls".
+ *
+ * That is, if div = floor(f/m), then check if the constraint is
+ *
+ *		f - m d >= 0
+ * or
+ *		-(f-(m-1)) + m d >= 0
+ */
+int isl_local_space_is_div_constraint(__isl_keep isl_local_space *ls,
+	isl_int *constraint, unsigned div)
+{
+	unsigned pos;
+
+	if (!ls)
+		return -1;
+
+	if (isl_int_is_zero(ls->div->row[div][0]))
+		return 0;
+
+	pos = isl_local_space_offset(ls, isl_dim_div) + div;
+
+	if (isl_int_eq(constraint[pos], ls->div->row[div][0])) {
+		int neg;
+		isl_int_sub(ls->div->row[div][1],
+				ls->div->row[div][1], ls->div->row[div][0]);
+		isl_int_add_ui(ls->div->row[div][1], ls->div->row[div][1], 1);
+		neg = isl_seq_is_neg(constraint, ls->div->row[div]+1, pos);
+		isl_int_sub_ui(ls->div->row[div][1], ls->div->row[div][1], 1);
+		isl_int_add(ls->div->row[div][1],
+				ls->div->row[div][1], ls->div->row[div][0]);
+		if (!neg)
+			return 0;
+		if (isl_seq_first_non_zero(constraint+pos+1,
+					    ls->div->n_row-div-1) != -1)
+			return 0;
+	} else if (isl_int_abs_eq(constraint[pos], ls->div->row[div][0])) {
+		if (!isl_seq_eq(constraint, ls->div->row[div]+1, pos))
+			return 0;
+		if (isl_seq_first_non_zero(constraint+pos+1,
+					    ls->div->n_row-div-1) != -1)
+			return 0;
+	} else
+		return 0;
+
+	return 1;
+}
+
+/*
+ * Set active[i] to 1 if the dimension at position i is involved
+ * in the linear expression l.
+ */
+int *isl_local_space_get_active(__isl_keep isl_local_space *ls, isl_int *l)
+{
+	int i, j;
+	isl_ctx *ctx;
+	int *active = NULL;
+	unsigned total;
+	unsigned offset;
+
+	ctx = isl_local_space_get_ctx(ls);
+	total = isl_local_space_dim(ls, isl_dim_all);
+	active = isl_calloc_array(ctx, int, total);
+	if (total && !active)
+		return NULL;
+
+	for (i = 0; i < total; ++i)
+		active[i] = !isl_int_is_zero(l[i]);
+
+	offset = isl_local_space_offset(ls, isl_dim_div) - 1;
+	for (i = ls->div->n_row - 1; i >= 0; --i) {
+		if (!active[offset + i])
+			continue;
+		for (j = 0; j < total; ++j)
+			active[j] |= !isl_int_is_zero(ls->div->row[i][2 + j]);
+	}
+
+	return active;
+}
+
+/* Given a local space "ls" of a set, create a local space
+ * for the lift of the set.  In particular, the result
+ * is of the form [dim -> local[..]], with ls->div->n_row variables in the
+ * range of the wrapped map.
+ */
+__isl_give isl_local_space *isl_local_space_lift(
+	__isl_take isl_local_space *ls)
+{
+	ls = isl_local_space_cow(ls);
+	if (!ls)
+		return NULL;
+
+	ls->dim = isl_space_lift(ls->dim, ls->div->n_row);
+	ls->div = isl_mat_drop_rows(ls->div, 0, ls->div->n_row);
+	if (!ls->dim || !ls->div)
+		return isl_local_space_free(ls);
+
+	return ls;
+}
+
+/* Construct a basic map that maps a set living in local space "ls"
+ * to the corresponding lifted local space.
+ */
+__isl_give isl_basic_map *isl_local_space_lifting(
+	__isl_take isl_local_space *ls)
+{
+	isl_basic_map *lifting;
+	isl_basic_set *bset;
+
+	if (!ls)
+		return NULL;
+	if (!isl_local_space_is_set(ls))
+		isl_die(isl_local_space_get_ctx(ls), isl_error_invalid,
+			"lifting only defined on set spaces", goto error);
+
+	bset = isl_basic_set_from_local_space(ls);
+	lifting = isl_basic_set_unwrap(isl_basic_set_lift(bset));
+	lifting = isl_basic_map_domain_map(lifting);
+	lifting = isl_basic_map_reverse(lifting);
+
+	return lifting;
+error:
+	isl_local_space_free(ls);
+	return NULL;
+}
+
+/* Compute the preimage of "ls" under the function represented by "ma".
+ * In other words, plug in "ma" in "ls".  The result is a local space
+ * that is part of the domain space of "ma".
+ *
+ * If the divs in "ls" are represented as
+ *
+ *	floor((a_i(p) + b_i x + c_i(divs))/n_i)
+ *
+ * and ma is represented by
+ *
+ *	x = D(p) + F(y) + G(divs')
+ *
+ * then the resulting divs are
+ *
+ *	floor((a_i(p) + b_i D(p) + b_i F(y) + B_i G(divs') + c_i(divs))/n_i)
+ *
+ * We first copy over the divs from "ma" and then
+ * we add the modified divs from "ls".
+ */
+__isl_give isl_local_space *isl_local_space_preimage_multi_aff(
+	__isl_take isl_local_space *ls, __isl_take isl_multi_aff *ma)
+{
+	int i;
+	isl_space *space;
+	isl_local_space *res = NULL;
+	int n_div_ls, n_div_ma;
+	isl_int f, c1, c2, g;
+
+	ma = isl_multi_aff_align_divs(ma);
+	if (!ls || !ma)
+		goto error;
+	if (!isl_space_is_range_internal(ls->dim, ma->space))
+		isl_die(isl_local_space_get_ctx(ls), isl_error_invalid,
+			"spaces don't match", goto error);
+
+	n_div_ls = isl_local_space_dim(ls, isl_dim_div);
+	n_div_ma = ma->n ? isl_aff_dim(ma->p[0], isl_dim_div) : 0;
+
+	space = isl_space_domain(isl_multi_aff_get_space(ma));
+	res = isl_local_space_alloc(space, n_div_ma + n_div_ls);
+	if (!res)
+		goto error;
+
+	if (n_div_ma) {
+		isl_mat_free(res->div);
+		res->div = isl_mat_copy(ma->p[0]->ls->div);
+		res->div = isl_mat_add_zero_cols(res->div, n_div_ls);
+		res->div = isl_mat_add_rows(res->div, n_div_ls);
+		if (!res->div)
+			goto error;
+	}
+
+	isl_int_init(f);
+	isl_int_init(c1);
+	isl_int_init(c2);
+	isl_int_init(g);
+
+	for (i = 0; i < ls->div->n_row; ++i) {
+		if (isl_int_is_zero(ls->div->row[i][0])) {
+			isl_int_set_si(res->div->row[n_div_ma + i][0], 0);
+			continue;
+		}
+		isl_seq_preimage(res->div->row[n_div_ma + i], ls->div->row[i],
+				ma, 0, 0, n_div_ma, n_div_ls, f, c1, c2, g, 1);
+		normalize_div(res, n_div_ma + i);
+	}
+
+	isl_int_clear(f);
+	isl_int_clear(c1);
+	isl_int_clear(c2);
+	isl_int_clear(g);
+
+	isl_local_space_free(ls);
+	isl_multi_aff_free(ma);
+	return res;
+error:
+	isl_local_space_free(ls);
+	isl_multi_aff_free(ma);
+	isl_local_space_free(res);
+	return NULL;
+}
+
+/* Move the "n" dimensions of "src_type" starting at "src_pos" of "ls"
+ * to dimensions of "dst_type" at "dst_pos".
+ *
+ * Moving to/from local dimensions is not allowed.
+ * We currently assume that the dimension type changes.
+ */
+__isl_give isl_local_space *isl_local_space_move_dims(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n)
+{
+	unsigned g_dst_pos;
+	unsigned g_src_pos;
+
+	if (!ls)
+		return NULL;
+	if (n == 0 &&
+	    !isl_local_space_is_named_or_nested(ls, src_type) &&
+	    !isl_local_space_is_named_or_nested(ls, dst_type))
+		return ls;
+
+	if (src_pos + n > isl_local_space_dim(ls, src_type))
+		isl_die(isl_local_space_get_ctx(ls), isl_error_invalid,
+			"range out of bounds", return isl_local_space_free(ls));
+	if (dst_pos > isl_local_space_dim(ls, dst_type))
+		isl_die(isl_local_space_get_ctx(ls), isl_error_invalid,
+			"position out of bounds",
+			return isl_local_space_free(ls));
+	if (src_type == isl_dim_div)
+		isl_die(isl_local_space_get_ctx(ls), isl_error_invalid,
+			"cannot move divs", return isl_local_space_free(ls));
+	if (dst_type == isl_dim_div)
+		isl_die(isl_local_space_get_ctx(ls), isl_error_invalid,
+			"cannot move to divs", return isl_local_space_free(ls));
+	if (dst_type == src_type && dst_pos == src_pos)
+		return ls;
+	if (dst_type == src_type)
+		isl_die(isl_local_space_get_ctx(ls), isl_error_unsupported,
+			"moving dims within the same type not supported",
+			return isl_local_space_free(ls));
+
+	ls = isl_local_space_cow(ls);
+	if (!ls)
+		return NULL;
+
+	g_src_pos = 1 + isl_local_space_offset(ls, src_type) + src_pos;
+	g_dst_pos = 1 + isl_local_space_offset(ls, dst_type) + dst_pos;
+	if (dst_type > src_type)
+		g_dst_pos -= n;
+	ls->div = isl_mat_move_cols(ls->div, g_dst_pos, g_src_pos, n);
+	if (!ls->div)
+		return isl_local_space_free(ls);
+	ls->dim = isl_space_move_dims(ls->dim, dst_type, dst_pos,
+					src_type, src_pos, n);
+	if (!ls->dim)
+		return isl_local_space_free(ls);
+
+	return ls;
+}
+
+/* Remove any internal structure of the domain of "ls".
+ * If there is any such internal structure in the input,
+ * then the name of the corresponding space is also removed.
+ */
+__isl_give isl_local_space *isl_local_space_flatten_domain(
+	__isl_take isl_local_space *ls)
+{
+	if (!ls)
+		return NULL;
+
+	if (!ls->dim->nested[0])
+		return ls;
+
+	ls = isl_local_space_cow(ls);
+	if (!ls)
+		return NULL;
+
+	ls->dim = isl_space_flatten_domain(ls->dim);
+	if (!ls->dim)
+		return isl_local_space_free(ls);
+
+	return ls;
+}
+
+/* Remove any internal structure of the range of "ls".
+ * If there is any such internal structure in the input,
+ * then the name of the corresponding space is also removed.
+ */
+__isl_give isl_local_space *isl_local_space_flatten_range(
+	__isl_take isl_local_space *ls)
+{
+	if (!ls)
+		return NULL;
+
+	if (!ls->dim->nested[1])
+		return ls;
+
+	ls = isl_local_space_cow(ls);
+	if (!ls)
+		return NULL;
+
+	ls->dim = isl_space_flatten_range(ls->dim);
+	if (!ls->dim)
+		return isl_local_space_free(ls);
+
+	return ls;
+}
diff --git a/polly/lib/External/isl/isl_local_space_private.h b/polly/lib/External/isl/isl_local_space_private.h
new file mode 100644
index 00000000000..4d7e69ff2b5
--- /dev/null
+++ b/polly/lib/External/isl/isl_local_space_private.h
@@ -0,0 +1,75 @@
+#ifndef ISL_LOCAL_SPACE_PRIVATE_H
+#define ISL_LOCAL_SPACE_PRIVATE_H
+
+#include <isl/mat.h>
+#include <isl/set.h>
+#include <isl/local_space.h>
+
+struct isl_local_space {
+	int ref;
+
+	isl_space *dim;
+	isl_mat *div;
+};
+
+__isl_give isl_local_space *isl_local_space_alloc(__isl_take isl_space *dim,
+	unsigned n_div);
+__isl_give isl_local_space *isl_local_space_alloc_div(__isl_take isl_space *dim,
+	__isl_take isl_mat *div);
+
+__isl_give isl_local_space *isl_local_space_swap_div(
+	__isl_take isl_local_space *ls, int a, int b);
+__isl_give isl_local_space *isl_local_space_add_div(
+	__isl_take isl_local_space *ls, __isl_take isl_vec *div);
+
+int isl_mat_cmp_div(__isl_keep isl_mat *div, int i, int j);
+__isl_give isl_mat *isl_merge_divs(__isl_keep isl_mat *div1,
+	__isl_keep isl_mat *div2, int *exp1, int *exp2);
+
+unsigned isl_local_space_offset(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type);
+
+__isl_give isl_local_space *isl_local_space_replace_divs(
+	__isl_take isl_local_space *ls, __isl_take isl_mat *div);
+int isl_local_space_div_is_known(__isl_keep isl_local_space *ls, int div);
+int isl_local_space_divs_known(__isl_keep isl_local_space *ls);
+
+__isl_give isl_local_space *isl_local_space_substitute_equalities(
+	__isl_take isl_local_space *ls, __isl_take isl_basic_set *eq);
+
+int isl_local_space_is_named_or_nested(__isl_keep isl_local_space *ls,
+	enum isl_dim_type type);
+
+__isl_give isl_local_space *isl_local_space_reset_space(
+	__isl_take isl_local_space *ls, __isl_take isl_space *dim);
+__isl_give isl_local_space *isl_local_space_realign(
+	__isl_take isl_local_space *ls, __isl_take isl_reordering *r);
+
+int isl_local_space_is_div_constraint(__isl_keep isl_local_space *ls,
+	isl_int *constraint, unsigned div);
+
+int *isl_local_space_get_active(__isl_keep isl_local_space *ls, isl_int *l);
+
+__isl_give isl_local_space *isl_local_space_substitute_seq(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos, isl_int *subs, int subs_len,
+	int first, int n);
+__isl_give isl_local_space *isl_local_space_substitute(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos, __isl_keep isl_aff *subs);
+
+__isl_give isl_local_space *isl_local_space_lift(
+	__isl_take isl_local_space *ls);
+
+__isl_give isl_local_space *isl_local_space_preimage_multi_aff(
+	__isl_take isl_local_space *ls, __isl_take isl_multi_aff *ma);
+
+__isl_give isl_local_space *isl_local_space_move_dims(
+	__isl_take isl_local_space *ls,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n);
+
+int isl_local_space_cmp(__isl_keep isl_local_space *ls1,
+	__isl_keep isl_local_space *ls2);
+
+#endif
diff --git a/polly/lib/External/isl/isl_lp.c b/polly/lib/External/isl/isl_lp.c
new file mode 100644
index 00000000000..1d5ae6ef862
--- /dev/null
+++ b/polly/lib/External/isl/isl_lp.c
@@ -0,0 +1,362 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl/lp.h>
+#include <isl_seq.h>
+#include "isl_tab.h"
+#include <isl_options_private.h>
+#include <isl_local_space_private.h>
+#include <isl_aff_private.h>
+#include <isl_mat_private.h>
+#include <isl_val_private.h>
+#include <isl_vec_private.h>
+
+enum isl_lp_result isl_tab_solve_lp(struct isl_basic_map *bmap, int maximize,
+				      isl_int *f, isl_int denom, isl_int *opt,
+				      isl_int *opt_denom,
+				      struct isl_vec **sol)
+{
+	struct isl_tab *tab;
+	enum isl_lp_result res;
+	unsigned dim = isl_basic_map_total_dim(bmap);
+
+	if (maximize)
+		isl_seq_neg(f, f, 1 + dim);
+
+	bmap = isl_basic_map_gauss(bmap, NULL);
+	tab = isl_tab_from_basic_map(bmap, 0);
+	res = isl_tab_min(tab, f, denom, opt, opt_denom, 0);
+	if (res == isl_lp_ok && sol) {
+		*sol = isl_tab_get_sample_value(tab);
+		if (!*sol)
+			res = isl_lp_error;
+	}
+	isl_tab_free(tab);
+
+	if (maximize)
+		isl_seq_neg(f, f, 1 + dim);
+	if (maximize && opt)
+		isl_int_neg(*opt, *opt);
+
+	return res;
+}
+
+/* Given a basic map "bmap" and an affine combination of the variables "f"
+ * with denominator "denom", set *opt / *opt_denom to the minimal
+ * (or maximal if "maximize" is true) value attained by f/d over "bmap",
+ * assuming the basic map is not empty and the expression cannot attain
+ * arbitrarily small (or large) values.
+ * If opt_denom is NULL, then *opt is rounded up (or down)
+ * to the nearest integer.
+ * The return value reflects the nature of the result (empty, unbounded,
+ * minmimal or maximal value returned in *opt).
+ */
+enum isl_lp_result isl_basic_map_solve_lp(struct isl_basic_map *bmap, int max,
+				      isl_int *f, isl_int d, isl_int *opt,
+				      isl_int *opt_denom,
+				      struct isl_vec **sol)
+{
+	if (sol)
+		*sol = NULL;
+
+	if (!bmap)
+		return isl_lp_error;
+
+	return isl_tab_solve_lp(bmap, max, f, d, opt, opt_denom, sol);
+}
+
+enum isl_lp_result isl_basic_set_solve_lp(struct isl_basic_set *bset, int max,
+				      isl_int *f, isl_int d, isl_int *opt,
+				      isl_int *opt_denom,
+				      struct isl_vec **sol)
+{
+	return isl_basic_map_solve_lp((struct isl_basic_map *)bset, max,
+					f, d, opt, opt_denom, sol);
+}
+
+enum isl_lp_result isl_map_solve_lp(__isl_keep isl_map *map, int max,
+				      isl_int *f, isl_int d, isl_int *opt,
+				      isl_int *opt_denom,
+				      struct isl_vec **sol)
+{
+	int i;
+	isl_int o;
+	isl_int t;
+	isl_int opt_i;
+	isl_int opt_denom_i;
+	enum isl_lp_result res;
+	int max_div;
+	isl_vec *v = NULL;
+
+	if (!map)
+		return isl_lp_error;
+	if (map->n == 0)
+		return isl_lp_empty;
+
+	max_div = 0;
+	for (i = 0; i < map->n; ++i)
+		if (map->p[i]->n_div > max_div)
+			max_div = map->p[i]->n_div;
+	if (max_div > 0) {
+		unsigned total = isl_space_dim(map->dim, isl_dim_all);
+		v = isl_vec_alloc(map->ctx, 1 + total + max_div);
+		if (!v)
+			return isl_lp_error;
+		isl_seq_cpy(v->el, f, 1 + total);
+		isl_seq_clr(v->el + 1 + total, max_div);
+		f = v->el;
+	}
+
+	if (!opt && map->n > 1 && sol) {
+		isl_int_init(o);
+		opt = &o;
+	}
+	if (map->n > 0)
+		isl_int_init(opt_i);
+	if (map->n > 0 && opt_denom) {
+		isl_int_init(opt_denom_i);
+		isl_int_init(t);
+	}
+
+	res = isl_basic_map_solve_lp(map->p[0], max, f, d,
+					opt, opt_denom, sol);
+	if (res == isl_lp_error || res == isl_lp_unbounded)
+		goto done;
+
+	if (sol)
+		*sol = NULL;
+
+	for (i = 1; i < map->n; ++i) {
+		isl_vec *sol_i = NULL;
+		enum isl_lp_result res_i;
+		int better;
+
+		res_i = isl_basic_map_solve_lp(map->p[i], max, f, d,
+					    &opt_i,
+					    opt_denom ? &opt_denom_i : NULL,
+					    sol ? &sol_i : NULL);
+		if (res_i == isl_lp_error || res_i == isl_lp_unbounded) {
+			res = res_i;
+			goto done;
+		}
+		if (res_i == isl_lp_empty)
+			continue;
+		if (res == isl_lp_empty) {
+			better = 1;
+		} else if (!opt_denom) {
+			if (max)
+				better = isl_int_gt(opt_i, *opt);
+			else
+				better = isl_int_lt(opt_i, *opt);
+		} else {
+			isl_int_mul(t, opt_i, *opt_denom);
+			isl_int_submul(t, *opt, opt_denom_i);
+			if (max)
+				better = isl_int_is_pos(t);
+			else
+				better = isl_int_is_neg(t);
+		}
+		if (better) {
+			res = res_i;
+			if (opt)
+				isl_int_set(*opt, opt_i);
+			if (opt_denom)
+				isl_int_set(*opt_denom, opt_denom_i);
+			if (sol) {
+				isl_vec_free(*sol);
+				*sol = sol_i;
+			}
+		} else
+			isl_vec_free(sol_i);
+	}
+
+done:
+	isl_vec_free(v);
+	if (map->n > 0 && opt_denom) {
+		isl_int_clear(opt_denom_i);
+		isl_int_clear(t);
+	}
+	if (map->n > 0)
+		isl_int_clear(opt_i);
+	if (opt == &o)
+		isl_int_clear(o);
+	return res;
+}
+
+enum isl_lp_result isl_set_solve_lp(__isl_keep isl_set *set, int max,
+				      isl_int *f, isl_int d, isl_int *opt,
+				      isl_int *opt_denom,
+				      struct isl_vec **sol)
+{
+	return isl_map_solve_lp((struct isl_map *)set, max,
+					f, d, opt, opt_denom, sol);
+}
+
+/* Return the optimal (rational) value of "obj" over "bset", assuming
+ * that "obj" and "bset" have aligned parameters and divs.
+ * If "max" is set, then the maximal value is computed.
+ * Otherwise, the minimal value is computed.
+ *
+ * Return infinity or negative infinity if the optimal value is unbounded and
+ * NaN if "bset" is empty.
+ *
+ * Call isl_basic_set_solve_lp and translate the results.
+ */
+static __isl_give isl_val *basic_set_opt_lp(
+	__isl_keep isl_basic_set *bset, int max, __isl_keep isl_aff *obj)
+{
+	isl_ctx *ctx;
+	isl_val *res;
+	enum isl_lp_result lp_res;
+
+	if (!bset || !obj)
+		return NULL;
+
+	ctx = isl_aff_get_ctx(obj);
+	res = isl_val_alloc(ctx);
+	if (!res)
+		return NULL;
+	lp_res = isl_basic_set_solve_lp(bset, max, obj->v->el + 1,
+					obj->v->el[0], &res->n, &res->d, NULL);
+	if (lp_res == isl_lp_ok)
+		return isl_val_normalize(res);
+	isl_val_free(res);
+	if (lp_res == isl_lp_error)
+		return NULL;
+	if (lp_res == isl_lp_empty)
+		return isl_val_nan(ctx);
+	if (max)
+		return isl_val_infty(ctx);
+	else
+		return isl_val_neginfty(ctx);
+}
+
+/* Return the optimal (rational) value of "obj" over "bset", assuming
+ * that "obj" and "bset" have aligned parameters.
+ * If "max" is set, then the maximal value is computed.
+ * Otherwise, the minimal value is computed.
+ *
+ * Return infinity or negative infinity if the optimal value is unbounded and
+ * NaN if "bset" is empty.
+ *
+ * Align the divs of "bset" and "obj" and call basic_set_opt_lp.
+ */
+static __isl_give isl_val *isl_basic_set_opt_lp_val_aligned(
+	__isl_keep isl_basic_set *bset, int max, __isl_keep isl_aff *obj)
+{
+	int *exp1 = NULL;
+	int *exp2 = NULL;
+	isl_ctx *ctx;
+	isl_mat *bset_div = NULL;
+	isl_mat *div = NULL;
+	isl_val *res;
+	int bset_n_div, obj_n_div;
+
+	if (!bset || !obj)
+		return NULL;
+
+	ctx = isl_aff_get_ctx(obj);
+	if (!isl_space_is_equal(bset->dim, obj->ls->dim))
+		isl_die(ctx, isl_error_invalid,
+			"spaces don't match", return NULL);
+
+	bset_n_div = isl_basic_set_dim(bset, isl_dim_div);
+	obj_n_div = isl_aff_dim(obj, isl_dim_div);
+	if (bset_n_div == 0 && obj_n_div == 0)
+		return basic_set_opt_lp(bset, max, obj);
+
+	bset = isl_basic_set_copy(bset);
+	obj = isl_aff_copy(obj);
+
+	bset_div = isl_basic_set_get_divs(bset);
+	exp1 = isl_alloc_array(ctx, int, bset_n_div);
+	exp2 = isl_alloc_array(ctx, int, obj_n_div);
+	if (!bset_div || (bset_n_div && !exp1) || (obj_n_div && !exp2))
+		goto error;
+
+	div = isl_merge_divs(bset_div, obj->ls->div, exp1, exp2);
+
+	bset = isl_basic_set_expand_divs(bset, isl_mat_copy(div), exp1);
+	obj = isl_aff_expand_divs(obj, isl_mat_copy(div), exp2);
+
+	res = basic_set_opt_lp(bset, max, obj);
+
+	isl_mat_free(bset_div);
+	isl_mat_free(div);
+	free(exp1);
+	free(exp2);
+	isl_basic_set_free(bset);
+	isl_aff_free(obj);
+
+	return res;
+error:
+	isl_mat_free(div);
+	isl_mat_free(bset_div);
+	free(exp1);
+	free(exp2);
+	isl_basic_set_free(bset);
+	isl_aff_free(obj);
+	return NULL;
+}
+
+/* Return the optimal (rational) value of "obj" over "bset".
+ * If "max" is set, then the maximal value is computed.
+ * Otherwise, the minimal value is computed.
+ *
+ * Return infinity or negative infinity if the optimal value is unbounded and
+ * NaN if "bset" is empty.
+ */
+static __isl_give isl_val *isl_basic_set_opt_lp_val(
+	__isl_keep isl_basic_set *bset, int max, __isl_keep isl_aff *obj)
+{
+	isl_val *res;
+
+	if (!bset || !obj)
+		return NULL;
+
+	if (isl_space_match(bset->dim, isl_dim_param,
+			    obj->ls->dim, isl_dim_param))
+		return isl_basic_set_opt_lp_val_aligned(bset, max, obj);
+
+	bset = isl_basic_set_copy(bset);
+	obj = isl_aff_copy(obj);
+	bset = isl_basic_set_align_params(bset, isl_aff_get_domain_space(obj));
+	obj = isl_aff_align_params(obj, isl_basic_set_get_space(bset));
+
+	res = isl_basic_set_opt_lp_val_aligned(bset, max, obj);
+
+	isl_basic_set_free(bset);
+	isl_aff_free(obj);
+
+	return res;
+}
+
+/* Return the minimal (rational) value of "obj" over "bset".
+ *
+ * Return negative infinity if the minimal value is unbounded and
+ * NaN if "bset" is empty.
+ */
+__isl_give isl_val *isl_basic_set_min_lp_val(__isl_keep isl_basic_set *bset,
+	__isl_keep isl_aff *obj)
+{
+	return isl_basic_set_opt_lp_val(bset, 0, obj);
+}
+
+/* Return the maximal (rational) value of "obj" over "bset".
+ *
+ * Return infinity if the maximal value is unbounded and
+ * NaN if "bset" is empty.
+ */
+__isl_give isl_val *isl_basic_set_max_lp_val(__isl_keep isl_basic_set *bset,
+	__isl_keep isl_aff *obj)
+{
+	return isl_basic_set_opt_lp_val(bset, 1, obj);
+}
diff --git a/polly/lib/External/isl/isl_lp_private.h b/polly/lib/External/isl/isl_lp_private.h
new file mode 100644
index 00000000000..ddc44c1eeda
--- /dev/null
+++ b/polly/lib/External/isl/isl_lp_private.h
@@ -0,0 +1,21 @@
+#ifndef ISL_LP_PRIVATE_H
+#define ISL_LP_PRIVATE_H
+
+#include <isl_int.h>
+#include <isl/lp.h>
+#include <isl/vec.h>
+
+enum isl_lp_result isl_basic_map_solve_lp(__isl_keep isl_basic_map *bmap,
+	int max, isl_int *f, isl_int denom, isl_int *opt, isl_int *opt_denom,
+	__isl_give isl_vec **sol);
+enum isl_lp_result isl_basic_set_solve_lp(__isl_keep isl_basic_set *bset,
+	int max, isl_int *f, isl_int denom, isl_int *opt, isl_int *opt_denom,
+	__isl_give isl_vec **sol);
+enum isl_lp_result isl_map_solve_lp(__isl_keep isl_map *map, int max,
+	isl_int *f, isl_int denom, isl_int *opt, isl_int *opt_denom,
+	__isl_give isl_vec **sol);
+enum isl_lp_result isl_set_solve_lp(__isl_keep isl_set *set, int max,
+	isl_int *f, isl_int denom, isl_int *opt, isl_int *opt_denom,
+	__isl_give isl_vec **sol);
+
+#endif
diff --git a/polly/lib/External/isl/isl_map.c b/polly/lib/External/isl/isl_map.c
new file mode 100644
index 00000000000..8f021c09e66
--- /dev/null
+++ b/polly/lib/External/isl/isl_map.c
@@ -0,0 +1,12888 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2010      INRIA Saclay
+ * Copyright 2012-2014 Ecole Normale Superieure
+ * Copyright 2014      INRIA Rocquencourt
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France 
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
+ * B.P. 105 - 78153 Le Chesnay, France
+ */
+
+#include <string.h>
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl_blk.h>
+#include "isl_space_private.h"
+#include "isl_equalities.h"
+#include <isl_lp_private.h>
+#include <isl_seq.h>
+#include <isl/set.h>
+#include <isl/map.h>
+#include <isl_reordering.h>
+#include "isl_sample.h"
+#include <isl_sort.h>
+#include "isl_tab.h"
+#include <isl/vec.h>
+#include <isl_mat_private.h>
+#include <isl_vec_private.h>
+#include <isl_dim_map.h>
+#include <isl_local_space_private.h>
+#include <isl_aff_private.h>
+#include <isl_options_private.h>
+#include <isl_morph.h>
+#include <isl_val_private.h>
+#include <isl/deprecated/map_int.h>
+#include <isl/deprecated/set_int.h>
+
+static unsigned n(__isl_keep isl_space *dim, enum isl_dim_type type)
+{
+	switch (type) {
+	case isl_dim_param:	return dim->nparam;
+	case isl_dim_in:	return dim->n_in;
+	case isl_dim_out:	return dim->n_out;
+	case isl_dim_all:	return dim->nparam + dim->n_in + dim->n_out;
+	default:		return 0;
+	}
+}
+
+static unsigned pos(__isl_keep isl_space *dim, enum isl_dim_type type)
+{
+	switch (type) {
+	case isl_dim_param:	return 1;
+	case isl_dim_in:	return 1 + dim->nparam;
+	case isl_dim_out:	return 1 + dim->nparam + dim->n_in;
+	default:		return 0;
+	}
+}
+
+unsigned isl_basic_map_dim(__isl_keep isl_basic_map *bmap,
+				enum isl_dim_type type)
+{
+	if (!bmap)
+		return 0;
+	switch (type) {
+	case isl_dim_cst:	return 1;
+	case isl_dim_param:
+	case isl_dim_in:
+	case isl_dim_out:	return isl_space_dim(bmap->dim, type);
+	case isl_dim_div:	return bmap->n_div;
+	case isl_dim_all:	return isl_basic_map_total_dim(bmap);
+	default:		return 0;
+	}
+}
+
+unsigned isl_map_dim(__isl_keep isl_map *map, enum isl_dim_type type)
+{
+	return map ? n(map->dim, type) : 0;
+}
+
+unsigned isl_set_dim(__isl_keep isl_set *set, enum isl_dim_type type)
+{
+	return set ? n(set->dim, type) : 0;
+}
+
+unsigned isl_basic_map_offset(struct isl_basic_map *bmap,
+					enum isl_dim_type type)
+{
+	isl_space *dim = bmap->dim;
+	switch (type) {
+	case isl_dim_cst:	return 0;
+	case isl_dim_param:	return 1;
+	case isl_dim_in:	return 1 + dim->nparam;
+	case isl_dim_out:	return 1 + dim->nparam + dim->n_in;
+	case isl_dim_div:	return 1 + dim->nparam + dim->n_in + dim->n_out;
+	default:		return 0;
+	}
+}
+
+unsigned isl_basic_set_offset(struct isl_basic_set *bset,
+					enum isl_dim_type type)
+{
+	return isl_basic_map_offset(bset, type);
+}
+
+static unsigned map_offset(struct isl_map *map, enum isl_dim_type type)
+{
+	return pos(map->dim, type);
+}
+
+unsigned isl_basic_set_dim(__isl_keep isl_basic_set *bset,
+				enum isl_dim_type type)
+{
+	return isl_basic_map_dim(bset, type);
+}
+
+unsigned isl_basic_set_n_dim(__isl_keep isl_basic_set *bset)
+{
+	return isl_basic_set_dim(bset, isl_dim_set);
+}
+
+unsigned isl_basic_set_n_param(__isl_keep isl_basic_set *bset)
+{
+	return isl_basic_set_dim(bset, isl_dim_param);
+}
+
+unsigned isl_basic_set_total_dim(const struct isl_basic_set *bset)
+{
+	if (!bset)
+		return 0;
+	return isl_space_dim(bset->dim, isl_dim_all) + bset->n_div;
+}
+
+unsigned isl_set_n_dim(__isl_keep isl_set *set)
+{
+	return isl_set_dim(set, isl_dim_set);
+}
+
+unsigned isl_set_n_param(__isl_keep isl_set *set)
+{
+	return isl_set_dim(set, isl_dim_param);
+}
+
+unsigned isl_basic_map_n_in(const struct isl_basic_map *bmap)
+{
+	return bmap ? bmap->dim->n_in : 0;
+}
+
+unsigned isl_basic_map_n_out(const struct isl_basic_map *bmap)
+{
+	return bmap ? bmap->dim->n_out : 0;
+}
+
+unsigned isl_basic_map_n_param(const struct isl_basic_map *bmap)
+{
+	return bmap ? bmap->dim->nparam : 0;
+}
+
+unsigned isl_basic_map_n_div(const struct isl_basic_map *bmap)
+{
+	return bmap ? bmap->n_div : 0;
+}
+
+unsigned isl_basic_map_total_dim(const struct isl_basic_map *bmap)
+{
+	return bmap ? isl_space_dim(bmap->dim, isl_dim_all) + bmap->n_div : 0;
+}
+
+unsigned isl_map_n_in(const struct isl_map *map)
+{
+	return map ? map->dim->n_in : 0;
+}
+
+unsigned isl_map_n_out(const struct isl_map *map)
+{
+	return map ? map->dim->n_out : 0;
+}
+
+unsigned isl_map_n_param(const struct isl_map *map)
+{
+	return map ? map->dim->nparam : 0;
+}
+
+int isl_map_compatible_domain(struct isl_map *map, struct isl_set *set)
+{
+	int m;
+	if (!map || !set)
+		return -1;
+	m = isl_space_match(map->dim, isl_dim_param, set->dim, isl_dim_param);
+	if (m < 0 || !m)
+		return m;
+	return isl_space_tuple_is_equal(map->dim, isl_dim_in,
+					set->dim, isl_dim_set);
+}
+
+int isl_basic_map_compatible_domain(struct isl_basic_map *bmap,
+		struct isl_basic_set *bset)
+{
+	int m;
+	if (!bmap || !bset)
+		return -1;
+	m = isl_space_match(bmap->dim, isl_dim_param, bset->dim, isl_dim_param);
+	if (m < 0 || !m)
+		return m;
+	return isl_space_tuple_is_equal(bmap->dim, isl_dim_in,
+					bset->dim, isl_dim_set);
+}
+
+int isl_map_compatible_range(__isl_keep isl_map *map, __isl_keep isl_set *set)
+{
+	int m;
+	if (!map || !set)
+		return -1;
+	m = isl_space_match(map->dim, isl_dim_param, set->dim, isl_dim_param);
+	if (m < 0 || !m)
+		return m;
+	return isl_space_tuple_is_equal(map->dim, isl_dim_out,
+					set->dim, isl_dim_set);
+}
+
+int isl_basic_map_compatible_range(struct isl_basic_map *bmap,
+		struct isl_basic_set *bset)
+{
+	int m;
+	if (!bmap || !bset)
+		return -1;
+	m = isl_space_match(bmap->dim, isl_dim_param, bset->dim, isl_dim_param);
+	if (m < 0 || !m)
+		return m;
+	return isl_space_tuple_is_equal(bmap->dim, isl_dim_out,
+					bset->dim, isl_dim_set);
+}
+
+isl_ctx *isl_basic_map_get_ctx(__isl_keep isl_basic_map *bmap)
+{
+	return bmap ? bmap->ctx : NULL;
+}
+
+isl_ctx *isl_basic_set_get_ctx(__isl_keep isl_basic_set *bset)
+{
+	return bset ? bset->ctx : NULL;
+}
+
+isl_ctx *isl_map_get_ctx(__isl_keep isl_map *map)
+{
+	return map ? map->ctx : NULL;
+}
+
+isl_ctx *isl_set_get_ctx(__isl_keep isl_set *set)
+{
+	return set ? set->ctx : NULL;
+}
+
+__isl_give isl_space *isl_basic_map_get_space(__isl_keep isl_basic_map *bmap)
+{
+	if (!bmap)
+		return NULL;
+	return isl_space_copy(bmap->dim);
+}
+
+__isl_give isl_space *isl_basic_set_get_space(__isl_keep isl_basic_set *bset)
+{
+	if (!bset)
+		return NULL;
+	return isl_space_copy(bset->dim);
+}
+
+/* Extract the divs in "bmap" as a matrix.
+ */
+__isl_give isl_mat *isl_basic_map_get_divs(__isl_keep isl_basic_map *bmap)
+{
+	int i;
+	isl_ctx *ctx;
+	isl_mat *div;
+	unsigned total;
+	unsigned cols;
+
+	if (!bmap)
+		return NULL;
+
+	ctx = isl_basic_map_get_ctx(bmap);
+	total = isl_space_dim(bmap->dim, isl_dim_all);
+	cols = 1 + 1 + total + bmap->n_div;
+	div = isl_mat_alloc(ctx, bmap->n_div, cols);
+	if (!div)
+		return NULL;
+
+	for (i = 0; i < bmap->n_div; ++i)
+		isl_seq_cpy(div->row[i], bmap->div[i], cols);
+
+	return div;
+}
+
+/* Extract the divs in "bset" as a matrix.
+ */
+__isl_give isl_mat *isl_basic_set_get_divs(__isl_keep isl_basic_set *bset)
+{
+	return isl_basic_map_get_divs(bset);
+}
+
+__isl_give isl_local_space *isl_basic_map_get_local_space(
+	__isl_keep isl_basic_map *bmap)
+{
+	isl_mat *div;
+
+	if (!bmap)
+		return NULL;
+
+	div = isl_basic_map_get_divs(bmap);
+	return isl_local_space_alloc_div(isl_space_copy(bmap->dim), div);
+}
+
+__isl_give isl_local_space *isl_basic_set_get_local_space(
+	__isl_keep isl_basic_set *bset)
+{
+	return isl_basic_map_get_local_space(bset);
+}
+
+__isl_give isl_basic_map *isl_basic_map_from_local_space(
+	__isl_take isl_local_space *ls)
+{
+	int i;
+	int n_div;
+	isl_basic_map *bmap;
+
+	if (!ls)
+		return NULL;
+
+	n_div = isl_local_space_dim(ls, isl_dim_div);
+	bmap = isl_basic_map_alloc_space(isl_local_space_get_space(ls),
+					n_div, 0, 2 * n_div);
+
+	for (i = 0; i < n_div; ++i)
+		if (isl_basic_map_alloc_div(bmap) < 0)
+			goto error;
+
+	for (i = 0; i < n_div; ++i) {
+		isl_seq_cpy(bmap->div[i], ls->div->row[i], ls->div->n_col);
+		if (isl_basic_map_add_div_constraints(bmap, i) < 0)
+			goto error;
+	}
+					
+	isl_local_space_free(ls);
+	return bmap;
+error:
+	isl_local_space_free(ls);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_from_local_space(
+	__isl_take isl_local_space *ls)
+{
+	return isl_basic_map_from_local_space(ls);
+}
+
+__isl_give isl_space *isl_map_get_space(__isl_keep isl_map *map)
+{
+	if (!map)
+		return NULL;
+	return isl_space_copy(map->dim);
+}
+
+__isl_give isl_space *isl_set_get_space(__isl_keep isl_set *set)
+{
+	if (!set)
+		return NULL;
+	return isl_space_copy(set->dim);
+}
+
+__isl_give isl_basic_map *isl_basic_map_set_tuple_name(
+	__isl_take isl_basic_map *bmap, enum isl_dim_type type, const char *s)
+{
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+	bmap->dim = isl_space_set_tuple_name(bmap->dim, type, s);
+	if (!bmap->dim)
+		goto error;
+	bmap = isl_basic_map_finalize(bmap);
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_set_tuple_name(
+	__isl_take isl_basic_set *bset, const char *s)
+{
+	return isl_basic_map_set_tuple_name(bset, isl_dim_set, s);
+}
+
+const char *isl_basic_map_get_tuple_name(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type)
+{
+	return bmap ? isl_space_get_tuple_name(bmap->dim, type) : NULL;
+}
+
+__isl_give isl_map *isl_map_set_tuple_name(__isl_take isl_map *map,
+	enum isl_dim_type type, const char *s)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	map->dim = isl_space_set_tuple_name(map->dim, type, s);
+	if (!map->dim)
+		goto error;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_set_tuple_name(map->p[i], type, s);
+		if (!map->p[i])
+			goto error;
+	}
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Replace the identifier of the tuple of type "type" by "id".
+ */
+__isl_give isl_basic_map *isl_basic_map_set_tuple_id(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, __isl_take isl_id *id)
+{
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		goto error;
+	bmap->dim = isl_space_set_tuple_id(bmap->dim, type, id);
+	if (!bmap->dim)
+		return isl_basic_map_free(bmap);
+	bmap = isl_basic_map_finalize(bmap);
+	return bmap;
+error:
+	isl_id_free(id);
+	return NULL;
+}
+
+/* Replace the identifier of the tuple by "id".
+ */
+__isl_give isl_basic_set *isl_basic_set_set_tuple_id(
+	__isl_take isl_basic_set *bset, __isl_take isl_id *id)
+{
+	return isl_basic_map_set_tuple_id(bset, isl_dim_set, id);
+}
+
+/* Does the input or output tuple have a name?
+ */
+int isl_map_has_tuple_name(__isl_keep isl_map *map, enum isl_dim_type type)
+{
+	return map ? isl_space_has_tuple_name(map->dim, type) : -1;
+}
+
+const char *isl_map_get_tuple_name(__isl_keep isl_map *map,
+	enum isl_dim_type type)
+{
+	return map ? isl_space_get_tuple_name(map->dim, type) : NULL;
+}
+
+__isl_give isl_set *isl_set_set_tuple_name(__isl_take isl_set *set,
+	const char *s)
+{
+	return (isl_set *)isl_map_set_tuple_name((isl_map *)set, isl_dim_set, s);
+}
+
+__isl_give isl_map *isl_map_set_tuple_id(__isl_take isl_map *map,
+	enum isl_dim_type type, __isl_take isl_id *id)
+{
+	map = isl_map_cow(map);
+	if (!map)
+		goto error;
+
+	map->dim = isl_space_set_tuple_id(map->dim, type, id);
+
+	return isl_map_reset_space(map, isl_space_copy(map->dim));
+error:
+	isl_id_free(id);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_set_tuple_id(__isl_take isl_set *set,
+	__isl_take isl_id *id)
+{
+	return isl_map_set_tuple_id(set, isl_dim_set, id);
+}
+
+__isl_give isl_map *isl_map_reset_tuple_id(__isl_take isl_map *map,
+	enum isl_dim_type type)
+{
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	map->dim = isl_space_reset_tuple_id(map->dim, type);
+
+	return isl_map_reset_space(map, isl_space_copy(map->dim));
+}
+
+__isl_give isl_set *isl_set_reset_tuple_id(__isl_take isl_set *set)
+{
+	return isl_map_reset_tuple_id(set, isl_dim_set);
+}
+
+int isl_map_has_tuple_id(__isl_keep isl_map *map, enum isl_dim_type type)
+{
+	return map ? isl_space_has_tuple_id(map->dim, type) : -1;
+}
+
+__isl_give isl_id *isl_map_get_tuple_id(__isl_keep isl_map *map,
+	enum isl_dim_type type)
+{
+	return map ? isl_space_get_tuple_id(map->dim, type) : NULL;
+}
+
+int isl_set_has_tuple_id(__isl_keep isl_set *set)
+{
+	return isl_map_has_tuple_id(set, isl_dim_set);
+}
+
+__isl_give isl_id *isl_set_get_tuple_id(__isl_keep isl_set *set)
+{
+	return isl_map_get_tuple_id(set, isl_dim_set);
+}
+
+/* Does the set tuple have a name?
+ */
+int isl_set_has_tuple_name(__isl_keep isl_set *set)
+{
+	return set ? isl_space_has_tuple_name(set->dim, isl_dim_set) : -1;
+}
+
+
+const char *isl_basic_set_get_tuple_name(__isl_keep isl_basic_set *bset)
+{
+	return bset ? isl_space_get_tuple_name(bset->dim, isl_dim_set) : NULL;
+}
+
+const char *isl_set_get_tuple_name(__isl_keep isl_set *set)
+{
+	return set ? isl_space_get_tuple_name(set->dim, isl_dim_set) : NULL;
+}
+
+const char *isl_basic_map_get_dim_name(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos)
+{
+	return bmap ? isl_space_get_dim_name(bmap->dim, type, pos) : NULL;
+}
+
+const char *isl_basic_set_get_dim_name(__isl_keep isl_basic_set *bset,
+	enum isl_dim_type type, unsigned pos)
+{
+	return bset ? isl_space_get_dim_name(bset->dim, type, pos) : NULL;
+}
+
+/* Does the given dimension have a name?
+ */
+int isl_map_has_dim_name(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos)
+{
+	return map ? isl_space_has_dim_name(map->dim, type, pos) : -1;
+}
+
+const char *isl_map_get_dim_name(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos)
+{
+	return map ? isl_space_get_dim_name(map->dim, type, pos) : NULL;
+}
+
+const char *isl_set_get_dim_name(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos)
+{
+	return set ? isl_space_get_dim_name(set->dim, type, pos) : NULL;
+}
+
+/* Does the given dimension have a name?
+ */
+int isl_set_has_dim_name(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos)
+{
+	return set ? isl_space_has_dim_name(set->dim, type, pos) : -1;
+}
+
+__isl_give isl_basic_map *isl_basic_map_set_dim_name(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos, const char *s)
+{
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+	bmap->dim = isl_space_set_dim_name(bmap->dim, type, pos, s);
+	if (!bmap->dim)
+		goto error;
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_set_dim_name(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned pos, const char *s)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	map->dim = isl_space_set_dim_name(map->dim, type, pos, s);
+	if (!map->dim)
+		goto error;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_set_dim_name(map->p[i], type, pos, s);
+		if (!map->p[i])
+			goto error;
+	}
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_set_dim_name(
+	__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned pos, const char *s)
+{
+	return (isl_basic_set *)isl_basic_map_set_dim_name(
+		(isl_basic_map *)bset, type, pos, s);
+}
+
+__isl_give isl_set *isl_set_set_dim_name(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, const char *s)
+{
+	return (isl_set *)isl_map_set_dim_name((isl_map *)set, type, pos, s);
+}
+
+int isl_basic_map_has_dim_id(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos)
+{
+	return bmap ? isl_space_has_dim_id(bmap->dim, type, pos) : -1;
+}
+
+__isl_give isl_id *isl_basic_set_get_dim_id(__isl_keep isl_basic_set *bset,
+	enum isl_dim_type type, unsigned pos)
+{
+	return bset ? isl_space_get_dim_id(bset->dim, type, pos) : NULL;
+}
+
+int isl_map_has_dim_id(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos)
+{
+	return map ? isl_space_has_dim_id(map->dim, type, pos) : -1;
+}
+
+__isl_give isl_id *isl_map_get_dim_id(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos)
+{
+	return map ? isl_space_get_dim_id(map->dim, type, pos) : NULL;
+}
+
+int isl_set_has_dim_id(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos)
+{
+	return isl_map_has_dim_id(set, type, pos);
+}
+
+__isl_give isl_id *isl_set_get_dim_id(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos)
+{
+	return isl_map_get_dim_id(set, type, pos);
+}
+
+__isl_give isl_map *isl_map_set_dim_id(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id)
+{
+	map = isl_map_cow(map);
+	if (!map)
+		goto error;
+
+	map->dim = isl_space_set_dim_id(map->dim, type, pos, id);
+
+	return isl_map_reset_space(map, isl_space_copy(map->dim));
+error:
+	isl_id_free(id);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_set_dim_id(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id)
+{
+	return isl_map_set_dim_id(set, type, pos, id);
+}
+
+int isl_map_find_dim_by_id(__isl_keep isl_map *map, enum isl_dim_type type,
+	__isl_keep isl_id *id)
+{
+	if (!map)
+		return -1;
+	return isl_space_find_dim_by_id(map->dim, type, id);
+}
+
+int isl_set_find_dim_by_id(__isl_keep isl_set *set, enum isl_dim_type type,
+	__isl_keep isl_id *id)
+{
+	return isl_map_find_dim_by_id(set, type, id);
+}
+
+/* Return the position of the dimension of the given type and name
+ * in "bmap".
+ * Return -1 if no such dimension can be found.
+ */
+int isl_basic_map_find_dim_by_name(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, const char *name)
+{
+	if (!bmap)
+		return -1;
+	return isl_space_find_dim_by_name(bmap->dim, type, name);
+}
+
+int isl_map_find_dim_by_name(__isl_keep isl_map *map, enum isl_dim_type type,
+	const char *name)
+{
+	if (!map)
+		return -1;
+	return isl_space_find_dim_by_name(map->dim, type, name);
+}
+
+int isl_set_find_dim_by_name(__isl_keep isl_set *set, enum isl_dim_type type,
+	const char *name)
+{
+	return isl_map_find_dim_by_name(set, type, name);
+}
+
+/* Reset the user pointer on all identifiers of parameters and tuples
+ * of the space of "map".
+ */
+__isl_give isl_map *isl_map_reset_user(__isl_take isl_map *map)
+{
+	isl_space *space;
+
+	space = isl_map_get_space(map);
+	space = isl_space_reset_user(space);
+	map = isl_map_reset_space(map, space);
+
+	return map;
+}
+
+/* Reset the user pointer on all identifiers of parameters and tuples
+ * of the space of "set".
+ */
+__isl_give isl_set *isl_set_reset_user(__isl_take isl_set *set)
+{
+	return isl_map_reset_user(set);
+}
+
+int isl_basic_map_is_rational(__isl_keep isl_basic_map *bmap)
+{
+	if (!bmap)
+		return -1;
+	return ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
+}
+
+int isl_basic_set_is_rational(__isl_keep isl_basic_set *bset)
+{
+	return isl_basic_map_is_rational(bset);
+}
+
+/* Does "bmap" contain any rational points?
+ *
+ * If "bmap" has an equality for each dimension, equating the dimension
+ * to an integer constant, then it has no rational points, even if it
+ * is marked as rational.
+ */
+int isl_basic_map_has_rational(__isl_keep isl_basic_map *bmap)
+{
+	int has_rational = 1;
+	unsigned total;
+
+	if (!bmap)
+		return -1;
+	if (isl_basic_map_plain_is_empty(bmap))
+		return 0;
+	if (!isl_basic_map_is_rational(bmap))
+		return 0;
+	bmap = isl_basic_map_copy(bmap);
+	bmap = isl_basic_map_implicit_equalities(bmap);
+	if (!bmap)
+		return -1;
+	total = isl_basic_map_total_dim(bmap);
+	if (bmap->n_eq == total) {
+		int i, j;
+		for (i = 0; i < bmap->n_eq; ++i) {
+			j = isl_seq_first_non_zero(bmap->eq[i] + 1, total);
+			if (j < 0)
+				break;
+			if (!isl_int_is_one(bmap->eq[i][1 + j]) &&
+			    !isl_int_is_negone(bmap->eq[i][1 + j]))
+				break;
+			j = isl_seq_first_non_zero(bmap->eq[i] + 1 + j + 1,
+						    total - j - 1);
+			if (j >= 0)
+				break;
+		}
+		if (i == bmap->n_eq)
+			has_rational = 0;
+	}
+	isl_basic_map_free(bmap);
+
+	return has_rational;
+}
+
+/* Does "map" contain any rational points?
+ */
+int isl_map_has_rational(__isl_keep isl_map *map)
+{
+	int i;
+	int has_rational;
+
+	if (!map)
+		return -1;
+	for (i = 0; i < map->n; ++i) {
+		has_rational = isl_basic_map_has_rational(map->p[i]);
+		if (has_rational < 0)
+			return -1;
+		if (has_rational)
+			return 1;
+	}
+	return 0;
+}
+
+/* Does "set" contain any rational points?
+ */
+int isl_set_has_rational(__isl_keep isl_set *set)
+{
+	return isl_map_has_rational(set);
+}
+
+/* Is this basic set a parameter domain?
+ */
+int isl_basic_set_is_params(__isl_keep isl_basic_set *bset)
+{
+	if (!bset)
+		return -1;
+	return isl_space_is_params(bset->dim);
+}
+
+/* Is this set a parameter domain?
+ */
+int isl_set_is_params(__isl_keep isl_set *set)
+{
+	if (!set)
+		return -1;
+	return isl_space_is_params(set->dim);
+}
+
+/* Is this map actually a parameter domain?
+ * Users should never call this function.  Outside of isl,
+ * a map can never be a parameter domain.
+ */
+int isl_map_is_params(__isl_keep isl_map *map)
+{
+	if (!map)
+		return -1;
+	return isl_space_is_params(map->dim);
+}
+
+static struct isl_basic_map *basic_map_init(struct isl_ctx *ctx,
+		struct isl_basic_map *bmap, unsigned extra,
+		unsigned n_eq, unsigned n_ineq)
+{
+	int i;
+	size_t row_size = 1 + isl_space_dim(bmap->dim, isl_dim_all) + extra;
+
+	bmap->ctx = ctx;
+	isl_ctx_ref(ctx);
+
+	bmap->block = isl_blk_alloc(ctx, (n_ineq + n_eq) * row_size);
+	if (isl_blk_is_error(bmap->block))
+		goto error;
+
+	bmap->ineq = isl_alloc_array(ctx, isl_int *, n_ineq + n_eq);
+	if ((n_ineq + n_eq) && !bmap->ineq)
+		goto error;
+
+	if (extra == 0) {
+		bmap->block2 = isl_blk_empty();
+		bmap->div = NULL;
+	} else {
+		bmap->block2 = isl_blk_alloc(ctx, extra * (1 + row_size));
+		if (isl_blk_is_error(bmap->block2))
+			goto error;
+
+		bmap->div = isl_alloc_array(ctx, isl_int *, extra);
+		if (!bmap->div)
+			goto error;
+	}
+
+	for (i = 0; i < n_ineq + n_eq; ++i)
+		bmap->ineq[i] = bmap->block.data + i * row_size;
+
+	for (i = 0; i < extra; ++i)
+		bmap->div[i] = bmap->block2.data + i * (1 + row_size);
+
+	bmap->ref = 1;
+	bmap->flags = 0;
+	bmap->c_size = n_eq + n_ineq;
+	bmap->eq = bmap->ineq + n_ineq;
+	bmap->extra = extra;
+	bmap->n_eq = 0;
+	bmap->n_ineq = 0;
+	bmap->n_div = 0;
+	bmap->sample = NULL;
+
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_alloc(struct isl_ctx *ctx,
+		unsigned nparam, unsigned dim, unsigned extra,
+		unsigned n_eq, unsigned n_ineq)
+{
+	struct isl_basic_map *bmap;
+	isl_space *space;
+
+	space = isl_space_set_alloc(ctx, nparam, dim);
+	if (!space)
+		return NULL;
+
+	bmap = isl_basic_map_alloc_space(space, extra, n_eq, n_ineq);
+	return (struct isl_basic_set *)bmap;
+}
+
+struct isl_basic_set *isl_basic_set_alloc_space(__isl_take isl_space *dim,
+		unsigned extra, unsigned n_eq, unsigned n_ineq)
+{
+	struct isl_basic_map *bmap;
+	if (!dim)
+		return NULL;
+	isl_assert(dim->ctx, dim->n_in == 0, goto error);
+	bmap = isl_basic_map_alloc_space(dim, extra, n_eq, n_ineq);
+	return (struct isl_basic_set *)bmap;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+struct isl_basic_map *isl_basic_map_alloc_space(__isl_take isl_space *dim,
+		unsigned extra, unsigned n_eq, unsigned n_ineq)
+{
+	struct isl_basic_map *bmap;
+
+	if (!dim)
+		return NULL;
+	bmap = isl_calloc_type(dim->ctx, struct isl_basic_map);
+	if (!bmap)
+		goto error;
+	bmap->dim = dim;
+
+	return basic_map_init(dim->ctx, bmap, extra, n_eq, n_ineq);
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+struct isl_basic_map *isl_basic_map_alloc(struct isl_ctx *ctx,
+		unsigned nparam, unsigned in, unsigned out, unsigned extra,
+		unsigned n_eq, unsigned n_ineq)
+{
+	struct isl_basic_map *bmap;
+	isl_space *dim;
+
+	dim = isl_space_alloc(ctx, nparam, in, out);
+	if (!dim)
+		return NULL;
+
+	bmap = isl_basic_map_alloc_space(dim, extra, n_eq, n_ineq);
+	return bmap;
+}
+
+static void dup_constraints(
+		struct isl_basic_map *dst, struct isl_basic_map *src)
+{
+	int i;
+	unsigned total = isl_basic_map_total_dim(src);
+
+	for (i = 0; i < src->n_eq; ++i) {
+		int j = isl_basic_map_alloc_equality(dst);
+		isl_seq_cpy(dst->eq[j], src->eq[i], 1+total);
+	}
+
+	for (i = 0; i < src->n_ineq; ++i) {
+		int j = isl_basic_map_alloc_inequality(dst);
+		isl_seq_cpy(dst->ineq[j], src->ineq[i], 1+total);
+	}
+
+	for (i = 0; i < src->n_div; ++i) {
+		int j = isl_basic_map_alloc_div(dst);
+		isl_seq_cpy(dst->div[j], src->div[i], 1+1+total);
+	}
+	ISL_F_SET(dst, ISL_BASIC_SET_FINAL);
+}
+
+struct isl_basic_map *isl_basic_map_dup(struct isl_basic_map *bmap)
+{
+	struct isl_basic_map *dup;
+
+	if (!bmap)
+		return NULL;
+	dup = isl_basic_map_alloc_space(isl_space_copy(bmap->dim),
+			bmap->n_div, bmap->n_eq, bmap->n_ineq);
+	if (!dup)
+		return NULL;
+	dup_constraints(dup, bmap);
+	dup->flags = bmap->flags;
+	dup->sample = isl_vec_copy(bmap->sample);
+	return dup;
+}
+
+struct isl_basic_set *isl_basic_set_dup(struct isl_basic_set *bset)
+{
+	struct isl_basic_map *dup;
+
+	dup = isl_basic_map_dup((struct isl_basic_map *)bset);
+	return (struct isl_basic_set *)dup;
+}
+
+struct isl_basic_set *isl_basic_set_copy(struct isl_basic_set *bset)
+{
+	if (!bset)
+		return NULL;
+
+	if (ISL_F_ISSET(bset, ISL_BASIC_SET_FINAL)) {
+		bset->ref++;
+		return bset;
+	}
+	return isl_basic_set_dup(bset);
+}
+
+struct isl_set *isl_set_copy(struct isl_set *set)
+{
+	if (!set)
+		return NULL;
+
+	set->ref++;
+	return set;
+}
+
+struct isl_basic_map *isl_basic_map_copy(struct isl_basic_map *bmap)
+{
+	if (!bmap)
+		return NULL;
+
+	if (ISL_F_ISSET(bmap, ISL_BASIC_SET_FINAL)) {
+		bmap->ref++;
+		return bmap;
+	}
+	bmap = isl_basic_map_dup(bmap);
+	if (bmap)
+		ISL_F_SET(bmap, ISL_BASIC_SET_FINAL);
+	return bmap;
+}
+
+struct isl_map *isl_map_copy(struct isl_map *map)
+{
+	if (!map)
+		return NULL;
+
+	map->ref++;
+	return map;
+}
+
+__isl_null isl_basic_map *isl_basic_map_free(__isl_take isl_basic_map *bmap)
+{
+	if (!bmap)
+		return NULL;
+
+	if (--bmap->ref > 0)
+		return NULL;
+
+	isl_ctx_deref(bmap->ctx);
+	free(bmap->div);
+	isl_blk_free(bmap->ctx, bmap->block2);
+	free(bmap->ineq);
+	isl_blk_free(bmap->ctx, bmap->block);
+	isl_vec_free(bmap->sample);
+	isl_space_free(bmap->dim);
+	free(bmap);
+
+	return NULL;
+}
+
+__isl_null isl_basic_set *isl_basic_set_free(__isl_take isl_basic_set *bset)
+{
+	return isl_basic_map_free((struct isl_basic_map *)bset);
+}
+
+static int room_for_con(struct isl_basic_map *bmap, unsigned n)
+{
+	return bmap->n_eq + bmap->n_ineq + n <= bmap->c_size;
+}
+
+__isl_give isl_map *isl_map_align_params_map_map_and(
+	__isl_take isl_map *map1, __isl_take isl_map *map2,
+	__isl_give isl_map *(*fn)(__isl_take isl_map *map1,
+				    __isl_take isl_map *map2))
+{
+	if (!map1 || !map2)
+		goto error;
+	if (isl_space_match(map1->dim, isl_dim_param, map2->dim, isl_dim_param))
+		return fn(map1, map2);
+	if (!isl_space_has_named_params(map1->dim) ||
+	    !isl_space_has_named_params(map2->dim))
+		isl_die(map1->ctx, isl_error_invalid,
+			"unaligned unnamed parameters", goto error);
+	map1 = isl_map_align_params(map1, isl_map_get_space(map2));
+	map2 = isl_map_align_params(map2, isl_map_get_space(map1));
+	return fn(map1, map2);
+error:
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return NULL;
+}
+
+int isl_map_align_params_map_map_and_test(__isl_keep isl_map *map1,
+	__isl_keep isl_map *map2,
+	int (*fn)(__isl_keep isl_map *map1, __isl_keep isl_map *map2))
+{
+	int r;
+
+	if (!map1 || !map2)
+		return -1;
+	if (isl_space_match(map1->dim, isl_dim_param, map2->dim, isl_dim_param))
+		return fn(map1, map2);
+	if (!isl_space_has_named_params(map1->dim) ||
+	    !isl_space_has_named_params(map2->dim))
+		isl_die(map1->ctx, isl_error_invalid,
+			"unaligned unnamed parameters", return -1);
+	map1 = isl_map_copy(map1);
+	map2 = isl_map_copy(map2);
+	map1 = isl_map_align_params(map1, isl_map_get_space(map2));
+	map2 = isl_map_align_params(map2, isl_map_get_space(map1));
+	r = fn(map1, map2);
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return r;
+}
+
+int isl_basic_map_alloc_equality(struct isl_basic_map *bmap)
+{
+	struct isl_ctx *ctx;
+	if (!bmap)
+		return -1;
+	ctx = bmap->ctx;
+	isl_assert(ctx, room_for_con(bmap, 1), return -1);
+	isl_assert(ctx, (bmap->eq - bmap->ineq) + bmap->n_eq <= bmap->c_size,
+			return -1);
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_ALL_EQUALITIES);
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS);
+	if ((bmap->eq - bmap->ineq) + bmap->n_eq == bmap->c_size) {
+		isl_int *t;
+		int j = isl_basic_map_alloc_inequality(bmap);
+		if (j < 0)
+			return -1;
+		t = bmap->ineq[j];
+		bmap->ineq[j] = bmap->ineq[bmap->n_ineq - 1];
+		bmap->ineq[bmap->n_ineq - 1] = bmap->eq[-1];
+		bmap->eq[-1] = t;
+		bmap->n_eq++;
+		bmap->n_ineq--;
+		bmap->eq--;
+		return 0;
+	}
+	isl_seq_clr(bmap->eq[bmap->n_eq] + 1 + isl_basic_map_total_dim(bmap),
+		      bmap->extra - bmap->n_div);
+	return bmap->n_eq++;
+}
+
+int isl_basic_set_alloc_equality(struct isl_basic_set *bset)
+{
+	return isl_basic_map_alloc_equality((struct isl_basic_map *)bset);
+}
+
+int isl_basic_map_free_equality(struct isl_basic_map *bmap, unsigned n)
+{
+	if (!bmap)
+		return -1;
+	isl_assert(bmap->ctx, n <= bmap->n_eq, return -1);
+	bmap->n_eq -= n;
+	return 0;
+}
+
+int isl_basic_set_free_equality(struct isl_basic_set *bset, unsigned n)
+{
+	return isl_basic_map_free_equality((struct isl_basic_map *)bset, n);
+}
+
+int isl_basic_map_drop_equality(struct isl_basic_map *bmap, unsigned pos)
+{
+	isl_int *t;
+	if (!bmap)
+		return -1;
+	isl_assert(bmap->ctx, pos < bmap->n_eq, return -1);
+
+	if (pos != bmap->n_eq - 1) {
+		t = bmap->eq[pos];
+		bmap->eq[pos] = bmap->eq[bmap->n_eq - 1];
+		bmap->eq[bmap->n_eq - 1] = t;
+	}
+	bmap->n_eq--;
+	return 0;
+}
+
+int isl_basic_set_drop_equality(struct isl_basic_set *bset, unsigned pos)
+{
+	return isl_basic_map_drop_equality((struct isl_basic_map *)bset, pos);
+}
+
+/* Turn inequality "pos" of "bmap" into an equality.
+ *
+ * In particular, we move the inequality in front of the equalities
+ * and move the last inequality in the position of the moved inequality.
+ * Note that isl_tab_make_equalities_explicit depends on this particular
+ * change in the ordering of the constraints.
+ */
+void isl_basic_map_inequality_to_equality(
+		struct isl_basic_map *bmap, unsigned pos)
+{
+	isl_int *t;
+
+	t = bmap->ineq[pos];
+	bmap->ineq[pos] = bmap->ineq[bmap->n_ineq - 1];
+	bmap->ineq[bmap->n_ineq - 1] = bmap->eq[-1];
+	bmap->eq[-1] = t;
+	bmap->n_eq++;
+	bmap->n_ineq--;
+	bmap->eq--;
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS);
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_ALL_EQUALITIES);
+}
+
+static int room_for_ineq(struct isl_basic_map *bmap, unsigned n)
+{
+	return bmap->n_ineq + n <= bmap->eq - bmap->ineq;
+}
+
+int isl_basic_map_alloc_inequality(struct isl_basic_map *bmap)
+{
+	struct isl_ctx *ctx;
+	if (!bmap)
+		return -1;
+	ctx = bmap->ctx;
+	isl_assert(ctx, room_for_ineq(bmap, 1), return -1);
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_ALL_EQUALITIES);
+	isl_seq_clr(bmap->ineq[bmap->n_ineq] +
+		      1 + isl_basic_map_total_dim(bmap),
+		      bmap->extra - bmap->n_div);
+	return bmap->n_ineq++;
+}
+
+int isl_basic_set_alloc_inequality(struct isl_basic_set *bset)
+{
+	return isl_basic_map_alloc_inequality((struct isl_basic_map *)bset);
+}
+
+int isl_basic_map_free_inequality(struct isl_basic_map *bmap, unsigned n)
+{
+	if (!bmap)
+		return -1;
+	isl_assert(bmap->ctx, n <= bmap->n_ineq, return -1);
+	bmap->n_ineq -= n;
+	return 0;
+}
+
+int isl_basic_set_free_inequality(struct isl_basic_set *bset, unsigned n)
+{
+	return isl_basic_map_free_inequality((struct isl_basic_map *)bset, n);
+}
+
+int isl_basic_map_drop_inequality(struct isl_basic_map *bmap, unsigned pos)
+{
+	isl_int *t;
+	if (!bmap)
+		return -1;
+	isl_assert(bmap->ctx, pos < bmap->n_ineq, return -1);
+
+	if (pos != bmap->n_ineq - 1) {
+		t = bmap->ineq[pos];
+		bmap->ineq[pos] = bmap->ineq[bmap->n_ineq - 1];
+		bmap->ineq[bmap->n_ineq - 1] = t;
+		ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
+	}
+	bmap->n_ineq--;
+	return 0;
+}
+
+int isl_basic_set_drop_inequality(struct isl_basic_set *bset, unsigned pos)
+{
+	return isl_basic_map_drop_inequality((struct isl_basic_map *)bset, pos);
+}
+
+__isl_give isl_basic_map *isl_basic_map_add_eq(__isl_take isl_basic_map *bmap,
+	isl_int *eq)
+{
+	int k;
+
+	bmap = isl_basic_map_extend_constraints(bmap, 1, 0);
+	if (!bmap)
+		return NULL;
+	k = isl_basic_map_alloc_equality(bmap);
+	if (k < 0)
+		goto error;
+	isl_seq_cpy(bmap->eq[k], eq, 1 + isl_basic_map_total_dim(bmap));
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_add_eq(__isl_take isl_basic_set *bset,
+	isl_int *eq)
+{
+	return (isl_basic_set *)
+		isl_basic_map_add_eq((isl_basic_map *)bset, eq);
+}
+
+__isl_give isl_basic_map *isl_basic_map_add_ineq(__isl_take isl_basic_map *bmap,
+	isl_int *ineq)
+{
+	int k;
+
+	bmap = isl_basic_map_extend_constraints(bmap, 0, 1);
+	if (!bmap)
+		return NULL;
+	k = isl_basic_map_alloc_inequality(bmap);
+	if (k < 0)
+		goto error;
+	isl_seq_cpy(bmap->ineq[k], ineq, 1 + isl_basic_map_total_dim(bmap));
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_add_ineq(__isl_take isl_basic_set *bset,
+	isl_int *ineq)
+{
+	return (isl_basic_set *)
+		isl_basic_map_add_ineq((isl_basic_map *)bset, ineq);
+}
+
+int isl_basic_map_alloc_div(struct isl_basic_map *bmap)
+{
+	if (!bmap)
+		return -1;
+	isl_assert(bmap->ctx, bmap->n_div < bmap->extra, return -1);
+	isl_seq_clr(bmap->div[bmap->n_div] +
+		      1 + 1 + isl_basic_map_total_dim(bmap),
+		      bmap->extra - bmap->n_div);
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS);
+	return bmap->n_div++;
+}
+
+int isl_basic_set_alloc_div(struct isl_basic_set *bset)
+{
+	return isl_basic_map_alloc_div((struct isl_basic_map *)bset);
+}
+
+int isl_basic_map_free_div(struct isl_basic_map *bmap, unsigned n)
+{
+	if (!bmap)
+		return -1;
+	isl_assert(bmap->ctx, n <= bmap->n_div, return -1);
+	bmap->n_div -= n;
+	return 0;
+}
+
+int isl_basic_set_free_div(struct isl_basic_set *bset, unsigned n)
+{
+	return isl_basic_map_free_div((struct isl_basic_map *)bset, n);
+}
+
+/* Copy constraint from src to dst, putting the vars of src at offset
+ * dim_off in dst and the divs of src at offset div_off in dst.
+ * If both sets are actually map, then dim_off applies to the input
+ * variables.
+ */
+static void copy_constraint(struct isl_basic_map *dst_map, isl_int *dst,
+			    struct isl_basic_map *src_map, isl_int *src,
+			    unsigned in_off, unsigned out_off, unsigned div_off)
+{
+	unsigned src_nparam = isl_basic_map_n_param(src_map);
+	unsigned dst_nparam = isl_basic_map_n_param(dst_map);
+	unsigned src_in = isl_basic_map_n_in(src_map);
+	unsigned dst_in = isl_basic_map_n_in(dst_map);
+	unsigned src_out = isl_basic_map_n_out(src_map);
+	unsigned dst_out = isl_basic_map_n_out(dst_map);
+	isl_int_set(dst[0], src[0]);
+	isl_seq_cpy(dst+1, src+1, isl_min(dst_nparam, src_nparam));
+	if (dst_nparam > src_nparam)
+		isl_seq_clr(dst+1+src_nparam,
+				dst_nparam - src_nparam);
+	isl_seq_clr(dst+1+dst_nparam, in_off);
+	isl_seq_cpy(dst+1+dst_nparam+in_off,
+		    src+1+src_nparam,
+		    isl_min(dst_in-in_off, src_in));
+	if (dst_in-in_off > src_in)
+		isl_seq_clr(dst+1+dst_nparam+in_off+src_in,
+				dst_in - in_off - src_in);
+	isl_seq_clr(dst+1+dst_nparam+dst_in, out_off);
+	isl_seq_cpy(dst+1+dst_nparam+dst_in+out_off,
+		    src+1+src_nparam+src_in,
+		    isl_min(dst_out-out_off, src_out));
+	if (dst_out-out_off > src_out)
+		isl_seq_clr(dst+1+dst_nparam+dst_in+out_off+src_out,
+				dst_out - out_off - src_out);
+	isl_seq_clr(dst+1+dst_nparam+dst_in+dst_out, div_off);
+	isl_seq_cpy(dst+1+dst_nparam+dst_in+dst_out+div_off,
+		    src+1+src_nparam+src_in+src_out,
+		    isl_min(dst_map->extra-div_off, src_map->n_div));
+	if (dst_map->n_div-div_off > src_map->n_div)
+		isl_seq_clr(dst+1+dst_nparam+dst_in+dst_out+
+				div_off+src_map->n_div,
+				dst_map->n_div - div_off - src_map->n_div);
+}
+
+static void copy_div(struct isl_basic_map *dst_map, isl_int *dst,
+		     struct isl_basic_map *src_map, isl_int *src,
+		     unsigned in_off, unsigned out_off, unsigned div_off)
+{
+	isl_int_set(dst[0], src[0]);
+	copy_constraint(dst_map, dst+1, src_map, src+1, in_off, out_off, div_off);
+}
+
+static struct isl_basic_map *add_constraints(struct isl_basic_map *bmap1,
+		struct isl_basic_map *bmap2, unsigned i_pos, unsigned o_pos)
+{
+	int i;
+	unsigned div_off;
+
+	if (!bmap1 || !bmap2)
+		goto error;
+
+	div_off = bmap1->n_div;
+
+	for (i = 0; i < bmap2->n_eq; ++i) {
+		int i1 = isl_basic_map_alloc_equality(bmap1);
+		if (i1 < 0)
+			goto error;
+		copy_constraint(bmap1, bmap1->eq[i1], bmap2, bmap2->eq[i],
+				i_pos, o_pos, div_off);
+	}
+
+	for (i = 0; i < bmap2->n_ineq; ++i) {
+		int i1 = isl_basic_map_alloc_inequality(bmap1);
+		if (i1 < 0)
+			goto error;
+		copy_constraint(bmap1, bmap1->ineq[i1], bmap2, bmap2->ineq[i],
+				i_pos, o_pos, div_off);
+	}
+
+	for (i = 0; i < bmap2->n_div; ++i) {
+		int i1 = isl_basic_map_alloc_div(bmap1);
+		if (i1 < 0)
+			goto error;
+		copy_div(bmap1, bmap1->div[i1], bmap2, bmap2->div[i],
+			 i_pos, o_pos, div_off);
+	}
+
+	isl_basic_map_free(bmap2);
+
+	return bmap1;
+
+error:
+	isl_basic_map_free(bmap1);
+	isl_basic_map_free(bmap2);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_add_constraints(struct isl_basic_set *bset1,
+		struct isl_basic_set *bset2, unsigned pos)
+{
+	return (struct isl_basic_set *)
+		add_constraints((struct isl_basic_map *)bset1,
+				(struct isl_basic_map *)bset2, 0, pos);
+}
+
+struct isl_basic_map *isl_basic_map_extend_space(struct isl_basic_map *base,
+		__isl_take isl_space *dim, unsigned extra,
+		unsigned n_eq, unsigned n_ineq)
+{
+	struct isl_basic_map *ext;
+	unsigned flags;
+	int dims_ok;
+
+	if (!dim)
+		goto error;
+
+	if (!base)
+		goto error;
+
+	dims_ok = isl_space_is_equal(base->dim, dim) &&
+		  base->extra >= base->n_div + extra;
+
+	if (dims_ok && room_for_con(base, n_eq + n_ineq) &&
+		       room_for_ineq(base, n_ineq)) {
+		isl_space_free(dim);
+		return base;
+	}
+
+	isl_assert(base->ctx, base->dim->nparam <= dim->nparam, goto error);
+	isl_assert(base->ctx, base->dim->n_in <= dim->n_in, goto error);
+	isl_assert(base->ctx, base->dim->n_out <= dim->n_out, goto error);
+	extra += base->extra;
+	n_eq += base->n_eq;
+	n_ineq += base->n_ineq;
+
+	ext = isl_basic_map_alloc_space(dim, extra, n_eq, n_ineq);
+	dim = NULL;
+	if (!ext)
+		goto error;
+
+	if (dims_ok)
+		ext->sample = isl_vec_copy(base->sample);
+	flags = base->flags;
+	ext = add_constraints(ext, base, 0, 0);
+	if (ext) {
+		ext->flags = flags;
+		ISL_F_CLR(ext, ISL_BASIC_SET_FINAL);
+	}
+
+	return ext;
+
+error:
+	isl_space_free(dim);
+	isl_basic_map_free(base);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_extend_space(struct isl_basic_set *base,
+		__isl_take isl_space *dim, unsigned extra,
+		unsigned n_eq, unsigned n_ineq)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_extend_space((struct isl_basic_map *)base, dim,
+							extra, n_eq, n_ineq);
+}
+
+struct isl_basic_map *isl_basic_map_extend_constraints(
+		struct isl_basic_map *base, unsigned n_eq, unsigned n_ineq)
+{
+	if (!base)
+		return NULL;
+	return isl_basic_map_extend_space(base, isl_space_copy(base->dim),
+					0, n_eq, n_ineq);
+}
+
+struct isl_basic_map *isl_basic_map_extend(struct isl_basic_map *base,
+		unsigned nparam, unsigned n_in, unsigned n_out, unsigned extra,
+		unsigned n_eq, unsigned n_ineq)
+{
+	struct isl_basic_map *bmap;
+	isl_space *dim;
+
+	if (!base)
+		return NULL;
+	dim = isl_space_alloc(base->ctx, nparam, n_in, n_out);
+	if (!dim)
+		goto error;
+
+	bmap = isl_basic_map_extend_space(base, dim, extra, n_eq, n_ineq);
+	return bmap;
+error:
+	isl_basic_map_free(base);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_extend(struct isl_basic_set *base,
+		unsigned nparam, unsigned dim, unsigned extra,
+		unsigned n_eq, unsigned n_ineq)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_extend((struct isl_basic_map *)base,
+					nparam, 0, dim, extra, n_eq, n_ineq);
+}
+
+struct isl_basic_set *isl_basic_set_extend_constraints(
+		struct isl_basic_set *base, unsigned n_eq, unsigned n_ineq)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_extend_constraints((struct isl_basic_map *)base,
+						    n_eq, n_ineq);
+}
+
+struct isl_basic_set *isl_basic_set_cow(struct isl_basic_set *bset)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_cow((struct isl_basic_map *)bset);
+}
+
+struct isl_basic_map *isl_basic_map_cow(struct isl_basic_map *bmap)
+{
+	if (!bmap)
+		return NULL;
+
+	if (bmap->ref > 1) {
+		bmap->ref--;
+		bmap = isl_basic_map_dup(bmap);
+	}
+	if (bmap)
+		ISL_F_CLR(bmap, ISL_BASIC_SET_FINAL);
+	return bmap;
+}
+
+struct isl_set *isl_set_cow(struct isl_set *set)
+{
+	if (!set)
+		return NULL;
+
+	if (set->ref == 1)
+		return set;
+	set->ref--;
+	return isl_set_dup(set);
+}
+
+struct isl_map *isl_map_cow(struct isl_map *map)
+{
+	if (!map)
+		return NULL;
+
+	if (map->ref == 1)
+		return map;
+	map->ref--;
+	return isl_map_dup(map);
+}
+
+static void swap_vars(struct isl_blk blk, isl_int *a,
+			unsigned a_len, unsigned b_len)
+{
+	isl_seq_cpy(blk.data, a+a_len, b_len);
+	isl_seq_cpy(blk.data+b_len, a, a_len);
+	isl_seq_cpy(a, blk.data, b_len+a_len);
+}
+
+static __isl_give isl_basic_map *isl_basic_map_swap_vars(
+	__isl_take isl_basic_map *bmap, unsigned pos, unsigned n1, unsigned n2)
+{
+	int i;
+	struct isl_blk blk;
+
+	if (!bmap)
+		goto error;
+
+	isl_assert(bmap->ctx,
+		pos + n1 + n2 <= 1 + isl_basic_map_total_dim(bmap), goto error);
+
+	if (n1 == 0 || n2 == 0)
+		return bmap;
+
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+
+	blk = isl_blk_alloc(bmap->ctx, n1 + n2);
+	if (isl_blk_is_error(blk))
+		goto error;
+
+	for (i = 0; i < bmap->n_eq; ++i)
+		swap_vars(blk,
+			  bmap->eq[i] + pos, n1, n2);
+
+	for (i = 0; i < bmap->n_ineq; ++i)
+		swap_vars(blk,
+			  bmap->ineq[i] + pos, n1, n2);
+
+	for (i = 0; i < bmap->n_div; ++i)
+		swap_vars(blk,
+			  bmap->div[i]+1 + pos, n1, n2);
+
+	isl_blk_free(bmap->ctx, blk);
+
+	ISL_F_CLR(bmap, ISL_BASIC_SET_NORMALIZED);
+	bmap = isl_basic_map_gauss(bmap, NULL);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+static __isl_give isl_basic_set *isl_basic_set_swap_vars(
+	__isl_take isl_basic_set *bset, unsigned n)
+{
+	unsigned dim;
+	unsigned nparam;
+
+	if (!bset)
+		return NULL;
+
+	nparam = isl_basic_set_n_param(bset);
+	dim = isl_basic_set_n_dim(bset);
+	isl_assert(bset->ctx, n <= dim, goto error);
+
+	return isl_basic_map_swap_vars(bset, 1 + nparam, n, dim - n);
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+struct isl_basic_map *isl_basic_map_set_to_empty(struct isl_basic_map *bmap)
+{
+	int i = 0;
+	unsigned total;
+	if (!bmap)
+		goto error;
+	total = isl_basic_map_total_dim(bmap);
+	isl_basic_map_free_div(bmap, bmap->n_div);
+	isl_basic_map_free_inequality(bmap, bmap->n_ineq);
+	if (bmap->n_eq > 0)
+		isl_basic_map_free_equality(bmap, bmap->n_eq-1);
+	else {
+		i = isl_basic_map_alloc_equality(bmap);
+		if (i < 0)
+			goto error;
+	}
+	isl_int_set_si(bmap->eq[i][0], 1);
+	isl_seq_clr(bmap->eq[i]+1, total);
+	ISL_F_SET(bmap, ISL_BASIC_MAP_EMPTY);
+	isl_vec_free(bmap->sample);
+	bmap->sample = NULL;
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_set_to_empty(struct isl_basic_set *bset)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_set_to_empty((struct isl_basic_map *)bset);
+}
+
+/* Swap divs "a" and "b" in "bmap" (without modifying any of the constraints
+ * of "bmap").
+ */
+static void swap_div(__isl_keep isl_basic_map *bmap, int a, int b)
+{
+	isl_int *t = bmap->div[a];
+	bmap->div[a] = bmap->div[b];
+	bmap->div[b] = t;
+}
+
+/* Swap divs "a" and "b" in "bmap" and adjust the constraints and
+ * div definitions accordingly.
+ */
+void isl_basic_map_swap_div(struct isl_basic_map *bmap, int a, int b)
+{
+	int i;
+	unsigned off = isl_space_dim(bmap->dim, isl_dim_all);
+
+	swap_div(bmap, a, b);
+
+	for (i = 0; i < bmap->n_eq; ++i)
+		isl_int_swap(bmap->eq[i][1+off+a], bmap->eq[i][1+off+b]);
+
+	for (i = 0; i < bmap->n_ineq; ++i)
+		isl_int_swap(bmap->ineq[i][1+off+a], bmap->ineq[i][1+off+b]);
+
+	for (i = 0; i < bmap->n_div; ++i)
+		isl_int_swap(bmap->div[i][1+1+off+a], bmap->div[i][1+1+off+b]);
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
+}
+
+/* Eliminate the specified n dimensions starting at first from the
+ * constraints, without removing the dimensions from the space.
+ * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
+ */
+__isl_give isl_map *isl_map_eliminate(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+	if (n == 0)
+		return map;
+
+	if (first + n > isl_map_dim(map, type) || first + n < first)
+		isl_die(map->ctx, isl_error_invalid,
+			"index out of bounds", goto error);
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_eliminate(map->p[i], type, first, n);
+		if (!map->p[i])
+			goto error;
+	}
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Eliminate the specified n dimensions starting at first from the
+ * constraints, without removing the dimensions from the space.
+ * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
+ */
+__isl_give isl_set *isl_set_eliminate(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return (isl_set *)isl_map_eliminate((isl_map *)set, type, first, n);
+}
+
+/* Eliminate the specified n dimensions starting at first from the
+ * constraints, without removing the dimensions from the space.
+ * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
+ */
+__isl_give isl_set *isl_set_eliminate_dims(__isl_take isl_set *set,
+	unsigned first, unsigned n)
+{
+	return isl_set_eliminate(set, isl_dim_set, first, n);
+}
+
+__isl_give isl_basic_map *isl_basic_map_remove_divs(
+	__isl_take isl_basic_map *bmap)
+{
+	if (!bmap)
+		return NULL;
+	bmap = isl_basic_map_eliminate_vars(bmap,
+			    isl_space_dim(bmap->dim, isl_dim_all), bmap->n_div);
+	if (!bmap)
+		return NULL;
+	bmap->n_div = 0;
+	return isl_basic_map_finalize(bmap);
+}
+
+__isl_give isl_basic_set *isl_basic_set_remove_divs(
+	__isl_take isl_basic_set *bset)
+{
+	return (struct isl_basic_set *)isl_basic_map_remove_divs(
+			(struct isl_basic_map *)bset);
+}
+
+__isl_give isl_map *isl_map_remove_divs(__isl_take isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+	if (map->n == 0)
+		return map;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+	
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_remove_divs(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_remove_divs(__isl_take isl_set *set)
+{
+	return isl_map_remove_divs(set);
+}
+
+struct isl_basic_map *isl_basic_map_remove_dims(struct isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	if (!bmap)
+		return NULL;
+	isl_assert(bmap->ctx, first + n <= isl_basic_map_dim(bmap, type),
+			goto error);
+	if (n == 0 && !isl_space_is_named_or_nested(bmap->dim, type))
+		return bmap;
+	bmap = isl_basic_map_eliminate_vars(bmap,
+			isl_basic_map_offset(bmap, type) - 1 + first, n);
+	if (!bmap)
+		return bmap;
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY) && type == isl_dim_div)
+		return bmap;
+	bmap = isl_basic_map_drop(bmap, type, first, n);
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Return true if the definition of the given div (recursively) involves
+ * any of the given variables.
+ */
+static int div_involves_vars(__isl_keep isl_basic_map *bmap, int div,
+	unsigned first, unsigned n)
+{
+	int i;
+	unsigned div_offset = isl_basic_map_offset(bmap, isl_dim_div);
+
+	if (isl_int_is_zero(bmap->div[div][0]))
+		return 0;
+	if (isl_seq_first_non_zero(bmap->div[div] + 1 + first, n) >= 0)
+		return 1;
+
+	for (i = bmap->n_div - 1; i >= 0; --i) {
+		if (isl_int_is_zero(bmap->div[div][1 + div_offset + i]))
+			continue;
+		if (div_involves_vars(bmap, i, first, n))
+			return 1;
+	}
+
+	return 0;
+}
+
+/* Try and add a lower and/or upper bound on "div" to "bmap"
+ * based on inequality "i".
+ * "total" is the total number of variables (excluding the divs).
+ * "v" is a temporary object that can be used during the calculations.
+ * If "lb" is set, then a lower bound should be constructed.
+ * If "ub" is set, then an upper bound should be constructed.
+ *
+ * The calling function has already checked that the inequality does not
+ * reference "div", but we still need to check that the inequality is
+ * of the right form.  We'll consider the case where we want to construct
+ * a lower bound.  The construction of upper bounds is similar.
+ *
+ * Let "div" be of the form
+ *
+ *	q = floor((a + f(x))/d)
+ *
+ * We essentially check if constraint "i" is of the form
+ *
+ *	b + f(x) >= 0
+ *
+ * so that we can use it to derive a lower bound on "div".
+ * However, we allow a slightly more general form
+ *
+ *	b + g(x) >= 0
+ *
+ * with the condition that the coefficients of g(x) - f(x) are all
+ * divisible by d.
+ * Rewriting this constraint as
+ *
+ *	0 >= -b - g(x)
+ *
+ * adding a + f(x) to both sides and dividing by d, we obtain
+ *
+ *	(a + f(x))/d >= (a-b)/d + (f(x)-g(x))/d
+ *
+ * Taking the floor on both sides, we obtain
+ *
+ *	q >= floor((a-b)/d) + (f(x)-g(x))/d
+ *
+ * or
+ *
+ *	(g(x)-f(x))/d + ceil((b-a)/d) + q >= 0
+ *
+ * In the case of an upper bound, we construct the constraint
+ *
+ *	(g(x)+f(x))/d + floor((b+a)/d) - q >= 0
+ *
+ */
+static __isl_give isl_basic_map *insert_bounds_on_div_from_ineq(
+	__isl_take isl_basic_map *bmap, int div, int i,
+	unsigned total, isl_int v, int lb, int ub)
+{
+	int j;
+
+	for (j = 0; (lb || ub) && j < total + bmap->n_div; ++j) {
+		if (lb) {
+			isl_int_sub(v, bmap->ineq[i][1 + j],
+					bmap->div[div][1 + 1 + j]);
+			lb = isl_int_is_divisible_by(v, bmap->div[div][0]);
+		}
+		if (ub) {
+			isl_int_add(v, bmap->ineq[i][1 + j],
+					bmap->div[div][1 + 1 + j]);
+			ub = isl_int_is_divisible_by(v, bmap->div[div][0]);
+		}
+	}
+	if (!lb && !ub)
+		return bmap;
+
+	bmap = isl_basic_map_cow(bmap);
+	bmap = isl_basic_map_extend_constraints(bmap, 0, lb + ub);
+	if (lb) {
+		int k = isl_basic_map_alloc_inequality(bmap);
+		if (k < 0)
+			goto error;
+		for (j = 0; j < 1 + total + bmap->n_div; ++j) {
+			isl_int_sub(bmap->ineq[k][j], bmap->ineq[i][j],
+					bmap->div[div][1 + j]);
+			isl_int_cdiv_q(bmap->ineq[k][j],
+					bmap->ineq[k][j], bmap->div[div][0]);
+		}
+		isl_int_set_si(bmap->ineq[k][1 + total + div], 1);
+	}
+	if (ub) {
+		int k = isl_basic_map_alloc_inequality(bmap);
+		if (k < 0)
+			goto error;
+		for (j = 0; j < 1 + total + bmap->n_div; ++j) {
+			isl_int_add(bmap->ineq[k][j], bmap->ineq[i][j],
+					bmap->div[div][1 + j]);
+			isl_int_fdiv_q(bmap->ineq[k][j],
+					bmap->ineq[k][j], bmap->div[div][0]);
+		}
+		isl_int_set_si(bmap->ineq[k][1 + total + div], -1);
+	}
+
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* This function is called right before "div" is eliminated from "bmap"
+ * using Fourier-Motzkin.
+ * Look through the constraints of "bmap" for constraints on the argument
+ * of the integer division and use them to construct constraints on the
+ * integer division itself.  These constraints can then be combined
+ * during the Fourier-Motzkin elimination.
+ * Note that it is only useful to introduce lower bounds on "div"
+ * if "bmap" already contains upper bounds on "div" as the newly
+ * introduce lower bounds can then be combined with the pre-existing
+ * upper bounds.  Similarly for upper bounds.
+ * We therefore first check if "bmap" contains any lower and/or upper bounds
+ * on "div".
+ *
+ * It is interesting to note that the introduction of these constraints
+ * can indeed lead to more accurate results, even when compared to
+ * deriving constraints on the argument of "div" from constraints on "div".
+ * Consider, for example, the set
+ *
+ *	{ [i,j,k] : 3 + i + 2j >= 0 and 2 * [(i+2j)/4] <= k }
+ *
+ * The second constraint can be rewritten as
+ *
+ *	2 * [(-i-2j+3)/4] + k >= 0
+ *
+ * from which we can derive
+ *
+ *	-i - 2j + 3 >= -2k
+ *
+ * or
+ *
+ *	i + 2j <= 3 + 2k
+ *
+ * Combined with the first constraint, we obtain
+ *
+ *	-3 <= 3 + 2k	or	k >= -3
+ *
+ * If, on the other hand we derive a constraint on [(i+2j)/4] from
+ * the first constraint, we obtain
+ *
+ *	[(i + 2j)/4] >= [-3/4] = -1
+ *
+ * Combining this constraint with the second constraint, we obtain
+ *
+ *	k >= -2
+ */
+static __isl_give isl_basic_map *insert_bounds_on_div(
+	__isl_take isl_basic_map *bmap, int div)
+{
+	int i;
+	int check_lb, check_ub;
+	isl_int v;
+	unsigned total;
+
+	if (!bmap)
+		return NULL;
+
+	if (isl_int_is_zero(bmap->div[div][0]))
+		return bmap;
+
+	total = isl_space_dim(bmap->dim, isl_dim_all);
+
+	check_lb = 0;
+	check_ub = 0;
+	for (i = 0; (!check_lb || !check_ub) && i < bmap->n_ineq; ++i) {
+		int s = isl_int_sgn(bmap->ineq[i][1 + total + div]);
+		if (s > 0)
+			check_ub = 1;
+		if (s < 0)
+			check_lb = 1;
+	}
+
+	if (!check_lb && !check_ub)
+		return bmap;
+
+	isl_int_init(v);
+
+	for (i = 0; bmap && i < bmap->n_ineq; ++i) {
+		if (!isl_int_is_zero(bmap->ineq[i][1 + total + div]))
+			continue;
+
+		bmap = insert_bounds_on_div_from_ineq(bmap, div, i, total, v,
+							check_lb, check_ub);
+	}
+
+	isl_int_clear(v);
+
+	return bmap;
+}
+
+/* Remove all divs (recursively) involving any of the given dimensions
+ * in their definitions.
+ */
+__isl_give isl_basic_map *isl_basic_map_remove_divs_involving_dims(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+
+	if (!bmap)
+		return NULL;
+	isl_assert(bmap->ctx, first + n <= isl_basic_map_dim(bmap, type),
+			goto error);
+	first += isl_basic_map_offset(bmap, type);
+
+	for (i = bmap->n_div - 1; i >= 0; --i) {
+		if (!div_involves_vars(bmap, i, first, n))
+			continue;
+		bmap = insert_bounds_on_div(bmap, i);
+		bmap = isl_basic_map_remove_dims(bmap, isl_dim_div, i, 1);
+		if (!bmap)
+			return NULL;
+		i = bmap->n_div;
+	}
+
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_remove_divs_involving_dims(
+	__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return isl_basic_map_remove_divs_involving_dims(bset, type, first, n);
+}
+
+__isl_give isl_map *isl_map_remove_divs_involving_dims(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+	if (map->n == 0)
+		return map;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_remove_divs_involving_dims(map->p[i],
+								type, first, n);
+		if (!map->p[i])
+			goto error;
+	}
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_remove_divs_involving_dims(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return (isl_set *)isl_map_remove_divs_involving_dims((isl_map *)set,
+							      type, first, n);
+}
+
+/* Does the desciption of "bmap" depend on the specified dimensions?
+ * We also check whether the dimensions appear in any of the div definitions.
+ * In principle there is no need for this check.  If the dimensions appear
+ * in a div definition, they also appear in the defining constraints of that
+ * div.
+ */
+int isl_basic_map_involves_dims(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+
+	if (!bmap)
+		return -1;
+
+	if (first + n > isl_basic_map_dim(bmap, type))
+		isl_die(bmap->ctx, isl_error_invalid,
+			"index out of bounds", return -1);
+
+	first += isl_basic_map_offset(bmap, type);
+	for (i = 0; i < bmap->n_eq; ++i)
+		if (isl_seq_first_non_zero(bmap->eq[i] + first, n) >= 0)
+			return 1;
+	for (i = 0; i < bmap->n_ineq; ++i)
+		if (isl_seq_first_non_zero(bmap->ineq[i] + first, n) >= 0)
+			return 1;
+	for (i = 0; i < bmap->n_div; ++i) {
+		if (isl_int_is_zero(bmap->div[i][0]))
+			continue;
+		if (isl_seq_first_non_zero(bmap->div[i] + 1 + first, n) >= 0)
+			return 1;
+	}
+
+	return 0;
+}
+
+int isl_map_involves_dims(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+
+	if (!map)
+		return -1;
+
+	if (first + n > isl_map_dim(map, type))
+		isl_die(map->ctx, isl_error_invalid,
+			"index out of bounds", return -1);
+
+	for (i = 0; i < map->n; ++i) {
+		int involves = isl_basic_map_involves_dims(map->p[i],
+							    type, first, n);
+		if (involves < 0 || involves)
+			return involves;
+	}
+
+	return 0;
+}
+
+int isl_basic_set_involves_dims(__isl_keep isl_basic_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return isl_basic_map_involves_dims(bset, type, first, n);
+}
+
+int isl_set_involves_dims(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return isl_map_involves_dims(set, type, first, n);
+}
+
+/* Return true if the definition of the given div is unknown or depends
+ * on unknown divs.
+ */
+static int div_is_unknown(__isl_keep isl_basic_map *bmap, int div)
+{
+	int i;
+	unsigned div_offset = isl_basic_map_offset(bmap, isl_dim_div);
+
+	if (isl_int_is_zero(bmap->div[div][0]))
+		return 1;
+
+	for (i = bmap->n_div - 1; i >= 0; --i) {
+		if (isl_int_is_zero(bmap->div[div][1 + div_offset + i]))
+			continue;
+		if (div_is_unknown(bmap, i))
+			return 1;
+	}
+
+	return 0;
+}
+
+/* Remove all divs that are unknown or defined in terms of unknown divs.
+ */
+__isl_give isl_basic_map *isl_basic_map_remove_unknown_divs(
+	__isl_take isl_basic_map *bmap)
+{
+	int i;
+
+	if (!bmap)
+		return NULL;
+
+	for (i = bmap->n_div - 1; i >= 0; --i) {
+		if (!div_is_unknown(bmap, i))
+			continue;
+		bmap = isl_basic_map_remove_dims(bmap, isl_dim_div, i, 1);
+		if (!bmap)
+			return NULL;
+		i = bmap->n_div;
+	}
+
+	return bmap;
+}
+
+/* Remove all divs that are unknown or defined in terms of unknown divs.
+ */
+__isl_give isl_basic_set *isl_basic_set_remove_unknown_divs(
+	__isl_take isl_basic_set *bset)
+{
+	return isl_basic_map_remove_unknown_divs(bset);
+}
+
+__isl_give isl_map *isl_map_remove_unknown_divs(__isl_take isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+	if (map->n == 0)
+		return map;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_remove_unknown_divs(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_remove_unknown_divs(__isl_take isl_set *set)
+{
+	return (isl_set *)isl_map_remove_unknown_divs((isl_map *)set);
+}
+
+__isl_give isl_basic_set *isl_basic_set_remove_dims(
+	__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return (isl_basic_set *)
+	    isl_basic_map_remove_dims((isl_basic_map *)bset, type, first, n);
+}
+
+struct isl_map *isl_map_remove_dims(struct isl_map *map,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+
+	if (n == 0)
+		return map;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+	isl_assert(map->ctx, first + n <= isl_map_dim(map, type), goto error);
+	
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_eliminate_vars(map->p[i],
+			isl_basic_map_offset(map->p[i], type) - 1 + first, n);
+		if (!map->p[i])
+			goto error;
+	}
+	map = isl_map_drop(map, type, first, n);
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_remove_dims(__isl_take isl_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return (isl_set *)isl_map_remove_dims((isl_map *)bset, type, first, n);
+}
+
+/* Project out n inputs starting at first using Fourier-Motzkin */
+struct isl_map *isl_map_remove_inputs(struct isl_map *map,
+	unsigned first, unsigned n)
+{
+	return isl_map_remove_dims(map, isl_dim_in, first, n);
+}
+
+static void dump_term(struct isl_basic_map *bmap,
+			isl_int c, int pos, FILE *out)
+{
+	const char *name;
+	unsigned in = isl_basic_map_n_in(bmap);
+	unsigned dim = in + isl_basic_map_n_out(bmap);
+	unsigned nparam = isl_basic_map_n_param(bmap);
+	if (!pos)
+		isl_int_print(out, c, 0);
+	else {
+		if (!isl_int_is_one(c))
+			isl_int_print(out, c, 0);
+		if (pos < 1 + nparam) {
+			name = isl_space_get_dim_name(bmap->dim,
+						isl_dim_param, pos - 1);
+			if (name)
+				fprintf(out, "%s", name);
+			else
+				fprintf(out, "p%d", pos - 1);
+		} else if (pos < 1 + nparam + in)
+			fprintf(out, "i%d", pos - 1 - nparam);
+		else if (pos < 1 + nparam + dim)
+			fprintf(out, "o%d", pos - 1 - nparam - in);
+		else
+			fprintf(out, "e%d", pos - 1 - nparam - dim);
+	}
+}
+
+static void dump_constraint_sign(struct isl_basic_map *bmap, isl_int *c,
+				int sign, FILE *out)
+{
+	int i;
+	int first;
+	unsigned len = 1 + isl_basic_map_total_dim(bmap);
+	isl_int v;
+
+	isl_int_init(v);
+	for (i = 0, first = 1; i < len; ++i) {
+		if (isl_int_sgn(c[i]) * sign <= 0)
+			continue;
+		if (!first)
+			fprintf(out, " + ");
+		first = 0;
+		isl_int_abs(v, c[i]);
+		dump_term(bmap, v, i, out);
+	}
+	isl_int_clear(v);
+	if (first)
+		fprintf(out, "0");
+}
+
+static void dump_constraint(struct isl_basic_map *bmap, isl_int *c,
+				const char *op, FILE *out, int indent)
+{
+	int i;
+
+	fprintf(out, "%*s", indent, "");
+
+	dump_constraint_sign(bmap, c, 1, out);
+	fprintf(out, " %s ", op);
+	dump_constraint_sign(bmap, c, -1, out);
+
+	fprintf(out, "\n");
+
+	for (i = bmap->n_div; i < bmap->extra; ++i) {
+		if (isl_int_is_zero(c[1+isl_space_dim(bmap->dim, isl_dim_all)+i]))
+			continue;
+		fprintf(out, "%*s", indent, "");
+		fprintf(out, "ERROR: unused div coefficient not zero\n");
+		abort();
+	}
+}
+
+static void dump_constraints(struct isl_basic_map *bmap,
+				isl_int **c, unsigned n,
+				const char *op, FILE *out, int indent)
+{
+	int i;
+
+	for (i = 0; i < n; ++i)
+		dump_constraint(bmap, c[i], op, out, indent);
+}
+
+static void dump_affine(struct isl_basic_map *bmap, isl_int *exp, FILE *out)
+{
+	int j;
+	int first = 1;
+	unsigned total = isl_basic_map_total_dim(bmap);
+
+	for (j = 0; j < 1 + total; ++j) {
+		if (isl_int_is_zero(exp[j]))
+			continue;
+		if (!first && isl_int_is_pos(exp[j]))
+			fprintf(out, "+");
+		dump_term(bmap, exp[j], j, out);
+		first = 0;
+	}
+}
+
+static void dump(struct isl_basic_map *bmap, FILE *out, int indent)
+{
+	int i;
+
+	dump_constraints(bmap, bmap->eq, bmap->n_eq, "=", out, indent);
+	dump_constraints(bmap, bmap->ineq, bmap->n_ineq, ">=", out, indent);
+
+	for (i = 0; i < bmap->n_div; ++i) {
+		fprintf(out, "%*s", indent, "");
+		fprintf(out, "e%d = [(", i);
+		dump_affine(bmap, bmap->div[i]+1, out);
+		fprintf(out, ")/");
+		isl_int_print(out, bmap->div[i][0], 0);
+		fprintf(out, "]\n");
+	}
+}
+
+void isl_basic_set_print_internal(struct isl_basic_set *bset,
+	FILE *out, int indent)
+{
+	if (!bset) {
+		fprintf(out, "null basic set\n");
+		return;
+	}
+
+	fprintf(out, "%*s", indent, "");
+	fprintf(out, "ref: %d, nparam: %d, dim: %d, extra: %d, flags: %x\n",
+			bset->ref, bset->dim->nparam, bset->dim->n_out,
+			bset->extra, bset->flags);
+	dump((struct isl_basic_map *)bset, out, indent);
+}
+
+void isl_basic_map_print_internal(struct isl_basic_map *bmap,
+	FILE *out, int indent)
+{
+	if (!bmap) {
+		fprintf(out, "null basic map\n");
+		return;
+	}
+
+	fprintf(out, "%*s", indent, "");
+	fprintf(out, "ref: %d, nparam: %d, in: %d, out: %d, extra: %d, "
+			"flags: %x, n_name: %d\n",
+		bmap->ref,
+		bmap->dim->nparam, bmap->dim->n_in, bmap->dim->n_out,
+		bmap->extra, bmap->flags, bmap->dim->n_id);
+	dump(bmap, out, indent);
+}
+
+int isl_inequality_negate(struct isl_basic_map *bmap, unsigned pos)
+{
+	unsigned total;
+	if (!bmap)
+		return -1;
+	total = isl_basic_map_total_dim(bmap);
+	isl_assert(bmap->ctx, pos < bmap->n_ineq, return -1);
+	isl_seq_neg(bmap->ineq[pos], bmap->ineq[pos], 1 + total);
+	isl_int_sub_ui(bmap->ineq[pos][0], bmap->ineq[pos][0], 1);
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
+	return 0;
+}
+
+__isl_give isl_set *isl_set_alloc_space(__isl_take isl_space *dim, int n,
+	unsigned flags)
+{
+	struct isl_set *set;
+
+	if (!dim)
+		return NULL;
+	isl_assert(dim->ctx, dim->n_in == 0, goto error);
+	isl_assert(dim->ctx, n >= 0, goto error);
+	set = isl_alloc(dim->ctx, struct isl_set,
+			sizeof(struct isl_set) +
+			(n - 1) * sizeof(struct isl_basic_set *));
+	if (!set)
+		goto error;
+
+	set->ctx = dim->ctx;
+	isl_ctx_ref(set->ctx);
+	set->ref = 1;
+	set->size = n;
+	set->n = 0;
+	set->dim = dim;
+	set->flags = flags;
+	return set;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+struct isl_set *isl_set_alloc(struct isl_ctx *ctx,
+		unsigned nparam, unsigned dim, int n, unsigned flags)
+{
+	struct isl_set *set;
+	isl_space *dims;
+
+	dims = isl_space_alloc(ctx, nparam, 0, dim);
+	if (!dims)
+		return NULL;
+
+	set = isl_set_alloc_space(dims, n, flags);
+	return set;
+}
+
+/* Make sure "map" has room for at least "n" more basic maps.
+ */
+struct isl_map *isl_map_grow(struct isl_map *map, int n)
+{
+	int i;
+	struct isl_map *grown = NULL;
+
+	if (!map)
+		return NULL;
+	isl_assert(map->ctx, n >= 0, goto error);
+	if (map->n + n <= map->size)
+		return map;
+	grown = isl_map_alloc_space(isl_map_get_space(map), map->n + n, map->flags);
+	if (!grown)
+		goto error;
+	for (i = 0; i < map->n; ++i) {
+		grown->p[i] = isl_basic_map_copy(map->p[i]);
+		if (!grown->p[i])
+			goto error;
+		grown->n++;
+	}
+	isl_map_free(map);
+	return grown;
+error:
+	isl_map_free(grown);
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Make sure "set" has room for at least "n" more basic sets.
+ */
+struct isl_set *isl_set_grow(struct isl_set *set, int n)
+{
+	return (struct isl_set *)isl_map_grow((struct isl_map *)set, n);
+}
+
+struct isl_set *isl_set_dup(struct isl_set *set)
+{
+	int i;
+	struct isl_set *dup;
+
+	if (!set)
+		return NULL;
+
+	dup = isl_set_alloc_space(isl_space_copy(set->dim), set->n, set->flags);
+	if (!dup)
+		return NULL;
+	for (i = 0; i < set->n; ++i)
+		dup = isl_set_add_basic_set(dup, isl_basic_set_copy(set->p[i]));
+	return dup;
+}
+
+struct isl_set *isl_set_from_basic_set(struct isl_basic_set *bset)
+{
+	return isl_map_from_basic_map(bset);
+}
+
+struct isl_map *isl_map_from_basic_map(struct isl_basic_map *bmap)
+{
+	struct isl_map *map;
+
+	if (!bmap)
+		return NULL;
+
+	map = isl_map_alloc_space(isl_space_copy(bmap->dim), 1, ISL_MAP_DISJOINT);
+	return isl_map_add_basic_map(map, bmap);
+}
+
+__isl_give isl_set *isl_set_add_basic_set(__isl_take isl_set *set,
+						__isl_take isl_basic_set *bset)
+{
+	return (struct isl_set *)isl_map_add_basic_map((struct isl_map *)set,
+						(struct isl_basic_map *)bset);
+}
+
+__isl_null isl_set *isl_set_free(__isl_take isl_set *set)
+{
+	int i;
+
+	if (!set)
+		return NULL;
+
+	if (--set->ref > 0)
+		return NULL;
+
+	isl_ctx_deref(set->ctx);
+	for (i = 0; i < set->n; ++i)
+		isl_basic_set_free(set->p[i]);
+	isl_space_free(set->dim);
+	free(set);
+
+	return NULL;
+}
+
+void isl_set_print_internal(struct isl_set *set, FILE *out, int indent)
+{
+	int i;
+
+	if (!set) {
+		fprintf(out, "null set\n");
+		return;
+	}
+
+	fprintf(out, "%*s", indent, "");
+	fprintf(out, "ref: %d, n: %d, nparam: %d, dim: %d, flags: %x\n",
+			set->ref, set->n, set->dim->nparam, set->dim->n_out,
+			set->flags);
+	for (i = 0; i < set->n; ++i) {
+		fprintf(out, "%*s", indent, "");
+		fprintf(out, "basic set %d:\n", i);
+		isl_basic_set_print_internal(set->p[i], out, indent+4);
+	}
+}
+
+void isl_map_print_internal(struct isl_map *map, FILE *out, int indent)
+{
+	int i;
+
+	if (!map) {
+		fprintf(out, "null map\n");
+		return;
+	}
+
+	fprintf(out, "%*s", indent, "");
+	fprintf(out, "ref: %d, n: %d, nparam: %d, in: %d, out: %d, "
+		     "flags: %x, n_name: %d\n",
+			map->ref, map->n, map->dim->nparam, map->dim->n_in,
+			map->dim->n_out, map->flags, map->dim->n_id);
+	for (i = 0; i < map->n; ++i) {
+		fprintf(out, "%*s", indent, "");
+		fprintf(out, "basic map %d:\n", i);
+		isl_basic_map_print_internal(map->p[i], out, indent+4);
+	}
+}
+
+struct isl_basic_map *isl_basic_map_intersect_domain(
+		struct isl_basic_map *bmap, struct isl_basic_set *bset)
+{
+	struct isl_basic_map *bmap_domain;
+
+	if (!bmap || !bset)
+		goto error;
+
+	isl_assert(bset->ctx, isl_space_match(bmap->dim, isl_dim_param,
+					bset->dim, isl_dim_param), goto error);
+
+	if (isl_space_dim(bset->dim, isl_dim_set) != 0)
+		isl_assert(bset->ctx,
+		    isl_basic_map_compatible_domain(bmap, bset), goto error);
+
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		goto error;
+	bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim),
+			bset->n_div, bset->n_eq, bset->n_ineq);
+	bmap_domain = isl_basic_map_from_domain(bset);
+	bmap = add_constraints(bmap, bmap_domain, 0, 0);
+
+	bmap = isl_basic_map_simplify(bmap);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+struct isl_basic_map *isl_basic_map_intersect_range(
+		struct isl_basic_map *bmap, struct isl_basic_set *bset)
+{
+	struct isl_basic_map *bmap_range;
+
+	if (!bmap || !bset)
+		goto error;
+
+	isl_assert(bset->ctx, isl_space_match(bmap->dim, isl_dim_param,
+					bset->dim, isl_dim_param), goto error);
+
+	if (isl_space_dim(bset->dim, isl_dim_set) != 0)
+		isl_assert(bset->ctx,
+		    isl_basic_map_compatible_range(bmap, bset), goto error);
+
+	if (isl_basic_set_is_universe(bset)) {
+		isl_basic_set_free(bset);
+		return bmap;
+	}
+
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		goto error;
+	bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim),
+			bset->n_div, bset->n_eq, bset->n_ineq);
+	bmap_range = isl_basic_map_from_basic_set(bset, isl_space_copy(bset->dim));
+	bmap = add_constraints(bmap, bmap_range, 0, 0);
+
+	bmap = isl_basic_map_simplify(bmap);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+int isl_basic_map_contains(struct isl_basic_map *bmap, struct isl_vec *vec)
+{
+	int i;
+	unsigned total;
+	isl_int s;
+
+	if (!bmap || !vec)
+		return -1;
+
+	total = 1 + isl_basic_map_total_dim(bmap);
+	if (total != vec->size)
+		return -1;
+
+	isl_int_init(s);
+
+	for (i = 0; i < bmap->n_eq; ++i) {
+		isl_seq_inner_product(vec->el, bmap->eq[i], total, &s);
+		if (!isl_int_is_zero(s)) {
+			isl_int_clear(s);
+			return 0;
+		}
+	}
+
+	for (i = 0; i < bmap->n_ineq; ++i) {
+		isl_seq_inner_product(vec->el, bmap->ineq[i], total, &s);
+		if (isl_int_is_neg(s)) {
+			isl_int_clear(s);
+			return 0;
+		}
+	}
+
+	isl_int_clear(s);
+
+	return 1;
+}
+
+int isl_basic_set_contains(struct isl_basic_set *bset, struct isl_vec *vec)
+{
+	return isl_basic_map_contains((struct isl_basic_map *)bset, vec);
+}
+
+struct isl_basic_map *isl_basic_map_intersect(
+		struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
+{
+	struct isl_vec *sample = NULL;
+
+	if (!bmap1 || !bmap2)
+		goto error;
+
+	isl_assert(bmap1->ctx, isl_space_match(bmap1->dim, isl_dim_param,
+				     bmap2->dim, isl_dim_param), goto error);
+	if (isl_space_dim(bmap1->dim, isl_dim_all) ==
+				isl_space_dim(bmap1->dim, isl_dim_param) &&
+	    isl_space_dim(bmap2->dim, isl_dim_all) !=
+				isl_space_dim(bmap2->dim, isl_dim_param))
+		return isl_basic_map_intersect(bmap2, bmap1);
+
+	if (isl_space_dim(bmap2->dim, isl_dim_all) !=
+					isl_space_dim(bmap2->dim, isl_dim_param))
+		isl_assert(bmap1->ctx,
+			    isl_space_is_equal(bmap1->dim, bmap2->dim), goto error);
+
+	if (bmap1->sample &&
+	    isl_basic_map_contains(bmap1, bmap1->sample) > 0 &&
+	    isl_basic_map_contains(bmap2, bmap1->sample) > 0)
+		sample = isl_vec_copy(bmap1->sample);
+	else if (bmap2->sample &&
+	    isl_basic_map_contains(bmap1, bmap2->sample) > 0 &&
+	    isl_basic_map_contains(bmap2, bmap2->sample) > 0)
+		sample = isl_vec_copy(bmap2->sample);
+
+	bmap1 = isl_basic_map_cow(bmap1);
+	if (!bmap1)
+		goto error;
+	bmap1 = isl_basic_map_extend_space(bmap1, isl_space_copy(bmap1->dim),
+			bmap2->n_div, bmap2->n_eq, bmap2->n_ineq);
+	bmap1 = add_constraints(bmap1, bmap2, 0, 0);
+
+	if (!bmap1)
+		isl_vec_free(sample);
+	else if (sample) {
+		isl_vec_free(bmap1->sample);
+		bmap1->sample = sample;
+	}
+
+	bmap1 = isl_basic_map_simplify(bmap1);
+	return isl_basic_map_finalize(bmap1);
+error:
+	if (sample)
+		isl_vec_free(sample);
+	isl_basic_map_free(bmap1);
+	isl_basic_map_free(bmap2);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_intersect(
+		struct isl_basic_set *bset1, struct isl_basic_set *bset2)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_intersect(
+			(struct isl_basic_map *)bset1,
+			(struct isl_basic_map *)bset2);
+}
+
+__isl_give isl_basic_set *isl_basic_set_intersect_params(
+	__isl_take isl_basic_set *bset1, __isl_take isl_basic_set *bset2)
+{
+	return isl_basic_set_intersect(bset1, bset2);
+}
+
+/* Special case of isl_map_intersect, where both map1 and map2
+ * are convex, without any divs and such that either map1 or map2
+ * contains a single constraint.  This constraint is then simply
+ * added to the other map.
+ */
+static __isl_give isl_map *map_intersect_add_constraint(
+	__isl_take isl_map *map1, __isl_take isl_map *map2)
+{
+	isl_assert(map1->ctx, map1->n == 1, goto error);
+	isl_assert(map2->ctx, map1->n == 1, goto error);
+	isl_assert(map1->ctx, map1->p[0]->n_div == 0, goto error);
+	isl_assert(map2->ctx, map1->p[0]->n_div == 0, goto error);
+
+	if (map2->p[0]->n_eq + map2->p[0]->n_ineq != 1)
+		return isl_map_intersect(map2, map1);
+
+	isl_assert(map2->ctx,
+		    map2->p[0]->n_eq + map2->p[0]->n_ineq == 1, goto error);
+
+	map1 = isl_map_cow(map1);
+	if (!map1)
+		goto error;
+	if (isl_map_plain_is_empty(map1)) {
+		isl_map_free(map2);
+		return map1;
+	}
+	map1->p[0] = isl_basic_map_cow(map1->p[0]);
+	if (map2->p[0]->n_eq == 1)
+		map1->p[0] = isl_basic_map_add_eq(map1->p[0], map2->p[0]->eq[0]);
+	else
+		map1->p[0] = isl_basic_map_add_ineq(map1->p[0],
+							map2->p[0]->ineq[0]);
+
+	map1->p[0] = isl_basic_map_simplify(map1->p[0]);
+	map1->p[0] = isl_basic_map_finalize(map1->p[0]);
+	if (!map1->p[0])
+		goto error;
+
+	if (isl_basic_map_plain_is_empty(map1->p[0])) {
+		isl_basic_map_free(map1->p[0]);
+		map1->n = 0;
+	}
+
+	isl_map_free(map2);
+
+	return map1;
+error:
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return NULL;
+}
+
+/* map2 may be either a parameter domain or a map living in the same
+ * space as map1.
+ */
+static __isl_give isl_map *map_intersect_internal(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	unsigned flags = 0;
+	isl_map *result;
+	int i, j;
+
+	if (!map1 || !map2)
+		goto error;
+
+	if ((isl_map_plain_is_empty(map1) ||
+	     isl_map_plain_is_universe(map2)) &&
+	    isl_space_is_equal(map1->dim, map2->dim)) {
+		isl_map_free(map2);
+		return map1;
+	}
+	if ((isl_map_plain_is_empty(map2) ||
+	     isl_map_plain_is_universe(map1)) &&
+	    isl_space_is_equal(map1->dim, map2->dim)) {
+		isl_map_free(map1);
+		return map2;
+	}
+
+	if (map1->n == 1 && map2->n == 1 &&
+	    map1->p[0]->n_div == 0 && map2->p[0]->n_div == 0 &&
+	    isl_space_is_equal(map1->dim, map2->dim) &&
+	    (map1->p[0]->n_eq + map1->p[0]->n_ineq == 1 ||
+	     map2->p[0]->n_eq + map2->p[0]->n_ineq == 1))
+		return map_intersect_add_constraint(map1, map2);
+
+	if (isl_space_dim(map2->dim, isl_dim_all) !=
+				isl_space_dim(map2->dim, isl_dim_param))
+		isl_assert(map1->ctx,
+			    isl_space_is_equal(map1->dim, map2->dim), goto error);
+
+	if (ISL_F_ISSET(map1, ISL_MAP_DISJOINT) &&
+	    ISL_F_ISSET(map2, ISL_MAP_DISJOINT))
+		ISL_FL_SET(flags, ISL_MAP_DISJOINT);
+
+	result = isl_map_alloc_space(isl_space_copy(map1->dim),
+				map1->n * map2->n, flags);
+	if (!result)
+		goto error;
+	for (i = 0; i < map1->n; ++i)
+		for (j = 0; j < map2->n; ++j) {
+			struct isl_basic_map *part;
+			part = isl_basic_map_intersect(
+				    isl_basic_map_copy(map1->p[i]),
+				    isl_basic_map_copy(map2->p[j]));
+			if (isl_basic_map_is_empty(part) < 0)
+				part = isl_basic_map_free(part);
+			result = isl_map_add_basic_map(result, part);
+			if (!result)
+				goto error;
+		}
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return result;
+error:
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return NULL;
+}
+
+static __isl_give isl_map *map_intersect(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	if (!map1 || !map2)
+		goto error;
+	if (!isl_space_is_equal(map1->dim, map2->dim))
+		isl_die(isl_map_get_ctx(map1), isl_error_invalid,
+			"spaces don't match", goto error);
+	return map_intersect_internal(map1, map2);
+error:
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_intersect(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	return isl_map_align_params_map_map_and(map1, map2, &map_intersect);
+}
+
+struct isl_set *isl_set_intersect(struct isl_set *set1, struct isl_set *set2)
+{
+	return (struct isl_set *)
+		isl_map_intersect((struct isl_map *)set1,
+				  (struct isl_map *)set2);
+}
+
+/* map_intersect_internal accepts intersections
+ * with parameter domains, so we can just call that function.
+ */
+static __isl_give isl_map *map_intersect_params(__isl_take isl_map *map,
+		__isl_take isl_set *params)
+{
+	return map_intersect_internal(map, params);
+}
+
+__isl_give isl_map *isl_map_intersect_params(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	return isl_map_align_params_map_map_and(map1, map2, &map_intersect_params);
+}
+
+__isl_give isl_set *isl_set_intersect_params(__isl_take isl_set *set,
+		__isl_take isl_set *params)
+{
+	return isl_map_intersect_params(set, params);
+}
+
+struct isl_basic_map *isl_basic_map_reverse(struct isl_basic_map *bmap)
+{
+	isl_space *dim;
+	struct isl_basic_set *bset;
+	unsigned in;
+
+	if (!bmap)
+		return NULL;
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+	dim = isl_space_reverse(isl_space_copy(bmap->dim));
+	in = isl_basic_map_n_in(bmap);
+	bset = isl_basic_set_from_basic_map(bmap);
+	bset = isl_basic_set_swap_vars(bset, in);
+	return isl_basic_map_from_basic_set(bset, dim);
+}
+
+static __isl_give isl_basic_map *basic_map_space_reset(
+	__isl_take isl_basic_map *bmap, enum isl_dim_type type)
+{
+	isl_space *space;
+
+	if (!bmap)
+		return NULL;
+	if (!isl_space_is_named_or_nested(bmap->dim, type))
+		return bmap;
+
+	space = isl_basic_map_get_space(bmap);
+	space = isl_space_reset(space, type);
+	bmap = isl_basic_map_reset_space(bmap, space);
+	return bmap;
+}
+
+__isl_give isl_basic_map *isl_basic_map_insert_dims(
+	__isl_take isl_basic_map *bmap, enum isl_dim_type type,
+	unsigned pos, unsigned n)
+{
+	isl_space *res_dim;
+	struct isl_basic_map *res;
+	struct isl_dim_map *dim_map;
+	unsigned total, off;
+	enum isl_dim_type t;
+
+	if (n == 0)
+		return basic_map_space_reset(bmap, type);
+
+	if (!bmap)
+		return NULL;
+
+	res_dim = isl_space_insert_dims(isl_basic_map_get_space(bmap), type, pos, n);
+
+	total = isl_basic_map_total_dim(bmap) + n;
+	dim_map = isl_dim_map_alloc(bmap->ctx, total);
+	off = 0;
+	for (t = isl_dim_param; t <= isl_dim_out; ++t) {
+		if (t != type) {
+			isl_dim_map_dim(dim_map, bmap->dim, t, off);
+		} else {
+			unsigned size = isl_basic_map_dim(bmap, t);
+			isl_dim_map_dim_range(dim_map, bmap->dim, t,
+						0, pos, off);
+			isl_dim_map_dim_range(dim_map, bmap->dim, t,
+						pos, size - pos, off + pos + n);
+		}
+		off += isl_space_dim(res_dim, t);
+	}
+	isl_dim_map_div(dim_map, bmap, off);
+
+	res = isl_basic_map_alloc_space(res_dim,
+			bmap->n_div, bmap->n_eq, bmap->n_ineq);
+	if (isl_basic_map_is_rational(bmap))
+		res = isl_basic_map_set_rational(res);
+	if (isl_basic_map_plain_is_empty(bmap)) {
+		isl_basic_map_free(bmap);
+		free(dim_map);
+		return isl_basic_map_set_to_empty(res);
+	}
+	res = isl_basic_map_add_constraints_dim_map(res, bmap, dim_map);
+	return isl_basic_map_finalize(res);
+}
+
+__isl_give isl_basic_set *isl_basic_set_insert_dims(
+	__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned pos, unsigned n)
+{
+	return isl_basic_map_insert_dims(bset, type, pos, n);
+}
+
+__isl_give isl_basic_map *isl_basic_map_add(__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned n)
+{
+	if (!bmap)
+		return NULL;
+	return isl_basic_map_insert_dims(bmap, type,
+					isl_basic_map_dim(bmap, type), n);
+}
+
+__isl_give isl_basic_set *isl_basic_set_add_dims(__isl_take isl_basic_set *bset,
+		enum isl_dim_type type, unsigned n)
+{
+	if (!bset)
+		return NULL;
+	isl_assert(bset->ctx, type != isl_dim_in, goto error);
+	return (isl_basic_set *)isl_basic_map_add((isl_basic_map *)bset, type, n);
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+static __isl_give isl_map *map_space_reset(__isl_take isl_map *map,
+	enum isl_dim_type type)
+{
+	isl_space *space;
+
+	if (!map || !isl_space_is_named_or_nested(map->dim, type))
+		return map;
+
+	space = isl_map_get_space(map);
+	space = isl_space_reset(space, type);
+	map = isl_map_reset_space(map, space);
+	return map;
+}
+
+__isl_give isl_map *isl_map_insert_dims(__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned pos, unsigned n)
+{
+	int i;
+
+	if (n == 0)
+		return map_space_reset(map, type);
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	map->dim = isl_space_insert_dims(map->dim, type, pos, n);
+	if (!map->dim)
+		goto error;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_insert_dims(map->p[i], type, pos, n);
+		if (!map->p[i])
+			goto error;
+	}
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_insert_dims(__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned pos, unsigned n)
+{
+	return isl_map_insert_dims(set, type, pos, n);
+}
+
+__isl_give isl_map *isl_map_add_dims(__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned n)
+{
+	if (!map)
+		return NULL;
+	return isl_map_insert_dims(map, type, isl_map_dim(map, type), n);
+}
+
+__isl_give isl_set *isl_set_add_dims(__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned n)
+{
+	if (!set)
+		return NULL;
+	isl_assert(set->ctx, type != isl_dim_in, goto error);
+	return (isl_set *)isl_map_add_dims((isl_map *)set, type, n);
+error:
+	isl_set_free(set);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_move_dims(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n)
+{
+	struct isl_dim_map *dim_map;
+	struct isl_basic_map *res;
+	enum isl_dim_type t;
+	unsigned total, off;
+
+	if (!bmap)
+		return NULL;
+	if (n == 0)
+		return bmap;
+
+	isl_assert(bmap->ctx, src_pos + n <= isl_basic_map_dim(bmap, src_type),
+		goto error);
+
+	if (dst_type == src_type && dst_pos == src_pos)
+		return bmap;
+
+	isl_assert(bmap->ctx, dst_type != src_type, goto error);
+
+	if (pos(bmap->dim, dst_type) + dst_pos ==
+	    pos(bmap->dim, src_type) + src_pos +
+					    ((src_type < dst_type) ? n : 0)) {
+		bmap = isl_basic_map_cow(bmap);
+		if (!bmap)
+			return NULL;
+
+		bmap->dim = isl_space_move_dims(bmap->dim, dst_type, dst_pos,
+						src_type, src_pos, n);
+		if (!bmap->dim)
+			goto error;
+
+		bmap = isl_basic_map_finalize(bmap);
+
+		return bmap;
+	}
+
+	total = isl_basic_map_total_dim(bmap);
+	dim_map = isl_dim_map_alloc(bmap->ctx, total);
+
+	off = 0;
+	for (t = isl_dim_param; t <= isl_dim_out; ++t) {
+		unsigned size = isl_space_dim(bmap->dim, t);
+		if (t == dst_type) {
+			isl_dim_map_dim_range(dim_map, bmap->dim, t,
+					    0, dst_pos, off);
+			off += dst_pos;
+			isl_dim_map_dim_range(dim_map, bmap->dim, src_type,
+					    src_pos, n, off);
+			off += n;
+			isl_dim_map_dim_range(dim_map, bmap->dim, t,
+					    dst_pos, size - dst_pos, off);
+			off += size - dst_pos;
+		} else if (t == src_type) {
+			isl_dim_map_dim_range(dim_map, bmap->dim, t,
+					    0, src_pos, off);
+			off += src_pos;
+			isl_dim_map_dim_range(dim_map, bmap->dim, t,
+					src_pos + n, size - src_pos - n, off);
+			off += size - src_pos - n;
+		} else {
+			isl_dim_map_dim(dim_map, bmap->dim, t, off);
+			off += size;
+		}
+	}
+	isl_dim_map_div(dim_map, bmap, off);
+
+	res = isl_basic_map_alloc_space(isl_basic_map_get_space(bmap),
+			bmap->n_div, bmap->n_eq, bmap->n_ineq);
+	bmap = isl_basic_map_add_constraints_dim_map(res, bmap, dim_map);
+	if (!bmap)
+		goto error;
+
+	bmap->dim = isl_space_move_dims(bmap->dim, dst_type, dst_pos,
+					src_type, src_pos, n);
+	if (!bmap->dim)
+		goto error;
+
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
+	bmap = isl_basic_map_gauss(bmap, NULL);
+	bmap = isl_basic_map_finalize(bmap);
+
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_move_dims(__isl_take isl_basic_set *bset,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n)
+{
+	return (isl_basic_set *)isl_basic_map_move_dims(
+		(isl_basic_map *)bset, dst_type, dst_pos, src_type, src_pos, n);
+}
+
+__isl_give isl_set *isl_set_move_dims(__isl_take isl_set *set,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n)
+{
+	if (!set)
+		return NULL;
+	isl_assert(set->ctx, dst_type != isl_dim_in, goto error);
+	return (isl_set *)isl_map_move_dims((isl_map *)set, dst_type, dst_pos,
+					src_type, src_pos, n);
+error:
+	isl_set_free(set);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_move_dims(__isl_take isl_map *map,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+	if (n == 0)
+		return map;
+
+	isl_assert(map->ctx, src_pos + n <= isl_map_dim(map, src_type),
+		goto error);
+
+	if (dst_type == src_type && dst_pos == src_pos)
+		return map;
+
+	isl_assert(map->ctx, dst_type != src_type, goto error);
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	map->dim = isl_space_move_dims(map->dim, dst_type, dst_pos, src_type, src_pos, n);
+	if (!map->dim)
+		goto error;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_move_dims(map->p[i],
+						dst_type, dst_pos,
+						src_type, src_pos, n);
+		if (!map->p[i])
+			goto error;
+	}
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Move the specified dimensions to the last columns right before
+ * the divs.  Don't change the dimension specification of bmap.
+ * That's the responsibility of the caller.
+ */
+static __isl_give isl_basic_map *move_last(__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	struct isl_dim_map *dim_map;
+	struct isl_basic_map *res;
+	enum isl_dim_type t;
+	unsigned total, off;
+
+	if (!bmap)
+		return NULL;
+	if (pos(bmap->dim, type) + first + n ==
+				1 + isl_space_dim(bmap->dim, isl_dim_all))
+		return bmap;
+
+	total = isl_basic_map_total_dim(bmap);
+	dim_map = isl_dim_map_alloc(bmap->ctx, total);
+
+	off = 0;
+	for (t = isl_dim_param; t <= isl_dim_out; ++t) {
+		unsigned size = isl_space_dim(bmap->dim, t);
+		if (t == type) {
+			isl_dim_map_dim_range(dim_map, bmap->dim, t,
+					    0, first, off);
+			off += first;
+			isl_dim_map_dim_range(dim_map, bmap->dim, t,
+					    first, n, total - bmap->n_div - n);
+			isl_dim_map_dim_range(dim_map, bmap->dim, t,
+					    first + n, size - (first + n), off);
+			off += size - (first + n);
+		} else {
+			isl_dim_map_dim(dim_map, bmap->dim, t, off);
+			off += size;
+		}
+	}
+	isl_dim_map_div(dim_map, bmap, off + n);
+
+	res = isl_basic_map_alloc_space(isl_basic_map_get_space(bmap),
+			bmap->n_div, bmap->n_eq, bmap->n_ineq);
+	res = isl_basic_map_add_constraints_dim_map(res, bmap, dim_map);
+	return res;
+}
+
+/* Insert "n" rows in the divs of "bmap".
+ *
+ * The number of columns is not changed, which means that the last
+ * dimensions of "bmap" are being reintepreted as the new divs.
+ * The space of "bmap" is not adjusted, however, which means
+ * that "bmap" is left in an inconsistent state.  Removing "n" dimensions
+ * from the space of "bmap" is the responsibility of the caller.
+ */
+static __isl_give isl_basic_map *insert_div_rows(__isl_take isl_basic_map *bmap,
+	int n)
+{
+	int i;
+	size_t row_size;
+	isl_int **new_div;
+	isl_int *old;
+
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+
+	row_size = 1 + isl_space_dim(bmap->dim, isl_dim_all) + bmap->extra;
+	old = bmap->block2.data;
+	bmap->block2 = isl_blk_extend(bmap->ctx, bmap->block2,
+					(bmap->extra + n) * (1 + row_size));
+	if (!bmap->block2.data)
+		return isl_basic_map_free(bmap);
+	new_div = isl_alloc_array(bmap->ctx, isl_int *, bmap->extra + n);
+	if (!new_div)
+		return isl_basic_map_free(bmap);
+	for (i = 0; i < n; ++i) {
+		new_div[i] = bmap->block2.data +
+				(bmap->extra + i) * (1 + row_size);
+		isl_seq_clr(new_div[i], 1 + row_size);
+	}
+	for (i = 0; i < bmap->extra; ++i)
+		new_div[n + i] = bmap->block2.data + (bmap->div[i] - old);
+	free(bmap->div);
+	bmap->div = new_div;
+	bmap->n_div += n;
+	bmap->extra += n;
+
+	return bmap;
+}
+
+/* Turn the n dimensions of type type, starting at first
+ * into existentially quantified variables.
+ */
+__isl_give isl_basic_map *isl_basic_map_project_out(
+		__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned first, unsigned n)
+{
+	if (n == 0)
+		return basic_map_space_reset(bmap, type);
+
+	if (!bmap)
+		return NULL;
+
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
+		return isl_basic_map_remove_dims(bmap, type, first, n);
+
+	isl_assert(bmap->ctx, first + n <= isl_basic_map_dim(bmap, type),
+			goto error);
+
+	bmap = move_last(bmap, type, first, n);
+	bmap = isl_basic_map_cow(bmap);
+	bmap = insert_div_rows(bmap, n);
+	if (!bmap)
+		return NULL;
+
+	bmap->dim = isl_space_drop_dims(bmap->dim, type, first, n);
+	if (!bmap->dim)
+		goto error;
+	bmap = isl_basic_map_simplify(bmap);
+	bmap = isl_basic_map_drop_redundant_divs(bmap);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Turn the n dimensions of type type, starting at first
+ * into existentially quantified variables.
+ */
+struct isl_basic_set *isl_basic_set_project_out(struct isl_basic_set *bset,
+		enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return (isl_basic_set *)isl_basic_map_project_out(
+			(isl_basic_map *)bset, type, first, n);
+}
+
+/* Turn the n dimensions of type type, starting at first
+ * into existentially quantified variables.
+ */
+__isl_give isl_map *isl_map_project_out(__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+
+	if (n == 0)
+		return map_space_reset(map, type);
+
+	isl_assert(map->ctx, first + n <= isl_map_dim(map, type), goto error);
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	map->dim = isl_space_drop_dims(map->dim, type, first, n);
+	if (!map->dim)
+		goto error;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_project_out(map->p[i], type, first, n);
+		if (!map->p[i])
+			goto error;
+	}
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Turn the n dimensions of type type, starting at first
+ * into existentially quantified variables.
+ */
+__isl_give isl_set *isl_set_project_out(__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return (isl_set *)isl_map_project_out((isl_map *)set, type, first, n);
+}
+
+static struct isl_basic_map *add_divs(struct isl_basic_map *bmap, unsigned n)
+{
+	int i, j;
+
+	for (i = 0; i < n; ++i) {
+		j = isl_basic_map_alloc_div(bmap);
+		if (j < 0)
+			goto error;
+		isl_seq_clr(bmap->div[j], 1+1+isl_basic_map_total_dim(bmap));
+	}
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+struct isl_basic_map *isl_basic_map_apply_range(
+		struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
+{
+	isl_space *dim_result = NULL;
+	struct isl_basic_map *bmap;
+	unsigned n_in, n_out, n, nparam, total, pos;
+	struct isl_dim_map *dim_map1, *dim_map2;
+
+	if (!bmap1 || !bmap2)
+		goto error;
+	if (!isl_space_match(bmap1->dim, isl_dim_param,
+				bmap2->dim, isl_dim_param))
+		isl_die(isl_basic_map_get_ctx(bmap1), isl_error_invalid,
+			"parameters don't match", goto error);
+	if (!isl_space_tuple_is_equal(bmap1->dim, isl_dim_out,
+				    bmap2->dim, isl_dim_in))
+		isl_die(isl_basic_map_get_ctx(bmap1), isl_error_invalid,
+			"spaces don't match", goto error);
+
+	dim_result = isl_space_join(isl_space_copy(bmap1->dim),
+				  isl_space_copy(bmap2->dim));
+
+	n_in = isl_basic_map_n_in(bmap1);
+	n_out = isl_basic_map_n_out(bmap2);
+	n = isl_basic_map_n_out(bmap1);
+	nparam = isl_basic_map_n_param(bmap1);
+
+	total = nparam + n_in + n_out + bmap1->n_div + bmap2->n_div + n;
+	dim_map1 = isl_dim_map_alloc(bmap1->ctx, total);
+	dim_map2 = isl_dim_map_alloc(bmap1->ctx, total);
+	isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_param, pos = 0);
+	isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_param, pos = 0);
+	isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_in, pos += nparam);
+	isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_out, pos += n_in);
+	isl_dim_map_div(dim_map1, bmap1, pos += n_out);
+	isl_dim_map_div(dim_map2, bmap2, pos += bmap1->n_div);
+	isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_out, pos += bmap2->n_div);
+	isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_in, pos);
+
+	bmap = isl_basic_map_alloc_space(dim_result,
+			bmap1->n_div + bmap2->n_div + n,
+			bmap1->n_eq + bmap2->n_eq,
+			bmap1->n_ineq + bmap2->n_ineq);
+	bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap1, dim_map1);
+	bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap2, dim_map2);
+	bmap = add_divs(bmap, n);
+	bmap = isl_basic_map_simplify(bmap);
+	bmap = isl_basic_map_drop_redundant_divs(bmap);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap1);
+	isl_basic_map_free(bmap2);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_apply(
+		struct isl_basic_set *bset, struct isl_basic_map *bmap)
+{
+	if (!bset || !bmap)
+		goto error;
+
+	isl_assert(bset->ctx, isl_basic_map_compatible_domain(bmap, bset),
+		    goto error);
+
+	return (struct isl_basic_set *)
+		isl_basic_map_apply_range((struct isl_basic_map *)bset, bmap);
+error:
+	isl_basic_set_free(bset);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+struct isl_basic_map *isl_basic_map_apply_domain(
+		struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
+{
+	if (!bmap1 || !bmap2)
+		goto error;
+
+	isl_assert(bmap1->ctx,
+	    isl_basic_map_n_in(bmap1) == isl_basic_map_n_in(bmap2), goto error);
+	isl_assert(bmap1->ctx,
+	    isl_basic_map_n_param(bmap1) == isl_basic_map_n_param(bmap2),
+	    goto error);
+
+	bmap1 = isl_basic_map_reverse(bmap1);
+	bmap1 = isl_basic_map_apply_range(bmap1, bmap2);
+	return isl_basic_map_reverse(bmap1);
+error:
+	isl_basic_map_free(bmap1);
+	isl_basic_map_free(bmap2);
+	return NULL;
+}
+
+/* Given two basic maps A -> f(A) and B -> g(B), construct a basic map
+ * A \cap B -> f(A) + f(B)
+ */
+struct isl_basic_map *isl_basic_map_sum(
+		struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
+{
+	unsigned n_in, n_out, nparam, total, pos;
+	struct isl_basic_map *bmap = NULL;
+	struct isl_dim_map *dim_map1, *dim_map2;
+	int i;
+
+	if (!bmap1 || !bmap2)
+		goto error;
+
+	isl_assert(bmap1->ctx, isl_space_is_equal(bmap1->dim, bmap2->dim),
+		goto error);
+
+	nparam = isl_basic_map_n_param(bmap1);
+	n_in = isl_basic_map_n_in(bmap1);
+	n_out = isl_basic_map_n_out(bmap1);
+
+	total = nparam + n_in + n_out + bmap1->n_div + bmap2->n_div + 2 * n_out;
+	dim_map1 = isl_dim_map_alloc(bmap1->ctx, total);
+	dim_map2 = isl_dim_map_alloc(bmap2->ctx, total);
+	isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_param, pos = 0);
+	isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_param, pos);
+	isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_in, pos += nparam);
+	isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_in, pos);
+	isl_dim_map_div(dim_map1, bmap1, pos += n_in + n_out);
+	isl_dim_map_div(dim_map2, bmap2, pos += bmap1->n_div);
+	isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_out, pos += bmap2->n_div);
+	isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_out, pos += n_out);
+
+	bmap = isl_basic_map_alloc_space(isl_space_copy(bmap1->dim),
+			bmap1->n_div + bmap2->n_div + 2 * n_out,
+			bmap1->n_eq + bmap2->n_eq + n_out,
+			bmap1->n_ineq + bmap2->n_ineq);
+	for (i = 0; i < n_out; ++i) {
+		int j = isl_basic_map_alloc_equality(bmap);
+		if (j < 0)
+			goto error;
+		isl_seq_clr(bmap->eq[j], 1+total);
+		isl_int_set_si(bmap->eq[j][1+nparam+n_in+i], -1);
+		isl_int_set_si(bmap->eq[j][1+pos+i], 1);
+		isl_int_set_si(bmap->eq[j][1+pos-n_out+i], 1);
+	}
+	bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap1, dim_map1);
+	bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap2, dim_map2);
+	bmap = add_divs(bmap, 2 * n_out);
+
+	bmap = isl_basic_map_simplify(bmap);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	isl_basic_map_free(bmap1);
+	isl_basic_map_free(bmap2);
+	return NULL;
+}
+
+/* Given two maps A -> f(A) and B -> g(B), construct a map
+ * A \cap B -> f(A) + f(B)
+ */
+struct isl_map *isl_map_sum(struct isl_map *map1, struct isl_map *map2)
+{
+	struct isl_map *result;
+	int i, j;
+
+	if (!map1 || !map2)
+		goto error;
+
+	isl_assert(map1->ctx, isl_space_is_equal(map1->dim, map2->dim), goto error);
+
+	result = isl_map_alloc_space(isl_space_copy(map1->dim),
+				map1->n * map2->n, 0);
+	if (!result)
+		goto error;
+	for (i = 0; i < map1->n; ++i)
+		for (j = 0; j < map2->n; ++j) {
+			struct isl_basic_map *part;
+			part = isl_basic_map_sum(
+				    isl_basic_map_copy(map1->p[i]),
+				    isl_basic_map_copy(map2->p[j]));
+			if (isl_basic_map_is_empty(part))
+				isl_basic_map_free(part);
+			else
+				result = isl_map_add_basic_map(result, part);
+			if (!result)
+				goto error;
+		}
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return result;
+error:
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_sum(__isl_take isl_set *set1,
+	__isl_take isl_set *set2)
+{
+	return (isl_set *)isl_map_sum((isl_map *)set1, (isl_map *)set2);
+}
+
+/* Given a basic map A -> f(A), construct A -> -f(A).
+ */
+struct isl_basic_map *isl_basic_map_neg(struct isl_basic_map *bmap)
+{
+	int i, j;
+	unsigned off, n;
+
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+
+	n = isl_basic_map_dim(bmap, isl_dim_out);
+	off = isl_basic_map_offset(bmap, isl_dim_out);
+	for (i = 0; i < bmap->n_eq; ++i)
+		for (j = 0; j < n; ++j)
+			isl_int_neg(bmap->eq[i][off+j], bmap->eq[i][off+j]);
+	for (i = 0; i < bmap->n_ineq; ++i)
+		for (j = 0; j < n; ++j)
+			isl_int_neg(bmap->ineq[i][off+j], bmap->ineq[i][off+j]);
+	for (i = 0; i < bmap->n_div; ++i)
+		for (j = 0; j < n; ++j)
+			isl_int_neg(bmap->div[i][1+off+j], bmap->div[i][1+off+j]);
+	bmap = isl_basic_map_gauss(bmap, NULL);
+	return isl_basic_map_finalize(bmap);
+}
+
+__isl_give isl_basic_set *isl_basic_set_neg(__isl_take isl_basic_set *bset)
+{
+	return isl_basic_map_neg(bset);
+}
+
+/* Given a map A -> f(A), construct A -> -f(A).
+ */
+struct isl_map *isl_map_neg(struct isl_map *map)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_neg(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_neg(__isl_take isl_set *set)
+{
+	return (isl_set *)isl_map_neg((isl_map *)set);
+}
+
+/* Given a basic map A -> f(A) and an integer d, construct a basic map
+ * A -> floor(f(A)/d).
+ */
+struct isl_basic_map *isl_basic_map_floordiv(struct isl_basic_map *bmap,
+		isl_int d)
+{
+	unsigned n_in, n_out, nparam, total, pos;
+	struct isl_basic_map *result = NULL;
+	struct isl_dim_map *dim_map;
+	int i;
+
+	if (!bmap)
+		return NULL;
+
+	nparam = isl_basic_map_n_param(bmap);
+	n_in = isl_basic_map_n_in(bmap);
+	n_out = isl_basic_map_n_out(bmap);
+
+	total = nparam + n_in + n_out + bmap->n_div + n_out;
+	dim_map = isl_dim_map_alloc(bmap->ctx, total);
+	isl_dim_map_dim(dim_map, bmap->dim, isl_dim_param, pos = 0);
+	isl_dim_map_dim(dim_map, bmap->dim, isl_dim_in, pos += nparam);
+	isl_dim_map_div(dim_map, bmap, pos += n_in + n_out);
+	isl_dim_map_dim(dim_map, bmap->dim, isl_dim_out, pos += bmap->n_div);
+
+	result = isl_basic_map_alloc_space(isl_space_copy(bmap->dim),
+			bmap->n_div + n_out,
+			bmap->n_eq, bmap->n_ineq + 2 * n_out);
+	result = isl_basic_map_add_constraints_dim_map(result, bmap, dim_map);
+	result = add_divs(result, n_out);
+	for (i = 0; i < n_out; ++i) {
+		int j;
+		j = isl_basic_map_alloc_inequality(result);
+		if (j < 0)
+			goto error;
+		isl_seq_clr(result->ineq[j], 1+total);
+		isl_int_neg(result->ineq[j][1+nparam+n_in+i], d);
+		isl_int_set_si(result->ineq[j][1+pos+i], 1);
+		j = isl_basic_map_alloc_inequality(result);
+		if (j < 0)
+			goto error;
+		isl_seq_clr(result->ineq[j], 1+total);
+		isl_int_set(result->ineq[j][1+nparam+n_in+i], d);
+		isl_int_set_si(result->ineq[j][1+pos+i], -1);
+		isl_int_sub_ui(result->ineq[j][0], d, 1);
+	}
+
+	result = isl_basic_map_simplify(result);
+	return isl_basic_map_finalize(result);
+error:
+	isl_basic_map_free(result);
+	return NULL;
+}
+
+/* Given a map A -> f(A) and an integer d, construct a map
+ * A -> floor(f(A)/d).
+ */
+struct isl_map *isl_map_floordiv(struct isl_map *map, isl_int d)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	ISL_F_CLR(map, ISL_MAP_DISJOINT);
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_floordiv(map->p[i], d);
+		if (!map->p[i])
+			goto error;
+	}
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Given a map A -> f(A) and an integer d, construct a map
+ * A -> floor(f(A)/d).
+ */
+__isl_give isl_map *isl_map_floordiv_val(__isl_take isl_map *map,
+	__isl_take isl_val *d)
+{
+	if (!map || !d)
+		goto error;
+	if (!isl_val_is_int(d))
+		isl_die(isl_val_get_ctx(d), isl_error_invalid,
+			"expecting integer denominator", goto error);
+	map = isl_map_floordiv(map, d->n);
+	isl_val_free(d);
+	return map;
+error:
+	isl_map_free(map);
+	isl_val_free(d);
+	return NULL;
+}
+
+static struct isl_basic_map *var_equal(struct isl_basic_map *bmap, unsigned pos)
+{
+	int i;
+	unsigned nparam;
+	unsigned n_in;
+
+	i = isl_basic_map_alloc_equality(bmap);
+	if (i < 0)
+		goto error;
+	nparam = isl_basic_map_n_param(bmap);
+	n_in = isl_basic_map_n_in(bmap);
+	isl_seq_clr(bmap->eq[i], 1 + isl_basic_map_total_dim(bmap));
+	isl_int_set_si(bmap->eq[i][1+nparam+pos], -1);
+	isl_int_set_si(bmap->eq[i][1+nparam+n_in+pos], 1);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Add a constraints to "bmap" expressing i_pos < o_pos
+ */
+static struct isl_basic_map *var_less(struct isl_basic_map *bmap, unsigned pos)
+{
+	int i;
+	unsigned nparam;
+	unsigned n_in;
+
+	i = isl_basic_map_alloc_inequality(bmap);
+	if (i < 0)
+		goto error;
+	nparam = isl_basic_map_n_param(bmap);
+	n_in = isl_basic_map_n_in(bmap);
+	isl_seq_clr(bmap->ineq[i], 1 + isl_basic_map_total_dim(bmap));
+	isl_int_set_si(bmap->ineq[i][0], -1);
+	isl_int_set_si(bmap->ineq[i][1+nparam+pos], -1);
+	isl_int_set_si(bmap->ineq[i][1+nparam+n_in+pos], 1);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Add a constraint to "bmap" expressing i_pos <= o_pos
+ */
+static __isl_give isl_basic_map *var_less_or_equal(
+	__isl_take isl_basic_map *bmap, unsigned pos)
+{
+	int i;
+	unsigned nparam;
+	unsigned n_in;
+
+	i = isl_basic_map_alloc_inequality(bmap);
+	if (i < 0)
+		goto error;
+	nparam = isl_basic_map_n_param(bmap);
+	n_in = isl_basic_map_n_in(bmap);
+	isl_seq_clr(bmap->ineq[i], 1 + isl_basic_map_total_dim(bmap));
+	isl_int_set_si(bmap->ineq[i][1+nparam+pos], -1);
+	isl_int_set_si(bmap->ineq[i][1+nparam+n_in+pos], 1);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Add a constraints to "bmap" expressing i_pos > o_pos
+ */
+static struct isl_basic_map *var_more(struct isl_basic_map *bmap, unsigned pos)
+{
+	int i;
+	unsigned nparam;
+	unsigned n_in;
+
+	i = isl_basic_map_alloc_inequality(bmap);
+	if (i < 0)
+		goto error;
+	nparam = isl_basic_map_n_param(bmap);
+	n_in = isl_basic_map_n_in(bmap);
+	isl_seq_clr(bmap->ineq[i], 1 + isl_basic_map_total_dim(bmap));
+	isl_int_set_si(bmap->ineq[i][0], -1);
+	isl_int_set_si(bmap->ineq[i][1+nparam+pos], 1);
+	isl_int_set_si(bmap->ineq[i][1+nparam+n_in+pos], -1);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Add a constraint to "bmap" expressing i_pos >= o_pos
+ */
+static __isl_give isl_basic_map *var_more_or_equal(
+	__isl_take isl_basic_map *bmap, unsigned pos)
+{
+	int i;
+	unsigned nparam;
+	unsigned n_in;
+
+	i = isl_basic_map_alloc_inequality(bmap);
+	if (i < 0)
+		goto error;
+	nparam = isl_basic_map_n_param(bmap);
+	n_in = isl_basic_map_n_in(bmap);
+	isl_seq_clr(bmap->ineq[i], 1 + isl_basic_map_total_dim(bmap));
+	isl_int_set_si(bmap->ineq[i][1+nparam+pos], 1);
+	isl_int_set_si(bmap->ineq[i][1+nparam+n_in+pos], -1);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_equal(
+	__isl_take isl_space *dim, unsigned n_equal)
+{
+	int i;
+	struct isl_basic_map *bmap;
+	bmap = isl_basic_map_alloc_space(dim, 0, n_equal, 0);
+	if (!bmap)
+		return NULL;
+	for (i = 0; i < n_equal && bmap; ++i)
+		bmap = var_equal(bmap, i);
+	return isl_basic_map_finalize(bmap);
+}
+
+/* Return a relation on of dimension "dim" expressing i_[0..pos] << o_[0..pos]
+ */
+__isl_give isl_basic_map *isl_basic_map_less_at(__isl_take isl_space *dim,
+	unsigned pos)
+{
+	int i;
+	struct isl_basic_map *bmap;
+	bmap = isl_basic_map_alloc_space(dim, 0, pos, 1);
+	if (!bmap)
+		return NULL;
+	for (i = 0; i < pos && bmap; ++i)
+		bmap = var_equal(bmap, i);
+	if (bmap)
+		bmap = var_less(bmap, pos);
+	return isl_basic_map_finalize(bmap);
+}
+
+/* Return a relation on of dimension "dim" expressing i_[0..pos] <<= o_[0..pos]
+ */
+__isl_give isl_basic_map *isl_basic_map_less_or_equal_at(
+	__isl_take isl_space *dim, unsigned pos)
+{
+	int i;
+	isl_basic_map *bmap;
+
+	bmap = isl_basic_map_alloc_space(dim, 0, pos, 1);
+	for (i = 0; i < pos; ++i)
+		bmap = var_equal(bmap, i);
+	bmap = var_less_or_equal(bmap, pos);
+	return isl_basic_map_finalize(bmap);
+}
+
+/* Return a relation on pairs of sets of dimension "dim" expressing i_pos > o_pos
+ */
+__isl_give isl_basic_map *isl_basic_map_more_at(__isl_take isl_space *dim,
+	unsigned pos)
+{
+	int i;
+	struct isl_basic_map *bmap;
+	bmap = isl_basic_map_alloc_space(dim, 0, pos, 1);
+	if (!bmap)
+		return NULL;
+	for (i = 0; i < pos && bmap; ++i)
+		bmap = var_equal(bmap, i);
+	if (bmap)
+		bmap = var_more(bmap, pos);
+	return isl_basic_map_finalize(bmap);
+}
+
+/* Return a relation on of dimension "dim" expressing i_[0..pos] >>= o_[0..pos]
+ */
+__isl_give isl_basic_map *isl_basic_map_more_or_equal_at(
+	__isl_take isl_space *dim, unsigned pos)
+{
+	int i;
+	isl_basic_map *bmap;
+
+	bmap = isl_basic_map_alloc_space(dim, 0, pos, 1);
+	for (i = 0; i < pos; ++i)
+		bmap = var_equal(bmap, i);
+	bmap = var_more_or_equal(bmap, pos);
+	return isl_basic_map_finalize(bmap);
+}
+
+static __isl_give isl_map *map_lex_lte_first(__isl_take isl_space *dims,
+	unsigned n, int equal)
+{
+	struct isl_map *map;
+	int i;
+
+	if (n == 0 && equal)
+		return isl_map_universe(dims);
+
+	map = isl_map_alloc_space(isl_space_copy(dims), n, ISL_MAP_DISJOINT);
+
+	for (i = 0; i + 1 < n; ++i)
+		map = isl_map_add_basic_map(map,
+				  isl_basic_map_less_at(isl_space_copy(dims), i));
+	if (n > 0) {
+		if (equal)
+			map = isl_map_add_basic_map(map,
+			      isl_basic_map_less_or_equal_at(dims, n - 1));
+		else
+			map = isl_map_add_basic_map(map,
+			      isl_basic_map_less_at(dims, n - 1));
+	} else
+		isl_space_free(dims);
+
+	return map;
+}
+
+static __isl_give isl_map *map_lex_lte(__isl_take isl_space *dims, int equal)
+{
+	if (!dims)
+		return NULL;
+	return map_lex_lte_first(dims, dims->n_out, equal);
+}
+
+__isl_give isl_map *isl_map_lex_lt_first(__isl_take isl_space *dim, unsigned n)
+{
+	return map_lex_lte_first(dim, n, 0);
+}
+
+__isl_give isl_map *isl_map_lex_le_first(__isl_take isl_space *dim, unsigned n)
+{
+	return map_lex_lte_first(dim, n, 1);
+}
+
+__isl_give isl_map *isl_map_lex_lt(__isl_take isl_space *set_dim)
+{
+	return map_lex_lte(isl_space_map_from_set(set_dim), 0);
+}
+
+__isl_give isl_map *isl_map_lex_le(__isl_take isl_space *set_dim)
+{
+	return map_lex_lte(isl_space_map_from_set(set_dim), 1);
+}
+
+static __isl_give isl_map *map_lex_gte_first(__isl_take isl_space *dims,
+	unsigned n, int equal)
+{
+	struct isl_map *map;
+	int i;
+
+	if (n == 0 && equal)
+		return isl_map_universe(dims);
+
+	map = isl_map_alloc_space(isl_space_copy(dims), n, ISL_MAP_DISJOINT);
+
+	for (i = 0; i + 1 < n; ++i)
+		map = isl_map_add_basic_map(map,
+				  isl_basic_map_more_at(isl_space_copy(dims), i));
+	if (n > 0) {
+		if (equal)
+			map = isl_map_add_basic_map(map,
+			      isl_basic_map_more_or_equal_at(dims, n - 1));
+		else
+			map = isl_map_add_basic_map(map,
+			      isl_basic_map_more_at(dims, n - 1));
+	} else
+		isl_space_free(dims);
+
+	return map;
+}
+
+static __isl_give isl_map *map_lex_gte(__isl_take isl_space *dims, int equal)
+{
+	if (!dims)
+		return NULL;
+	return map_lex_gte_first(dims, dims->n_out, equal);
+}
+
+__isl_give isl_map *isl_map_lex_gt_first(__isl_take isl_space *dim, unsigned n)
+{
+	return map_lex_gte_first(dim, n, 0);
+}
+
+__isl_give isl_map *isl_map_lex_ge_first(__isl_take isl_space *dim, unsigned n)
+{
+	return map_lex_gte_first(dim, n, 1);
+}
+
+__isl_give isl_map *isl_map_lex_gt(__isl_take isl_space *set_dim)
+{
+	return map_lex_gte(isl_space_map_from_set(set_dim), 0);
+}
+
+__isl_give isl_map *isl_map_lex_ge(__isl_take isl_space *set_dim)
+{
+	return map_lex_gte(isl_space_map_from_set(set_dim), 1);
+}
+
+__isl_give isl_map *isl_set_lex_le_set(__isl_take isl_set *set1,
+	__isl_take isl_set *set2)
+{
+	isl_map *map;
+	map = isl_map_lex_le(isl_set_get_space(set1));
+	map = isl_map_intersect_domain(map, set1);
+	map = isl_map_intersect_range(map, set2);
+	return map;
+}
+
+__isl_give isl_map *isl_set_lex_lt_set(__isl_take isl_set *set1,
+	__isl_take isl_set *set2)
+{
+	isl_map *map;
+	map = isl_map_lex_lt(isl_set_get_space(set1));
+	map = isl_map_intersect_domain(map, set1);
+	map = isl_map_intersect_range(map, set2);
+	return map;
+}
+
+__isl_give isl_map *isl_set_lex_ge_set(__isl_take isl_set *set1,
+	__isl_take isl_set *set2)
+{
+	isl_map *map;
+	map = isl_map_lex_ge(isl_set_get_space(set1));
+	map = isl_map_intersect_domain(map, set1);
+	map = isl_map_intersect_range(map, set2);
+	return map;
+}
+
+__isl_give isl_map *isl_set_lex_gt_set(__isl_take isl_set *set1,
+	__isl_take isl_set *set2)
+{
+	isl_map *map;
+	map = isl_map_lex_gt(isl_set_get_space(set1));
+	map = isl_map_intersect_domain(map, set1);
+	map = isl_map_intersect_range(map, set2);
+	return map;
+}
+
+__isl_give isl_map *isl_map_lex_le_map(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	isl_map *map;
+	map = isl_map_lex_le(isl_space_range(isl_map_get_space(map1)));
+	map = isl_map_apply_domain(map, isl_map_reverse(map1));
+	map = isl_map_apply_range(map, isl_map_reverse(map2));
+	return map;
+}
+
+__isl_give isl_map *isl_map_lex_lt_map(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	isl_map *map;
+	map = isl_map_lex_lt(isl_space_range(isl_map_get_space(map1)));
+	map = isl_map_apply_domain(map, isl_map_reverse(map1));
+	map = isl_map_apply_range(map, isl_map_reverse(map2));
+	return map;
+}
+
+__isl_give isl_map *isl_map_lex_ge_map(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	isl_map *map;
+	map = isl_map_lex_ge(isl_space_range(isl_map_get_space(map1)));
+	map = isl_map_apply_domain(map, isl_map_reverse(map1));
+	map = isl_map_apply_range(map, isl_map_reverse(map2));
+	return map;
+}
+
+__isl_give isl_map *isl_map_lex_gt_map(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	isl_map *map;
+	map = isl_map_lex_gt(isl_space_range(isl_map_get_space(map1)));
+	map = isl_map_apply_domain(map, isl_map_reverse(map1));
+	map = isl_map_apply_range(map, isl_map_reverse(map2));
+	return map;
+}
+
+__isl_give isl_basic_map *isl_basic_map_from_basic_set(
+	__isl_take isl_basic_set *bset, __isl_take isl_space *dim)
+{
+	struct isl_basic_map *bmap;
+
+	bset = isl_basic_set_cow(bset);
+	if (!bset || !dim)
+		goto error;
+
+	isl_assert(bset->ctx, isl_space_compatible(bset->dim, dim), goto error);
+	isl_space_free(bset->dim);
+	bmap = (struct isl_basic_map *) bset;
+	bmap->dim = dim;
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_set_free(bset);
+	isl_space_free(dim);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_from_basic_map(struct isl_basic_map *bmap)
+{
+	if (!bmap)
+		goto error;
+	if (bmap->dim->n_in == 0)
+		return (struct isl_basic_set *)bmap;
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		goto error;
+	bmap->dim = isl_space_as_set_space(bmap->dim);
+	if (!bmap->dim)
+		goto error;
+	bmap = isl_basic_map_finalize(bmap);
+	return (struct isl_basic_set *)bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* For a div d = floor(f/m), add the constraint
+ *
+ *		f - m d >= 0
+ */
+static int add_upper_div_constraint(__isl_keep isl_basic_map *bmap,
+	unsigned pos, isl_int *div)
+{
+	int i;
+	unsigned total = isl_basic_map_total_dim(bmap);
+
+	i = isl_basic_map_alloc_inequality(bmap);
+	if (i < 0)
+		return -1;
+	isl_seq_cpy(bmap->ineq[i], div + 1, 1 + total);
+	isl_int_neg(bmap->ineq[i][1 + pos], div[0]);
+
+	return 0;
+}
+
+/* For a div d = floor(f/m), add the constraint
+ *
+ *		-(f-(n-1)) + m d >= 0
+ */
+static int add_lower_div_constraint(__isl_keep isl_basic_map *bmap,
+	unsigned pos, isl_int *div)
+{
+	int i;
+	unsigned total = isl_basic_map_total_dim(bmap);
+
+	i = isl_basic_map_alloc_inequality(bmap);
+	if (i < 0)
+		return -1;
+	isl_seq_neg(bmap->ineq[i], div + 1, 1 + total);
+	isl_int_set(bmap->ineq[i][1 + pos], div[0]);
+	isl_int_add(bmap->ineq[i][0], bmap->ineq[i][0], bmap->ineq[i][1 + pos]);
+	isl_int_sub_ui(bmap->ineq[i][0], bmap->ineq[i][0], 1);
+
+	return 0;
+}
+
+/* For a div d = floor(f/m), add the constraints
+ *
+ *		f - m d >= 0
+ *		-(f-(n-1)) + m d >= 0
+ *
+ * Note that the second constraint is the negation of
+ *
+ *		f - m d >= n
+ */
+int isl_basic_map_add_div_constraints_var(__isl_keep isl_basic_map *bmap,
+	unsigned pos, isl_int *div)
+{
+	if (add_upper_div_constraint(bmap, pos, div) < 0)
+		return -1;
+	if (add_lower_div_constraint(bmap, pos, div) < 0)
+		return -1;
+	return 0;
+}
+
+int isl_basic_set_add_div_constraints_var(__isl_keep isl_basic_set *bset,
+	unsigned pos, isl_int *div)
+{
+	return isl_basic_map_add_div_constraints_var((isl_basic_map *)bset,
+							pos, div);
+}
+
+int isl_basic_map_add_div_constraints(struct isl_basic_map *bmap, unsigned div)
+{
+	unsigned total = isl_basic_map_total_dim(bmap);
+	unsigned div_pos = total - bmap->n_div + div;
+
+	return isl_basic_map_add_div_constraints_var(bmap, div_pos,
+							bmap->div[div]);
+}
+
+/* For each known div d = floor(f/m), add the constraints
+ *
+ *		f - m d >= 0
+ *		-(f-(n-1)) + m d >= 0
+ */
+__isl_give isl_basic_map *isl_basic_map_add_known_div_constraints(
+	__isl_take isl_basic_map *bmap)
+{
+	int i;
+	unsigned n_div;
+
+	if (!bmap)
+		return NULL;
+	n_div = isl_basic_map_dim(bmap, isl_dim_div);
+	if (n_div == 0)
+		return bmap;
+	bmap = isl_basic_map_cow(bmap);
+	bmap = isl_basic_map_extend_constraints(bmap, 0, 2 * n_div);
+	if (!bmap)
+		return NULL;
+	for (i = 0; i < n_div; ++i) {
+		if (isl_int_is_zero(bmap->div[i][0]))
+			continue;
+		if (isl_basic_map_add_div_constraints(bmap, i) < 0)
+			return isl_basic_map_free(bmap);
+	}
+
+	bmap = isl_basic_map_remove_duplicate_constraints(bmap, NULL, 0);
+	bmap = isl_basic_map_finalize(bmap);
+	return bmap;
+}
+
+/* Add the div constraint of sign "sign" for div "div" of "bmap".
+ *
+ * In particular, if this div is of the form d = floor(f/m),
+ * then add the constraint
+ *
+ *		f - m d >= 0
+ *
+ * if sign < 0 or the constraint
+ *
+ *		-(f-(n-1)) + m d >= 0
+ *
+ * if sign > 0.
+ */
+int isl_basic_map_add_div_constraint(__isl_keep isl_basic_map *bmap,
+	unsigned div, int sign)
+{
+	unsigned total;
+	unsigned div_pos;
+
+	if (!bmap)
+		return -1;
+
+	total = isl_basic_map_total_dim(bmap);
+	div_pos = total - bmap->n_div + div;
+
+	if (sign < 0)
+		return add_upper_div_constraint(bmap, div_pos, bmap->div[div]);
+	else
+		return add_lower_div_constraint(bmap, div_pos, bmap->div[div]);
+}
+
+int isl_basic_set_add_div_constraints(struct isl_basic_set *bset, unsigned div)
+{
+	return isl_basic_map_add_div_constraints(bset, div);
+}
+
+struct isl_basic_set *isl_basic_map_underlying_set(
+		struct isl_basic_map *bmap)
+{
+	if (!bmap)
+		goto error;
+	if (bmap->dim->nparam == 0 && bmap->dim->n_in == 0 &&
+	    bmap->n_div == 0 &&
+	    !isl_space_is_named_or_nested(bmap->dim, isl_dim_in) &&
+	    !isl_space_is_named_or_nested(bmap->dim, isl_dim_out))
+		return (struct isl_basic_set *)bmap;
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		goto error;
+	bmap->dim = isl_space_underlying(bmap->dim, bmap->n_div);
+	if (!bmap->dim)
+		goto error;
+	bmap->extra -= bmap->n_div;
+	bmap->n_div = 0;
+	bmap = isl_basic_map_finalize(bmap);
+	return (struct isl_basic_set *)bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_underlying_set(
+		__isl_take isl_basic_set *bset)
+{
+	return isl_basic_map_underlying_set((isl_basic_map *)bset);
+}
+
+/* Replace each element in "list" by the result of applying
+ * isl_basic_set_underlying_set to the element.
+ */
+__isl_give isl_basic_set_list *isl_basic_set_list_underlying_set(
+	__isl_take isl_basic_set_list *list)
+{
+	int i, n;
+
+	if (!list)
+		return NULL;
+
+	n = isl_basic_set_list_n_basic_set(list);
+	for (i = 0; i < n; ++i) {
+		isl_basic_set *bset;
+
+		bset = isl_basic_set_list_get_basic_set(list, i);
+		bset = isl_basic_set_underlying_set(bset);
+		list = isl_basic_set_list_set_basic_set(list, i, bset);
+	}
+
+	return list;
+}
+
+struct isl_basic_map *isl_basic_map_overlying_set(
+	struct isl_basic_set *bset, struct isl_basic_map *like)
+{
+	struct isl_basic_map *bmap;
+	struct isl_ctx *ctx;
+	unsigned total;
+	int i;
+
+	if (!bset || !like)
+		goto error;
+	ctx = bset->ctx;
+	isl_assert(ctx, bset->n_div == 0, goto error);
+	isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
+	isl_assert(ctx, bset->dim->n_out == isl_basic_map_total_dim(like),
+			goto error);
+	if (isl_space_is_equal(bset->dim, like->dim) && like->n_div == 0) {
+		isl_basic_map_free(like);
+		return (struct isl_basic_map *)bset;
+	}
+	bset = isl_basic_set_cow(bset);
+	if (!bset)
+		goto error;
+	total = bset->dim->n_out + bset->extra;
+	bmap = (struct isl_basic_map *)bset;
+	isl_space_free(bmap->dim);
+	bmap->dim = isl_space_copy(like->dim);
+	if (!bmap->dim)
+		goto error;
+	bmap->n_div = like->n_div;
+	bmap->extra += like->n_div;
+	if (bmap->extra) {
+		unsigned ltotal;
+		isl_int **div;
+		ltotal = total - bmap->extra + like->extra;
+		if (ltotal > total)
+			ltotal = total;
+		bmap->block2 = isl_blk_extend(ctx, bmap->block2,
+					bmap->extra * (1 + 1 + total));
+		if (isl_blk_is_error(bmap->block2))
+			goto error;
+		div = isl_realloc_array(ctx, bmap->div, isl_int *, bmap->extra);
+		if (!div)
+			goto error;
+		bmap->div = div;
+		for (i = 0; i < bmap->extra; ++i)
+			bmap->div[i] = bmap->block2.data + i * (1 + 1 + total);
+		for (i = 0; i < like->n_div; ++i) {
+			isl_seq_cpy(bmap->div[i], like->div[i], 1 + 1 + ltotal);
+			isl_seq_clr(bmap->div[i]+1+1+ltotal, total - ltotal);
+		}
+		bmap = isl_basic_map_add_known_div_constraints(bmap);
+	}
+	isl_basic_map_free(like);
+	bmap = isl_basic_map_simplify(bmap);
+	bmap = isl_basic_map_finalize(bmap);
+	return bmap;
+error:
+	isl_basic_map_free(like);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_from_underlying_set(
+	struct isl_basic_set *bset, struct isl_basic_set *like)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_overlying_set(bset, (struct isl_basic_map *)like);
+}
+
+struct isl_set *isl_set_from_underlying_set(
+	struct isl_set *set, struct isl_basic_set *like)
+{
+	int i;
+
+	if (!set || !like)
+		goto error;
+	isl_assert(set->ctx, set->dim->n_out == isl_basic_set_total_dim(like),
+		    goto error);
+	if (isl_space_is_equal(set->dim, like->dim) && like->n_div == 0) {
+		isl_basic_set_free(like);
+		return set;
+	}
+	set = isl_set_cow(set);
+	if (!set)
+		goto error;
+	for (i = 0; i < set->n; ++i) {
+		set->p[i] = isl_basic_set_from_underlying_set(set->p[i],
+						      isl_basic_set_copy(like));
+		if (!set->p[i])
+			goto error;
+	}
+	isl_space_free(set->dim);
+	set->dim = isl_space_copy(like->dim);
+	if (!set->dim)
+		goto error;
+	isl_basic_set_free(like);
+	return set;
+error:
+	isl_basic_set_free(like);
+	isl_set_free(set);
+	return NULL;
+}
+
+struct isl_set *isl_map_underlying_set(struct isl_map *map)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+	map->dim = isl_space_cow(map->dim);
+	if (!map->dim)
+		goto error;
+
+	for (i = 1; i < map->n; ++i)
+		isl_assert(map->ctx, map->p[0]->n_div == map->p[i]->n_div,
+				goto error);
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = (struct isl_basic_map *)
+				isl_basic_map_underlying_set(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+	if (map->n == 0)
+		map->dim = isl_space_underlying(map->dim, 0);
+	else {
+		isl_space_free(map->dim);
+		map->dim = isl_space_copy(map->p[0]->dim);
+	}
+	if (!map->dim)
+		goto error;
+	return (struct isl_set *)map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+struct isl_set *isl_set_to_underlying_set(struct isl_set *set)
+{
+	return (struct isl_set *)isl_map_underlying_set((struct isl_map *)set);
+}
+
+__isl_give isl_basic_map *isl_basic_map_reset_space(
+	__isl_take isl_basic_map *bmap, __isl_take isl_space *dim)
+{
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap || !dim)
+		goto error;
+
+	isl_space_free(bmap->dim);
+	bmap->dim = dim;
+
+	bmap = isl_basic_map_finalize(bmap);
+
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	isl_space_free(dim);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_reset_space(
+	__isl_take isl_basic_set *bset, __isl_take isl_space *dim)
+{
+	return (isl_basic_set *)isl_basic_map_reset_space((isl_basic_map *)bset,
+							dim);
+}
+
+__isl_give isl_map *isl_map_reset_space(__isl_take isl_map *map,
+	__isl_take isl_space *dim)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map || !dim)
+		goto error;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_reset_space(map->p[i],
+						    isl_space_copy(dim));
+		if (!map->p[i])
+			goto error;
+	}
+	isl_space_free(map->dim);
+	map->dim = dim;
+
+	return map;
+error:
+	isl_map_free(map);
+	isl_space_free(dim);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_reset_space(__isl_take isl_set *set,
+	__isl_take isl_space *dim)
+{
+	return (struct isl_set *) isl_map_reset_space((struct isl_map *)set, dim);
+}
+
+/* Compute the parameter domain of the given basic set.
+ */
+__isl_give isl_basic_set *isl_basic_set_params(__isl_take isl_basic_set *bset)
+{
+	isl_space *space;
+	unsigned n;
+
+	if (isl_basic_set_is_params(bset))
+		return bset;
+
+	n = isl_basic_set_dim(bset, isl_dim_set);
+	bset = isl_basic_set_project_out(bset, isl_dim_set, 0, n);
+	space = isl_basic_set_get_space(bset);
+	space = isl_space_params(space);
+	bset = isl_basic_set_reset_space(bset, space);
+	return bset;
+}
+
+/* Construct a zero-dimensional basic set with the given parameter domain.
+ */
+__isl_give isl_basic_set *isl_basic_set_from_params(
+	__isl_take isl_basic_set *bset)
+{
+	isl_space *space;
+	space = isl_basic_set_get_space(bset);
+	space = isl_space_set_from_params(space);
+	bset = isl_basic_set_reset_space(bset, space);
+	return bset;
+}
+
+/* Compute the parameter domain of the given set.
+ */
+__isl_give isl_set *isl_set_params(__isl_take isl_set *set)
+{
+	isl_space *space;
+	unsigned n;
+
+	if (isl_set_is_params(set))
+		return set;
+
+	n = isl_set_dim(set, isl_dim_set);
+	set = isl_set_project_out(set, isl_dim_set, 0, n);
+	space = isl_set_get_space(set);
+	space = isl_space_params(space);
+	set = isl_set_reset_space(set, space);
+	return set;
+}
+
+/* Construct a zero-dimensional set with the given parameter domain.
+ */
+__isl_give isl_set *isl_set_from_params(__isl_take isl_set *set)
+{
+	isl_space *space;
+	space = isl_set_get_space(set);
+	space = isl_space_set_from_params(space);
+	set = isl_set_reset_space(set, space);
+	return set;
+}
+
+/* Compute the parameter domain of the given map.
+ */
+__isl_give isl_set *isl_map_params(__isl_take isl_map *map)
+{
+	isl_space *space;
+	unsigned n;
+
+	n = isl_map_dim(map, isl_dim_in);
+	map = isl_map_project_out(map, isl_dim_in, 0, n);
+	n = isl_map_dim(map, isl_dim_out);
+	map = isl_map_project_out(map, isl_dim_out, 0, n);
+	space = isl_map_get_space(map);
+	space = isl_space_params(space);
+	map = isl_map_reset_space(map, space);
+	return map;
+}
+
+struct isl_basic_set *isl_basic_map_domain(struct isl_basic_map *bmap)
+{
+	isl_space *dim;
+	struct isl_basic_set *domain;
+	unsigned n_in;
+	unsigned n_out;
+
+	if (!bmap)
+		return NULL;
+	dim = isl_space_domain(isl_basic_map_get_space(bmap));
+
+	n_in = isl_basic_map_n_in(bmap);
+	n_out = isl_basic_map_n_out(bmap);
+	domain = isl_basic_set_from_basic_map(bmap);
+	domain = isl_basic_set_project_out(domain, isl_dim_set, n_in, n_out);
+
+	domain = isl_basic_set_reset_space(domain, dim);
+
+	return domain;
+}
+
+int isl_basic_map_may_be_set(__isl_keep isl_basic_map *bmap)
+{
+	if (!bmap)
+		return -1;
+	return isl_space_may_be_set(bmap->dim);
+}
+
+/* Is this basic map actually a set?
+ * Users should never call this function.  Outside of isl,
+ * the type should indicate whether something is a set or a map.
+ */
+int isl_basic_map_is_set(__isl_keep isl_basic_map *bmap)
+{
+	if (!bmap)
+		return -1;
+	return isl_space_is_set(bmap->dim);
+}
+
+struct isl_basic_set *isl_basic_map_range(struct isl_basic_map *bmap)
+{
+	if (!bmap)
+		return NULL;
+	if (isl_basic_map_is_set(bmap))
+		return bmap;
+	return isl_basic_map_domain(isl_basic_map_reverse(bmap));
+}
+
+__isl_give isl_basic_map *isl_basic_map_domain_map(
+	__isl_take isl_basic_map *bmap)
+{
+	int i, k;
+	isl_space *dim;
+	isl_basic_map *domain;
+	int nparam, n_in, n_out;
+	unsigned total;
+
+	nparam = isl_basic_map_dim(bmap, isl_dim_param);
+	n_in = isl_basic_map_dim(bmap, isl_dim_in);
+	n_out = isl_basic_map_dim(bmap, isl_dim_out);
+
+	dim = isl_space_from_range(isl_space_domain(isl_basic_map_get_space(bmap)));
+	domain = isl_basic_map_universe(dim);
+
+	bmap = isl_basic_map_from_domain(isl_basic_map_wrap(bmap));
+	bmap = isl_basic_map_apply_range(bmap, domain);
+	bmap = isl_basic_map_extend_constraints(bmap, n_in, 0);
+
+	total = isl_basic_map_total_dim(bmap);
+
+	for (i = 0; i < n_in; ++i) {
+		k = isl_basic_map_alloc_equality(bmap);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(bmap->eq[k], 1 + total);
+		isl_int_set_si(bmap->eq[k][1 + nparam + i], -1);
+		isl_int_set_si(bmap->eq[k][1 + nparam + n_in + n_out + i], 1);
+	}
+
+	bmap = isl_basic_map_gauss(bmap, NULL);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_range_map(
+	__isl_take isl_basic_map *bmap)
+{
+	int i, k;
+	isl_space *dim;
+	isl_basic_map *range;
+	int nparam, n_in, n_out;
+	unsigned total;
+
+	nparam = isl_basic_map_dim(bmap, isl_dim_param);
+	n_in = isl_basic_map_dim(bmap, isl_dim_in);
+	n_out = isl_basic_map_dim(bmap, isl_dim_out);
+
+	dim = isl_space_from_range(isl_space_range(isl_basic_map_get_space(bmap)));
+	range = isl_basic_map_universe(dim);
+
+	bmap = isl_basic_map_from_domain(isl_basic_map_wrap(bmap));
+	bmap = isl_basic_map_apply_range(bmap, range);
+	bmap = isl_basic_map_extend_constraints(bmap, n_out, 0);
+
+	total = isl_basic_map_total_dim(bmap);
+
+	for (i = 0; i < n_out; ++i) {
+		k = isl_basic_map_alloc_equality(bmap);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(bmap->eq[k], 1 + total);
+		isl_int_set_si(bmap->eq[k][1 + nparam + n_in + i], -1);
+		isl_int_set_si(bmap->eq[k][1 + nparam + n_in + n_out + i], 1);
+	}
+
+	bmap = isl_basic_map_gauss(bmap, NULL);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+int isl_map_may_be_set(__isl_keep isl_map *map)
+{
+	if (!map)
+		return -1;
+	return isl_space_may_be_set(map->dim);
+}
+
+/* Is this map actually a set?
+ * Users should never call this function.  Outside of isl,
+ * the type should indicate whether something is a set or a map.
+ */
+int isl_map_is_set(__isl_keep isl_map *map)
+{
+	if (!map)
+		return -1;
+	return isl_space_is_set(map->dim);
+}
+
+struct isl_set *isl_map_range(struct isl_map *map)
+{
+	int i;
+	struct isl_set *set;
+
+	if (!map)
+		goto error;
+	if (isl_map_is_set(map))
+		return (isl_set *)map;
+
+	map = isl_map_cow(map);
+	if (!map)
+		goto error;
+
+	set = (struct isl_set *) map;
+	set->dim = isl_space_range(set->dim);
+	if (!set->dim)
+		goto error;
+	for (i = 0; i < map->n; ++i) {
+		set->p[i] = isl_basic_map_range(map->p[i]);
+		if (!set->p[i])
+			goto error;
+	}
+	ISL_F_CLR(set, ISL_MAP_DISJOINT);
+	ISL_F_CLR(set, ISL_SET_NORMALIZED);
+	return set;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_domain_map(__isl_take isl_map *map)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	map->dim = isl_space_domain_map(map->dim);
+	if (!map->dim)
+		goto error;
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_domain_map(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+	ISL_F_CLR(map, ISL_MAP_DISJOINT);
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_range_map(__isl_take isl_map *map)
+{
+	int i;
+	isl_space *range_dim;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	range_dim = isl_space_range(isl_map_get_space(map));
+	range_dim = isl_space_from_range(range_dim);
+	map->dim = isl_space_from_domain(isl_space_wrap(map->dim));
+	map->dim = isl_space_join(map->dim, range_dim);
+	if (!map->dim)
+		goto error;
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_range_map(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+	ISL_F_CLR(map, ISL_MAP_DISJOINT);
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Given a wrapped map of the form A[B -> C],
+ * return the map A[B -> C] -> B.
+ */
+__isl_give isl_map *isl_set_wrapped_domain_map(__isl_take isl_set *set)
+{
+	isl_id *id;
+	isl_map *map;
+
+	if (!set)
+		return NULL;
+	if (!isl_set_has_tuple_id(set))
+		return isl_map_domain_map(isl_set_unwrap(set));
+
+	id = isl_set_get_tuple_id(set);
+	map = isl_map_domain_map(isl_set_unwrap(set));
+	map = isl_map_set_tuple_id(map, isl_dim_in, id);
+
+	return map;
+}
+
+__isl_give isl_map *isl_map_from_set(__isl_take isl_set *set,
+	__isl_take isl_space *dim)
+{
+	int i;
+	struct isl_map *map = NULL;
+
+	set = isl_set_cow(set);
+	if (!set || !dim)
+		goto error;
+	isl_assert(set->ctx, isl_space_compatible(set->dim, dim), goto error);
+	map = (struct isl_map *)set;
+	for (i = 0; i < set->n; ++i) {
+		map->p[i] = isl_basic_map_from_basic_set(
+				set->p[i], isl_space_copy(dim));
+		if (!map->p[i])
+			goto error;
+	}
+	isl_space_free(map->dim);
+	map->dim = dim;
+	return map;
+error:
+	isl_space_free(dim);
+	isl_set_free(set);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_from_domain(
+	__isl_take isl_basic_set *bset)
+{
+	return isl_basic_map_reverse(isl_basic_map_from_range(bset));
+}
+
+__isl_give isl_basic_map *isl_basic_map_from_range(
+	__isl_take isl_basic_set *bset)
+{
+	isl_space *space;
+	space = isl_basic_set_get_space(bset);
+	space = isl_space_from_range(space);
+	bset = isl_basic_set_reset_space(bset, space);
+	return (isl_basic_map *)bset;
+}
+
+/* Create a relation with the given set as range.
+ * The domain of the created relation is a zero-dimensional
+ * flat anonymous space.
+ */
+__isl_give isl_map *isl_map_from_range(__isl_take isl_set *set)
+{
+	isl_space *space;
+	space = isl_set_get_space(set);
+	space = isl_space_from_range(space);
+	set = isl_set_reset_space(set, space);
+	return (struct isl_map *)set;
+}
+
+/* Create a relation with the given set as domain.
+ * The range of the created relation is a zero-dimensional
+ * flat anonymous space.
+ */
+__isl_give isl_map *isl_map_from_domain(__isl_take isl_set *set)
+{
+	return isl_map_reverse(isl_map_from_range(set));
+}
+
+__isl_give isl_basic_map *isl_basic_map_from_domain_and_range(
+	__isl_take isl_basic_set *domain, __isl_take isl_basic_set *range)
+{
+	return isl_basic_map_apply_range(isl_basic_map_reverse(domain), range);
+}
+
+__isl_give isl_map *isl_map_from_domain_and_range(__isl_take isl_set *domain,
+	__isl_take isl_set *range)
+{
+	return isl_map_apply_range(isl_map_reverse(domain), range);
+}
+
+struct isl_set *isl_set_from_map(struct isl_map *map)
+{
+	int i;
+	struct isl_set *set = NULL;
+
+	if (!map)
+		return NULL;
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+	map->dim = isl_space_as_set_space(map->dim);
+	if (!map->dim)
+		goto error;
+	set = (struct isl_set *)map;
+	for (i = 0; i < map->n; ++i) {
+		set->p[i] = isl_basic_set_from_basic_map(map->p[i]);
+		if (!set->p[i])
+			goto error;
+	}
+	return set;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_alloc_space(__isl_take isl_space *dim, int n,
+	unsigned flags)
+{
+	struct isl_map *map;
+
+	if (!dim)
+		return NULL;
+	if (n < 0)
+		isl_die(dim->ctx, isl_error_internal,
+			"negative number of basic maps", goto error);
+	map = isl_alloc(dim->ctx, struct isl_map,
+			sizeof(struct isl_map) +
+			(n - 1) * sizeof(struct isl_basic_map *));
+	if (!map)
+		goto error;
+
+	map->ctx = dim->ctx;
+	isl_ctx_ref(map->ctx);
+	map->ref = 1;
+	map->size = n;
+	map->n = 0;
+	map->dim = dim;
+	map->flags = flags;
+	return map;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+struct isl_map *isl_map_alloc(struct isl_ctx *ctx,
+		unsigned nparam, unsigned in, unsigned out, int n,
+		unsigned flags)
+{
+	struct isl_map *map;
+	isl_space *dims;
+
+	dims = isl_space_alloc(ctx, nparam, in, out);
+	if (!dims)
+		return NULL;
+
+	map = isl_map_alloc_space(dims, n, flags);
+	return map;
+}
+
+__isl_give isl_basic_map *isl_basic_map_empty(__isl_take isl_space *dim)
+{
+	struct isl_basic_map *bmap;
+	bmap = isl_basic_map_alloc_space(dim, 0, 1, 0);
+	bmap = isl_basic_map_set_to_empty(bmap);
+	return bmap;
+}
+
+__isl_give isl_basic_set *isl_basic_set_empty(__isl_take isl_space *dim)
+{
+	struct isl_basic_set *bset;
+	bset = isl_basic_set_alloc_space(dim, 0, 1, 0);
+	bset = isl_basic_set_set_to_empty(bset);
+	return bset;
+}
+
+struct isl_basic_map *isl_basic_map_empty_like(struct isl_basic_map *model)
+{
+	struct isl_basic_map *bmap;
+	if (!model)
+		return NULL;
+	bmap = isl_basic_map_alloc_space(isl_space_copy(model->dim), 0, 1, 0);
+	bmap = isl_basic_map_set_to_empty(bmap);
+	return bmap;
+}
+
+struct isl_basic_map *isl_basic_map_empty_like_map(struct isl_map *model)
+{
+	struct isl_basic_map *bmap;
+	if (!model)
+		return NULL;
+	bmap = isl_basic_map_alloc_space(isl_space_copy(model->dim), 0, 1, 0);
+	bmap = isl_basic_map_set_to_empty(bmap);
+	return bmap;
+}
+
+struct isl_basic_set *isl_basic_set_empty_like(struct isl_basic_set *model)
+{
+	struct isl_basic_set *bset;
+	if (!model)
+		return NULL;
+	bset = isl_basic_set_alloc_space(isl_space_copy(model->dim), 0, 1, 0);
+	bset = isl_basic_set_set_to_empty(bset);
+	return bset;
+}
+
+__isl_give isl_basic_map *isl_basic_map_universe(__isl_take isl_space *dim)
+{
+	struct isl_basic_map *bmap;
+	bmap = isl_basic_map_alloc_space(dim, 0, 0, 0);
+	bmap = isl_basic_map_finalize(bmap);
+	return bmap;
+}
+
+__isl_give isl_basic_set *isl_basic_set_universe(__isl_take isl_space *dim)
+{
+	struct isl_basic_set *bset;
+	bset = isl_basic_set_alloc_space(dim, 0, 0, 0);
+	bset = isl_basic_set_finalize(bset);
+	return bset;
+}
+
+__isl_give isl_basic_map *isl_basic_map_nat_universe(__isl_take isl_space *dim)
+{
+	int i;
+	unsigned total = isl_space_dim(dim, isl_dim_all);
+	isl_basic_map *bmap;
+
+	bmap= isl_basic_map_alloc_space(dim, 0, 0, total);
+	for (i = 0; i < total; ++i) {
+		int k = isl_basic_map_alloc_inequality(bmap);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(bmap->ineq[k], 1 + total);
+		isl_int_set_si(bmap->ineq[k][1 + i], 1);
+	}
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_nat_universe(__isl_take isl_space *dim)
+{
+	return isl_basic_map_nat_universe(dim);
+}
+
+__isl_give isl_map *isl_map_nat_universe(__isl_take isl_space *dim)
+{
+	return isl_map_from_basic_map(isl_basic_map_nat_universe(dim));
+}
+
+__isl_give isl_set *isl_set_nat_universe(__isl_take isl_space *dim)
+{
+	return isl_map_nat_universe(dim);
+}
+
+__isl_give isl_basic_map *isl_basic_map_universe_like(
+		__isl_keep isl_basic_map *model)
+{
+	if (!model)
+		return NULL;
+	return isl_basic_map_alloc_space(isl_space_copy(model->dim), 0, 0, 0);
+}
+
+struct isl_basic_set *isl_basic_set_universe_like(struct isl_basic_set *model)
+{
+	if (!model)
+		return NULL;
+	return isl_basic_set_alloc_space(isl_space_copy(model->dim), 0, 0, 0);
+}
+
+__isl_give isl_basic_set *isl_basic_set_universe_like_set(
+	__isl_keep isl_set *model)
+{
+	if (!model)
+		return NULL;
+	return isl_basic_set_alloc_space(isl_space_copy(model->dim), 0, 0, 0);
+}
+
+__isl_give isl_map *isl_map_empty(__isl_take isl_space *dim)
+{
+	return isl_map_alloc_space(dim, 0, ISL_MAP_DISJOINT);
+}
+
+struct isl_map *isl_map_empty_like(struct isl_map *model)
+{
+	if (!model)
+		return NULL;
+	return isl_map_alloc_space(isl_space_copy(model->dim), 0, ISL_MAP_DISJOINT);
+}
+
+struct isl_map *isl_map_empty_like_basic_map(struct isl_basic_map *model)
+{
+	if (!model)
+		return NULL;
+	return isl_map_alloc_space(isl_space_copy(model->dim), 0, ISL_MAP_DISJOINT);
+}
+
+__isl_give isl_set *isl_set_empty(__isl_take isl_space *dim)
+{
+	return isl_set_alloc_space(dim, 0, ISL_MAP_DISJOINT);
+}
+
+struct isl_set *isl_set_empty_like(struct isl_set *model)
+{
+	if (!model)
+		return NULL;
+	return isl_set_empty(isl_space_copy(model->dim));
+}
+
+__isl_give isl_map *isl_map_universe(__isl_take isl_space *dim)
+{
+	struct isl_map *map;
+	if (!dim)
+		return NULL;
+	map = isl_map_alloc_space(isl_space_copy(dim), 1, ISL_MAP_DISJOINT);
+	map = isl_map_add_basic_map(map, isl_basic_map_universe(dim));
+	return map;
+}
+
+__isl_give isl_set *isl_set_universe(__isl_take isl_space *dim)
+{
+	struct isl_set *set;
+	if (!dim)
+		return NULL;
+	set = isl_set_alloc_space(isl_space_copy(dim), 1, ISL_MAP_DISJOINT);
+	set = isl_set_add_basic_set(set, isl_basic_set_universe(dim));
+	return set;
+}
+
+__isl_give isl_set *isl_set_universe_like(__isl_keep isl_set *model)
+{
+	if (!model)
+		return NULL;
+	return isl_set_universe(isl_space_copy(model->dim));
+}
+
+struct isl_map *isl_map_dup(struct isl_map *map)
+{
+	int i;
+	struct isl_map *dup;
+
+	if (!map)
+		return NULL;
+	dup = isl_map_alloc_space(isl_space_copy(map->dim), map->n, map->flags);
+	for (i = 0; i < map->n; ++i)
+		dup = isl_map_add_basic_map(dup, isl_basic_map_copy(map->p[i]));
+	return dup;
+}
+
+__isl_give isl_map *isl_map_add_basic_map(__isl_take isl_map *map,
+						__isl_take isl_basic_map *bmap)
+{
+	if (!bmap || !map)
+		goto error;
+	if (isl_basic_map_plain_is_empty(bmap)) {
+		isl_basic_map_free(bmap);
+		return map;
+	}
+	isl_assert(map->ctx, isl_space_is_equal(map->dim, bmap->dim), goto error);
+	isl_assert(map->ctx, map->n < map->size, goto error);
+	map->p[map->n] = bmap;
+	map->n++;
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+	return map;
+error:
+	if (map)
+		isl_map_free(map);
+	if (bmap)
+		isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_null isl_map *isl_map_free(__isl_take isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+
+	if (--map->ref > 0)
+		return NULL;
+
+	isl_ctx_deref(map->ctx);
+	for (i = 0; i < map->n; ++i)
+		isl_basic_map_free(map->p[i]);
+	isl_space_free(map->dim);
+	free(map);
+
+	return NULL;
+}
+
+struct isl_map *isl_map_extend(struct isl_map *base,
+		unsigned nparam, unsigned n_in, unsigned n_out)
+{
+	int i;
+
+	base = isl_map_cow(base);
+	if (!base)
+		return NULL;
+
+	base->dim = isl_space_extend(base->dim, nparam, n_in, n_out);
+	if (!base->dim)
+		goto error;
+	for (i = 0; i < base->n; ++i) {
+		base->p[i] = isl_basic_map_extend_space(base->p[i],
+				isl_space_copy(base->dim), 0, 0, 0);
+		if (!base->p[i])
+			goto error;
+	}
+	return base;
+error:
+	isl_map_free(base);
+	return NULL;
+}
+
+struct isl_set *isl_set_extend(struct isl_set *base,
+		unsigned nparam, unsigned dim)
+{
+	return (struct isl_set *)isl_map_extend((struct isl_map *)base,
+							nparam, 0, dim);
+}
+
+static struct isl_basic_map *isl_basic_map_fix_pos_si(
+	struct isl_basic_map *bmap, unsigned pos, int value)
+{
+	int j;
+
+	bmap = isl_basic_map_cow(bmap);
+	bmap = isl_basic_map_extend_constraints(bmap, 1, 0);
+	j = isl_basic_map_alloc_equality(bmap);
+	if (j < 0)
+		goto error;
+	isl_seq_clr(bmap->eq[j] + 1, isl_basic_map_total_dim(bmap));
+	isl_int_set_si(bmap->eq[j][pos], -1);
+	isl_int_set_si(bmap->eq[j][0], value);
+	bmap = isl_basic_map_simplify(bmap);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+static __isl_give isl_basic_map *isl_basic_map_fix_pos(
+	__isl_take isl_basic_map *bmap, unsigned pos, isl_int value)
+{
+	int j;
+
+	bmap = isl_basic_map_cow(bmap);
+	bmap = isl_basic_map_extend_constraints(bmap, 1, 0);
+	j = isl_basic_map_alloc_equality(bmap);
+	if (j < 0)
+		goto error;
+	isl_seq_clr(bmap->eq[j] + 1, isl_basic_map_total_dim(bmap));
+	isl_int_set_si(bmap->eq[j][pos], -1);
+	isl_int_set(bmap->eq[j][0], value);
+	bmap = isl_basic_map_simplify(bmap);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+struct isl_basic_map *isl_basic_map_fix_si(struct isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned pos, int value)
+{
+	if (!bmap)
+		return NULL;
+	isl_assert(bmap->ctx, pos < isl_basic_map_dim(bmap, type), goto error);
+	return isl_basic_map_fix_pos_si(bmap,
+		isl_basic_map_offset(bmap, type) + pos, value);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_fix(__isl_take isl_basic_map *bmap,
+		enum isl_dim_type type, unsigned pos, isl_int value)
+{
+	if (!bmap)
+		return NULL;
+	isl_assert(bmap->ctx, pos < isl_basic_map_dim(bmap, type), goto error);
+	return isl_basic_map_fix_pos(bmap,
+		isl_basic_map_offset(bmap, type) + pos, value);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Fix the value of the variable at position "pos" of type "type" of "bmap"
+ * to be equal to "v".
+ */
+__isl_give isl_basic_map *isl_basic_map_fix_val(__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_val *v)
+{
+	if (!bmap || !v)
+		goto error;
+	if (!isl_val_is_int(v))
+		isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
+			"expecting integer value", goto error);
+	if (pos >= isl_basic_map_dim(bmap, type))
+		isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
+			"index out of bounds", goto error);
+	pos += isl_basic_map_offset(bmap, type);
+	bmap = isl_basic_map_fix_pos(bmap, pos, v->n);
+	isl_val_free(v);
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	isl_val_free(v);
+	return NULL;
+}
+
+/* Fix the value of the variable at position "pos" of type "type" of "bset"
+ * to be equal to "v".
+ */
+__isl_give isl_basic_set *isl_basic_set_fix_val(__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_val *v)
+{
+	return isl_basic_map_fix_val(bset, type, pos, v);
+}
+
+struct isl_basic_set *isl_basic_set_fix_si(struct isl_basic_set *bset,
+		enum isl_dim_type type, unsigned pos, int value)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_fix_si((struct isl_basic_map *)bset,
+					type, pos, value);
+}
+
+__isl_give isl_basic_set *isl_basic_set_fix(__isl_take isl_basic_set *bset,
+		enum isl_dim_type type, unsigned pos, isl_int value)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_fix((struct isl_basic_map *)bset,
+					type, pos, value);
+}
+
+struct isl_basic_map *isl_basic_map_fix_input_si(struct isl_basic_map *bmap,
+		unsigned input, int value)
+{
+	return isl_basic_map_fix_si(bmap, isl_dim_in, input, value);
+}
+
+struct isl_basic_set *isl_basic_set_fix_dim_si(struct isl_basic_set *bset,
+		unsigned dim, int value)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_fix_si((struct isl_basic_map *)bset,
+					isl_dim_set, dim, value);
+}
+
+static int remove_if_empty(__isl_keep isl_map *map, int i)
+{
+	int empty = isl_basic_map_plain_is_empty(map->p[i]);
+
+	if (empty < 0)
+		return -1;
+	if (!empty)
+		return 0;
+
+	isl_basic_map_free(map->p[i]);
+	if (i != map->n - 1) {
+		ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+		map->p[i] = map->p[map->n - 1];
+	}
+	map->n--;
+
+	return 0;
+}
+
+/* Perform "fn" on each basic map of "map", where we may not be holding
+ * the only reference to "map".
+ * In particular, "fn" should be a semantics preserving operation
+ * that we want to apply to all copies of "map".  We therefore need
+ * to be careful not to modify "map" in a way that breaks "map"
+ * in case anything goes wrong.
+ */
+__isl_give isl_map *isl_map_inline_foreach_basic_map(__isl_take isl_map *map,
+	__isl_give isl_basic_map *(*fn)(__isl_take isl_basic_map *bmap))
+{
+	struct isl_basic_map *bmap;
+	int i;
+
+	if (!map)
+		return NULL;
+
+	for (i = map->n - 1; i >= 0; --i) {
+		bmap = isl_basic_map_copy(map->p[i]);
+		bmap = fn(bmap);
+		if (!bmap)
+			goto error;
+		isl_basic_map_free(map->p[i]);
+		map->p[i] = bmap;
+		if (remove_if_empty(map, i) < 0)
+			goto error;
+	}
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+struct isl_map *isl_map_fix_si(struct isl_map *map,
+		enum isl_dim_type type, unsigned pos, int value)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	isl_assert(map->ctx, pos < isl_map_dim(map, type), goto error);
+	for (i = map->n - 1; i >= 0; --i) {
+		map->p[i] = isl_basic_map_fix_si(map->p[i], type, pos, value);
+		if (remove_if_empty(map, i) < 0)
+			goto error;
+	}
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_fix_si(__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned pos, int value)
+{
+	return (struct isl_set *)
+		isl_map_fix_si((struct isl_map *)set, type, pos, value);
+}
+
+__isl_give isl_map *isl_map_fix(__isl_take isl_map *map,
+		enum isl_dim_type type, unsigned pos, isl_int value)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	isl_assert(map->ctx, pos < isl_map_dim(map, type), goto error);
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_fix(map->p[i], type, pos, value);
+		if (!map->p[i])
+			goto error;
+	}
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_fix(__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned pos, isl_int value)
+{
+	return (struct isl_set *)isl_map_fix((isl_map *)set, type, pos, value);
+}
+
+/* Fix the value of the variable at position "pos" of type "type" of "map"
+ * to be equal to "v".
+ */
+__isl_give isl_map *isl_map_fix_val(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_val *v)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map || !v)
+		goto error;
+
+	if (!isl_val_is_int(v))
+		isl_die(isl_map_get_ctx(map), isl_error_invalid,
+			"expecting integer value", goto error);
+	if (pos >= isl_map_dim(map, type))
+		isl_die(isl_map_get_ctx(map), isl_error_invalid,
+			"index out of bounds", goto error);
+	for (i = map->n - 1; i >= 0; --i) {
+		map->p[i] = isl_basic_map_fix_val(map->p[i], type, pos,
+							isl_val_copy(v));
+		if (remove_if_empty(map, i) < 0)
+			goto error;
+	}
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+	isl_val_free(v);
+	return map;
+error:
+	isl_map_free(map);
+	isl_val_free(v);
+	return NULL;
+}
+
+/* Fix the value of the variable at position "pos" of type "type" of "set"
+ * to be equal to "v".
+ */
+__isl_give isl_set *isl_set_fix_val(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_val *v)
+{
+	return isl_map_fix_val(set, type, pos, v);
+}
+
+struct isl_map *isl_map_fix_input_si(struct isl_map *map,
+		unsigned input, int value)
+{
+	return isl_map_fix_si(map, isl_dim_in, input, value);
+}
+
+struct isl_set *isl_set_fix_dim_si(struct isl_set *set, unsigned dim, int value)
+{
+	return (struct isl_set *)
+		isl_map_fix_si((struct isl_map *)set, isl_dim_set, dim, value);
+}
+
+static __isl_give isl_basic_map *basic_map_bound_si(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos, int value, int upper)
+{
+	int j;
+
+	if (!bmap)
+		return NULL;
+	isl_assert(bmap->ctx, pos < isl_basic_map_dim(bmap, type), goto error);
+	pos += isl_basic_map_offset(bmap, type);
+	bmap = isl_basic_map_cow(bmap);
+	bmap = isl_basic_map_extend_constraints(bmap, 0, 1);
+	j = isl_basic_map_alloc_inequality(bmap);
+	if (j < 0)
+		goto error;
+	isl_seq_clr(bmap->ineq[j], 1 + isl_basic_map_total_dim(bmap));
+	if (upper) {
+		isl_int_set_si(bmap->ineq[j][pos], -1);
+		isl_int_set_si(bmap->ineq[j][0], value);
+	} else {
+		isl_int_set_si(bmap->ineq[j][pos], 1);
+		isl_int_set_si(bmap->ineq[j][0], -value);
+	}
+	bmap = isl_basic_map_simplify(bmap);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_lower_bound_si(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos, int value)
+{
+	return basic_map_bound_si(bmap, type, pos, value, 0);
+}
+
+/* Constrain the values of the given dimension to be no greater than "value".
+ */
+__isl_give isl_basic_map *isl_basic_map_upper_bound_si(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos, int value)
+{
+	return basic_map_bound_si(bmap, type, pos, value, 1);
+}
+
+struct isl_basic_set *isl_basic_set_lower_bound_dim(struct isl_basic_set *bset,
+	unsigned dim, isl_int value)
+{
+	int j;
+
+	bset = isl_basic_set_cow(bset);
+	bset = isl_basic_set_extend_constraints(bset, 0, 1);
+	j = isl_basic_set_alloc_inequality(bset);
+	if (j < 0)
+		goto error;
+	isl_seq_clr(bset->ineq[j], 1 + isl_basic_set_total_dim(bset));
+	isl_int_set_si(bset->ineq[j][1 + isl_basic_set_n_param(bset) + dim], 1);
+	isl_int_neg(bset->ineq[j][0], value);
+	bset = isl_basic_set_simplify(bset);
+	return isl_basic_set_finalize(bset);
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+static __isl_give isl_map *map_bound_si(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned pos, int value, int upper)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	isl_assert(map->ctx, pos < isl_map_dim(map, type), goto error);
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = basic_map_bound_si(map->p[i],
+						 type, pos, value, upper);
+		if (!map->p[i])
+			goto error;
+	}
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_lower_bound_si(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned pos, int value)
+{
+	return map_bound_si(map, type, pos, value, 0);
+}
+
+__isl_give isl_map *isl_map_upper_bound_si(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned pos, int value)
+{
+	return map_bound_si(map, type, pos, value, 1);
+}
+
+__isl_give isl_set *isl_set_lower_bound_si(__isl_take isl_set *set,
+		enum isl_dim_type type, unsigned pos, int value)
+{
+	return (struct isl_set *)
+		isl_map_lower_bound_si((struct isl_map *)set, type, pos, value);
+}
+
+__isl_give isl_set *isl_set_upper_bound_si(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, int value)
+{
+	return isl_map_upper_bound_si(set, type, pos, value);
+}
+
+/* Bound the given variable of "bmap" from below (or above is "upper"
+ * is set) to "value".
+ */
+static __isl_give isl_basic_map *basic_map_bound(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos, isl_int value, int upper)
+{
+	int j;
+
+	if (!bmap)
+		return NULL;
+	if (pos >= isl_basic_map_dim(bmap, type))
+		isl_die(bmap->ctx, isl_error_invalid,
+			"index out of bounds", goto error);
+	pos += isl_basic_map_offset(bmap, type);
+	bmap = isl_basic_map_cow(bmap);
+	bmap = isl_basic_map_extend_constraints(bmap, 0, 1);
+	j = isl_basic_map_alloc_inequality(bmap);
+	if (j < 0)
+		goto error;
+	isl_seq_clr(bmap->ineq[j], 1 + isl_basic_map_total_dim(bmap));
+	if (upper) {
+		isl_int_set_si(bmap->ineq[j][pos], -1);
+		isl_int_set(bmap->ineq[j][0], value);
+	} else {
+		isl_int_set_si(bmap->ineq[j][pos], 1);
+		isl_int_neg(bmap->ineq[j][0], value);
+	}
+	bmap = isl_basic_map_simplify(bmap);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Bound the given variable of "map" from below (or above is "upper"
+ * is set) to "value".
+ */
+static __isl_give isl_map *map_bound(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned pos, isl_int value, int upper)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	if (pos >= isl_map_dim(map, type))
+		isl_die(map->ctx, isl_error_invalid,
+			"index out of bounds", goto error);
+	for (i = map->n - 1; i >= 0; --i) {
+		map->p[i] = basic_map_bound(map->p[i], type, pos, value, upper);
+		if (remove_if_empty(map, i) < 0)
+			goto error;
+	}
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_lower_bound(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned pos, isl_int value)
+{
+	return map_bound(map, type, pos, value, 0);
+}
+
+__isl_give isl_map *isl_map_upper_bound(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned pos, isl_int value)
+{
+	return map_bound(map, type, pos, value, 1);
+}
+
+__isl_give isl_set *isl_set_lower_bound(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, isl_int value)
+{
+	return isl_map_lower_bound(set, type, pos, value);
+}
+
+__isl_give isl_set *isl_set_upper_bound(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, isl_int value)
+{
+	return isl_map_upper_bound(set, type, pos, value);
+}
+
+/* Force the values of the variable at position "pos" of type "type" of "set"
+ * to be no smaller than "value".
+ */
+__isl_give isl_set *isl_set_lower_bound_val(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_val *value)
+{
+	if (!value)
+		goto error;
+	if (!isl_val_is_int(value))
+		isl_die(isl_set_get_ctx(set), isl_error_invalid,
+			"expecting integer value", goto error);
+	set = isl_set_lower_bound(set, type, pos, value->n);
+	isl_val_free(value);
+	return set;
+error:
+	isl_val_free(value);
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Force the values of the variable at position "pos" of type "type" of "set"
+ * to be no greater than "value".
+ */
+__isl_give isl_set *isl_set_upper_bound_val(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_val *value)
+{
+	if (!value)
+		goto error;
+	if (!isl_val_is_int(value))
+		isl_die(isl_set_get_ctx(set), isl_error_invalid,
+			"expecting integer value", goto error);
+	set = isl_set_upper_bound(set, type, pos, value->n);
+	isl_val_free(value);
+	return set;
+error:
+	isl_val_free(value);
+	isl_set_free(set);
+	return NULL;
+}
+
+struct isl_set *isl_set_lower_bound_dim(struct isl_set *set, unsigned dim,
+					isl_int value)
+{
+	int i;
+
+	set = isl_set_cow(set);
+	if (!set)
+		return NULL;
+
+	isl_assert(set->ctx, dim < isl_set_n_dim(set), goto error);
+	for (i = 0; i < set->n; ++i) {
+		set->p[i] = isl_basic_set_lower_bound_dim(set->p[i], dim, value);
+		if (!set->p[i])
+			goto error;
+	}
+	return set;
+error:
+	isl_set_free(set);
+	return NULL;
+}
+
+struct isl_map *isl_map_reverse(struct isl_map *map)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	map->dim = isl_space_reverse(map->dim);
+	if (!map->dim)
+		goto error;
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_reverse(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+static struct isl_map *isl_basic_map_partial_lexopt(
+		struct isl_basic_map *bmap, struct isl_basic_set *dom,
+		struct isl_set **empty, int max)
+{
+	return isl_tab_basic_map_partial_lexopt(bmap, dom, empty, max);
+}
+
+struct isl_map *isl_basic_map_partial_lexmax(
+		struct isl_basic_map *bmap, struct isl_basic_set *dom,
+		struct isl_set **empty)
+{
+	return isl_basic_map_partial_lexopt(bmap, dom, empty, 1);
+}
+
+struct isl_map *isl_basic_map_partial_lexmin(
+		struct isl_basic_map *bmap, struct isl_basic_set *dom,
+		struct isl_set **empty)
+{
+	return isl_basic_map_partial_lexopt(bmap, dom, empty, 0);
+}
+
+struct isl_set *isl_basic_set_partial_lexmin(
+		struct isl_basic_set *bset, struct isl_basic_set *dom,
+		struct isl_set **empty)
+{
+	return (struct isl_set *)
+		isl_basic_map_partial_lexmin((struct isl_basic_map *)bset,
+			dom, empty);
+}
+
+struct isl_set *isl_basic_set_partial_lexmax(
+		struct isl_basic_set *bset, struct isl_basic_set *dom,
+		struct isl_set **empty)
+{
+	return (struct isl_set *)
+		isl_basic_map_partial_lexmax((struct isl_basic_map *)bset,
+			dom, empty);
+}
+
+__isl_give isl_pw_multi_aff *isl_basic_map_partial_lexmin_pw_multi_aff(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty)
+{
+	return isl_basic_map_partial_lexopt_pw_multi_aff(bmap, dom, empty, 0);
+}
+
+__isl_give isl_pw_multi_aff *isl_basic_map_partial_lexmax_pw_multi_aff(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty)
+{
+	return isl_basic_map_partial_lexopt_pw_multi_aff(bmap, dom, empty, 1);
+}
+
+__isl_give isl_pw_multi_aff *isl_basic_set_partial_lexmin_pw_multi_aff(
+	__isl_take isl_basic_set *bset, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty)
+{
+	return isl_basic_map_partial_lexmin_pw_multi_aff(bset, dom, empty);
+}
+
+__isl_give isl_pw_multi_aff *isl_basic_set_partial_lexmax_pw_multi_aff(
+	__isl_take isl_basic_set *bset, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty)
+{
+	return isl_basic_map_partial_lexmax_pw_multi_aff(bset, dom, empty);
+}
+
+__isl_give isl_pw_multi_aff *isl_basic_map_lexopt_pw_multi_aff(
+	__isl_take isl_basic_map *bmap, int max)
+{
+	isl_basic_set *dom = NULL;
+	isl_space *dom_space;
+
+	if (!bmap)
+		goto error;
+	dom_space = isl_space_domain(isl_space_copy(bmap->dim));
+	dom = isl_basic_set_universe(dom_space);
+	return isl_basic_map_partial_lexopt_pw_multi_aff(bmap, dom, NULL, max);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_pw_multi_aff *isl_basic_map_lexmin_pw_multi_aff(
+	__isl_take isl_basic_map *bmap)
+{
+	return isl_basic_map_lexopt_pw_multi_aff(bmap, 0);
+}
+
+#undef TYPE
+#define TYPE	isl_pw_multi_aff
+#undef SUFFIX
+#define SUFFIX	_pw_multi_aff
+#undef EMPTY
+#define EMPTY	isl_pw_multi_aff_empty
+#undef ADD
+#define ADD	isl_pw_multi_aff_union_add
+#include "isl_map_lexopt_templ.c"
+
+/* Given a map "map", compute the lexicographically minimal
+ * (or maximal) image element for each domain element in dom,
+ * in the form of an isl_pw_multi_aff.
+ * Set *empty to those elements in dom that do not have an image element.
+ *
+ * We first compute the lexicographically minimal or maximal element
+ * in the first basic map.  This results in a partial solution "res"
+ * and a subset "todo" of dom that still need to be handled.
+ * We then consider each of the remaining maps in "map" and successively
+ * update both "res" and "todo".
+ */
+static __isl_give isl_pw_multi_aff *isl_map_partial_lexopt_aligned_pw_multi_aff(
+	__isl_take isl_map *map, __isl_take isl_set *dom,
+	__isl_give isl_set **empty, int max)
+{
+	int i;
+	isl_pw_multi_aff *res;
+	isl_set *todo;
+
+	if (!map || !dom)
+		goto error;
+
+	if (isl_map_plain_is_empty(map)) {
+		if (empty)
+			*empty = dom;
+		else
+			isl_set_free(dom);
+		return isl_pw_multi_aff_from_map(map);
+	}
+
+	res = basic_map_partial_lexopt_pw_multi_aff(
+					    isl_basic_map_copy(map->p[0]),
+					    isl_set_copy(dom), &todo, max);
+
+	for (i = 1; i < map->n; ++i) {
+		isl_pw_multi_aff *res_i;
+		isl_set *todo_i;
+
+		res_i = basic_map_partial_lexopt_pw_multi_aff(
+					    isl_basic_map_copy(map->p[i]),
+					    isl_set_copy(dom), &todo_i, max);
+
+		if (max)
+			res = isl_pw_multi_aff_union_lexmax(res, res_i);
+		else
+			res = isl_pw_multi_aff_union_lexmin(res, res_i);
+
+		todo = isl_set_intersect(todo, todo_i);
+	}
+
+	isl_set_free(dom);
+	isl_map_free(map);
+
+	if (empty)
+		*empty = todo;
+	else
+		isl_set_free(todo);
+
+	return res;
+error:
+	if (empty)
+		*empty = NULL;
+	isl_set_free(dom);
+	isl_map_free(map);
+	return NULL;
+}
+
+#undef TYPE
+#define TYPE	isl_map
+#undef SUFFIX
+#define SUFFIX
+#undef EMPTY
+#define EMPTY	isl_map_empty
+#undef ADD
+#define ADD	isl_map_union_disjoint
+#include "isl_map_lexopt_templ.c"
+
+/* Given a map "map", compute the lexicographically minimal
+ * (or maximal) image element for each domain element in dom.
+ * Set *empty to those elements in dom that do not have an image element.
+ *
+ * We first compute the lexicographically minimal or maximal element
+ * in the first basic map.  This results in a partial solution "res"
+ * and a subset "todo" of dom that still need to be handled.
+ * We then consider each of the remaining maps in "map" and successively
+ * update both "res" and "todo".
+ *
+ * Let res^k and todo^k be the results after k steps and let i = k + 1.
+ * Assume we are computing the lexicographical maximum.
+ * We first compute the lexicographically maximal element in basic map i.
+ * This results in a partial solution res_i and a subset todo_i.
+ * Then we combine these results with those obtain for the first k basic maps
+ * to obtain a result that is valid for the first k+1 basic maps.
+ * In particular, the set where there is no solution is the set where
+ * there is no solution for the first k basic maps and also no solution
+ * for the ith basic map, i.e.,
+ *
+ *	todo^i = todo^k * todo_i
+ *
+ * On dom(res^k) * dom(res_i), we need to pick the larger of the two
+ * solutions, arbitrarily breaking ties in favor of res^k.
+ * That is, when res^k(a) >= res_i(a), we pick res^k and
+ * when res^k(a) < res_i(a), we pick res_i.  (Here, ">=" and "<" denote
+ * the lexicographic order.)
+ * In practice, we compute
+ *
+ *	res^k * (res_i . "<=")
+ *
+ * and
+ *
+ *	res_i * (res^k . "<")
+ *
+ * Finally, we consider the symmetric difference of dom(res^k) and dom(res_i),
+ * where only one of res^k and res_i provides a solution and we simply pick
+ * that one, i.e.,
+ *
+ *	res^k * todo_i
+ * and
+ *	res_i * todo^k
+ *
+ * Note that we only compute these intersections when dom(res^k) intersects
+ * dom(res_i).  Otherwise, the only effect of these intersections is to
+ * potentially break up res^k and res_i into smaller pieces.
+ * We want to avoid such splintering as much as possible.
+ * In fact, an earlier implementation of this function would look for
+ * better results in the domain of res^k and for extra results in todo^k,
+ * but this would always result in a splintering according to todo^k,
+ * even when the domain of basic map i is disjoint from the domains of
+ * the previous basic maps.
+ */
+static __isl_give isl_map *isl_map_partial_lexopt_aligned(
+		__isl_take isl_map *map, __isl_take isl_set *dom,
+		__isl_give isl_set **empty, int max)
+{
+	int i;
+	struct isl_map *res;
+	struct isl_set *todo;
+
+	if (!map || !dom)
+		goto error;
+
+	if (isl_map_plain_is_empty(map)) {
+		if (empty)
+			*empty = dom;
+		else
+			isl_set_free(dom);
+		return map;
+	}
+
+	res = basic_map_partial_lexopt(isl_basic_map_copy(map->p[0]),
+					isl_set_copy(dom), &todo, max);
+
+	for (i = 1; i < map->n; ++i) {
+		isl_map *lt, *le;
+		isl_map *res_i;
+		isl_set *todo_i;
+		isl_space *dim = isl_space_range(isl_map_get_space(res));
+
+		res_i = basic_map_partial_lexopt(isl_basic_map_copy(map->p[i]),
+					isl_set_copy(dom), &todo_i, max);
+
+		if (max) {
+			lt = isl_map_lex_lt(isl_space_copy(dim));
+			le = isl_map_lex_le(dim);
+		} else {
+			lt = isl_map_lex_gt(isl_space_copy(dim));
+			le = isl_map_lex_ge(dim);
+		}
+		lt = isl_map_apply_range(isl_map_copy(res), lt);
+		lt = isl_map_intersect(lt, isl_map_copy(res_i));
+		le = isl_map_apply_range(isl_map_copy(res_i), le);
+		le = isl_map_intersect(le, isl_map_copy(res));
+
+		if (!isl_map_is_empty(lt) || !isl_map_is_empty(le)) {
+			res = isl_map_intersect_domain(res,
+							isl_set_copy(todo_i));
+			res_i = isl_map_intersect_domain(res_i,
+							isl_set_copy(todo));
+		}
+
+		res = isl_map_union_disjoint(res, res_i);
+		res = isl_map_union_disjoint(res, lt);
+		res = isl_map_union_disjoint(res, le);
+
+		todo = isl_set_intersect(todo, todo_i);
+	}
+
+	isl_set_free(dom);
+	isl_map_free(map);
+
+	if (empty)
+		*empty = todo;
+	else
+		isl_set_free(todo);
+
+	return res;
+error:
+	if (empty)
+		*empty = NULL;
+	isl_set_free(dom);
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_partial_lexmax(
+		__isl_take isl_map *map, __isl_take isl_set *dom,
+		__isl_give isl_set **empty)
+{
+	return isl_map_partial_lexopt(map, dom, empty, 1);
+}
+
+__isl_give isl_map *isl_map_partial_lexmin(
+		__isl_take isl_map *map, __isl_take isl_set *dom,
+		__isl_give isl_set **empty)
+{
+	return isl_map_partial_lexopt(map, dom, empty, 0);
+}
+
+__isl_give isl_set *isl_set_partial_lexmin(
+		__isl_take isl_set *set, __isl_take isl_set *dom,
+		__isl_give isl_set **empty)
+{
+	return (struct isl_set *)
+		isl_map_partial_lexmin((struct isl_map *)set,
+			dom, empty);
+}
+
+__isl_give isl_set *isl_set_partial_lexmax(
+		__isl_take isl_set *set, __isl_take isl_set *dom,
+		__isl_give isl_set **empty)
+{
+	return (struct isl_set *)
+		isl_map_partial_lexmax((struct isl_map *)set,
+			dom, empty);
+}
+
+/* Compute the lexicographic minimum (or maximum if "max" is set)
+ * of "bmap" over its domain.
+ *
+ * Since we are not interested in the part of the domain space where
+ * there is no solution, we initialize the domain to those constraints
+ * of "bmap" that only involve the parameters and the input dimensions.
+ * This relieves the parametric programming engine from detecting those
+ * inequalities and transferring them to the context.  More importantly,
+ * it ensures that those inequalities are transferred first and not
+ * intermixed with inequalities that actually split the domain.
+ */
+__isl_give isl_map *isl_basic_map_lexopt(__isl_take isl_basic_map *bmap, int max)
+{
+	int n_div;
+	int n_out;
+	isl_basic_map *copy;
+	isl_basic_set *dom;
+
+	n_div = isl_basic_map_dim(bmap, isl_dim_div);
+	n_out = isl_basic_map_dim(bmap, isl_dim_out);
+	copy = isl_basic_map_copy(bmap);
+	copy = isl_basic_map_drop_constraints_involving_dims(copy,
+							isl_dim_div, 0, n_div);
+	copy = isl_basic_map_drop_constraints_involving_dims(copy,
+							isl_dim_out, 0, n_out);
+	dom = isl_basic_map_domain(copy);
+	return isl_basic_map_partial_lexopt(bmap, dom, NULL, max);
+}
+
+__isl_give isl_map *isl_basic_map_lexmin(__isl_take isl_basic_map *bmap)
+{
+	return isl_basic_map_lexopt(bmap, 0);
+}
+
+__isl_give isl_map *isl_basic_map_lexmax(__isl_take isl_basic_map *bmap)
+{
+	return isl_basic_map_lexopt(bmap, 1);
+}
+
+__isl_give isl_set *isl_basic_set_lexmin(__isl_take isl_basic_set *bset)
+{
+	return (isl_set *)isl_basic_map_lexmin((isl_basic_map *)bset);
+}
+
+__isl_give isl_set *isl_basic_set_lexmax(__isl_take isl_basic_set *bset)
+{
+	return (isl_set *)isl_basic_map_lexmax((isl_basic_map *)bset);
+}
+
+/* Extract the first and only affine expression from list
+ * and then add it to *pwaff with the given dom.
+ * This domain is known to be disjoint from other domains
+ * because of the way isl_basic_map_foreach_lexmax works.
+ */
+static int update_dim_opt(__isl_take isl_basic_set *dom,
+	__isl_take isl_aff_list *list, void *user)
+{
+	isl_ctx *ctx = isl_basic_set_get_ctx(dom);
+	isl_aff *aff;
+	isl_pw_aff **pwaff = user;
+	isl_pw_aff *pwaff_i;
+
+	if (!list)
+		goto error;
+	if (isl_aff_list_n_aff(list) != 1)
+		isl_die(ctx, isl_error_internal,
+			"expecting single element list", goto error);
+
+	aff = isl_aff_list_get_aff(list, 0);
+	pwaff_i = isl_pw_aff_alloc(isl_set_from_basic_set(dom), aff);
+
+	*pwaff = isl_pw_aff_add_disjoint(*pwaff, pwaff_i);
+
+	isl_aff_list_free(list);
+
+	return 0;
+error:
+	isl_basic_set_free(dom);
+	isl_aff_list_free(list);
+	return -1;
+}
+
+/* Given a basic map with one output dimension, compute the minimum or
+ * maximum of that dimension as an isl_pw_aff.
+ *
+ * The isl_pw_aff is constructed by having isl_basic_map_foreach_lexopt
+ * call update_dim_opt on each leaf of the result.
+ */
+static __isl_give isl_pw_aff *basic_map_dim_opt(__isl_keep isl_basic_map *bmap,
+	int max)
+{
+	isl_space *dim = isl_basic_map_get_space(bmap);
+	isl_pw_aff *pwaff;
+	int r;
+
+	dim = isl_space_from_domain(isl_space_domain(dim));
+	dim = isl_space_add_dims(dim, isl_dim_out, 1);
+	pwaff = isl_pw_aff_empty(dim);
+
+	r = isl_basic_map_foreach_lexopt(bmap, max, &update_dim_opt, &pwaff);
+	if (r < 0)
+		return isl_pw_aff_free(pwaff);
+
+	return pwaff;
+}
+
+/* Compute the minimum or maximum of the given output dimension
+ * as a function of the parameters and the input dimensions,
+ * but independently of the other output dimensions.
+ *
+ * We first project out the other output dimension and then compute
+ * the "lexicographic" maximum in each basic map, combining the results
+ * using isl_pw_aff_union_max.
+ */
+static __isl_give isl_pw_aff *map_dim_opt(__isl_take isl_map *map, int pos,
+	int max)
+{
+	int i;
+	isl_pw_aff *pwaff;
+	unsigned n_out;
+
+	n_out = isl_map_dim(map, isl_dim_out);
+	map = isl_map_project_out(map, isl_dim_out, pos + 1, n_out - (pos + 1));
+	map = isl_map_project_out(map, isl_dim_out, 0, pos);
+	if (!map)
+		return NULL;
+
+	if (map->n == 0) {
+		isl_space *dim = isl_map_get_space(map);
+		isl_map_free(map);
+		return isl_pw_aff_empty(dim);
+	}
+
+	pwaff = basic_map_dim_opt(map->p[0], max);
+	for (i = 1; i < map->n; ++i) {
+		isl_pw_aff *pwaff_i;
+
+		pwaff_i = basic_map_dim_opt(map->p[i], max);
+		pwaff = isl_pw_aff_union_opt(pwaff, pwaff_i, max);
+	}
+
+	isl_map_free(map);
+
+	return pwaff;
+}
+
+/* Compute the maximum of the given output dimension as a function of the
+ * parameters and input dimensions, but independently of
+ * the other output dimensions.
+ */
+__isl_give isl_pw_aff *isl_map_dim_max(__isl_take isl_map *map, int pos)
+{
+	return map_dim_opt(map, pos, 1);
+}
+
+/* Compute the minimum or maximum of the given set dimension
+ * as a function of the parameters,
+ * but independently of the other set dimensions.
+ */
+static __isl_give isl_pw_aff *set_dim_opt(__isl_take isl_set *set, int pos,
+	int max)
+{
+	return map_dim_opt(set, pos, max);
+}
+
+/* Compute the maximum of the given set dimension as a function of the
+ * parameters, but independently of the other set dimensions.
+ */
+__isl_give isl_pw_aff *isl_set_dim_max(__isl_take isl_set *set, int pos)
+{
+	return set_dim_opt(set, pos, 1);
+}
+
+/* Compute the minimum of the given set dimension as a function of the
+ * parameters, but independently of the other set dimensions.
+ */
+__isl_give isl_pw_aff *isl_set_dim_min(__isl_take isl_set *set, int pos)
+{
+	return set_dim_opt(set, pos, 0);
+}
+
+/* Apply a preimage specified by "mat" on the parameters of "bset".
+ * bset is assumed to have only parameters and divs.
+ */
+static struct isl_basic_set *basic_set_parameter_preimage(
+	struct isl_basic_set *bset, struct isl_mat *mat)
+{
+	unsigned nparam;
+
+	if (!bset || !mat)
+		goto error;
+
+	bset->dim = isl_space_cow(bset->dim);
+	if (!bset->dim)
+		goto error;
+
+	nparam = isl_basic_set_dim(bset, isl_dim_param);
+
+	isl_assert(bset->ctx, mat->n_row == 1 + nparam, goto error);
+
+	bset->dim->nparam = 0;
+	bset->dim->n_out = nparam;
+	bset = isl_basic_set_preimage(bset, mat);
+	if (bset) {
+		bset->dim->nparam = bset->dim->n_out;
+		bset->dim->n_out = 0;
+	}
+	return bset;
+error:
+	isl_mat_free(mat);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Apply a preimage specified by "mat" on the parameters of "set".
+ * set is assumed to have only parameters and divs.
+ */
+static struct isl_set *set_parameter_preimage(
+	struct isl_set *set, struct isl_mat *mat)
+{
+	isl_space *dim = NULL;
+	unsigned nparam;
+
+	if (!set || !mat)
+		goto error;
+
+	dim = isl_space_copy(set->dim);
+	dim = isl_space_cow(dim);
+	if (!dim)
+		goto error;
+
+	nparam = isl_set_dim(set, isl_dim_param);
+
+	isl_assert(set->ctx, mat->n_row == 1 + nparam, goto error);
+
+	dim->nparam = 0;
+	dim->n_out = nparam;
+	isl_set_reset_space(set, dim);
+	set = isl_set_preimage(set, mat);
+	if (!set)
+		goto error2;
+	dim = isl_space_copy(set->dim);
+	dim = isl_space_cow(dim);
+	if (!dim)
+		goto error2;
+	dim->nparam = dim->n_out;
+	dim->n_out = 0;
+	isl_set_reset_space(set, dim);
+	return set;
+error:
+	isl_space_free(dim);
+	isl_mat_free(mat);
+error2:
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Intersect the basic set "bset" with the affine space specified by the
+ * equalities in "eq".
+ */
+static struct isl_basic_set *basic_set_append_equalities(
+	struct isl_basic_set *bset, struct isl_mat *eq)
+{
+	int i, k;
+	unsigned len;
+
+	if (!bset || !eq)
+		goto error;
+
+	bset = isl_basic_set_extend_space(bset, isl_space_copy(bset->dim), 0,
+					eq->n_row, 0);
+	if (!bset)
+		goto error;
+
+	len = 1 + isl_space_dim(bset->dim, isl_dim_all) + bset->extra;
+	for (i = 0; i < eq->n_row; ++i) {
+		k = isl_basic_set_alloc_equality(bset);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(bset->eq[k], eq->row[i], eq->n_col);
+		isl_seq_clr(bset->eq[k] + eq->n_col, len - eq->n_col);
+	}
+	isl_mat_free(eq);
+
+	bset = isl_basic_set_gauss(bset, NULL);
+	bset = isl_basic_set_finalize(bset);
+
+	return bset;
+error:
+	isl_mat_free(eq);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Intersect the set "set" with the affine space specified by the
+ * equalities in "eq".
+ */
+static struct isl_set *set_append_equalities(struct isl_set *set,
+	struct isl_mat *eq)
+{
+	int i;
+
+	if (!set || !eq)
+		goto error;
+
+	for (i = 0; i < set->n; ++i) {
+		set->p[i] = basic_set_append_equalities(set->p[i],
+					isl_mat_copy(eq));
+		if (!set->p[i])
+			goto error;
+	}
+	isl_mat_free(eq);
+	return set;
+error:
+	isl_mat_free(eq);
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Given a basic set "bset" that only involves parameters and existentially
+ * quantified variables, return the index of the first equality
+ * that only involves parameters.  If there is no such equality then
+ * return bset->n_eq.
+ *
+ * This function assumes that isl_basic_set_gauss has been called on "bset".
+ */
+static int first_parameter_equality(__isl_keep isl_basic_set *bset)
+{
+	int i, j;
+	unsigned nparam, n_div;
+
+	if (!bset)
+		return -1;
+
+	nparam = isl_basic_set_dim(bset, isl_dim_param);
+	n_div = isl_basic_set_dim(bset, isl_dim_div);
+
+	for (i = 0, j = n_div - 1; i < bset->n_eq && j >= 0; --j) {
+		if (!isl_int_is_zero(bset->eq[i][1 + nparam + j]))
+			++i;
+	}
+
+	return i;
+}
+
+/* Compute an explicit representation for the existentially quantified
+ * variables in "bset" by computing the "minimal value" of the set
+ * variables.  Since there are no set variables, the computation of
+ * the minimal value essentially computes an explicit representation
+ * of the non-empty part(s) of "bset".
+ *
+ * The input only involves parameters and existentially quantified variables.
+ * All equalities among parameters have been removed.
+ *
+ * Since the existentially quantified variables in the result are in general
+ * going to be different from those in the input, we first replace
+ * them by the minimal number of variables based on their equalities.
+ * This should simplify the parametric integer programming.
+ */
+static __isl_give isl_set *base_compute_divs(__isl_take isl_basic_set *bset)
+{
+	isl_morph *morph1, *morph2;
+	isl_set *set;
+	unsigned n;
+
+	if (!bset)
+		return NULL;
+	if (bset->n_eq == 0)
+		return isl_basic_set_lexmin(bset);
+
+	morph1 = isl_basic_set_parameter_compression(bset);
+	bset = isl_morph_basic_set(isl_morph_copy(morph1), bset);
+	bset = isl_basic_set_lift(bset);
+	morph2 = isl_basic_set_variable_compression(bset, isl_dim_set);
+	bset = isl_morph_basic_set(morph2, bset);
+	n = isl_basic_set_dim(bset, isl_dim_set);
+	bset = isl_basic_set_project_out(bset, isl_dim_set, 0, n);
+
+	set = isl_basic_set_lexmin(bset);
+
+	set = isl_morph_set(isl_morph_inverse(morph1), set);
+
+	return set;
+}
+
+/* Project the given basic set onto its parameter domain, possibly introducing
+ * new, explicit, existential variables in the constraints.
+ * The input has parameters and (possibly implicit) existential variables.
+ * The output has the same parameters, but only
+ * explicit existentially quantified variables.
+ *
+ * The actual projection is performed by pip, but pip doesn't seem
+ * to like equalities very much, so we first remove the equalities
+ * among the parameters by performing a variable compression on
+ * the parameters.  Afterward, an inverse transformation is performed
+ * and the equalities among the parameters are inserted back in.
+ *
+ * The variable compression on the parameters may uncover additional
+ * equalities that were only implicit before.  We therefore check
+ * if there are any new parameter equalities in the result and
+ * if so recurse.  The removal of parameter equalities is required
+ * for the parameter compression performed by base_compute_divs.
+ */
+static struct isl_set *parameter_compute_divs(struct isl_basic_set *bset)
+{
+	int i;
+	struct isl_mat *eq;
+	struct isl_mat *T, *T2;
+	struct isl_set *set;
+	unsigned nparam;
+
+	bset = isl_basic_set_cow(bset);
+	if (!bset)
+		return NULL;
+
+	if (bset->n_eq == 0)
+		return base_compute_divs(bset);
+
+	bset = isl_basic_set_gauss(bset, NULL);
+	if (!bset)
+		return NULL;
+	if (isl_basic_set_plain_is_empty(bset))
+		return isl_set_from_basic_set(bset);
+
+	i = first_parameter_equality(bset);
+	if (i == bset->n_eq)
+		return base_compute_divs(bset);
+
+	nparam = isl_basic_set_dim(bset, isl_dim_param);
+	eq = isl_mat_sub_alloc6(bset->ctx, bset->eq, i, bset->n_eq - i,
+		0, 1 + nparam);
+	eq = isl_mat_cow(eq);
+	T = isl_mat_variable_compression(isl_mat_copy(eq), &T2);
+	if (T && T->n_col == 0) {
+		isl_mat_free(T);
+		isl_mat_free(T2);
+		isl_mat_free(eq);
+		bset = isl_basic_set_set_to_empty(bset);
+		return isl_set_from_basic_set(bset);
+	}
+	bset = basic_set_parameter_preimage(bset, T);
+
+	i = first_parameter_equality(bset);
+	if (!bset)
+		set = NULL;
+	else if (i == bset->n_eq)
+		set = base_compute_divs(bset);
+	else
+		set = parameter_compute_divs(bset);
+	set = set_parameter_preimage(set, T2);
+	set = set_append_equalities(set, eq);
+	return set;
+}
+
+/* Insert the divs from "ls" before those of "bmap".
+ *
+ * The number of columns is not changed, which means that the last
+ * dimensions of "bmap" are being reintepreted as the divs from "ls".
+ * The caller is responsible for removing the same number of dimensions
+ * from the space of "bmap".
+ */
+static __isl_give isl_basic_map *insert_divs_from_local_space(
+	__isl_take isl_basic_map *bmap, __isl_keep isl_local_space *ls)
+{
+	int i;
+	int n_div;
+	int old_n_div;
+
+	n_div = isl_local_space_dim(ls, isl_dim_div);
+	if (n_div == 0)
+		return bmap;
+
+	old_n_div = bmap->n_div;
+	bmap = insert_div_rows(bmap, n_div);
+	if (!bmap)
+		return NULL;
+
+	for (i = 0; i < n_div; ++i) {
+		isl_seq_cpy(bmap->div[i], ls->div->row[i], ls->div->n_col);
+		isl_seq_clr(bmap->div[i] + ls->div->n_col, old_n_div);
+	}
+
+	return bmap;
+}
+
+/* Replace the space of "bmap" by the space and divs of "ls".
+ *
+ * If "ls" has any divs, then we simplify the result since we may
+ * have discovered some additional equalities that could simplify
+ * the div expressions.
+ */
+static __isl_give isl_basic_map *basic_replace_space_by_local_space(
+	__isl_take isl_basic_map *bmap, __isl_take isl_local_space *ls)
+{
+	int n_div;
+
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap || !ls)
+		goto error;
+
+	n_div = isl_local_space_dim(ls, isl_dim_div);
+	bmap = insert_divs_from_local_space(bmap, ls);
+	if (!bmap)
+		goto error;
+
+	isl_space_free(bmap->dim);
+	bmap->dim = isl_local_space_get_space(ls);
+	if (!bmap->dim)
+		goto error;
+
+	isl_local_space_free(ls);
+	if (n_div > 0)
+		bmap = isl_basic_map_simplify(bmap);
+	bmap = isl_basic_map_finalize(bmap);
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	isl_local_space_free(ls);
+	return NULL;
+}
+
+/* Replace the space of "map" by the space and divs of "ls".
+ */
+static __isl_give isl_map *replace_space_by_local_space(__isl_take isl_map *map,
+	__isl_take isl_local_space *ls)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map || !ls)
+		goto error;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = basic_replace_space_by_local_space(map->p[i],
+						    isl_local_space_copy(ls));
+		if (!map->p[i])
+			goto error;
+	}
+	isl_space_free(map->dim);
+	map->dim = isl_local_space_get_space(ls);
+	if (!map->dim)
+		goto error;
+
+	isl_local_space_free(ls);
+	return map;
+error:
+	isl_local_space_free(ls);
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Compute an explicit representation for the existentially
+ * quantified variables for which do not know any explicit representation yet.
+ *
+ * We first sort the existentially quantified variables so that the
+ * existentially quantified variables for which we already have an explicit
+ * representation are placed before those for which we do not.
+ * The input dimensions, the output dimensions and the existentially
+ * quantified variables for which we already have an explicit
+ * representation are then turned into parameters.
+ * compute_divs returns a map with the same parameters and
+ * no input or output dimensions and the dimension specification
+ * is reset to that of the input, including the existentially quantified
+ * variables for which we already had an explicit representation.
+ */
+static struct isl_map *compute_divs(struct isl_basic_map *bmap)
+{
+	struct isl_basic_set *bset;
+	struct isl_set *set;
+	struct isl_map *map;
+	isl_space *dim;
+	isl_local_space *ls;
+	unsigned	 nparam;
+	unsigned	 n_in;
+	unsigned	 n_out;
+	unsigned n_known;
+	int i;
+
+	bmap = isl_basic_map_sort_divs(bmap);
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+
+	for (n_known = 0; n_known < bmap->n_div; ++n_known)
+		if (isl_int_is_zero(bmap->div[n_known][0]))
+			break;
+
+	nparam = isl_basic_map_dim(bmap, isl_dim_param);
+	n_in = isl_basic_map_dim(bmap, isl_dim_in);
+	n_out = isl_basic_map_dim(bmap, isl_dim_out);
+	dim = isl_space_set_alloc(bmap->ctx,
+				    nparam + n_in + n_out + n_known, 0);
+	if (!dim)
+		goto error;
+
+	ls = isl_basic_map_get_local_space(bmap);
+	ls = isl_local_space_drop_dims(ls, isl_dim_div,
+					n_known, bmap->n_div - n_known);
+	if (n_known > 0) {
+		for (i = n_known; i < bmap->n_div; ++i)
+			swap_div(bmap, i - n_known, i);
+		bmap->n_div -= n_known;
+		bmap->extra -= n_known;
+	}
+	bmap = isl_basic_map_reset_space(bmap, dim);
+	bset = (struct isl_basic_set *)bmap;
+
+	set = parameter_compute_divs(bset);
+	map = (struct isl_map *)set;
+	map = replace_space_by_local_space(map, ls);
+
+	return map;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+int isl_basic_map_divs_known(__isl_keep isl_basic_map *bmap)
+{
+	int i;
+	unsigned off;
+
+	if (!bmap)
+		return -1;
+
+	off = isl_space_dim(bmap->dim, isl_dim_all);
+	for (i = 0; i < bmap->n_div; ++i) {
+		if (isl_int_is_zero(bmap->div[i][0]))
+			return 0;
+		isl_assert(bmap->ctx, isl_int_is_zero(bmap->div[i][1+1+off+i]),
+				return -1);
+	}
+	return 1;
+}
+
+static int map_divs_known(__isl_keep isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return -1;
+
+	for (i = 0; i < map->n; ++i) {
+		int known = isl_basic_map_divs_known(map->p[i]);
+		if (known <= 0)
+			return known;
+	}
+
+	return 1;
+}
+
+/* If bmap contains any unknown divs, then compute explicit
+ * expressions for them.  However, this computation may be
+ * quite expensive, so first try to remove divs that aren't
+ * strictly needed.
+ */
+struct isl_map *isl_basic_map_compute_divs(struct isl_basic_map *bmap)
+{
+	int known;
+	struct isl_map *map;
+
+	known = isl_basic_map_divs_known(bmap);
+	if (known < 0)
+		goto error;
+	if (known)
+		return isl_map_from_basic_map(bmap);
+
+	bmap = isl_basic_map_drop_redundant_divs(bmap);
+
+	known = isl_basic_map_divs_known(bmap);
+	if (known < 0)
+		goto error;
+	if (known)
+		return isl_map_from_basic_map(bmap);
+
+	map = compute_divs(bmap);
+	return map;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+struct isl_map *isl_map_compute_divs(struct isl_map *map)
+{
+	int i;
+	int known;
+	struct isl_map *res;
+
+	if (!map)
+		return NULL;
+	if (map->n == 0)
+		return map;
+
+	known = map_divs_known(map);
+	if (known < 0) {
+		isl_map_free(map);
+		return NULL;
+	}
+	if (known)
+		return map;
+
+	res = isl_basic_map_compute_divs(isl_basic_map_copy(map->p[0]));
+	for (i = 1 ; i < map->n; ++i) {
+		struct isl_map *r2;
+		r2 = isl_basic_map_compute_divs(isl_basic_map_copy(map->p[i]));
+		if (ISL_F_ISSET(map, ISL_MAP_DISJOINT))
+			res = isl_map_union_disjoint(res, r2);
+		else
+			res = isl_map_union(res, r2);
+	}
+	isl_map_free(map);
+
+	return res;
+}
+
+struct isl_set *isl_basic_set_compute_divs(struct isl_basic_set *bset)
+{
+	return (struct isl_set *)
+		isl_basic_map_compute_divs((struct isl_basic_map *)bset);
+}
+
+struct isl_set *isl_set_compute_divs(struct isl_set *set)
+{
+	return (struct isl_set *)
+		isl_map_compute_divs((struct isl_map *)set);
+}
+
+struct isl_set *isl_map_domain(struct isl_map *map)
+{
+	int i;
+	struct isl_set *set;
+
+	if (!map)
+		goto error;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	set = (struct isl_set *)map;
+	set->dim = isl_space_domain(set->dim);
+	if (!set->dim)
+		goto error;
+	for (i = 0; i < map->n; ++i) {
+		set->p[i] = isl_basic_map_domain(map->p[i]);
+		if (!set->p[i])
+			goto error;
+	}
+	ISL_F_CLR(set, ISL_MAP_DISJOINT);
+	ISL_F_CLR(set, ISL_SET_NORMALIZED);
+	return set;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Return the union of "map1" and "map2", where we assume for now that
+ * "map1" and "map2" are disjoint.  Note that the basic maps inside
+ * "map1" or "map2" may not be disjoint from each other.
+ * Also note that this function is also called from isl_map_union,
+ * which takes care of handling the situation where "map1" and "map2"
+ * may not be disjoint.
+ *
+ * If one of the inputs is empty, we can simply return the other input.
+ * Similarly, if one of the inputs is universal, then it is equal to the union.
+ */
+static __isl_give isl_map *map_union_disjoint(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	int i;
+	unsigned flags = 0;
+	struct isl_map *map = NULL;
+	int is_universe;
+
+	if (!map1 || !map2)
+		goto error;
+
+	if (!isl_space_is_equal(map1->dim, map2->dim))
+		isl_die(isl_map_get_ctx(map1), isl_error_invalid,
+			"spaces don't match", goto error);
+
+	if (map1->n == 0) {
+		isl_map_free(map1);
+		return map2;
+	}
+	if (map2->n == 0) {
+		isl_map_free(map2);
+		return map1;
+	}
+
+	is_universe = isl_map_plain_is_universe(map1);
+	if (is_universe < 0)
+		goto error;
+	if (is_universe) {
+		isl_map_free(map2);
+		return map1;
+	}
+
+	is_universe = isl_map_plain_is_universe(map2);
+	if (is_universe < 0)
+		goto error;
+	if (is_universe) {
+		isl_map_free(map1);
+		return map2;
+	}
+
+	if (ISL_F_ISSET(map1, ISL_MAP_DISJOINT) &&
+	    ISL_F_ISSET(map2, ISL_MAP_DISJOINT))
+		ISL_FL_SET(flags, ISL_MAP_DISJOINT);
+
+	map = isl_map_alloc_space(isl_space_copy(map1->dim),
+				map1->n + map2->n, flags);
+	if (!map)
+		goto error;
+	for (i = 0; i < map1->n; ++i) {
+		map = isl_map_add_basic_map(map,
+				  isl_basic_map_copy(map1->p[i]));
+		if (!map)
+			goto error;
+	}
+	for (i = 0; i < map2->n; ++i) {
+		map = isl_map_add_basic_map(map,
+				  isl_basic_map_copy(map2->p[i]));
+		if (!map)
+			goto error;
+	}
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return map;
+error:
+	isl_map_free(map);
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return NULL;
+}
+
+/* Return the union of "map1" and "map2", where "map1" and "map2" are
+ * guaranteed to be disjoint by the caller.
+ *
+ * Note that this functions is called from within isl_map_make_disjoint,
+ * so we have to be careful not to touch the constraints of the inputs
+ * in any way.
+ */
+__isl_give isl_map *isl_map_union_disjoint(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	return isl_map_align_params_map_map_and(map1, map2, &map_union_disjoint);
+}
+
+/* Return the union of "map1" and "map2", where "map1" and "map2" may
+ * not be disjoint.  The parameters are assumed to have been aligned.
+ *
+ * We currently simply call map_union_disjoint, the internal operation
+ * of which does not really depend on the inputs being disjoint.
+ * If the result contains more than one basic map, then we clear
+ * the disjoint flag since the result may contain basic maps from
+ * both inputs and these are not guaranteed to be disjoint.
+ *
+ * As a special case, if "map1" and "map2" are obviously equal,
+ * then we simply return "map1".
+ */
+static __isl_give isl_map *map_union_aligned(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	int equal;
+
+	if (!map1 || !map2)
+		goto error;
+
+	equal = isl_map_plain_is_equal(map1, map2);
+	if (equal < 0)
+		goto error;
+	if (equal) {
+		isl_map_free(map2);
+		return map1;
+	}
+
+	map1 = map_union_disjoint(map1, map2);
+	if (!map1)
+		return NULL;
+	if (map1->n > 1)
+		ISL_F_CLR(map1, ISL_MAP_DISJOINT);
+	return map1;
+error:
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return NULL;
+}
+
+/* Return the union of "map1" and "map2", where "map1" and "map2" may
+ * not be disjoint.
+ */
+__isl_give isl_map *isl_map_union(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	return isl_map_align_params_map_map_and(map1, map2, &map_union_aligned);
+}
+
+struct isl_set *isl_set_union_disjoint(
+			struct isl_set *set1, struct isl_set *set2)
+{
+	return (struct isl_set *)
+		isl_map_union_disjoint(
+			(struct isl_map *)set1, (struct isl_map *)set2);
+}
+
+struct isl_set *isl_set_union(struct isl_set *set1, struct isl_set *set2)
+{
+	return (struct isl_set *)
+		isl_map_union((struct isl_map *)set1, (struct isl_map *)set2);
+}
+
+/* Apply "fn" to pairs of elements from "map" and "set" and collect
+ * the results.
+ *
+ * "map" and "set" are assumed to be compatible and non-NULL.
+ */
+static __isl_give isl_map *map_intersect_set(__isl_take isl_map *map,
+	__isl_take isl_set *set,
+	__isl_give isl_basic_map *fn(__isl_take isl_basic_map *bmap,
+		__isl_take isl_basic_set *bset))
+{
+	unsigned flags = 0;
+	struct isl_map *result;
+	int i, j;
+
+	if (isl_set_plain_is_universe(set)) {
+		isl_set_free(set);
+		return map;
+	}
+
+	if (ISL_F_ISSET(map, ISL_MAP_DISJOINT) &&
+	    ISL_F_ISSET(set, ISL_MAP_DISJOINT))
+		ISL_FL_SET(flags, ISL_MAP_DISJOINT);
+
+	result = isl_map_alloc_space(isl_space_copy(map->dim),
+					map->n * set->n, flags);
+	for (i = 0; result && i < map->n; ++i)
+		for (j = 0; j < set->n; ++j) {
+			result = isl_map_add_basic_map(result,
+					fn(isl_basic_map_copy(map->p[i]),
+					    isl_basic_set_copy(set->p[j])));
+			if (!result)
+				break;
+		}
+
+	isl_map_free(map);
+	isl_set_free(set);
+	return result;
+}
+
+static __isl_give isl_map *map_intersect_range(__isl_take isl_map *map,
+	__isl_take isl_set *set)
+{
+	if (!map || !set)
+		goto error;
+
+	if (!isl_map_compatible_range(map, set))
+		isl_die(set->ctx, isl_error_invalid,
+			"incompatible spaces", goto error);
+
+	return map_intersect_set(map, set, &isl_basic_map_intersect_range);
+error:
+	isl_map_free(map);
+	isl_set_free(set);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_intersect_range(__isl_take isl_map *map,
+	__isl_take isl_set *set)
+{
+	return isl_map_align_params_map_map_and(map, set, &map_intersect_range);
+}
+
+static __isl_give isl_map *map_intersect_domain(__isl_take isl_map *map,
+	__isl_take isl_set *set)
+{
+	if (!map || !set)
+		goto error;
+
+	if (!isl_map_compatible_domain(map, set))
+		isl_die(set->ctx, isl_error_invalid,
+			"incompatible spaces", goto error);
+
+	return map_intersect_set(map, set, &isl_basic_map_intersect_domain);
+error:
+	isl_map_free(map);
+	isl_set_free(set);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_intersect_domain(__isl_take isl_map *map,
+	__isl_take isl_set *set)
+{
+	return isl_map_align_params_map_map_and(map, set,
+						&map_intersect_domain);
+}
+
+static __isl_give isl_map *map_apply_domain(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	if (!map1 || !map2)
+		goto error;
+	map1 = isl_map_reverse(map1);
+	map1 = isl_map_apply_range(map1, map2);
+	return isl_map_reverse(map1);
+error:
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_apply_domain(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	return isl_map_align_params_map_map_and(map1, map2, &map_apply_domain);
+}
+
+static __isl_give isl_map *map_apply_range(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	isl_space *dim_result;
+	struct isl_map *result;
+	int i, j;
+
+	if (!map1 || !map2)
+		goto error;
+
+	dim_result = isl_space_join(isl_space_copy(map1->dim),
+				  isl_space_copy(map2->dim));
+
+	result = isl_map_alloc_space(dim_result, map1->n * map2->n, 0);
+	if (!result)
+		goto error;
+	for (i = 0; i < map1->n; ++i)
+		for (j = 0; j < map2->n; ++j) {
+			result = isl_map_add_basic_map(result,
+			    isl_basic_map_apply_range(
+				isl_basic_map_copy(map1->p[i]),
+				isl_basic_map_copy(map2->p[j])));
+			if (!result)
+				goto error;
+		}
+	isl_map_free(map1);
+	isl_map_free(map2);
+	if (result && result->n <= 1)
+		ISL_F_SET(result, ISL_MAP_DISJOINT);
+	return result;
+error:
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_apply_range(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	return isl_map_align_params_map_map_and(map1, map2, &map_apply_range);
+}
+
+/*
+ * returns range - domain
+ */
+struct isl_basic_set *isl_basic_map_deltas(struct isl_basic_map *bmap)
+{
+	isl_space *dims, *target_dim;
+	struct isl_basic_set *bset;
+	unsigned dim;
+	unsigned nparam;
+	int i;
+
+	if (!bmap)
+		goto error;
+	isl_assert(bmap->ctx, isl_space_tuple_is_equal(bmap->dim, isl_dim_in,
+						  bmap->dim, isl_dim_out),
+		   goto error);
+	target_dim = isl_space_domain(isl_basic_map_get_space(bmap));
+	dim = isl_basic_map_n_in(bmap);
+	nparam = isl_basic_map_n_param(bmap);
+	bset = isl_basic_set_from_basic_map(bmap);
+	bset = isl_basic_set_cow(bset);
+	dims = isl_basic_set_get_space(bset);
+	dims = isl_space_add_dims(dims, isl_dim_set, dim);
+	bset = isl_basic_set_extend_space(bset, dims, 0, dim, 0);
+	bset = isl_basic_set_swap_vars(bset, 2*dim);
+	for (i = 0; i < dim; ++i) {
+		int j = isl_basic_map_alloc_equality(
+					    (struct isl_basic_map *)bset);
+		if (j < 0) {
+			bset = isl_basic_set_free(bset);
+			break;
+		}
+		isl_seq_clr(bset->eq[j], 1 + isl_basic_set_total_dim(bset));
+		isl_int_set_si(bset->eq[j][1+nparam+i], 1);
+		isl_int_set_si(bset->eq[j][1+nparam+dim+i], 1);
+		isl_int_set_si(bset->eq[j][1+nparam+2*dim+i], -1);
+	}
+	bset = isl_basic_set_project_out(bset, isl_dim_set, dim, 2*dim);
+	bset = isl_basic_set_reset_space(bset, target_dim);
+	return bset;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/*
+ * returns range - domain
+ */
+__isl_give isl_set *isl_map_deltas(__isl_take isl_map *map)
+{
+	int i;
+	isl_space *dim;
+	struct isl_set *result;
+
+	if (!map)
+		return NULL;
+
+	isl_assert(map->ctx, isl_space_tuple_is_equal(map->dim, isl_dim_in,
+						 map->dim, isl_dim_out),
+		   goto error);
+	dim = isl_map_get_space(map);
+	dim = isl_space_domain(dim);
+	result = isl_set_alloc_space(dim, map->n, 0);
+	if (!result)
+		goto error;
+	for (i = 0; i < map->n; ++i)
+		result = isl_set_add_basic_set(result,
+			  isl_basic_map_deltas(isl_basic_map_copy(map->p[i])));
+	isl_map_free(map);
+	return result;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/*
+ * returns [domain -> range] -> range - domain
+ */
+__isl_give isl_basic_map *isl_basic_map_deltas_map(
+	__isl_take isl_basic_map *bmap)
+{
+	int i, k;
+	isl_space *dim;
+	isl_basic_map *domain;
+	int nparam, n;
+	unsigned total;
+
+	if (!isl_space_tuple_is_equal(bmap->dim, isl_dim_in,
+					bmap->dim, isl_dim_out))
+		isl_die(bmap->ctx, isl_error_invalid,
+			"domain and range don't match", goto error);
+
+	nparam = isl_basic_map_dim(bmap, isl_dim_param);
+	n = isl_basic_map_dim(bmap, isl_dim_in);
+
+	dim = isl_space_from_range(isl_space_domain(isl_basic_map_get_space(bmap)));
+	domain = isl_basic_map_universe(dim);
+
+	bmap = isl_basic_map_from_domain(isl_basic_map_wrap(bmap));
+	bmap = isl_basic_map_apply_range(bmap, domain);
+	bmap = isl_basic_map_extend_constraints(bmap, n, 0);
+
+	total = isl_basic_map_total_dim(bmap);
+
+	for (i = 0; i < n; ++i) {
+		k = isl_basic_map_alloc_equality(bmap);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(bmap->eq[k], 1 + total);
+		isl_int_set_si(bmap->eq[k][1 + nparam + i], 1);
+		isl_int_set_si(bmap->eq[k][1 + nparam + n + i], -1);
+		isl_int_set_si(bmap->eq[k][1 + nparam + n + n + i], 1);
+	}
+
+	bmap = isl_basic_map_gauss(bmap, NULL);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/*
+ * returns [domain -> range] -> range - domain
+ */
+__isl_give isl_map *isl_map_deltas_map(__isl_take isl_map *map)
+{
+	int i;
+	isl_space *domain_dim;
+
+	if (!map)
+		return NULL;
+
+	if (!isl_space_tuple_is_equal(map->dim, isl_dim_in,
+					map->dim, isl_dim_out))
+		isl_die(map->ctx, isl_error_invalid,
+			"domain and range don't match", goto error);
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	domain_dim = isl_space_from_range(isl_space_domain(isl_map_get_space(map)));
+	map->dim = isl_space_from_domain(isl_space_wrap(map->dim));
+	map->dim = isl_space_join(map->dim, domain_dim);
+	if (!map->dim)
+		goto error;
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_deltas_map(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+static __isl_give isl_basic_map *basic_map_identity(__isl_take isl_space *dims)
+{
+	struct isl_basic_map *bmap;
+	unsigned nparam;
+	unsigned dim;
+	int i;
+
+	if (!dims)
+		return NULL;
+
+	nparam = dims->nparam;
+	dim = dims->n_out;
+	bmap = isl_basic_map_alloc_space(dims, 0, dim, 0);
+	if (!bmap)
+		goto error;
+
+	for (i = 0; i < dim; ++i) {
+		int j = isl_basic_map_alloc_equality(bmap);
+		if (j < 0)
+			goto error;
+		isl_seq_clr(bmap->eq[j], 1 + isl_basic_map_total_dim(bmap));
+		isl_int_set_si(bmap->eq[j][1+nparam+i], 1);
+		isl_int_set_si(bmap->eq[j][1+nparam+dim+i], -1);
+	}
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_identity(__isl_take isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+	if (dim->n_in != dim->n_out)
+		isl_die(dim->ctx, isl_error_invalid,
+			"number of input and output dimensions needs to be "
+			"the same", goto error);
+	return basic_map_identity(dim);
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+struct isl_basic_map *isl_basic_map_identity_like(struct isl_basic_map *model)
+{
+	if (!model || !model->dim)
+		return NULL;
+	return isl_basic_map_identity(isl_space_copy(model->dim));
+}
+
+__isl_give isl_map *isl_map_identity(__isl_take isl_space *dim)
+{
+	return isl_map_from_basic_map(isl_basic_map_identity(dim));
+}
+
+struct isl_map *isl_map_identity_like(struct isl_map *model)
+{
+	if (!model || !model->dim)
+		return NULL;
+	return isl_map_identity(isl_space_copy(model->dim));
+}
+
+struct isl_map *isl_map_identity_like_basic_map(struct isl_basic_map *model)
+{
+	if (!model || !model->dim)
+		return NULL;
+	return isl_map_identity(isl_space_copy(model->dim));
+}
+
+__isl_give isl_map *isl_set_identity(__isl_take isl_set *set)
+{
+	isl_space *dim = isl_set_get_space(set);
+	isl_map *id;
+	id = isl_map_identity(isl_space_map_from_set(dim));
+	return isl_map_intersect_range(id, set);
+}
+
+/* Construct a basic set with all set dimensions having only non-negative
+ * values.
+ */
+__isl_give isl_basic_set *isl_basic_set_positive_orthant(
+	__isl_take isl_space *space)
+{
+	int i;
+	unsigned nparam;
+	unsigned dim;
+	struct isl_basic_set *bset;
+
+	if (!space)
+		return NULL;
+	nparam = space->nparam;
+	dim = space->n_out;
+	bset = isl_basic_set_alloc_space(space, 0, 0, dim);
+	if (!bset)
+		return NULL;
+	for (i = 0; i < dim; ++i) {
+		int k = isl_basic_set_alloc_inequality(bset);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(bset->ineq[k], 1 + isl_basic_set_total_dim(bset));
+		isl_int_set_si(bset->ineq[k][1 + nparam + i], 1);
+	}
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Construct the half-space x_pos >= 0.
+ */
+static __isl_give isl_basic_set *nonneg_halfspace(__isl_take isl_space *dim,
+	int pos)
+{
+	int k;
+	isl_basic_set *nonneg;
+
+	nonneg = isl_basic_set_alloc_space(dim, 0, 0, 1);
+	k = isl_basic_set_alloc_inequality(nonneg);
+	if (k < 0)
+		goto error;
+	isl_seq_clr(nonneg->ineq[k], 1 + isl_basic_set_total_dim(nonneg));
+	isl_int_set_si(nonneg->ineq[k][pos], 1);
+
+	return isl_basic_set_finalize(nonneg);
+error:
+	isl_basic_set_free(nonneg);
+	return NULL;
+}
+
+/* Construct the half-space x_pos <= -1.
+ */
+static __isl_give isl_basic_set *neg_halfspace(__isl_take isl_space *dim, int pos)
+{
+	int k;
+	isl_basic_set *neg;
+
+	neg = isl_basic_set_alloc_space(dim, 0, 0, 1);
+	k = isl_basic_set_alloc_inequality(neg);
+	if (k < 0)
+		goto error;
+	isl_seq_clr(neg->ineq[k], 1 + isl_basic_set_total_dim(neg));
+	isl_int_set_si(neg->ineq[k][0], -1);
+	isl_int_set_si(neg->ineq[k][pos], -1);
+
+	return isl_basic_set_finalize(neg);
+error:
+	isl_basic_set_free(neg);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_split_dims(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+	isl_basic_set *nonneg;
+	isl_basic_set *neg;
+
+	if (!set)
+		return NULL;
+	if (n == 0)
+		return set;
+
+	isl_assert(set->ctx, first + n <= isl_set_dim(set, type), goto error);
+
+	for (i = 0; i < n; ++i) {
+		nonneg = nonneg_halfspace(isl_set_get_space(set),
+					  pos(set->dim, type) + first + i);
+		neg = neg_halfspace(isl_set_get_space(set),
+					  pos(set->dim, type) + first + i);
+
+		set = isl_set_intersect(set, isl_basic_set_union(nonneg, neg));
+	}
+
+	return set;
+error:
+	isl_set_free(set);
+	return NULL;
+}
+
+static int foreach_orthant(__isl_take isl_set *set, int *signs, int first,
+	int len, int (*fn)(__isl_take isl_set *orthant, int *signs, void *user),
+	void *user)
+{
+	isl_set *half;
+
+	if (!set)
+		return -1;
+	if (isl_set_plain_is_empty(set)) {
+		isl_set_free(set);
+		return 0;
+	}
+	if (first == len)
+		return fn(set, signs, user);
+
+	signs[first] = 1;
+	half = isl_set_from_basic_set(nonneg_halfspace(isl_set_get_space(set),
+							1 + first));
+	half = isl_set_intersect(half, isl_set_copy(set));
+	if (foreach_orthant(half, signs, first + 1, len, fn, user) < 0)
+		goto error;
+
+	signs[first] = -1;
+	half = isl_set_from_basic_set(neg_halfspace(isl_set_get_space(set),
+							1 + first));
+	half = isl_set_intersect(half, set);
+	return foreach_orthant(half, signs, first + 1, len, fn, user);
+error:
+	isl_set_free(set);
+	return -1;
+}
+
+/* Call "fn" on the intersections of "set" with each of the orthants
+ * (except for obviously empty intersections).  The orthant is identified
+ * by the signs array, with each entry having value 1 or -1 according
+ * to the sign of the corresponding variable.
+ */
+int isl_set_foreach_orthant(__isl_keep isl_set *set,
+	int (*fn)(__isl_take isl_set *orthant, int *signs, void *user),
+	void *user)
+{
+	unsigned nparam;
+	unsigned nvar;
+	int *signs;
+	int r;
+
+	if (!set)
+		return -1;
+	if (isl_set_plain_is_empty(set))
+		return 0;
+
+	nparam = isl_set_dim(set, isl_dim_param);
+	nvar = isl_set_dim(set, isl_dim_set);
+
+	signs = isl_alloc_array(set->ctx, int, nparam + nvar);
+
+	r = foreach_orthant(isl_set_copy(set), signs, 0, nparam + nvar,
+			    fn, user);
+
+	free(signs);
+
+	return r;
+}
+
+int isl_set_is_equal(struct isl_set *set1, struct isl_set *set2)
+{
+	return isl_map_is_equal((struct isl_map *)set1, (struct isl_map *)set2);
+}
+
+int isl_basic_map_is_subset(
+		struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
+{
+	int is_subset;
+	struct isl_map *map1;
+	struct isl_map *map2;
+
+	if (!bmap1 || !bmap2)
+		return -1;
+
+	map1 = isl_map_from_basic_map(isl_basic_map_copy(bmap1));
+	map2 = isl_map_from_basic_map(isl_basic_map_copy(bmap2));
+
+	is_subset = isl_map_is_subset(map1, map2);
+
+	isl_map_free(map1);
+	isl_map_free(map2);
+
+	return is_subset;
+}
+
+int isl_basic_set_is_subset(__isl_keep isl_basic_set *bset1,
+	__isl_keep isl_basic_set *bset2)
+{
+	return isl_basic_map_is_subset(bset1, bset2);
+}
+
+int isl_basic_map_is_equal(
+		struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
+{
+	int is_subset;
+
+	if (!bmap1 || !bmap2)
+		return -1;
+	is_subset = isl_basic_map_is_subset(bmap1, bmap2);
+	if (is_subset != 1)
+		return is_subset;
+	is_subset = isl_basic_map_is_subset(bmap2, bmap1);
+	return is_subset;
+}
+
+int isl_basic_set_is_equal(
+		struct isl_basic_set *bset1, struct isl_basic_set *bset2)
+{
+	return isl_basic_map_is_equal(
+		(struct isl_basic_map *)bset1, (struct isl_basic_map *)bset2);
+}
+
+int isl_map_is_empty(struct isl_map *map)
+{
+	int i;
+	int is_empty;
+
+	if (!map)
+		return -1;
+	for (i = 0; i < map->n; ++i) {
+		is_empty = isl_basic_map_is_empty(map->p[i]);
+		if (is_empty < 0)
+			return -1;
+		if (!is_empty)
+			return 0;
+	}
+	return 1;
+}
+
+int isl_map_plain_is_empty(__isl_keep isl_map *map)
+{
+	return map ? map->n == 0 : -1;
+}
+
+int isl_map_fast_is_empty(__isl_keep isl_map *map)
+{
+	return isl_map_plain_is_empty(map);
+}
+
+int isl_set_plain_is_empty(struct isl_set *set)
+{
+	return set ? set->n == 0 : -1;
+}
+
+int isl_set_fast_is_empty(__isl_keep isl_set *set)
+{
+	return isl_set_plain_is_empty(set);
+}
+
+int isl_set_is_empty(struct isl_set *set)
+{
+	return isl_map_is_empty((struct isl_map *)set);
+}
+
+int isl_map_has_equal_space(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
+{
+	if (!map1 || !map2)
+		return -1;
+
+	return isl_space_is_equal(map1->dim, map2->dim);
+}
+
+int isl_set_has_equal_space(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
+{
+	if (!set1 || !set2)
+		return -1;
+
+	return isl_space_is_equal(set1->dim, set2->dim);
+}
+
+static int map_is_equal(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
+{
+	int is_subset;
+
+	if (!map1 || !map2)
+		return -1;
+	is_subset = isl_map_is_subset(map1, map2);
+	if (is_subset != 1)
+		return is_subset;
+	is_subset = isl_map_is_subset(map2, map1);
+	return is_subset;
+}
+
+int isl_map_is_equal(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
+{
+	return isl_map_align_params_map_map_and_test(map1, map2, &map_is_equal);
+}
+
+int isl_basic_map_is_strict_subset(
+		struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
+{
+	int is_subset;
+
+	if (!bmap1 || !bmap2)
+		return -1;
+	is_subset = isl_basic_map_is_subset(bmap1, bmap2);
+	if (is_subset != 1)
+		return is_subset;
+	is_subset = isl_basic_map_is_subset(bmap2, bmap1);
+	if (is_subset == -1)
+		return is_subset;
+	return !is_subset;
+}
+
+int isl_map_is_strict_subset(struct isl_map *map1, struct isl_map *map2)
+{
+	int is_subset;
+
+	if (!map1 || !map2)
+		return -1;
+	is_subset = isl_map_is_subset(map1, map2);
+	if (is_subset != 1)
+		return is_subset;
+	is_subset = isl_map_is_subset(map2, map1);
+	if (is_subset == -1)
+		return is_subset;
+	return !is_subset;
+}
+
+int isl_set_is_strict_subset(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
+{
+	return isl_map_is_strict_subset((isl_map *)set1, (isl_map *)set2);
+}
+
+int isl_basic_map_is_universe(struct isl_basic_map *bmap)
+{
+	if (!bmap)
+		return -1;
+	return bmap->n_eq == 0 && bmap->n_ineq == 0;
+}
+
+int isl_basic_set_is_universe(struct isl_basic_set *bset)
+{
+	if (!bset)
+		return -1;
+	return bset->n_eq == 0 && bset->n_ineq == 0;
+}
+
+int isl_map_plain_is_universe(__isl_keep isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return -1;
+
+	for (i = 0; i < map->n; ++i) {
+		int r = isl_basic_map_is_universe(map->p[i]);
+		if (r < 0 || r)
+			return r;
+	}
+
+	return 0;
+}
+
+int isl_set_plain_is_universe(__isl_keep isl_set *set)
+{
+	return isl_map_plain_is_universe((isl_map *) set);
+}
+
+int isl_set_fast_is_universe(__isl_keep isl_set *set)
+{
+	return isl_set_plain_is_universe(set);
+}
+
+int isl_basic_map_is_empty(struct isl_basic_map *bmap)
+{
+	struct isl_basic_set *bset = NULL;
+	struct isl_vec *sample = NULL;
+	int empty;
+	unsigned total;
+
+	if (!bmap)
+		return -1;
+
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
+		return 1;
+
+	if (isl_basic_map_is_universe(bmap))
+		return 0;
+
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL)) {
+		struct isl_basic_map *copy = isl_basic_map_copy(bmap);
+		copy = isl_basic_map_remove_redundancies(copy);
+		empty = isl_basic_map_plain_is_empty(copy);
+		isl_basic_map_free(copy);
+		return empty;
+	}
+
+	total = 1 + isl_basic_map_total_dim(bmap);
+	if (bmap->sample && bmap->sample->size == total) {
+		int contains = isl_basic_map_contains(bmap, bmap->sample);
+		if (contains < 0)
+			return -1;
+		if (contains)
+			return 0;
+	}
+	isl_vec_free(bmap->sample);
+	bmap->sample = NULL;
+	bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
+	if (!bset)
+		return -1;
+	sample = isl_basic_set_sample_vec(bset);
+	if (!sample)
+		return -1;
+	empty = sample->size == 0;
+	isl_vec_free(bmap->sample);
+	bmap->sample = sample;
+	if (empty)
+		ISL_F_SET(bmap, ISL_BASIC_MAP_EMPTY);
+
+	return empty;
+}
+
+int isl_basic_map_plain_is_empty(__isl_keep isl_basic_map *bmap)
+{
+	if (!bmap)
+		return -1;
+	return ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY);
+}
+
+int isl_basic_map_fast_is_empty(__isl_keep isl_basic_map *bmap)
+{
+	return isl_basic_map_plain_is_empty(bmap);
+}
+
+int isl_basic_set_plain_is_empty(__isl_keep isl_basic_set *bset)
+{
+	if (!bset)
+		return -1;
+	return ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY);
+}
+
+int isl_basic_set_fast_is_empty(__isl_keep isl_basic_set *bset)
+{
+	return isl_basic_set_plain_is_empty(bset);
+}
+
+int isl_basic_set_is_empty(struct isl_basic_set *bset)
+{
+	return isl_basic_map_is_empty((struct isl_basic_map *)bset);
+}
+
+struct isl_map *isl_basic_map_union(
+	struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
+{
+	struct isl_map *map;
+	if (!bmap1 || !bmap2)
+		goto error;
+
+	isl_assert(bmap1->ctx, isl_space_is_equal(bmap1->dim, bmap2->dim), goto error);
+
+	map = isl_map_alloc_space(isl_space_copy(bmap1->dim), 2, 0);
+	if (!map)
+		goto error;
+	map = isl_map_add_basic_map(map, bmap1);
+	map = isl_map_add_basic_map(map, bmap2);
+	return map;
+error:
+	isl_basic_map_free(bmap1);
+	isl_basic_map_free(bmap2);
+	return NULL;
+}
+
+struct isl_set *isl_basic_set_union(
+		struct isl_basic_set *bset1, struct isl_basic_set *bset2)
+{
+	return (struct isl_set *)isl_basic_map_union(
+					    (struct isl_basic_map *)bset1,
+					    (struct isl_basic_map *)bset2);
+}
+
+/* Order divs such that any div only depends on previous divs */
+struct isl_basic_map *isl_basic_map_order_divs(struct isl_basic_map *bmap)
+{
+	int i;
+	unsigned off;
+
+	if (!bmap)
+		return NULL;
+
+	off = isl_space_dim(bmap->dim, isl_dim_all);
+
+	for (i = 0; i < bmap->n_div; ++i) {
+		int pos;
+		if (isl_int_is_zero(bmap->div[i][0]))
+			continue;
+		pos = isl_seq_first_non_zero(bmap->div[i]+1+1+off+i,
+							    bmap->n_div-i);
+		if (pos == -1)
+			continue;
+		isl_basic_map_swap_div(bmap, i, i + pos);
+		--i;
+	}
+	return bmap;
+}
+
+struct isl_basic_set *isl_basic_set_order_divs(struct isl_basic_set *bset)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_order_divs((struct isl_basic_map *)bset);
+}
+
+__isl_give isl_map *isl_map_order_divs(__isl_take isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return 0;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_order_divs(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Apply the expansion computed by isl_merge_divs.
+ * The expansion itself is given by "exp" while the resulting
+ * list of divs is given by "div".
+ */
+__isl_give isl_basic_set *isl_basic_set_expand_divs(
+	__isl_take isl_basic_set *bset, __isl_take isl_mat *div, int *exp)
+{
+	int i, j;
+	int n_div;
+
+	bset = isl_basic_set_cow(bset);
+	if (!bset || !div)
+		goto error;
+
+	if (div->n_row < bset->n_div)
+		isl_die(isl_mat_get_ctx(div), isl_error_invalid,
+			"not an expansion", goto error);
+
+	n_div = bset->n_div;
+	bset = isl_basic_map_extend_space(bset, isl_space_copy(bset->dim),
+					    div->n_row - n_div, 0,
+					    2 * (div->n_row - n_div));
+
+	for (i = n_div; i < div->n_row; ++i)
+		if (isl_basic_set_alloc_div(bset) < 0)
+			goto error;
+
+	j = n_div - 1;
+	for (i = div->n_row - 1; i >= 0; --i) {
+		if (j >= 0 && exp[j] == i) {
+			if (i != j)
+				isl_basic_map_swap_div(bset, i, j);
+			j--;
+		} else {
+			isl_seq_cpy(bset->div[i], div->row[i], div->n_col);
+			if (isl_basic_map_add_div_constraints(bset, i) < 0)
+				goto error;
+		}
+	}
+
+	isl_mat_free(div);
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	isl_mat_free(div);
+	return NULL;
+}
+
+/* Look for a div in dst that corresponds to the div "div" in src.
+ * The divs before "div" in src and dst are assumed to be the same.
+ * 
+ * Returns -1 if no corresponding div was found and the position
+ * of the corresponding div in dst otherwise.
+ */
+static int find_div(struct isl_basic_map *dst,
+			struct isl_basic_map *src, unsigned div)
+{
+	int i;
+
+	unsigned total = isl_space_dim(src->dim, isl_dim_all);
+
+	isl_assert(dst->ctx, div <= dst->n_div, return -1);
+	for (i = div; i < dst->n_div; ++i)
+		if (isl_seq_eq(dst->div[i], src->div[div], 1+1+total+div) &&
+		    isl_seq_first_non_zero(dst->div[i]+1+1+total+div,
+						dst->n_div - div) == -1)
+			return i;
+	return -1;
+}
+
+/* Align the divs of "dst" to those of "src", adding divs from "src"
+ * if needed.  That is, make sure that the first src->n_div divs
+ * of the result are equal to those of src.
+ *
+ * The result is not finalized as by design it will have redundant
+ * divs if any divs from "src" were copied.
+ */
+__isl_give isl_basic_map *isl_basic_map_align_divs(
+	__isl_take isl_basic_map *dst, __isl_keep isl_basic_map *src)
+{
+	int i;
+	int known, extended;
+	unsigned total;
+
+	if (!dst || !src)
+		return isl_basic_map_free(dst);
+
+	if (src->n_div == 0)
+		return dst;
+
+	known = isl_basic_map_divs_known(src);
+	if (known < 0)
+		return isl_basic_map_free(dst);
+	if (!known)
+		isl_die(isl_basic_map_get_ctx(src), isl_error_invalid,
+			"some src divs are unknown",
+			return isl_basic_map_free(dst));
+
+	src = isl_basic_map_order_divs(src);
+
+	extended = 0;
+	total = isl_space_dim(src->dim, isl_dim_all);
+	for (i = 0; i < src->n_div; ++i) {
+		int j = find_div(dst, src, i);
+		if (j < 0) {
+			if (!extended) {
+				int extra = src->n_div - i;
+				dst = isl_basic_map_cow(dst);
+				if (!dst)
+					return NULL;
+				dst = isl_basic_map_extend_space(dst,
+						isl_space_copy(dst->dim),
+						extra, 0, 2 * extra);
+				extended = 1;
+			}
+			j = isl_basic_map_alloc_div(dst);
+			if (j < 0)
+				return isl_basic_map_free(dst);
+			isl_seq_cpy(dst->div[j], src->div[i], 1+1+total+i);
+			isl_seq_clr(dst->div[j]+1+1+total+i, dst->n_div - i);
+			if (isl_basic_map_add_div_constraints(dst, j) < 0)
+				return isl_basic_map_free(dst);
+		}
+		if (j != i)
+			isl_basic_map_swap_div(dst, i, j);
+	}
+	return dst;
+}
+
+struct isl_basic_set *isl_basic_set_align_divs(
+		struct isl_basic_set *dst, struct isl_basic_set *src)
+{
+	return (struct isl_basic_set *)isl_basic_map_align_divs(
+		(struct isl_basic_map *)dst, (struct isl_basic_map *)src);
+}
+
+struct isl_map *isl_map_align_divs(struct isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+	if (map->n == 0)
+		return map;
+	map = isl_map_compute_divs(map);
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	for (i = 1; i < map->n; ++i)
+		map->p[0] = isl_basic_map_align_divs(map->p[0], map->p[i]);
+	for (i = 1; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_align_divs(map->p[i], map->p[0]);
+		if (!map->p[i])
+			return isl_map_free(map);
+	}
+
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+	return map;
+}
+
+struct isl_set *isl_set_align_divs(struct isl_set *set)
+{
+	return (struct isl_set *)isl_map_align_divs((struct isl_map *)set);
+}
+
+/* Align the divs of the basic sets in "set" to those
+ * of the basic sets in "list", as well as to the other basic sets in "set".
+ * The elements in "list" are assumed to have known divs.
+ */
+__isl_give isl_set *isl_set_align_divs_to_basic_set_list(
+	__isl_take isl_set *set, __isl_keep isl_basic_set_list *list)
+{
+	int i, n;
+
+	set = isl_set_compute_divs(set);
+	set = isl_set_cow(set);
+	if (!set || !list)
+		return isl_set_free(set);
+	if (set->n == 0)
+		return set;
+
+	n = isl_basic_set_list_n_basic_set(list);
+	for (i = 0; i < n; ++i) {
+		isl_basic_set *bset;
+
+		bset = isl_basic_set_list_get_basic_set(list, i);
+		set->p[0] = isl_basic_set_align_divs(set->p[0], bset);
+		isl_basic_set_free(bset);
+	}
+	if (!set->p[0])
+		return isl_set_free(set);
+
+	return isl_set_align_divs(set);
+}
+
+/* Align the divs of each element of "list" to those of "bset".
+ * Both "bset" and the elements of "list" are assumed to have known divs.
+ */
+__isl_give isl_basic_set_list *isl_basic_set_list_align_divs_to_basic_set(
+	__isl_take isl_basic_set_list *list, __isl_keep isl_basic_set *bset)
+{
+	int i, n;
+
+	if (!list || !bset)
+		return isl_basic_set_list_free(list);
+
+	n = isl_basic_set_list_n_basic_set(list);
+	for (i = 0; i < n; ++i) {
+		isl_basic_set *bset_i;
+
+		bset_i = isl_basic_set_list_get_basic_set(list, i);
+		bset_i = isl_basic_set_align_divs(bset_i, bset);
+		list = isl_basic_set_list_set_basic_set(list, i, bset_i);
+	}
+
+	return list;
+}
+
+static __isl_give isl_set *set_apply( __isl_take isl_set *set,
+	__isl_take isl_map *map)
+{
+	if (!set || !map)
+		goto error;
+	isl_assert(set->ctx, isl_map_compatible_domain(map, set), goto error);
+	map = isl_map_intersect_domain(map, set);
+	set = isl_map_range(map);
+	return set;
+error:
+	isl_set_free(set);
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_apply( __isl_take isl_set *set,
+	__isl_take isl_map *map)
+{
+	return isl_map_align_params_map_map_and(set, map, &set_apply);
+}
+
+/* There is no need to cow as removing empty parts doesn't change
+ * the meaning of the set.
+ */
+struct isl_map *isl_map_remove_empty_parts(struct isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+
+	for (i = map->n - 1; i >= 0; --i)
+		remove_if_empty(map, i);
+
+	return map;
+}
+
+struct isl_set *isl_set_remove_empty_parts(struct isl_set *set)
+{
+	return (struct isl_set *)
+		isl_map_remove_empty_parts((struct isl_map *)set);
+}
+
+struct isl_basic_map *isl_map_copy_basic_map(struct isl_map *map)
+{
+	struct isl_basic_map *bmap;
+	if (!map || map->n == 0)
+		return NULL;
+	bmap = map->p[map->n-1];
+	isl_assert(map->ctx, ISL_F_ISSET(bmap, ISL_BASIC_SET_FINAL), return NULL);
+	return isl_basic_map_copy(bmap);
+}
+
+struct isl_basic_set *isl_set_copy_basic_set(struct isl_set *set)
+{
+	return (struct isl_basic_set *)
+		isl_map_copy_basic_map((struct isl_map *)set);
+}
+
+__isl_give isl_map *isl_map_drop_basic_map(__isl_take isl_map *map,
+						__isl_keep isl_basic_map *bmap)
+{
+	int i;
+
+	if (!map || !bmap)
+		goto error;
+	for (i = map->n-1; i >= 0; --i) {
+		if (map->p[i] != bmap)
+			continue;
+		map = isl_map_cow(map);
+		if (!map)
+			goto error;
+		isl_basic_map_free(map->p[i]);
+		if (i != map->n-1) {
+			ISL_F_CLR(map, ISL_SET_NORMALIZED);
+			map->p[i] = map->p[map->n-1];
+		}
+		map->n--;
+		return map;
+	}
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+struct isl_set *isl_set_drop_basic_set(struct isl_set *set,
+						struct isl_basic_set *bset)
+{
+	return (struct isl_set *)isl_map_drop_basic_map((struct isl_map *)set,
+						(struct isl_basic_map *)bset);
+}
+
+/* Given two basic sets bset1 and bset2, compute the maximal difference
+ * between the values of dimension pos in bset1 and those in bset2
+ * for any common value of the parameters and dimensions preceding pos.
+ */
+static enum isl_lp_result basic_set_maximal_difference_at(
+	__isl_keep isl_basic_set *bset1, __isl_keep isl_basic_set *bset2,
+	int pos, isl_int *opt)
+{
+	isl_space *dims;
+	struct isl_basic_map *bmap1 = NULL;
+	struct isl_basic_map *bmap2 = NULL;
+	struct isl_ctx *ctx;
+	struct isl_vec *obj;
+	unsigned total;
+	unsigned nparam;
+	unsigned dim1, dim2;
+	enum isl_lp_result res;
+
+	if (!bset1 || !bset2)
+		return isl_lp_error;
+
+	nparam = isl_basic_set_n_param(bset1);
+	dim1 = isl_basic_set_n_dim(bset1);
+	dim2 = isl_basic_set_n_dim(bset2);
+	dims = isl_space_alloc(bset1->ctx, nparam, pos, dim1 - pos);
+	bmap1 = isl_basic_map_from_basic_set(isl_basic_set_copy(bset1), dims);
+	dims = isl_space_alloc(bset2->ctx, nparam, pos, dim2 - pos);
+	bmap2 = isl_basic_map_from_basic_set(isl_basic_set_copy(bset2), dims);
+	if (!bmap1 || !bmap2)
+		goto error;
+	bmap1 = isl_basic_map_cow(bmap1);
+	bmap1 = isl_basic_map_extend(bmap1, nparam,
+			pos, (dim1 - pos) + (dim2 - pos),
+			bmap2->n_div, bmap2->n_eq, bmap2->n_ineq);
+	bmap1 = add_constraints(bmap1, bmap2, 0, dim1 - pos);
+	if (!bmap1)
+		goto error2;
+	total = isl_basic_map_total_dim(bmap1);
+	ctx = bmap1->ctx;
+	obj = isl_vec_alloc(ctx, 1 + total);
+	if (!obj)
+		goto error2;
+	isl_seq_clr(obj->block.data, 1 + total);
+	isl_int_set_si(obj->block.data[1+nparam+pos], 1);
+	isl_int_set_si(obj->block.data[1+nparam+pos+(dim1-pos)], -1);
+	res = isl_basic_map_solve_lp(bmap1, 1, obj->block.data, ctx->one,
+					opt, NULL, NULL);
+	isl_basic_map_free(bmap1);
+	isl_vec_free(obj);
+	return res;
+error:
+	isl_basic_map_free(bmap2);
+error2:
+	isl_basic_map_free(bmap1);
+	return isl_lp_error;
+}
+
+/* Given two _disjoint_ basic sets bset1 and bset2, check whether
+ * for any common value of the parameters and dimensions preceding pos
+ * in both basic sets, the values of dimension pos in bset1 are
+ * smaller or larger than those in bset2.
+ *
+ * Returns
+ *	 1 if bset1 follows bset2
+ *	-1 if bset1 precedes bset2
+ *	 0 if bset1 and bset2 are incomparable
+ *	-2 if some error occurred.
+ */
+int isl_basic_set_compare_at(struct isl_basic_set *bset1,
+	struct isl_basic_set *bset2, int pos)
+{
+	isl_int opt;
+	enum isl_lp_result res;
+	int cmp;
+
+	isl_int_init(opt);
+
+	res = basic_set_maximal_difference_at(bset1, bset2, pos, &opt);
+
+	if (res == isl_lp_empty)
+		cmp = 0;
+	else if ((res == isl_lp_ok && isl_int_is_pos(opt)) ||
+		  res == isl_lp_unbounded)
+		cmp = 1;
+	else if (res == isl_lp_ok && isl_int_is_neg(opt))
+		cmp = -1;
+	else
+		cmp = -2;
+
+	isl_int_clear(opt);
+	return cmp;
+}
+
+/* Given two basic sets bset1 and bset2, check whether
+ * for any common value of the parameters and dimensions preceding pos
+ * there is a value of dimension pos in bset1 that is larger
+ * than a value of the same dimension in bset2.
+ *
+ * Return
+ *	 1 if there exists such a pair
+ *	 0 if there is no such pair, but there is a pair of equal values
+ *	-1 otherwise
+ *	-2 if some error occurred.
+ */
+int isl_basic_set_follows_at(__isl_keep isl_basic_set *bset1,
+	__isl_keep isl_basic_set *bset2, int pos)
+{
+	isl_int opt;
+	enum isl_lp_result res;
+	int cmp;
+
+	isl_int_init(opt);
+
+	res = basic_set_maximal_difference_at(bset1, bset2, pos, &opt);
+
+	if (res == isl_lp_empty)
+		cmp = -1;
+	else if ((res == isl_lp_ok && isl_int_is_pos(opt)) ||
+		  res == isl_lp_unbounded)
+		cmp = 1;
+	else if (res == isl_lp_ok && isl_int_is_neg(opt))
+		cmp = -1;
+	else if (res == isl_lp_ok)
+		cmp = 0;
+	else
+		cmp = -2;
+
+	isl_int_clear(opt);
+	return cmp;
+}
+
+/* Given two sets set1 and set2, check whether
+ * for any common value of the parameters and dimensions preceding pos
+ * there is a value of dimension pos in set1 that is larger
+ * than a value of the same dimension in set2.
+ *
+ * Return
+ *	 1 if there exists such a pair
+ *	 0 if there is no such pair, but there is a pair of equal values
+ *	-1 otherwise
+ *	-2 if some error occurred.
+ */
+int isl_set_follows_at(__isl_keep isl_set *set1,
+	__isl_keep isl_set *set2, int pos)
+{
+	int i, j;
+	int follows = -1;
+
+	if (!set1 || !set2)
+		return -2;
+
+	for (i = 0; i < set1->n; ++i)
+		for (j = 0; j < set2->n; ++j) {
+			int f;
+			f = isl_basic_set_follows_at(set1->p[i], set2->p[j], pos);
+			if (f == 1 || f == -2)
+				return f;
+			if (f > follows)
+				follows = f;
+		}
+
+	return follows;
+}
+
+static int isl_basic_map_plain_has_fixed_var(__isl_keep isl_basic_map *bmap,
+	unsigned pos, isl_int *val)
+{
+	int i;
+	int d;
+	unsigned total;
+
+	if (!bmap)
+		return -1;
+	total = isl_basic_map_total_dim(bmap);
+	for (i = 0, d = total-1; i < bmap->n_eq && d+1 > pos; ++i) {
+		for (; d+1 > pos; --d)
+			if (!isl_int_is_zero(bmap->eq[i][1+d]))
+				break;
+		if (d != pos)
+			continue;
+		if (isl_seq_first_non_zero(bmap->eq[i]+1, d) != -1)
+			return 0;
+		if (isl_seq_first_non_zero(bmap->eq[i]+1+d+1, total-d-1) != -1)
+			return 0;
+		if (!isl_int_is_one(bmap->eq[i][1+d]))
+			return 0;
+		if (val)
+			isl_int_neg(*val, bmap->eq[i][0]);
+		return 1;
+	}
+	return 0;
+}
+
+static int isl_map_plain_has_fixed_var(__isl_keep isl_map *map,
+	unsigned pos, isl_int *val)
+{
+	int i;
+	isl_int v;
+	isl_int tmp;
+	int fixed;
+
+	if (!map)
+		return -1;
+	if (map->n == 0)
+		return 0;
+	if (map->n == 1)
+		return isl_basic_map_plain_has_fixed_var(map->p[0], pos, val); 
+	isl_int_init(v);
+	isl_int_init(tmp);
+	fixed = isl_basic_map_plain_has_fixed_var(map->p[0], pos, &v); 
+	for (i = 1; fixed == 1 && i < map->n; ++i) {
+		fixed = isl_basic_map_plain_has_fixed_var(map->p[i], pos, &tmp); 
+		if (fixed == 1 && isl_int_ne(tmp, v))
+			fixed = 0;
+	}
+	if (val)
+		isl_int_set(*val, v);
+	isl_int_clear(tmp);
+	isl_int_clear(v);
+	return fixed;
+}
+
+static int isl_basic_set_plain_has_fixed_var(__isl_keep isl_basic_set *bset,
+	unsigned pos, isl_int *val)
+{
+	return isl_basic_map_plain_has_fixed_var((struct isl_basic_map *)bset,
+						pos, val);
+}
+
+static int isl_set_plain_has_fixed_var(__isl_keep isl_set *set, unsigned pos,
+	isl_int *val)
+{
+	return isl_map_plain_has_fixed_var((struct isl_map *)set, pos, val);
+}
+
+int isl_basic_map_plain_is_fixed(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos, isl_int *val)
+{
+	if (pos >= isl_basic_map_dim(bmap, type))
+		return -1;
+	return isl_basic_map_plain_has_fixed_var(bmap,
+		isl_basic_map_offset(bmap, type) - 1 + pos, val);
+}
+
+/* If "bmap" obviously lies on a hyperplane where the given dimension
+ * has a fixed value, then return that value.
+ * Otherwise return NaN.
+ */
+__isl_give isl_val *isl_basic_map_plain_get_val_if_fixed(
+	__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos)
+{
+	isl_ctx *ctx;
+	isl_val *v;
+	int fixed;
+
+	if (!bmap)
+		return NULL;
+	ctx = isl_basic_map_get_ctx(bmap);
+	v = isl_val_alloc(ctx);
+	if (!v)
+		return NULL;
+	fixed = isl_basic_map_plain_is_fixed(bmap, type, pos, &v->n);
+	if (fixed < 0)
+		return isl_val_free(v);
+	if (fixed) {
+		isl_int_set_si(v->d, 1);
+		return v;
+	}
+	isl_val_free(v);
+	return isl_val_nan(ctx);
+}
+
+int isl_map_plain_is_fixed(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos, isl_int *val)
+{
+	if (pos >= isl_map_dim(map, type))
+		return -1;
+	return isl_map_plain_has_fixed_var(map,
+		map_offset(map, type) - 1 + pos, val);
+}
+
+/* If "map" obviously lies on a hyperplane where the given dimension
+ * has a fixed value, then return that value.
+ * Otherwise return NaN.
+ */
+__isl_give isl_val *isl_map_plain_get_val_if_fixed(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos)
+{
+	isl_ctx *ctx;
+	isl_val *v;
+	int fixed;
+
+	if (!map)
+		return NULL;
+	ctx = isl_map_get_ctx(map);
+	v = isl_val_alloc(ctx);
+	if (!v)
+		return NULL;
+	fixed = isl_map_plain_is_fixed(map, type, pos, &v->n);
+	if (fixed < 0)
+		return isl_val_free(v);
+	if (fixed) {
+		isl_int_set_si(v->d, 1);
+		return v;
+	}
+	isl_val_free(v);
+	return isl_val_nan(ctx);
+}
+
+/* If "set" obviously lies on a hyperplane where the given dimension
+ * has a fixed value, then return that value.
+ * Otherwise return NaN.
+ */
+__isl_give isl_val *isl_set_plain_get_val_if_fixed(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos)
+{
+	return isl_map_plain_get_val_if_fixed(set, type, pos);
+}
+
+int isl_set_plain_is_fixed(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos, isl_int *val)
+{
+	return isl_map_plain_is_fixed(set, type, pos, val);
+}
+
+int isl_map_fast_is_fixed(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos, isl_int *val)
+{
+	return isl_map_plain_is_fixed(map, type, pos, val);
+}
+
+/* Check if dimension dim has fixed value and if so and if val is not NULL,
+ * then return this fixed value in *val.
+ */
+int isl_basic_set_plain_dim_is_fixed(__isl_keep isl_basic_set *bset,
+	unsigned dim, isl_int *val)
+{
+	return isl_basic_set_plain_has_fixed_var(bset,
+					isl_basic_set_n_param(bset) + dim, val);
+}
+
+/* Check if dimension dim has fixed value and if so and if val is not NULL,
+ * then return this fixed value in *val.
+ */
+int isl_set_plain_dim_is_fixed(__isl_keep isl_set *set,
+	unsigned dim, isl_int *val)
+{
+	return isl_set_plain_has_fixed_var(set, isl_set_n_param(set) + dim, val);
+}
+
+int isl_set_fast_dim_is_fixed(__isl_keep isl_set *set,
+	unsigned dim, isl_int *val)
+{
+	return isl_set_plain_dim_is_fixed(set, dim, val);
+}
+
+/* Check if input variable in has fixed value and if so and if val is not NULL,
+ * then return this fixed value in *val.
+ */
+int isl_map_plain_input_is_fixed(__isl_keep isl_map *map,
+	unsigned in, isl_int *val)
+{
+	return isl_map_plain_has_fixed_var(map, isl_map_n_param(map) + in, val);
+}
+
+/* Check if dimension dim has an (obvious) fixed lower bound and if so
+ * and if val is not NULL, then return this lower bound in *val.
+ */
+int isl_basic_set_plain_dim_has_fixed_lower_bound(
+	__isl_keep isl_basic_set *bset, unsigned dim, isl_int *val)
+{
+	int i, i_eq = -1, i_ineq = -1;
+	isl_int *c;
+	unsigned total;
+	unsigned nparam;
+
+	if (!bset)
+		return -1;
+	total = isl_basic_set_total_dim(bset);
+	nparam = isl_basic_set_n_param(bset);
+	for (i = 0; i < bset->n_eq; ++i) {
+		if (isl_int_is_zero(bset->eq[i][1+nparam+dim]))
+			continue;
+		if (i_eq != -1)
+			return 0;
+		i_eq = i;
+	}
+	for (i = 0; i < bset->n_ineq; ++i) {
+		if (!isl_int_is_pos(bset->ineq[i][1+nparam+dim]))
+			continue;
+		if (i_eq != -1 || i_ineq != -1)
+			return 0;
+		i_ineq = i;
+	}
+	if (i_eq == -1 && i_ineq == -1)
+		return 0;
+	c = i_eq != -1 ? bset->eq[i_eq] : bset->ineq[i_ineq];
+	/* The coefficient should always be one due to normalization. */
+	if (!isl_int_is_one(c[1+nparam+dim]))
+		return 0;
+	if (isl_seq_first_non_zero(c+1, nparam+dim) != -1)
+		return 0;
+	if (isl_seq_first_non_zero(c+1+nparam+dim+1,
+					total - nparam - dim - 1) != -1)
+		return 0;
+	if (val)
+		isl_int_neg(*val, c[0]);
+	return 1;
+}
+
+int isl_set_plain_dim_has_fixed_lower_bound(__isl_keep isl_set *set,
+	unsigned dim, isl_int *val)
+{
+	int i;
+	isl_int v;
+	isl_int tmp;
+	int fixed;
+
+	if (!set)
+		return -1;
+	if (set->n == 0)
+		return 0;
+	if (set->n == 1)
+		return isl_basic_set_plain_dim_has_fixed_lower_bound(set->p[0],
+								dim, val);
+	isl_int_init(v);
+	isl_int_init(tmp);
+	fixed = isl_basic_set_plain_dim_has_fixed_lower_bound(set->p[0],
+								dim, &v);
+	for (i = 1; fixed == 1 && i < set->n; ++i) {
+		fixed = isl_basic_set_plain_dim_has_fixed_lower_bound(set->p[i],
+								dim, &tmp);
+		if (fixed == 1 && isl_int_ne(tmp, v))
+			fixed = 0;
+	}
+	if (val)
+		isl_int_set(*val, v);
+	isl_int_clear(tmp);
+	isl_int_clear(v);
+	return fixed;
+}
+
+/* uset_gist depends on constraints without existentially quantified
+ * variables sorting first.
+ */
+static int sort_constraint_cmp(const void *p1, const void *p2, void *arg)
+{
+	isl_int **c1 = (isl_int **) p1;
+	isl_int **c2 = (isl_int **) p2;
+	int l1, l2;
+	unsigned size = *(unsigned *) arg;
+
+	l1 = isl_seq_last_non_zero(*c1 + 1, size);
+	l2 = isl_seq_last_non_zero(*c2 + 1, size);
+
+	if (l1 != l2)
+		return l1 - l2;
+
+	return isl_seq_cmp(*c1 + 1, *c2 + 1, size);
+}
+
+static struct isl_basic_map *isl_basic_map_sort_constraints(
+	struct isl_basic_map *bmap)
+{
+	unsigned total;
+
+	if (!bmap)
+		return NULL;
+	if (bmap->n_ineq == 0)
+		return bmap;
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NORMALIZED))
+		return bmap;
+	total = isl_basic_map_total_dim(bmap);
+	if (isl_sort(bmap->ineq, bmap->n_ineq, sizeof(isl_int *),
+		    &sort_constraint_cmp, &total) < 0)
+		return isl_basic_map_free(bmap);
+	return bmap;
+}
+
+__isl_give isl_basic_set *isl_basic_set_sort_constraints(
+	__isl_take isl_basic_set *bset)
+{
+	return (struct isl_basic_set *)isl_basic_map_sort_constraints(
+						(struct isl_basic_map *)bset);
+}
+
+struct isl_basic_map *isl_basic_map_normalize(struct isl_basic_map *bmap)
+{
+	if (!bmap)
+		return NULL;
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NORMALIZED))
+		return bmap;
+	bmap = isl_basic_map_remove_redundancies(bmap);
+	bmap = isl_basic_map_sort_constraints(bmap);
+	if (bmap)
+		ISL_F_SET(bmap, ISL_BASIC_MAP_NORMALIZED);
+	return bmap;
+}
+
+struct isl_basic_set *isl_basic_set_normalize(struct isl_basic_set *bset)
+{
+	return (struct isl_basic_set *)isl_basic_map_normalize(
+						(struct isl_basic_map *)bset);
+}
+
+int isl_basic_map_plain_cmp(const __isl_keep isl_basic_map *bmap1,
+	const __isl_keep isl_basic_map *bmap2)
+{
+	int i, cmp;
+	unsigned total;
+
+	if (bmap1 == bmap2)
+		return 0;
+	if (ISL_F_ISSET(bmap1, ISL_BASIC_MAP_RATIONAL) !=
+	    ISL_F_ISSET(bmap2, ISL_BASIC_MAP_RATIONAL))
+		return ISL_F_ISSET(bmap1, ISL_BASIC_MAP_RATIONAL) ? -1 : 1;
+	if (isl_basic_map_n_param(bmap1) != isl_basic_map_n_param(bmap2))
+		return isl_basic_map_n_param(bmap1) - isl_basic_map_n_param(bmap2);
+	if (isl_basic_map_n_in(bmap1) != isl_basic_map_n_in(bmap2))
+		return isl_basic_map_n_out(bmap1) - isl_basic_map_n_out(bmap2);
+	if (isl_basic_map_n_out(bmap1) != isl_basic_map_n_out(bmap2))
+		return isl_basic_map_n_out(bmap1) - isl_basic_map_n_out(bmap2);
+	if (ISL_F_ISSET(bmap1, ISL_BASIC_MAP_EMPTY) &&
+	    ISL_F_ISSET(bmap2, ISL_BASIC_MAP_EMPTY))
+		return 0;
+	if (ISL_F_ISSET(bmap1, ISL_BASIC_MAP_EMPTY))
+		return 1;
+	if (ISL_F_ISSET(bmap2, ISL_BASIC_MAP_EMPTY))
+		return -1;
+	if (bmap1->n_eq != bmap2->n_eq)
+		return bmap1->n_eq - bmap2->n_eq;
+	if (bmap1->n_ineq != bmap2->n_ineq)
+		return bmap1->n_ineq - bmap2->n_ineq;
+	if (bmap1->n_div != bmap2->n_div)
+		return bmap1->n_div - bmap2->n_div;
+	total = isl_basic_map_total_dim(bmap1);
+	for (i = 0; i < bmap1->n_eq; ++i) {
+		cmp = isl_seq_cmp(bmap1->eq[i], bmap2->eq[i], 1+total);
+		if (cmp)
+			return cmp;
+	}
+	for (i = 0; i < bmap1->n_ineq; ++i) {
+		cmp = isl_seq_cmp(bmap1->ineq[i], bmap2->ineq[i], 1+total);
+		if (cmp)
+			return cmp;
+	}
+	for (i = 0; i < bmap1->n_div; ++i) {
+		cmp = isl_seq_cmp(bmap1->div[i], bmap2->div[i], 1+1+total);
+		if (cmp)
+			return cmp;
+	}
+	return 0;
+}
+
+int isl_basic_set_plain_cmp(const __isl_keep isl_basic_set *bset1,
+	const __isl_keep isl_basic_set *bset2)
+{
+	return isl_basic_map_plain_cmp(bset1, bset2);
+}
+
+int isl_set_plain_cmp(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
+{
+	int i, cmp;
+
+	if (set1 == set2)
+		return 0;
+	if (set1->n != set2->n)
+		return set1->n - set2->n;
+
+	for (i = 0; i < set1->n; ++i) {
+		cmp = isl_basic_set_plain_cmp(set1->p[i], set2->p[i]);
+		if (cmp)
+			return cmp;
+	}
+
+	return 0;
+}
+
+int isl_basic_map_plain_is_equal(__isl_keep isl_basic_map *bmap1,
+	__isl_keep isl_basic_map *bmap2)
+{
+	return isl_basic_map_plain_cmp(bmap1, bmap2) == 0;
+}
+
+int isl_basic_set_plain_is_equal(__isl_keep isl_basic_set *bset1,
+	__isl_keep isl_basic_set *bset2)
+{
+	return isl_basic_map_plain_is_equal((isl_basic_map *)bset1,
+					    (isl_basic_map *)bset2);
+}
+
+static int qsort_bmap_cmp(const void *p1, const void *p2)
+{
+	const struct isl_basic_map *bmap1 = *(const struct isl_basic_map **)p1;
+	const struct isl_basic_map *bmap2 = *(const struct isl_basic_map **)p2;
+
+	return isl_basic_map_plain_cmp(bmap1, bmap2);
+}
+
+/* Sort the basic maps of "map" and remove duplicate basic maps.
+ *
+ * While removing basic maps, we make sure that the basic maps remain
+ * sorted because isl_map_normalize expects the basic maps of the result
+ * to be sorted.
+ */
+static __isl_give isl_map *sort_and_remove_duplicates(__isl_take isl_map *map)
+{
+	int i, j;
+
+	map = isl_map_remove_empty_parts(map);
+	if (!map)
+		return NULL;
+	qsort(map->p, map->n, sizeof(struct isl_basic_map *), qsort_bmap_cmp);
+	for (i = map->n - 1; i >= 1; --i) {
+		if (!isl_basic_map_plain_is_equal(map->p[i - 1], map->p[i]))
+			continue;
+		isl_basic_map_free(map->p[i-1]);
+		for (j = i; j < map->n; ++j)
+			map->p[j - 1] = map->p[j];
+		map->n--;
+	}
+
+	return map;
+}
+
+/* Remove obvious duplicates among the basic maps of "map".
+ *
+ * Unlike isl_map_normalize, this function does not remove redundant
+ * constraints and only removes duplicates that have exactly the same
+ * constraints in the input.  It does sort the constraints and
+ * the basic maps to ease the detection of duplicates.
+ *
+ * If "map" has already been normalized or if the basic maps are
+ * disjoint, then there can be no duplicates.
+ */
+__isl_give isl_map *isl_map_remove_obvious_duplicates(__isl_take isl_map *map)
+{
+	int i;
+	isl_basic_map *bmap;
+
+	if (!map)
+		return NULL;
+	if (map->n <= 1)
+		return map;
+	if (ISL_F_ISSET(map, ISL_MAP_NORMALIZED | ISL_MAP_DISJOINT))
+		return map;
+	for (i = 0; i < map->n; ++i) {
+		bmap = isl_basic_map_copy(map->p[i]);
+		bmap = isl_basic_map_sort_constraints(bmap);
+		if (!bmap)
+			return isl_map_free(map);
+		isl_basic_map_free(map->p[i]);
+		map->p[i] = bmap;
+	}
+
+	map = sort_and_remove_duplicates(map);
+	return map;
+}
+
+/* We normalize in place, but if anything goes wrong we need
+ * to return NULL, so we need to make sure we don't change the
+ * meaning of any possible other copies of map.
+ */
+__isl_give isl_map *isl_map_normalize(__isl_take isl_map *map)
+{
+	int i;
+	struct isl_basic_map *bmap;
+
+	if (!map)
+		return NULL;
+	if (ISL_F_ISSET(map, ISL_MAP_NORMALIZED))
+		return map;
+	for (i = 0; i < map->n; ++i) {
+		bmap = isl_basic_map_normalize(isl_basic_map_copy(map->p[i]));
+		if (!bmap)
+			goto error;
+		isl_basic_map_free(map->p[i]);
+		map->p[i] = bmap;
+	}
+
+	map = sort_and_remove_duplicates(map);
+	if (map)
+		ISL_F_SET(map, ISL_MAP_NORMALIZED);
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+struct isl_set *isl_set_normalize(struct isl_set *set)
+{
+	return (struct isl_set *)isl_map_normalize((struct isl_map *)set);
+}
+
+int isl_map_plain_is_equal(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
+{
+	int i;
+	int equal;
+
+	if (!map1 || !map2)
+		return -1;
+
+	if (map1 == map2)
+		return 1;
+	if (!isl_space_is_equal(map1->dim, map2->dim))
+		return 0;
+
+	map1 = isl_map_copy(map1);
+	map2 = isl_map_copy(map2);
+	map1 = isl_map_normalize(map1);
+	map2 = isl_map_normalize(map2);
+	if (!map1 || !map2)
+		goto error;
+	equal = map1->n == map2->n;
+	for (i = 0; equal && i < map1->n; ++i) {
+		equal = isl_basic_map_plain_is_equal(map1->p[i], map2->p[i]);
+		if (equal < 0)
+			goto error;
+	}
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return equal;
+error:
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return -1;
+}
+
+int isl_map_fast_is_equal(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
+{
+	return isl_map_plain_is_equal(map1, map2);
+}
+
+int isl_set_plain_is_equal(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
+{
+	return isl_map_plain_is_equal((struct isl_map *)set1,
+						(struct isl_map *)set2);
+}
+
+int isl_set_fast_is_equal(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
+{
+	return isl_set_plain_is_equal(set1, set2);
+}
+
+/* Return an interval that ranges from min to max (inclusive)
+ */
+struct isl_basic_set *isl_basic_set_interval(struct isl_ctx *ctx,
+	isl_int min, isl_int max)
+{
+	int k;
+	struct isl_basic_set *bset = NULL;
+
+	bset = isl_basic_set_alloc(ctx, 0, 1, 0, 0, 2);
+	if (!bset)
+		goto error;
+
+	k = isl_basic_set_alloc_inequality(bset);
+	if (k < 0)
+		goto error;
+	isl_int_set_si(bset->ineq[k][1], 1);
+	isl_int_neg(bset->ineq[k][0], min);
+
+	k = isl_basic_set_alloc_inequality(bset);
+	if (k < 0)
+		goto error;
+	isl_int_set_si(bset->ineq[k][1], -1);
+	isl_int_set(bset->ineq[k][0], max);
+
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Return the basic sets in "set" as a list.
+ */
+__isl_give isl_basic_set_list *isl_set_get_basic_set_list(
+	__isl_keep isl_set *set)
+{
+	int i;
+	isl_ctx *ctx;
+	isl_basic_set_list *list;
+
+	if (!set)
+		return NULL;
+	ctx = isl_set_get_ctx(set);
+	list = isl_basic_set_list_alloc(ctx, set->n);
+
+	for (i = 0; i < set->n; ++i) {
+		isl_basic_set *bset;
+
+		bset = isl_basic_set_copy(set->p[i]);
+		list = isl_basic_set_list_add(list, bset);
+	}
+
+	return list;
+}
+
+/* Return the intersection of the elements in the non-empty list "list".
+ * All elements are assumed to live in the same space.
+ */
+__isl_give isl_basic_set *isl_basic_set_list_intersect(
+	__isl_take struct isl_basic_set_list *list)
+{
+	int i, n;
+	isl_basic_set *bset;
+
+	if (!list)
+		return NULL;
+	n = isl_basic_set_list_n_basic_set(list);
+	if (n < 1)
+		isl_die(isl_basic_set_list_get_ctx(list), isl_error_invalid,
+			"expecting non-empty list", goto error);
+	
+	bset = isl_basic_set_list_get_basic_set(list, 0);
+	for (i = 1; i < n; ++i) {
+		isl_basic_set *bset_i;
+
+		bset_i = isl_basic_set_list_get_basic_set(list, i);
+		bset = isl_basic_set_intersect(bset, bset_i);
+	}
+
+	isl_basic_set_list_free(list);
+	return bset;
+error:
+	isl_basic_set_list_free(list);
+	return NULL;
+}
+
+/* Return the Cartesian product of the basic sets in list (in the given order).
+ */
+__isl_give isl_basic_set *isl_basic_set_list_product(
+	__isl_take struct isl_basic_set_list *list)
+{
+	int i;
+	unsigned dim;
+	unsigned nparam;
+	unsigned extra;
+	unsigned n_eq;
+	unsigned n_ineq;
+	struct isl_basic_set *product = NULL;
+
+	if (!list)
+		goto error;
+	isl_assert(list->ctx, list->n > 0, goto error);
+	isl_assert(list->ctx, list->p[0], goto error);
+	nparam = isl_basic_set_n_param(list->p[0]);
+	dim = isl_basic_set_n_dim(list->p[0]);
+	extra = list->p[0]->n_div;
+	n_eq = list->p[0]->n_eq;
+	n_ineq = list->p[0]->n_ineq;
+	for (i = 1; i < list->n; ++i) {
+		isl_assert(list->ctx, list->p[i], goto error);
+		isl_assert(list->ctx,
+		    nparam == isl_basic_set_n_param(list->p[i]), goto error);
+		dim += isl_basic_set_n_dim(list->p[i]);
+		extra += list->p[i]->n_div;
+		n_eq += list->p[i]->n_eq;
+		n_ineq += list->p[i]->n_ineq;
+	}
+	product = isl_basic_set_alloc(list->ctx, nparam, dim, extra,
+					n_eq, n_ineq);
+	if (!product)
+		goto error;
+	dim = 0;
+	for (i = 0; i < list->n; ++i) {
+		isl_basic_set_add_constraints(product,
+					isl_basic_set_copy(list->p[i]), dim);
+		dim += isl_basic_set_n_dim(list->p[i]);
+	}
+	isl_basic_set_list_free(list);
+	return product;
+error:
+	isl_basic_set_free(product);
+	isl_basic_set_list_free(list);
+	return NULL;
+}
+
+struct isl_basic_map *isl_basic_map_product(
+		struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
+{
+	isl_space *dim_result = NULL;
+	struct isl_basic_map *bmap;
+	unsigned in1, in2, out1, out2, nparam, total, pos;
+	struct isl_dim_map *dim_map1, *dim_map2;
+
+	if (!bmap1 || !bmap2)
+		goto error;
+
+	isl_assert(bmap1->ctx, isl_space_match(bmap1->dim, isl_dim_param,
+				     bmap2->dim, isl_dim_param), goto error);
+	dim_result = isl_space_product(isl_space_copy(bmap1->dim),
+						   isl_space_copy(bmap2->dim));
+
+	in1 = isl_basic_map_n_in(bmap1);
+	in2 = isl_basic_map_n_in(bmap2);
+	out1 = isl_basic_map_n_out(bmap1);
+	out2 = isl_basic_map_n_out(bmap2);
+	nparam = isl_basic_map_n_param(bmap1);
+
+	total = nparam + in1 + in2 + out1 + out2 + bmap1->n_div + bmap2->n_div;
+	dim_map1 = isl_dim_map_alloc(bmap1->ctx, total);
+	dim_map2 = isl_dim_map_alloc(bmap1->ctx, total);
+	isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_param, pos = 0);
+	isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_param, pos = 0);
+	isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_in, pos += nparam);
+	isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_in, pos += in1);
+	isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_out, pos += in2);
+	isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_out, pos += out1);
+	isl_dim_map_div(dim_map1, bmap1, pos += out2);
+	isl_dim_map_div(dim_map2, bmap2, pos += bmap1->n_div);
+
+	bmap = isl_basic_map_alloc_space(dim_result,
+			bmap1->n_div + bmap2->n_div,
+			bmap1->n_eq + bmap2->n_eq,
+			bmap1->n_ineq + bmap2->n_ineq);
+	bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap1, dim_map1);
+	bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap2, dim_map2);
+	bmap = isl_basic_map_simplify(bmap);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap1);
+	isl_basic_map_free(bmap2);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_flat_product(
+	__isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2)
+{
+	isl_basic_map *prod;
+
+	prod = isl_basic_map_product(bmap1, bmap2);
+	prod = isl_basic_map_flatten(prod);
+	return prod;
+}
+
+__isl_give isl_basic_set *isl_basic_set_flat_product(
+	__isl_take isl_basic_set *bset1, __isl_take isl_basic_set *bset2)
+{
+	return isl_basic_map_flat_range_product(bset1, bset2);
+}
+
+__isl_give isl_basic_map *isl_basic_map_domain_product(
+	__isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2)
+{
+	isl_space *space_result = NULL;
+	isl_basic_map *bmap;
+	unsigned in1, in2, out, nparam, total, pos;
+	struct isl_dim_map *dim_map1, *dim_map2;
+
+	if (!bmap1 || !bmap2)
+		goto error;
+
+	space_result = isl_space_domain_product(isl_space_copy(bmap1->dim),
+						isl_space_copy(bmap2->dim));
+
+	in1 = isl_basic_map_dim(bmap1, isl_dim_in);
+	in2 = isl_basic_map_dim(bmap2, isl_dim_in);
+	out = isl_basic_map_dim(bmap1, isl_dim_out);
+	nparam = isl_basic_map_dim(bmap1, isl_dim_param);
+
+	total = nparam + in1 + in2 + out + bmap1->n_div + bmap2->n_div;
+	dim_map1 = isl_dim_map_alloc(bmap1->ctx, total);
+	dim_map2 = isl_dim_map_alloc(bmap1->ctx, total);
+	isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_param, pos = 0);
+	isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_param, pos = 0);
+	isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_in, pos += nparam);
+	isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_in, pos += in1);
+	isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_out, pos += in2);
+	isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_out, pos);
+	isl_dim_map_div(dim_map1, bmap1, pos += out);
+	isl_dim_map_div(dim_map2, bmap2, pos += bmap1->n_div);
+
+	bmap = isl_basic_map_alloc_space(space_result,
+			bmap1->n_div + bmap2->n_div,
+			bmap1->n_eq + bmap2->n_eq,
+			bmap1->n_ineq + bmap2->n_ineq);
+	bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap1, dim_map1);
+	bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap2, dim_map2);
+	bmap = isl_basic_map_simplify(bmap);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap1);
+	isl_basic_map_free(bmap2);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_range_product(
+	__isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2)
+{
+	isl_space *dim_result = NULL;
+	isl_basic_map *bmap;
+	unsigned in, out1, out2, nparam, total, pos;
+	struct isl_dim_map *dim_map1, *dim_map2;
+
+	if (!bmap1 || !bmap2)
+		goto error;
+
+	if (!isl_space_match(bmap1->dim, isl_dim_param,
+			    bmap2->dim, isl_dim_param))
+		isl_die(isl_basic_map_get_ctx(bmap1), isl_error_invalid,
+			"parameters don't match", goto error);
+
+	dim_result = isl_space_range_product(isl_space_copy(bmap1->dim),
+					   isl_space_copy(bmap2->dim));
+
+	in = isl_basic_map_dim(bmap1, isl_dim_in);
+	out1 = isl_basic_map_n_out(bmap1);
+	out2 = isl_basic_map_n_out(bmap2);
+	nparam = isl_basic_map_n_param(bmap1);
+
+	total = nparam + in + out1 + out2 + bmap1->n_div + bmap2->n_div;
+	dim_map1 = isl_dim_map_alloc(bmap1->ctx, total);
+	dim_map2 = isl_dim_map_alloc(bmap1->ctx, total);
+	isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_param, pos = 0);
+	isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_param, pos = 0);
+	isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_in, pos += nparam);
+	isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_in, pos);
+	isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_out, pos += in);
+	isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_out, pos += out1);
+	isl_dim_map_div(dim_map1, bmap1, pos += out2);
+	isl_dim_map_div(dim_map2, bmap2, pos += bmap1->n_div);
+
+	bmap = isl_basic_map_alloc_space(dim_result,
+			bmap1->n_div + bmap2->n_div,
+			bmap1->n_eq + bmap2->n_eq,
+			bmap1->n_ineq + bmap2->n_ineq);
+	bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap1, dim_map1);
+	bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap2, dim_map2);
+	bmap = isl_basic_map_simplify(bmap);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap1);
+	isl_basic_map_free(bmap2);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_flat_range_product(
+	__isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2)
+{
+	isl_basic_map *prod;
+
+	prod = isl_basic_map_range_product(bmap1, bmap2);
+	prod = isl_basic_map_flatten_range(prod);
+	return prod;
+}
+
+/* Apply "basic_map_product" to each pair of basic maps in "map1" and "map2"
+ * and collect the results.
+ * The result live in the space obtained by calling "space_product"
+ * on the spaces of "map1" and "map2".
+ * If "remove_duplicates" is set then the result may contain duplicates
+ * (even if the inputs do not) and so we try and remove the obvious
+ * duplicates.
+ */
+static __isl_give isl_map *map_product(__isl_take isl_map *map1,
+	__isl_take isl_map *map2,
+	__isl_give isl_space *(*space_product)(__isl_take isl_space *left,
+					   __isl_take isl_space *right),
+	__isl_give isl_basic_map *(*basic_map_product)(
+		__isl_take isl_basic_map *left,
+		__isl_take isl_basic_map *right),
+	int remove_duplicates)
+{
+	unsigned flags = 0;
+	struct isl_map *result;
+	int i, j;
+
+	if (!map1 || !map2)
+		goto error;
+
+	isl_assert(map1->ctx, isl_space_match(map1->dim, isl_dim_param,
+					 map2->dim, isl_dim_param), goto error);
+
+	if (ISL_F_ISSET(map1, ISL_MAP_DISJOINT) &&
+	    ISL_F_ISSET(map2, ISL_MAP_DISJOINT))
+		ISL_FL_SET(flags, ISL_MAP_DISJOINT);
+
+	result = isl_map_alloc_space(space_product(isl_space_copy(map1->dim),
+					       isl_space_copy(map2->dim)),
+				map1->n * map2->n, flags);
+	if (!result)
+		goto error;
+	for (i = 0; i < map1->n; ++i)
+		for (j = 0; j < map2->n; ++j) {
+			struct isl_basic_map *part;
+			part = basic_map_product(isl_basic_map_copy(map1->p[i]),
+						 isl_basic_map_copy(map2->p[j]));
+			if (isl_basic_map_is_empty(part))
+				isl_basic_map_free(part);
+			else
+				result = isl_map_add_basic_map(result, part);
+			if (!result)
+				goto error;
+		}
+	if (remove_duplicates)
+		result = isl_map_remove_obvious_duplicates(result);
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return result;
+error:
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return NULL;
+}
+
+/* Given two maps A -> B and C -> D, construct a map [A -> C] -> [B -> D]
+ */
+static __isl_give isl_map *map_product_aligned(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	return map_product(map1, map2, &isl_space_product,
+			&isl_basic_map_product, 0);
+}
+
+__isl_give isl_map *isl_map_product(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	return isl_map_align_params_map_map_and(map1, map2, &map_product_aligned);
+}
+
+/* Given two maps A -> B and C -> D, construct a map (A, C) -> (B, D)
+ */
+__isl_give isl_map *isl_map_flat_product(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	isl_map *prod;
+
+	prod = isl_map_product(map1, map2);
+	prod = isl_map_flatten(prod);
+	return prod;
+}
+
+/* Given two set A and B, construct its Cartesian product A x B.
+ */
+struct isl_set *isl_set_product(struct isl_set *set1, struct isl_set *set2)
+{
+	return isl_map_range_product(set1, set2);
+}
+
+__isl_give isl_set *isl_set_flat_product(__isl_take isl_set *set1,
+	__isl_take isl_set *set2)
+{
+	return isl_map_flat_range_product(set1, set2);
+}
+
+/* Given two maps A -> B and C -> D, construct a map [A -> C] -> (B * D)
+ */
+static __isl_give isl_map *map_domain_product_aligned(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	return map_product(map1, map2, &isl_space_domain_product,
+				&isl_basic_map_domain_product, 1);
+}
+
+/* Given two maps A -> B and C -> D, construct a map (A * C) -> [B -> D]
+ */
+static __isl_give isl_map *map_range_product_aligned(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	return map_product(map1, map2, &isl_space_range_product,
+				&isl_basic_map_range_product, 1);
+}
+
+__isl_give isl_map *isl_map_domain_product(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	return isl_map_align_params_map_map_and(map1, map2,
+						&map_domain_product_aligned);
+}
+
+__isl_give isl_map *isl_map_range_product(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	return isl_map_align_params_map_map_and(map1, map2,
+						&map_range_product_aligned);
+}
+
+/* Given a map of the form [A -> B] -> [C -> D], return the map A -> C.
+ */
+__isl_give isl_map *isl_map_factor_domain(__isl_take isl_map *map)
+{
+	isl_space *space;
+	int total1, keep1, total2, keep2;
+
+	if (!map)
+		return NULL;
+	if (!isl_space_domain_is_wrapping(map->dim) ||
+	    !isl_space_range_is_wrapping(map->dim))
+		isl_die(isl_map_get_ctx(map), isl_error_invalid,
+			"not a product", return isl_map_free(map));
+
+	space = isl_map_get_space(map);
+	total1 = isl_space_dim(space, isl_dim_in);
+	total2 = isl_space_dim(space, isl_dim_out);
+	space = isl_space_factor_domain(space);
+	keep1 = isl_space_dim(space, isl_dim_in);
+	keep2 = isl_space_dim(space, isl_dim_out);
+	map = isl_map_project_out(map, isl_dim_in, keep1, total1 - keep1);
+	map = isl_map_project_out(map, isl_dim_out, keep2, total2 - keep2);
+	map = isl_map_reset_space(map, space);
+
+	return map;
+}
+
+/* Given a map of the form [A -> B] -> [C -> D], return the map B -> D.
+ */
+__isl_give isl_map *isl_map_factor_range(__isl_take isl_map *map)
+{
+	isl_space *space;
+	int total1, keep1, total2, keep2;
+
+	if (!map)
+		return NULL;
+	if (!isl_space_domain_is_wrapping(map->dim) ||
+	    !isl_space_range_is_wrapping(map->dim))
+		isl_die(isl_map_get_ctx(map), isl_error_invalid,
+			"not a product", return isl_map_free(map));
+
+	space = isl_map_get_space(map);
+	total1 = isl_space_dim(space, isl_dim_in);
+	total2 = isl_space_dim(space, isl_dim_out);
+	space = isl_space_factor_range(space);
+	keep1 = isl_space_dim(space, isl_dim_in);
+	keep2 = isl_space_dim(space, isl_dim_out);
+	map = isl_map_project_out(map, isl_dim_in, 0, total1 - keep1);
+	map = isl_map_project_out(map, isl_dim_out, 0, total2 - keep2);
+	map = isl_map_reset_space(map, space);
+
+	return map;
+}
+
+/* Given a map of the form [A -> B] -> C, return the map A -> C.
+ */
+__isl_give isl_map *isl_map_domain_factor_domain(__isl_take isl_map *map)
+{
+	isl_space *space;
+	int total, keep;
+
+	if (!map)
+		return NULL;
+	if (!isl_space_domain_is_wrapping(map->dim))
+		isl_die(isl_map_get_ctx(map), isl_error_invalid,
+			"domain is not a product", return isl_map_free(map));
+
+	space = isl_map_get_space(map);
+	total = isl_space_dim(space, isl_dim_in);
+	space = isl_space_domain_factor_domain(space);
+	keep = isl_space_dim(space, isl_dim_in);
+	map = isl_map_project_out(map, isl_dim_in, keep, total - keep);
+	map = isl_map_reset_space(map, space);
+
+	return map;
+}
+
+/* Given a map of the form [A -> B] -> C, return the map B -> C.
+ */
+__isl_give isl_map *isl_map_domain_factor_range(__isl_take isl_map *map)
+{
+	isl_space *space;
+	int total, keep;
+
+	if (!map)
+		return NULL;
+	if (!isl_space_domain_is_wrapping(map->dim))
+		isl_die(isl_map_get_ctx(map), isl_error_invalid,
+			"domain is not a product", return isl_map_free(map));
+
+	space = isl_map_get_space(map);
+	total = isl_space_dim(space, isl_dim_in);
+	space = isl_space_domain_factor_range(space);
+	keep = isl_space_dim(space, isl_dim_in);
+	map = isl_map_project_out(map, isl_dim_in, 0, total - keep);
+	map = isl_map_reset_space(map, space);
+
+	return map;
+}
+
+/* Given a map A -> [B -> C], extract the map A -> B.
+ */
+__isl_give isl_map *isl_map_range_factor_domain(__isl_take isl_map *map)
+{
+	isl_space *space;
+	int total, keep;
+
+	if (!map)
+		return NULL;
+	if (!isl_space_range_is_wrapping(map->dim))
+		isl_die(isl_map_get_ctx(map), isl_error_invalid,
+			"range is not a product", return isl_map_free(map));
+
+	space = isl_map_get_space(map);
+	total = isl_space_dim(space, isl_dim_out);
+	space = isl_space_range_factor_domain(space);
+	keep = isl_space_dim(space, isl_dim_out);
+	map = isl_map_project_out(map, isl_dim_out, keep, total - keep);
+	map = isl_map_reset_space(map, space);
+
+	return map;
+}
+
+/* Given a map A -> [B -> C], extract the map A -> C.
+ */
+__isl_give isl_map *isl_map_range_factor_range(__isl_take isl_map *map)
+{
+	isl_space *space;
+	int total, keep;
+
+	if (!map)
+		return NULL;
+	if (!isl_space_range_is_wrapping(map->dim))
+		isl_die(isl_map_get_ctx(map), isl_error_invalid,
+			"range is not a product", return isl_map_free(map));
+
+	space = isl_map_get_space(map);
+	total = isl_space_dim(space, isl_dim_out);
+	space = isl_space_range_factor_range(space);
+	keep = isl_space_dim(space, isl_dim_out);
+	map = isl_map_project_out(map, isl_dim_out, 0, total - keep);
+	map = isl_map_reset_space(map, space);
+
+	return map;
+}
+
+/* Given two maps A -> B and C -> D, construct a map (A, C) -> (B * D)
+ */
+__isl_give isl_map *isl_map_flat_domain_product(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	isl_map *prod;
+
+	prod = isl_map_domain_product(map1, map2);
+	prod = isl_map_flatten_domain(prod);
+	return prod;
+}
+
+/* Given two maps A -> B and C -> D, construct a map (A * C) -> (B, D)
+ */
+__isl_give isl_map *isl_map_flat_range_product(__isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	isl_map *prod;
+
+	prod = isl_map_range_product(map1, map2);
+	prod = isl_map_flatten_range(prod);
+	return prod;
+}
+
+uint32_t isl_basic_map_get_hash(__isl_keep isl_basic_map *bmap)
+{
+	int i;
+	uint32_t hash = isl_hash_init();
+	unsigned total;
+
+	if (!bmap)
+		return 0;
+	bmap = isl_basic_map_copy(bmap);
+	bmap = isl_basic_map_normalize(bmap);
+	if (!bmap)
+		return 0;
+	total = isl_basic_map_total_dim(bmap);
+	isl_hash_byte(hash, bmap->n_eq & 0xFF);
+	for (i = 0; i < bmap->n_eq; ++i) {
+		uint32_t c_hash;
+		c_hash = isl_seq_get_hash(bmap->eq[i], 1 + total);
+		isl_hash_hash(hash, c_hash);
+	}
+	isl_hash_byte(hash, bmap->n_ineq & 0xFF);
+	for (i = 0; i < bmap->n_ineq; ++i) {
+		uint32_t c_hash;
+		c_hash = isl_seq_get_hash(bmap->ineq[i], 1 + total);
+		isl_hash_hash(hash, c_hash);
+	}
+	isl_hash_byte(hash, bmap->n_div & 0xFF);
+	for (i = 0; i < bmap->n_div; ++i) {
+		uint32_t c_hash;
+		if (isl_int_is_zero(bmap->div[i][0]))
+			continue;
+		isl_hash_byte(hash, i & 0xFF);
+		c_hash = isl_seq_get_hash(bmap->div[i], 1 + 1 + total);
+		isl_hash_hash(hash, c_hash);
+	}
+	isl_basic_map_free(bmap);
+	return hash;
+}
+
+uint32_t isl_basic_set_get_hash(__isl_keep isl_basic_set *bset)
+{
+	return isl_basic_map_get_hash((isl_basic_map *)bset);
+}
+
+uint32_t isl_map_get_hash(__isl_keep isl_map *map)
+{
+	int i;
+	uint32_t hash;
+
+	if (!map)
+		return 0;
+	map = isl_map_copy(map);
+	map = isl_map_normalize(map);
+	if (!map)
+		return 0;
+
+	hash = isl_hash_init();
+	for (i = 0; i < map->n; ++i) {
+		uint32_t bmap_hash;
+		bmap_hash = isl_basic_map_get_hash(map->p[i]);
+		isl_hash_hash(hash, bmap_hash);
+	}
+		
+	isl_map_free(map);
+
+	return hash;
+}
+
+uint32_t isl_set_get_hash(__isl_keep isl_set *set)
+{
+	return isl_map_get_hash((isl_map *)set);
+}
+
+/* Check if the value for dimension dim is completely determined
+ * by the values of the other parameters and variables.
+ * That is, check if dimension dim is involved in an equality.
+ */
+int isl_basic_set_dim_is_unique(struct isl_basic_set *bset, unsigned dim)
+{
+	int i;
+	unsigned nparam;
+
+	if (!bset)
+		return -1;
+	nparam = isl_basic_set_n_param(bset);
+	for (i = 0; i < bset->n_eq; ++i)
+		if (!isl_int_is_zero(bset->eq[i][1 + nparam + dim]))
+			return 1;
+	return 0;
+}
+
+/* Check if the value for dimension dim is completely determined
+ * by the values of the other parameters and variables.
+ * That is, check if dimension dim is involved in an equality
+ * for each of the subsets.
+ */
+int isl_set_dim_is_unique(struct isl_set *set, unsigned dim)
+{
+	int i;
+
+	if (!set)
+		return -1;
+	for (i = 0; i < set->n; ++i) {
+		int unique;
+		unique = isl_basic_set_dim_is_unique(set->p[i], dim);
+		if (unique != 1)
+			return unique;
+	}
+	return 1;
+}
+
+/* Return the number of basic maps in the (current) representation of "map".
+ */
+int isl_map_n_basic_map(__isl_keep isl_map *map)
+{
+	return map ? map->n : 0;
+}
+
+int isl_set_n_basic_set(__isl_keep isl_set *set)
+{
+	return set ? set->n : 0;
+}
+
+int isl_map_foreach_basic_map(__isl_keep isl_map *map,
+	int (*fn)(__isl_take isl_basic_map *bmap, void *user), void *user)
+{
+	int i;
+
+	if (!map)
+		return -1;
+
+	for (i = 0; i < map->n; ++i)
+		if (fn(isl_basic_map_copy(map->p[i]), user) < 0)
+			return -1;
+
+	return 0;
+}
+
+int isl_set_foreach_basic_set(__isl_keep isl_set *set,
+	int (*fn)(__isl_take isl_basic_set *bset, void *user), void *user)
+{
+	int i;
+
+	if (!set)
+		return -1;
+
+	for (i = 0; i < set->n; ++i)
+		if (fn(isl_basic_set_copy(set->p[i]), user) < 0)
+			return -1;
+
+	return 0;
+}
+
+__isl_give isl_basic_set *isl_basic_set_lift(__isl_take isl_basic_set *bset)
+{
+	isl_space *dim;
+
+	if (!bset)
+		return NULL;
+
+	bset = isl_basic_set_cow(bset);
+	if (!bset)
+		return NULL;
+
+	dim = isl_basic_set_get_space(bset);
+	dim = isl_space_lift(dim, bset->n_div);
+	if (!dim)
+		goto error;
+	isl_space_free(bset->dim);
+	bset->dim = dim;
+	bset->extra -= bset->n_div;
+	bset->n_div = 0;
+
+	bset = isl_basic_set_finalize(bset);
+
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_lift(__isl_take isl_set *set)
+{
+	int i;
+	isl_space *dim;
+	unsigned n_div;
+
+	set = isl_set_align_divs(set);
+
+	if (!set)
+		return NULL;
+
+	set = isl_set_cow(set);
+	if (!set)
+		return NULL;
+
+	n_div = set->p[0]->n_div;
+	dim = isl_set_get_space(set);
+	dim = isl_space_lift(dim, n_div);
+	if (!dim)
+		goto error;
+	isl_space_free(set->dim);
+	set->dim = dim;
+
+	for (i = 0; i < set->n; ++i) {
+		set->p[i] = isl_basic_set_lift(set->p[i]);
+		if (!set->p[i])
+			goto error;
+	}
+
+	return set;
+error:
+	isl_set_free(set);
+	return NULL;
+}
+
+__isl_give isl_map *isl_set_lifting(__isl_take isl_set *set)
+{
+	isl_space *dim;
+	struct isl_basic_map *bmap;
+	unsigned n_set;
+	unsigned n_div;
+	unsigned n_param;
+	unsigned total;
+	int i, k, l;
+
+	set = isl_set_align_divs(set);
+
+	if (!set)
+		return NULL;
+
+	dim = isl_set_get_space(set);
+	if (set->n == 0 || set->p[0]->n_div == 0) {
+		isl_set_free(set);
+		return isl_map_identity(isl_space_map_from_set(dim));
+	}
+
+	n_div = set->p[0]->n_div;
+	dim = isl_space_map_from_set(dim);
+	n_param = isl_space_dim(dim, isl_dim_param);
+	n_set = isl_space_dim(dim, isl_dim_in);
+	dim = isl_space_extend(dim, n_param, n_set, n_set + n_div);
+	bmap = isl_basic_map_alloc_space(dim, 0, n_set, 2 * n_div);
+	for (i = 0; i < n_set; ++i)
+		bmap = var_equal(bmap, i);
+
+	total = n_param + n_set + n_set + n_div;
+	for (i = 0; i < n_div; ++i) {
+		k = isl_basic_map_alloc_inequality(bmap);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(bmap->ineq[k], set->p[0]->div[i]+1, 1+n_param);
+		isl_seq_clr(bmap->ineq[k]+1+n_param, n_set);
+		isl_seq_cpy(bmap->ineq[k]+1+n_param+n_set,
+			    set->p[0]->div[i]+1+1+n_param, n_set + n_div);
+		isl_int_neg(bmap->ineq[k][1+n_param+n_set+n_set+i],
+			    set->p[0]->div[i][0]);
+
+		l = isl_basic_map_alloc_inequality(bmap);
+		if (l < 0)
+			goto error;
+		isl_seq_neg(bmap->ineq[l], bmap->ineq[k], 1 + total);
+		isl_int_add(bmap->ineq[l][0], bmap->ineq[l][0],
+			    set->p[0]->div[i][0]);
+		isl_int_sub_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
+	}
+
+	isl_set_free(set);
+	bmap = isl_basic_map_simplify(bmap);
+	bmap = isl_basic_map_finalize(bmap);
+	return isl_map_from_basic_map(bmap);
+error:
+	isl_set_free(set);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+int isl_basic_set_size(__isl_keep isl_basic_set *bset)
+{
+	unsigned dim;
+	int size = 0;
+
+	if (!bset)
+		return -1;
+
+	dim = isl_basic_set_total_dim(bset);
+	size += bset->n_eq * (1 + dim);
+	size += bset->n_ineq * (1 + dim);
+	size += bset->n_div * (2 + dim);
+
+	return size;
+}
+
+int isl_set_size(__isl_keep isl_set *set)
+{
+	int i;
+	int size = 0;
+
+	if (!set)
+		return -1;
+
+	for (i = 0; i < set->n; ++i)
+		size += isl_basic_set_size(set->p[i]);
+
+	return size;
+}
+
+/* Check if there is any lower bound (if lower == 0) and/or upper
+ * bound (if upper == 0) on the specified dim.
+ */
+static int basic_map_dim_is_bounded(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos, int lower, int upper)
+{
+	int i;
+
+	if (!bmap)
+		return -1;
+
+	isl_assert(bmap->ctx, pos < isl_basic_map_dim(bmap, type), return -1);
+
+	pos += isl_basic_map_offset(bmap, type);
+
+	for (i = 0; i < bmap->n_div; ++i) {
+		if (isl_int_is_zero(bmap->div[i][0]))
+			continue;
+		if (!isl_int_is_zero(bmap->div[i][1 + pos]))
+			return 1;
+	}
+
+	for (i = 0; i < bmap->n_eq; ++i)
+		if (!isl_int_is_zero(bmap->eq[i][pos]))
+			return 1;
+
+	for (i = 0; i < bmap->n_ineq; ++i) {
+		int sgn = isl_int_sgn(bmap->ineq[i][pos]);
+		if (sgn > 0)
+			lower = 1;
+		if (sgn < 0)
+			upper = 1;
+	}
+
+	return lower && upper;
+}
+
+int isl_basic_map_dim_is_bounded(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos)
+{
+	return basic_map_dim_is_bounded(bmap, type, pos, 0, 0);
+}
+
+int isl_basic_map_dim_has_lower_bound(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos)
+{
+	return basic_map_dim_is_bounded(bmap, type, pos, 0, 1);
+}
+
+int isl_basic_map_dim_has_upper_bound(__isl_keep isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned pos)
+{
+	return basic_map_dim_is_bounded(bmap, type, pos, 1, 0);
+}
+
+int isl_map_dim_is_bounded(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos)
+{
+	int i;
+
+	if (!map)
+		return -1;
+
+	for (i = 0; i < map->n; ++i) {
+		int bounded;
+		bounded = isl_basic_map_dim_is_bounded(map->p[i], type, pos);
+		if (bounded < 0 || !bounded)
+			return bounded;
+	}
+
+	return 1;
+}
+
+/* Return 1 if the specified dim is involved in both an upper bound
+ * and a lower bound.
+ */
+int isl_set_dim_is_bounded(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos)
+{
+	return isl_map_dim_is_bounded((isl_map *)set, type, pos);
+}
+
+/* Does "map" have a bound (according to "fn") for any of its basic maps?
+ */
+static int has_any_bound(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos,
+	int (*fn)(__isl_keep isl_basic_map *bmap,
+		  enum isl_dim_type type, unsigned pos))
+{
+	int i;
+
+	if (!map)
+		return -1;
+
+	for (i = 0; i < map->n; ++i) {
+		int bounded;
+		bounded = fn(map->p[i], type, pos);
+		if (bounded < 0 || bounded)
+			return bounded;
+	}
+
+	return 0;
+}
+
+/* Return 1 if the specified dim is involved in any lower bound.
+ */
+int isl_set_dim_has_any_lower_bound(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos)
+{
+	return has_any_bound(set, type, pos,
+				&isl_basic_map_dim_has_lower_bound);
+}
+
+/* Return 1 if the specified dim is involved in any upper bound.
+ */
+int isl_set_dim_has_any_upper_bound(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos)
+{
+	return has_any_bound(set, type, pos,
+				&isl_basic_map_dim_has_upper_bound);
+}
+
+/* Does "map" have a bound (according to "fn") for all of its basic maps?
+ */
+static int has_bound(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos,
+	int (*fn)(__isl_keep isl_basic_map *bmap,
+		  enum isl_dim_type type, unsigned pos))
+{
+	int i;
+
+	if (!map)
+		return -1;
+
+	for (i = 0; i < map->n; ++i) {
+		int bounded;
+		bounded = fn(map->p[i], type, pos);
+		if (bounded < 0 || !bounded)
+			return bounded;
+	}
+
+	return 1;
+}
+
+/* Return 1 if the specified dim has a lower bound (in each of its basic sets).
+ */
+int isl_set_dim_has_lower_bound(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos)
+{
+	return has_bound(set, type, pos, &isl_basic_map_dim_has_lower_bound);
+}
+
+/* Return 1 if the specified dim has an upper bound (in each of its basic sets).
+ */
+int isl_set_dim_has_upper_bound(__isl_keep isl_set *set,
+	enum isl_dim_type type, unsigned pos)
+{
+	return has_bound(set, type, pos, &isl_basic_map_dim_has_upper_bound);
+}
+
+/* For each of the "n" variables starting at "first", determine
+ * the sign of the variable and put the results in the first "n"
+ * elements of the array "signs".
+ * Sign
+ *	1 means that the variable is non-negative
+ *	-1 means that the variable is non-positive
+ *	0 means the variable attains both positive and negative values.
+ */
+int isl_basic_set_vars_get_sign(__isl_keep isl_basic_set *bset,
+	unsigned first, unsigned n, int *signs)
+{
+	isl_vec *bound = NULL;
+	struct isl_tab *tab = NULL;
+	struct isl_tab_undo *snap;
+	int i;
+
+	if (!bset || !signs)
+		return -1;
+
+	bound = isl_vec_alloc(bset->ctx, 1 + isl_basic_set_total_dim(bset));
+	tab = isl_tab_from_basic_set(bset, 0);
+	if (!bound || !tab)
+		goto error;
+
+	isl_seq_clr(bound->el, bound->size);
+	isl_int_set_si(bound->el[0], -1);
+
+	snap = isl_tab_snap(tab);
+	for (i = 0; i < n; ++i) {
+		int empty;
+
+		isl_int_set_si(bound->el[1 + first + i], -1);
+		if (isl_tab_add_ineq(tab, bound->el) < 0)
+			goto error;
+		empty = tab->empty;
+		isl_int_set_si(bound->el[1 + first + i], 0);
+		if (isl_tab_rollback(tab, snap) < 0)
+			goto error;
+
+		if (empty) {
+			signs[i] = 1;
+			continue;
+		}
+
+		isl_int_set_si(bound->el[1 + first + i], 1);
+		if (isl_tab_add_ineq(tab, bound->el) < 0)
+			goto error;
+		empty = tab->empty;
+		isl_int_set_si(bound->el[1 + first + i], 0);
+		if (isl_tab_rollback(tab, snap) < 0)
+			goto error;
+
+		signs[i] = empty ? -1 : 0;
+	}
+
+	isl_tab_free(tab);
+	isl_vec_free(bound);
+	return 0;
+error:
+	isl_tab_free(tab);
+	isl_vec_free(bound);
+	return -1;
+}
+
+int isl_basic_set_dims_get_sign(__isl_keep isl_basic_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n, int *signs)
+{
+	if (!bset || !signs)
+		return -1;
+	isl_assert(bset->ctx, first + n <= isl_basic_set_dim(bset, type),
+		return -1);
+
+	first += pos(bset->dim, type) - 1;
+	return isl_basic_set_vars_get_sign(bset, first, n, signs);
+}
+
+/* Is it possible for the integer division "div" to depend (possibly
+ * indirectly) on any output dimensions?
+ *
+ * If the div is undefined, then we conservatively assume that it
+ * may depend on them.
+ * Otherwise, we check if it actually depends on them or on any integer
+ * divisions that may depend on them.
+ */
+static int div_may_involve_output(__isl_keep isl_basic_map *bmap, int div)
+{
+	int i;
+	unsigned n_out, o_out;
+	unsigned n_div, o_div;
+
+	if (isl_int_is_zero(bmap->div[div][0]))
+		return 1;
+
+	n_out = isl_basic_map_dim(bmap, isl_dim_out);
+	o_out = isl_basic_map_offset(bmap, isl_dim_out);
+
+	if (isl_seq_first_non_zero(bmap->div[div] + 1 + o_out, n_out) != -1)
+		return 1;
+
+	n_div = isl_basic_map_dim(bmap, isl_dim_div);
+	o_div = isl_basic_map_offset(bmap, isl_dim_div);
+
+	for (i = 0; i < n_div; ++i) {
+		if (isl_int_is_zero(bmap->div[div][1 + o_div + i]))
+			continue;
+		if (div_may_involve_output(bmap, i))
+			return 1;
+	}
+
+	return 0;
+}
+
+/* Return the index of the equality of "bmap" that defines
+ * the output dimension "pos" in terms of earlier dimensions.
+ * The equality may also involve integer divisions, as long
+ * as those integer divisions are defined in terms of
+ * parameters or input dimensions.
+ * Return bmap->n_eq if there is no such equality.
+ * Return -1 on error.
+ */
+int isl_basic_map_output_defining_equality(__isl_keep isl_basic_map *bmap,
+	int pos)
+{
+	int j, k;
+	unsigned n_out, o_out;
+	unsigned n_div, o_div;
+
+	if (!bmap)
+		return -1;
+
+	n_out = isl_basic_map_dim(bmap, isl_dim_out);
+	o_out = isl_basic_map_offset(bmap, isl_dim_out);
+	n_div = isl_basic_map_dim(bmap, isl_dim_div);
+	o_div = isl_basic_map_offset(bmap, isl_dim_div);
+
+	for (j = 0; j < bmap->n_eq; ++j) {
+		if (isl_int_is_zero(bmap->eq[j][o_out + pos]))
+			continue;
+		if (isl_seq_first_non_zero(bmap->eq[j] + o_out + pos + 1,
+					n_out - (pos + 1)) != -1)
+			continue;
+		for (k = 0; k < n_div; ++k) {
+			if (isl_int_is_zero(bmap->eq[j][o_div + k]))
+				continue;
+			if (div_may_involve_output(bmap, k))
+				break;
+		}
+		if (k >= n_div)
+			return j;
+	}
+
+	return bmap->n_eq;
+}
+
+/* Check if the given basic map is obviously single-valued.
+ * In particular, for each output dimension, check that there is
+ * an equality that defines the output dimension in terms of
+ * earlier dimensions.
+ */
+int isl_basic_map_plain_is_single_valued(__isl_keep isl_basic_map *bmap)
+{
+	int i;
+	unsigned n_out;
+
+	if (!bmap)
+		return -1;
+
+	n_out = isl_basic_map_dim(bmap, isl_dim_out);
+
+	for (i = 0; i < n_out; ++i) {
+		int eq;
+
+		eq = isl_basic_map_output_defining_equality(bmap, i);
+		if (eq < 0)
+			return -1;
+		if (eq >= bmap->n_eq)
+			return 0;
+	}
+
+	return 1;
+}
+
+/* Check if the given basic map is single-valued.
+ * We simply compute
+ *
+ *	M \circ M^-1
+ *
+ * and check if the result is a subset of the identity mapping.
+ */
+int isl_basic_map_is_single_valued(__isl_keep isl_basic_map *bmap)
+{
+	isl_space *space;
+	isl_basic_map *test;
+	isl_basic_map *id;
+	int sv;
+
+	sv = isl_basic_map_plain_is_single_valued(bmap);
+	if (sv < 0 || sv)
+		return sv;
+
+	test = isl_basic_map_reverse(isl_basic_map_copy(bmap));
+	test = isl_basic_map_apply_range(test, isl_basic_map_copy(bmap));
+
+	space = isl_basic_map_get_space(bmap);
+	space = isl_space_map_from_set(isl_space_range(space));
+	id = isl_basic_map_identity(space);
+
+	sv = isl_basic_map_is_subset(test, id);
+
+	isl_basic_map_free(test);
+	isl_basic_map_free(id);
+
+	return sv;
+}
+
+/* Check if the given map is obviously single-valued.
+ */
+int isl_map_plain_is_single_valued(__isl_keep isl_map *map)
+{
+	if (!map)
+		return -1;
+	if (map->n == 0)
+		return 1;
+	if (map->n >= 2)
+		return 0;
+
+	return isl_basic_map_plain_is_single_valued(map->p[0]);
+}
+
+/* Check if the given map is single-valued.
+ * We simply compute
+ *
+ *	M \circ M^-1
+ *
+ * and check if the result is a subset of the identity mapping.
+ */
+int isl_map_is_single_valued(__isl_keep isl_map *map)
+{
+	isl_space *dim;
+	isl_map *test;
+	isl_map *id;
+	int sv;
+
+	sv = isl_map_plain_is_single_valued(map);
+	if (sv < 0 || sv)
+		return sv;
+
+	test = isl_map_reverse(isl_map_copy(map));
+	test = isl_map_apply_range(test, isl_map_copy(map));
+
+	dim = isl_space_map_from_set(isl_space_range(isl_map_get_space(map)));
+	id = isl_map_identity(dim);
+
+	sv = isl_map_is_subset(test, id);
+
+	isl_map_free(test);
+	isl_map_free(id);
+
+	return sv;
+}
+
+int isl_map_is_injective(__isl_keep isl_map *map)
+{
+	int in;
+
+	map = isl_map_copy(map);
+	map = isl_map_reverse(map);
+	in = isl_map_is_single_valued(map);
+	isl_map_free(map);
+
+	return in;
+}
+
+/* Check if the given map is obviously injective.
+ */
+int isl_map_plain_is_injective(__isl_keep isl_map *map)
+{
+	int in;
+
+	map = isl_map_copy(map);
+	map = isl_map_reverse(map);
+	in = isl_map_plain_is_single_valued(map);
+	isl_map_free(map);
+
+	return in;
+}
+
+int isl_map_is_bijective(__isl_keep isl_map *map)
+{
+	int sv;
+
+	sv = isl_map_is_single_valued(map);
+	if (sv < 0 || !sv)
+		return sv;
+
+	return isl_map_is_injective(map);
+}
+
+int isl_set_is_singleton(__isl_keep isl_set *set)
+{
+	return isl_map_is_single_valued((isl_map *)set);
+}
+
+int isl_map_is_translation(__isl_keep isl_map *map)
+{
+	int ok;
+	isl_set *delta;
+
+	delta = isl_map_deltas(isl_map_copy(map));
+	ok = isl_set_is_singleton(delta);
+	isl_set_free(delta);
+
+	return ok;
+}
+
+static int unique(isl_int *p, unsigned pos, unsigned len)
+{
+	if (isl_seq_first_non_zero(p, pos) != -1)
+		return 0;
+	if (isl_seq_first_non_zero(p + pos + 1, len - pos - 1) != -1)
+		return 0;
+	return 1;
+}
+
+int isl_basic_set_is_box(__isl_keep isl_basic_set *bset)
+{
+	int i, j;
+	unsigned nvar;
+	unsigned ovar;
+
+	if (!bset)
+		return -1;
+
+	if (isl_basic_set_dim(bset, isl_dim_div) != 0)
+		return 0;
+
+	nvar = isl_basic_set_dim(bset, isl_dim_set);
+	ovar = isl_space_offset(bset->dim, isl_dim_set);
+	for (j = 0; j < nvar; ++j) {
+		int lower = 0, upper = 0;
+		for (i = 0; i < bset->n_eq; ++i) {
+			if (isl_int_is_zero(bset->eq[i][1 + ovar + j]))
+				continue;
+			if (!unique(bset->eq[i] + 1 + ovar, j, nvar))
+				return 0;
+			break;
+		}
+		if (i < bset->n_eq)
+			continue;
+		for (i = 0; i < bset->n_ineq; ++i) {
+			if (isl_int_is_zero(bset->ineq[i][1 + ovar + j]))
+				continue;
+			if (!unique(bset->ineq[i] + 1 + ovar, j, nvar))
+				return 0;
+			if (isl_int_is_pos(bset->ineq[i][1 + ovar + j]))
+				lower = 1;
+			else
+				upper = 1;
+		}
+		if (!lower || !upper)
+			return 0;
+	}
+
+	return 1;
+}
+
+int isl_set_is_box(__isl_keep isl_set *set)
+{
+	if (!set)
+		return -1;
+	if (set->n != 1)
+		return 0;
+
+	return isl_basic_set_is_box(set->p[0]);
+}
+
+int isl_basic_set_is_wrapping(__isl_keep isl_basic_set *bset)
+{
+	if (!bset)
+		return -1;
+	
+	return isl_space_is_wrapping(bset->dim);
+}
+
+int isl_set_is_wrapping(__isl_keep isl_set *set)
+{
+	if (!set)
+		return -1;
+	
+	return isl_space_is_wrapping(set->dim);
+}
+
+/* Is the domain of "map" a wrapped relation?
+ */
+int isl_map_domain_is_wrapping(__isl_keep isl_map *map)
+{
+	if (!map)
+		return -1;
+
+	return isl_space_domain_is_wrapping(map->dim);
+}
+
+/* Is the range of "map" a wrapped relation?
+ */
+int isl_map_range_is_wrapping(__isl_keep isl_map *map)
+{
+	if (!map)
+		return -1;
+
+	return isl_space_range_is_wrapping(map->dim);
+}
+
+__isl_give isl_basic_set *isl_basic_map_wrap(__isl_take isl_basic_map *bmap)
+{
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+
+	bmap->dim = isl_space_wrap(bmap->dim);
+	if (!bmap->dim)
+		goto error;
+
+	bmap = isl_basic_map_finalize(bmap);
+
+	return (isl_basic_set *)bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_set *isl_map_wrap(__isl_take isl_map *map)
+{
+	int i;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = (isl_basic_map *)isl_basic_map_wrap(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+	map->dim = isl_space_wrap(map->dim);
+	if (!map->dim)
+		goto error;
+
+	return (isl_set *)map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_set_unwrap(__isl_take isl_basic_set *bset)
+{
+	bset = isl_basic_set_cow(bset);
+	if (!bset)
+		return NULL;
+
+	bset->dim = isl_space_unwrap(bset->dim);
+	if (!bset->dim)
+		goto error;
+
+	bset = isl_basic_set_finalize(bset);
+
+	return (isl_basic_map *)bset;
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+__isl_give isl_map *isl_set_unwrap(__isl_take isl_set *set)
+{
+	int i;
+
+	if (!set)
+		return NULL;
+
+	if (!isl_set_is_wrapping(set))
+		isl_die(set->ctx, isl_error_invalid, "not a wrapping set",
+			goto error);
+
+	set = isl_set_cow(set);
+	if (!set)
+		return NULL;
+
+	for (i = 0; i < set->n; ++i) {
+		set->p[i] = (isl_basic_set *)isl_basic_set_unwrap(set->p[i]);
+		if (!set->p[i])
+			goto error;
+	}
+
+	set->dim = isl_space_unwrap(set->dim);
+	if (!set->dim)
+		goto error;
+
+	return (isl_map *)set;
+error:
+	isl_set_free(set);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_reset(__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type)
+{
+	if (!bmap)
+		return NULL;
+
+	if (!isl_space_is_named_or_nested(bmap->dim, type))
+		return bmap;
+
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+
+	bmap->dim = isl_space_reset(bmap->dim, type);
+	if (!bmap->dim)
+		goto error;
+
+	bmap = isl_basic_map_finalize(bmap);
+
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_reset(__isl_take isl_map *map,
+	enum isl_dim_type type)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+
+	if (!isl_space_is_named_or_nested(map->dim, type))
+		return map;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_reset(map->p[i], type);
+		if (!map->p[i])
+			goto error;
+	}
+	map->dim = isl_space_reset(map->dim, type);
+	if (!map->dim)
+		goto error;
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_flatten(__isl_take isl_basic_map *bmap)
+{
+	if (!bmap)
+		return NULL;
+
+	if (!bmap->dim->nested[0] && !bmap->dim->nested[1])
+		return bmap;
+
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+
+	bmap->dim = isl_space_flatten(bmap->dim);
+	if (!bmap->dim)
+		goto error;
+
+	bmap = isl_basic_map_finalize(bmap);
+
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_flatten(__isl_take isl_basic_set *bset)
+{
+	return (isl_basic_set *)isl_basic_map_flatten((isl_basic_map *)bset);
+}
+
+__isl_give isl_basic_map *isl_basic_map_flatten_domain(
+	__isl_take isl_basic_map *bmap)
+{
+	if (!bmap)
+		return NULL;
+
+	if (!bmap->dim->nested[0])
+		return bmap;
+
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+
+	bmap->dim = isl_space_flatten_domain(bmap->dim);
+	if (!bmap->dim)
+		goto error;
+
+	bmap = isl_basic_map_finalize(bmap);
+
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_flatten_range(
+	__isl_take isl_basic_map *bmap)
+{
+	if (!bmap)
+		return NULL;
+
+	if (!bmap->dim->nested[1])
+		return bmap;
+
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+
+	bmap->dim = isl_space_flatten_range(bmap->dim);
+	if (!bmap->dim)
+		goto error;
+
+	bmap = isl_basic_map_finalize(bmap);
+
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_flatten(__isl_take isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+
+	if (!map->dim->nested[0] && !map->dim->nested[1])
+		return map;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_flatten(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+	map->dim = isl_space_flatten(map->dim);
+	if (!map->dim)
+		goto error;
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_flatten(__isl_take isl_set *set)
+{
+	return (isl_set *)isl_map_flatten((isl_map *)set);
+}
+
+__isl_give isl_map *isl_set_flatten_map(__isl_take isl_set *set)
+{
+	isl_space *dim, *flat_dim;
+	isl_map *map;
+
+	dim = isl_set_get_space(set);
+	flat_dim = isl_space_flatten(isl_space_copy(dim));
+	map = isl_map_identity(isl_space_join(isl_space_reverse(dim), flat_dim));
+	map = isl_map_intersect_domain(map, set);
+
+	return map;
+}
+
+__isl_give isl_map *isl_map_flatten_domain(__isl_take isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+
+	if (!map->dim->nested[0])
+		return map;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_flatten_domain(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+	map->dim = isl_space_flatten_domain(map->dim);
+	if (!map->dim)
+		goto error;
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_flatten_range(__isl_take isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+
+	if (!map->dim->nested[1])
+		return map;
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_flatten_range(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+	map->dim = isl_space_flatten_range(map->dim);
+	if (!map->dim)
+		goto error;
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Reorder the dimensions of "bmap" according to the given dim_map
+ * and set the dimension specification to "dim".
+ */
+__isl_give isl_basic_map *isl_basic_map_realign(__isl_take isl_basic_map *bmap,
+	__isl_take isl_space *dim, __isl_take struct isl_dim_map *dim_map)
+{
+	isl_basic_map *res;
+	unsigned flags;
+
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap || !dim || !dim_map)
+		goto error;
+
+	flags = bmap->flags;
+	ISL_FL_CLR(flags, ISL_BASIC_MAP_FINAL);
+	ISL_FL_CLR(flags, ISL_BASIC_MAP_NORMALIZED);
+	ISL_FL_CLR(flags, ISL_BASIC_MAP_NORMALIZED_DIVS);
+	res = isl_basic_map_alloc_space(dim,
+			bmap->n_div, bmap->n_eq, bmap->n_ineq);
+	res = isl_basic_map_add_constraints_dim_map(res, bmap, dim_map);
+	if (res)
+		res->flags = flags;
+	res = isl_basic_map_finalize(res);
+	return res;
+error:
+	free(dim_map);
+	isl_basic_map_free(bmap);
+	isl_space_free(dim);
+	return NULL;
+}
+
+/* Reorder the dimensions of "map" according to given reordering.
+ */
+__isl_give isl_map *isl_map_realign(__isl_take isl_map *map,
+	__isl_take isl_reordering *r)
+{
+	int i;
+	struct isl_dim_map *dim_map;
+
+	map = isl_map_cow(map);
+	dim_map = isl_dim_map_from_reordering(r);
+	if (!map || !r || !dim_map)
+		goto error;
+
+	for (i = 0; i < map->n; ++i) {
+		struct isl_dim_map *dim_map_i;
+
+		dim_map_i = isl_dim_map_extend(dim_map, map->p[i]);
+
+		map->p[i] = isl_basic_map_realign(map->p[i],
+					    isl_space_copy(r->dim), dim_map_i);
+
+		if (!map->p[i])
+			goto error;
+	}
+
+	map = isl_map_reset_space(map, isl_space_copy(r->dim));
+
+	isl_reordering_free(r);
+	free(dim_map);
+	return map;
+error:
+	free(dim_map);
+	isl_map_free(map);
+	isl_reordering_free(r);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_realign(__isl_take isl_set *set,
+	__isl_take isl_reordering *r)
+{
+	return (isl_set *)isl_map_realign((isl_map *)set, r);
+}
+
+__isl_give isl_map *isl_map_align_params(__isl_take isl_map *map,
+	__isl_take isl_space *model)
+{
+	isl_ctx *ctx;
+
+	if (!map || !model)
+		goto error;
+
+	ctx = isl_space_get_ctx(model);
+	if (!isl_space_has_named_params(model))
+		isl_die(ctx, isl_error_invalid,
+			"model has unnamed parameters", goto error);
+	if (!isl_space_has_named_params(map->dim))
+		isl_die(ctx, isl_error_invalid,
+			"relation has unnamed parameters", goto error);
+	if (!isl_space_match(map->dim, isl_dim_param, model, isl_dim_param)) {
+		isl_reordering *exp;
+
+		model = isl_space_drop_dims(model, isl_dim_in,
+					0, isl_space_dim(model, isl_dim_in));
+		model = isl_space_drop_dims(model, isl_dim_out,
+					0, isl_space_dim(model, isl_dim_out));
+		exp = isl_parameter_alignment_reordering(map->dim, model);
+		exp = isl_reordering_extend_space(exp, isl_map_get_space(map));
+		map = isl_map_realign(map, exp);
+	}
+
+	isl_space_free(model);
+	return map;
+error:
+	isl_space_free(model);
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_align_params(__isl_take isl_set *set,
+	__isl_take isl_space *model)
+{
+	return isl_map_align_params(set, model);
+}
+
+/* Align the parameters of "bmap" to those of "model", introducing
+ * additional parameters if needed.
+ */
+__isl_give isl_basic_map *isl_basic_map_align_params(
+	__isl_take isl_basic_map *bmap, __isl_take isl_space *model)
+{
+	isl_ctx *ctx;
+
+	if (!bmap || !model)
+		goto error;
+
+	ctx = isl_space_get_ctx(model);
+	if (!isl_space_has_named_params(model))
+		isl_die(ctx, isl_error_invalid,
+			"model has unnamed parameters", goto error);
+	if (!isl_space_has_named_params(bmap->dim))
+		isl_die(ctx, isl_error_invalid,
+			"relation has unnamed parameters", goto error);
+	if (!isl_space_match(bmap->dim, isl_dim_param, model, isl_dim_param)) {
+		isl_reordering *exp;
+		struct isl_dim_map *dim_map;
+
+		model = isl_space_drop_dims(model, isl_dim_in,
+					0, isl_space_dim(model, isl_dim_in));
+		model = isl_space_drop_dims(model, isl_dim_out,
+					0, isl_space_dim(model, isl_dim_out));
+		exp = isl_parameter_alignment_reordering(bmap->dim, model);
+		exp = isl_reordering_extend_space(exp,
+					isl_basic_map_get_space(bmap));
+		dim_map = isl_dim_map_from_reordering(exp);
+		bmap = isl_basic_map_realign(bmap,
+				    exp ? isl_space_copy(exp->dim) : NULL,
+				    isl_dim_map_extend(dim_map, bmap));
+		isl_reordering_free(exp);
+		free(dim_map);
+	}
+
+	isl_space_free(model);
+	return bmap;
+error:
+	isl_space_free(model);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Align the parameters of "bset" to those of "model", introducing
+ * additional parameters if needed.
+ */
+__isl_give isl_basic_set *isl_basic_set_align_params(
+	__isl_take isl_basic_set *bset, __isl_take isl_space *model)
+{
+	return isl_basic_map_align_params(bset, model);
+}
+
+__isl_give isl_mat *isl_basic_map_equalities_matrix(
+		__isl_keep isl_basic_map *bmap, enum isl_dim_type c1,
+		enum isl_dim_type c2, enum isl_dim_type c3,
+		enum isl_dim_type c4, enum isl_dim_type c5)
+{
+	enum isl_dim_type c[5] = { c1, c2, c3, c4, c5 };
+	struct isl_mat *mat;
+	int i, j, k;
+	int pos;
+
+	if (!bmap)
+		return NULL;
+	mat = isl_mat_alloc(bmap->ctx, bmap->n_eq,
+				isl_basic_map_total_dim(bmap) + 1);
+	if (!mat)
+		return NULL;
+	for (i = 0; i < bmap->n_eq; ++i)
+		for (j = 0, pos = 0; j < 5; ++j) {
+			int off = isl_basic_map_offset(bmap, c[j]);
+			for (k = 0; k < isl_basic_map_dim(bmap, c[j]); ++k) {
+				isl_int_set(mat->row[i][pos],
+					    bmap->eq[i][off + k]);
+				++pos;
+			}
+		}
+
+	return mat;
+}
+
+__isl_give isl_mat *isl_basic_map_inequalities_matrix(
+		__isl_keep isl_basic_map *bmap, enum isl_dim_type c1,
+		enum isl_dim_type c2, enum isl_dim_type c3,
+		enum isl_dim_type c4, enum isl_dim_type c5)
+{
+	enum isl_dim_type c[5] = { c1, c2, c3, c4, c5 };
+	struct isl_mat *mat;
+	int i, j, k;
+	int pos;
+
+	if (!bmap)
+		return NULL;
+	mat = isl_mat_alloc(bmap->ctx, bmap->n_ineq,
+				isl_basic_map_total_dim(bmap) + 1);
+	if (!mat)
+		return NULL;
+	for (i = 0; i < bmap->n_ineq; ++i)
+		for (j = 0, pos = 0; j < 5; ++j) {
+			int off = isl_basic_map_offset(bmap, c[j]);
+			for (k = 0; k < isl_basic_map_dim(bmap, c[j]); ++k) {
+				isl_int_set(mat->row[i][pos],
+					    bmap->ineq[i][off + k]);
+				++pos;
+			}
+		}
+
+	return mat;
+}
+
+__isl_give isl_basic_map *isl_basic_map_from_constraint_matrices(
+	__isl_take isl_space *dim,
+	__isl_take isl_mat *eq, __isl_take isl_mat *ineq, enum isl_dim_type c1,
+	enum isl_dim_type c2, enum isl_dim_type c3,
+	enum isl_dim_type c4, enum isl_dim_type c5)
+{
+	enum isl_dim_type c[5] = { c1, c2, c3, c4, c5 };
+	isl_basic_map *bmap;
+	unsigned total;
+	unsigned extra;
+	int i, j, k, l;
+	int pos;
+
+	if (!dim || !eq || !ineq)
+		goto error;
+
+	if (eq->n_col != ineq->n_col)
+		isl_die(dim->ctx, isl_error_invalid,
+			"equalities and inequalities matrices should have "
+			"same number of columns", goto error);
+
+	total = 1 + isl_space_dim(dim, isl_dim_all);
+
+	if (eq->n_col < total)
+		isl_die(dim->ctx, isl_error_invalid,
+			"number of columns too small", goto error);
+
+	extra = eq->n_col - total;
+
+	bmap = isl_basic_map_alloc_space(isl_space_copy(dim), extra,
+				       eq->n_row, ineq->n_row);
+	if (!bmap)
+		goto error;
+	for (i = 0; i < extra; ++i) {
+		k = isl_basic_map_alloc_div(bmap);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(bmap->div[k][0], 0);
+	}
+	for (i = 0; i < eq->n_row; ++i) {
+		l = isl_basic_map_alloc_equality(bmap);
+		if (l < 0)
+			goto error;
+		for (j = 0, pos = 0; j < 5; ++j) {
+			int off = isl_basic_map_offset(bmap, c[j]);
+			for (k = 0; k < isl_basic_map_dim(bmap, c[j]); ++k) {
+				isl_int_set(bmap->eq[l][off + k], 
+					    eq->row[i][pos]);
+				++pos;
+			}
+		}
+	}
+	for (i = 0; i < ineq->n_row; ++i) {
+		l = isl_basic_map_alloc_inequality(bmap);
+		if (l < 0)
+			goto error;
+		for (j = 0, pos = 0; j < 5; ++j) {
+			int off = isl_basic_map_offset(bmap, c[j]);
+			for (k = 0; k < isl_basic_map_dim(bmap, c[j]); ++k) {
+				isl_int_set(bmap->ineq[l][off + k], 
+					    ineq->row[i][pos]);
+				++pos;
+			}
+		}
+	}
+
+	isl_space_free(dim);
+	isl_mat_free(eq);
+	isl_mat_free(ineq);
+
+	bmap = isl_basic_map_simplify(bmap);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_space_free(dim);
+	isl_mat_free(eq);
+	isl_mat_free(ineq);
+	return NULL;
+}
+
+__isl_give isl_mat *isl_basic_set_equalities_matrix(
+	__isl_keep isl_basic_set *bset, enum isl_dim_type c1,
+	enum isl_dim_type c2, enum isl_dim_type c3, enum isl_dim_type c4)
+{
+	return isl_basic_map_equalities_matrix((isl_basic_map *)bset,
+						c1, c2, c3, c4, isl_dim_in);
+}
+
+__isl_give isl_mat *isl_basic_set_inequalities_matrix(
+	__isl_keep isl_basic_set *bset, enum isl_dim_type c1,
+	enum isl_dim_type c2, enum isl_dim_type c3, enum isl_dim_type c4)
+{
+	return isl_basic_map_inequalities_matrix((isl_basic_map *)bset,
+						 c1, c2, c3, c4, isl_dim_in);
+}
+
+__isl_give isl_basic_set *isl_basic_set_from_constraint_matrices(
+	__isl_take isl_space *dim,
+	__isl_take isl_mat *eq, __isl_take isl_mat *ineq, enum isl_dim_type c1,
+	enum isl_dim_type c2, enum isl_dim_type c3, enum isl_dim_type c4)
+{
+	return (isl_basic_set*)
+	    isl_basic_map_from_constraint_matrices(dim, eq, ineq,
+						   c1, c2, c3, c4, isl_dim_in);
+}
+
+int isl_basic_map_can_zip(__isl_keep isl_basic_map *bmap)
+{
+	if (!bmap)
+		return -1;
+	
+	return isl_space_can_zip(bmap->dim);
+}
+
+int isl_map_can_zip(__isl_keep isl_map *map)
+{
+	if (!map)
+		return -1;
+	
+	return isl_space_can_zip(map->dim);
+}
+
+/* Given a basic map (A -> B) -> (C -> D), return the corresponding basic map
+ * (A -> C) -> (B -> D).
+ */
+__isl_give isl_basic_map *isl_basic_map_zip(__isl_take isl_basic_map *bmap)
+{
+	unsigned pos;
+	unsigned n1;
+	unsigned n2;
+
+	if (!bmap)
+		return NULL;
+
+	if (!isl_basic_map_can_zip(bmap))
+		isl_die(bmap->ctx, isl_error_invalid,
+			"basic map cannot be zipped", goto error);
+	pos = isl_basic_map_offset(bmap, isl_dim_in) +
+		isl_space_dim(bmap->dim->nested[0], isl_dim_in);
+	n1 = isl_space_dim(bmap->dim->nested[0], isl_dim_out);
+	n2 = isl_space_dim(bmap->dim->nested[1], isl_dim_in);
+	bmap = isl_basic_map_cow(bmap);
+	bmap = isl_basic_map_swap_vars(bmap, pos, n1, n2);
+	if (!bmap)
+		return NULL;
+	bmap->dim = isl_space_zip(bmap->dim);
+	if (!bmap->dim)
+		goto error;
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Given a map (A -> B) -> (C -> D), return the corresponding map
+ * (A -> C) -> (B -> D).
+ */
+__isl_give isl_map *isl_map_zip(__isl_take isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+
+	if (!isl_map_can_zip(map))
+		isl_die(map->ctx, isl_error_invalid, "map cannot be zipped",
+			goto error);
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_zip(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+
+	map->dim = isl_space_zip(map->dim);
+	if (!map->dim)
+		goto error;
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Can we apply isl_basic_map_curry to "bmap"?
+ * That is, does it have a nested relation in its domain?
+ */
+int isl_basic_map_can_curry(__isl_keep isl_basic_map *bmap)
+{
+	if (!bmap)
+		return -1;
+
+	return isl_space_can_curry(bmap->dim);
+}
+
+/* Can we apply isl_map_curry to "map"?
+ * That is, does it have a nested relation in its domain?
+ */
+int isl_map_can_curry(__isl_keep isl_map *map)
+{
+	if (!map)
+		return -1;
+
+	return isl_space_can_curry(map->dim);
+}
+
+/* Given a basic map (A -> B) -> C, return the corresponding basic map
+ * A -> (B -> C).
+ */
+__isl_give isl_basic_map *isl_basic_map_curry(__isl_take isl_basic_map *bmap)
+{
+
+	if (!bmap)
+		return NULL;
+
+	if (!isl_basic_map_can_curry(bmap))
+		isl_die(bmap->ctx, isl_error_invalid,
+			"basic map cannot be curried", goto error);
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+	bmap->dim = isl_space_curry(bmap->dim);
+	if (!bmap->dim)
+		goto error;
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Given a map (A -> B) -> C, return the corresponding map
+ * A -> (B -> C).
+ */
+__isl_give isl_map *isl_map_curry(__isl_take isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+
+	if (!isl_map_can_curry(map))
+		isl_die(map->ctx, isl_error_invalid, "map cannot be curried",
+			goto error);
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_curry(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+
+	map->dim = isl_space_curry(map->dim);
+	if (!map->dim)
+		goto error;
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Can we apply isl_basic_map_uncurry to "bmap"?
+ * That is, does it have a nested relation in its domain?
+ */
+int isl_basic_map_can_uncurry(__isl_keep isl_basic_map *bmap)
+{
+	if (!bmap)
+		return -1;
+
+	return isl_space_can_uncurry(bmap->dim);
+}
+
+/* Can we apply isl_map_uncurry to "map"?
+ * That is, does it have a nested relation in its domain?
+ */
+int isl_map_can_uncurry(__isl_keep isl_map *map)
+{
+	if (!map)
+		return -1;
+
+	return isl_space_can_uncurry(map->dim);
+}
+
+/* Given a basic map A -> (B -> C), return the corresponding basic map
+ * (A -> B) -> C.
+ */
+__isl_give isl_basic_map *isl_basic_map_uncurry(__isl_take isl_basic_map *bmap)
+{
+
+	if (!bmap)
+		return NULL;
+
+	if (!isl_basic_map_can_uncurry(bmap))
+		isl_die(bmap->ctx, isl_error_invalid,
+			"basic map cannot be uncurried",
+			return isl_basic_map_free(bmap));
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+	bmap->dim = isl_space_uncurry(bmap->dim);
+	if (!bmap->dim)
+		return isl_basic_map_free(bmap);
+	return bmap;
+}
+
+/* Given a map A -> (B -> C), return the corresponding map
+ * (A -> B) -> C.
+ */
+__isl_give isl_map *isl_map_uncurry(__isl_take isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+
+	if (!isl_map_can_uncurry(map))
+		isl_die(map->ctx, isl_error_invalid, "map cannot be uncurried",
+			return isl_map_free(map));
+
+	map = isl_map_cow(map);
+	if (!map)
+		return NULL;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_uncurry(map->p[i]);
+		if (!map->p[i])
+			return isl_map_free(map);
+	}
+
+	map->dim = isl_space_uncurry(map->dim);
+	if (!map->dim)
+		return isl_map_free(map);
+
+	return map;
+}
+
+/* Construct a basic map mapping the domain of the affine expression
+ * to a one-dimensional range prescribed by the affine expression.
+ */
+__isl_give isl_basic_map *isl_basic_map_from_aff(__isl_take isl_aff *aff)
+{
+	int k;
+	int pos;
+	isl_local_space *ls;
+	isl_basic_map *bmap;
+
+	if (!aff)
+		return NULL;
+
+	ls = isl_aff_get_local_space(aff);
+	bmap = isl_basic_map_from_local_space(ls);
+	bmap = isl_basic_map_extend_constraints(bmap, 1, 0);
+	k = isl_basic_map_alloc_equality(bmap);
+	if (k < 0)
+		goto error;
+
+	pos = isl_basic_map_offset(bmap, isl_dim_out);
+	isl_seq_cpy(bmap->eq[k], aff->v->el + 1, pos);
+	isl_int_neg(bmap->eq[k][pos], aff->v->el[0]);
+	isl_seq_cpy(bmap->eq[k] + pos + 1, aff->v->el + 1 + pos,
+		    aff->v->size - (pos + 1));
+
+	isl_aff_free(aff);
+	bmap = isl_basic_map_finalize(bmap);
+	return bmap;
+error:
+	isl_aff_free(aff);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Construct a map mapping the domain of the affine expression
+ * to a one-dimensional range prescribed by the affine expression.
+ */
+__isl_give isl_map *isl_map_from_aff(__isl_take isl_aff *aff)
+{
+	isl_basic_map *bmap;
+
+	bmap = isl_basic_map_from_aff(aff);
+	return isl_map_from_basic_map(bmap);
+}
+
+/* Construct a basic map mapping the domain the multi-affine expression
+ * to its range, with each dimension in the range equated to the
+ * corresponding affine expression.
+ */
+__isl_give isl_basic_map *isl_basic_map_from_multi_aff(
+	__isl_take isl_multi_aff *maff)
+{
+	int i;
+	isl_space *space;
+	isl_basic_map *bmap;
+
+	if (!maff)
+		return NULL;
+
+	if (isl_space_dim(maff->space, isl_dim_out) != maff->n)
+		isl_die(isl_multi_aff_get_ctx(maff), isl_error_internal,
+			"invalid space", goto error);
+
+	space = isl_space_domain(isl_multi_aff_get_space(maff));
+	bmap = isl_basic_map_universe(isl_space_from_domain(space));
+
+	for (i = 0; i < maff->n; ++i) {
+		isl_aff *aff;
+		isl_basic_map *bmap_i;
+
+		aff = isl_aff_copy(maff->p[i]);
+		bmap_i = isl_basic_map_from_aff(aff);
+
+		bmap = isl_basic_map_flat_range_product(bmap, bmap_i);
+	}
+
+	bmap = isl_basic_map_reset_space(bmap, isl_multi_aff_get_space(maff));
+
+	isl_multi_aff_free(maff);
+	return bmap;
+error:
+	isl_multi_aff_free(maff);
+	return NULL;
+}
+
+/* Construct a map mapping the domain the multi-affine expression
+ * to its range, with each dimension in the range equated to the
+ * corresponding affine expression.
+ */
+__isl_give isl_map *isl_map_from_multi_aff(__isl_take isl_multi_aff *maff)
+{
+	isl_basic_map *bmap;
+
+	bmap = isl_basic_map_from_multi_aff(maff);
+	return isl_map_from_basic_map(bmap);
+}
+
+/* Construct a basic map mapping a domain in the given space to
+ * to an n-dimensional range, with n the number of elements in the list,
+ * where each coordinate in the range is prescribed by the
+ * corresponding affine expression.
+ * The domains of all affine expressions in the list are assumed to match
+ * domain_dim.
+ */
+__isl_give isl_basic_map *isl_basic_map_from_aff_list(
+	__isl_take isl_space *domain_dim, __isl_take isl_aff_list *list)
+{
+	int i;
+	isl_space *dim;
+	isl_basic_map *bmap;
+
+	if (!list)
+		return NULL;
+
+	dim = isl_space_from_domain(domain_dim);
+	bmap = isl_basic_map_universe(dim);
+
+	for (i = 0; i < list->n; ++i) {
+		isl_aff *aff;
+		isl_basic_map *bmap_i;
+
+		aff = isl_aff_copy(list->p[i]);
+		bmap_i = isl_basic_map_from_aff(aff);
+
+		bmap = isl_basic_map_flat_range_product(bmap, bmap_i);
+	}
+
+	isl_aff_list_free(list);
+	return bmap;
+}
+
+__isl_give isl_set *isl_set_equate(__isl_take isl_set *set,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
+{
+	return isl_map_equate(set, type1, pos1, type2, pos2);
+}
+
+/* Construct a basic map where the given dimensions are equal to each other.
+ */
+static __isl_give isl_basic_map *equator(__isl_take isl_space *space,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
+{
+	isl_basic_map *bmap = NULL;
+	int i;
+
+	if (!space)
+		return NULL;
+
+	if (pos1 >= isl_space_dim(space, type1))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"index out of bounds", goto error);
+	if (pos2 >= isl_space_dim(space, type2))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"index out of bounds", goto error);
+
+	if (type1 == type2 && pos1 == pos2)
+		return isl_basic_map_universe(space);
+
+	bmap = isl_basic_map_alloc_space(isl_space_copy(space), 0, 1, 0);
+	i = isl_basic_map_alloc_equality(bmap);
+	if (i < 0)
+		goto error;
+	isl_seq_clr(bmap->eq[i], 1 + isl_basic_map_total_dim(bmap));
+	pos1 += isl_basic_map_offset(bmap, type1);
+	pos2 += isl_basic_map_offset(bmap, type2);
+	isl_int_set_si(bmap->eq[i][pos1], -1);
+	isl_int_set_si(bmap->eq[i][pos2], 1);
+	bmap = isl_basic_map_finalize(bmap);
+	isl_space_free(space);
+	return bmap;
+error:
+	isl_space_free(space);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Add a constraint imposing that the given two dimensions are equal.
+ */
+__isl_give isl_basic_map *isl_basic_map_equate(__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
+{
+	isl_basic_map *eq;
+
+	eq = equator(isl_basic_map_get_space(bmap), type1, pos1, type2, pos2);
+
+	bmap = isl_basic_map_intersect(bmap, eq);
+
+	return bmap;
+}
+
+/* Add a constraint imposing that the given two dimensions are equal.
+ */
+__isl_give isl_map *isl_map_equate(__isl_take isl_map *map,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
+{
+	isl_basic_map *bmap;
+
+	bmap = equator(isl_map_get_space(map), type1, pos1, type2, pos2);
+
+	map = isl_map_intersect(map, isl_map_from_basic_map(bmap));
+
+	return map;
+}
+
+/* Add a constraint imposing that the given two dimensions have opposite values.
+ */
+__isl_give isl_map *isl_map_oppose(__isl_take isl_map *map,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
+{
+	isl_basic_map *bmap = NULL;
+	int i;
+
+	if (!map)
+		return NULL;
+
+	if (pos1 >= isl_map_dim(map, type1))
+		isl_die(map->ctx, isl_error_invalid,
+			"index out of bounds", goto error);
+	if (pos2 >= isl_map_dim(map, type2))
+		isl_die(map->ctx, isl_error_invalid,
+			"index out of bounds", goto error);
+
+	bmap = isl_basic_map_alloc_space(isl_map_get_space(map), 0, 1, 0);
+	i = isl_basic_map_alloc_equality(bmap);
+	if (i < 0)
+		goto error;
+	isl_seq_clr(bmap->eq[i], 1 + isl_basic_map_total_dim(bmap));
+	pos1 += isl_basic_map_offset(bmap, type1);
+	pos2 += isl_basic_map_offset(bmap, type2);
+	isl_int_set_si(bmap->eq[i][pos1], 1);
+	isl_int_set_si(bmap->eq[i][pos2], 1);
+	bmap = isl_basic_map_finalize(bmap);
+
+	map = isl_map_intersect(map, isl_map_from_basic_map(bmap));
+
+	return map;
+error:
+	isl_basic_map_free(bmap);
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Construct a constraint imposing that the value of the first dimension is
+ * greater than or equal to that of the second.
+ */
+static __isl_give isl_constraint *constraint_order_ge(
+	__isl_take isl_space *space, enum isl_dim_type type1, int pos1,
+	enum isl_dim_type type2, int pos2)
+{
+	isl_constraint *c;
+
+	if (!space)
+		return NULL;
+
+	c = isl_inequality_alloc(isl_local_space_from_space(space));
+
+	if (pos1 >= isl_constraint_dim(c, type1))
+		isl_die(isl_constraint_get_ctx(c), isl_error_invalid,
+			"index out of bounds", return isl_constraint_free(c));
+	if (pos2 >= isl_constraint_dim(c, type2))
+		isl_die(isl_constraint_get_ctx(c), isl_error_invalid,
+			"index out of bounds", return isl_constraint_free(c));
+
+	if (type1 == type2 && pos1 == pos2)
+		return c;
+
+	c = isl_constraint_set_coefficient_si(c, type1, pos1, 1);
+	c = isl_constraint_set_coefficient_si(c, type2, pos2, -1);
+
+	return c;
+}
+
+/* Add a constraint imposing that the value of the first dimension is
+ * greater than or equal to that of the second.
+ */
+__isl_give isl_basic_map *isl_basic_map_order_ge(__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
+{
+	isl_constraint *c;
+	isl_space *space;
+
+	if (type1 == type2 && pos1 == pos2)
+		return bmap;
+	space = isl_basic_map_get_space(bmap);
+	c = constraint_order_ge(space, type1, pos1, type2, pos2);
+	bmap = isl_basic_map_add_constraint(bmap, c);
+
+	return bmap;
+}
+
+/* Add a constraint imposing that the value of the first dimension is
+ * greater than or equal to that of the second.
+ */
+__isl_give isl_map *isl_map_order_ge(__isl_take isl_map *map,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
+{
+	isl_constraint *c;
+	isl_space *space;
+
+	if (type1 == type2 && pos1 == pos2)
+		return map;
+	space = isl_map_get_space(map);
+	c = constraint_order_ge(space, type1, pos1, type2, pos2);
+	map = isl_map_add_constraint(map, c);
+
+	return map;
+}
+
+/* Add a constraint imposing that the value of the first dimension is
+ * less than or equal to that of the second.
+ */
+__isl_give isl_map *isl_map_order_le(__isl_take isl_map *map,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
+{
+	return isl_map_order_ge(map, type2, pos2, type1, pos1);
+}
+
+/* Construct a basic map where the value of the first dimension is
+ * greater than that of the second.
+ */
+static __isl_give isl_basic_map *greator(__isl_take isl_space *space,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
+{
+	isl_basic_map *bmap = NULL;
+	int i;
+
+	if (!space)
+		return NULL;
+
+	if (pos1 >= isl_space_dim(space, type1))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"index out of bounds", goto error);
+	if (pos2 >= isl_space_dim(space, type2))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"index out of bounds", goto error);
+
+	if (type1 == type2 && pos1 == pos2)
+		return isl_basic_map_empty(space);
+
+	bmap = isl_basic_map_alloc_space(space, 0, 0, 1);
+	i = isl_basic_map_alloc_inequality(bmap);
+	if (i < 0)
+		return isl_basic_map_free(bmap);
+	isl_seq_clr(bmap->ineq[i], 1 + isl_basic_map_total_dim(bmap));
+	pos1 += isl_basic_map_offset(bmap, type1);
+	pos2 += isl_basic_map_offset(bmap, type2);
+	isl_int_set_si(bmap->ineq[i][pos1], 1);
+	isl_int_set_si(bmap->ineq[i][pos2], -1);
+	isl_int_set_si(bmap->ineq[i][0], -1);
+	bmap = isl_basic_map_finalize(bmap);
+
+	return bmap;
+error:
+	isl_space_free(space);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Add a constraint imposing that the value of the first dimension is
+ * greater than that of the second.
+ */
+__isl_give isl_basic_map *isl_basic_map_order_gt(__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
+{
+	isl_basic_map *gt;
+
+	gt = greator(isl_basic_map_get_space(bmap), type1, pos1, type2, pos2);
+
+	bmap = isl_basic_map_intersect(bmap, gt);
+
+	return bmap;
+}
+
+/* Add a constraint imposing that the value of the first dimension is
+ * greater than that of the second.
+ */
+__isl_give isl_map *isl_map_order_gt(__isl_take isl_map *map,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
+{
+	isl_basic_map *bmap;
+
+	bmap = greator(isl_map_get_space(map), type1, pos1, type2, pos2);
+
+	map = isl_map_intersect(map, isl_map_from_basic_map(bmap));
+
+	return map;
+}
+
+/* Add a constraint imposing that the value of the first dimension is
+ * smaller than that of the second.
+ */
+__isl_give isl_map *isl_map_order_lt(__isl_take isl_map *map,
+	enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
+{
+	return isl_map_order_gt(map, type2, pos2, type1, pos1);
+}
+
+__isl_give isl_aff *isl_basic_map_get_div(__isl_keep isl_basic_map *bmap,
+	int pos)
+{
+	isl_aff *div;
+	isl_local_space *ls;
+
+	if (!bmap)
+		return NULL;
+
+	if (!isl_basic_map_divs_known(bmap))
+		isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
+			"some divs are unknown", return NULL);
+
+	ls = isl_basic_map_get_local_space(bmap);
+	div = isl_local_space_get_div(ls, pos);
+	isl_local_space_free(ls);
+
+	return div;
+}
+
+__isl_give isl_aff *isl_basic_set_get_div(__isl_keep isl_basic_set *bset,
+	int pos)
+{
+	return isl_basic_map_get_div(bset, pos);
+}
+
+/* Plug in "subs" for dimension "type", "pos" of "bset".
+ *
+ * Let i be the dimension to replace and let "subs" be of the form
+ *
+ *	f/d
+ *
+ * Any integer division with a non-zero coefficient for i,
+ *
+ *	floor((a i + g)/m)
+ *
+ * is replaced by
+ *
+ *	floor((a f + d g)/(m d))
+ *
+ * Constraints of the form
+ *
+ *	a i + g
+ *
+ * are replaced by
+ *
+ *	a f + d g
+ *
+ * We currently require that "subs" is an integral expression.
+ * Handling rational expressions may require us to add stride constraints
+ * as we do in isl_basic_set_preimage_multi_aff.
+ */
+__isl_give isl_basic_set *isl_basic_set_substitute(
+	__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned pos, __isl_keep isl_aff *subs)
+{
+	int i;
+	isl_int v;
+	isl_ctx *ctx;
+
+	if (bset && isl_basic_set_plain_is_empty(bset))
+		return bset;
+
+	bset = isl_basic_set_cow(bset);
+	if (!bset || !subs)
+		goto error;
+
+	ctx = isl_basic_set_get_ctx(bset);
+	if (!isl_space_is_equal(bset->dim, subs->ls->dim))
+		isl_die(ctx, isl_error_invalid,
+			"spaces don't match", goto error);
+	if (isl_local_space_dim(subs->ls, isl_dim_div) != 0)
+		isl_die(ctx, isl_error_unsupported,
+			"cannot handle divs yet", goto error);
+	if (!isl_int_is_one(subs->v->el[0]))
+		isl_die(ctx, isl_error_invalid,
+			"can only substitute integer expressions", goto error);
+
+	pos += isl_basic_set_offset(bset, type);
+
+	isl_int_init(v);
+
+	for (i = 0; i < bset->n_eq; ++i) {
+		if (isl_int_is_zero(bset->eq[i][pos]))
+			continue;
+		isl_int_set(v, bset->eq[i][pos]);
+		isl_int_set_si(bset->eq[i][pos], 0);
+		isl_seq_combine(bset->eq[i], subs->v->el[0], bset->eq[i],
+				v, subs->v->el + 1, subs->v->size - 1);
+	}
+
+	for (i = 0; i < bset->n_ineq; ++i) {
+		if (isl_int_is_zero(bset->ineq[i][pos]))
+			continue;
+		isl_int_set(v, bset->ineq[i][pos]);
+		isl_int_set_si(bset->ineq[i][pos], 0);
+		isl_seq_combine(bset->ineq[i], subs->v->el[0], bset->ineq[i],
+				v, subs->v->el + 1, subs->v->size - 1);
+	}
+
+	for (i = 0; i < bset->n_div; ++i) {
+		if (isl_int_is_zero(bset->div[i][1 + pos]))
+			continue;
+		isl_int_set(v, bset->div[i][1 + pos]);
+		isl_int_set_si(bset->div[i][1 + pos], 0);
+		isl_seq_combine(bset->div[i] + 1,
+				subs->v->el[0], bset->div[i] + 1,
+				v, subs->v->el + 1, subs->v->size - 1);
+		isl_int_mul(bset->div[i][0], bset->div[i][0], subs->v->el[0]);
+	}
+
+	isl_int_clear(v);
+
+	bset = isl_basic_set_simplify(bset);
+	return isl_basic_set_finalize(bset);
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Plug in "subs" for dimension "type", "pos" of "set".
+ */
+__isl_give isl_set *isl_set_substitute(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, __isl_keep isl_aff *subs)
+{
+	int i;
+
+	if (set && isl_set_plain_is_empty(set))
+		return set;
+
+	set = isl_set_cow(set);
+	if (!set || !subs)
+		goto error;
+
+	for (i = set->n - 1; i >= 0; --i) {
+		set->p[i] = isl_basic_set_substitute(set->p[i], type, pos, subs);
+		if (remove_if_empty(set, i) < 0)
+			goto error;
+	}
+
+	return set;
+error:
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Check if the range of "ma" is compatible with the domain or range
+ * (depending on "type") of "bmap".
+ * Return -1 if anything is wrong.
+ */
+static int check_basic_map_compatible_range_multi_aff(
+	__isl_keep isl_basic_map *bmap, enum isl_dim_type type,
+	__isl_keep isl_multi_aff *ma)
+{
+	int m;
+	isl_space *ma_space;
+
+	ma_space = isl_multi_aff_get_space(ma);
+
+	m = isl_space_match(bmap->dim, isl_dim_param, ma_space, isl_dim_param);
+	if (m < 0)
+		goto error;
+	if (!m)
+		isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
+			"parameters don't match", goto error);
+	m = isl_space_tuple_is_equal(bmap->dim, type, ma_space, isl_dim_out);
+	if (m < 0)
+		goto error;
+	if (!m)
+		isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
+			"spaces don't match", goto error);
+
+	isl_space_free(ma_space);
+	return m;
+error:
+	isl_space_free(ma_space);
+	return -1;
+}
+
+/* Copy the divs from "ma" to "bmap", adding zeros for the "n_before"
+ * coefficients before the transformed range of dimensions,
+ * the "n_after" coefficients after the transformed range of dimensions
+ * and the coefficients of the other divs in "bmap".
+ */
+static int set_ma_divs(__isl_keep isl_basic_map *bmap,
+	__isl_keep isl_multi_aff *ma, int n_before, int n_after, int n_div)
+{
+	int i;
+	int n_param;
+	int n_set;
+	isl_local_space *ls;
+
+	if (n_div == 0)
+		return 0;
+
+	ls = isl_aff_get_domain_local_space(ma->p[0]);
+	if (!ls)
+		return -1;
+
+	n_param = isl_local_space_dim(ls, isl_dim_param);
+	n_set = isl_local_space_dim(ls, isl_dim_set);
+	for (i = 0; i < n_div; ++i) {
+		int o_bmap = 0, o_ls = 0;
+
+		isl_seq_cpy(bmap->div[i], ls->div->row[i], 1 + 1 + n_param);
+		o_bmap += 1 + 1 + n_param;
+		o_ls += 1 + 1 + n_param;
+		isl_seq_clr(bmap->div[i] + o_bmap, n_before);
+		o_bmap += n_before;
+		isl_seq_cpy(bmap->div[i] + o_bmap,
+			    ls->div->row[i] + o_ls, n_set);
+		o_bmap += n_set;
+		o_ls += n_set;
+		isl_seq_clr(bmap->div[i] + o_bmap, n_after);
+		o_bmap += n_after;
+		isl_seq_cpy(bmap->div[i] + o_bmap,
+			    ls->div->row[i] + o_ls, n_div);
+		o_bmap += n_div;
+		o_ls += n_div;
+		isl_seq_clr(bmap->div[i] + o_bmap, bmap->n_div - n_div);
+		if (isl_basic_set_add_div_constraints(bmap, i) < 0)
+			goto error;
+	}
+
+	isl_local_space_free(ls);
+	return 0;
+error:
+	isl_local_space_free(ls);
+	return -1;
+}
+
+/* How many stride constraints does "ma" enforce?
+ * That is, how many of the affine expressions have a denominator
+ * different from one?
+ */
+static int multi_aff_strides(__isl_keep isl_multi_aff *ma)
+{
+	int i;
+	int strides = 0;
+
+	for (i = 0; i < ma->n; ++i)
+		if (!isl_int_is_one(ma->p[i]->v->el[0]))
+			strides++;
+
+	return strides;
+}
+
+/* For each affine expression in ma of the form
+ *
+ *	x_i = (f_i y + h_i)/m_i
+ *
+ * with m_i different from one, add a constraint to "bmap"
+ * of the form
+ *
+ *	f_i y + h_i = m_i alpha_i
+ *
+ * with alpha_i an additional existentially quantified variable.
+ */
+static __isl_give isl_basic_map *add_ma_strides(
+	__isl_take isl_basic_map *bmap, __isl_keep isl_multi_aff *ma,
+	int n_before, int n_after)
+{
+	int i, k;
+	int div;
+	int total;
+	int n_param;
+	int n_in;
+	int n_div;
+
+	total = isl_basic_map_total_dim(bmap);
+	n_param = isl_multi_aff_dim(ma, isl_dim_param);
+	n_in = isl_multi_aff_dim(ma, isl_dim_in);
+	n_div = isl_multi_aff_dim(ma, isl_dim_div);
+	for (i = 0; i < ma->n; ++i) {
+		int o_bmap = 0, o_ma = 1;
+
+		if (isl_int_is_one(ma->p[i]->v->el[0]))
+			continue;
+		div = isl_basic_map_alloc_div(bmap);
+		k = isl_basic_map_alloc_equality(bmap);
+		if (div < 0 || k < 0)
+			goto error;
+		isl_int_set_si(bmap->div[div][0], 0);
+		isl_seq_cpy(bmap->eq[k] + o_bmap,
+			    ma->p[i]->v->el + o_ma, 1 + n_param);
+		o_bmap += 1 + n_param;
+		o_ma += 1 + n_param;
+		isl_seq_clr(bmap->eq[k] + o_bmap, n_before);
+		o_bmap += n_before;
+		isl_seq_cpy(bmap->eq[k] + o_bmap,
+			    ma->p[i]->v->el + o_ma, n_in);
+		o_bmap += n_in;
+		o_ma += n_in;
+		isl_seq_clr(bmap->eq[k] + o_bmap, n_after);
+		o_bmap += n_after;
+		isl_seq_cpy(bmap->eq[k] + o_bmap,
+			    ma->p[i]->v->el + o_ma, n_div);
+		o_bmap += n_div;
+		o_ma += n_div;
+		isl_seq_clr(bmap->eq[k] + o_bmap, 1 + total - o_bmap);
+		isl_int_neg(bmap->eq[k][1 + total], ma->p[i]->v->el[0]);
+		total++;
+	}
+
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Replace the domain or range space (depending on "type) of "space" by "set".
+ */
+static __isl_give isl_space *isl_space_set(__isl_take isl_space *space,
+	enum isl_dim_type type, __isl_take isl_space *set)
+{
+	if (type == isl_dim_in) {
+		space = isl_space_range(space);
+		space = isl_space_map_from_domain_and_range(set, space);
+	} else {
+		space = isl_space_domain(space);
+		space = isl_space_map_from_domain_and_range(space, set);
+	}
+
+	return space;
+}
+
+/* Compute the preimage of the domain or range (depending on "type")
+ * of "bmap" under the function represented by "ma".
+ * In other words, plug in "ma" in the domain or range of "bmap".
+ * The result is a basic map that lives in the same space as "bmap"
+ * except that the domain or range has been replaced by
+ * the domain space of "ma".
+ *
+ * If bmap is represented by
+ *
+ *	A(p) + S u + B x + T v + C(divs) >= 0,
+ *
+ * where u and x are input and output dimensions if type == isl_dim_out
+ * while x and v are input and output dimensions if type == isl_dim_in,
+ * and ma is represented by
+ *
+ *	x = D(p) + F(y) + G(divs')
+ *
+ * then the result is
+ *
+ *	A(p) + B D(p) + S u + B F(y) + T v + B G(divs') + C(divs) >= 0
+ *
+ * The divs in the input set are similarly adjusted.
+ * In particular
+ *
+ *	floor((a_i(p) + s u + b_i x + t v + c_i(divs))/n_i)
+ *
+ * becomes
+ *
+ *	floor((a_i(p) + b_i D(p) + s u + b_i F(y) + t v +
+ *		B_i G(divs') + c_i(divs))/n_i)
+ *
+ * If bmap is not a rational map and if F(y) involves any denominators
+ *
+ *	x_i = (f_i y + h_i)/m_i
+ *
+ * then additional constraints are added to ensure that we only
+ * map back integer points.  That is we enforce
+ *
+ *	f_i y + h_i = m_i alpha_i
+ *
+ * with alpha_i an additional existentially quantified variable.
+ *
+ * We first copy over the divs from "ma".
+ * Then we add the modified constraints and divs from "bmap".
+ * Finally, we add the stride constraints, if needed.
+ */
+__isl_give isl_basic_map *isl_basic_map_preimage_multi_aff(
+	__isl_take isl_basic_map *bmap, enum isl_dim_type type,
+	__isl_take isl_multi_aff *ma)
+{
+	int i, k;
+	isl_space *space;
+	isl_basic_map *res = NULL;
+	int n_before, n_after, n_div_bmap, n_div_ma;
+	isl_int f, c1, c2, g;
+	int rational, strides;
+
+	isl_int_init(f);
+	isl_int_init(c1);
+	isl_int_init(c2);
+	isl_int_init(g);
+
+	ma = isl_multi_aff_align_divs(ma);
+	if (!bmap || !ma)
+		goto error;
+	if (check_basic_map_compatible_range_multi_aff(bmap, type, ma) < 0)
+		goto error;
+
+	if (type == isl_dim_in) {
+		n_before = 0;
+		n_after = isl_basic_map_dim(bmap, isl_dim_out);
+	} else {
+		n_before = isl_basic_map_dim(bmap, isl_dim_in);
+		n_after = 0;
+	}
+	n_div_bmap = isl_basic_map_dim(bmap, isl_dim_div);
+	n_div_ma = ma->n ? isl_aff_dim(ma->p[0], isl_dim_div) : 0;
+
+	space = isl_multi_aff_get_domain_space(ma);
+	space = isl_space_set(isl_basic_map_get_space(bmap), type, space);
+	rational = isl_basic_map_is_rational(bmap);
+	strides = rational ? 0 : multi_aff_strides(ma);
+	res = isl_basic_map_alloc_space(space, n_div_ma + n_div_bmap + strides,
+			    bmap->n_eq + strides, bmap->n_ineq + 2 * n_div_ma);
+	if (rational)
+		res = isl_basic_map_set_rational(res);
+
+	for (i = 0; i < n_div_ma + n_div_bmap; ++i)
+		if (isl_basic_map_alloc_div(res) < 0)
+			goto error;
+
+	if (set_ma_divs(res, ma, n_before, n_after, n_div_ma) < 0)
+		goto error;
+
+	for (i = 0; i < bmap->n_eq; ++i) {
+		k = isl_basic_map_alloc_equality(res);
+		if (k < 0)
+			goto error;
+		isl_seq_preimage(res->eq[k], bmap->eq[i], ma, n_before,
+				n_after, n_div_ma, n_div_bmap, f, c1, c2, g, 0);
+	}
+
+	for (i = 0; i < bmap->n_ineq; ++i) {
+		k = isl_basic_map_alloc_inequality(res);
+		if (k < 0)
+			goto error;
+		isl_seq_preimage(res->ineq[k], bmap->ineq[i], ma, n_before,
+				n_after, n_div_ma, n_div_bmap, f, c1, c2, g, 0);
+	}
+
+	for (i = 0; i < bmap->n_div; ++i) {
+		if (isl_int_is_zero(bmap->div[i][0])) {
+			isl_int_set_si(res->div[n_div_ma + i][0], 0);
+			continue;
+		}
+		isl_seq_preimage(res->div[n_div_ma + i], bmap->div[i], ma,
+				    n_before, n_after, n_div_ma, n_div_bmap,
+				    f, c1, c2, g, 1);
+	}
+
+	if (strides)
+		res = add_ma_strides(res, ma, n_before, n_after);
+
+	isl_int_clear(f);
+	isl_int_clear(c1);
+	isl_int_clear(c2);
+	isl_int_clear(g);
+	isl_basic_map_free(bmap);
+	isl_multi_aff_free(ma);
+	res = isl_basic_set_simplify(res);
+	return isl_basic_map_finalize(res);
+error:
+	isl_int_clear(f);
+	isl_int_clear(c1);
+	isl_int_clear(c2);
+	isl_int_clear(g);
+	isl_basic_map_free(bmap);
+	isl_multi_aff_free(ma);
+	isl_basic_map_free(res);
+	return NULL;
+}
+
+/* Compute the preimage of "bset" under the function represented by "ma".
+ * In other words, plug in "ma" in "bset".  The result is a basic set
+ * that lives in the domain space of "ma".
+ */
+__isl_give isl_basic_set *isl_basic_set_preimage_multi_aff(
+	__isl_take isl_basic_set *bset, __isl_take isl_multi_aff *ma)
+{
+	return isl_basic_map_preimage_multi_aff(bset, isl_dim_set, ma);
+}
+
+/* Compute the preimage of the domain of "bmap" under the function
+ * represented by "ma".
+ * In other words, plug in "ma" in the domain of "bmap".
+ * The result is a basic map that lives in the same space as "bmap"
+ * except that the domain has been replaced by the domain space of "ma".
+ */
+__isl_give isl_basic_map *isl_basic_map_preimage_domain_multi_aff(
+	__isl_take isl_basic_map *bmap, __isl_take isl_multi_aff *ma)
+{
+	return isl_basic_map_preimage_multi_aff(bmap, isl_dim_in, ma);
+}
+
+/* Compute the preimage of the range of "bmap" under the function
+ * represented by "ma".
+ * In other words, plug in "ma" in the range of "bmap".
+ * The result is a basic map that lives in the same space as "bmap"
+ * except that the range has been replaced by the domain space of "ma".
+ */
+__isl_give isl_basic_map *isl_basic_map_preimage_range_multi_aff(
+	__isl_take isl_basic_map *bmap, __isl_take isl_multi_aff *ma)
+{
+	return isl_basic_map_preimage_multi_aff(bmap, isl_dim_out, ma);
+}
+
+/* Check if the range of "ma" is compatible with the domain or range
+ * (depending on "type") of "map".
+ * Return -1 if anything is wrong.
+ */
+static int check_map_compatible_range_multi_aff(
+	__isl_keep isl_map *map, enum isl_dim_type type,
+	__isl_keep isl_multi_aff *ma)
+{
+	int m;
+	isl_space *ma_space;
+
+	ma_space = isl_multi_aff_get_space(ma);
+	m = isl_space_tuple_is_equal(map->dim, type, ma_space, isl_dim_out);
+	isl_space_free(ma_space);
+	if (m >= 0 && !m)
+		isl_die(isl_map_get_ctx(map), isl_error_invalid,
+			"spaces don't match", return -1);
+	return m;
+}
+
+/* Compute the preimage of the domain or range (depending on "type")
+ * of "map" under the function represented by "ma".
+ * In other words, plug in "ma" in the domain or range of "map".
+ * The result is a map that lives in the same space as "map"
+ * except that the domain or range has been replaced by
+ * the domain space of "ma".
+ *
+ * The parameters are assumed to have been aligned.
+ */
+static __isl_give isl_map *map_preimage_multi_aff(__isl_take isl_map *map,
+	enum isl_dim_type type, __isl_take isl_multi_aff *ma)
+{
+	int i;
+	isl_space *space;
+
+	map = isl_map_cow(map);
+	ma = isl_multi_aff_align_divs(ma);
+	if (!map || !ma)
+		goto error;
+	if (check_map_compatible_range_multi_aff(map, type, ma) < 0)
+		goto error;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_preimage_multi_aff(map->p[i], type,
+							isl_multi_aff_copy(ma));
+		if (!map->p[i])
+			goto error;
+	}
+
+	space = isl_multi_aff_get_domain_space(ma);
+	space = isl_space_set(isl_map_get_space(map), type, space);
+
+	isl_space_free(map->dim);
+	map->dim = space;
+	if (!map->dim)
+		goto error;
+
+	isl_multi_aff_free(ma);
+	if (map->n > 1)
+		ISL_F_CLR(map, ISL_MAP_DISJOINT);
+	ISL_F_CLR(map, ISL_SET_NORMALIZED);
+	return map;
+error:
+	isl_multi_aff_free(ma);
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Compute the preimage of the domain or range (depending on "type")
+ * of "map" under the function represented by "ma".
+ * In other words, plug in "ma" in the domain or range of "map".
+ * The result is a map that lives in the same space as "map"
+ * except that the domain or range has been replaced by
+ * the domain space of "ma".
+ */
+__isl_give isl_map *isl_map_preimage_multi_aff(__isl_take isl_map *map,
+	enum isl_dim_type type, __isl_take isl_multi_aff *ma)
+{
+	if (!map || !ma)
+		goto error;
+
+	if (isl_space_match(map->dim, isl_dim_param, ma->space, isl_dim_param))
+		return map_preimage_multi_aff(map, type, ma);
+
+	if (!isl_space_has_named_params(map->dim) ||
+	    !isl_space_has_named_params(ma->space))
+		isl_die(map->ctx, isl_error_invalid,
+			"unaligned unnamed parameters", goto error);
+	map = isl_map_align_params(map, isl_multi_aff_get_space(ma));
+	ma = isl_multi_aff_align_params(ma, isl_map_get_space(map));
+
+	return map_preimage_multi_aff(map, type, ma);
+error:
+	isl_multi_aff_free(ma);
+	return isl_map_free(map);
+}
+
+/* Compute the preimage of "set" under the function represented by "ma".
+ * In other words, plug in "ma" in "set".  The result is a set
+ * that lives in the domain space of "ma".
+ */
+__isl_give isl_set *isl_set_preimage_multi_aff(__isl_take isl_set *set,
+	__isl_take isl_multi_aff *ma)
+{
+	return isl_map_preimage_multi_aff(set, isl_dim_set, ma);
+}
+
+/* Compute the preimage of the domain of "map" under the function
+ * represented by "ma".
+ * In other words, plug in "ma" in the domain of "map".
+ * The result is a map that lives in the same space as "map"
+ * except that the domain has been replaced by the domain space of "ma".
+ */
+__isl_give isl_map *isl_map_preimage_domain_multi_aff(__isl_take isl_map *map,
+	__isl_take isl_multi_aff *ma)
+{
+	return isl_map_preimage_multi_aff(map, isl_dim_in, ma);
+}
+
+/* Compute the preimage of the range of "map" under the function
+ * represented by "ma".
+ * In other words, plug in "ma" in the range of "map".
+ * The result is a map that lives in the same space as "map"
+ * except that the range has been replaced by the domain space of "ma".
+ */
+__isl_give isl_map *isl_map_preimage_range_multi_aff(__isl_take isl_map *map,
+	__isl_take isl_multi_aff *ma)
+{
+	return isl_map_preimage_multi_aff(map, isl_dim_out, ma);
+}
+
+/* Compute the preimage of "map" under the function represented by "pma".
+ * In other words, plug in "pma" in the domain or range of "map".
+ * The result is a map that lives in the same space as "map",
+ * except that the space of type "type" has been replaced by
+ * the domain space of "pma".
+ *
+ * The parameters of "map" and "pma" are assumed to have been aligned.
+ */
+static __isl_give isl_map *isl_map_preimage_pw_multi_aff_aligned(
+	__isl_take isl_map *map, enum isl_dim_type type,
+	__isl_take isl_pw_multi_aff *pma)
+{
+	int i;
+	isl_map *res;
+
+	if (!pma)
+		goto error;
+
+	if (pma->n == 0) {
+		isl_pw_multi_aff_free(pma);
+		res = isl_map_empty(isl_map_get_space(map));
+		isl_map_free(map);
+		return res;
+	}
+
+	res = isl_map_preimage_multi_aff(isl_map_copy(map), type,
+					isl_multi_aff_copy(pma->p[0].maff));
+	if (type == isl_dim_in)
+		res = isl_map_intersect_domain(res,
+						isl_map_copy(pma->p[0].set));
+	else
+		res = isl_map_intersect_range(res,
+						isl_map_copy(pma->p[0].set));
+
+	for (i = 1; i < pma->n; ++i) {
+		isl_map *res_i;
+
+		res_i = isl_map_preimage_multi_aff(isl_map_copy(map), type,
+					isl_multi_aff_copy(pma->p[i].maff));
+		if (type == isl_dim_in)
+			res_i = isl_map_intersect_domain(res_i,
+						isl_map_copy(pma->p[i].set));
+		else
+			res_i = isl_map_intersect_range(res_i,
+						isl_map_copy(pma->p[i].set));
+		res = isl_map_union(res, res_i);
+	}
+
+	isl_pw_multi_aff_free(pma);
+	isl_map_free(map);
+	return res;
+error:
+	isl_pw_multi_aff_free(pma);
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Compute the preimage of "map" under the function represented by "pma".
+ * In other words, plug in "pma" in the domain or range of "map".
+ * The result is a map that lives in the same space as "map",
+ * except that the space of type "type" has been replaced by
+ * the domain space of "pma".
+ */
+__isl_give isl_map *isl_map_preimage_pw_multi_aff(__isl_take isl_map *map,
+	enum isl_dim_type type, __isl_take isl_pw_multi_aff *pma)
+{
+	if (!map || !pma)
+		goto error;
+
+	if (isl_space_match(map->dim, isl_dim_param, pma->dim, isl_dim_param))
+		return isl_map_preimage_pw_multi_aff_aligned(map, type, pma);
+
+	if (!isl_space_has_named_params(map->dim) ||
+	    !isl_space_has_named_params(pma->dim))
+		isl_die(map->ctx, isl_error_invalid,
+			"unaligned unnamed parameters", goto error);
+	map = isl_map_align_params(map, isl_pw_multi_aff_get_space(pma));
+	pma = isl_pw_multi_aff_align_params(pma, isl_map_get_space(map));
+
+	return isl_map_preimage_pw_multi_aff_aligned(map, type, pma);
+error:
+	isl_pw_multi_aff_free(pma);
+	return isl_map_free(map);
+}
+
+/* Compute the preimage of "set" under the function represented by "pma".
+ * In other words, plug in "pma" in "set".  The result is a set
+ * that lives in the domain space of "pma".
+ */
+__isl_give isl_set *isl_set_preimage_pw_multi_aff(__isl_take isl_set *set,
+	__isl_take isl_pw_multi_aff *pma)
+{
+	return isl_map_preimage_pw_multi_aff(set, isl_dim_set, pma);
+}
+
+/* Compute the preimage of the domain of "map" under the function
+ * represented by "pma".
+ * In other words, plug in "pma" in the domain of "map".
+ * The result is a map that lives in the same space as "map",
+ * except that domain space has been replaced by the domain space of "pma".
+ */
+__isl_give isl_map *isl_map_preimage_domain_pw_multi_aff(
+	__isl_take isl_map *map, __isl_take isl_pw_multi_aff *pma)
+{
+	return isl_map_preimage_pw_multi_aff(map, isl_dim_in, pma);
+}
+
+/* Compute the preimage of the range of "map" under the function
+ * represented by "pma".
+ * In other words, plug in "pma" in the range of "map".
+ * The result is a map that lives in the same space as "map",
+ * except that range space has been replaced by the domain space of "pma".
+ */
+__isl_give isl_map *isl_map_preimage_range_pw_multi_aff(
+	__isl_take isl_map *map, __isl_take isl_pw_multi_aff *pma)
+{
+	return isl_map_preimage_pw_multi_aff(map, isl_dim_out, pma);
+}
+
+/* Compute the preimage of "map" under the function represented by "mpa".
+ * In other words, plug in "mpa" in the domain or range of "map".
+ * The result is a map that lives in the same space as "map",
+ * except that the space of type "type" has been replaced by
+ * the domain space of "mpa".
+ *
+ * If the map does not involve any constraints that refer to the
+ * dimensions of the substituted space, then the only possible
+ * effect of "mpa" on the map is to map the space to a different space.
+ * We create a separate isl_multi_aff to effectuate this change
+ * in order to avoid spurious splitting of the map along the pieces
+ * of "mpa".
+ */
+__isl_give isl_map *isl_map_preimage_multi_pw_aff(__isl_take isl_map *map,
+	enum isl_dim_type type, __isl_take isl_multi_pw_aff *mpa)
+{
+	int n;
+	isl_pw_multi_aff *pma;
+
+	if (!map || !mpa)
+		goto error;
+
+	n = isl_map_dim(map, type);
+	if (!isl_map_involves_dims(map, type, 0, n)) {
+		isl_space *space;
+		isl_multi_aff *ma;
+
+		space = isl_multi_pw_aff_get_space(mpa);
+		isl_multi_pw_aff_free(mpa);
+		ma = isl_multi_aff_zero(space);
+		return isl_map_preimage_multi_aff(map, type, ma);
+	}
+
+	pma = isl_pw_multi_aff_from_multi_pw_aff(mpa);
+	return isl_map_preimage_pw_multi_aff(map, type, pma);
+error:
+	isl_map_free(map);
+	isl_multi_pw_aff_free(mpa);
+	return NULL;
+}
+
+/* Compute the preimage of "map" under the function represented by "mpa".
+ * In other words, plug in "mpa" in the domain "map".
+ * The result is a map that lives in the same space as "map",
+ * except that domain space has been replaced by the domain space of "mpa".
+ */
+__isl_give isl_map *isl_map_preimage_domain_multi_pw_aff(
+	__isl_take isl_map *map, __isl_take isl_multi_pw_aff *mpa)
+{
+	return isl_map_preimage_multi_pw_aff(map, isl_dim_in, mpa);
+}
+
+/* Compute the preimage of "set" by the function represented by "mpa".
+ * In other words, plug in "mpa" in "set".
+ */
+__isl_give isl_set *isl_set_preimage_multi_pw_aff(__isl_take isl_set *set,
+	__isl_take isl_multi_pw_aff *mpa)
+{
+	return isl_map_preimage_multi_pw_aff(set, isl_dim_set, mpa);
+}
diff --git a/polly/lib/External/isl/isl_map_lexopt_templ.c b/polly/lib/External/isl/isl_map_lexopt_templ.c
new file mode 100644
index 00000000000..1196a87f7ea
--- /dev/null
+++ b/polly/lib/External/isl/isl_map_lexopt_templ.c
@@ -0,0 +1,142 @@
+/*
+ * Copyright 2010      INRIA Saclay
+ * Copyright 2012      Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege,
+ * INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+/* Function for computing the lexicographic optimum of a map
+ * in the form of either an isl_map or an isl_pw_multi_aff.
+ */
+
+#define xSF(TYPE,SUFFIX) TYPE ## SUFFIX
+#define SF(TYPE,SUFFIX) xSF(TYPE,SUFFIX)
+
+/* Given a basic map "bmap", compute the lexicographically minimal
+ * (or maximal) image element for each domain element in dom.
+ * Set *empty to those elements in dom that do not have an image element.
+ *
+ * We first make sure the basic sets in dom are disjoint and then
+ * simply collect the results over each of the basic sets separately.
+ * We could probably improve the efficiency a bit by moving the union
+ * domain down into the parametric integer programming.
+ */
+static __isl_give TYPE *SF(basic_map_partial_lexopt,SUFFIX)(
+	__isl_take isl_basic_map *bmap, __isl_take isl_set *dom,
+	__isl_give isl_set **empty, int max)
+{
+	int i;
+	TYPE *res;
+
+	dom = isl_set_make_disjoint(dom);
+	if (!dom)
+		goto error;
+
+	if (isl_set_plain_is_empty(dom)) {
+		isl_space *space = isl_basic_map_get_space(bmap);
+		if (empty)
+			*empty = dom;
+		else
+			isl_set_free(dom);
+		isl_basic_map_free(bmap);
+		return EMPTY(space);
+	}
+
+	res = SF(isl_basic_map_partial_lexopt,SUFFIX)(isl_basic_map_copy(bmap),
+			isl_basic_set_copy(dom->p[0]), empty, max);
+		
+	for (i = 1; i < dom->n; ++i) {
+		TYPE *res_i;
+		isl_set *empty_i;
+
+		res_i = SF(isl_basic_map_partial_lexopt,SUFFIX)(
+				isl_basic_map_copy(bmap),
+				isl_basic_set_copy(dom->p[i]), &empty_i, max);
+
+		res = ADD(res, res_i);
+		*empty = isl_set_union_disjoint(*empty, empty_i);
+	}
+
+	isl_set_free(dom);
+	isl_basic_map_free(bmap);
+	return res;
+error:
+	*empty = NULL;
+	isl_set_free(dom);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+static __isl_give TYPE *SF(isl_map_partial_lexopt_aligned,SUFFIX)(
+	__isl_take isl_map *map, __isl_take isl_set *dom,
+	__isl_give isl_set **empty, int max);
+
+/* Given a map "map", compute the lexicographically minimal
+ * (or maximal) image element for each domain element in dom.
+ * Set *empty to those elements in dom that do not have an image element.
+ *
+ * Align parameters if needed and then call isl_map_partial_lexopt_aligned.
+ */
+static __isl_give TYPE *SF(isl_map_partial_lexopt,SUFFIX)(
+	__isl_take isl_map *map, __isl_take isl_set *dom,
+	__isl_give isl_set **empty, int max)
+{
+	if (!map || !dom)
+		goto error;
+	if (isl_space_match(map->dim, isl_dim_param, dom->dim, isl_dim_param))
+		return SF(isl_map_partial_lexopt_aligned,SUFFIX)(map, dom,
+								empty, max);
+	if (!isl_space_has_named_params(map->dim) ||
+	    !isl_space_has_named_params(dom->dim))
+		isl_die(map->ctx, isl_error_invalid,
+			"unaligned unnamed parameters", goto error);
+	map = isl_map_align_params(map, isl_map_get_space(dom));
+	dom = isl_map_align_params(dom, isl_map_get_space(map));
+	return SF(isl_map_partial_lexopt_aligned,SUFFIX)(map, dom, empty, max);
+error:
+	if (empty)
+		*empty = NULL;
+	isl_set_free(dom);
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give TYPE *SF(isl_map_lexopt,SUFFIX)(__isl_take isl_map *map, int max)
+{
+	isl_set *dom = NULL;
+	isl_space *dom_space;
+
+	if (!map)
+		goto error;
+	dom_space = isl_space_domain(isl_space_copy(map->dim));
+	dom = isl_set_universe(dom_space);
+	return SF(isl_map_partial_lexopt,SUFFIX)(map, dom, NULL, max);
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give TYPE *SF(isl_map_lexmin,SUFFIX)(__isl_take isl_map *map)
+{
+	return SF(isl_map_lexopt,SUFFIX)(map, 0);
+}
+
+__isl_give TYPE *SF(isl_map_lexmax,SUFFIX)(__isl_take isl_map *map)
+{
+	return SF(isl_map_lexopt,SUFFIX)(map, 1);
+}
+
+__isl_give TYPE *SF(isl_set_lexmin,SUFFIX)(__isl_take isl_set *set)
+{
+	return SF(isl_map_lexmin,SUFFIX)(set);
+}
+
+__isl_give TYPE *SF(isl_set_lexmax,SUFFIX)(__isl_take isl_set *set)
+{
+	return SF(isl_map_lexmax,SUFFIX)(set);
+}
diff --git a/polly/lib/External/isl/isl_map_private.h b/polly/lib/External/isl/isl_map_private.h
new file mode 100644
index 00000000000..7f1cd589b56
--- /dev/null
+++ b/polly/lib/External/isl/isl_map_private.h
@@ -0,0 +1,408 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_MAP_PRIVATE_H
+#define ISL_MAP_PRIVATE_H
+
+#define isl_basic_set	isl_basic_map
+#define isl_set		isl_map
+#define isl_basic_set_list	isl_basic_map_list
+#define isl_set_list	isl_map_list
+#include <isl/list.h>
+ISL_DECLARE_LIST(basic_map)
+ISL_DECLARE_LIST(map)
+#include <isl/set.h>
+#include <isl/map.h>
+#include <isl_reordering.h>
+#include <isl/vec.h>
+#include <isl_blk.h>
+
+/* A "basic map" is a relation between two sets of variables,
+ * called the "in" and "out" variables.
+ * A "basic set" is a basic map with a zero-dimensional
+ * domain.
+ *
+ * It is implemented as a set with two extra fields:
+ * n_in is the number of in variables
+ * n_out is the number of out variables
+ * n_in + n_out should be equal to set.dim
+ */
+struct isl_basic_map {
+	int ref;
+#define ISL_BASIC_MAP_FINAL		(1 << 0)
+#define ISL_BASIC_MAP_EMPTY		(1 << 1)
+#define ISL_BASIC_MAP_NO_IMPLICIT	(1 << 2)
+#define ISL_BASIC_MAP_NO_REDUNDANT	(1 << 3)
+#define ISL_BASIC_MAP_RATIONAL		(1 << 4)
+#define ISL_BASIC_MAP_NORMALIZED	(1 << 5)
+#define ISL_BASIC_MAP_NORMALIZED_DIVS	(1 << 6)
+#define ISL_BASIC_MAP_ALL_EQUALITIES	(1 << 7)
+#define ISL_BASIC_SET_FINAL		(1 << 0)
+#define ISL_BASIC_SET_EMPTY		(1 << 1)
+#define ISL_BASIC_SET_NO_IMPLICIT	(1 << 2)
+#define ISL_BASIC_SET_NO_REDUNDANT	(1 << 3)
+#define ISL_BASIC_SET_RATIONAL		(1 << 4)
+#define ISL_BASIC_SET_NORMALIZED	(1 << 5)
+#define ISL_BASIC_SET_NORMALIZED_DIVS	(1 << 6)
+#define ISL_BASIC_SET_ALL_EQUALITIES	(1 << 7)
+	unsigned flags;
+
+	struct isl_ctx *ctx;
+
+	isl_space *dim;
+	unsigned extra;
+
+	unsigned n_eq;
+	unsigned n_ineq;
+
+	size_t c_size;
+	isl_int **eq;
+	isl_int **ineq;
+
+	unsigned n_div;
+
+	isl_int **div;
+
+	struct isl_vec *sample;
+
+	struct isl_blk block;
+	struct isl_blk block2;
+};
+
+#undef EL
+#define EL isl_basic_set
+
+#include <isl_list_templ.h>
+
+/* A "map" is a (possibly disjoint) union of basic maps.
+ * A "set" is a (possibly disjoint) union of basic sets.
+ *
+ * Currently, the isl_set structure is identical to the isl_map structure
+ * and the library depends on this correspondence internally.
+ * However, users should not depend on this correspondence.
+ */
+struct isl_map {
+	int ref;
+#define ISL_MAP_DISJOINT		(1 << 0)
+#define ISL_MAP_NORMALIZED		(1 << 1)
+#define ISL_SET_DISJOINT		(1 << 0)
+#define ISL_SET_NORMALIZED		(1 << 1)
+	unsigned flags;
+
+	struct isl_ctx *ctx;
+
+	isl_space *dim;
+
+	int n;
+
+	size_t size;
+	struct isl_basic_map *p[1];
+};
+
+#undef EL
+#define EL isl_set
+
+#include <isl_list_templ.h>
+
+__isl_give isl_map *isl_map_realign(__isl_take isl_map *map,
+	__isl_take isl_reordering *r);
+__isl_give isl_set *isl_set_realign(__isl_take isl_set *set,
+	__isl_take isl_reordering *r);
+
+__isl_give isl_map *isl_map_reset(__isl_take isl_map *map,
+	enum isl_dim_type type);
+
+__isl_give isl_basic_set *isl_basic_set_reset_space(
+	__isl_take isl_basic_set *bset, __isl_take isl_space *dim);
+__isl_give isl_basic_map *isl_basic_map_reset_space(
+	__isl_take isl_basic_map *bmap, __isl_take isl_space *dim);
+__isl_give isl_map *isl_map_reset_space(__isl_take isl_map *map,
+	__isl_take isl_space *dim);
+
+unsigned isl_basic_map_offset(struct isl_basic_map *bmap,
+					enum isl_dim_type type);
+unsigned isl_basic_set_offset(struct isl_basic_set *bset,
+					enum isl_dim_type type);
+
+int isl_basic_map_may_be_set(__isl_keep isl_basic_map *bmap);
+int isl_map_may_be_set(__isl_keep isl_map *map);
+int isl_map_compatible_domain(struct isl_map *map, struct isl_set *set);
+int isl_basic_map_compatible_domain(struct isl_basic_map *bmap,
+		struct isl_basic_set *bset);
+int isl_basic_map_compatible_range(struct isl_basic_map *bmap,
+		struct isl_basic_set *bset);
+
+struct isl_basic_map *isl_basic_map_extend_space(struct isl_basic_map *base,
+		__isl_take isl_space *dim, unsigned extra,
+		unsigned n_eq, unsigned n_ineq);
+struct isl_basic_set *isl_basic_set_extend_space(struct isl_basic_set *base,
+		__isl_take isl_space *dim, unsigned extra,
+		unsigned n_eq, unsigned n_ineq);
+struct isl_basic_set *isl_basic_set_add_constraints(struct isl_basic_set *bset1,
+		struct isl_basic_set *bset2, unsigned pos);
+
+struct isl_map *isl_map_grow(struct isl_map *map, int n);
+struct isl_set *isl_set_grow(struct isl_set *set, int n);
+
+int isl_basic_set_contains(struct isl_basic_set *bset, struct isl_vec *vec);
+int isl_basic_map_contains(struct isl_basic_map *bmap, struct isl_vec *vec);
+
+__isl_give isl_basic_set *isl_basic_set_alloc_space(__isl_take isl_space *dim,
+		unsigned extra, unsigned n_eq, unsigned n_ineq);
+__isl_give isl_set *isl_set_alloc_space(__isl_take isl_space *dim, int n,
+	unsigned flags);
+__isl_give isl_basic_map *isl_basic_map_alloc_space(__isl_take isl_space *dim,
+		unsigned extra, unsigned n_eq, unsigned n_ineq);
+__isl_give isl_map *isl_map_alloc_space(__isl_take isl_space *dim, int n,
+	unsigned flags);
+
+unsigned isl_basic_map_total_dim(const struct isl_basic_map *bmap);
+
+int isl_basic_map_alloc_equality(struct isl_basic_map *bmap);
+int isl_basic_set_alloc_equality(struct isl_basic_set *bset);
+int isl_basic_set_free_inequality(struct isl_basic_set *bset, unsigned n);
+int isl_basic_map_free_equality(struct isl_basic_map *bmap, unsigned n);
+int isl_basic_set_free_equality(struct isl_basic_set *bset, unsigned n);
+int isl_basic_set_alloc_inequality(struct isl_basic_set *bset);
+int isl_basic_map_alloc_inequality(struct isl_basic_map *bmap);
+int isl_basic_map_free_inequality(struct isl_basic_map *bmap, unsigned n);
+int isl_basic_map_alloc_div(struct isl_basic_map *bmap);
+int isl_basic_set_alloc_div(struct isl_basic_set *bset);
+int isl_basic_map_free_div(struct isl_basic_map *bmap, unsigned n);
+int isl_basic_set_free_div(struct isl_basic_set *bset, unsigned n);
+void isl_basic_map_inequality_to_equality(
+		struct isl_basic_map *bmap, unsigned pos);
+int isl_basic_map_drop_equality(struct isl_basic_map *bmap, unsigned pos);
+int isl_basic_set_drop_equality(struct isl_basic_set *bset, unsigned pos);
+int isl_basic_set_drop_inequality(struct isl_basic_set *bset, unsigned pos);
+int isl_basic_map_drop_inequality(struct isl_basic_map *bmap, unsigned pos);
+__isl_give isl_basic_set *isl_basic_set_add_eq(__isl_take isl_basic_set *bset,
+	isl_int *eq);
+__isl_give isl_basic_map *isl_basic_map_add_eq(__isl_take isl_basic_map *bmap,
+	isl_int *eq);
+__isl_give isl_basic_set *isl_basic_set_add_ineq(__isl_take isl_basic_set *bset,
+	isl_int *ineq);
+__isl_give isl_basic_map *isl_basic_map_add_ineq(__isl_take isl_basic_map *bmap,
+	isl_int *ineq);
+
+int isl_inequality_negate(struct isl_basic_map *bmap, unsigned pos);
+
+struct isl_basic_set *isl_basic_set_cow(struct isl_basic_set *bset);
+struct isl_basic_map *isl_basic_map_cow(struct isl_basic_map *bmap);
+struct isl_set *isl_set_cow(struct isl_set *set);
+struct isl_map *isl_map_cow(struct isl_map *map);
+
+struct isl_basic_map *isl_basic_map_set_to_empty(struct isl_basic_map *bmap);
+struct isl_basic_set *isl_basic_set_set_to_empty(struct isl_basic_set *bset);
+struct isl_basic_set *isl_basic_set_order_divs(struct isl_basic_set *bset);
+void isl_basic_map_swap_div(struct isl_basic_map *bmap, int a, int b);
+struct isl_basic_map *isl_basic_map_order_divs(struct isl_basic_map *bmap);
+__isl_give isl_map *isl_map_order_divs(__isl_take isl_map *map);
+struct isl_basic_map *isl_basic_map_align_divs(
+		struct isl_basic_map *dst, struct isl_basic_map *src);
+struct isl_basic_set *isl_basic_set_align_divs(
+		struct isl_basic_set *dst, struct isl_basic_set *src);
+__isl_give isl_set *isl_set_align_divs_to_basic_set_list(
+	__isl_take isl_set *set, __isl_keep isl_basic_set_list *list);
+__isl_give isl_basic_set_list *isl_basic_set_list_align_divs_to_basic_set(
+	__isl_take isl_basic_set_list *list, __isl_keep isl_basic_set *bset);
+__isl_give isl_basic_map *isl_basic_map_sort_divs(
+	__isl_take isl_basic_map *bmap);
+__isl_give isl_map *isl_map_sort_divs(__isl_take isl_map *map);
+struct isl_basic_map *isl_basic_map_gauss(
+	struct isl_basic_map *bmap, int *progress);
+struct isl_basic_set *isl_basic_set_gauss(
+	struct isl_basic_set *bset, int *progress);
+__isl_give isl_basic_set *isl_basic_set_sort_constraints(
+	__isl_take isl_basic_set *bset);
+int isl_basic_map_plain_cmp(const __isl_keep isl_basic_map *bmap1,
+	const __isl_keep isl_basic_map *bmap2);
+int isl_basic_map_plain_is_equal(__isl_keep isl_basic_map *bmap1,
+	__isl_keep isl_basic_map *bmap2);
+struct isl_basic_map *isl_basic_map_normalize_constraints(
+	struct isl_basic_map *bmap);
+struct isl_basic_set *isl_basic_set_normalize_constraints(
+	struct isl_basic_set *bset);
+struct isl_basic_map *isl_basic_map_implicit_equalities(
+						struct isl_basic_map *bmap);
+struct isl_basic_set *isl_basic_map_underlying_set(struct isl_basic_map *bmap);
+__isl_give isl_basic_set *isl_basic_set_underlying_set(
+		__isl_take isl_basic_set *bset);
+__isl_give isl_basic_set_list *isl_basic_set_list_underlying_set(
+	__isl_take isl_basic_set_list *list);
+struct isl_set *isl_map_underlying_set(struct isl_map *map);
+struct isl_basic_map *isl_basic_map_overlying_set(struct isl_basic_set *bset,
+	struct isl_basic_map *like);
+__isl_give isl_basic_set *isl_basic_set_drop_constraints_involving(
+	__isl_take isl_basic_set *bset, unsigned first, unsigned n);
+__isl_give isl_basic_set *isl_basic_set_drop(__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n);
+struct isl_basic_map *isl_basic_map_drop(struct isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned first, unsigned n);
+struct isl_set *isl_set_drop(struct isl_set *set,
+	enum isl_dim_type type, unsigned first, unsigned n);
+struct isl_basic_set *isl_basic_set_drop_dims(
+		struct isl_basic_set *bset, unsigned first, unsigned n);
+struct isl_set *isl_set_drop_dims(
+		struct isl_set *set, unsigned first, unsigned n);
+struct isl_map *isl_map_drop_inputs(
+		struct isl_map *map, unsigned first, unsigned n);
+struct isl_map *isl_map_drop(struct isl_map *map,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+__isl_give isl_basic_map *isl_basic_map_remove_duplicate_constraints(
+	__isl_take isl_basic_map *bmap, int *progress, int detect_divs);
+
+struct isl_map *isl_map_remove_empty_parts(struct isl_map *map);
+struct isl_set *isl_set_remove_empty_parts(struct isl_set *set);
+__isl_give isl_map *isl_map_remove_obvious_duplicates(__isl_take isl_map *map);
+
+struct isl_set *isl_set_normalize(struct isl_set *set);
+
+struct isl_set *isl_set_drop_vars(
+		struct isl_set *set, unsigned first, unsigned n);
+
+struct isl_basic_map *isl_basic_map_eliminate_vars(
+	struct isl_basic_map *bmap, unsigned pos, unsigned n);
+struct isl_basic_set *isl_basic_set_eliminate_vars(
+	struct isl_basic_set *bset, unsigned pos, unsigned n);
+
+__isl_give isl_map *isl_map_eliminate(__isl_take isl_map *map,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_set *isl_set_eliminate(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+int isl_basic_set_constraint_is_redundant(struct isl_basic_set **bset,
+	isl_int *c, isl_int *opt_n, isl_int *opt_d);
+
+int isl_basic_map_add_div_constraint(__isl_keep isl_basic_map *bmap,
+	unsigned div, int sign);
+int isl_basic_map_add_div_constraints(struct isl_basic_map *bmap, unsigned div);
+__isl_give isl_basic_map *isl_basic_map_add_known_div_constraints(
+	__isl_take isl_basic_map *bmap);
+struct isl_basic_map *isl_basic_map_drop_redundant_divs(
+	struct isl_basic_map *bmap);
+struct isl_basic_set *isl_basic_set_drop_redundant_divs(
+	struct isl_basic_set *bset);
+
+struct isl_basic_set *isl_basic_set_recession_cone(struct isl_basic_set *bset);
+struct isl_basic_set *isl_basic_set_lineality_space(struct isl_basic_set *bset);
+
+struct isl_basic_set *isl_basic_set_set_rational(struct isl_basic_set *bset);
+__isl_give isl_set *isl_set_set_rational(__isl_take isl_set *set);
+__isl_give isl_basic_map *isl_basic_map_set_rational(
+	__isl_take isl_basic_map *bmap);
+__isl_give isl_map *isl_map_set_rational(__isl_take isl_map *map);
+
+int isl_map_has_rational(__isl_keep isl_map *map);
+int isl_set_has_rational(__isl_keep isl_set *set);
+
+struct isl_mat;
+
+struct isl_basic_set *isl_basic_set_preimage(struct isl_basic_set *bset,
+	struct isl_mat *mat);
+struct isl_set *isl_set_preimage(struct isl_set *set, struct isl_mat *mat);
+
+__isl_give isl_basic_set *isl_basic_set_transform_dims(
+	__isl_take isl_basic_set *bset, enum isl_dim_type type, unsigned first,
+	__isl_take isl_mat *trans);
+
+isl_int *isl_set_wrap_facet(__isl_keep isl_set *set,
+	isl_int *facet, isl_int *ridge);
+
+int isl_basic_map_contains_point(__isl_keep isl_basic_map *bmap,
+	__isl_keep isl_point *point);
+int isl_set_contains_point(__isl_keep isl_set *set, __isl_keep isl_point *point);
+
+int isl_basic_set_vars_get_sign(__isl_keep isl_basic_set *bset,
+	unsigned first, unsigned n, int *signs);
+int isl_set_foreach_orthant(__isl_keep isl_set *set,
+	int (*fn)(__isl_take isl_set *orthant, int *signs, void *user),
+	void *user);
+
+int isl_basic_map_add_div_constraints_var(__isl_keep isl_basic_map *bmap,
+	unsigned pos, isl_int *div);
+int isl_basic_set_add_div_constraints_var(__isl_keep isl_basic_set *bset,
+	unsigned pos, isl_int *div);
+int isl_basic_map_is_div_constraint(__isl_keep isl_basic_map *bmap,
+	isl_int *constraint, unsigned div);
+int isl_basic_set_is_div_constraint(__isl_keep isl_basic_set *bset,
+	isl_int *constraint, unsigned div);
+
+__isl_give isl_basic_set *isl_basic_set_from_local_space(
+	__isl_take isl_local_space *ls);
+__isl_give isl_basic_map *isl_basic_map_from_local_space(
+	__isl_take isl_local_space *ls);
+__isl_give isl_basic_set *isl_basic_set_expand_divs(
+	__isl_take isl_basic_set *bset, __isl_take isl_mat *div, int *exp);
+
+int isl_basic_map_divs_known(__isl_keep isl_basic_map *bmap);
+__isl_give isl_mat *isl_basic_set_get_divs(__isl_keep isl_basic_set *bset);
+__isl_give isl_mat *isl_basic_map_get_divs(__isl_keep isl_basic_map *bmap);
+
+__isl_give isl_map *isl_map_inline_foreach_basic_map(__isl_take isl_map *map,
+	__isl_give isl_basic_map *(*fn)(__isl_take isl_basic_map *bmap));
+
+__isl_give isl_map *isl_map_align_params_map_map_and(
+	__isl_take isl_map *map1, __isl_take isl_map *map2,
+	__isl_give isl_map *(*fn)(__isl_take isl_map *map1,
+				    __isl_take isl_map *map2));
+int isl_map_align_params_map_map_and_test(__isl_keep isl_map *map1,
+	__isl_keep isl_map *map2,
+	int (*fn)(__isl_keep isl_map *map1, __isl_keep isl_map *map2));
+
+int isl_basic_map_foreach_lexopt(__isl_keep isl_basic_map *bmap, int max,
+	int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_aff_list *list,
+		  void *user),
+	void *user);
+int isl_basic_set_foreach_lexopt(__isl_keep isl_basic_set *bset, int max,
+	int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_aff_list *list,
+		  void *user),
+	void *user);
+
+__isl_give isl_set *isl_set_substitute(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, __isl_keep isl_aff *subs);
+
+__isl_give isl_set *isl_set_gist_params_basic_set(__isl_take isl_set *set,
+	__isl_take isl_basic_set *context);
+
+int isl_map_compatible_range(__isl_keep isl_map *map, __isl_keep isl_set *set);
+
+int isl_basic_map_plain_is_single_valued(__isl_keep isl_basic_map *bmap);
+
+int isl_map_is_set(__isl_keep isl_map *map);
+
+int isl_basic_set_plain_dim_is_fixed(__isl_keep isl_basic_set *bset,
+	unsigned dim, isl_int *val);
+
+int isl_basic_set_dim_residue_class(struct isl_basic_set *bset,
+	int pos, isl_int *modulo, isl_int *residue);
+int isl_set_dim_residue_class(struct isl_set *set,
+	int pos, isl_int *modulo, isl_int *residue);
+
+__isl_give isl_basic_set *isl_basic_set_fix(__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned pos, isl_int value);
+__isl_give isl_set *isl_set_fix(__isl_take isl_set *set,
+	enum isl_dim_type type, unsigned pos, isl_int value);
+int isl_map_plain_is_fixed(__isl_keep isl_map *map,
+	enum isl_dim_type type, unsigned pos, isl_int *val);
+
+int isl_basic_map_output_defining_equality(__isl_keep isl_basic_map *bmap,
+	int pos);
+
+__isl_give isl_basic_set_list *isl_set_get_basic_set_list(
+	__isl_keep isl_set *set);
+
+__isl_give isl_map *isl_map_fixed_power(__isl_take isl_map *map, isl_int exp);
+
+int isl_basic_set_count_upto(__isl_keep isl_basic_set *bset,
+	isl_int max, isl_int *count);
+int isl_set_count_upto(__isl_keep isl_set *set, isl_int max, isl_int *count);
+
+#endif
diff --git a/polly/lib/External/isl/isl_map_simplify.c b/polly/lib/External/isl/isl_map_simplify.c
new file mode 100644
index 00000000000..78815bd98c7
--- /dev/null
+++ b/polly/lib/External/isl/isl_map_simplify.c
@@ -0,0 +1,3257 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2012      Ecole Normale Superieure
+ * Copyright 2014      INRIA Rocquencourt
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
+ * B.P. 105 - 78153 Le Chesnay, France
+ */
+
+#include <strings.h>
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include "isl_equalities.h"
+#include <isl/map.h>
+#include <isl_seq.h>
+#include "isl_tab.h"
+#include <isl_space_private.h>
+#include <isl_mat_private.h>
+#include <isl_vec_private.h>
+
+static void swap_equality(struct isl_basic_map *bmap, int a, int b)
+{
+	isl_int *t = bmap->eq[a];
+	bmap->eq[a] = bmap->eq[b];
+	bmap->eq[b] = t;
+}
+
+static void swap_inequality(struct isl_basic_map *bmap, int a, int b)
+{
+	if (a != b) {
+		isl_int *t = bmap->ineq[a];
+		bmap->ineq[a] = bmap->ineq[b];
+		bmap->ineq[b] = t;
+	}
+}
+
+static void constraint_drop_vars(isl_int *c, unsigned n, unsigned rem)
+{
+	isl_seq_cpy(c, c + n, rem);
+	isl_seq_clr(c + rem, n);
+}
+
+/* Drop n dimensions starting at first.
+ *
+ * In principle, this frees up some extra variables as the number
+ * of columns remains constant, but we would have to extend
+ * the div array too as the number of rows in this array is assumed
+ * to be equal to extra.
+ */
+struct isl_basic_set *isl_basic_set_drop_dims(
+		struct isl_basic_set *bset, unsigned first, unsigned n)
+{
+	int i;
+
+	if (!bset)
+		goto error;
+
+	isl_assert(bset->ctx, first + n <= bset->dim->n_out, goto error);
+
+	if (n == 0 && !isl_space_get_tuple_name(bset->dim, isl_dim_set))
+		return bset;
+
+	bset = isl_basic_set_cow(bset);
+	if (!bset)
+		return NULL;
+
+	for (i = 0; i < bset->n_eq; ++i)
+		constraint_drop_vars(bset->eq[i]+1+bset->dim->nparam+first, n,
+				     (bset->dim->n_out-first-n)+bset->extra);
+
+	for (i = 0; i < bset->n_ineq; ++i)
+		constraint_drop_vars(bset->ineq[i]+1+bset->dim->nparam+first, n,
+				     (bset->dim->n_out-first-n)+bset->extra);
+
+	for (i = 0; i < bset->n_div; ++i)
+		constraint_drop_vars(bset->div[i]+1+1+bset->dim->nparam+first, n,
+				     (bset->dim->n_out-first-n)+bset->extra);
+
+	bset->dim = isl_space_drop_outputs(bset->dim, first, n);
+	if (!bset->dim)
+		goto error;
+
+	ISL_F_CLR(bset, ISL_BASIC_SET_NORMALIZED);
+	bset = isl_basic_set_simplify(bset);
+	return isl_basic_set_finalize(bset);
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+struct isl_set *isl_set_drop_dims(
+		struct isl_set *set, unsigned first, unsigned n)
+{
+	int i;
+
+	if (!set)
+		goto error;
+
+	isl_assert(set->ctx, first + n <= set->dim->n_out, goto error);
+
+	if (n == 0 && !isl_space_get_tuple_name(set->dim, isl_dim_set))
+		return set;
+	set = isl_set_cow(set);
+	if (!set)
+		goto error;
+	set->dim = isl_space_drop_outputs(set->dim, first, n);
+	if (!set->dim)
+		goto error;
+
+	for (i = 0; i < set->n; ++i) {
+		set->p[i] = isl_basic_set_drop_dims(set->p[i], first, n);
+		if (!set->p[i])
+			goto error;
+	}
+
+	ISL_F_CLR(set, ISL_SET_NORMALIZED);
+	return set;
+error:
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Move "n" divs starting at "first" to the end of the list of divs.
+ */
+static struct isl_basic_map *move_divs_last(struct isl_basic_map *bmap,
+	unsigned first, unsigned n)
+{
+	isl_int **div;
+	int i;
+
+	if (first + n == bmap->n_div)
+		return bmap;
+
+	div = isl_alloc_array(bmap->ctx, isl_int *, n);
+	if (!div)
+		goto error;
+	for (i = 0; i < n; ++i)
+		div[i] = bmap->div[first + i];
+	for (i = 0; i < bmap->n_div - first - n; ++i)
+		bmap->div[first + i] = bmap->div[first + n + i];
+	for (i = 0; i < n; ++i)
+		bmap->div[bmap->n_div - n + i] = div[i];
+	free(div);
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Drop "n" dimensions of type "type" starting at "first".
+ *
+ * In principle, this frees up some extra variables as the number
+ * of columns remains constant, but we would have to extend
+ * the div array too as the number of rows in this array is assumed
+ * to be equal to extra.
+ */
+struct isl_basic_map *isl_basic_map_drop(struct isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+	unsigned dim;
+	unsigned offset;
+	unsigned left;
+
+	if (!bmap)
+		goto error;
+
+	dim = isl_basic_map_dim(bmap, type);
+	isl_assert(bmap->ctx, first + n <= dim, goto error);
+
+	if (n == 0 && !isl_space_is_named_or_nested(bmap->dim, type))
+		return bmap;
+
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+
+	offset = isl_basic_map_offset(bmap, type) + first;
+	left = isl_basic_map_total_dim(bmap) - (offset - 1) - n;
+	for (i = 0; i < bmap->n_eq; ++i)
+		constraint_drop_vars(bmap->eq[i]+offset, n, left);
+
+	for (i = 0; i < bmap->n_ineq; ++i)
+		constraint_drop_vars(bmap->ineq[i]+offset, n, left);
+
+	for (i = 0; i < bmap->n_div; ++i)
+		constraint_drop_vars(bmap->div[i]+1+offset, n, left);
+
+	if (type == isl_dim_div) {
+		bmap = move_divs_last(bmap, first, n);
+		if (!bmap)
+			goto error;
+		isl_basic_map_free_div(bmap, n);
+	} else
+		bmap->dim = isl_space_drop_dims(bmap->dim, type, first, n);
+	if (!bmap->dim)
+		goto error;
+
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
+	bmap = isl_basic_map_simplify(bmap);
+	return isl_basic_map_finalize(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_drop(__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return (isl_basic_set *)isl_basic_map_drop((isl_basic_map *)bset,
+							type, first, n);
+}
+
+struct isl_basic_map *isl_basic_map_drop_inputs(
+		struct isl_basic_map *bmap, unsigned first, unsigned n)
+{
+	return isl_basic_map_drop(bmap, isl_dim_in, first, n);
+}
+
+struct isl_map *isl_map_drop(struct isl_map *map,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+
+	if (!map)
+		goto error;
+
+	isl_assert(map->ctx, first + n <= isl_map_dim(map, type), goto error);
+
+	if (n == 0 && !isl_space_get_tuple_name(map->dim, type))
+		return map;
+	map = isl_map_cow(map);
+	if (!map)
+		goto error;
+	map->dim = isl_space_drop_dims(map->dim, type, first, n);
+	if (!map->dim)
+		goto error;
+
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_drop(map->p[i], type, first, n);
+		if (!map->p[i])
+			goto error;
+	}
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+struct isl_set *isl_set_drop(struct isl_set *set,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return (isl_set *)isl_map_drop((isl_map *)set, type, first, n);
+}
+
+struct isl_map *isl_map_drop_inputs(
+		struct isl_map *map, unsigned first, unsigned n)
+{
+	return isl_map_drop(map, isl_dim_in, first, n);
+}
+
+/*
+ * We don't cow, as the div is assumed to be redundant.
+ */
+static struct isl_basic_map *isl_basic_map_drop_div(
+		struct isl_basic_map *bmap, unsigned div)
+{
+	int i;
+	unsigned pos;
+
+	if (!bmap)
+		goto error;
+
+	pos = 1 + isl_space_dim(bmap->dim, isl_dim_all) + div;
+
+	isl_assert(bmap->ctx, div < bmap->n_div, goto error);
+
+	for (i = 0; i < bmap->n_eq; ++i)
+		constraint_drop_vars(bmap->eq[i]+pos, 1, bmap->extra-div-1);
+
+	for (i = 0; i < bmap->n_ineq; ++i) {
+		if (!isl_int_is_zero(bmap->ineq[i][pos])) {
+			isl_basic_map_drop_inequality(bmap, i);
+			--i;
+			continue;
+		}
+		constraint_drop_vars(bmap->ineq[i]+pos, 1, bmap->extra-div-1);
+	}
+
+	for (i = 0; i < bmap->n_div; ++i)
+		constraint_drop_vars(bmap->div[i]+1+pos, 1, bmap->extra-div-1);
+
+	if (div != bmap->n_div - 1) {
+		int j;
+		isl_int *t = bmap->div[div];
+
+		for (j = div; j < bmap->n_div - 1; ++j)
+			bmap->div[j] = bmap->div[j+1];
+
+		bmap->div[bmap->n_div - 1] = t;
+	}
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
+	isl_basic_map_free_div(bmap, 1);
+
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+struct isl_basic_map *isl_basic_map_normalize_constraints(
+	struct isl_basic_map *bmap)
+{
+	int i;
+	isl_int gcd;
+	unsigned total = isl_basic_map_total_dim(bmap);
+
+	if (!bmap)
+		return NULL;
+
+	isl_int_init(gcd);
+	for (i = bmap->n_eq - 1; i >= 0; --i) {
+		isl_seq_gcd(bmap->eq[i]+1, total, &gcd);
+		if (isl_int_is_zero(gcd)) {
+			if (!isl_int_is_zero(bmap->eq[i][0])) {
+				bmap = isl_basic_map_set_to_empty(bmap);
+				break;
+			}
+			isl_basic_map_drop_equality(bmap, i);
+			continue;
+		}
+		if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
+			isl_int_gcd(gcd, gcd, bmap->eq[i][0]);
+		if (isl_int_is_one(gcd))
+			continue;
+		if (!isl_int_is_divisible_by(bmap->eq[i][0], gcd)) {
+			bmap = isl_basic_map_set_to_empty(bmap);
+			break;
+		}
+		isl_seq_scale_down(bmap->eq[i], bmap->eq[i], gcd, 1+total);
+	}
+
+	for (i = bmap->n_ineq - 1; i >= 0; --i) {
+		isl_seq_gcd(bmap->ineq[i]+1, total, &gcd);
+		if (isl_int_is_zero(gcd)) {
+			if (isl_int_is_neg(bmap->ineq[i][0])) {
+				bmap = isl_basic_map_set_to_empty(bmap);
+				break;
+			}
+			isl_basic_map_drop_inequality(bmap, i);
+			continue;
+		}
+		if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
+			isl_int_gcd(gcd, gcd, bmap->ineq[i][0]);
+		if (isl_int_is_one(gcd))
+			continue;
+		isl_int_fdiv_q(bmap->ineq[i][0], bmap->ineq[i][0], gcd);
+		isl_seq_scale_down(bmap->ineq[i]+1, bmap->ineq[i]+1, gcd, total);
+	}
+	isl_int_clear(gcd);
+
+	return bmap;
+}
+
+struct isl_basic_set *isl_basic_set_normalize_constraints(
+	struct isl_basic_set *bset)
+{
+	return (struct isl_basic_set *)isl_basic_map_normalize_constraints(
+		(struct isl_basic_map *)bset);
+}
+
+/* Remove any common factor in numerator and denominator of the div expression,
+ * not taking into account the constant term.
+ * That is, if the div is of the form
+ *
+ *	floor((a + m f(x))/(m d))
+ *
+ * then replace it by
+ *
+ *	floor((floor(a/m) + f(x))/d)
+ *
+ * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
+ * and can therefore not influence the result of the floor.
+ */
+static void normalize_div_expression(__isl_keep isl_basic_map *bmap, int div)
+{
+	unsigned total = isl_basic_map_total_dim(bmap);
+	isl_ctx *ctx = bmap->ctx;
+
+	if (isl_int_is_zero(bmap->div[div][0]))
+		return;
+	isl_seq_gcd(bmap->div[div] + 2, total, &ctx->normalize_gcd);
+	isl_int_gcd(ctx->normalize_gcd, ctx->normalize_gcd, bmap->div[div][0]);
+	if (isl_int_is_one(ctx->normalize_gcd))
+		return;
+	isl_int_fdiv_q(bmap->div[div][1], bmap->div[div][1],
+			ctx->normalize_gcd);
+	isl_int_divexact(bmap->div[div][0], bmap->div[div][0],
+			ctx->normalize_gcd);
+	isl_seq_scale_down(bmap->div[div] + 2, bmap->div[div] + 2,
+			ctx->normalize_gcd, total);
+}
+
+/* Remove any common factor in numerator and denominator of a div expression,
+ * not taking into account the constant term.
+ * That is, look for any div of the form
+ *
+ *	floor((a + m f(x))/(m d))
+ *
+ * and replace it by
+ *
+ *	floor((floor(a/m) + f(x))/d)
+ *
+ * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
+ * and can therefore not influence the result of the floor.
+ */
+static __isl_give isl_basic_map *normalize_div_expressions(
+	__isl_take isl_basic_map *bmap)
+{
+	int i;
+
+	if (!bmap)
+		return NULL;
+	if (bmap->n_div == 0)
+		return bmap;
+
+	for (i = 0; i < bmap->n_div; ++i)
+		normalize_div_expression(bmap, i);
+
+	return bmap;
+}
+
+/* Assumes divs have been ordered if keep_divs is set.
+ */
+static void eliminate_var_using_equality(struct isl_basic_map *bmap,
+	unsigned pos, isl_int *eq, int keep_divs, int *progress)
+{
+	unsigned total;
+	unsigned space_total;
+	int k;
+	int last_div;
+
+	total = isl_basic_map_total_dim(bmap);
+	space_total = isl_space_dim(bmap->dim, isl_dim_all);
+	last_div = isl_seq_last_non_zero(eq + 1 + space_total, bmap->n_div);
+	for (k = 0; k < bmap->n_eq; ++k) {
+		if (bmap->eq[k] == eq)
+			continue;
+		if (isl_int_is_zero(bmap->eq[k][1+pos]))
+			continue;
+		if (progress)
+			*progress = 1;
+		isl_seq_elim(bmap->eq[k], eq, 1+pos, 1+total, NULL);
+		isl_seq_normalize(bmap->ctx, bmap->eq[k], 1 + total);
+	}
+
+	for (k = 0; k < bmap->n_ineq; ++k) {
+		if (isl_int_is_zero(bmap->ineq[k][1+pos]))
+			continue;
+		if (progress)
+			*progress = 1;
+		isl_seq_elim(bmap->ineq[k], eq, 1+pos, 1+total, NULL);
+		isl_seq_normalize(bmap->ctx, bmap->ineq[k], 1 + total);
+		ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
+	}
+
+	for (k = 0; k < bmap->n_div; ++k) {
+		if (isl_int_is_zero(bmap->div[k][0]))
+			continue;
+		if (isl_int_is_zero(bmap->div[k][1+1+pos]))
+			continue;
+		if (progress)
+			*progress = 1;
+		/* We need to be careful about circular definitions,
+		 * so for now we just remove the definition of div k
+		 * if the equality contains any divs.
+		 * If keep_divs is set, then the divs have been ordered
+		 * and we can keep the definition as long as the result
+		 * is still ordered.
+		 */
+		if (last_div == -1 || (keep_divs && last_div < k)) {
+			isl_seq_elim(bmap->div[k]+1, eq,
+					1+pos, 1+total, &bmap->div[k][0]);
+			normalize_div_expression(bmap, k);
+		} else
+			isl_seq_clr(bmap->div[k], 1 + total);
+		ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
+	}
+}
+
+/* Assumes divs have been ordered if keep_divs is set.
+ */
+static void eliminate_div(struct isl_basic_map *bmap, isl_int *eq,
+	unsigned div, int keep_divs)
+{
+	unsigned pos = isl_space_dim(bmap->dim, isl_dim_all) + div;
+
+	eliminate_var_using_equality(bmap, pos, eq, keep_divs, NULL);
+
+	isl_basic_map_drop_div(bmap, div);
+}
+
+/* Check if elimination of div "div" using equality "eq" would not
+ * result in a div depending on a later div.
+ */
+static int ok_to_eliminate_div(struct isl_basic_map *bmap, isl_int *eq,
+	unsigned div)
+{
+	int k;
+	int last_div;
+	unsigned space_total = isl_space_dim(bmap->dim, isl_dim_all);
+	unsigned pos = space_total + div;
+
+	last_div = isl_seq_last_non_zero(eq + 1 + space_total, bmap->n_div);
+	if (last_div < 0 || last_div <= div)
+		return 1;
+
+	for (k = 0; k <= last_div; ++k) {
+		if (isl_int_is_zero(bmap->div[k][0]))
+			return 1;
+		if (!isl_int_is_zero(bmap->div[k][1 + 1 + pos]))
+			return 0;
+	}
+
+	return 1;
+}
+
+/* Elimininate divs based on equalities
+ */
+static struct isl_basic_map *eliminate_divs_eq(
+		struct isl_basic_map *bmap, int *progress)
+{
+	int d;
+	int i;
+	int modified = 0;
+	unsigned off;
+
+	bmap = isl_basic_map_order_divs(bmap);
+
+	if (!bmap)
+		return NULL;
+
+	off = 1 + isl_space_dim(bmap->dim, isl_dim_all);
+
+	for (d = bmap->n_div - 1; d >= 0 ; --d) {
+		for (i = 0; i < bmap->n_eq; ++i) {
+			if (!isl_int_is_one(bmap->eq[i][off + d]) &&
+			    !isl_int_is_negone(bmap->eq[i][off + d]))
+				continue;
+			if (!ok_to_eliminate_div(bmap, bmap->eq[i], d))
+				continue;
+			modified = 1;
+			*progress = 1;
+			eliminate_div(bmap, bmap->eq[i], d, 1);
+			isl_basic_map_drop_equality(bmap, i);
+			break;
+		}
+	}
+	if (modified)
+		return eliminate_divs_eq(bmap, progress);
+	return bmap;
+}
+
+/* Elimininate divs based on inequalities
+ */
+static struct isl_basic_map *eliminate_divs_ineq(
+		struct isl_basic_map *bmap, int *progress)
+{
+	int d;
+	int i;
+	unsigned off;
+	struct isl_ctx *ctx;
+
+	if (!bmap)
+		return NULL;
+
+	ctx = bmap->ctx;
+	off = 1 + isl_space_dim(bmap->dim, isl_dim_all);
+
+	for (d = bmap->n_div - 1; d >= 0 ; --d) {
+		for (i = 0; i < bmap->n_eq; ++i)
+			if (!isl_int_is_zero(bmap->eq[i][off + d]))
+				break;
+		if (i < bmap->n_eq)
+			continue;
+		for (i = 0; i < bmap->n_ineq; ++i)
+			if (isl_int_abs_gt(bmap->ineq[i][off + d], ctx->one))
+				break;
+		if (i < bmap->n_ineq)
+			continue;
+		*progress = 1;
+		bmap = isl_basic_map_eliminate_vars(bmap, (off-1)+d, 1);
+		if (!bmap || ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
+			break;
+		bmap = isl_basic_map_drop_div(bmap, d);
+		if (!bmap)
+			break;
+	}
+	return bmap;
+}
+
+struct isl_basic_map *isl_basic_map_gauss(
+	struct isl_basic_map *bmap, int *progress)
+{
+	int k;
+	int done;
+	int last_var;
+	unsigned total_var;
+	unsigned total;
+
+	bmap = isl_basic_map_order_divs(bmap);
+
+	if (!bmap)
+		return NULL;
+
+	total = isl_basic_map_total_dim(bmap);
+	total_var = total - bmap->n_div;
+
+	last_var = total - 1;
+	for (done = 0; done < bmap->n_eq; ++done) {
+		for (; last_var >= 0; --last_var) {
+			for (k = done; k < bmap->n_eq; ++k)
+				if (!isl_int_is_zero(bmap->eq[k][1+last_var]))
+					break;
+			if (k < bmap->n_eq)
+				break;
+		}
+		if (last_var < 0)
+			break;
+		if (k != done)
+			swap_equality(bmap, k, done);
+		if (isl_int_is_neg(bmap->eq[done][1+last_var]))
+			isl_seq_neg(bmap->eq[done], bmap->eq[done], 1+total);
+
+		eliminate_var_using_equality(bmap, last_var, bmap->eq[done], 1,
+						progress);
+
+		if (last_var >= total_var &&
+		    isl_int_is_zero(bmap->div[last_var - total_var][0])) {
+			unsigned div = last_var - total_var;
+			isl_seq_neg(bmap->div[div]+1, bmap->eq[done], 1+total);
+			isl_int_set_si(bmap->div[div][1+1+last_var], 0);
+			isl_int_set(bmap->div[div][0],
+				    bmap->eq[done][1+last_var]);
+			if (progress)
+				*progress = 1;
+			ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
+		}
+	}
+	if (done == bmap->n_eq)
+		return bmap;
+	for (k = done; k < bmap->n_eq; ++k) {
+		if (isl_int_is_zero(bmap->eq[k][0]))
+			continue;
+		return isl_basic_map_set_to_empty(bmap);
+	}
+	isl_basic_map_free_equality(bmap, bmap->n_eq-done);
+	return bmap;
+}
+
+struct isl_basic_set *isl_basic_set_gauss(
+	struct isl_basic_set *bset, int *progress)
+{
+	return (struct isl_basic_set*)isl_basic_map_gauss(
+			(struct isl_basic_map *)bset, progress);
+}
+
+
+static unsigned int round_up(unsigned int v)
+{
+	int old_v = v;
+
+	while (v) {
+		old_v = v;
+		v ^= v & -v;
+	}
+	return old_v << 1;
+}
+
+static int hash_index(isl_int ***index, unsigned int size, int bits,
+			struct isl_basic_map *bmap, int k)
+{
+	int h;
+	unsigned total = isl_basic_map_total_dim(bmap);
+	uint32_t hash = isl_seq_get_hash_bits(bmap->ineq[k]+1, total, bits);
+	for (h = hash; index[h]; h = (h+1) % size)
+		if (&bmap->ineq[k] != index[h] &&
+		    isl_seq_eq(bmap->ineq[k]+1, index[h][0]+1, total))
+			break;
+	return h;
+}
+
+static int set_hash_index(isl_int ***index, unsigned int size, int bits,
+			  struct isl_basic_set *bset, int k)
+{
+	return hash_index(index, size, bits, (struct isl_basic_map *)bset, k);
+}
+
+/* If we can eliminate more than one div, then we need to make
+ * sure we do it from last div to first div, in order not to
+ * change the position of the other divs that still need to
+ * be removed.
+ */
+static struct isl_basic_map *remove_duplicate_divs(
+	struct isl_basic_map *bmap, int *progress)
+{
+	unsigned int size;
+	int *index;
+	int *elim_for;
+	int k, l, h;
+	int bits;
+	struct isl_blk eq;
+	unsigned total_var;
+	unsigned total;
+	struct isl_ctx *ctx;
+
+	bmap = isl_basic_map_order_divs(bmap);
+	if (!bmap || bmap->n_div <= 1)
+		return bmap;
+
+	total_var = isl_space_dim(bmap->dim, isl_dim_all);
+	total = total_var + bmap->n_div;
+
+	ctx = bmap->ctx;
+	for (k = bmap->n_div - 1; k >= 0; --k)
+		if (!isl_int_is_zero(bmap->div[k][0]))
+			break;
+	if (k <= 0)
+		return bmap;
+
+	elim_for = isl_calloc_array(ctx, int, bmap->n_div);
+	size = round_up(4 * bmap->n_div / 3 - 1);
+	bits = ffs(size) - 1;
+	index = isl_calloc_array(ctx, int, size);
+	if (!index)
+		return bmap;
+	eq = isl_blk_alloc(ctx, 1+total);
+	if (isl_blk_is_error(eq))
+		goto out;
+
+	isl_seq_clr(eq.data, 1+total);
+	index[isl_seq_get_hash_bits(bmap->div[k], 2+total, bits)] = k + 1;
+	for (--k; k >= 0; --k) {
+		uint32_t hash;
+
+		if (isl_int_is_zero(bmap->div[k][0]))
+			continue;
+
+		hash = isl_seq_get_hash_bits(bmap->div[k], 2+total, bits);
+		for (h = hash; index[h]; h = (h+1) % size)
+			if (isl_seq_eq(bmap->div[k],
+				       bmap->div[index[h]-1], 2+total))
+				break;
+		if (index[h]) {
+			*progress = 1;
+			l = index[h] - 1;
+			elim_for[l] = k + 1;
+		}
+		index[h] = k+1;
+	}
+	for (l = bmap->n_div - 1; l >= 0; --l) {
+		if (!elim_for[l])
+			continue;
+		k = elim_for[l] - 1;
+		isl_int_set_si(eq.data[1+total_var+k], -1);
+		isl_int_set_si(eq.data[1+total_var+l], 1);
+		eliminate_div(bmap, eq.data, l, 1);
+		isl_int_set_si(eq.data[1+total_var+k], 0);
+		isl_int_set_si(eq.data[1+total_var+l], 0);
+	}
+
+	isl_blk_free(ctx, eq);
+out:
+	free(index);
+	free(elim_for);
+	return bmap;
+}
+
+static int n_pure_div_eq(struct isl_basic_map *bmap)
+{
+	int i, j;
+	unsigned total;
+
+	total = isl_space_dim(bmap->dim, isl_dim_all);
+	for (i = 0, j = bmap->n_div-1; i < bmap->n_eq; ++i) {
+		while (j >= 0 && isl_int_is_zero(bmap->eq[i][1 + total + j]))
+			--j;
+		if (j < 0)
+			break;
+		if (isl_seq_first_non_zero(bmap->eq[i] + 1 + total, j) != -1)
+			return 0;
+	}
+	return i;
+}
+
+/* Normalize divs that appear in equalities.
+ *
+ * In particular, we assume that bmap contains some equalities
+ * of the form
+ *
+ *	a x = m * e_i
+ *
+ * and we want to replace the set of e_i by a minimal set and
+ * such that the new e_i have a canonical representation in terms
+ * of the vector x.
+ * If any of the equalities involves more than one divs, then
+ * we currently simply bail out.
+ *
+ * Let us first additionally assume that all equalities involve
+ * a div.  The equalities then express modulo constraints on the
+ * remaining variables and we can use "parameter compression"
+ * to find a minimal set of constraints.  The result is a transformation
+ *
+ *	x = T(x') = x_0 + G x'
+ *
+ * with G a lower-triangular matrix with all elements below the diagonal
+ * non-negative and smaller than the diagonal element on the same row.
+ * We first normalize x_0 by making the same property hold in the affine
+ * T matrix.
+ * The rows i of G with a 1 on the diagonal do not impose any modulo
+ * constraint and simply express x_i = x'_i.
+ * For each of the remaining rows i, we introduce a div and a corresponding
+ * equality.  In particular
+ *
+ *	g_ii e_j = x_i - g_i(x')
+ *
+ * where each x'_k is replaced either by x_k (if g_kk = 1) or the
+ * corresponding div (if g_kk != 1).
+ *
+ * If there are any equalities not involving any div, then we
+ * first apply a variable compression on the variables x:
+ *
+ *	x = C x''	x'' = C_2 x
+ *
+ * and perform the above parameter compression on A C instead of on A.
+ * The resulting compression is then of the form
+ *
+ *	x'' = T(x') = x_0 + G x'
+ *
+ * and in constructing the new divs and the corresponding equalities,
+ * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
+ * by the corresponding row from C_2.
+ */
+static struct isl_basic_map *normalize_divs(
+	struct isl_basic_map *bmap, int *progress)
+{
+	int i, j, k;
+	int total;
+	int div_eq;
+	struct isl_mat *B;
+	struct isl_vec *d;
+	struct isl_mat *T = NULL;
+	struct isl_mat *C = NULL;
+	struct isl_mat *C2 = NULL;
+	isl_int v;
+	int *pos;
+	int dropped, needed;
+
+	if (!bmap)
+		return NULL;
+
+	if (bmap->n_div == 0)
+		return bmap;
+
+	if (bmap->n_eq == 0)
+		return bmap;
+
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS))
+		return bmap;
+
+	total = isl_space_dim(bmap->dim, isl_dim_all);
+	div_eq = n_pure_div_eq(bmap);
+	if (div_eq == 0)
+		return bmap;
+
+	if (div_eq < bmap->n_eq) {
+		B = isl_mat_sub_alloc6(bmap->ctx, bmap->eq, div_eq,
+					bmap->n_eq - div_eq, 0, 1 + total);
+		C = isl_mat_variable_compression(B, &C2);
+		if (!C || !C2)
+			goto error;
+		if (C->n_col == 0) {
+			bmap = isl_basic_map_set_to_empty(bmap);
+			isl_mat_free(C);
+			isl_mat_free(C2);
+			goto done;
+		}
+	}
+
+	d = isl_vec_alloc(bmap->ctx, div_eq);
+	if (!d)
+		goto error;
+	for (i = 0, j = bmap->n_div-1; i < div_eq; ++i) {
+		while (j >= 0 && isl_int_is_zero(bmap->eq[i][1 + total + j]))
+			--j;
+		isl_int_set(d->block.data[i], bmap->eq[i][1 + total + j]);
+	}
+	B = isl_mat_sub_alloc6(bmap->ctx, bmap->eq, 0, div_eq, 0, 1 + total);
+
+	if (C) {
+		B = isl_mat_product(B, C);
+		C = NULL;
+	}
+
+	T = isl_mat_parameter_compression(B, d);
+	if (!T)
+		goto error;
+	if (T->n_col == 0) {
+		bmap = isl_basic_map_set_to_empty(bmap);
+		isl_mat_free(C2);
+		isl_mat_free(T);
+		goto done;
+	}
+	isl_int_init(v);
+	for (i = 0; i < T->n_row - 1; ++i) {
+		isl_int_fdiv_q(v, T->row[1 + i][0], T->row[1 + i][1 + i]);
+		if (isl_int_is_zero(v))
+			continue;
+		isl_mat_col_submul(T, 0, v, 1 + i);
+	}
+	isl_int_clear(v);
+	pos = isl_alloc_array(bmap->ctx, int, T->n_row);
+	if (!pos)
+		goto error;
+	/* We have to be careful because dropping equalities may reorder them */
+	dropped = 0;
+	for (j = bmap->n_div - 1; j >= 0; --j) {
+		for (i = 0; i < bmap->n_eq; ++i)
+			if (!isl_int_is_zero(bmap->eq[i][1 + total + j]))
+				break;
+		if (i < bmap->n_eq) {
+			bmap = isl_basic_map_drop_div(bmap, j);
+			isl_basic_map_drop_equality(bmap, i);
+			++dropped;
+		}
+	}
+	pos[0] = 0;
+	needed = 0;
+	for (i = 1; i < T->n_row; ++i) {
+		if (isl_int_is_one(T->row[i][i]))
+			pos[i] = i;
+		else
+			needed++;
+	}
+	if (needed > dropped) {
+		bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim),
+				needed, needed, 0);
+		if (!bmap)
+			goto error;
+	}
+	for (i = 1; i < T->n_row; ++i) {
+		if (isl_int_is_one(T->row[i][i]))
+			continue;
+		k = isl_basic_map_alloc_div(bmap);
+		pos[i] = 1 + total + k;
+		isl_seq_clr(bmap->div[k] + 1, 1 + total + bmap->n_div);
+		isl_int_set(bmap->div[k][0], T->row[i][i]);
+		if (C2)
+			isl_seq_cpy(bmap->div[k] + 1, C2->row[i], 1 + total);
+		else
+			isl_int_set_si(bmap->div[k][1 + i], 1);
+		for (j = 0; j < i; ++j) {
+			if (isl_int_is_zero(T->row[i][j]))
+				continue;
+			if (pos[j] < T->n_row && C2)
+				isl_seq_submul(bmap->div[k] + 1, T->row[i][j],
+						C2->row[pos[j]], 1 + total);
+			else
+				isl_int_neg(bmap->div[k][1 + pos[j]],
+								T->row[i][j]);
+		}
+		j = isl_basic_map_alloc_equality(bmap);
+		isl_seq_neg(bmap->eq[j], bmap->div[k]+1, 1+total+bmap->n_div);
+		isl_int_set(bmap->eq[j][pos[i]], bmap->div[k][0]);
+	}
+	free(pos);
+	isl_mat_free(C2);
+	isl_mat_free(T);
+
+	if (progress)
+		*progress = 1;
+done:
+	ISL_F_SET(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS);
+
+	return bmap;
+error:
+	isl_mat_free(C);
+	isl_mat_free(C2);
+	isl_mat_free(T);
+	return bmap;
+}
+
+static struct isl_basic_map *set_div_from_lower_bound(
+	struct isl_basic_map *bmap, int div, int ineq)
+{
+	unsigned total = 1 + isl_space_dim(bmap->dim, isl_dim_all);
+
+	isl_seq_neg(bmap->div[div] + 1, bmap->ineq[ineq], total + bmap->n_div);
+	isl_int_set(bmap->div[div][0], bmap->ineq[ineq][total + div]);
+	isl_int_add(bmap->div[div][1], bmap->div[div][1], bmap->div[div][0]);
+	isl_int_sub_ui(bmap->div[div][1], bmap->div[div][1], 1);
+	isl_int_set_si(bmap->div[div][1 + total + div], 0);
+
+	return bmap;
+}
+
+/* Check whether it is ok to define a div based on an inequality.
+ * To avoid the introduction of circular definitions of divs, we
+ * do not allow such a definition if the resulting expression would refer to
+ * any other undefined divs or if any known div is defined in
+ * terms of the unknown div.
+ */
+static int ok_to_set_div_from_bound(struct isl_basic_map *bmap,
+	int div, int ineq)
+{
+	int j;
+	unsigned total = 1 + isl_space_dim(bmap->dim, isl_dim_all);
+
+	/* Not defined in terms of unknown divs */
+	for (j = 0; j < bmap->n_div; ++j) {
+		if (div == j)
+			continue;
+		if (isl_int_is_zero(bmap->ineq[ineq][total + j]))
+			continue;
+		if (isl_int_is_zero(bmap->div[j][0]))
+			return 0;
+	}
+
+	/* No other div defined in terms of this one => avoid loops */
+	for (j = 0; j < bmap->n_div; ++j) {
+		if (div == j)
+			continue;
+		if (isl_int_is_zero(bmap->div[j][0]))
+			continue;
+		if (!isl_int_is_zero(bmap->div[j][1 + total + div]))
+			return 0;
+	}
+
+	return 1;
+}
+
+/* Would an expression for div "div" based on inequality "ineq" of "bmap"
+ * be a better expression than the current one?
+ *
+ * If we do not have any expression yet, then any expression would be better.
+ * Otherwise we check if the last variable involved in the inequality
+ * (disregarding the div that it would define) is in an earlier position
+ * than the last variable involved in the current div expression.
+ */
+static int better_div_constraint(__isl_keep isl_basic_map *bmap,
+	int div, int ineq)
+{
+	unsigned total = 1 + isl_space_dim(bmap->dim, isl_dim_all);
+	int last_div;
+	int last_ineq;
+
+	if (isl_int_is_zero(bmap->div[div][0]))
+		return 1;
+
+	if (isl_seq_last_non_zero(bmap->ineq[ineq] + total + div + 1,
+				  bmap->n_div - (div + 1)) >= 0)
+		return 0;
+
+	last_ineq = isl_seq_last_non_zero(bmap->ineq[ineq], total + div);
+	last_div = isl_seq_last_non_zero(bmap->div[div] + 1,
+					 total + bmap->n_div);
+
+	return last_ineq < last_div;
+}
+
+/* Given two constraints "k" and "l" that are opposite to each other,
+ * except for the constant term, check if we can use them
+ * to obtain an expression for one of the hitherto unknown divs or
+ * a "better" expression for a div for which we already have an expression.
+ * "sum" is the sum of the constant terms of the constraints.
+ * If this sum is strictly smaller than the coefficient of one
+ * of the divs, then this pair can be used define the div.
+ * To avoid the introduction of circular definitions of divs, we
+ * do not use the pair if the resulting expression would refer to
+ * any other undefined divs or if any known div is defined in
+ * terms of the unknown div.
+ */
+static struct isl_basic_map *check_for_div_constraints(
+	struct isl_basic_map *bmap, int k, int l, isl_int sum, int *progress)
+{
+	int i;
+	unsigned total = 1 + isl_space_dim(bmap->dim, isl_dim_all);
+
+	for (i = 0; i < bmap->n_div; ++i) {
+		if (isl_int_is_zero(bmap->ineq[k][total + i]))
+			continue;
+		if (isl_int_abs_ge(sum, bmap->ineq[k][total + i]))
+			continue;
+		if (!better_div_constraint(bmap, i, k))
+			continue;
+		if (!ok_to_set_div_from_bound(bmap, i, k))
+			break;
+		if (isl_int_is_pos(bmap->ineq[k][total + i]))
+			bmap = set_div_from_lower_bound(bmap, i, k);
+		else
+			bmap = set_div_from_lower_bound(bmap, i, l);
+		if (progress)
+			*progress = 1;
+		break;
+	}
+	return bmap;
+}
+
+__isl_give isl_basic_map *isl_basic_map_remove_duplicate_constraints(
+	__isl_take isl_basic_map *bmap, int *progress, int detect_divs)
+{
+	unsigned int size;
+	isl_int ***index;
+	int k, l, h;
+	int bits;
+	unsigned total = isl_basic_map_total_dim(bmap);
+	isl_int sum;
+	isl_ctx *ctx;
+
+	if (!bmap || bmap->n_ineq <= 1)
+		return bmap;
+
+	size = round_up(4 * (bmap->n_ineq+1) / 3 - 1);
+	bits = ffs(size) - 1;
+	ctx = isl_basic_map_get_ctx(bmap);
+	index = isl_calloc_array(ctx, isl_int **, size);
+	if (!index)
+		return bmap;
+
+	index[isl_seq_get_hash_bits(bmap->ineq[0]+1, total, bits)] = &bmap->ineq[0];
+	for (k = 1; k < bmap->n_ineq; ++k) {
+		h = hash_index(index, size, bits, bmap, k);
+		if (!index[h]) {
+			index[h] = &bmap->ineq[k];
+			continue;
+		}
+		if (progress)
+			*progress = 1;
+		l = index[h] - &bmap->ineq[0];
+		if (isl_int_lt(bmap->ineq[k][0], bmap->ineq[l][0]))
+			swap_inequality(bmap, k, l);
+		isl_basic_map_drop_inequality(bmap, k);
+		--k;
+	}
+	isl_int_init(sum);
+	for (k = 0; k < bmap->n_ineq-1; ++k) {
+		isl_seq_neg(bmap->ineq[k]+1, bmap->ineq[k]+1, total);
+		h = hash_index(index, size, bits, bmap, k);
+		isl_seq_neg(bmap->ineq[k]+1, bmap->ineq[k]+1, total);
+		if (!index[h])
+			continue;
+		l = index[h] - &bmap->ineq[0];
+		isl_int_add(sum, bmap->ineq[k][0], bmap->ineq[l][0]);
+		if (isl_int_is_pos(sum)) {
+			if (detect_divs)
+				bmap = check_for_div_constraints(bmap, k, l,
+								 sum, progress);
+			continue;
+		}
+		if (isl_int_is_zero(sum)) {
+			/* We need to break out of the loop after these
+			 * changes since the contents of the hash
+			 * will no longer be valid.
+			 * Plus, we probably we want to regauss first.
+			 */
+			if (progress)
+				*progress = 1;
+			isl_basic_map_drop_inequality(bmap, l);
+			isl_basic_map_inequality_to_equality(bmap, k);
+		} else
+			bmap = isl_basic_map_set_to_empty(bmap);
+		break;
+	}
+	isl_int_clear(sum);
+
+	free(index);
+	return bmap;
+}
+
+
+/* Eliminate knowns divs from constraints where they appear with
+ * a (positive or negative) unit coefficient.
+ *
+ * That is, replace
+ *
+ *	floor(e/m) + f >= 0
+ *
+ * by
+ *
+ *	e + m f >= 0
+ *
+ * and
+ *
+ *	-floor(e/m) + f >= 0
+ *
+ * by
+ *
+ *	-e + m f + m - 1 >= 0
+ *
+ * The first conversion is valid because floor(e/m) >= -f is equivalent
+ * to e/m >= -f because -f is an integral expression.
+ * The second conversion follows from the fact that
+ *
+ *	-floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
+ *
+ *
+ * Note that one of the div constraints may have been eliminated
+ * due to being redundant with respect to the constraint that is
+ * being modified by this function.  The modified constraint may
+ * no longer imply this div constraint, so we add it back to make
+ * sure we do not lose any information.
+ *
+ * We skip integral divs, i.e., those with denominator 1, as we would
+ * risk eliminating the div from the div constraints.  We do not need
+ * to handle those divs here anyway since the div constraints will turn
+ * out to form an equality and this equality can then be use to eliminate
+ * the div from all constraints.
+ */
+static __isl_give isl_basic_map *eliminate_unit_divs(
+	__isl_take isl_basic_map *bmap, int *progress)
+{
+	int i, j;
+	isl_ctx *ctx;
+	unsigned total;
+
+	if (!bmap)
+		return NULL;
+
+	ctx = isl_basic_map_get_ctx(bmap);
+	total = 1 + isl_space_dim(bmap->dim, isl_dim_all);
+
+	for (i = 0; i < bmap->n_div; ++i) {
+		if (isl_int_is_zero(bmap->div[i][0]))
+			continue;
+		if (isl_int_is_one(bmap->div[i][0]))
+			continue;
+		for (j = 0; j < bmap->n_ineq; ++j) {
+			int s;
+
+			if (!isl_int_is_one(bmap->ineq[j][total + i]) &&
+			    !isl_int_is_negone(bmap->ineq[j][total + i]))
+				continue;
+
+			*progress = 1;
+
+			s = isl_int_sgn(bmap->ineq[j][total + i]);
+			isl_int_set_si(bmap->ineq[j][total + i], 0);
+			if (s < 0)
+				isl_seq_combine(bmap->ineq[j],
+					ctx->negone, bmap->div[i] + 1,
+					bmap->div[i][0], bmap->ineq[j],
+					total + bmap->n_div);
+			else
+				isl_seq_combine(bmap->ineq[j],
+					ctx->one, bmap->div[i] + 1,
+					bmap->div[i][0], bmap->ineq[j],
+					total + bmap->n_div);
+			if (s < 0) {
+				isl_int_add(bmap->ineq[j][0],
+					bmap->ineq[j][0], bmap->div[i][0]);
+				isl_int_sub_ui(bmap->ineq[j][0],
+					bmap->ineq[j][0], 1);
+			}
+
+			bmap = isl_basic_map_extend_constraints(bmap, 0, 1);
+			if (isl_basic_map_add_div_constraint(bmap, i, s) < 0)
+				return isl_basic_map_free(bmap);
+		}
+	}
+
+	return bmap;
+}
+
+struct isl_basic_map *isl_basic_map_simplify(struct isl_basic_map *bmap)
+{
+	int progress = 1;
+	if (!bmap)
+		return NULL;
+	while (progress) {
+		progress = 0;
+		if (!bmap)
+			break;
+		if (isl_basic_map_plain_is_empty(bmap))
+			break;
+		bmap = isl_basic_map_normalize_constraints(bmap);
+		bmap = normalize_div_expressions(bmap);
+		bmap = remove_duplicate_divs(bmap, &progress);
+		bmap = eliminate_unit_divs(bmap, &progress);
+		bmap = eliminate_divs_eq(bmap, &progress);
+		bmap = eliminate_divs_ineq(bmap, &progress);
+		bmap = isl_basic_map_gauss(bmap, &progress);
+		/* requires equalities in normal form */
+		bmap = normalize_divs(bmap, &progress);
+		bmap = isl_basic_map_remove_duplicate_constraints(bmap,
+								&progress, 1);
+	}
+	return bmap;
+}
+
+struct isl_basic_set *isl_basic_set_simplify(struct isl_basic_set *bset)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_simplify((struct isl_basic_map *)bset);
+}
+
+
+int isl_basic_map_is_div_constraint(__isl_keep isl_basic_map *bmap,
+	isl_int *constraint, unsigned div)
+{
+	unsigned pos;
+
+	if (!bmap)
+		return -1;
+
+	pos = 1 + isl_space_dim(bmap->dim, isl_dim_all) + div;
+
+	if (isl_int_eq(constraint[pos], bmap->div[div][0])) {
+		int neg;
+		isl_int_sub(bmap->div[div][1],
+				bmap->div[div][1], bmap->div[div][0]);
+		isl_int_add_ui(bmap->div[div][1], bmap->div[div][1], 1);
+		neg = isl_seq_is_neg(constraint, bmap->div[div]+1, pos);
+		isl_int_sub_ui(bmap->div[div][1], bmap->div[div][1], 1);
+		isl_int_add(bmap->div[div][1],
+				bmap->div[div][1], bmap->div[div][0]);
+		if (!neg)
+			return 0;
+		if (isl_seq_first_non_zero(constraint+pos+1,
+					    bmap->n_div-div-1) != -1)
+			return 0;
+	} else if (isl_int_abs_eq(constraint[pos], bmap->div[div][0])) {
+		if (!isl_seq_eq(constraint, bmap->div[div]+1, pos))
+			return 0;
+		if (isl_seq_first_non_zero(constraint+pos+1,
+					    bmap->n_div-div-1) != -1)
+			return 0;
+	} else
+		return 0;
+
+	return 1;
+}
+
+int isl_basic_set_is_div_constraint(__isl_keep isl_basic_set *bset,
+	isl_int *constraint, unsigned div)
+{
+	return isl_basic_map_is_div_constraint(bset, constraint, div);
+}
+
+
+/* If the only constraints a div d=floor(f/m)
+ * appears in are its two defining constraints
+ *
+ *	f - m d >=0
+ *	-(f - (m - 1)) + m d >= 0
+ *
+ * then it can safely be removed.
+ */
+static int div_is_redundant(struct isl_basic_map *bmap, int div)
+{
+	int i;
+	unsigned pos = 1 + isl_space_dim(bmap->dim, isl_dim_all) + div;
+
+	for (i = 0; i < bmap->n_eq; ++i)
+		if (!isl_int_is_zero(bmap->eq[i][pos]))
+			return 0;
+
+	for (i = 0; i < bmap->n_ineq; ++i) {
+		if (isl_int_is_zero(bmap->ineq[i][pos]))
+			continue;
+		if (!isl_basic_map_is_div_constraint(bmap, bmap->ineq[i], div))
+			return 0;
+	}
+
+	for (i = 0; i < bmap->n_div; ++i) {
+		if (isl_int_is_zero(bmap->div[i][0]))
+			continue;
+		if (!isl_int_is_zero(bmap->div[i][1+pos]))
+			return 0;
+	}
+
+	return 1;
+}
+
+/*
+ * Remove divs that don't occur in any of the constraints or other divs.
+ * These can arise when dropping constraints from a basic map or
+ * when the divs of a basic map have been temporarily aligned
+ * with the divs of another basic map.
+ */
+static struct isl_basic_map *remove_redundant_divs(struct isl_basic_map *bmap)
+{
+	int i;
+
+	if (!bmap)
+		return NULL;
+
+	for (i = bmap->n_div-1; i >= 0; --i) {
+		if (!div_is_redundant(bmap, i))
+			continue;
+		bmap = isl_basic_map_drop_div(bmap, i);
+	}
+	return bmap;
+}
+
+struct isl_basic_map *isl_basic_map_finalize(struct isl_basic_map *bmap)
+{
+	bmap = remove_redundant_divs(bmap);
+	if (!bmap)
+		return NULL;
+	ISL_F_SET(bmap, ISL_BASIC_SET_FINAL);
+	return bmap;
+}
+
+struct isl_basic_set *isl_basic_set_finalize(struct isl_basic_set *bset)
+{
+	return (struct isl_basic_set *)
+		isl_basic_map_finalize((struct isl_basic_map *)bset);
+}
+
+struct isl_set *isl_set_finalize(struct isl_set *set)
+{
+	int i;
+
+	if (!set)
+		return NULL;
+	for (i = 0; i < set->n; ++i) {
+		set->p[i] = isl_basic_set_finalize(set->p[i]);
+		if (!set->p[i])
+			goto error;
+	}
+	return set;
+error:
+	isl_set_free(set);
+	return NULL;
+}
+
+struct isl_map *isl_map_finalize(struct isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_finalize(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+
+/* Remove definition of any div that is defined in terms of the given variable.
+ * The div itself is not removed.  Functions such as
+ * eliminate_divs_ineq depend on the other divs remaining in place.
+ */
+static struct isl_basic_map *remove_dependent_vars(struct isl_basic_map *bmap,
+									int pos)
+{
+	int i;
+
+	if (!bmap)
+		return NULL;
+
+	for (i = 0; i < bmap->n_div; ++i) {
+		if (isl_int_is_zero(bmap->div[i][0]))
+			continue;
+		if (isl_int_is_zero(bmap->div[i][1+1+pos]))
+			continue;
+		isl_int_set_si(bmap->div[i][0], 0);
+	}
+	return bmap;
+}
+
+/* Eliminate the specified variables from the constraints using
+ * Fourier-Motzkin.  The variables themselves are not removed.
+ */
+struct isl_basic_map *isl_basic_map_eliminate_vars(
+	struct isl_basic_map *bmap, unsigned pos, unsigned n)
+{
+	int d;
+	int i, j, k;
+	unsigned total;
+	int need_gauss = 0;
+
+	if (n == 0)
+		return bmap;
+	if (!bmap)
+		return NULL;
+	total = isl_basic_map_total_dim(bmap);
+
+	bmap = isl_basic_map_cow(bmap);
+	for (d = pos + n - 1; d >= 0 && d >= pos; --d)
+		bmap = remove_dependent_vars(bmap, d);
+	if (!bmap)
+		return NULL;
+
+	for (d = pos + n - 1;
+	     d >= 0 && d >= total - bmap->n_div && d >= pos; --d)
+		isl_seq_clr(bmap->div[d-(total-bmap->n_div)], 2+total);
+	for (d = pos + n - 1; d >= 0 && d >= pos; --d) {
+		int n_lower, n_upper;
+		if (!bmap)
+			return NULL;
+		for (i = 0; i < bmap->n_eq; ++i) {
+			if (isl_int_is_zero(bmap->eq[i][1+d]))
+				continue;
+			eliminate_var_using_equality(bmap, d, bmap->eq[i], 0, NULL);
+			isl_basic_map_drop_equality(bmap, i);
+			need_gauss = 1;
+			break;
+		}
+		if (i < bmap->n_eq)
+			continue;
+		n_lower = 0;
+		n_upper = 0;
+		for (i = 0; i < bmap->n_ineq; ++i) {
+			if (isl_int_is_pos(bmap->ineq[i][1+d]))
+				n_lower++;
+			else if (isl_int_is_neg(bmap->ineq[i][1+d]))
+				n_upper++;
+		}
+		bmap = isl_basic_map_extend_constraints(bmap,
+				0, n_lower * n_upper);
+		if (!bmap)
+			goto error;
+		for (i = bmap->n_ineq - 1; i >= 0; --i) {
+			int last;
+			if (isl_int_is_zero(bmap->ineq[i][1+d]))
+				continue;
+			last = -1;
+			for (j = 0; j < i; ++j) {
+				if (isl_int_is_zero(bmap->ineq[j][1+d]))
+					continue;
+				last = j;
+				if (isl_int_sgn(bmap->ineq[i][1+d]) ==
+				    isl_int_sgn(bmap->ineq[j][1+d]))
+					continue;
+				k = isl_basic_map_alloc_inequality(bmap);
+				if (k < 0)
+					goto error;
+				isl_seq_cpy(bmap->ineq[k], bmap->ineq[i],
+						1+total);
+				isl_seq_elim(bmap->ineq[k], bmap->ineq[j],
+						1+d, 1+total, NULL);
+			}
+			isl_basic_map_drop_inequality(bmap, i);
+			i = last + 1;
+		}
+		if (n_lower > 0 && n_upper > 0) {
+			bmap = isl_basic_map_normalize_constraints(bmap);
+			bmap = isl_basic_map_remove_duplicate_constraints(bmap,
+								    NULL, 0);
+			bmap = isl_basic_map_gauss(bmap, NULL);
+			bmap = isl_basic_map_remove_redundancies(bmap);
+			need_gauss = 0;
+			if (!bmap)
+				goto error;
+			if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
+				break;
+		}
+	}
+	ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
+	if (need_gauss)
+		bmap = isl_basic_map_gauss(bmap, NULL);
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_eliminate_vars(
+	struct isl_basic_set *bset, unsigned pos, unsigned n)
+{
+	return (struct isl_basic_set *)isl_basic_map_eliminate_vars(
+			(struct isl_basic_map *)bset, pos, n);
+}
+
+/* Eliminate the specified n dimensions starting at first from the
+ * constraints, without removing the dimensions from the space.
+ * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
+ * Otherwise, they are projected out and the original space is restored.
+ */
+__isl_give isl_basic_map *isl_basic_map_eliminate(
+	__isl_take isl_basic_map *bmap,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	isl_space *space;
+
+	if (!bmap)
+		return NULL;
+	if (n == 0)
+		return bmap;
+
+	if (first + n > isl_basic_map_dim(bmap, type) || first + n < first)
+		isl_die(bmap->ctx, isl_error_invalid,
+			"index out of bounds", goto error);
+
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL)) {
+		first += isl_basic_map_offset(bmap, type) - 1;
+		bmap = isl_basic_map_eliminate_vars(bmap, first, n);
+		return isl_basic_map_finalize(bmap);
+	}
+
+	space = isl_basic_map_get_space(bmap);
+	bmap = isl_basic_map_project_out(bmap, type, first, n);
+	bmap = isl_basic_map_insert_dims(bmap, type, first, n);
+	bmap = isl_basic_map_reset_space(bmap, space);
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_eliminate(
+	__isl_take isl_basic_set *bset,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return isl_basic_map_eliminate(bset, type, first, n);
+}
+
+/* Don't assume equalities are in order, because align_divs
+ * may have changed the order of the divs.
+ */
+static void compute_elimination_index(struct isl_basic_map *bmap, int *elim)
+{
+	int d, i;
+	unsigned total;
+
+	total = isl_space_dim(bmap->dim, isl_dim_all);
+	for (d = 0; d < total; ++d)
+		elim[d] = -1;
+	for (i = 0; i < bmap->n_eq; ++i) {
+		for (d = total - 1; d >= 0; --d) {
+			if (isl_int_is_zero(bmap->eq[i][1+d]))
+				continue;
+			elim[d] = i;
+			break;
+		}
+	}
+}
+
+static void set_compute_elimination_index(struct isl_basic_set *bset, int *elim)
+{
+	compute_elimination_index((struct isl_basic_map *)bset, elim);
+}
+
+static int reduced_using_equalities(isl_int *dst, isl_int *src,
+	struct isl_basic_map *bmap, int *elim)
+{
+	int d;
+	int copied = 0;
+	unsigned total;
+
+	total = isl_space_dim(bmap->dim, isl_dim_all);
+	for (d = total - 1; d >= 0; --d) {
+		if (isl_int_is_zero(src[1+d]))
+			continue;
+		if (elim[d] == -1)
+			continue;
+		if (!copied) {
+			isl_seq_cpy(dst, src, 1 + total);
+			copied = 1;
+		}
+		isl_seq_elim(dst, bmap->eq[elim[d]], 1 + d, 1 + total, NULL);
+	}
+	return copied;
+}
+
+static int set_reduced_using_equalities(isl_int *dst, isl_int *src,
+	struct isl_basic_set *bset, int *elim)
+{
+	return reduced_using_equalities(dst, src,
+					(struct isl_basic_map *)bset, elim);
+}
+
+static struct isl_basic_set *isl_basic_set_reduce_using_equalities(
+	struct isl_basic_set *bset, struct isl_basic_set *context)
+{
+	int i;
+	int *elim;
+
+	if (!bset || !context)
+		goto error;
+
+	if (context->n_eq == 0) {
+		isl_basic_set_free(context);
+		return bset;
+	}
+
+	bset = isl_basic_set_cow(bset);
+	if (!bset)
+		goto error;
+
+	elim = isl_alloc_array(bset->ctx, int, isl_basic_set_n_dim(bset));
+	if (!elim)
+		goto error;
+	set_compute_elimination_index(context, elim);
+	for (i = 0; i < bset->n_eq; ++i)
+		set_reduced_using_equalities(bset->eq[i], bset->eq[i],
+							context, elim);
+	for (i = 0; i < bset->n_ineq; ++i)
+		set_reduced_using_equalities(bset->ineq[i], bset->ineq[i],
+							context, elim);
+	isl_basic_set_free(context);
+	free(elim);
+	bset = isl_basic_set_simplify(bset);
+	bset = isl_basic_set_finalize(bset);
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	isl_basic_set_free(context);
+	return NULL;
+}
+
+static struct isl_basic_set *remove_shifted_constraints(
+	struct isl_basic_set *bset, struct isl_basic_set *context)
+{
+	unsigned int size;
+	isl_int ***index;
+	int bits;
+	int k, h, l;
+	isl_ctx *ctx;
+
+	if (!bset)
+		return NULL;
+
+	size = round_up(4 * (context->n_ineq+1) / 3 - 1);
+	bits = ffs(size) - 1;
+	ctx = isl_basic_set_get_ctx(bset);
+	index = isl_calloc_array(ctx, isl_int **, size);
+	if (!index)
+		return bset;
+
+	for (k = 0; k < context->n_ineq; ++k) {
+		h = set_hash_index(index, size, bits, context, k);
+		index[h] = &context->ineq[k];
+	}
+	for (k = 0; k < bset->n_ineq; ++k) {
+		h = set_hash_index(index, size, bits, bset, k);
+		if (!index[h])
+			continue;
+		l = index[h] - &context->ineq[0];
+		if (isl_int_lt(bset->ineq[k][0], context->ineq[l][0]))
+			continue;
+		bset = isl_basic_set_cow(bset);
+		if (!bset)
+			goto error;
+		isl_basic_set_drop_inequality(bset, k);
+		--k;
+	}
+	free(index);
+	return bset;
+error:
+	free(index);
+	return bset;
+}
+
+/* Remove constraints from "bmap" that are identical to constraints
+ * in "context" or that are more relaxed (greater constant term).
+ *
+ * We perform the test for shifted copies on the pure constraints
+ * in remove_shifted_constraints.
+ */
+static __isl_give isl_basic_map *isl_basic_map_remove_shifted_constraints(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_map *context)
+{
+	isl_basic_set *bset, *bset_context;
+
+	if (!bmap || !context)
+		goto error;
+
+	if (bmap->n_ineq == 0 || context->n_ineq == 0) {
+		isl_basic_map_free(context);
+		return bmap;
+	}
+
+	context = isl_basic_map_align_divs(context, bmap);
+	bmap = isl_basic_map_align_divs(bmap, context);
+
+	bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
+	bset_context = isl_basic_map_underlying_set(context);
+	bset = remove_shifted_constraints(bset, bset_context);
+	isl_basic_set_free(bset_context);
+
+	bmap = isl_basic_map_overlying_set(bset, bmap);
+
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	isl_basic_map_free(context);
+	return NULL;
+}
+
+/* Does the (linear part of a) constraint "c" involve any of the "len"
+ * "relevant" dimensions?
+ */
+static int is_related(isl_int *c, int len, int *relevant)
+{
+	int i;
+
+	for (i = 0; i < len; ++i) {
+		if (!relevant[i])
+			continue;
+		if (!isl_int_is_zero(c[i]))
+			return 1;
+	}
+
+	return 0;
+}
+
+/* Drop constraints from "bset" that do not involve any of
+ * the dimensions marked "relevant".
+ */
+static __isl_give isl_basic_set *drop_unrelated_constraints(
+	__isl_take isl_basic_set *bset, int *relevant)
+{
+	int i, dim;
+
+	dim = isl_basic_set_dim(bset, isl_dim_set);
+	for (i = 0; i < dim; ++i)
+		if (!relevant[i])
+			break;
+	if (i >= dim)
+		return bset;
+
+	for (i = bset->n_eq - 1; i >= 0; --i)
+		if (!is_related(bset->eq[i] + 1, dim, relevant))
+			isl_basic_set_drop_equality(bset, i);
+
+	for (i = bset->n_ineq - 1; i >= 0; --i)
+		if (!is_related(bset->ineq[i] + 1, dim, relevant))
+			isl_basic_set_drop_inequality(bset, i);
+
+	return bset;
+}
+
+/* Update the groups in "group" based on the (linear part of a) constraint "c".
+ *
+ * In particular, for any variable involved in the constraint,
+ * find the actual group id from before and replace the group
+ * of the corresponding variable by the minimal group of all
+ * the variables involved in the constraint considered so far
+ * (if this minimum is smaller) or replace the minimum by this group
+ * (if the minimum is larger).
+ *
+ * At the end, all the variables in "c" will (indirectly) point
+ * to the minimal of the groups that they referred to originally.
+ */
+static void update_groups(int dim, int *group, isl_int *c)
+{
+	int j;
+	int min = dim;
+
+	for (j = 0; j < dim; ++j) {
+		if (isl_int_is_zero(c[j]))
+			continue;
+		while (group[j] >= 0 && group[group[j]] != group[j])
+			group[j] = group[group[j]];
+		if (group[j] == min)
+			continue;
+		if (group[j] < min) {
+			if (min >= 0 && min < dim)
+				group[min] = group[j];
+			min = group[j];
+		} else
+			group[group[j]] = min;
+	}
+}
+
+/* Drop constraints from "context" that are irrelevant for computing
+ * the gist of "bset".
+ *
+ * In particular, drop constraints in variables that are not related
+ * to any of the variables involved in the constraints of "bset"
+ * in the sense that there is no sequence of constraints that connects them.
+ *
+ * We construct groups of variables that collect variables that
+ * (indirectly) appear in some common constraint of "context".
+ * Each group is identified by the first variable in the group,
+ * except for the special group of variables that appear in "bset"
+ * (or are related to those variables), which is identified by -1.
+ * If group[i] is equal to i (or -1), then the group of i is i (or -1),
+ * otherwise the group of i is the group of group[i].
+ *
+ * We first initialize the -1 group with the variables that appear in "bset".
+ * Then we initialize groups for the remaining variables.
+ * Then we iterate over the constraints of "context" and update the
+ * group of the variables in the constraint by the smallest group.
+ * Finally, we resolve indirect references to groups by running over
+ * the variables.
+ *
+ * After computing the groups, we drop constraints that do not involve
+ * any variables in the -1 group.
+ */
+static __isl_give isl_basic_set *drop_irrelevant_constraints(
+	__isl_take isl_basic_set *context, __isl_keep isl_basic_set *bset)
+{
+	isl_ctx *ctx;
+	int *group;
+	int dim;
+	int i, j;
+	int last;
+
+	if (!context || !bset)
+		return isl_basic_set_free(context);
+
+	dim = isl_basic_set_dim(bset, isl_dim_set);
+	ctx = isl_basic_set_get_ctx(bset);
+	group = isl_calloc_array(ctx, int, dim);
+
+	if (!group)
+		goto error;
+
+	for (i = 0; i < dim; ++i) {
+		for (j = 0; j < bset->n_eq; ++j)
+			if (!isl_int_is_zero(bset->eq[j][1 + i]))
+				break;
+		if (j < bset->n_eq) {
+			group[i] = -1;
+			continue;
+		}
+		for (j = 0; j < bset->n_ineq; ++j)
+			if (!isl_int_is_zero(bset->ineq[j][1 + i]))
+				break;
+		if (j < bset->n_ineq)
+			group[i] = -1;
+	}
+
+	last = -1;
+	for (i = 0; i < dim; ++i)
+		if (group[i] >= 0)
+			last = group[i] = i;
+	if (last < 0) {
+		free(group);
+		return context;
+	}
+
+	for (i = 0; i < context->n_eq; ++i)
+		update_groups(dim, group, context->eq[i] + 1);
+	for (i = 0; i < context->n_ineq; ++i)
+		update_groups(dim, group, context->ineq[i] + 1);
+
+	for (i = 0; i < dim; ++i)
+		if (group[i] >= 0)
+			group[i] = group[group[i]];
+
+	for (i = 0; i < dim; ++i)
+		group[i] = group[i] == -1;
+
+	context = drop_unrelated_constraints(context, group);
+
+	free(group);
+	return context;
+error:
+	free(group);
+	return isl_basic_set_free(context);
+}
+
+/* Remove all information from bset that is redundant in the context
+ * of context.  Both bset and context are assumed to be full-dimensional.
+ *
+ * We first remove the inequalities from "bset"
+ * that are obviously redundant with respect to some inequality in "context".
+ * Then we remove those constraints from "context" that have become
+ * irrelevant for computing the gist of "bset".
+ * Note that this removal of constraints cannot be replaced by
+ * a factorization because factors in "bset" may still be connected
+ * to each other through constraints in "context".
+ *
+ * If there are any inequalities left, we construct a tableau for
+ * the context and then add the inequalities of "bset".
+ * Before adding these inequalities, we freeze all constraints such that
+ * they won't be considered redundant in terms of the constraints of "bset".
+ * Then we detect all redundant constraints (among the
+ * constraints that weren't frozen), first by checking for redundancy in the
+ * the tableau and then by checking if replacing a constraint by its negation
+ * would lead to an empty set.  This last step is fairly expensive
+ * and could be optimized by more reuse of the tableau.
+ * Finally, we update bset according to the results.
+ */
+static __isl_give isl_basic_set *uset_gist_full(__isl_take isl_basic_set *bset,
+	__isl_take isl_basic_set *context)
+{
+	int i, k;
+	isl_basic_set *combined = NULL;
+	struct isl_tab *tab = NULL;
+	unsigned context_ineq;
+	unsigned total;
+
+	if (!bset || !context)
+		goto error;
+
+	if (isl_basic_set_is_universe(bset)) {
+		isl_basic_set_free(context);
+		return bset;
+	}
+
+	if (isl_basic_set_is_universe(context)) {
+		isl_basic_set_free(context);
+		return bset;
+	}
+
+	bset = remove_shifted_constraints(bset, context);
+	if (!bset)
+		goto error;
+	if (bset->n_ineq == 0)
+		goto done;
+
+	context = drop_irrelevant_constraints(context, bset);
+	if (!context)
+		goto error;
+	if (isl_basic_set_is_universe(context)) {
+		isl_basic_set_free(context);
+		return bset;
+	}
+
+	context_ineq = context->n_ineq;
+	combined = isl_basic_set_cow(isl_basic_set_copy(context));
+	combined = isl_basic_set_extend_constraints(combined, 0, bset->n_ineq);
+	tab = isl_tab_from_basic_set(combined, 0);
+	for (i = 0; i < context_ineq; ++i)
+		if (isl_tab_freeze_constraint(tab, i) < 0)
+			goto error;
+	if (isl_tab_extend_cons(tab, bset->n_ineq) < 0)
+		goto error;
+	for (i = 0; i < bset->n_ineq; ++i)
+		if (isl_tab_add_ineq(tab, bset->ineq[i]) < 0)
+			goto error;
+	bset = isl_basic_set_add_constraints(combined, bset, 0);
+	combined = NULL;
+	if (!bset)
+		goto error;
+	if (isl_tab_detect_redundant(tab) < 0)
+		goto error;
+	total = isl_basic_set_total_dim(bset);
+	for (i = context_ineq; i < bset->n_ineq; ++i) {
+		int is_empty;
+		if (tab->con[i].is_redundant)
+			continue;
+		tab->con[i].is_redundant = 1;
+		combined = isl_basic_set_dup(bset);
+		combined = isl_basic_set_update_from_tab(combined, tab);
+		combined = isl_basic_set_extend_constraints(combined, 0, 1);
+		k = isl_basic_set_alloc_inequality(combined);
+		if (k < 0)
+			goto error;
+		isl_seq_neg(combined->ineq[k], bset->ineq[i], 1 + total);
+		isl_int_sub_ui(combined->ineq[k][0], combined->ineq[k][0], 1);
+		is_empty = isl_basic_set_is_empty(combined);
+		if (is_empty < 0)
+			goto error;
+		isl_basic_set_free(combined);
+		combined = NULL;
+		if (!is_empty)
+			tab->con[i].is_redundant = 0;
+	}
+	for (i = 0; i < context_ineq; ++i)
+		tab->con[i].is_redundant = 1;
+	bset = isl_basic_set_update_from_tab(bset, tab);
+	if (bset) {
+		ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT);
+		ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT);
+	}
+
+	isl_tab_free(tab);
+done:
+	bset = isl_basic_set_simplify(bset);
+	bset = isl_basic_set_finalize(bset);
+	isl_basic_set_free(context);
+	return bset;
+error:
+	isl_tab_free(tab);
+	isl_basic_set_free(combined);
+	isl_basic_set_free(context);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Remove all information from bset that is redundant in the context
+ * of context.  In particular, equalities that are linear combinations
+ * of those in context are removed.  Then the inequalities that are
+ * redundant in the context of the equalities and inequalities of
+ * context are removed.
+ *
+ * First of all, we drop those constraints from "context"
+ * that are irrelevant for computing the gist of "bset".
+ * Alternatively, we could factorize the intersection of "context" and "bset".
+ *
+ * We first compute the integer affine hull of the intersection,
+ * compute the gist inside this affine hull and then add back
+ * those equalities that are not implied by the context.
+ *
+ * If two constraints are mutually redundant, then uset_gist_full
+ * will remove the second of those constraints.  We therefore first
+ * sort the constraints so that constraints not involving existentially
+ * quantified variables are given precedence over those that do.
+ * We have to perform this sorting before the variable compression,
+ * because that may effect the order of the variables.
+ */
+static __isl_give isl_basic_set *uset_gist(__isl_take isl_basic_set *bset,
+	__isl_take isl_basic_set *context)
+{
+	isl_mat *eq;
+	isl_mat *T, *T2;
+	isl_basic_set *aff;
+	isl_basic_set *aff_context;
+	unsigned total;
+
+	if (!bset || !context)
+		goto error;
+
+	context = drop_irrelevant_constraints(context, bset);
+
+	aff = isl_basic_set_copy(bset);
+	aff = isl_basic_set_intersect(aff, isl_basic_set_copy(context));
+	aff = isl_basic_set_affine_hull(aff);
+	if (!aff)
+		goto error;
+	if (isl_basic_set_plain_is_empty(aff)) {
+		isl_basic_set_free(bset);
+		isl_basic_set_free(context);
+		return aff;
+	}
+	bset = isl_basic_set_sort_constraints(bset);
+	if (aff->n_eq == 0) {
+		isl_basic_set_free(aff);
+		return uset_gist_full(bset, context);
+	}
+	total = isl_basic_set_total_dim(bset);
+	eq = isl_mat_sub_alloc6(bset->ctx, aff->eq, 0, aff->n_eq, 0, 1 + total);
+	eq = isl_mat_cow(eq);
+	T = isl_mat_variable_compression(eq, &T2);
+	if (T && T->n_col == 0) {
+		isl_mat_free(T);
+		isl_mat_free(T2);
+		isl_basic_set_free(context);
+		isl_basic_set_free(aff);
+		return isl_basic_set_set_to_empty(bset);
+	}
+
+	aff_context = isl_basic_set_affine_hull(isl_basic_set_copy(context));
+
+	bset = isl_basic_set_preimage(bset, isl_mat_copy(T));
+	context = isl_basic_set_preimage(context, T);
+
+	bset = uset_gist_full(bset, context);
+	bset = isl_basic_set_preimage(bset, T2);
+	bset = isl_basic_set_intersect(bset, aff);
+	bset = isl_basic_set_reduce_using_equalities(bset, aff_context);
+
+	if (bset) {
+		ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT);
+		ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT);
+	}
+
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	isl_basic_set_free(context);
+	return NULL;
+}
+
+/* Normalize the divs in "bmap" in the context of the equalities in "context".
+ * We simply add the equalities in context to bmap and then do a regular
+ * div normalizations.  Better results can be obtained by normalizing
+ * only the divs in bmap than do not also appear in context.
+ * We need to be careful to reduce the divs using the equalities
+ * so that later calls to isl_basic_map_overlying_set wouldn't introduce
+ * spurious constraints.
+ */
+static struct isl_basic_map *normalize_divs_in_context(
+	struct isl_basic_map *bmap, struct isl_basic_map *context)
+{
+	int i;
+	unsigned total_context;
+	int div_eq;
+
+	div_eq = n_pure_div_eq(bmap);
+	if (div_eq == 0)
+		return bmap;
+
+	bmap = isl_basic_map_cow(bmap);
+	if (context->n_div > 0)
+		bmap = isl_basic_map_align_divs(bmap, context);
+
+	total_context = isl_basic_map_total_dim(context);
+	bmap = isl_basic_map_extend_constraints(bmap, context->n_eq, 0);
+	for (i = 0; i < context->n_eq; ++i) {
+		int k;
+		k = isl_basic_map_alloc_equality(bmap);
+		if (k < 0)
+			return isl_basic_map_free(bmap);
+		isl_seq_cpy(bmap->eq[k], context->eq[i], 1 + total_context);
+		isl_seq_clr(bmap->eq[k] + 1 + total_context,
+				isl_basic_map_total_dim(bmap) - total_context);
+	}
+	bmap = isl_basic_map_gauss(bmap, NULL);
+	bmap = normalize_divs(bmap, NULL);
+	bmap = isl_basic_map_gauss(bmap, NULL);
+	return bmap;
+}
+
+/* Return a basic map that has the same intersection with "context" as "bmap"
+ * and that is as "simple" as possible.
+ *
+ * The core computation is performed on the pure constraints.
+ * When we add back the meaning of the integer divisions, we need
+ * to (re)introduce the div constraints.  If we happen to have
+ * discovered that some of these integer divisions are equal to
+ * some affine combination of other variables, then these div
+ * constraints may end up getting simplified in terms of the equalities,
+ * resulting in extra inequalities on the other variables that
+ * may have been removed already or that may not even have been
+ * part of the input.  We try and remove those constraints of
+ * this form that are most obviously redundant with respect to
+ * the context.  We also remove those div constraints that are
+ * redundant with respect to the other constraints in the result.
+ */
+struct isl_basic_map *isl_basic_map_gist(struct isl_basic_map *bmap,
+	struct isl_basic_map *context)
+{
+	isl_basic_set *bset, *eq;
+	isl_basic_map *eq_bmap;
+	unsigned n_div, n_eq, n_ineq;
+
+	if (!bmap || !context)
+		goto error;
+
+	if (isl_basic_map_is_universe(bmap)) {
+		isl_basic_map_free(context);
+		return bmap;
+	}
+	if (isl_basic_map_plain_is_empty(context)) {
+		isl_space *space = isl_basic_map_get_space(bmap);
+		isl_basic_map_free(bmap);
+		isl_basic_map_free(context);
+		return isl_basic_map_universe(space);
+	}
+	if (isl_basic_map_plain_is_empty(bmap)) {
+		isl_basic_map_free(context);
+		return bmap;
+	}
+
+	bmap = isl_basic_map_remove_redundancies(bmap);
+	context = isl_basic_map_remove_redundancies(context);
+	if (!context)
+		goto error;
+
+	if (context->n_eq)
+		bmap = normalize_divs_in_context(bmap, context);
+
+	context = isl_basic_map_align_divs(context, bmap);
+	bmap = isl_basic_map_align_divs(bmap, context);
+	n_div = isl_basic_map_dim(bmap, isl_dim_div);
+
+	bset = uset_gist(isl_basic_map_underlying_set(isl_basic_map_copy(bmap)),
+		    isl_basic_map_underlying_set(isl_basic_map_copy(context)));
+
+	if (!bset || bset->n_eq == 0 || n_div == 0 ||
+	    isl_basic_set_plain_is_empty(bset)) {
+		isl_basic_map_free(context);
+		return isl_basic_map_overlying_set(bset, bmap);
+	}
+
+	n_eq = bset->n_eq;
+	n_ineq = bset->n_ineq;
+	eq = isl_basic_set_copy(bset);
+	eq = isl_basic_set_cow(bset);
+	if (isl_basic_set_free_inequality(eq, n_ineq) < 0)
+		eq = isl_basic_set_free(eq);
+	if (isl_basic_set_free_equality(bset, n_eq) < 0)
+		bset = isl_basic_set_free(bset);
+
+	eq_bmap = isl_basic_map_overlying_set(eq, isl_basic_map_copy(bmap));
+	eq_bmap = isl_basic_map_remove_shifted_constraints(eq_bmap, context);
+	bmap = isl_basic_map_overlying_set(bset, bmap);
+	bmap = isl_basic_map_intersect(bmap, eq_bmap);
+	bmap = isl_basic_map_remove_redundancies(bmap);
+
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	isl_basic_map_free(context);
+	return NULL;
+}
+
+/*
+ * Assumes context has no implicit divs.
+ */
+__isl_give isl_map *isl_map_gist_basic_map(__isl_take isl_map *map,
+	__isl_take isl_basic_map *context)
+{
+	int i;
+
+	if (!map || !context)
+		goto error;
+
+	if (isl_basic_map_plain_is_empty(context)) {
+		isl_space *space = isl_map_get_space(map);
+		isl_map_free(map);
+		isl_basic_map_free(context);
+		return isl_map_universe(space);
+	}
+
+	context = isl_basic_map_remove_redundancies(context);
+	map = isl_map_cow(map);
+	if (!map || !context)
+		goto error;
+	isl_assert(map->ctx, isl_space_is_equal(map->dim, context->dim), goto error);
+	map = isl_map_compute_divs(map);
+	if (!map)
+		goto error;
+	for (i = map->n - 1; i >= 0; --i) {
+		map->p[i] = isl_basic_map_gist(map->p[i],
+						isl_basic_map_copy(context));
+		if (!map->p[i])
+			goto error;
+		if (isl_basic_map_plain_is_empty(map->p[i])) {
+			isl_basic_map_free(map->p[i]);
+			if (i != map->n - 1)
+				map->p[i] = map->p[map->n - 1];
+			map->n--;
+		}
+	}
+	isl_basic_map_free(context);
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+	return map;
+error:
+	isl_map_free(map);
+	isl_basic_map_free(context);
+	return NULL;
+}
+
+/* Return a map that has the same intersection with "context" as "map"
+ * and that is as "simple" as possible.
+ *
+ * If "map" is already the universe, then we cannot make it any simpler.
+ * Similarly, if "context" is the universe, then we cannot exploit it
+ * to simplify "map"
+ * If "map" and "context" are identical to each other, then we can
+ * return the corresponding universe.
+ *
+ * If none of these cases apply, we have to work a bit harder.
+ */
+static __isl_give isl_map *map_gist(__isl_take isl_map *map,
+	__isl_take isl_map *context)
+{
+	int equal;
+	int is_universe;
+
+	is_universe = isl_map_plain_is_universe(map);
+	if (is_universe >= 0 && !is_universe)
+		is_universe = isl_map_plain_is_universe(context);
+	if (is_universe < 0)
+		goto error;
+	if (is_universe) {
+		isl_map_free(context);
+		return map;
+	}
+
+	equal = isl_map_plain_is_equal(map, context);
+	if (equal < 0)
+		goto error;
+	if (equal) {
+		isl_map *res = isl_map_universe(isl_map_get_space(map));
+		isl_map_free(map);
+		isl_map_free(context);
+		return res;
+	}
+
+	context = isl_map_compute_divs(context);
+	return isl_map_gist_basic_map(map, isl_map_simple_hull(context));
+error:
+	isl_map_free(map);
+	isl_map_free(context);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_gist(__isl_take isl_map *map,
+	__isl_take isl_map *context)
+{
+	return isl_map_align_params_map_map_and(map, context, &map_gist);
+}
+
+struct isl_basic_set *isl_basic_set_gist(struct isl_basic_set *bset,
+						struct isl_basic_set *context)
+{
+	return (struct isl_basic_set *)isl_basic_map_gist(
+		(struct isl_basic_map *)bset, (struct isl_basic_map *)context);
+}
+
+__isl_give isl_set *isl_set_gist_basic_set(__isl_take isl_set *set,
+	__isl_take isl_basic_set *context)
+{
+	return (struct isl_set *)isl_map_gist_basic_map((struct isl_map *)set,
+					(struct isl_basic_map *)context);
+}
+
+__isl_give isl_set *isl_set_gist_params_basic_set(__isl_take isl_set *set,
+	__isl_take isl_basic_set *context)
+{
+	isl_space *space = isl_set_get_space(set);
+	isl_basic_set *dom_context = isl_basic_set_universe(space);
+	dom_context = isl_basic_set_intersect_params(dom_context, context);
+	return isl_set_gist_basic_set(set, dom_context);
+}
+
+__isl_give isl_set *isl_set_gist(__isl_take isl_set *set,
+	__isl_take isl_set *context)
+{
+	return (struct isl_set *)isl_map_gist((struct isl_map *)set,
+					(struct isl_map *)context);
+}
+
+__isl_give isl_map *isl_map_gist_domain(__isl_take isl_map *map,
+	__isl_take isl_set *context)
+{
+	isl_map *map_context = isl_map_universe(isl_map_get_space(map));
+	map_context = isl_map_intersect_domain(map_context, context);
+	return isl_map_gist(map, map_context);
+}
+
+__isl_give isl_map *isl_map_gist_range(__isl_take isl_map *map,
+	__isl_take isl_set *context)
+{
+	isl_map *map_context = isl_map_universe(isl_map_get_space(map));
+	map_context = isl_map_intersect_range(map_context, context);
+	return isl_map_gist(map, map_context);
+}
+
+__isl_give isl_map *isl_map_gist_params(__isl_take isl_map *map,
+	__isl_take isl_set *context)
+{
+	isl_map *map_context = isl_map_universe(isl_map_get_space(map));
+	map_context = isl_map_intersect_params(map_context, context);
+	return isl_map_gist(map, map_context);
+}
+
+__isl_give isl_set *isl_set_gist_params(__isl_take isl_set *set,
+	__isl_take isl_set *context)
+{
+	return isl_map_gist_params(set, context);
+}
+
+/* Quick check to see if two basic maps are disjoint.
+ * In particular, we reduce the equalities and inequalities of
+ * one basic map in the context of the equalities of the other
+ * basic map and check if we get a contradiction.
+ */
+int isl_basic_map_plain_is_disjoint(__isl_keep isl_basic_map *bmap1,
+	__isl_keep isl_basic_map *bmap2)
+{
+	struct isl_vec *v = NULL;
+	int *elim = NULL;
+	unsigned total;
+	int i;
+
+	if (!bmap1 || !bmap2)
+		return -1;
+	isl_assert(bmap1->ctx, isl_space_is_equal(bmap1->dim, bmap2->dim),
+			return -1);
+	if (bmap1->n_div || bmap2->n_div)
+		return 0;
+	if (!bmap1->n_eq && !bmap2->n_eq)
+		return 0;
+
+	total = isl_space_dim(bmap1->dim, isl_dim_all);
+	if (total == 0)
+		return 0;
+	v = isl_vec_alloc(bmap1->ctx, 1 + total);
+	if (!v)
+		goto error;
+	elim = isl_alloc_array(bmap1->ctx, int, total);
+	if (!elim)
+		goto error;
+	compute_elimination_index(bmap1, elim);
+	for (i = 0; i < bmap2->n_eq; ++i) {
+		int reduced;
+		reduced = reduced_using_equalities(v->block.data, bmap2->eq[i],
+							bmap1, elim);
+		if (reduced && !isl_int_is_zero(v->block.data[0]) &&
+		    isl_seq_first_non_zero(v->block.data + 1, total) == -1)
+			goto disjoint;
+	}
+	for (i = 0; i < bmap2->n_ineq; ++i) {
+		int reduced;
+		reduced = reduced_using_equalities(v->block.data,
+						bmap2->ineq[i], bmap1, elim);
+		if (reduced && isl_int_is_neg(v->block.data[0]) &&
+		    isl_seq_first_non_zero(v->block.data + 1, total) == -1)
+			goto disjoint;
+	}
+	compute_elimination_index(bmap2, elim);
+	for (i = 0; i < bmap1->n_ineq; ++i) {
+		int reduced;
+		reduced = reduced_using_equalities(v->block.data,
+						bmap1->ineq[i], bmap2, elim);
+		if (reduced && isl_int_is_neg(v->block.data[0]) &&
+		    isl_seq_first_non_zero(v->block.data + 1, total) == -1)
+			goto disjoint;
+	}
+	isl_vec_free(v);
+	free(elim);
+	return 0;
+disjoint:
+	isl_vec_free(v);
+	free(elim);
+	return 1;
+error:
+	isl_vec_free(v);
+	free(elim);
+	return -1;
+}
+
+int isl_basic_set_plain_is_disjoint(__isl_keep isl_basic_set *bset1,
+	__isl_keep isl_basic_set *bset2)
+{
+	return isl_basic_map_plain_is_disjoint((struct isl_basic_map *)bset1,
+					      (struct isl_basic_map *)bset2);
+}
+
+/* Are "map1" and "map2" obviously disjoint?
+ *
+ * If one of them is empty or if they live in different spaces (ignoring
+ * parameters), then they are clearly disjoint.
+ *
+ * If they have different parameters, then we skip any further tests.
+ *
+ * If they are obviously equal, but not obviously empty, then we will
+ * not be able to detect if they are disjoint.
+ *
+ * Otherwise we check if each basic map in "map1" is obviously disjoint
+ * from each basic map in "map2".
+ */
+int isl_map_plain_is_disjoint(__isl_keep isl_map *map1,
+	__isl_keep isl_map *map2)
+{
+	int i, j;
+	int disjoint;
+	int intersect;
+	int match;
+
+	if (!map1 || !map2)
+		return -1;
+
+	disjoint = isl_map_plain_is_empty(map1);
+	if (disjoint < 0 || disjoint)
+		return disjoint;
+
+	disjoint = isl_map_plain_is_empty(map2);
+	if (disjoint < 0 || disjoint)
+		return disjoint;
+
+	match = isl_space_tuple_is_equal(map1->dim, isl_dim_in,
+				map2->dim, isl_dim_in);
+	if (match < 0 || !match)
+		return match < 0 ? -1 : 1;
+
+	match = isl_space_tuple_is_equal(map1->dim, isl_dim_out,
+				map2->dim, isl_dim_out);
+	if (match < 0 || !match)
+		return match < 0 ? -1 : 1;
+
+	match = isl_space_match(map1->dim, isl_dim_param,
+				map2->dim, isl_dim_param);
+	if (match < 0 || !match)
+		return match < 0 ? -1 : 0;
+
+	intersect = isl_map_plain_is_equal(map1, map2);
+	if (intersect < 0 || intersect)
+		return intersect < 0 ? -1 : 0;
+
+	for (i = 0; i < map1->n; ++i) {
+		for (j = 0; j < map2->n; ++j) {
+			int d = isl_basic_map_plain_is_disjoint(map1->p[i],
+							       map2->p[j]);
+			if (d != 1)
+				return d;
+		}
+	}
+	return 1;
+}
+
+/* Are "map1" and "map2" disjoint?
+ *
+ * They are disjoint if they are "obviously disjoint" or if one of them
+ * is empty.  Otherwise, they are not disjoint if one of them is universal.
+ * If none of these cases apply, we compute the intersection and see if
+ * the result is empty.
+ */
+int isl_map_is_disjoint(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
+{
+	int disjoint;
+	int intersect;
+	isl_map *test;
+
+	disjoint = isl_map_plain_is_disjoint(map1, map2);
+	if (disjoint < 0 || disjoint)
+		return disjoint;
+
+	disjoint = isl_map_is_empty(map1);
+	if (disjoint < 0 || disjoint)
+		return disjoint;
+
+	disjoint = isl_map_is_empty(map2);
+	if (disjoint < 0 || disjoint)
+		return disjoint;
+
+	intersect = isl_map_plain_is_universe(map1);
+	if (intersect < 0 || intersect)
+		return intersect < 0 ? -1 : 0;
+
+	intersect = isl_map_plain_is_universe(map2);
+	if (intersect < 0 || intersect)
+		return intersect < 0 ? -1 : 0;
+
+	test = isl_map_intersect(isl_map_copy(map1), isl_map_copy(map2));
+	disjoint = isl_map_is_empty(test);
+	isl_map_free(test);
+
+	return disjoint;
+}
+
+/* Are "bmap1" and "bmap2" disjoint?
+ *
+ * They are disjoint if they are "obviously disjoint" or if one of them
+ * is empty.  Otherwise, they are not disjoint if one of them is universal.
+ * If none of these cases apply, we compute the intersection and see if
+ * the result is empty.
+ */
+int isl_basic_map_is_disjoint(__isl_keep isl_basic_map *bmap1,
+	__isl_keep isl_basic_map *bmap2)
+{
+	int disjoint;
+	int intersect;
+	isl_basic_map *test;
+
+	disjoint = isl_basic_map_plain_is_disjoint(bmap1, bmap2);
+	if (disjoint < 0 || disjoint)
+		return disjoint;
+
+	disjoint = isl_basic_map_is_empty(bmap1);
+	if (disjoint < 0 || disjoint)
+		return disjoint;
+
+	disjoint = isl_basic_map_is_empty(bmap2);
+	if (disjoint < 0 || disjoint)
+		return disjoint;
+
+	intersect = isl_basic_map_is_universe(bmap1);
+	if (intersect < 0 || intersect)
+		return intersect < 0 ? -1 : 0;
+
+	intersect = isl_basic_map_is_universe(bmap2);
+	if (intersect < 0 || intersect)
+		return intersect < 0 ? -1 : 0;
+
+	test = isl_basic_map_intersect(isl_basic_map_copy(bmap1),
+		isl_basic_map_copy(bmap2));
+	disjoint = isl_basic_map_is_empty(test);
+	isl_basic_map_free(test);
+
+	return disjoint;
+}
+
+/* Are "bset1" and "bset2" disjoint?
+ */
+int isl_basic_set_is_disjoint(__isl_keep isl_basic_set *bset1,
+	__isl_keep isl_basic_set *bset2)
+{
+	return isl_basic_map_is_disjoint(bset1, bset2);
+}
+
+int isl_set_plain_is_disjoint(__isl_keep isl_set *set1,
+	__isl_keep isl_set *set2)
+{
+	return isl_map_plain_is_disjoint((struct isl_map *)set1,
+					(struct isl_map *)set2);
+}
+
+/* Are "set1" and "set2" disjoint?
+ */
+int isl_set_is_disjoint(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
+{
+	return isl_map_is_disjoint(set1, set2);
+}
+
+int isl_set_fast_is_disjoint(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
+{
+	return isl_set_plain_is_disjoint(set1, set2);
+}
+
+/* Check if we can combine a given div with lower bound l and upper
+ * bound u with some other div and if so return that other div.
+ * Otherwise return -1.
+ *
+ * We first check that
+ *	- the bounds are opposites of each other (except for the constant
+ *	  term)
+ *	- the bounds do not reference any other div
+ *	- no div is defined in terms of this div
+ *
+ * Let m be the size of the range allowed on the div by the bounds.
+ * That is, the bounds are of the form
+ *
+ *	e <= a <= e + m - 1
+ *
+ * with e some expression in the other variables.
+ * We look for another div b such that no third div is defined in terms
+ * of this second div b and such that in any constraint that contains
+ * a (except for the given lower and upper bound), also contains b
+ * with a coefficient that is m times that of b.
+ * That is, all constraints (execpt for the lower and upper bound)
+ * are of the form
+ *
+ *	e + f (a + m b) >= 0
+ *
+ * If so, we return b so that "a + m b" can be replaced by
+ * a single div "c = a + m b".
+ */
+static int div_find_coalesce(struct isl_basic_map *bmap, int *pairs,
+	unsigned div, unsigned l, unsigned u)
+{
+	int i, j;
+	unsigned dim;
+	int coalesce = -1;
+
+	if (bmap->n_div <= 1)
+		return -1;
+	dim = isl_space_dim(bmap->dim, isl_dim_all);
+	if (isl_seq_first_non_zero(bmap->ineq[l] + 1 + dim, div) != -1)
+		return -1;
+	if (isl_seq_first_non_zero(bmap->ineq[l] + 1 + dim + div + 1,
+				   bmap->n_div - div - 1) != -1)
+		return -1;
+	if (!isl_seq_is_neg(bmap->ineq[l] + 1, bmap->ineq[u] + 1,
+			    dim + bmap->n_div))
+		return -1;
+
+	for (i = 0; i < bmap->n_div; ++i) {
+		if (isl_int_is_zero(bmap->div[i][0]))
+			continue;
+		if (!isl_int_is_zero(bmap->div[i][1 + 1 + dim + div]))
+			return -1;
+	}
+
+	isl_int_add(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]);
+	if (isl_int_is_neg(bmap->ineq[l][0])) {
+		isl_int_sub(bmap->ineq[l][0],
+			    bmap->ineq[l][0], bmap->ineq[u][0]);
+		bmap = isl_basic_map_copy(bmap);
+		bmap = isl_basic_map_set_to_empty(bmap);
+		isl_basic_map_free(bmap);
+		return -1;
+	}
+	isl_int_add_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
+	for (i = 0; i < bmap->n_div; ++i) {
+		if (i == div)
+			continue;
+		if (!pairs[i])
+			continue;
+		for (j = 0; j < bmap->n_div; ++j) {
+			if (isl_int_is_zero(bmap->div[j][0]))
+				continue;
+			if (!isl_int_is_zero(bmap->div[j][1 + 1 + dim + i]))
+				break;
+		}
+		if (j < bmap->n_div)
+			continue;
+		for (j = 0; j < bmap->n_ineq; ++j) {
+			int valid;
+			if (j == l || j == u)
+				continue;
+			if (isl_int_is_zero(bmap->ineq[j][1 + dim + div]))
+				continue;
+			if (isl_int_is_zero(bmap->ineq[j][1 + dim + i]))
+				break;
+			isl_int_mul(bmap->ineq[j][1 + dim + div],
+				    bmap->ineq[j][1 + dim + div],
+				    bmap->ineq[l][0]);
+			valid = isl_int_eq(bmap->ineq[j][1 + dim + div],
+					   bmap->ineq[j][1 + dim + i]);
+			isl_int_divexact(bmap->ineq[j][1 + dim + div],
+					 bmap->ineq[j][1 + dim + div],
+					 bmap->ineq[l][0]);
+			if (!valid)
+				break;
+		}
+		if (j < bmap->n_ineq)
+			continue;
+		coalesce = i;
+		break;
+	}
+	isl_int_sub_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
+	isl_int_sub(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]);
+	return coalesce;
+}
+
+/* Given a lower and an upper bound on div i, construct an inequality
+ * that when nonnegative ensures that this pair of bounds always allows
+ * for an integer value of the given div.
+ * The lower bound is inequality l, while the upper bound is inequality u.
+ * The constructed inequality is stored in ineq.
+ * g, fl, fu are temporary scalars.
+ *
+ * Let the upper bound be
+ *
+ *	-n_u a + e_u >= 0
+ *
+ * and the lower bound
+ *
+ *	n_l a + e_l >= 0
+ *
+ * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
+ * We have
+ *
+ *	- f_u e_l <= f_u f_l g a <= f_l e_u
+ *
+ * Since all variables are integer valued, this is equivalent to
+ *
+ *	- f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
+ *
+ * If this interval is at least f_u f_l g, then it contains at least
+ * one integer value for a.
+ * That is, the test constraint is
+ *
+ *	f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
+ */
+static void construct_test_ineq(struct isl_basic_map *bmap, int i,
+	int l, int u, isl_int *ineq, isl_int g, isl_int fl, isl_int fu)
+{
+	unsigned dim;
+	dim = isl_space_dim(bmap->dim, isl_dim_all);
+
+	isl_int_gcd(g, bmap->ineq[l][1 + dim + i], bmap->ineq[u][1 + dim + i]);
+	isl_int_divexact(fl, bmap->ineq[l][1 + dim + i], g);
+	isl_int_divexact(fu, bmap->ineq[u][1 + dim + i], g);
+	isl_int_neg(fu, fu);
+	isl_seq_combine(ineq, fl, bmap->ineq[u], fu, bmap->ineq[l],
+			1 + dim + bmap->n_div);
+	isl_int_add(ineq[0], ineq[0], fl);
+	isl_int_add(ineq[0], ineq[0], fu);
+	isl_int_sub_ui(ineq[0], ineq[0], 1);
+	isl_int_mul(g, g, fl);
+	isl_int_mul(g, g, fu);
+	isl_int_sub(ineq[0], ineq[0], g);
+}
+
+/* Remove more kinds of divs that are not strictly needed.
+ * In particular, if all pairs of lower and upper bounds on a div
+ * are such that they allow at least one integer value of the div,
+ * the we can eliminate the div using Fourier-Motzkin without
+ * introducing any spurious solutions.
+ */
+static struct isl_basic_map *drop_more_redundant_divs(
+	struct isl_basic_map *bmap, int *pairs, int n)
+{
+	struct isl_tab *tab = NULL;
+	struct isl_vec *vec = NULL;
+	unsigned dim;
+	int remove = -1;
+	isl_int g, fl, fu;
+
+	isl_int_init(g);
+	isl_int_init(fl);
+	isl_int_init(fu);
+
+	if (!bmap)
+		goto error;
+
+	dim = isl_space_dim(bmap->dim, isl_dim_all);
+	vec = isl_vec_alloc(bmap->ctx, 1 + dim + bmap->n_div);
+	if (!vec)
+		goto error;
+
+	tab = isl_tab_from_basic_map(bmap, 0);
+
+	while (n > 0) {
+		int i, l, u;
+		int best = -1;
+		enum isl_lp_result res;
+
+		for (i = 0; i < bmap->n_div; ++i) {
+			if (!pairs[i])
+				continue;
+			if (best >= 0 && pairs[best] <= pairs[i])
+				continue;
+			best = i;
+		}
+
+		i = best;
+		for (l = 0; l < bmap->n_ineq; ++l) {
+			if (!isl_int_is_pos(bmap->ineq[l][1 + dim + i]))
+				continue;
+			for (u = 0; u < bmap->n_ineq; ++u) {
+				if (!isl_int_is_neg(bmap->ineq[u][1 + dim + i]))
+					continue;
+				construct_test_ineq(bmap, i, l, u,
+						    vec->el, g, fl, fu);
+				res = isl_tab_min(tab, vec->el,
+						  bmap->ctx->one, &g, NULL, 0);
+				if (res == isl_lp_error)
+					goto error;
+				if (res == isl_lp_empty) {
+					bmap = isl_basic_map_set_to_empty(bmap);
+					break;
+				}
+				if (res != isl_lp_ok || isl_int_is_neg(g))
+					break;
+			}
+			if (u < bmap->n_ineq)
+				break;
+		}
+		if (l == bmap->n_ineq) {
+			remove = i;
+			break;
+		}
+		pairs[i] = 0;
+		--n;
+	}
+
+	isl_tab_free(tab);
+	isl_vec_free(vec);
+
+	isl_int_clear(g);
+	isl_int_clear(fl);
+	isl_int_clear(fu);
+
+	free(pairs);
+
+	if (remove < 0)
+		return bmap;
+
+	bmap = isl_basic_map_remove_dims(bmap, isl_dim_div, remove, 1);
+	return isl_basic_map_drop_redundant_divs(bmap);
+error:
+	free(pairs);
+	isl_basic_map_free(bmap);
+	isl_tab_free(tab);
+	isl_vec_free(vec);
+	isl_int_clear(g);
+	isl_int_clear(fl);
+	isl_int_clear(fu);
+	return NULL;
+}
+
+/* Given a pair of divs div1 and div2 such that, expect for the lower bound l
+ * and the upper bound u, div1 always occurs together with div2 in the form 
+ * (div1 + m div2), where m is the constant range on the variable div1
+ * allowed by l and u, replace the pair div1 and div2 by a single
+ * div that is equal to div1 + m div2.
+ *
+ * The new div will appear in the location that contains div2.
+ * We need to modify all constraints that contain
+ * div2 = (div - div1) / m
+ * (If a constraint does not contain div2, it will also not contain div1.)
+ * If the constraint also contains div1, then we know they appear
+ * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
+ * i.e., the coefficient of div is f.
+ *
+ * Otherwise, we first need to introduce div1 into the constraint.
+ * Let the l be
+ *
+ *	div1 + f >=0
+ *
+ * and u
+ *
+ *	-div1 + f' >= 0
+ *
+ * A lower bound on div2
+ *
+ *	n div2 + t >= 0
+ *
+ * can be replaced by
+ *
+ *	(n * (m div 2 + div1) + m t + n f)/g >= 0
+ *
+ * with g = gcd(m,n).
+ * An upper bound
+ *
+ *	-n div2 + t >= 0
+ *
+ * can be replaced by
+ *
+ *	(-n * (m div2 + div1) + m t + n f')/g >= 0
+ *
+ * These constraint are those that we would obtain from eliminating
+ * div1 using Fourier-Motzkin.
+ *
+ * After all constraints have been modified, we drop the lower and upper
+ * bound and then drop div1.
+ */
+static struct isl_basic_map *coalesce_divs(struct isl_basic_map *bmap,
+	unsigned div1, unsigned div2, unsigned l, unsigned u)
+{
+	isl_int a;
+	isl_int b;
+	isl_int m;
+	unsigned dim, total;
+	int i;
+
+	dim = isl_space_dim(bmap->dim, isl_dim_all);
+	total = 1 + dim + bmap->n_div;
+
+	isl_int_init(a);
+	isl_int_init(b);
+	isl_int_init(m);
+	isl_int_add(m, bmap->ineq[l][0], bmap->ineq[u][0]);
+	isl_int_add_ui(m, m, 1);
+
+	for (i = 0; i < bmap->n_ineq; ++i) {
+		if (i == l || i == u)
+			continue;
+		if (isl_int_is_zero(bmap->ineq[i][1 + dim + div2]))
+			continue;
+		if (isl_int_is_zero(bmap->ineq[i][1 + dim + div1])) {
+			isl_int_gcd(b, m, bmap->ineq[i][1 + dim + div2]);
+			isl_int_divexact(a, m, b);
+			isl_int_divexact(b, bmap->ineq[i][1 + dim + div2], b);
+			if (isl_int_is_pos(b)) {
+				isl_seq_combine(bmap->ineq[i], a, bmap->ineq[i],
+						b, bmap->ineq[l], total);
+			} else {
+				isl_int_neg(b, b);
+				isl_seq_combine(bmap->ineq[i], a, bmap->ineq[i],
+						b, bmap->ineq[u], total);
+			}
+		}
+		isl_int_set(bmap->ineq[i][1 + dim + div2],
+			    bmap->ineq[i][1 + dim + div1]);
+		isl_int_set_si(bmap->ineq[i][1 + dim + div1], 0);
+	}
+
+	isl_int_clear(a);
+	isl_int_clear(b);
+	isl_int_clear(m);
+	if (l > u) {
+		isl_basic_map_drop_inequality(bmap, l);
+		isl_basic_map_drop_inequality(bmap, u);
+	} else {
+		isl_basic_map_drop_inequality(bmap, u);
+		isl_basic_map_drop_inequality(bmap, l);
+	}
+	bmap = isl_basic_map_drop_div(bmap, div1);
+	return bmap;
+}
+
+/* First check if we can coalesce any pair of divs and
+ * then continue with dropping more redundant divs.
+ *
+ * We loop over all pairs of lower and upper bounds on a div
+ * with coefficient 1 and -1, respectively, check if there
+ * is any other div "c" with which we can coalesce the div
+ * and if so, perform the coalescing.
+ */
+static struct isl_basic_map *coalesce_or_drop_more_redundant_divs(
+	struct isl_basic_map *bmap, int *pairs, int n)
+{
+	int i, l, u;
+	unsigned dim;
+
+	dim = isl_space_dim(bmap->dim, isl_dim_all);
+
+	for (i = 0; i < bmap->n_div; ++i) {
+		if (!pairs[i])
+			continue;
+		for (l = 0; l < bmap->n_ineq; ++l) {
+			if (!isl_int_is_one(bmap->ineq[l][1 + dim + i]))
+				continue;
+			for (u = 0; u < bmap->n_ineq; ++u) {
+				int c;
+
+				if (!isl_int_is_negone(bmap->ineq[u][1+dim+i]))
+					continue;
+				c = div_find_coalesce(bmap, pairs, i, l, u);
+				if (c < 0)
+					continue;
+				free(pairs);
+				bmap = coalesce_divs(bmap, i, c, l, u);
+				return isl_basic_map_drop_redundant_divs(bmap);
+			}
+		}
+	}
+
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
+		return bmap;
+
+	return drop_more_redundant_divs(bmap, pairs, n);
+}
+
+/* Remove divs that are not strictly needed.
+ * In particular, if a div only occurs positively (or negatively)
+ * in constraints, then it can simply be dropped.
+ * Also, if a div occurs in only two constraints and if moreover
+ * those two constraints are opposite to each other, except for the constant
+ * term and if the sum of the constant terms is such that for any value
+ * of the other values, there is always at least one integer value of the
+ * div, i.e., if one plus this sum is greater than or equal to
+ * the (absolute value) of the coefficent of the div in the constraints,
+ * then we can also simply drop the div.
+ *
+ * We skip divs that appear in equalities or in the definition of other divs.
+ * Divs that appear in the definition of other divs usually occur in at least
+ * 4 constraints, but the constraints may have been simplified.
+ *
+ * If any divs are left after these simple checks then we move on
+ * to more complicated cases in drop_more_redundant_divs.
+ */
+struct isl_basic_map *isl_basic_map_drop_redundant_divs(
+	struct isl_basic_map *bmap)
+{
+	int i, j;
+	unsigned off;
+	int *pairs = NULL;
+	int n = 0;
+
+	if (!bmap)
+		goto error;
+	if (bmap->n_div == 0)
+		return bmap;
+
+	off = isl_space_dim(bmap->dim, isl_dim_all);
+	pairs = isl_calloc_array(bmap->ctx, int, bmap->n_div);
+	if (!pairs)
+		goto error;
+
+	for (i = 0; i < bmap->n_div; ++i) {
+		int pos, neg;
+		int last_pos, last_neg;
+		int redundant;
+		int defined;
+
+		defined = !isl_int_is_zero(bmap->div[i][0]);
+		for (j = i; j < bmap->n_div; ++j)
+			if (!isl_int_is_zero(bmap->div[j][1 + 1 + off + i]))
+				break;
+		if (j < bmap->n_div)
+			continue;
+		for (j = 0; j < bmap->n_eq; ++j)
+			if (!isl_int_is_zero(bmap->eq[j][1 + off + i]))
+				break;
+		if (j < bmap->n_eq)
+			continue;
+		++n;
+		pos = neg = 0;
+		for (j = 0; j < bmap->n_ineq; ++j) {
+			if (isl_int_is_pos(bmap->ineq[j][1 + off + i])) {
+				last_pos = j;
+				++pos;
+			}
+			if (isl_int_is_neg(bmap->ineq[j][1 + off + i])) {
+				last_neg = j;
+				++neg;
+			}
+		}
+		pairs[i] = pos * neg;
+		if (pairs[i] == 0) {
+			for (j = bmap->n_ineq - 1; j >= 0; --j)
+				if (!isl_int_is_zero(bmap->ineq[j][1+off+i]))
+					isl_basic_map_drop_inequality(bmap, j);
+			bmap = isl_basic_map_drop_div(bmap, i);
+			free(pairs);
+			return isl_basic_map_drop_redundant_divs(bmap);
+		}
+		if (pairs[i] != 1)
+			continue;
+		if (!isl_seq_is_neg(bmap->ineq[last_pos] + 1,
+				    bmap->ineq[last_neg] + 1,
+				    off + bmap->n_div))
+			continue;
+
+		isl_int_add(bmap->ineq[last_pos][0],
+			    bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]);
+		isl_int_add_ui(bmap->ineq[last_pos][0],
+			       bmap->ineq[last_pos][0], 1);
+		redundant = isl_int_ge(bmap->ineq[last_pos][0],
+				bmap->ineq[last_pos][1+off+i]);
+		isl_int_sub_ui(bmap->ineq[last_pos][0],
+			       bmap->ineq[last_pos][0], 1);
+		isl_int_sub(bmap->ineq[last_pos][0],
+			    bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]);
+		if (!redundant) {
+			if (defined ||
+			    !ok_to_set_div_from_bound(bmap, i, last_pos)) {
+				pairs[i] = 0;
+				--n;
+				continue;
+			}
+			bmap = set_div_from_lower_bound(bmap, i, last_pos);
+			bmap = isl_basic_map_simplify(bmap);
+			free(pairs);
+			return isl_basic_map_drop_redundant_divs(bmap);
+		}
+		if (last_pos > last_neg) {
+			isl_basic_map_drop_inequality(bmap, last_pos);
+			isl_basic_map_drop_inequality(bmap, last_neg);
+		} else {
+			isl_basic_map_drop_inequality(bmap, last_neg);
+			isl_basic_map_drop_inequality(bmap, last_pos);
+		}
+		bmap = isl_basic_map_drop_div(bmap, i);
+		free(pairs);
+		return isl_basic_map_drop_redundant_divs(bmap);
+	}
+
+	if (n > 0)
+		return coalesce_or_drop_more_redundant_divs(bmap, pairs, n);
+
+	free(pairs);
+	return bmap;
+error:
+	free(pairs);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+struct isl_basic_set *isl_basic_set_drop_redundant_divs(
+	struct isl_basic_set *bset)
+{
+	return (struct isl_basic_set *)
+	    isl_basic_map_drop_redundant_divs((struct isl_basic_map *)bset);
+}
+
+struct isl_map *isl_map_drop_redundant_divs(struct isl_map *map)
+{
+	int i;
+
+	if (!map)
+		return NULL;
+	for (i = 0; i < map->n; ++i) {
+		map->p[i] = isl_basic_map_drop_redundant_divs(map->p[i]);
+		if (!map->p[i])
+			goto error;
+	}
+	ISL_F_CLR(map, ISL_MAP_NORMALIZED);
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+struct isl_set *isl_set_drop_redundant_divs(struct isl_set *set)
+{
+	return (struct isl_set *)
+	    isl_map_drop_redundant_divs((struct isl_map *)set);
+}
diff --git a/polly/lib/External/isl/isl_map_subtract.c b/polly/lib/External/isl/isl_map_subtract.c
new file mode 100644
index 00000000000..4d10cab807d
--- /dev/null
+++ b/polly/lib/External/isl/isl_map_subtract.c
@@ -0,0 +1,900 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <isl_map_private.h>
+#include <isl_seq.h>
+#include <isl/set.h>
+#include <isl/map.h>
+#include "isl_tab.h"
+#include <isl_point_private.h>
+#include <isl_vec_private.h>
+
+/* Expand the constraint "c" into "v".  The initial "dim" dimensions
+ * are the same, but "v" may have more divs than "c" and the divs of "c"
+ * may appear in different positions in "v".
+ * The number of divs in "c" is given by "n_div" and the mapping
+ * of divs in "c" to divs in "v" is given by "div_map".
+ *
+ * Although it shouldn't happen in practice, it is theoretically
+ * possible that two or more divs in "c" are mapped to the same div in "v".
+ * These divs are then necessarily the same, so we simply add their
+ * coefficients.
+ */
+static void expand_constraint(isl_vec *v, unsigned dim,
+	isl_int *c, int *div_map, unsigned n_div)
+{
+	int i;
+
+	isl_seq_cpy(v->el, c, 1 + dim);
+	isl_seq_clr(v->el + 1 + dim, v->size - (1 + dim));
+
+	for (i = 0; i < n_div; ++i) {
+		int pos = 1 + dim + div_map[i];
+		isl_int_add(v->el[pos], v->el[pos], c[1 + dim + i]);
+	}
+}
+
+/* Add all constraints of bmap to tab.  The equalities of bmap
+ * are added as a pair of inequalities.
+ */
+static int tab_add_constraints(struct isl_tab *tab,
+	__isl_keep isl_basic_map *bmap, int *div_map)
+{
+	int i;
+	unsigned dim;
+	unsigned tab_total;
+	unsigned bmap_total;
+	isl_vec *v;
+
+	if (!tab || !bmap)
+		return -1;
+
+	tab_total = isl_basic_map_total_dim(tab->bmap);
+	bmap_total = isl_basic_map_total_dim(bmap);
+	dim = isl_space_dim(tab->bmap->dim, isl_dim_all);
+
+	if (isl_tab_extend_cons(tab, 2 * bmap->n_eq + bmap->n_ineq) < 0)
+		return -1;
+
+	v = isl_vec_alloc(bmap->ctx, 1 + tab_total);
+	if (!v)
+		return -1;
+
+	for (i = 0; i < bmap->n_eq; ++i) {
+		expand_constraint(v, dim, bmap->eq[i], div_map, bmap->n_div);
+		if (isl_tab_add_ineq(tab, v->el) < 0)
+			goto error;
+		isl_seq_neg(bmap->eq[i], bmap->eq[i], 1 + bmap_total);
+		expand_constraint(v, dim, bmap->eq[i], div_map, bmap->n_div);
+		if (isl_tab_add_ineq(tab, v->el) < 0)
+			goto error;
+		isl_seq_neg(bmap->eq[i], bmap->eq[i], 1 + bmap_total);
+		if (tab->empty)
+			break;
+	}
+
+	for (i = 0; i < bmap->n_ineq; ++i) {
+		expand_constraint(v, dim, bmap->ineq[i], div_map, bmap->n_div);
+		if (isl_tab_add_ineq(tab, v->el) < 0)
+			goto error;
+		if (tab->empty)
+			break;
+	}
+
+	isl_vec_free(v);
+	return 0;
+error:
+	isl_vec_free(v);
+	return -1;
+}
+
+/* Add a specific constraint of bmap (or its opposite) to tab.
+ * The position of the constraint is specified by "c", where
+ * the equalities of bmap are counted twice, once for the inequality
+ * that is equal to the equality, and once for its negation.
+ */
+static int tab_add_constraint(struct isl_tab *tab,
+	__isl_keep isl_basic_map *bmap, int *div_map, int c, int oppose)
+{
+	unsigned dim;
+	unsigned tab_total;
+	unsigned bmap_total;
+	isl_vec *v;
+	int r;
+
+	if (!tab || !bmap)
+		return -1;
+
+	tab_total = isl_basic_map_total_dim(tab->bmap);
+	bmap_total = isl_basic_map_total_dim(bmap);
+	dim = isl_space_dim(tab->bmap->dim, isl_dim_all);
+
+	v = isl_vec_alloc(bmap->ctx, 1 + tab_total);
+	if (!v)
+		return -1;
+
+	if (c < 2 * bmap->n_eq) {
+		if ((c % 2) != oppose)
+			isl_seq_neg(bmap->eq[c/2], bmap->eq[c/2],
+					1 + bmap_total);
+		if (oppose)
+			isl_int_sub_ui(bmap->eq[c/2][0], bmap->eq[c/2][0], 1);
+		expand_constraint(v, dim, bmap->eq[c/2], div_map, bmap->n_div);
+		r = isl_tab_add_ineq(tab, v->el);
+		if (oppose)
+			isl_int_add_ui(bmap->eq[c/2][0], bmap->eq[c/2][0], 1);
+		if ((c % 2) != oppose)
+			isl_seq_neg(bmap->eq[c/2], bmap->eq[c/2],
+					1 + bmap_total);
+	} else {
+		c -= 2 * bmap->n_eq;
+		if (oppose) {
+			isl_seq_neg(bmap->ineq[c], bmap->ineq[c],
+					1 + bmap_total);
+			isl_int_sub_ui(bmap->ineq[c][0], bmap->ineq[c][0], 1);
+		}
+		expand_constraint(v, dim, bmap->ineq[c], div_map, bmap->n_div);
+		r = isl_tab_add_ineq(tab, v->el);
+		if (oppose) {
+			isl_int_add_ui(bmap->ineq[c][0], bmap->ineq[c][0], 1);
+			isl_seq_neg(bmap->ineq[c], bmap->ineq[c],
+					1 + bmap_total);
+		}
+	}
+
+	isl_vec_free(v);
+	return r;
+}
+
+static int tab_add_divs(struct isl_tab *tab, __isl_keep isl_basic_map *bmap,
+	int **div_map)
+{
+	int i, j;
+	struct isl_vec *vec;
+	unsigned total;
+	unsigned dim;
+
+	if (!bmap)
+		return -1;
+	if (!bmap->n_div)
+		return 0;
+
+	if (!*div_map)
+		*div_map = isl_alloc_array(bmap->ctx, int, bmap->n_div);
+	if (!*div_map)
+		return -1;
+
+	total = isl_basic_map_total_dim(tab->bmap);
+	dim = total - tab->bmap->n_div;
+	vec = isl_vec_alloc(bmap->ctx, 2 + total + bmap->n_div);
+	if (!vec)
+		return -1;
+
+	for (i = 0; i < bmap->n_div; ++i) {
+		isl_seq_cpy(vec->el, bmap->div[i], 2 + dim);
+		isl_seq_clr(vec->el + 2 + dim, tab->bmap->n_div);
+		for (j = 0; j < i; ++j)
+			isl_int_set(vec->el[2 + dim + (*div_map)[j]],
+					bmap->div[i][2 + dim + j]);
+		for (j = 0; j < tab->bmap->n_div; ++j)
+			if (isl_seq_eq(tab->bmap->div[j],
+					vec->el, 2 + dim + tab->bmap->n_div))
+				break;
+		(*div_map)[i] = j;
+		if (j == tab->bmap->n_div) {
+			vec->size = 2 + dim + tab->bmap->n_div;
+			if (isl_tab_add_div(tab, vec, NULL, NULL) < 0)
+				goto error;
+		}
+	}
+
+	isl_vec_free(vec);
+
+	return 0;
+error:
+	isl_vec_free(vec);
+
+	return -1;
+}
+
+/* Freeze all constraints of tableau tab.
+ */
+static int tab_freeze_constraints(struct isl_tab *tab)
+{
+	int i;
+
+	for (i = 0; i < tab->n_con; ++i)
+		if (isl_tab_freeze_constraint(tab, i) < 0)
+			return -1;
+
+	return 0;
+}
+
+/* Check for redundant constraints starting at offset.
+ * Put the indices of the redundant constraints in index
+ * and return the number of redundant constraints.
+ */
+static int n_non_redundant(isl_ctx *ctx, struct isl_tab *tab,
+	int offset, int **index)
+{
+	int i, n;
+	int n_test = tab->n_con - offset;
+
+	if (isl_tab_detect_redundant(tab) < 0)
+		return -1;
+
+	if (n_test == 0)
+		return 0;
+	if (!*index)
+		*index = isl_alloc_array(ctx, int, n_test);
+	if (!*index)
+		return -1;
+
+	for (n = 0, i = 0; i < n_test; ++i) {
+		int r;
+		r = isl_tab_is_redundant(tab, offset + i);
+		if (r < 0)
+			return -1;
+		if (r)
+			continue;
+		(*index)[n++] = i;
+	}
+
+	return n;
+}
+
+/* basic_map_collect_diff calls add on each of the pieces of
+ * the set difference between bmap and map until the add method
+ * return a negative value.
+ */
+struct isl_diff_collector {
+	int (*add)(struct isl_diff_collector *dc,
+		    __isl_take isl_basic_map *bmap);
+};
+
+/* Compute the set difference between bmap and map and call
+ * dc->add on each of the piece until this function returns
+ * a negative value.
+ * Return 0 on success and -1 on error.  dc->add returning
+ * a negative value is treated as an error, but the calling
+ * function can interpret the results based on the state of dc.
+ *
+ * Assumes that map has known divs.
+ *
+ * The difference is computed by a backtracking algorithm.
+ * Each level corresponds to a basic map in "map".
+ * When a node in entered for the first time, we check
+ * if the corresonding basic map intersects the current piece
+ * of "bmap".  If not, we move to the next level.
+ * Otherwise, we split the current piece into as many
+ * pieces as there are non-redundant constraints of the current
+ * basic map in the intersection.  Each of these pieces is
+ * handled by a child of the current node.
+ * In particular, if there are n non-redundant constraints,
+ * then for each 0 <= i < n, a piece is cut off by adding
+ * constraints 0 <= j < i and adding the opposite of constraint i.
+ * If there are no non-redundant constraints, meaning that the current
+ * piece is a subset of the current basic map, then we simply backtrack.
+ *
+ * In the leaves, we check if the remaining piece has any integer points
+ * and if so, pass it along to dc->add.  As a special case, if nothing
+ * has been removed when we end up in a leaf, we simply pass along
+ * the original basic map.
+ */
+static int basic_map_collect_diff(__isl_take isl_basic_map *bmap,
+	__isl_take isl_map *map, struct isl_diff_collector *dc)
+{
+	int i;
+	int modified;
+	int level;
+	int init;
+	int empty;
+	isl_ctx *ctx;
+	struct isl_tab *tab = NULL;
+	struct isl_tab_undo **snap = NULL;
+	int *k = NULL;
+	int *n = NULL;
+	int **index = NULL;
+	int **div_map = NULL;
+
+	empty = isl_basic_map_is_empty(bmap);
+	if (empty) {
+		isl_basic_map_free(bmap);
+		isl_map_free(map);
+		return empty < 0 ? -1 : 0;
+	}
+
+	bmap = isl_basic_map_cow(bmap);
+	map = isl_map_cow(map);
+
+	if (!bmap || !map)
+		goto error;
+
+	ctx = map->ctx;
+	snap = isl_alloc_array(map->ctx, struct isl_tab_undo *, map->n);
+	k = isl_alloc_array(map->ctx, int, map->n);
+	n = isl_alloc_array(map->ctx, int, map->n);
+	index = isl_calloc_array(map->ctx, int *, map->n);
+	div_map = isl_calloc_array(map->ctx, int *, map->n);
+	if (!snap || !k || !n || !index || !div_map)
+		goto error;
+
+	bmap = isl_basic_map_order_divs(bmap);
+	map = isl_map_order_divs(map);
+
+	tab = isl_tab_from_basic_map(bmap, 1);
+	if (!tab)
+		goto error;
+
+	modified = 0;
+	level = 0;
+	init = 1;
+
+	while (level >= 0) {
+		if (level >= map->n) {
+			int empty;
+			struct isl_basic_map *bm;
+			if (!modified) {
+				if (dc->add(dc, isl_basic_map_copy(bmap)) < 0)
+					goto error;
+				break;
+			}
+			bm = isl_basic_map_copy(tab->bmap);
+			bm = isl_basic_map_cow(bm);
+			bm = isl_basic_map_update_from_tab(bm, tab);
+			bm = isl_basic_map_simplify(bm);
+			bm = isl_basic_map_finalize(bm);
+			empty = isl_basic_map_is_empty(bm);
+			if (empty)
+				isl_basic_map_free(bm);
+			else if (dc->add(dc, bm) < 0)
+				goto error;
+			if (empty < 0)
+				goto error;
+			level--;
+			init = 0;
+			continue;
+		}
+		if (init) {
+			int offset;
+			struct isl_tab_undo *snap2;
+			snap2 = isl_tab_snap(tab);
+			if (tab_add_divs(tab, map->p[level],
+					 &div_map[level]) < 0)
+				goto error;
+			offset = tab->n_con;
+			snap[level] = isl_tab_snap(tab);
+			if (tab_freeze_constraints(tab) < 0)
+				goto error;
+			if (tab_add_constraints(tab, map->p[level],
+						div_map[level]) < 0)
+				goto error;
+			k[level] = 0;
+			n[level] = 0;
+			if (tab->empty) {
+				if (isl_tab_rollback(tab, snap2) < 0)
+					goto error;
+				level++;
+				continue;
+			}
+			modified = 1;
+			n[level] = n_non_redundant(ctx, tab, offset,
+						    &index[level]);
+			if (n[level] < 0)
+				goto error;
+			if (n[level] == 0) {
+				level--;
+				init = 0;
+				continue;
+			}
+			if (isl_tab_rollback(tab, snap[level]) < 0)
+				goto error;
+			if (tab_add_constraint(tab, map->p[level],
+					div_map[level], index[level][0], 1) < 0)
+				goto error;
+			level++;
+			continue;
+		} else {
+			if (k[level] + 1 >= n[level]) {
+				level--;
+				continue;
+			}
+			if (isl_tab_rollback(tab, snap[level]) < 0)
+				goto error;
+			if (tab_add_constraint(tab, map->p[level],
+						div_map[level],
+						index[level][k[level]], 0) < 0)
+				goto error;
+			snap[level] = isl_tab_snap(tab);
+			k[level]++;
+			if (tab_add_constraint(tab, map->p[level],
+						div_map[level],
+						index[level][k[level]], 1) < 0)
+				goto error;
+			level++;
+			init = 1;
+			continue;
+		}
+	}
+
+	isl_tab_free(tab);
+	free(snap);
+	free(n);
+	free(k);
+	for (i = 0; index && i < map->n; ++i)
+		free(index[i]);
+	free(index);
+	for (i = 0; div_map && i < map->n; ++i)
+		free(div_map[i]);
+	free(div_map);
+
+	isl_basic_map_free(bmap);
+	isl_map_free(map);
+
+	return 0;
+error:
+	isl_tab_free(tab);
+	free(snap);
+	free(n);
+	free(k);
+	for (i = 0; index && i < map->n; ++i)
+		free(index[i]);
+	free(index);
+	for (i = 0; div_map && i < map->n; ++i)
+		free(div_map[i]);
+	free(div_map);
+	isl_basic_map_free(bmap);
+	isl_map_free(map);
+	return -1;
+}
+
+/* A diff collector that actually collects all parts of the
+ * set difference in the field diff.
+ */
+struct isl_subtract_diff_collector {
+	struct isl_diff_collector dc;
+	struct isl_map *diff;
+};
+
+/* isl_subtract_diff_collector callback.
+ */
+static int basic_map_subtract_add(struct isl_diff_collector *dc,
+			    __isl_take isl_basic_map *bmap)
+{
+	struct isl_subtract_diff_collector *sdc;
+	sdc = (struct isl_subtract_diff_collector *)dc;
+
+	sdc->diff = isl_map_union_disjoint(sdc->diff,
+			isl_map_from_basic_map(bmap));
+
+	return sdc->diff ? 0 : -1;
+}
+
+/* Return the set difference between bmap and map.
+ */
+static __isl_give isl_map *basic_map_subtract(__isl_take isl_basic_map *bmap,
+	__isl_take isl_map *map)
+{
+	struct isl_subtract_diff_collector sdc;
+	sdc.dc.add = &basic_map_subtract_add;
+	sdc.diff = isl_map_empty_like_basic_map(bmap);
+	if (basic_map_collect_diff(bmap, map, &sdc.dc) < 0) {
+		isl_map_free(sdc.diff);
+		sdc.diff = NULL;
+	}
+	return sdc.diff;
+}
+
+/* Return the set difference between map1 and map2.
+ * (U_i A_i) \ (U_j B_j) is computed as U_i (A_i \ (U_j B_j))
+ *
+ * If "map1" and "map2" are disjoint, then simply return "map1".
+ */
+static __isl_give isl_map *map_subtract( __isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	int i;
+	int disjoint;
+	struct isl_map *diff;
+
+	if (!map1 || !map2)
+		goto error;
+
+	isl_assert(map1->ctx, isl_space_is_equal(map1->dim, map2->dim), goto error);
+
+	disjoint = isl_map_is_disjoint(map1, map2);
+	if (disjoint < 0)
+		goto error;
+	if (disjoint) {
+		isl_map_free(map2);
+		return map1;
+	}
+
+	map1 = isl_map_compute_divs(map1);
+	map2 = isl_map_compute_divs(map2);
+	if (!map1 || !map2)
+		goto error;
+
+	map1 = isl_map_remove_empty_parts(map1);
+	map2 = isl_map_remove_empty_parts(map2);
+
+	diff = isl_map_empty_like(map1);
+	for (i = 0; i < map1->n; ++i) {
+		struct isl_map *d;
+		d = basic_map_subtract(isl_basic_map_copy(map1->p[i]),
+				       isl_map_copy(map2));
+		if (ISL_F_ISSET(map1, ISL_MAP_DISJOINT))
+			diff = isl_map_union_disjoint(diff, d);
+		else
+			diff = isl_map_union(diff, d);
+	}
+
+	isl_map_free(map1);
+	isl_map_free(map2);
+
+	return diff;
+error:
+	isl_map_free(map1);
+	isl_map_free(map2);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_subtract( __isl_take isl_map *map1,
+	__isl_take isl_map *map2)
+{
+	return isl_map_align_params_map_map_and(map1, map2, &map_subtract);
+}
+
+struct isl_set *isl_set_subtract(struct isl_set *set1, struct isl_set *set2)
+{
+	return (struct isl_set *)
+		isl_map_subtract(
+			(struct isl_map *)set1, (struct isl_map *)set2);
+}
+
+/* Remove the elements of "dom" from the domain of "map".
+ */
+static __isl_give isl_map *map_subtract_domain(__isl_take isl_map *map,
+	__isl_take isl_set *dom)
+{
+	isl_map *ext_dom;
+
+	if (!isl_map_compatible_domain(map, dom))
+		isl_die(isl_set_get_ctx(dom), isl_error_invalid,
+			"incompatible spaces", goto error);
+	
+	ext_dom = isl_map_universe(isl_map_get_space(map));
+	ext_dom = isl_map_intersect_domain(ext_dom, dom);
+	return isl_map_subtract(map, ext_dom);
+error:
+	isl_map_free(map);
+	isl_set_free(dom);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_subtract_domain(__isl_take isl_map *map,
+	__isl_take isl_set *dom)
+{
+	return isl_map_align_params_map_map_and(map, dom, &map_subtract_domain);
+}
+
+/* Remove the elements of "dom" from the range of "map".
+ */
+static __isl_give isl_map *map_subtract_range(__isl_take isl_map *map,
+	__isl_take isl_set *dom)
+{
+	isl_map *ext_dom;
+
+	if (!isl_map_compatible_range(map, dom))
+		isl_die(isl_set_get_ctx(dom), isl_error_invalid,
+			"incompatible spaces", goto error);
+	
+	ext_dom = isl_map_universe(isl_map_get_space(map));
+	ext_dom = isl_map_intersect_range(ext_dom, dom);
+	return isl_map_subtract(map, ext_dom);
+error:
+	isl_map_free(map);
+	isl_set_free(dom);
+	return NULL;
+}
+
+__isl_give isl_map *isl_map_subtract_range(__isl_take isl_map *map,
+	__isl_take isl_set *dom)
+{
+	return isl_map_align_params_map_map_and(map, dom, &map_subtract_range);
+}
+
+/* A diff collector that aborts as soon as its add function is called,
+ * setting empty to 0.
+ */
+struct isl_is_empty_diff_collector {
+	struct isl_diff_collector dc;
+	int empty;
+};
+
+/* isl_is_empty_diff_collector callback.
+ */
+static int basic_map_is_empty_add(struct isl_diff_collector *dc,
+			    __isl_take isl_basic_map *bmap)
+{
+	struct isl_is_empty_diff_collector *edc;
+	edc = (struct isl_is_empty_diff_collector *)dc;
+
+	edc->empty = 0;
+
+	isl_basic_map_free(bmap);
+	return -1;
+}
+
+/* Check if bmap \ map is empty by computing this set difference
+ * and breaking off as soon as the difference is known to be non-empty.
+ */
+static int basic_map_diff_is_empty(__isl_keep isl_basic_map *bmap,
+	__isl_keep isl_map *map)
+{
+	int r;
+	struct isl_is_empty_diff_collector edc;
+
+	r = isl_basic_map_plain_is_empty(bmap);
+	if (r)
+		return r;
+
+	edc.dc.add = &basic_map_is_empty_add;
+	edc.empty = 1;
+	r = basic_map_collect_diff(isl_basic_map_copy(bmap),
+				   isl_map_copy(map), &edc.dc);
+	if (!edc.empty)
+		return 0;
+
+	return r < 0 ? -1 : 1;
+}
+
+/* Check if map1 \ map2 is empty by checking if the set difference is empty
+ * for each of the basic maps in map1.
+ */
+static int map_diff_is_empty(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
+{
+	int i;
+	int is_empty = 1;
+
+	if (!map1 || !map2)
+		return -1;
+	
+	for (i = 0; i < map1->n; ++i) {
+		is_empty = basic_map_diff_is_empty(map1->p[i], map2);
+		if (is_empty < 0 || !is_empty)
+			 break;
+	}
+
+	return is_empty;
+}
+
+/* Return 1 if "bmap" contains a single element.
+ */
+int isl_basic_map_plain_is_singleton(__isl_keep isl_basic_map *bmap)
+{
+	if (!bmap)
+		return -1;
+	if (bmap->n_div)
+		return 0;
+	if (bmap->n_ineq)
+		return 0;
+	return bmap->n_eq == isl_basic_map_total_dim(bmap);
+}
+
+/* Return 1 if "map" contains a single element.
+ */
+int isl_map_plain_is_singleton(__isl_keep isl_map *map)
+{
+	if (!map)
+		return -1;
+	if (map->n != 1)
+		return 0;
+
+	return isl_basic_map_plain_is_singleton(map->p[0]);
+}
+
+/* Given a singleton basic map, extract the single element
+ * as an isl_point.
+ */
+static __isl_give isl_point *singleton_extract_point(
+	__isl_keep isl_basic_map *bmap)
+{
+	int j;
+	unsigned dim;
+	struct isl_vec *point;
+	isl_int m;
+
+	if (!bmap)
+		return NULL;
+
+	dim = isl_basic_map_total_dim(bmap);
+	isl_assert(bmap->ctx, bmap->n_eq == dim, return NULL);
+	point = isl_vec_alloc(bmap->ctx, 1 + dim);
+	if (!point)
+		return NULL;
+
+	isl_int_init(m);
+
+	isl_int_set_si(point->el[0], 1);
+	for (j = 0; j < bmap->n_eq; ++j) {
+		int i = dim - 1 - j;
+		isl_assert(bmap->ctx,
+		    isl_seq_first_non_zero(bmap->eq[j] + 1, i) == -1,
+		    goto error);
+		isl_assert(bmap->ctx,
+		    isl_int_is_one(bmap->eq[j][1 + i]) ||
+		    isl_int_is_negone(bmap->eq[j][1 + i]),
+		    goto error);
+		isl_assert(bmap->ctx,
+		    isl_seq_first_non_zero(bmap->eq[j]+1+i+1, dim-i-1) == -1,
+		    goto error);
+
+		isl_int_gcd(m, point->el[0], bmap->eq[j][1 + i]);
+		isl_int_divexact(m, bmap->eq[j][1 + i], m);
+		isl_int_abs(m, m);
+		isl_seq_scale(point->el, point->el, m, 1 + i);
+		isl_int_divexact(m, point->el[0], bmap->eq[j][1 + i]);
+		isl_int_neg(m, m);
+		isl_int_mul(point->el[1 + i], m, bmap->eq[j][0]);
+	}
+
+	isl_int_clear(m);
+	return isl_point_alloc(isl_basic_map_get_space(bmap), point);
+error:
+	isl_int_clear(m);
+	isl_vec_free(point);
+	return NULL;
+}
+
+/* Return 1 is the singleton map "map1" is a subset of "map2",
+ * i.e., if the single element of "map1" is also an element of "map2".
+ * Assumes "map2" has known divs.
+ */
+static int map_is_singleton_subset(__isl_keep isl_map *map1,
+	__isl_keep isl_map *map2)
+{
+	int i;
+	int is_subset = 0;
+	struct isl_point *point;
+
+	if (!map1 || !map2)
+		return -1;
+	if (map1->n != 1)
+		return -1;
+
+	point = singleton_extract_point(map1->p[0]);
+	if (!point)
+		return -1;
+
+	for (i = 0; i < map2->n; ++i) {
+		is_subset = isl_basic_map_contains_point(map2->p[i], point);
+		if (is_subset)
+			break;
+	}
+
+	isl_point_free(point);
+	return is_subset;
+}
+
+static int map_is_subset(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
+{
+	int is_subset = 0;
+	int empty;
+	int rat1, rat2;
+
+	if (!map1 || !map2)
+		return -1;
+
+	if (!isl_map_has_equal_space(map1, map2))
+		return 0;
+
+	empty = isl_map_is_empty(map1);
+	if (empty < 0)
+		return -1;
+	if (empty)
+		return 1;
+
+	empty = isl_map_is_empty(map2);
+	if (empty < 0)
+		return -1;
+	if (empty)
+		return 0;
+
+	rat1 = isl_map_has_rational(map1);
+	rat2 = isl_map_has_rational(map2);
+	if (rat1 < 0 || rat2 < 0)
+		return -1;
+	if (rat1 && !rat2)
+		return 0;
+
+	if (isl_map_plain_is_universe(map2))
+		return 1;
+
+	map2 = isl_map_compute_divs(isl_map_copy(map2));
+	if (isl_map_plain_is_singleton(map1)) {
+		is_subset = map_is_singleton_subset(map1, map2);
+		isl_map_free(map2);
+		return is_subset;
+	}
+	is_subset = map_diff_is_empty(map1, map2);
+	isl_map_free(map2);
+
+	return is_subset;
+}
+
+int isl_map_is_subset(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
+{
+	return isl_map_align_params_map_map_and_test(map1, map2,
+							&map_is_subset);
+}
+
+int isl_set_is_subset(struct isl_set *set1, struct isl_set *set2)
+{
+	return isl_map_is_subset(
+			(struct isl_map *)set1, (struct isl_map *)set2);
+}
+
+__isl_give isl_map *isl_map_make_disjoint(__isl_take isl_map *map)
+{
+	int i;
+	struct isl_subtract_diff_collector sdc;
+	sdc.dc.add = &basic_map_subtract_add;
+
+	if (!map)
+		return NULL;
+	if (ISL_F_ISSET(map, ISL_MAP_DISJOINT))
+		return map;
+	if (map->n <= 1)
+		return map;
+
+	map = isl_map_compute_divs(map);
+	map = isl_map_remove_empty_parts(map);
+
+	if (!map || map->n <= 1)
+		return map;
+
+	sdc.diff = isl_map_from_basic_map(isl_basic_map_copy(map->p[0]));
+
+	for (i = 1; i < map->n; ++i) {
+		struct isl_basic_map *bmap = isl_basic_map_copy(map->p[i]);
+		struct isl_map *copy = isl_map_copy(sdc.diff);
+		if (basic_map_collect_diff(bmap, copy, &sdc.dc) < 0) {
+			isl_map_free(sdc.diff);
+			sdc.diff = NULL;
+			break;
+		}
+	}
+
+	isl_map_free(map);
+
+	return sdc.diff;
+}
+
+__isl_give isl_set *isl_set_make_disjoint(__isl_take isl_set *set)
+{
+	return (struct isl_set *)isl_map_make_disjoint((struct isl_map *)set);
+}
+
+__isl_give isl_map *isl_map_complement(__isl_take isl_map *map)
+{
+	isl_map *universe;
+
+	if (!map)
+		return NULL;
+
+	universe = isl_map_universe(isl_map_get_space(map));
+
+	return isl_map_subtract(universe, map);
+}
+
+__isl_give isl_set *isl_set_complement(__isl_take isl_set *set)
+{
+	return isl_map_complement(set);
+}
diff --git a/polly/lib/External/isl/isl_map_to_basic_set.c b/polly/lib/External/isl/isl_map_to_basic_set.c
new file mode 100644
index 00000000000..6755d52bcdd
--- /dev/null
+++ b/polly/lib/External/isl/isl_map_to_basic_set.c
@@ -0,0 +1,10 @@
+#include <isl/map_to_basic_set.h>
+#include <isl/map.h>
+#include <isl/set.h>
+
+#define KEY_BASE	map
+#define KEY_EQUAL	isl_map_plain_is_equal
+#define VAL_BASE	basic_set
+#define VAL_EQUAL	isl_basic_set_plain_is_equal
+
+#include <isl_hmap_templ.c>
diff --git a/polly/lib/External/isl/isl_mat.c b/polly/lib/External/isl/isl_mat.c
new file mode 100644
index 00000000000..3a8d31034be
--- /dev/null
+++ b/polly/lib/External/isl/isl_mat.c
@@ -0,0 +1,1742 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2014      Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl/space.h>
+#include <isl_seq.h>
+#include <isl_mat_private.h>
+#include <isl_vec_private.h>
+#include <isl_space_private.h>
+#include <isl_val_private.h>
+#include <isl/deprecated/mat_int.h>
+
+isl_ctx *isl_mat_get_ctx(__isl_keep isl_mat *mat)
+{
+	return mat ? mat->ctx : NULL;
+}
+
+struct isl_mat *isl_mat_alloc(struct isl_ctx *ctx,
+	unsigned n_row, unsigned n_col)
+{
+	int i;
+	struct isl_mat *mat;
+
+	mat = isl_alloc_type(ctx, struct isl_mat);
+	if (!mat)
+		return NULL;
+
+	mat->row = NULL;
+	mat->block = isl_blk_alloc(ctx, n_row * n_col);
+	if (isl_blk_is_error(mat->block))
+		goto error;
+	mat->row = isl_alloc_array(ctx, isl_int *, n_row);
+	if (n_row && !mat->row)
+		goto error;
+
+	for (i = 0; i < n_row; ++i)
+		mat->row[i] = mat->block.data + i * n_col;
+
+	mat->ctx = ctx;
+	isl_ctx_ref(ctx);
+	mat->ref = 1;
+	mat->n_row = n_row;
+	mat->n_col = n_col;
+	mat->max_col = n_col;
+	mat->flags = 0;
+
+	return mat;
+error:
+	isl_blk_free(ctx, mat->block);
+	free(mat);
+	return NULL;
+}
+
+struct isl_mat *isl_mat_extend(struct isl_mat *mat,
+	unsigned n_row, unsigned n_col)
+{
+	int i;
+	isl_int *old;
+	isl_int **row;
+
+	if (!mat)
+		return NULL;
+
+	if (mat->max_col >= n_col && mat->n_row >= n_row) {
+		if (mat->n_col < n_col)
+			mat->n_col = n_col;
+		return mat;
+	}
+
+	if (mat->max_col < n_col) {
+		struct isl_mat *new_mat;
+
+		if (n_row < mat->n_row)
+			n_row = mat->n_row;
+		new_mat = isl_mat_alloc(mat->ctx, n_row, n_col);
+		if (!new_mat)
+			goto error;
+		for (i = 0; i < mat->n_row; ++i)
+			isl_seq_cpy(new_mat->row[i], mat->row[i], mat->n_col);
+		isl_mat_free(mat);
+		return new_mat;
+	}
+
+	mat = isl_mat_cow(mat);
+	if (!mat)
+		goto error;
+
+	old = mat->block.data;
+	mat->block = isl_blk_extend(mat->ctx, mat->block, n_row * mat->max_col);
+	if (isl_blk_is_error(mat->block))
+		goto error;
+	row = isl_realloc_array(mat->ctx, mat->row, isl_int *, n_row);
+	if (n_row && !row)
+		goto error;
+	mat->row = row;
+
+	for (i = 0; i < mat->n_row; ++i)
+		mat->row[i] = mat->block.data + (mat->row[i] - old);
+	for (i = mat->n_row; i < n_row; ++i)
+		mat->row[i] = mat->block.data + i * mat->max_col;
+	mat->n_row = n_row;
+	if (mat->n_col < n_col)
+		mat->n_col = n_col;
+
+	return mat;
+error:
+	isl_mat_free(mat);
+	return NULL;
+}
+
+__isl_give isl_mat *isl_mat_sub_alloc6(isl_ctx *ctx, isl_int **row,
+	unsigned first_row, unsigned n_row, unsigned first_col, unsigned n_col)
+{
+	int i;
+	struct isl_mat *mat;
+
+	mat = isl_alloc_type(ctx, struct isl_mat);
+	if (!mat)
+		return NULL;
+	mat->row = isl_alloc_array(ctx, isl_int *, n_row);
+	if (n_row && !mat->row)
+		goto error;
+	for (i = 0; i < n_row; ++i)
+		mat->row[i] = row[first_row+i] + first_col;
+	mat->ctx = ctx;
+	isl_ctx_ref(ctx);
+	mat->ref = 1;
+	mat->n_row = n_row;
+	mat->n_col = n_col;
+	mat->block = isl_blk_empty();
+	mat->flags = ISL_MAT_BORROWED;
+	return mat;
+error:
+	free(mat);
+	return NULL;
+}
+
+__isl_give isl_mat *isl_mat_sub_alloc(__isl_keep isl_mat *mat,
+	unsigned first_row, unsigned n_row, unsigned first_col, unsigned n_col)
+{
+	if (!mat)
+		return NULL;
+	return isl_mat_sub_alloc6(mat->ctx, mat->row, first_row, n_row,
+				  first_col, n_col);
+}
+
+void isl_mat_sub_copy(struct isl_ctx *ctx, isl_int **dst, isl_int **src,
+	unsigned n_row, unsigned dst_col, unsigned src_col, unsigned n_col)
+{
+	int i;
+
+	for (i = 0; i < n_row; ++i)
+		isl_seq_cpy(dst[i]+dst_col, src[i]+src_col, n_col);
+}
+
+void isl_mat_sub_neg(struct isl_ctx *ctx, isl_int **dst, isl_int **src,
+	unsigned n_row, unsigned dst_col, unsigned src_col, unsigned n_col)
+{
+	int i;
+
+	for (i = 0; i < n_row; ++i)
+		isl_seq_neg(dst[i]+dst_col, src[i]+src_col, n_col);
+}
+
+struct isl_mat *isl_mat_copy(struct isl_mat *mat)
+{
+	if (!mat)
+		return NULL;
+
+	mat->ref++;
+	return mat;
+}
+
+struct isl_mat *isl_mat_dup(struct isl_mat *mat)
+{
+	int i;
+	struct isl_mat *mat2;
+
+	if (!mat)
+		return NULL;
+	mat2 = isl_mat_alloc(mat->ctx, mat->n_row, mat->n_col);
+	if (!mat2)
+		return NULL;
+	for (i = 0; i < mat->n_row; ++i)
+		isl_seq_cpy(mat2->row[i], mat->row[i], mat->n_col);
+	return mat2;
+}
+
+struct isl_mat *isl_mat_cow(struct isl_mat *mat)
+{
+	struct isl_mat *mat2;
+	if (!mat)
+		return NULL;
+
+	if (mat->ref == 1 && !ISL_F_ISSET(mat, ISL_MAT_BORROWED))
+		return mat;
+
+	mat2 = isl_mat_dup(mat);
+	isl_mat_free(mat);
+	return mat2;
+}
+
+__isl_null isl_mat *isl_mat_free(__isl_take isl_mat *mat)
+{
+	if (!mat)
+		return NULL;
+
+	if (--mat->ref > 0)
+		return NULL;
+
+	if (!ISL_F_ISSET(mat, ISL_MAT_BORROWED))
+		isl_blk_free(mat->ctx, mat->block);
+	isl_ctx_deref(mat->ctx);
+	free(mat->row);
+	free(mat);
+
+	return NULL;
+}
+
+int isl_mat_rows(__isl_keep isl_mat *mat)
+{
+	return mat ? mat->n_row : -1;
+}
+
+int isl_mat_cols(__isl_keep isl_mat *mat)
+{
+	return mat ? mat->n_col : -1;
+}
+
+int isl_mat_get_element(__isl_keep isl_mat *mat, int row, int col, isl_int *v)
+{
+	if (!mat)
+		return -1;
+	if (row < 0 || row >= mat->n_row)
+		isl_die(mat->ctx, isl_error_invalid, "row out of range",
+			return -1);
+	if (col < 0 || col >= mat->n_col)
+		isl_die(mat->ctx, isl_error_invalid, "column out of range",
+			return -1);
+	isl_int_set(*v, mat->row[row][col]);
+	return 0;
+}
+
+/* Extract the element at row "row", oolumn "col" of "mat".
+ */
+__isl_give isl_val *isl_mat_get_element_val(__isl_keep isl_mat *mat,
+	int row, int col)
+{
+	isl_ctx *ctx;
+
+	if (!mat)
+		return NULL;
+	ctx = isl_mat_get_ctx(mat);
+	if (row < 0 || row >= mat->n_row)
+		isl_die(ctx, isl_error_invalid, "row out of range",
+			return NULL);
+	if (col < 0 || col >= mat->n_col)
+		isl_die(ctx, isl_error_invalid, "column out of range",
+			return NULL);
+	return isl_val_int_from_isl_int(ctx, mat->row[row][col]);
+}
+
+__isl_give isl_mat *isl_mat_set_element(__isl_take isl_mat *mat,
+	int row, int col, isl_int v)
+{
+	mat = isl_mat_cow(mat);
+	if (!mat)
+		return NULL;
+	if (row < 0 || row >= mat->n_row)
+		isl_die(mat->ctx, isl_error_invalid, "row out of range",
+			goto error);
+	if (col < 0 || col >= mat->n_col)
+		isl_die(mat->ctx, isl_error_invalid, "column out of range",
+			goto error);
+	isl_int_set(mat->row[row][col], v);
+	return mat;
+error:
+	isl_mat_free(mat);
+	return NULL;
+}
+
+__isl_give isl_mat *isl_mat_set_element_si(__isl_take isl_mat *mat,
+	int row, int col, int v)
+{
+	mat = isl_mat_cow(mat);
+	if (!mat)
+		return NULL;
+	if (row < 0 || row >= mat->n_row)
+		isl_die(mat->ctx, isl_error_invalid, "row out of range",
+			goto error);
+	if (col < 0 || col >= mat->n_col)
+		isl_die(mat->ctx, isl_error_invalid, "column out of range",
+			goto error);
+	isl_int_set_si(mat->row[row][col], v);
+	return mat;
+error:
+	isl_mat_free(mat);
+	return NULL;
+}
+
+/* Replace the element at row "row", column "col" of "mat" by "v".
+ */
+__isl_give isl_mat *isl_mat_set_element_val(__isl_take isl_mat *mat,
+	int row, int col, __isl_take isl_val *v)
+{
+	if (!v)
+		return isl_mat_free(mat);
+	if (!isl_val_is_int(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting integer value", goto error);
+	mat = isl_mat_set_element(mat, row, col, v->n);
+	isl_val_free(v);
+	return mat;
+error:
+	isl_val_free(v);
+	return isl_mat_free(mat);
+}
+
+__isl_give isl_mat *isl_mat_diag(isl_ctx *ctx, unsigned n_row, isl_int d)
+{
+	int i;
+	struct isl_mat *mat;
+
+	mat = isl_mat_alloc(ctx, n_row, n_row);
+	if (!mat)
+		return NULL;
+	for (i = 0; i < n_row; ++i) {
+		isl_seq_clr(mat->row[i], i);
+		isl_int_set(mat->row[i][i], d);
+		isl_seq_clr(mat->row[i]+i+1, n_row-(i+1));
+	}
+
+	return mat;
+}
+
+__isl_give isl_mat *isl_mat_identity(isl_ctx *ctx, unsigned n_row)
+{
+	if (!ctx)
+		return NULL;
+	return isl_mat_diag(ctx, n_row, ctx->one);
+}
+
+/* Is "mat" a (possibly scaled) identity matrix?
+ */
+int isl_mat_is_scaled_identity(__isl_keep isl_mat *mat)
+{
+	int i;
+
+	if (!mat)
+		return -1;
+	if (mat->n_row != mat->n_col)
+		return 0;
+
+	for (i = 0; i < mat->n_row; ++i) {
+		if (isl_seq_first_non_zero(mat->row[i], i) != -1)
+			return 0;
+		if (isl_int_ne(mat->row[0][0], mat->row[i][i]))
+			return 0;
+		if (isl_seq_first_non_zero(mat->row[i] + i + 1,
+					    mat->n_col - (i + 1)) != -1)
+			return 0;
+	}
+
+	return 1;
+}
+
+struct isl_vec *isl_mat_vec_product(struct isl_mat *mat, struct isl_vec *vec)
+{
+	int i;
+	struct isl_vec *prod;
+
+	if (!mat || !vec)
+		goto error;
+
+	isl_assert(mat->ctx, mat->n_col == vec->size, goto error);
+
+	prod = isl_vec_alloc(mat->ctx, mat->n_row);
+	if (!prod)
+		goto error;
+
+	for (i = 0; i < prod->size; ++i)
+		isl_seq_inner_product(mat->row[i], vec->el, vec->size,
+					&prod->block.data[i]);
+	isl_mat_free(mat);
+	isl_vec_free(vec);
+	return prod;
+error:
+	isl_mat_free(mat);
+	isl_vec_free(vec);
+	return NULL;
+}
+
+__isl_give isl_vec *isl_mat_vec_inverse_product(__isl_take isl_mat *mat,
+	__isl_take isl_vec *vec)
+{
+	struct isl_mat *vec_mat;
+	int i;
+
+	if (!mat || !vec)
+		goto error;
+	vec_mat = isl_mat_alloc(vec->ctx, vec->size, 1);
+	if (!vec_mat)
+		goto error;
+	for (i = 0; i < vec->size; ++i)
+		isl_int_set(vec_mat->row[i][0], vec->el[i]);
+	vec_mat = isl_mat_inverse_product(mat, vec_mat);
+	isl_vec_free(vec);
+	if (!vec_mat)
+		return NULL;
+	vec = isl_vec_alloc(vec_mat->ctx, vec_mat->n_row);
+	if (vec)
+		for (i = 0; i < vec->size; ++i)
+			isl_int_set(vec->el[i], vec_mat->row[i][0]);
+	isl_mat_free(vec_mat);
+	return vec;
+error:
+	isl_mat_free(mat);
+	isl_vec_free(vec);
+	return NULL;
+}
+
+struct isl_vec *isl_vec_mat_product(struct isl_vec *vec, struct isl_mat *mat)
+{
+	int i, j;
+	struct isl_vec *prod;
+
+	if (!mat || !vec)
+		goto error;
+
+	isl_assert(mat->ctx, mat->n_row == vec->size, goto error);
+
+	prod = isl_vec_alloc(mat->ctx, mat->n_col);
+	if (!prod)
+		goto error;
+
+	for (i = 0; i < prod->size; ++i) {
+		isl_int_set_si(prod->el[i], 0);
+		for (j = 0; j < vec->size; ++j)
+			isl_int_addmul(prod->el[i], vec->el[j], mat->row[j][i]);
+	}
+	isl_mat_free(mat);
+	isl_vec_free(vec);
+	return prod;
+error:
+	isl_mat_free(mat);
+	isl_vec_free(vec);
+	return NULL;
+}
+
+struct isl_mat *isl_mat_aff_direct_sum(struct isl_mat *left,
+	struct isl_mat *right)
+{
+	int i;
+	struct isl_mat *sum;
+
+	if (!left || !right)
+		goto error;
+
+	isl_assert(left->ctx, left->n_row == right->n_row, goto error);
+	isl_assert(left->ctx, left->n_row >= 1, goto error);
+	isl_assert(left->ctx, left->n_col >= 1, goto error);
+	isl_assert(left->ctx, right->n_col >= 1, goto error);
+	isl_assert(left->ctx,
+	    isl_seq_first_non_zero(left->row[0]+1, left->n_col-1) == -1,
+	    goto error);
+	isl_assert(left->ctx,
+	    isl_seq_first_non_zero(right->row[0]+1, right->n_col-1) == -1,
+	    goto error);
+
+	sum = isl_mat_alloc(left->ctx, left->n_row, left->n_col + right->n_col - 1);
+	if (!sum)
+		goto error;
+	isl_int_lcm(sum->row[0][0], left->row[0][0], right->row[0][0]);
+	isl_int_divexact(left->row[0][0], sum->row[0][0], left->row[0][0]);
+	isl_int_divexact(right->row[0][0], sum->row[0][0], right->row[0][0]);
+
+	isl_seq_clr(sum->row[0]+1, sum->n_col-1);
+	for (i = 1; i < sum->n_row; ++i) {
+		isl_int_mul(sum->row[i][0], left->row[0][0], left->row[i][0]);
+		isl_int_addmul(sum->row[i][0],
+				right->row[0][0], right->row[i][0]);
+		isl_seq_scale(sum->row[i]+1, left->row[i]+1, left->row[0][0],
+				left->n_col-1);
+		isl_seq_scale(sum->row[i]+left->n_col,
+				right->row[i]+1, right->row[0][0],
+				right->n_col-1);
+	}
+
+	isl_int_divexact(left->row[0][0], sum->row[0][0], left->row[0][0]);
+	isl_int_divexact(right->row[0][0], sum->row[0][0], right->row[0][0]);
+	isl_mat_free(left);
+	isl_mat_free(right);
+	return sum;
+error:
+	isl_mat_free(left);
+	isl_mat_free(right);
+	return NULL;
+}
+
+static void exchange(struct isl_mat *M, struct isl_mat **U,
+	struct isl_mat **Q, unsigned row, unsigned i, unsigned j)
+{
+	int r;
+	for (r = row; r < M->n_row; ++r)
+		isl_int_swap(M->row[r][i], M->row[r][j]);
+	if (U) {
+		for (r = 0; r < (*U)->n_row; ++r)
+			isl_int_swap((*U)->row[r][i], (*U)->row[r][j]);
+	}
+	if (Q)
+		isl_mat_swap_rows(*Q, i, j);
+}
+
+static void subtract(struct isl_mat *M, struct isl_mat **U,
+	struct isl_mat **Q, unsigned row, unsigned i, unsigned j, isl_int m)
+{
+	int r;
+	for (r = row; r < M->n_row; ++r)
+		isl_int_submul(M->row[r][j], m, M->row[r][i]);
+	if (U) {
+		for (r = 0; r < (*U)->n_row; ++r)
+			isl_int_submul((*U)->row[r][j], m, (*U)->row[r][i]);
+	}
+	if (Q) {
+		for (r = 0; r < (*Q)->n_col; ++r)
+			isl_int_addmul((*Q)->row[i][r], m, (*Q)->row[j][r]);
+	}
+}
+
+static void oppose(struct isl_mat *M, struct isl_mat **U,
+	struct isl_mat **Q, unsigned row, unsigned col)
+{
+	int r;
+	for (r = row; r < M->n_row; ++r)
+		isl_int_neg(M->row[r][col], M->row[r][col]);
+	if (U) {
+		for (r = 0; r < (*U)->n_row; ++r)
+			isl_int_neg((*U)->row[r][col], (*U)->row[r][col]);
+	}
+	if (Q)
+		isl_seq_neg((*Q)->row[col], (*Q)->row[col], (*Q)->n_col);
+}
+
+/* Given matrix M, compute
+ *
+ *		M U = H
+ *		M   = H Q
+ *
+ * with U and Q unimodular matrices and H a matrix in column echelon form
+ * such that on each echelon row the entries in the non-echelon column
+ * are non-negative (if neg == 0) or non-positive (if neg == 1)
+ * and strictly smaller (in absolute value) than the entries in the echelon
+ * column.
+ * If U or Q are NULL, then these matrices are not computed.
+ */
+struct isl_mat *isl_mat_left_hermite(struct isl_mat *M, int neg,
+	struct isl_mat **U, struct isl_mat **Q)
+{
+	isl_int c;
+	int row, col;
+
+	if (U)
+		*U = NULL;
+	if (Q)
+		*Q = NULL;
+	if (!M)
+		goto error;
+	M = isl_mat_cow(M);
+	if (!M)
+		goto error;
+	if (U) {
+		*U = isl_mat_identity(M->ctx, M->n_col);
+		if (!*U)
+			goto error;
+	}
+	if (Q) {
+		*Q = isl_mat_identity(M->ctx, M->n_col);
+		if (!*Q)
+			goto error;
+	}
+
+	col = 0;
+	isl_int_init(c);
+	for (row = 0; row < M->n_row; ++row) {
+		int first, i, off;
+		first = isl_seq_abs_min_non_zero(M->row[row]+col, M->n_col-col);
+		if (first == -1)
+			continue;
+		first += col;
+		if (first != col)
+			exchange(M, U, Q, row, first, col);
+		if (isl_int_is_neg(M->row[row][col]))
+			oppose(M, U, Q, row, col);
+		first = col+1;
+		while ((off = isl_seq_first_non_zero(M->row[row]+first,
+						       M->n_col-first)) != -1) {
+			first += off;
+			isl_int_fdiv_q(c, M->row[row][first], M->row[row][col]);
+			subtract(M, U, Q, row, col, first, c);
+			if (!isl_int_is_zero(M->row[row][first]))
+				exchange(M, U, Q, row, first, col);
+			else
+				++first;
+		}
+		for (i = 0; i < col; ++i) {
+			if (isl_int_is_zero(M->row[row][i]))
+				continue;
+			if (neg)
+				isl_int_cdiv_q(c, M->row[row][i], M->row[row][col]);
+			else
+				isl_int_fdiv_q(c, M->row[row][i], M->row[row][col]);
+			if (isl_int_is_zero(c))
+				continue;
+			subtract(M, U, Q, row, col, i, c);
+		}
+		++col;
+	}
+	isl_int_clear(c);
+
+	return M;
+error:
+	if (Q) {
+		isl_mat_free(*Q);
+		*Q = NULL;
+	}
+	if (U) {
+		isl_mat_free(*U);
+		*U = NULL;
+	}
+	isl_mat_free(M);
+	return NULL;
+}
+
+struct isl_mat *isl_mat_right_kernel(struct isl_mat *mat)
+{
+	int i, rank;
+	struct isl_mat *U = NULL;
+	struct isl_mat *K;
+
+	mat = isl_mat_left_hermite(mat, 0, &U, NULL);
+	if (!mat || !U)
+		goto error;
+
+	for (i = 0, rank = 0; rank < mat->n_col; ++rank) {
+		while (i < mat->n_row && isl_int_is_zero(mat->row[i][rank]))
+			++i;
+		if (i >= mat->n_row)
+			break;
+	}
+	K = isl_mat_alloc(U->ctx, U->n_row, U->n_col - rank);
+	if (!K)
+		goto error;
+	isl_mat_sub_copy(K->ctx, K->row, U->row, U->n_row, 0, rank, U->n_col-rank);
+	isl_mat_free(mat);
+	isl_mat_free(U);
+	return K;
+error:
+	isl_mat_free(mat);
+	isl_mat_free(U);
+	return NULL;
+}
+
+struct isl_mat *isl_mat_lin_to_aff(struct isl_mat *mat)
+{
+	int i;
+	struct isl_mat *mat2;
+
+	if (!mat)
+		return NULL;
+	mat2 = isl_mat_alloc(mat->ctx, 1+mat->n_row, 1+mat->n_col);
+	if (!mat2)
+		goto error;
+	isl_int_set_si(mat2->row[0][0], 1);
+	isl_seq_clr(mat2->row[0]+1, mat->n_col);
+	for (i = 0; i < mat->n_row; ++i) {
+		isl_int_set_si(mat2->row[1+i][0], 0);
+		isl_seq_cpy(mat2->row[1+i]+1, mat->row[i], mat->n_col);
+	}
+	isl_mat_free(mat);
+	return mat2;
+error:
+	isl_mat_free(mat);
+	return NULL;
+}
+
+/* Given two matrices M1 and M2, return the block matrix
+ *
+ *	[ M1  0  ]
+ *	[ 0   M2 ]
+ */
+__isl_give isl_mat *isl_mat_diagonal(__isl_take isl_mat *mat1,
+	__isl_take isl_mat *mat2)
+{
+	int i;
+	isl_mat *mat;
+
+	if (!mat1 || !mat2)
+		goto error;
+
+	mat = isl_mat_alloc(mat1->ctx, mat1->n_row + mat2->n_row,
+				       mat1->n_col + mat2->n_col);
+	if (!mat)
+		goto error;
+	for (i = 0; i < mat1->n_row; ++i) {
+		isl_seq_cpy(mat->row[i], mat1->row[i], mat1->n_col);
+		isl_seq_clr(mat->row[i] + mat1->n_col, mat2->n_col);
+	}
+	for (i = 0; i < mat2->n_row; ++i) {
+		isl_seq_clr(mat->row[mat1->n_row + i], mat1->n_col);
+		isl_seq_cpy(mat->row[mat1->n_row + i] + mat1->n_col,
+						    mat2->row[i], mat2->n_col);
+	}
+	isl_mat_free(mat1);
+	isl_mat_free(mat2);
+	return mat;
+error:
+	isl_mat_free(mat1);
+	isl_mat_free(mat2);
+	return NULL;
+}
+
+static int row_first_non_zero(isl_int **row, unsigned n_row, unsigned col)
+{
+	int i;
+
+	for (i = 0; i < n_row; ++i)
+		if (!isl_int_is_zero(row[i][col]))
+			return i;
+	return -1;
+}
+
+static int row_abs_min_non_zero(isl_int **row, unsigned n_row, unsigned col)
+{
+	int i, min = row_first_non_zero(row, n_row, col);
+	if (min < 0)
+		return -1;
+	for (i = min + 1; i < n_row; ++i) {
+		if (isl_int_is_zero(row[i][col]))
+			continue;
+		if (isl_int_abs_lt(row[i][col], row[min][col]))
+			min = i;
+	}
+	return min;
+}
+
+static void inv_exchange(struct isl_mat *left, struct isl_mat *right,
+	unsigned i, unsigned j)
+{
+	left = isl_mat_swap_rows(left, i, j);
+	right = isl_mat_swap_rows(right, i, j);
+}
+
+static void inv_oppose(
+	struct isl_mat *left, struct isl_mat *right, unsigned row)
+{
+	isl_seq_neg(left->row[row]+row, left->row[row]+row, left->n_col-row);
+	isl_seq_neg(right->row[row], right->row[row], right->n_col);
+}
+
+static void inv_subtract(struct isl_mat *left, struct isl_mat *right,
+	unsigned row, unsigned i, isl_int m)
+{
+	isl_int_neg(m, m);
+	isl_seq_combine(left->row[i]+row,
+			left->ctx->one, left->row[i]+row,
+			m, left->row[row]+row,
+			left->n_col-row);
+	isl_seq_combine(right->row[i], right->ctx->one, right->row[i],
+			m, right->row[row], right->n_col);
+}
+
+/* Compute inv(left)*right
+ */
+struct isl_mat *isl_mat_inverse_product(struct isl_mat *left,
+	struct isl_mat *right)
+{
+	int row;
+	isl_int a, b;
+
+	if (!left || !right)
+		goto error;
+
+	isl_assert(left->ctx, left->n_row == left->n_col, goto error);
+	isl_assert(left->ctx, left->n_row == right->n_row, goto error);
+
+	if (left->n_row == 0) {
+		isl_mat_free(left);
+		return right;
+	}
+
+	left = isl_mat_cow(left);
+	right = isl_mat_cow(right);
+	if (!left || !right)
+		goto error;
+
+	isl_int_init(a);
+	isl_int_init(b);
+	for (row = 0; row < left->n_row; ++row) {
+		int pivot, first, i, off;
+		pivot = row_abs_min_non_zero(left->row+row, left->n_row-row, row);
+		if (pivot < 0) {
+			isl_int_clear(a);
+			isl_int_clear(b);
+			isl_assert(left->ctx, pivot >= 0, goto error);
+		}
+		pivot += row;
+		if (pivot != row)
+			inv_exchange(left, right, pivot, row);
+		if (isl_int_is_neg(left->row[row][row]))
+			inv_oppose(left, right, row);
+		first = row+1;
+		while ((off = row_first_non_zero(left->row+first,
+					left->n_row-first, row)) != -1) {
+			first += off;
+			isl_int_fdiv_q(a, left->row[first][row],
+					left->row[row][row]);
+			inv_subtract(left, right, row, first, a);
+			if (!isl_int_is_zero(left->row[first][row]))
+				inv_exchange(left, right, row, first);
+			else
+				++first;
+		}
+		for (i = 0; i < row; ++i) {
+			if (isl_int_is_zero(left->row[i][row]))
+				continue;
+			isl_int_gcd(a, left->row[row][row], left->row[i][row]);
+			isl_int_divexact(b, left->row[i][row], a);
+			isl_int_divexact(a, left->row[row][row], a);
+			isl_int_neg(b, b);
+			isl_seq_combine(left->row[i] + i,
+					a, left->row[i] + i,
+					b, left->row[row] + i,
+					left->n_col - i);
+			isl_seq_combine(right->row[i], a, right->row[i],
+					b, right->row[row], right->n_col);
+		}
+	}
+	isl_int_clear(b);
+
+	isl_int_set(a, left->row[0][0]);
+	for (row = 1; row < left->n_row; ++row)
+		isl_int_lcm(a, a, left->row[row][row]);
+	if (isl_int_is_zero(a)){
+		isl_int_clear(a);
+		isl_assert(left->ctx, 0, goto error);
+	}
+	for (row = 0; row < left->n_row; ++row) {
+		isl_int_divexact(left->row[row][row], a, left->row[row][row]);
+		if (isl_int_is_one(left->row[row][row]))
+			continue;
+		isl_seq_scale(right->row[row], right->row[row],
+				left->row[row][row], right->n_col);
+	}
+	isl_int_clear(a);
+
+	isl_mat_free(left);
+	return right;
+error:
+	isl_mat_free(left);
+	isl_mat_free(right);
+	return NULL;
+}
+
+void isl_mat_col_scale(struct isl_mat *mat, unsigned col, isl_int m)
+{
+	int i;
+
+	for (i = 0; i < mat->n_row; ++i)
+		isl_int_mul(mat->row[i][col], mat->row[i][col], m);
+}
+
+void isl_mat_col_combine(struct isl_mat *mat, unsigned dst,
+	isl_int m1, unsigned src1, isl_int m2, unsigned src2)
+{
+	int i;
+	isl_int tmp;
+
+	isl_int_init(tmp);
+	for (i = 0; i < mat->n_row; ++i) {
+		isl_int_mul(tmp, m1, mat->row[i][src1]);
+		isl_int_addmul(tmp, m2, mat->row[i][src2]);
+		isl_int_set(mat->row[i][dst], tmp);
+	}
+	isl_int_clear(tmp);
+}
+
+struct isl_mat *isl_mat_right_inverse(struct isl_mat *mat)
+{
+	struct isl_mat *inv;
+	int row;
+	isl_int a, b;
+
+	mat = isl_mat_cow(mat);
+	if (!mat)
+		return NULL;
+
+	inv = isl_mat_identity(mat->ctx, mat->n_col);
+	inv = isl_mat_cow(inv);
+	if (!inv)
+		goto error;
+
+	isl_int_init(a);
+	isl_int_init(b);
+	for (row = 0; row < mat->n_row; ++row) {
+		int pivot, first, i, off;
+		pivot = isl_seq_abs_min_non_zero(mat->row[row]+row, mat->n_col-row);
+		if (pivot < 0) {
+			isl_int_clear(a);
+			isl_int_clear(b);
+			isl_assert(mat->ctx, pivot >= 0, goto error);
+		}
+		pivot += row;
+		if (pivot != row)
+			exchange(mat, &inv, NULL, row, pivot, row);
+		if (isl_int_is_neg(mat->row[row][row]))
+			oppose(mat, &inv, NULL, row, row);
+		first = row+1;
+		while ((off = isl_seq_first_non_zero(mat->row[row]+first,
+						    mat->n_col-first)) != -1) {
+			first += off;
+			isl_int_fdiv_q(a, mat->row[row][first],
+						    mat->row[row][row]);
+			subtract(mat, &inv, NULL, row, row, first, a);
+			if (!isl_int_is_zero(mat->row[row][first]))
+				exchange(mat, &inv, NULL, row, row, first);
+			else
+				++first;
+		}
+		for (i = 0; i < row; ++i) {
+			if (isl_int_is_zero(mat->row[row][i]))
+				continue;
+			isl_int_gcd(a, mat->row[row][row], mat->row[row][i]);
+			isl_int_divexact(b, mat->row[row][i], a);
+			isl_int_divexact(a, mat->row[row][row], a);
+			isl_int_neg(a, a);
+			isl_mat_col_combine(mat, i, a, i, b, row);
+			isl_mat_col_combine(inv, i, a, i, b, row);
+		}
+	}
+	isl_int_clear(b);
+
+	isl_int_set(a, mat->row[0][0]);
+	for (row = 1; row < mat->n_row; ++row)
+		isl_int_lcm(a, a, mat->row[row][row]);
+	if (isl_int_is_zero(a)){
+		isl_int_clear(a);
+		goto error;
+	}
+	for (row = 0; row < mat->n_row; ++row) {
+		isl_int_divexact(mat->row[row][row], a, mat->row[row][row]);
+		if (isl_int_is_one(mat->row[row][row]))
+			continue;
+		isl_mat_col_scale(inv, row, mat->row[row][row]);
+	}
+	isl_int_clear(a);
+
+	isl_mat_free(mat);
+
+	return inv;
+error:
+	isl_mat_free(mat);
+	isl_mat_free(inv);
+	return NULL;
+}
+
+struct isl_mat *isl_mat_transpose(struct isl_mat *mat)
+{
+	struct isl_mat *transpose = NULL;
+	int i, j;
+
+	if (!mat)
+		return NULL;
+
+	if (mat->n_col == mat->n_row) {
+		mat = isl_mat_cow(mat);
+		if (!mat)
+			return NULL;
+		for (i = 0; i < mat->n_row; ++i)
+			for (j = i + 1; j < mat->n_col; ++j)
+				isl_int_swap(mat->row[i][j], mat->row[j][i]);
+		return mat;
+	}
+	transpose = isl_mat_alloc(mat->ctx, mat->n_col, mat->n_row);
+	if (!transpose)
+		goto error;
+	for (i = 0; i < mat->n_row; ++i)
+		for (j = 0; j < mat->n_col; ++j)
+			isl_int_set(transpose->row[j][i], mat->row[i][j]);
+	isl_mat_free(mat);
+	return transpose;
+error:
+	isl_mat_free(mat);
+	return NULL;
+}
+
+struct isl_mat *isl_mat_swap_cols(struct isl_mat *mat, unsigned i, unsigned j)
+{
+	int r;
+
+	mat = isl_mat_cow(mat);
+	if (!mat)
+		return NULL;
+	isl_assert(mat->ctx, i < mat->n_col, goto error);
+	isl_assert(mat->ctx, j < mat->n_col, goto error);
+
+	for (r = 0; r < mat->n_row; ++r)
+		isl_int_swap(mat->row[r][i], mat->row[r][j]);
+	return mat;
+error:
+	isl_mat_free(mat);
+	return NULL;
+}
+
+struct isl_mat *isl_mat_swap_rows(struct isl_mat *mat, unsigned i, unsigned j)
+{
+	isl_int *t;
+
+	if (!mat)
+		return NULL;
+	mat = isl_mat_cow(mat);
+	if (!mat)
+		return NULL;
+	t = mat->row[i];
+	mat->row[i] = mat->row[j];
+	mat->row[j] = t;
+	return mat;
+}
+
+__isl_give isl_mat *isl_mat_product(__isl_take isl_mat *left,
+	__isl_take isl_mat *right)
+{
+	int i, j, k;
+	struct isl_mat *prod;
+
+	if (!left || !right)
+		goto error;
+	isl_assert(left->ctx, left->n_col == right->n_row, goto error);
+	prod = isl_mat_alloc(left->ctx, left->n_row, right->n_col);
+	if (!prod)
+		goto error;
+	if (left->n_col == 0) {
+		for (i = 0; i < prod->n_row; ++i)
+			isl_seq_clr(prod->row[i], prod->n_col);
+		isl_mat_free(left);
+		isl_mat_free(right);
+		return prod;
+	}
+	for (i = 0; i < prod->n_row; ++i) {
+		for (j = 0; j < prod->n_col; ++j) {
+			isl_int_mul(prod->row[i][j],
+				    left->row[i][0], right->row[0][j]);
+			for (k = 1; k < left->n_col; ++k)
+				isl_int_addmul(prod->row[i][j],
+					    left->row[i][k], right->row[k][j]);
+		}
+	}
+	isl_mat_free(left);
+	isl_mat_free(right);
+	return prod;
+error:
+	isl_mat_free(left);
+	isl_mat_free(right);
+	return NULL;
+}
+
+/* Replace the variables x in the rows q by x' given by x = M x',
+ * with M the matrix mat.
+ *
+ * If the number of new variables is greater than the original
+ * number of variables, then the rows q have already been
+ * preextended.  If the new number is smaller, then the coefficients
+ * of the divs, which are not changed, need to be shifted down.
+ * The row q may be the equalities, the inequalities or the
+ * div expressions.  In the latter case, has_div is true and
+ * we need to take into account the extra denominator column.
+ */
+static int preimage(struct isl_ctx *ctx, isl_int **q, unsigned n,
+	unsigned n_div, int has_div, struct isl_mat *mat)
+{
+	int i;
+	struct isl_mat *t;
+	int e;
+
+	if (mat->n_col >= mat->n_row)
+		e = 0;
+	else
+		e = mat->n_row - mat->n_col;
+	if (has_div)
+		for (i = 0; i < n; ++i)
+			isl_int_mul(q[i][0], q[i][0], mat->row[0][0]);
+	t = isl_mat_sub_alloc6(mat->ctx, q, 0, n, has_div, mat->n_row);
+	t = isl_mat_product(t, mat);
+	if (!t)
+		return -1;
+	for (i = 0; i < n; ++i) {
+		isl_seq_swp_or_cpy(q[i] + has_div, t->row[i], t->n_col);
+		isl_seq_cpy(q[i] + has_div + t->n_col,
+			    q[i] + has_div + t->n_col + e, n_div);
+		isl_seq_clr(q[i] + has_div + t->n_col + n_div, e);
+	}
+	isl_mat_free(t);
+	return 0;
+}
+
+/* Replace the variables x in bset by x' given by x = M x', with
+ * M the matrix mat.
+ *
+ * If there are fewer variables x' then there are x, then we perform
+ * the transformation in place, which that, in principle,
+ * this frees up some extra variables as the number
+ * of columns remains constant, but we would have to extend
+ * the div array too as the number of rows in this array is assumed
+ * to be equal to extra.
+ */
+struct isl_basic_set *isl_basic_set_preimage(struct isl_basic_set *bset,
+	struct isl_mat *mat)
+{
+	struct isl_ctx *ctx;
+
+	if (!bset || !mat)
+		goto error;
+
+	ctx = bset->ctx;
+	bset = isl_basic_set_cow(bset);
+	if (!bset)
+		goto error;
+
+	isl_assert(ctx, bset->dim->nparam == 0, goto error);
+	isl_assert(ctx, 1+bset->dim->n_out == mat->n_row, goto error);
+	isl_assert(ctx, mat->n_col > 0, goto error);
+
+	if (mat->n_col > mat->n_row) {
+		bset = isl_basic_set_extend(bset, 0, mat->n_col-1, 0, 0, 0);
+		if (!bset)
+			goto error;
+	} else if (mat->n_col < mat->n_row) {
+		bset->dim = isl_space_cow(bset->dim);
+		if (!bset->dim)
+			goto error;
+		bset->dim->n_out -= mat->n_row - mat->n_col;
+	}
+
+	if (preimage(ctx, bset->eq, bset->n_eq, bset->n_div, 0,
+			isl_mat_copy(mat)) < 0)
+		goto error;
+
+	if (preimage(ctx, bset->ineq, bset->n_ineq, bset->n_div, 0,
+			isl_mat_copy(mat)) < 0)
+		goto error;
+
+	if (preimage(ctx, bset->div, bset->n_div, bset->n_div, 1, mat) < 0)
+		goto error2;
+
+	ISL_F_CLR(bset, ISL_BASIC_SET_NO_IMPLICIT);
+	ISL_F_CLR(bset, ISL_BASIC_SET_NO_REDUNDANT);
+	ISL_F_CLR(bset, ISL_BASIC_SET_NORMALIZED);
+	ISL_F_CLR(bset, ISL_BASIC_SET_NORMALIZED_DIVS);
+	ISL_F_CLR(bset, ISL_BASIC_SET_ALL_EQUALITIES);
+
+	bset = isl_basic_set_simplify(bset);
+	bset = isl_basic_set_finalize(bset);
+
+	return bset;
+error:
+	isl_mat_free(mat);
+error2:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+struct isl_set *isl_set_preimage(struct isl_set *set, struct isl_mat *mat)
+{
+	struct isl_ctx *ctx;
+	int i;
+
+	set = isl_set_cow(set);
+	if (!set)
+		return NULL;
+
+	ctx = set->ctx;
+	for (i = 0; i < set->n; ++i) {
+		set->p[i] = isl_basic_set_preimage(set->p[i],
+						    isl_mat_copy(mat));
+		if (!set->p[i])
+			goto error;
+	}
+	if (mat->n_col != mat->n_row) {
+		set->dim = isl_space_cow(set->dim);
+		if (!set->dim)
+			goto error;
+		set->dim->n_out += mat->n_col;
+		set->dim->n_out -= mat->n_row;
+	}
+	isl_mat_free(mat);
+	ISL_F_CLR(set, ISL_SET_NORMALIZED);
+	return set;
+error:
+	isl_set_free(set);
+	isl_mat_free(mat);
+	return NULL;
+}
+
+/* Replace the variables x starting at pos in the rows q
+ * by x' with x = M x' with M the matrix mat.
+ * That is, replace the corresponding coefficients c by c M.
+ */
+static int transform(isl_ctx *ctx, isl_int **q, unsigned n,
+	unsigned pos, __isl_take isl_mat *mat)
+{
+	int i;
+	isl_mat *t;
+
+	t = isl_mat_sub_alloc6(ctx, q, 0, n, pos, mat->n_row);
+	t = isl_mat_product(t, mat);
+	if (!t)
+		return -1;
+	for (i = 0; i < n; ++i)
+		isl_seq_swp_or_cpy(q[i] + pos, t->row[i], t->n_col);
+	isl_mat_free(t);
+	return 0;
+}
+
+/* Replace the variables x of type "type" starting at "first" in "bset"
+ * by x' with x = M x' with M the matrix trans.
+ * That is, replace the corresponding coefficients c by c M.
+ *
+ * The transformation matrix should be a square matrix.
+ */
+__isl_give isl_basic_set *isl_basic_set_transform_dims(
+	__isl_take isl_basic_set *bset, enum isl_dim_type type, unsigned first,
+	__isl_take isl_mat *trans)
+{
+	isl_ctx *ctx;
+	unsigned pos;
+
+	bset = isl_basic_set_cow(bset);
+	if (!bset || !trans)
+		goto error;
+
+	ctx = isl_basic_set_get_ctx(bset);
+	if (trans->n_row != trans->n_col)
+		isl_die(trans->ctx, isl_error_invalid,
+			"expecting square transformation matrix", goto error);
+	if (first + trans->n_row > isl_basic_set_dim(bset, type))
+		isl_die(trans->ctx, isl_error_invalid,
+			"oversized transformation matrix", goto error);
+
+	pos = isl_basic_set_offset(bset, type) + first;
+
+	if (transform(ctx, bset->eq, bset->n_eq, pos, isl_mat_copy(trans)) < 0)
+		goto error;
+	if (transform(ctx, bset->ineq, bset->n_ineq, pos,
+		      isl_mat_copy(trans)) < 0)
+		goto error;
+	if (transform(ctx, bset->div, bset->n_div, 1 + pos,
+		      isl_mat_copy(trans)) < 0)
+		goto error;
+
+	ISL_F_CLR(bset, ISL_BASIC_SET_NORMALIZED);
+	ISL_F_CLR(bset, ISL_BASIC_SET_NORMALIZED_DIVS);
+
+	isl_mat_free(trans);
+	return bset;
+error:
+	isl_mat_free(trans);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+void isl_mat_print_internal(__isl_keep isl_mat *mat, FILE *out, int indent)
+{
+	int i, j;
+
+	if (!mat) {
+		fprintf(out, "%*snull mat\n", indent, "");
+		return;
+	}
+
+	if (mat->n_row == 0)
+		fprintf(out, "%*s[]\n", indent, "");
+
+	for (i = 0; i < mat->n_row; ++i) {
+		if (!i)
+			fprintf(out, "%*s[[", indent, "");
+		else
+			fprintf(out, "%*s[", indent+1, "");
+		for (j = 0; j < mat->n_col; ++j) {
+			if (j)
+			    fprintf(out, ",");
+			isl_int_print(out, mat->row[i][j], 0);
+		}
+		if (i == mat->n_row-1)
+			fprintf(out, "]]\n");
+		else
+			fprintf(out, "]\n");
+	}
+}
+
+void isl_mat_dump(__isl_keep isl_mat *mat)
+{
+	isl_mat_print_internal(mat, stderr, 0);
+}
+
+struct isl_mat *isl_mat_drop_cols(struct isl_mat *mat, unsigned col, unsigned n)
+{
+	int r;
+
+	if (n == 0)
+		return mat;
+
+	mat = isl_mat_cow(mat);
+	if (!mat)
+		return NULL;
+
+	if (col != mat->n_col-n) {
+		for (r = 0; r < mat->n_row; ++r)
+			isl_seq_cpy(mat->row[r]+col, mat->row[r]+col+n,
+					mat->n_col - col - n);
+	}
+	mat->n_col -= n;
+	return mat;
+}
+
+struct isl_mat *isl_mat_drop_rows(struct isl_mat *mat, unsigned row, unsigned n)
+{
+	int r;
+
+	mat = isl_mat_cow(mat);
+	if (!mat)
+		return NULL;
+
+	for (r = row; r+n < mat->n_row; ++r)
+		mat->row[r] = mat->row[r+n];
+
+	mat->n_row -= n;
+	return mat;
+}
+
+__isl_give isl_mat *isl_mat_insert_cols(__isl_take isl_mat *mat,
+				unsigned col, unsigned n)
+{
+	isl_mat *ext;
+
+	if (!mat)
+		return NULL;
+	if (n == 0)
+		return mat;
+
+	ext = isl_mat_alloc(mat->ctx, mat->n_row, mat->n_col + n);
+	if (!ext)
+		goto error;
+
+	isl_mat_sub_copy(mat->ctx, ext->row, mat->row, mat->n_row, 0, 0, col);
+	isl_mat_sub_copy(mat->ctx, ext->row, mat->row, mat->n_row,
+				col + n, col, mat->n_col - col);
+
+	isl_mat_free(mat);
+	return ext;
+error:
+	isl_mat_free(mat);
+	return NULL;
+}
+
+__isl_give isl_mat *isl_mat_insert_zero_cols(__isl_take isl_mat *mat,
+	unsigned first, unsigned n)
+{
+	int i;
+
+	if (!mat)
+		return NULL;
+	mat = isl_mat_insert_cols(mat, first, n);
+	if (!mat)
+		return NULL;
+
+	for (i = 0; i < mat->n_row; ++i)
+		isl_seq_clr(mat->row[i] + first, n);
+
+	return mat;
+}
+
+__isl_give isl_mat *isl_mat_add_zero_cols(__isl_take isl_mat *mat, unsigned n)
+{
+	if (!mat)
+		return NULL;
+
+	return isl_mat_insert_zero_cols(mat, mat->n_col, n);
+}
+
+__isl_give isl_mat *isl_mat_insert_rows(__isl_take isl_mat *mat,
+				unsigned row, unsigned n)
+{
+	isl_mat *ext;
+
+	if (!mat)
+		return NULL;
+	if (n == 0)
+		return mat;
+
+	ext = isl_mat_alloc(mat->ctx, mat->n_row + n, mat->n_col);
+	if (!ext)
+		goto error;
+
+	isl_mat_sub_copy(mat->ctx, ext->row, mat->row, row, 0, 0, mat->n_col);
+	isl_mat_sub_copy(mat->ctx, ext->row + row + n, mat->row + row,
+				mat->n_row - row, 0, 0, mat->n_col);
+
+	isl_mat_free(mat);
+	return ext;
+error:
+	isl_mat_free(mat);
+	return NULL;
+}
+
+__isl_give isl_mat *isl_mat_add_rows(__isl_take isl_mat *mat, unsigned n)
+{
+	if (!mat)
+		return NULL;
+
+	return isl_mat_insert_rows(mat, mat->n_row, n);
+}
+
+__isl_give isl_mat *isl_mat_insert_zero_rows(__isl_take isl_mat *mat,
+	unsigned row, unsigned n)
+{
+	int i;
+
+	mat = isl_mat_insert_rows(mat, row, n);
+	if (!mat)
+		return NULL;
+	
+	for (i = 0; i < n; ++i)
+		isl_seq_clr(mat->row[row + i], mat->n_col);
+
+	return mat;
+}
+
+__isl_give isl_mat *isl_mat_add_zero_rows(__isl_take isl_mat *mat, unsigned n)
+{
+	if (!mat)
+		return NULL;
+
+	return isl_mat_insert_zero_rows(mat, mat->n_row, n);
+}
+
+void isl_mat_col_submul(struct isl_mat *mat,
+			int dst_col, isl_int f, int src_col)
+{
+	int i;
+
+	for (i = 0; i < mat->n_row; ++i)
+		isl_int_submul(mat->row[i][dst_col], f, mat->row[i][src_col]);
+}
+
+void isl_mat_col_add(__isl_keep isl_mat *mat, int dst_col, int src_col)
+{
+	int i;
+
+	if (!mat)
+		return;
+
+	for (i = 0; i < mat->n_row; ++i)
+		isl_int_add(mat->row[i][dst_col],
+			    mat->row[i][dst_col], mat->row[i][src_col]);
+}
+
+void isl_mat_col_mul(struct isl_mat *mat, int dst_col, isl_int f, int src_col)
+{
+	int i;
+
+	for (i = 0; i < mat->n_row; ++i)
+		isl_int_mul(mat->row[i][dst_col], f, mat->row[i][src_col]);
+}
+
+struct isl_mat *isl_mat_unimodular_complete(struct isl_mat *M, int row)
+{
+	int r;
+	struct isl_mat *H = NULL, *Q = NULL;
+
+	if (!M)
+		return NULL;
+
+	isl_assert(M->ctx, M->n_row == M->n_col, goto error);
+	M->n_row = row;
+	H = isl_mat_left_hermite(isl_mat_copy(M), 0, NULL, &Q);
+	M->n_row = M->n_col;
+	if (!H)
+		goto error;
+	for (r = 0; r < row; ++r)
+		isl_assert(M->ctx, isl_int_is_one(H->row[r][r]), goto error);
+	for (r = row; r < M->n_row; ++r)
+		isl_seq_cpy(M->row[r], Q->row[r], M->n_col);
+	isl_mat_free(H);
+	isl_mat_free(Q);
+	return M;
+error:
+	isl_mat_free(H);
+	isl_mat_free(Q);
+	isl_mat_free(M);
+	return NULL;
+}
+
+__isl_give isl_mat *isl_mat_concat(__isl_take isl_mat *top,
+	__isl_take isl_mat *bot)
+{
+	struct isl_mat *mat;
+
+	if (!top || !bot)
+		goto error;
+
+	isl_assert(top->ctx, top->n_col == bot->n_col, goto error);
+	if (top->n_row == 0) {
+		isl_mat_free(top);
+		return bot;
+	}
+	if (bot->n_row == 0) {
+		isl_mat_free(bot);
+		return top;
+	}
+
+	mat = isl_mat_alloc(top->ctx, top->n_row + bot->n_row, top->n_col);
+	if (!mat)
+		goto error;
+	isl_mat_sub_copy(mat->ctx, mat->row, top->row, top->n_row,
+			 0, 0, mat->n_col);
+	isl_mat_sub_copy(mat->ctx, mat->row + top->n_row, bot->row, bot->n_row,
+			 0, 0, mat->n_col);
+	isl_mat_free(top);
+	isl_mat_free(bot);
+	return mat;
+error:
+	isl_mat_free(top);
+	isl_mat_free(bot);
+	return NULL;
+}
+
+int isl_mat_is_equal(__isl_keep isl_mat *mat1, __isl_keep isl_mat *mat2)
+{
+	int i;
+
+	if (!mat1 || !mat2)
+		return -1;
+
+	if (mat1->n_row != mat2->n_row)
+		return 0;
+
+	if (mat1->n_col != mat2->n_col)
+		return 0;
+
+	for (i = 0; i < mat1->n_row; ++i)
+		if (!isl_seq_eq(mat1->row[i], mat2->row[i], mat1->n_col))
+			return 0;
+
+	return 1;
+}
+
+__isl_give isl_mat *isl_mat_from_row_vec(__isl_take isl_vec *vec)
+{
+	struct isl_mat *mat;
+
+	if (!vec)
+		return NULL;
+	mat = isl_mat_alloc(vec->ctx, 1, vec->size);
+	if (!mat)
+		goto error;
+
+	isl_seq_cpy(mat->row[0], vec->el, vec->size);
+
+	isl_vec_free(vec);
+	return mat;
+error:
+	isl_vec_free(vec);
+	return NULL;
+}
+
+/* Return a copy of row "row" of "mat" as an isl_vec.
+ */
+__isl_give isl_vec *isl_mat_get_row(__isl_keep isl_mat *mat, unsigned row)
+{
+	isl_vec *v;
+
+	if (!mat)
+		return NULL;
+	if (row >= mat->n_row)
+		isl_die(mat->ctx, isl_error_invalid, "row out of range",
+			return NULL);
+
+	v = isl_vec_alloc(isl_mat_get_ctx(mat), mat->n_col);
+	if (!v)
+		return NULL;
+	isl_seq_cpy(v->el, mat->row[row], mat->n_col);
+
+	return v;
+}
+
+__isl_give isl_mat *isl_mat_vec_concat(__isl_take isl_mat *top,
+	__isl_take isl_vec *bot)
+{
+	return isl_mat_concat(top, isl_mat_from_row_vec(bot));
+}
+
+__isl_give isl_mat *isl_mat_move_cols(__isl_take isl_mat *mat,
+	unsigned dst_col, unsigned src_col, unsigned n)
+{
+	isl_mat *res;
+
+	if (!mat)
+		return NULL;
+	if (n == 0 || dst_col == src_col)
+		return mat;
+
+	res = isl_mat_alloc(mat->ctx, mat->n_row, mat->n_col);
+	if (!res)
+		goto error;
+
+	if (dst_col < src_col) {
+		isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
+				 0, 0, dst_col);
+		isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
+				 dst_col, src_col, n);
+		isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
+				 dst_col + n, dst_col, src_col - dst_col);
+		isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
+				 src_col + n, src_col + n,
+				 res->n_col - src_col - n);
+	} else {
+		isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
+				 0, 0, src_col);
+		isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
+				 src_col, src_col + n, dst_col - src_col);
+		isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
+				 dst_col, src_col, n);
+		isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
+				 dst_col + n, dst_col + n,
+				 res->n_col - dst_col - n);
+	}
+	isl_mat_free(mat);
+
+	return res;
+error:
+	isl_mat_free(mat);
+	return NULL;
+}
+
+void isl_mat_gcd(__isl_keep isl_mat *mat, isl_int *gcd)
+{
+	int i;
+	isl_int g;
+
+	isl_int_set_si(*gcd, 0);
+	if (!mat)
+		return;
+
+	isl_int_init(g);
+	for (i = 0; i < mat->n_row; ++i) {
+		isl_seq_gcd(mat->row[i], mat->n_col, &g);
+		isl_int_gcd(*gcd, *gcd, g);
+	}
+	isl_int_clear(g);
+}
+
+__isl_give isl_mat *isl_mat_scale_down(__isl_take isl_mat *mat, isl_int m)
+{
+	int i;
+
+	if (isl_int_is_one(m))
+		return mat;
+
+	mat = isl_mat_cow(mat);
+	if (!mat)
+		return NULL;
+
+	for (i = 0; i < mat->n_row; ++i)
+		isl_seq_scale_down(mat->row[i], mat->row[i], m, mat->n_col);
+
+	return mat;
+}
+
+__isl_give isl_mat *isl_mat_scale_down_row(__isl_take isl_mat *mat, int row,
+	isl_int m)
+{
+	if (isl_int_is_one(m))
+		return mat;
+
+	mat = isl_mat_cow(mat);
+	if (!mat)
+		return NULL;
+
+	isl_seq_scale_down(mat->row[row], mat->row[row], m, mat->n_col);
+
+	return mat;
+}
+
+__isl_give isl_mat *isl_mat_normalize(__isl_take isl_mat *mat)
+{
+	isl_int gcd;
+
+	if (!mat)
+		return NULL;
+
+	isl_int_init(gcd);
+	isl_mat_gcd(mat, &gcd);
+	mat = isl_mat_scale_down(mat, gcd);
+	isl_int_clear(gcd);
+
+	return mat;
+}
+
+__isl_give isl_mat *isl_mat_normalize_row(__isl_take isl_mat *mat, int row)
+{
+	mat = isl_mat_cow(mat);
+	if (!mat)
+		return NULL;
+
+	isl_seq_normalize(mat->ctx, mat->row[row], mat->n_col);
+
+	return mat;
+}
+
+/* Number of initial non-zero columns.
+ */
+int isl_mat_initial_non_zero_cols(__isl_keep isl_mat *mat)
+{
+	int i;
+
+	if (!mat)
+		return -1;
+
+	for (i = 0; i < mat->n_col; ++i)
+		if (row_first_non_zero(mat->row, mat->n_row, i) < 0)
+			break;
+
+	return i;
+}
diff --git a/polly/lib/External/isl/isl_mat_private.h b/polly/lib/External/isl/isl_mat_private.h
new file mode 100644
index 00000000000..566634d352b
--- /dev/null
+++ b/polly/lib/External/isl/isl_mat_private.h
@@ -0,0 +1,46 @@
+#include <isl/mat.h>
+#include <isl_blk.h>
+
+struct isl_mat {
+	int ref;
+
+	struct isl_ctx *ctx;
+
+#define ISL_MAT_BORROWED		(1 << 0)
+	unsigned flags;
+
+	unsigned n_row;
+	unsigned n_col;
+
+	isl_int **row;
+
+	/* actual size of the rows in memory; n_col <= max_col */
+	unsigned max_col;
+
+	struct isl_blk block;
+};
+
+__isl_give isl_mat *isl_mat_sub_alloc(__isl_keep isl_mat *mat,
+	unsigned first_row, unsigned n_row, unsigned first_col, unsigned n_col);
+__isl_give isl_mat *isl_mat_sub_alloc6(isl_ctx *ctx, isl_int **row,
+	unsigned first_row, unsigned n_row, unsigned first_col, unsigned n_col);
+void isl_mat_sub_copy(struct isl_ctx *ctx, isl_int **dst, isl_int **src,
+	unsigned n_row, unsigned dst_col, unsigned src_col, unsigned n_col);
+void isl_mat_sub_neg(struct isl_ctx *ctx, isl_int **dst, isl_int **src,
+	unsigned n_row, unsigned dst_col, unsigned src_col, unsigned n_col);
+__isl_give isl_mat *isl_mat_diag(isl_ctx *ctx, unsigned n_row, isl_int d);
+
+__isl_give isl_mat *isl_mat_scale_down_row(__isl_take isl_mat *mat, int row,
+	isl_int m);
+
+__isl_give isl_vec *isl_mat_get_row(__isl_keep isl_mat *mat, unsigned row);
+
+int isl_mat_is_scaled_identity(__isl_keep isl_mat *mat);
+
+void isl_mat_col_mul(struct isl_mat *mat, int dst_col, isl_int f, int src_col);
+void isl_mat_col_submul(struct isl_mat *mat,
+			int dst_col, isl_int f, int src_col);
+
+int isl_mat_get_element(__isl_keep isl_mat *mat, int row, int col, isl_int *v);
+__isl_give isl_mat *isl_mat_set_element(__isl_take isl_mat *mat,
+	int row, int col, isl_int v);
diff --git a/polly/lib/External/isl/isl_morph.c b/polly/lib/External/isl/isl_morph.c
new file mode 100644
index 00000000000..9030cab1bda
--- /dev/null
+++ b/polly/lib/External/isl/isl_morph.c
@@ -0,0 +1,917 @@
+/*
+ * Copyright 2010-2011 INRIA Saclay
+ * Copyright 2014      Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France 
+ * and Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France
+ */
+
+#include <isl_map_private.h>
+#include <isl_aff_private.h>
+#include <isl_morph.h>
+#include <isl_seq.h>
+#include <isl_mat_private.h>
+#include <isl_space_private.h>
+#include <isl_equalities.h>
+
+isl_ctx *isl_morph_get_ctx(__isl_keep isl_morph *morph)
+{
+	if (!morph)
+		return NULL;
+	return isl_basic_set_get_ctx(morph->dom);
+}
+
+__isl_give isl_morph *isl_morph_alloc(
+	__isl_take isl_basic_set *dom, __isl_take isl_basic_set *ran,
+	__isl_take isl_mat *map, __isl_take isl_mat *inv)
+{
+	isl_morph *morph;
+
+	if (!dom || !ran || !map || !inv)
+		goto error;
+
+	morph = isl_alloc_type(dom->ctx, struct isl_morph);
+	if (!morph)
+		goto error;
+
+	morph->ref = 1;
+	morph->dom = dom;
+	morph->ran = ran;
+	morph->map = map;
+	morph->inv = inv;
+
+	return morph;
+error:
+	isl_basic_set_free(dom);
+	isl_basic_set_free(ran);
+	isl_mat_free(map);
+	isl_mat_free(inv);
+	return NULL;
+}
+
+__isl_give isl_morph *isl_morph_copy(__isl_keep isl_morph *morph)
+{
+	if (!morph)
+		return NULL;
+
+	morph->ref++;
+	return morph;
+}
+
+__isl_give isl_morph *isl_morph_dup(__isl_keep isl_morph *morph)
+{
+	if (!morph)
+		return NULL;
+
+	return isl_morph_alloc(isl_basic_set_copy(morph->dom),
+		isl_basic_set_copy(morph->ran),
+		isl_mat_copy(morph->map), isl_mat_copy(morph->inv));
+}
+
+__isl_give isl_morph *isl_morph_cow(__isl_take isl_morph *morph)
+{
+	if (!morph)
+		return NULL;
+
+	if (morph->ref == 1)
+		return morph;
+	morph->ref--;
+	return isl_morph_dup(morph);
+}
+
+void isl_morph_free(__isl_take isl_morph *morph)
+{
+	if (!morph)
+		return;
+
+	if (--morph->ref > 0)
+		return;
+
+	isl_basic_set_free(morph->dom);
+	isl_basic_set_free(morph->ran);
+	isl_mat_free(morph->map);
+	isl_mat_free(morph->inv);
+	free(morph);
+}
+
+/* Is "morph" an identity on the parameters?
+ */
+static int identity_on_parameters(__isl_keep isl_morph *morph)
+{
+	int is_identity;
+	unsigned nparam;
+	isl_mat *sub;
+
+	nparam = isl_morph_dom_dim(morph, isl_dim_param);
+	if (nparam != isl_morph_ran_dim(morph, isl_dim_param))
+		return 0;
+	if (nparam == 0)
+		return 1;
+	sub = isl_mat_sub_alloc(morph->map, 0, 1 + nparam, 0, 1 + nparam);
+	is_identity = isl_mat_is_scaled_identity(sub);
+	isl_mat_free(sub);
+
+	return is_identity;
+}
+
+/* Return an affine expression of the variables of the range of "morph"
+ * in terms of the parameters and the variables of the domain on "morph".
+ *
+ * In order for the space manipulations to make sense, we require
+ * that the parameters are not modified by "morph".
+ */
+__isl_give isl_multi_aff *isl_morph_get_var_multi_aff(
+	__isl_keep isl_morph *morph)
+{
+	isl_space *dom, *ran, *space;
+	isl_local_space *ls;
+	isl_multi_aff *ma;
+	unsigned nparam, nvar;
+	int i;
+	int is_identity;
+
+	if (!morph)
+		return NULL;
+
+	is_identity = identity_on_parameters(morph);
+	if (is_identity < 0)
+		return NULL;
+	if (!is_identity)
+		isl_die(isl_morph_get_ctx(morph), isl_error_invalid,
+			"cannot handle parameter compression", return NULL);
+
+	dom = isl_morph_get_dom_space(morph);
+	ls = isl_local_space_from_space(isl_space_copy(dom));
+	ran = isl_morph_get_ran_space(morph);
+	space = isl_space_map_from_domain_and_range(dom, ran);
+	ma = isl_multi_aff_zero(space);
+
+	nparam = isl_multi_aff_dim(ma, isl_dim_param);
+	nvar = isl_multi_aff_dim(ma, isl_dim_out);
+	for (i = 0; i < nvar; ++i) {
+		isl_val *val;
+		isl_vec *v;
+		isl_aff *aff;
+
+		v = isl_mat_get_row(morph->map, 1 + nparam + i);
+		v = isl_vec_insert_els(v, 0, 1);
+		val = isl_mat_get_element_val(morph->map, 0, 0);
+		v = isl_vec_set_element_val(v, 0, val);
+		aff = isl_aff_alloc_vec(isl_local_space_copy(ls), v);
+		ma = isl_multi_aff_set_aff(ma, i, aff);
+	}
+
+	isl_local_space_free(ls);
+	return ma;
+}
+
+/* Return the domain space of "morph".
+ */
+__isl_give isl_space *isl_morph_get_dom_space(__isl_keep isl_morph *morph)
+{
+	if (!morph)
+		return NULL;
+
+	return isl_basic_set_get_space(morph->dom);
+}
+
+__isl_give isl_space *isl_morph_get_ran_space(__isl_keep isl_morph *morph)
+{
+	if (!morph)
+		return NULL;
+	
+	return isl_space_copy(morph->ran->dim);
+}
+
+unsigned isl_morph_dom_dim(__isl_keep isl_morph *morph, enum isl_dim_type type)
+{
+	if (!morph)
+		return 0;
+
+	return isl_basic_set_dim(morph->dom, type);
+}
+
+unsigned isl_morph_ran_dim(__isl_keep isl_morph *morph, enum isl_dim_type type)
+{
+	if (!morph)
+		return 0;
+
+	return isl_basic_set_dim(morph->ran, type);
+}
+
+__isl_give isl_morph *isl_morph_remove_dom_dims(__isl_take isl_morph *morph,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	unsigned dom_offset;
+
+	if (n == 0)
+		return morph;
+
+	morph = isl_morph_cow(morph);
+	if (!morph)
+		return NULL;
+
+	dom_offset = 1 + isl_space_offset(morph->dom->dim, type);
+
+	morph->dom = isl_basic_set_remove_dims(morph->dom, type, first, n);
+
+	morph->map = isl_mat_drop_cols(morph->map, dom_offset + first, n);
+
+	morph->inv = isl_mat_drop_rows(morph->inv, dom_offset + first, n);
+
+	if (morph->dom && morph->ran && morph->map && morph->inv)
+		return morph;
+
+	isl_morph_free(morph);
+	return NULL;
+}
+
+__isl_give isl_morph *isl_morph_remove_ran_dims(__isl_take isl_morph *morph,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	unsigned ran_offset;
+
+	if (n == 0)
+		return morph;
+
+	morph = isl_morph_cow(morph);
+	if (!morph)
+		return NULL;
+
+	ran_offset = 1 + isl_space_offset(morph->ran->dim, type);
+
+	morph->ran = isl_basic_set_remove_dims(morph->ran, type, first, n);
+
+	morph->map = isl_mat_drop_rows(morph->map, ran_offset + first, n);
+
+	morph->inv = isl_mat_drop_cols(morph->inv, ran_offset + first, n);
+
+	if (morph->dom && morph->ran && morph->map && morph->inv)
+		return morph;
+
+	isl_morph_free(morph);
+	return NULL;
+}
+
+/* Project domain of morph onto its parameter domain.
+ */
+__isl_give isl_morph *isl_morph_dom_params(__isl_take isl_morph *morph)
+{
+	unsigned n;
+
+	morph = isl_morph_cow(morph);
+	if (!morph)
+		return NULL;
+	n = isl_basic_set_dim(morph->dom, isl_dim_set);
+	morph = isl_morph_remove_dom_dims(morph, isl_dim_set, 0, n);
+	if (!morph)
+		return NULL;
+	morph->dom = isl_basic_set_params(morph->dom);
+	if (morph->dom)
+		return morph;
+
+	isl_morph_free(morph);
+	return NULL;
+}
+
+/* Project range of morph onto its parameter domain.
+ */
+__isl_give isl_morph *isl_morph_ran_params(__isl_take isl_morph *morph)
+{
+	unsigned n;
+
+	morph = isl_morph_cow(morph);
+	if (!morph)
+		return NULL;
+	n = isl_basic_set_dim(morph->ran, isl_dim_set);
+	morph = isl_morph_remove_ran_dims(morph, isl_dim_set, 0, n);
+	if (!morph)
+		return NULL;
+	morph->ran = isl_basic_set_params(morph->ran);
+	if (morph->ran)
+		return morph;
+
+	isl_morph_free(morph);
+	return NULL;
+}
+
+void isl_morph_print_internal(__isl_take isl_morph *morph, FILE *out)
+{
+	if (!morph)
+		return;
+
+	isl_basic_set_print(morph->dom, out, 0, "", "", ISL_FORMAT_ISL);
+	isl_basic_set_print(morph->ran, out, 0, "", "", ISL_FORMAT_ISL);
+	isl_mat_print_internal(morph->map, out, 4);
+	isl_mat_print_internal(morph->inv, out, 4);
+}
+
+void isl_morph_dump(__isl_take isl_morph *morph)
+{
+	isl_morph_print_internal(morph, stderr);
+}
+
+__isl_give isl_morph *isl_morph_identity(__isl_keep isl_basic_set *bset)
+{
+	isl_mat *id;
+	isl_basic_set *universe;
+	unsigned total;
+
+	if (!bset)
+		return NULL;
+
+	total = isl_basic_set_total_dim(bset);
+	id = isl_mat_identity(bset->ctx, 1 + total);
+	universe = isl_basic_set_universe(isl_space_copy(bset->dim));
+
+	return isl_morph_alloc(universe, isl_basic_set_copy(universe),
+		id, isl_mat_copy(id));
+}
+
+/* Create a(n identity) morphism between empty sets of the same dimension
+ * a "bset".
+ */
+__isl_give isl_morph *isl_morph_empty(__isl_keep isl_basic_set *bset)
+{
+	isl_mat *id;
+	isl_basic_set *empty;
+	unsigned total;
+
+	if (!bset)
+		return NULL;
+
+	total = isl_basic_set_total_dim(bset);
+	id = isl_mat_identity(bset->ctx, 1 + total);
+	empty = isl_basic_set_empty(isl_space_copy(bset->dim));
+
+	return isl_morph_alloc(empty, isl_basic_set_copy(empty),
+		id, isl_mat_copy(id));
+}
+
+/* Given a matrix that maps a (possibly) parametric domain to
+ * a parametric domain, add in rows that map the "nparam" parameters onto
+ * themselves.
+ */
+static __isl_give isl_mat *insert_parameter_rows(__isl_take isl_mat *mat,
+	unsigned nparam)
+{
+	int i;
+
+	if (nparam == 0)
+		return mat;
+	if (!mat)
+		return NULL;
+
+	mat = isl_mat_insert_rows(mat, 1, nparam);
+	if (!mat)
+		return NULL;
+
+	for (i = 0; i < nparam; ++i) {
+		isl_seq_clr(mat->row[1 + i], mat->n_col);
+		isl_int_set(mat->row[1 + i][1 + i], mat->row[0][0]);
+	}
+
+	return mat;
+}
+
+/* Construct a basic set described by the "n" equalities of "bset" starting
+ * at "first".
+ */
+static __isl_give isl_basic_set *copy_equalities(__isl_keep isl_basic_set *bset,
+	unsigned first, unsigned n)
+{
+	int i, k;
+	isl_basic_set *eq;
+	unsigned total;
+
+	isl_assert(bset->ctx, bset->n_div == 0, return NULL);
+
+	total = isl_basic_set_total_dim(bset);
+	eq = isl_basic_set_alloc_space(isl_space_copy(bset->dim), 0, n, 0);
+	if (!eq)
+		return NULL;
+	for (i = 0; i < n; ++i) {
+		k = isl_basic_set_alloc_equality(eq);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(eq->eq[k], bset->eq[first + k], 1 + total);
+	}
+
+	return eq;
+error:
+	isl_basic_set_free(eq);
+	return NULL;
+}
+
+/* Given a basic set, exploit the equalties in the basic set to construct
+ * a morphishm that maps the basic set to a lower-dimensional space.
+ * Specifically, the morphism reduces the number of dimensions of type "type".
+ *
+ * This function is a slight generalization of isl_mat_variable_compression
+ * in that it allows the input to be parametric and that it allows for the
+ * compression of either parameters or set variables.
+ *
+ * We first select the equalities of interest, that is those that involve
+ * variables of type "type" and no later variables.
+ * Denote those equalities as
+ *
+ *		-C(p) + M x = 0
+ *
+ * where C(p) depends on the parameters if type == isl_dim_set and
+ * is a constant if type == isl_dim_param.
+ *
+ * First compute the (left) Hermite normal form of M,
+ *
+ *		M [U1 U2] = M U = H = [H1 0]
+ * or
+ *		              M = H Q = [H1 0] [Q1]
+ *                                             [Q2]
+ *
+ * with U, Q unimodular, Q = U^{-1} (and H lower triangular).
+ * Define the transformed variables as
+ *
+ *		x = [U1 U2] [ x1' ] = [U1 U2] [Q1] x
+ *		            [ x2' ]           [Q2]
+ *
+ * The equalities then become
+ *
+ *		-C(p) + H1 x1' = 0   or   x1' = H1^{-1} C(p) = C'(p)
+ *
+ * If the denominator of the constant term does not divide the
+ * the common denominator of the parametric terms, then every
+ * integer point is mapped to a non-integer point and then the original set has no
+ * integer solutions (since the x' are a unimodular transformation
+ * of the x).  In this case, an empty morphism is returned.
+ * Otherwise, the transformation is given by
+ *
+ *		x = U1 H1^{-1} C(p) + U2 x2'
+ *
+ * The inverse transformation is simply
+ *
+ *		x2' = Q2 x
+ *
+ * Both matrices are extended to map the full original space to the full
+ * compressed space.
+ */
+__isl_give isl_morph *isl_basic_set_variable_compression(
+	__isl_keep isl_basic_set *bset, enum isl_dim_type type)
+{
+	unsigned otype;
+	unsigned ntype;
+	unsigned orest;
+	unsigned nrest;
+	int f_eq, n_eq;
+	isl_space *dim;
+	isl_mat *H, *U, *Q, *C = NULL, *H1, *U1, *U2;
+	isl_basic_set *dom, *ran;
+
+	if (!bset)
+		return NULL;
+
+	if (isl_basic_set_plain_is_empty(bset))
+		return isl_morph_empty(bset);
+
+	isl_assert(bset->ctx, bset->n_div == 0, return NULL);
+
+	otype = 1 + isl_space_offset(bset->dim, type);
+	ntype = isl_basic_set_dim(bset, type);
+	orest = otype + ntype;
+	nrest = isl_basic_set_total_dim(bset) - (orest - 1);
+
+	for (f_eq = 0; f_eq < bset->n_eq; ++f_eq)
+		if (isl_seq_first_non_zero(bset->eq[f_eq] + orest, nrest) == -1)
+			break;
+	for (n_eq = 0; f_eq + n_eq < bset->n_eq; ++n_eq)
+		if (isl_seq_first_non_zero(bset->eq[f_eq + n_eq] + otype, ntype) == -1)
+			break;
+	if (n_eq == 0)
+		return isl_morph_identity(bset);
+
+	H = isl_mat_sub_alloc6(bset->ctx, bset->eq, f_eq, n_eq, otype, ntype);
+	H = isl_mat_left_hermite(H, 0, &U, &Q);
+	if (!H || !U || !Q)
+		goto error;
+	Q = isl_mat_drop_rows(Q, 0, n_eq);
+	Q = isl_mat_diagonal(isl_mat_identity(bset->ctx, otype), Q);
+	Q = isl_mat_diagonal(Q, isl_mat_identity(bset->ctx, nrest));
+	C = isl_mat_alloc(bset->ctx, 1 + n_eq, otype);
+	if (!C)
+		goto error;
+	isl_int_set_si(C->row[0][0], 1);
+	isl_seq_clr(C->row[0] + 1, otype - 1);
+	isl_mat_sub_neg(C->ctx, C->row + 1, bset->eq + f_eq, n_eq, 0, 0, otype);
+	H1 = isl_mat_sub_alloc(H, 0, H->n_row, 0, H->n_row);
+	H1 = isl_mat_lin_to_aff(H1);
+	C = isl_mat_inverse_product(H1, C);
+	if (!C)
+		goto error;
+	isl_mat_free(H);
+
+	if (!isl_int_is_one(C->row[0][0])) {
+		int i;
+		isl_int g;
+
+		isl_int_init(g);
+		for (i = 0; i < n_eq; ++i) {
+			isl_seq_gcd(C->row[1 + i] + 1, otype - 1, &g);
+			isl_int_gcd(g, g, C->row[0][0]);
+			if (!isl_int_is_divisible_by(C->row[1 + i][0], g))
+				break;
+		}
+		isl_int_clear(g);
+
+		if (i < n_eq) {
+			isl_mat_free(C);
+			isl_mat_free(U);
+			isl_mat_free(Q);
+			return isl_morph_empty(bset);
+		}
+
+		C = isl_mat_normalize(C);
+	}
+
+	U1 = isl_mat_sub_alloc(U, 0, U->n_row, 0, n_eq);
+	U1 = isl_mat_lin_to_aff(U1);
+	U2 = isl_mat_sub_alloc(U, 0, U->n_row, n_eq, U->n_row - n_eq);
+	U2 = isl_mat_lin_to_aff(U2);
+	isl_mat_free(U);
+
+	C = isl_mat_product(U1, C);
+	C = isl_mat_aff_direct_sum(C, U2);
+	C = insert_parameter_rows(C, otype - 1);
+	C = isl_mat_diagonal(C, isl_mat_identity(bset->ctx, nrest));
+
+	dim = isl_space_copy(bset->dim);
+	dim = isl_space_drop_dims(dim, type, 0, ntype);
+	dim = isl_space_add_dims(dim, type, ntype - n_eq);
+	ran = isl_basic_set_universe(dim);
+	dom = copy_equalities(bset, f_eq, n_eq);
+
+	return isl_morph_alloc(dom, ran, Q, C);
+error:
+	isl_mat_free(C);
+	isl_mat_free(H);
+	isl_mat_free(U);
+	isl_mat_free(Q);
+	return NULL;
+}
+
+/* Construct a parameter compression for "bset".
+ * We basically just call isl_mat_parameter_compression with the right input
+ * and then extend the resulting matrix to include the variables.
+ *
+ * The implementation assumes that "bset" does not have any equalities
+ * that only involve the parameters and that isl_basic_set_gauss has
+ * been applied to "bset".
+ *
+ * Let the equalities be given as
+ *
+ *	B(p) + A x = 0.
+ *
+ * We use isl_mat_parameter_compression_ext to compute the compression
+ *
+ *	p = T p'.
+ */
+__isl_give isl_morph *isl_basic_set_parameter_compression(
+	__isl_keep isl_basic_set *bset)
+{
+	unsigned nparam;
+	unsigned nvar;
+	unsigned n_div;
+	int n_eq;
+	isl_mat *H, *B;
+	isl_mat *map, *inv;
+	isl_basic_set *dom, *ran;
+
+	if (!bset)
+		return NULL;
+
+	if (isl_basic_set_plain_is_empty(bset))
+		return isl_morph_empty(bset);
+	if (bset->n_eq == 0)
+		return isl_morph_identity(bset);
+
+	n_eq = bset->n_eq;
+	nparam = isl_basic_set_dim(bset, isl_dim_param);
+	nvar = isl_basic_set_dim(bset, isl_dim_set);
+	n_div = isl_basic_set_dim(bset, isl_dim_div);
+
+	if (isl_seq_first_non_zero(bset->eq[bset->n_eq - 1] + 1 + nparam,
+				    nvar + n_div) == -1)
+		isl_die(isl_basic_set_get_ctx(bset), isl_error_invalid,
+			"input not allowed to have parameter equalities",
+			return NULL);
+	if (n_eq > nvar + n_div)
+		isl_die(isl_basic_set_get_ctx(bset), isl_error_invalid,
+			"input not gaussed", return NULL);
+
+	B = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, n_eq, 0, 1 + nparam);
+	H = isl_mat_sub_alloc6(bset->ctx, bset->eq,
+				0, n_eq, 1 + nparam, nvar + n_div);
+	inv = isl_mat_parameter_compression_ext(B, H);
+	inv = isl_mat_diagonal(inv, isl_mat_identity(bset->ctx, nvar));
+	map = isl_mat_right_inverse(isl_mat_copy(inv));
+
+	dom = isl_basic_set_universe(isl_space_copy(bset->dim));
+	ran = isl_basic_set_universe(isl_space_copy(bset->dim));
+
+	return isl_morph_alloc(dom, ran, map, inv);
+}
+
+/* Add stride constraints to "bset" based on the inverse mapping
+ * that was plugged in.  In particular, if morph maps x' to x,
+ * the the constraints of the original input
+ *
+ *	A x' + b >= 0
+ *
+ * have been rewritten to
+ *
+ *	A inv x + b >= 0
+ *
+ * However, this substitution may loose information on the integrality of x',
+ * so we need to impose that
+ *
+ *	inv x
+ *
+ * is integral.  If inv = B/d, this means that we need to impose that
+ *
+ *	B x = 0		mod d
+ *
+ * or
+ *
+ *	exists alpha in Z^m: B x = d alpha
+ *
+ * This function is similar to add_strides in isl_affine_hull.c
+ */
+static __isl_give isl_basic_set *add_strides(__isl_take isl_basic_set *bset,
+	__isl_keep isl_morph *morph)
+{
+	int i, div, k;
+	isl_int gcd;
+
+	if (isl_int_is_one(morph->inv->row[0][0]))
+		return bset;
+
+	isl_int_init(gcd);
+
+	for (i = 0; 1 + i < morph->inv->n_row; ++i) {
+		isl_seq_gcd(morph->inv->row[1 + i], morph->inv->n_col, &gcd);
+		if (isl_int_is_divisible_by(gcd, morph->inv->row[0][0]))
+			continue;
+		div = isl_basic_set_alloc_div(bset);
+		if (div < 0)
+			goto error;
+		isl_int_set_si(bset->div[div][0], 0);
+		k = isl_basic_set_alloc_equality(bset);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(bset->eq[k], morph->inv->row[1 + i],
+			    morph->inv->n_col);
+		isl_seq_clr(bset->eq[k] + morph->inv->n_col, bset->n_div);
+		isl_int_set(bset->eq[k][morph->inv->n_col + div],
+			    morph->inv->row[0][0]);
+	}
+
+	isl_int_clear(gcd);
+
+	return bset;
+error:
+	isl_int_clear(gcd);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Apply the morphism to the basic set.
+ * We basically just compute the preimage of "bset" under the inverse mapping
+ * in morph, add in stride constraints and intersect with the range
+ * of the morphism.
+ */
+__isl_give isl_basic_set *isl_morph_basic_set(__isl_take isl_morph *morph,
+	__isl_take isl_basic_set *bset)
+{
+	isl_basic_set *res = NULL;
+	isl_mat *mat = NULL;
+	int i, k;
+	int max_stride;
+
+	if (!morph || !bset)
+		goto error;
+
+	isl_assert(bset->ctx, isl_space_is_equal(bset->dim, morph->dom->dim),
+		    goto error);
+
+	max_stride = morph->inv->n_row - 1;
+	if (isl_int_is_one(morph->inv->row[0][0]))
+		max_stride = 0;
+	res = isl_basic_set_alloc_space(isl_space_copy(morph->ran->dim),
+		bset->n_div + max_stride, bset->n_eq + max_stride, bset->n_ineq);
+
+	for (i = 0; i < bset->n_div; ++i)
+		if (isl_basic_set_alloc_div(res) < 0)
+			goto error;
+
+	mat = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, bset->n_eq,
+					0, morph->inv->n_row);
+	mat = isl_mat_product(mat, isl_mat_copy(morph->inv));
+	if (!mat)
+		goto error;
+	for (i = 0; i < bset->n_eq; ++i) {
+		k = isl_basic_set_alloc_equality(res);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(res->eq[k], mat->row[i], mat->n_col);
+		isl_seq_scale(res->eq[k] + mat->n_col, bset->eq[i] + mat->n_col,
+				morph->inv->row[0][0], bset->n_div);
+	}
+	isl_mat_free(mat);
+
+	mat = isl_mat_sub_alloc6(bset->ctx, bset->ineq, 0, bset->n_ineq,
+					0, morph->inv->n_row);
+	mat = isl_mat_product(mat, isl_mat_copy(morph->inv));
+	if (!mat)
+		goto error;
+	for (i = 0; i < bset->n_ineq; ++i) {
+		k = isl_basic_set_alloc_inequality(res);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(res->ineq[k], mat->row[i], mat->n_col);
+		isl_seq_scale(res->ineq[k] + mat->n_col,
+				bset->ineq[i] + mat->n_col,
+				morph->inv->row[0][0], bset->n_div);
+	}
+	isl_mat_free(mat);
+
+	mat = isl_mat_sub_alloc6(bset->ctx, bset->div, 0, bset->n_div,
+					1, morph->inv->n_row);
+	mat = isl_mat_product(mat, isl_mat_copy(morph->inv));
+	if (!mat)
+		goto error;
+	for (i = 0; i < bset->n_div; ++i) {
+		isl_int_mul(res->div[i][0],
+				morph->inv->row[0][0], bset->div[i][0]);
+		isl_seq_cpy(res->div[i] + 1, mat->row[i], mat->n_col);
+		isl_seq_scale(res->div[i] + 1 + mat->n_col,
+				bset->div[i] + 1 + mat->n_col,
+				morph->inv->row[0][0], bset->n_div);
+	}
+	isl_mat_free(mat);
+
+	res = add_strides(res, morph);
+
+	if (isl_basic_set_is_rational(bset))
+		res = isl_basic_set_set_rational(res);
+
+	res = isl_basic_set_simplify(res);
+	res = isl_basic_set_finalize(res);
+
+	res = isl_basic_set_intersect(res, isl_basic_set_copy(morph->ran));
+
+	isl_morph_free(morph);
+	isl_basic_set_free(bset);
+	return res;
+error:
+	isl_mat_free(mat);
+	isl_morph_free(morph);
+	isl_basic_set_free(bset);
+	isl_basic_set_free(res);
+	return NULL;
+}
+
+/* Apply the morphism to the set.
+ */
+__isl_give isl_set *isl_morph_set(__isl_take isl_morph *morph,
+	__isl_take isl_set *set)
+{
+	int i;
+
+	if (!morph || !set)
+		goto error;
+
+	isl_assert(set->ctx, isl_space_is_equal(set->dim, morph->dom->dim), goto error);
+
+	set = isl_set_cow(set);
+	if (!set)
+		goto error;
+
+	isl_space_free(set->dim);
+	set->dim = isl_space_copy(morph->ran->dim);
+	if (!set->dim)
+		goto error;
+
+	for (i = 0; i < set->n; ++i) {
+		set->p[i] = isl_morph_basic_set(isl_morph_copy(morph), set->p[i]);
+		if (!set->p[i])
+			goto error;
+	}
+
+	isl_morph_free(morph);
+
+	ISL_F_CLR(set, ISL_SET_NORMALIZED);
+
+	return set;
+error:
+	isl_set_free(set);
+	isl_morph_free(morph);
+	return NULL;
+}
+
+/* Construct a morphism that first does morph2 and then morph1.
+ */
+__isl_give isl_morph *isl_morph_compose(__isl_take isl_morph *morph1,
+	__isl_take isl_morph *morph2)
+{
+	isl_mat *map, *inv;
+	isl_basic_set *dom, *ran;
+
+	if (!morph1 || !morph2)
+		goto error;
+
+	map = isl_mat_product(isl_mat_copy(morph1->map), isl_mat_copy(morph2->map));
+	inv = isl_mat_product(isl_mat_copy(morph2->inv), isl_mat_copy(morph1->inv));
+	dom = isl_morph_basic_set(isl_morph_inverse(isl_morph_copy(morph2)),
+				  isl_basic_set_copy(morph1->dom));
+	dom = isl_basic_set_intersect(dom, isl_basic_set_copy(morph2->dom));
+	ran = isl_morph_basic_set(isl_morph_copy(morph1),
+				  isl_basic_set_copy(morph2->ran));
+	ran = isl_basic_set_intersect(ran, isl_basic_set_copy(morph1->ran));
+
+	isl_morph_free(morph1);
+	isl_morph_free(morph2);
+
+	return isl_morph_alloc(dom, ran, map, inv);
+error:
+	isl_morph_free(morph1);
+	isl_morph_free(morph2);
+	return NULL;
+}
+
+__isl_give isl_morph *isl_morph_inverse(__isl_take isl_morph *morph)
+{
+	isl_basic_set *bset;
+	isl_mat *mat;
+
+	morph = isl_morph_cow(morph);
+	if (!morph)
+		return NULL;
+
+	bset = morph->dom;
+	morph->dom = morph->ran;
+	morph->ran = bset;
+
+	mat = morph->map;
+	morph->map = morph->inv;
+	morph->inv = mat;
+
+	return morph;
+}
+
+/* We detect all the equalities first to avoid implicit equalties
+ * being discovered during the computations.  In particular,
+ * the compression on the variables could expose additional stride
+ * constraints on the parameters.  This would result in existentially
+ * quantified variables after applying the resulting morph, which
+ * in turn could break invariants of the calling functions.
+ */
+__isl_give isl_morph *isl_basic_set_full_compression(
+	__isl_keep isl_basic_set *bset)
+{
+	isl_morph *morph, *morph2;
+
+	bset = isl_basic_set_copy(bset);
+	bset = isl_basic_set_detect_equalities(bset);
+
+	morph = isl_basic_set_variable_compression(bset, isl_dim_param);
+	bset = isl_morph_basic_set(isl_morph_copy(morph), bset);
+
+	morph2 = isl_basic_set_parameter_compression(bset);
+	bset = isl_morph_basic_set(isl_morph_copy(morph2), bset);
+
+	morph = isl_morph_compose(morph2, morph);
+
+	morph2 = isl_basic_set_variable_compression(bset, isl_dim_set);
+	isl_basic_set_free(bset);
+
+	morph = isl_morph_compose(morph2, morph);
+
+	return morph;
+}
+
+__isl_give isl_vec *isl_morph_vec(__isl_take isl_morph *morph,
+	__isl_take isl_vec *vec)
+{
+	if (!morph)
+		goto error;
+
+	vec = isl_mat_vec_product(isl_mat_copy(morph->map), vec);
+
+	isl_morph_free(morph);
+	return vec;
+error:
+	isl_morph_free(morph);
+	isl_vec_free(vec);
+	return NULL;
+}
diff --git a/polly/lib/External/isl/isl_morph.h b/polly/lib/External/isl/isl_morph.h
new file mode 100644
index 00000000000..4f6b8d491be
--- /dev/null
+++ b/polly/lib/External/isl/isl_morph.h
@@ -0,0 +1,86 @@
+/*
+ * Copyright 2010      INRIA Saclay
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France 
+ */
+
+#ifndef ISL_MORHP_H
+#define ISL_MORHP_H
+
+#include <stdio.h>
+#include <isl/space.h>
+#include <isl/mat.h>
+#include <isl/set.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+/* An isl_morph is a "morphism" on (basic) sets.
+ * "map" is an affine mapping from "dom" to "ran"
+ * and "inv" is the inverse mapping.
+ */
+struct isl_morph {
+	int ref;
+
+	isl_basic_set *dom;
+	isl_basic_set *ran;
+
+	isl_mat *map;
+	isl_mat *inv;
+};
+typedef struct isl_morph isl_morph;
+
+isl_ctx *isl_morph_get_ctx(__isl_keep isl_morph *morph);
+
+__isl_give isl_morph *isl_morph_alloc(
+	__isl_take isl_basic_set *dom, __isl_take isl_basic_set *ran,
+	__isl_take isl_mat *map, __isl_take isl_mat *inv);
+__isl_give isl_morph *isl_morph_copy(__isl_keep isl_morph *morph);
+__isl_give isl_morph *isl_morph_identity(__isl_keep isl_basic_set *bset);
+void isl_morph_free(__isl_take isl_morph *morph);
+
+__isl_give isl_space *isl_morph_get_dom_space(__isl_keep isl_morph *morph);
+__isl_give isl_space *isl_morph_get_ran_space(__isl_keep isl_morph *morph);
+__isl_give isl_multi_aff *isl_morph_get_var_multi_aff(
+	__isl_keep isl_morph *morph);
+unsigned isl_morph_dom_dim(__isl_keep isl_morph *morph, enum isl_dim_type type);
+unsigned isl_morph_ran_dim(__isl_keep isl_morph *morph, enum isl_dim_type type);
+
+__isl_give isl_morph *isl_morph_remove_dom_dims(__isl_take isl_morph *morph,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_morph *isl_morph_remove_ran_dims(__isl_take isl_morph *morph,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_morph *isl_morph_dom_params(__isl_take isl_morph *morph);
+__isl_give isl_morph *isl_morph_ran_params(__isl_take isl_morph *morph);
+
+__isl_give isl_morph *isl_morph_compose(__isl_take isl_morph *morph1,
+	__isl_take isl_morph *morph2);
+__isl_give isl_morph *isl_morph_inverse(__isl_take isl_morph *morph);
+
+void isl_morph_print_internal(__isl_take isl_morph *morph, FILE *out);
+void isl_morph_dump(__isl_take isl_morph *morph);
+
+__isl_give isl_morph *isl_basic_set_variable_compression(
+	__isl_keep isl_basic_set *bset, enum isl_dim_type type);
+__isl_give isl_morph *isl_basic_set_parameter_compression(
+	__isl_keep isl_basic_set *bset);
+__isl_give isl_morph *isl_basic_set_full_compression(
+	__isl_keep isl_basic_set *bset);
+
+__isl_give isl_basic_set *isl_morph_basic_set(__isl_take isl_morph *morph,
+	__isl_take isl_basic_set *bset);
+__isl_give isl_set *isl_morph_set(__isl_take isl_morph *morph,
+	__isl_take isl_set *set);
+__isl_give isl_vec *isl_morph_vec(__isl_take isl_morph *morph,
+	__isl_take isl_vec *vec);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/isl_multi_templ.c b/polly/lib/External/isl/isl_multi_templ.c
new file mode 100644
index 00000000000..69dc93c3007
--- /dev/null
+++ b/polly/lib/External/isl/isl_multi_templ.c
@@ -0,0 +1,1498 @@
+/*
+ * Copyright 2011      Sven Verdoolaege
+ * Copyright 2012-2013 Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege,
+ * Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+#include <isl_space_private.h>
+#include <isl/set.h>
+#include <isl_reordering.h>
+
+#define xCAT(A,B) A ## B
+#define CAT(A,B) xCAT(A,B)
+#undef EL
+#define EL CAT(isl_,BASE)
+#define xFN(TYPE,NAME) TYPE ## _ ## NAME
+#define FN(TYPE,NAME) xFN(TYPE,NAME)
+#define xMULTI(BASE) isl_multi_ ## BASE
+#define MULTI(BASE) xMULTI(BASE)
+#define MULTI_NAME(BASE) "isl_multi_" #BASE
+#define xLIST(EL) EL ## _list
+#define LIST(EL) xLIST(EL)
+
+isl_ctx *FN(MULTI(BASE),get_ctx)(__isl_keep MULTI(BASE) *multi)
+{
+	return multi ? isl_space_get_ctx(multi->space) : NULL;
+}
+
+__isl_give isl_space *FN(MULTI(BASE),get_space)(__isl_keep MULTI(BASE) *multi)
+{
+	return multi ? isl_space_copy(multi->space) : NULL;
+}
+
+/* Return the position of the dimension of the given type and name
+ * in "multi".
+ * Return -1 if no such dimension can be found.
+ */
+int FN(MULTI(BASE),find_dim_by_name)(__isl_keep MULTI(BASE) *multi,
+	enum isl_dim_type type, const char *name)
+{
+	if (!multi)
+		return -1;
+	return isl_space_find_dim_by_name(multi->space, type, name);
+}
+
+__isl_give isl_space *FN(MULTI(BASE),get_domain_space)(
+	__isl_keep MULTI(BASE) *multi)
+{
+	return multi ? isl_space_domain(isl_space_copy(multi->space)) : NULL;
+}
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),alloc)(__isl_take isl_space *space)
+{
+	isl_ctx *ctx;
+	int n;
+	MULTI(BASE) *multi;
+
+	if (!space)
+		return NULL;
+
+	ctx = isl_space_get_ctx(space);
+	n = isl_space_dim(space, isl_dim_out);
+	multi = isl_calloc(ctx, MULTI(BASE),
+			 sizeof(MULTI(BASE)) + (n - 1) * sizeof(struct EL *));
+	if (!multi)
+		goto error;
+
+	multi->space = space;
+	multi->n = n;
+	multi->ref = 1;
+	return multi;
+error:
+	isl_space_free(space);
+	return NULL;
+}
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),dup)(__isl_keep MULTI(BASE) *multi)
+{
+	int i;
+	MULTI(BASE) *dup;
+
+	if (!multi)
+		return NULL;
+
+	dup = FN(MULTI(BASE),alloc)(isl_space_copy(multi->space));
+	if (!dup)
+		return NULL;
+
+	for (i = 0; i < multi->n; ++i)
+		dup = FN(FN(MULTI(BASE),set),BASE)(dup, i,
+						    FN(EL,copy)(multi->p[i]));
+
+	return dup;
+}
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),cow)(__isl_take MULTI(BASE) *multi)
+{
+	if (!multi)
+		return NULL;
+
+	if (multi->ref == 1)
+		return multi;
+
+	multi->ref--;
+	return FN(MULTI(BASE),dup)(multi);
+}
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),copy)(__isl_keep MULTI(BASE) *multi)
+{
+	if (!multi)
+		return NULL;
+
+	multi->ref++;
+	return multi;
+}
+
+__isl_null MULTI(BASE) *FN(MULTI(BASE),free)(__isl_take MULTI(BASE) *multi)
+{
+	int i;
+
+	if (!multi)
+		return NULL;
+
+	if (--multi->ref > 0)
+		return NULL;
+
+	isl_space_free(multi->space);
+	for (i = 0; i < multi->n; ++i)
+		FN(EL,free)(multi->p[i]);
+	free(multi);
+
+	return NULL;
+}
+
+/* Check whether "multi" has non-zero coefficients for any dimension
+ * in the given range or if any of these dimensions appear
+ * with non-zero coefficients in any of the integer divisions involved.
+ */
+int FN(MULTI(BASE),involves_dims)(__isl_keep MULTI(BASE) *multi,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+
+	if (!multi)
+		return -1;
+	if (multi->n == 0 || n == 0)
+		return 0;
+
+	for (i = 0; i < multi->n; ++i) {
+		int involves;
+
+		involves = FN(EL,involves_dims)(multi->p[i], type, first, n);
+		if (involves < 0 || involves)
+			return involves;
+	}
+
+	return 0;
+}
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),insert_dims)(
+	__isl_take MULTI(BASE) *multi,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+
+	if (!multi)
+		return NULL;
+	if (type == isl_dim_out)
+		isl_die(FN(MULTI(BASE),get_ctx)(multi), isl_error_invalid,
+			"cannot insert output/set dimensions",
+			return FN(MULTI(BASE),free)(multi));
+	if (n == 0 && !isl_space_is_named_or_nested(multi->space, type))
+		return multi;
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi)
+		return NULL;
+
+	multi->space = isl_space_insert_dims(multi->space, type, first, n);
+	if (!multi->space)
+		return FN(MULTI(BASE),free)(multi);
+
+	for (i = 0; i < multi->n; ++i) {
+		multi->p[i] = FN(EL,insert_dims)(multi->p[i], type, first, n);
+		if (!multi->p[i])
+			return FN(MULTI(BASE),free)(multi);
+	}
+
+	return multi;
+}
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),add_dims)(__isl_take MULTI(BASE) *multi,
+	enum isl_dim_type type, unsigned n)
+{
+	unsigned pos;
+
+	pos = FN(MULTI(BASE),dim)(multi, type);
+
+	return FN(MULTI(BASE),insert_dims)(multi, type, pos, n);
+}
+
+unsigned FN(MULTI(BASE),dim)(__isl_keep MULTI(BASE) *multi,
+	enum isl_dim_type type)
+{
+	return multi ? isl_space_dim(multi->space, type) : 0;
+}
+
+/* Return the position of the first dimension of "type" with id "id".
+ * Return -1 if there is no such dimension.
+ */
+int FN(MULTI(BASE),find_dim_by_id)(__isl_keep MULTI(BASE) *multi,
+	enum isl_dim_type type, __isl_keep isl_id *id)
+{
+	if (!multi)
+		return -1;
+	return isl_space_find_dim_by_id(multi->space, type, id);
+}
+
+/* Return the id of the given dimension.
+ */
+__isl_give isl_id *FN(MULTI(BASE),get_dim_id)(__isl_keep MULTI(BASE) *multi,
+	enum isl_dim_type type, unsigned pos)
+{
+	return multi ? isl_space_get_dim_id(multi->space, type, pos) : NULL;
+}
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),set_dim_name)(
+	__isl_take MULTI(BASE) *multi,
+	enum isl_dim_type type, unsigned pos, const char *s)
+{
+	int i;
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi)
+		return NULL;
+
+	multi->space = isl_space_set_dim_name(multi->space, type, pos, s);
+	if (!multi->space)
+		return FN(MULTI(BASE),free)(multi);
+
+	if (type == isl_dim_out)
+		return multi;
+	for (i = 0; i < multi->n; ++i) {
+		multi->p[i] = FN(EL,set_dim_name)(multi->p[i], type, pos, s);
+		if (!multi->p[i])
+			return FN(MULTI(BASE),free)(multi);
+	}
+
+	return multi;
+}
+
+const char *FN(MULTI(BASE),get_tuple_name)(__isl_keep MULTI(BASE) *multi,
+	enum isl_dim_type type)
+{
+	return multi ? isl_space_get_tuple_name(multi->space, type) : NULL;
+}
+
+/* Does the specified tuple have an id?
+ */
+int FN(MULTI(BASE),has_tuple_id)(__isl_keep MULTI(BASE) *multi,
+	enum isl_dim_type type)
+{
+	return multi ? isl_space_has_tuple_id(multi->space, type) : -1;
+}
+
+/* Return the id of the specified tuple.
+ */
+__isl_give isl_id *FN(MULTI(BASE),get_tuple_id)(__isl_keep MULTI(BASE) *multi,
+	enum isl_dim_type type)
+{
+	return multi ? isl_space_get_tuple_id(multi->space, type) : NULL;
+}
+
+__isl_give EL *FN(FN(MULTI(BASE),get),BASE)(__isl_keep MULTI(BASE) *multi,
+	int pos)
+{
+	isl_ctx *ctx;
+
+	if (!multi)
+		return NULL;
+	ctx = FN(MULTI(BASE),get_ctx)(multi);
+	if (pos < 0 || pos >= multi->n)
+		isl_die(ctx, isl_error_invalid,
+			"index out of bounds", return NULL);
+	return FN(EL,copy)(multi->p[pos]);
+}
+
+__isl_give MULTI(BASE) *FN(FN(MULTI(BASE),set),BASE)(
+	__isl_take MULTI(BASE) *multi, int pos, __isl_take EL *el)
+{
+	isl_space *multi_space = NULL;
+	isl_space *el_space = NULL;
+	int match;
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi || !el)
+		goto error;
+
+	multi_space = FN(MULTI(BASE),get_space)(multi);
+	match = FN(EL,matching_params)(el, multi_space);
+	if (match < 0)
+		goto error;
+	if (!match) {
+		multi = FN(MULTI(BASE),align_params)(multi,
+						    FN(EL,get_space)(el));
+		isl_space_free(multi_space);
+		multi_space = FN(MULTI(BASE),get_space)(multi);
+		el = FN(EL,align_params)(el, isl_space_copy(multi_space));
+	}
+	if (FN(EL,check_match_domain_space)(el, multi_space) < 0)
+		goto error;
+
+	if (pos < 0 || pos >= multi->n)
+		isl_die(FN(MULTI(BASE),get_ctx)(multi), isl_error_invalid,
+			"index out of bounds", goto error);
+
+	FN(EL,free)(multi->p[pos]);
+	multi->p[pos] = el;
+
+	isl_space_free(multi_space);
+	isl_space_free(el_space);
+
+	return multi;
+error:
+	FN(MULTI(BASE),free)(multi);
+	FN(EL,free)(el);
+	isl_space_free(multi_space);
+	isl_space_free(el_space);
+	return NULL;
+}
+
+/* Reset the space of "multi".  This function is called from isl_pw_templ.c
+ * and doesn't know if the space of an element object is represented
+ * directly or through its domain.  It therefore passes along both,
+ * which we pass along to the element function since we don't how
+ * that is represented either.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),reset_space_and_domain)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_space *space,
+	__isl_take isl_space *domain)
+{
+	int i;
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi || !space || !domain)
+		goto error;
+
+	for (i = 0; i < multi->n; ++i) {
+		multi->p[i] = FN(EL,reset_domain_space)(multi->p[i],
+				 isl_space_copy(domain));
+		if (!multi->p[i])
+			goto error;
+	}
+	isl_space_free(domain);
+	isl_space_free(multi->space);
+	multi->space = space;
+
+	return multi;
+error:
+	isl_space_free(domain);
+	isl_space_free(space);
+	FN(MULTI(BASE),free)(multi);
+	return NULL;
+}
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),reset_domain_space)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_space *domain)
+{
+	isl_space *space;
+
+	space = isl_space_extend_domain_with_range(isl_space_copy(domain),
+						isl_space_copy(multi->space));
+	return FN(MULTI(BASE),reset_space_and_domain)(multi, space, domain);
+}
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),reset_space)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_space *space)
+{
+	isl_space *domain;
+
+	domain = isl_space_domain(isl_space_copy(space));
+	return FN(MULTI(BASE),reset_space_and_domain)(multi, space, domain);
+}
+
+/* Set the id of the given dimension of "multi" to "id".
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),set_dim_id)(
+	__isl_take MULTI(BASE) *multi,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id)
+{
+	isl_space *space;
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi || !id)
+		goto error;
+
+	space = FN(MULTI(BASE),get_space)(multi);
+	space = isl_space_set_dim_id(space, type, pos, id);
+
+	return FN(MULTI(BASE),reset_space)(multi, space);
+error:
+	isl_id_free(id);
+	FN(MULTI(BASE),free)(multi);
+	return NULL;
+}
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),set_tuple_name)(
+	__isl_keep MULTI(BASE) *multi, enum isl_dim_type type,
+	const char *s)
+{
+	isl_space *space;
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi)
+		return NULL;
+
+	space = FN(MULTI(BASE),get_space)(multi);
+	space = isl_space_set_tuple_name(space, type, s);
+
+	return FN(MULTI(BASE),reset_space)(multi, space);
+}
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),set_tuple_id)(
+	__isl_take MULTI(BASE) *multi, enum isl_dim_type type,
+	__isl_take isl_id *id)
+{
+	isl_space *space;
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi)
+		goto error;
+
+	space = FN(MULTI(BASE),get_space)(multi);
+	space = isl_space_set_tuple_id(space, type, id);
+
+	return FN(MULTI(BASE),reset_space)(multi, space);
+error:
+	isl_id_free(id);
+	return NULL;
+}
+
+/* Drop the id on the specified tuple.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),reset_tuple_id)(
+	__isl_take MULTI(BASE) *multi, enum isl_dim_type type)
+{
+	isl_space *space;
+
+	if (!multi)
+		return NULL;
+	if (!FN(MULTI(BASE),has_tuple_id)(multi, type))
+		return multi;
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi)
+		return NULL;
+
+	space = FN(MULTI(BASE),get_space)(multi);
+	space = isl_space_reset_tuple_id(space, type);
+
+	return FN(MULTI(BASE),reset_space)(multi, space);
+}
+
+/* Reset the user pointer on all identifiers of parameters and tuples
+ * of the space of "multi".
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),reset_user)(
+	__isl_take MULTI(BASE) *multi)
+{
+	isl_space *space;
+
+	space = FN(MULTI(BASE),get_space)(multi);
+	space = isl_space_reset_user(space);
+
+	return FN(MULTI(BASE),reset_space)(multi, space);
+}
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),realign_domain)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_reordering *exp)
+{
+	int i;
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi || !exp)
+		goto error;
+
+	for (i = 0; i < multi->n; ++i) {
+		multi->p[i] = FN(EL,realign_domain)(multi->p[i],
+						isl_reordering_copy(exp));
+		if (!multi->p[i])
+			goto error;
+	}
+
+	multi = FN(MULTI(BASE),reset_domain_space)(multi,
+						    isl_space_copy(exp->dim));
+
+	isl_reordering_free(exp);
+	return multi;
+error:
+	isl_reordering_free(exp);
+	FN(MULTI(BASE),free)(multi);
+	return NULL;
+}
+
+/* Align the parameters of "multi" to those of "model".
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),align_params)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_space *model)
+{
+	isl_ctx *ctx;
+	isl_reordering *exp;
+
+	if (!multi || !model)
+		goto error;
+
+	if (isl_space_match(multi->space, isl_dim_param,
+			     model, isl_dim_param)) {
+		isl_space_free(model);
+		return multi;
+	}
+
+	ctx = isl_space_get_ctx(model);
+	if (!isl_space_has_named_params(model))
+		isl_die(ctx, isl_error_invalid,
+			"model has unnamed parameters", goto error);
+	if (!isl_space_has_named_params(multi->space))
+		isl_die(ctx, isl_error_invalid,
+			"input has unnamed parameters", goto error);
+
+	model = isl_space_params(model);
+	exp = isl_parameter_alignment_reordering(multi->space, model);
+	exp = isl_reordering_extend_space(exp,
+				    FN(MULTI(BASE),get_domain_space)(multi));
+	multi = FN(MULTI(BASE),realign_domain)(multi, exp);
+
+	isl_space_free(model);
+	return multi;
+error:
+	isl_space_free(model);
+	FN(MULTI(BASE),free)(multi);
+	return NULL;
+}
+
+#if !defined(NO_GIST) || !defined(NO_INTERSECT_DOMAIN)
+static __isl_give MULTI(BASE) *FN(MULTI(BASE),align_params_multi_set_and)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_set *set,
+	__isl_give MULTI(BASE) *(*fn)(__isl_take MULTI(BASE) *multi,
+					__isl_take isl_set *set))
+{
+	isl_ctx *ctx;
+
+	if (!multi || !set)
+		goto error;
+	if (isl_space_match(multi->space, isl_dim_param,
+			    set->dim, isl_dim_param))
+		return fn(multi, set);
+	ctx = FN(MULTI(BASE),get_ctx)(multi);
+	if (!isl_space_has_named_params(multi->space) ||
+	    !isl_space_has_named_params(set->dim))
+		isl_die(ctx, isl_error_invalid,
+			"unaligned unnamed parameters", goto error);
+	multi = FN(MULTI(BASE),align_params)(multi, isl_set_get_space(set));
+	set = isl_set_align_params(set, FN(MULTI(BASE),get_space)(multi));
+	return fn(multi, set);
+error:
+	FN(MULTI(BASE),free)(multi);
+	isl_set_free(set);
+	return NULL;
+}
+#endif
+
+#ifndef NO_GIST
+__isl_give MULTI(BASE) *FN(MULTI(BASE),gist_aligned)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_set *context)
+{
+	int i;
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi || !context)
+		goto error;
+
+	for (i = 0; i < multi->n; ++i) {
+		multi->p[i] = FN(EL,gist)(multi->p[i], isl_set_copy(context));
+		if (!multi->p[i])
+			goto error;
+	}
+
+	isl_set_free(context);
+	return multi;
+error:
+	isl_set_free(context);
+	FN(MULTI(BASE),free)(multi);
+	return NULL;
+}
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),gist)(__isl_take MULTI(BASE) *multi,
+	__isl_take isl_set *context)
+{
+	return FN(MULTI(BASE),align_params_multi_set_and)(multi, context,
+						&FN(MULTI(BASE),gist_aligned));
+}
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),gist_params)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_set *context)
+{
+	isl_space *space = FN(MULTI(BASE),get_domain_space)(multi);
+	isl_set *dom_context = isl_set_universe(space);
+	dom_context = isl_set_intersect_params(dom_context, context);
+	return FN(MULTI(BASE),gist)(multi, dom_context);
+}
+#endif
+
+#ifndef NO_INTERSECT_DOMAIN
+/* Transform the domain of "multi" by combining it with "domain"
+ * using "fn".
+ *
+ * The parameters of "multi" and "domain" are assumed to have been aligned.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),intersect_aligned)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_set *domain,
+	__isl_give EL *(*fn)(EL *el, __isl_take isl_set *set2))
+{
+	int i;
+
+	if (!multi || !domain)
+		goto error;
+
+	if (multi->n == 0) {
+		isl_set_free(domain);
+		return multi;
+	}
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi)
+		goto error;
+
+	for (i = 0; i < multi->n; ++i) {
+		multi->p[i] = fn(multi->p[i], isl_set_copy(domain));
+		if (!multi->p[i])
+			goto error;
+	}
+
+	isl_set_free(domain);
+	return multi;
+error:
+	isl_set_free(domain);
+	FN(MULTI(BASE),free)(multi);
+	return NULL;
+}
+
+/* Intersect the domain of "multi" with "domain".
+ *
+ * The parameters of "multi" and "domain" are assumed to have been aligned.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),intersect_domain_aligned)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_set *domain)
+{
+	return FN(MULTI(BASE),intersect_aligned)(multi, domain,
+						&FN(EL,intersect_domain));
+}
+
+/* Intersect the domain of "multi" with "domain".
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),intersect_domain)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_set *domain)
+{
+	return FN(MULTI(BASE),align_params_multi_set_and)(multi, domain,
+				    &FN(MULTI(BASE),intersect_domain_aligned));
+}
+
+/* Intersect the parameter domain of "multi" with "domain".
+ *
+ * The parameters of "multi" and "domain" are assumed to have been aligned.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),intersect_params_aligned)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_set *domain)
+{
+	return FN(MULTI(BASE),intersect_aligned)(multi, domain,
+						&FN(EL,intersect_params));
+}
+
+/* Intersect the parameter domain of "multi" with "domain".
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),intersect_params)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_set *domain)
+{
+	return FN(MULTI(BASE),align_params_multi_set_and)(multi, domain,
+				    &FN(MULTI(BASE),intersect_params_aligned));
+}
+#endif
+
+__isl_give MULTI(BASE) *FN(FN(MULTI(BASE),from),LIST(BASE))(
+	__isl_take isl_space *space, __isl_take LIST(EL) *list)
+{
+	int i;
+	int n;
+	isl_ctx *ctx;
+	MULTI(BASE) *multi;
+
+	if (!space || !list)
+		goto error;
+
+	ctx = isl_space_get_ctx(space);
+	n = FN(FN(LIST(EL),n),BASE)(list);
+	if (n != isl_space_dim(space, isl_dim_out))
+		isl_die(ctx, isl_error_invalid,
+			"invalid number of elements in list", goto error);
+
+	multi = FN(MULTI(BASE),alloc)(isl_space_copy(space));
+	for (i = 0; i < n; ++i) {
+		multi = FN(FN(MULTI(BASE),set),BASE)(multi, i,
+					FN(FN(LIST(EL),get),BASE)(list, i));
+	}
+
+	isl_space_free(space);
+	FN(LIST(EL),free)(list);
+	return multi;
+error:
+	isl_space_free(space);
+	FN(LIST(EL),free)(list);
+	return NULL;
+}
+
+#ifndef NO_IDENTITY
+/* Create a multi expression in the given space that maps each
+ * input dimension to the corresponding output dimension.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),identity)(__isl_take isl_space *space)
+{
+	int i, n;
+	isl_local_space *ls;
+	MULTI(BASE) *multi;
+
+	if (!space)
+		return NULL;
+
+	if (isl_space_is_set(space))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"expecting map space", goto error);
+
+	n = isl_space_dim(space, isl_dim_out);
+	if (n != isl_space_dim(space, isl_dim_in))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"number of input and output dimensions needs to be "
+			"the same", goto error);
+
+	multi = FN(MULTI(BASE),alloc)(isl_space_copy(space));
+
+	if (!n) {
+		isl_space_free(space);
+		return multi;
+	}
+
+	space = isl_space_domain(space);
+	ls = isl_local_space_from_space(space);
+
+	for (i = 0; i < n; ++i) {
+		EL *el;
+		el = FN(EL,var_on_domain)(isl_local_space_copy(ls),
+						isl_dim_set, i);
+		multi = FN(FN(MULTI(BASE),set),BASE)(multi, i, el);
+	}
+
+	isl_local_space_free(ls);
+
+	return multi;
+error:
+	isl_space_free(space);
+	return NULL;
+}
+#endif
+
+/* Construct a multi expression in the given space with value zero in
+ * each of the output dimensions.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),zero)(__isl_take isl_space *space)
+{
+	int n;
+	MULTI(BASE) *multi;
+
+	if (!space)
+		return NULL;
+
+	n = isl_space_dim(space , isl_dim_out);
+	multi = FN(MULTI(BASE),alloc)(isl_space_copy(space));
+
+	if (!n)
+		isl_space_free(space);
+	else {
+		int i;
+		isl_local_space *ls;
+		EL *el;
+
+		space = isl_space_domain(space);
+		ls = isl_local_space_from_space(space);
+		el = FN(EL,zero_on_domain)(ls);
+
+		for (i = 0; i < n; ++i)
+			multi = FN(FN(MULTI(BASE),set),BASE)(multi, i,
+							    FN(EL,copy)(el));
+
+		FN(EL,free)(el);
+	}
+
+	return multi;
+}
+
+#ifndef NO_FROM_BASE
+__isl_give MULTI(BASE) *FN(FN(MULTI(BASE),from),BASE)(__isl_take EL *el)
+{
+	MULTI(BASE) *multi;
+
+	multi = FN(MULTI(BASE),alloc)(FN(EL,get_space)(el));
+	multi = FN(FN(MULTI(BASE),set),BASE)(multi, 0, el);
+
+	return multi;
+}
+#endif
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),drop_dims)(
+	__isl_take MULTI(BASE) *multi,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+	unsigned dim;
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi)
+		return NULL;
+
+	dim = FN(MULTI(BASE),dim)(multi, type);
+	if (first + n > dim || first + n < first)
+		isl_die(FN(MULTI(BASE),get_ctx)(multi), isl_error_invalid,
+			"index out of bounds",
+			return FN(MULTI(BASE),cow)(multi));
+
+	multi->space = isl_space_drop_dims(multi->space, type, first, n);
+	if (!multi->space)
+		return FN(MULTI(BASE),cow)(multi);
+
+	if (type == isl_dim_out) {
+		for (i = 0; i < n; ++i)
+			FN(EL,free)(multi->p[first + i]);
+		for (i = first; i + n < multi->n; ++i)
+			multi->p[i] = multi->p[i + n];
+		multi->n -= n;
+
+		return multi;
+	}
+
+	for (i = 0; i < multi->n; ++i) {
+		multi->p[i] = FN(EL,drop_dims)(multi->p[i], type, first, n);
+		if (!multi->p[i])
+			return FN(MULTI(BASE),cow)(multi);
+	}
+
+	return multi;
+}
+
+/* Align the parameters of "multi1" and "multi2" (if needed) and call "fn".
+ */
+static __isl_give MULTI(BASE) *FN(MULTI(BASE),align_params_multi_multi_and)(
+	__isl_take MULTI(BASE) *multi1, __isl_take MULTI(BASE) *multi2,
+	__isl_give MULTI(BASE) *(*fn)(__isl_take MULTI(BASE) *multi1,
+		__isl_take MULTI(BASE) *multi2))
+{
+	isl_ctx *ctx;
+
+	if (!multi1 || !multi2)
+		goto error;
+	if (isl_space_match(multi1->space, isl_dim_param,
+			    multi2->space, isl_dim_param))
+		return fn(multi1, multi2);
+	ctx = FN(MULTI(BASE),get_ctx)(multi1);
+	if (!isl_space_has_named_params(multi1->space) ||
+	    !isl_space_has_named_params(multi2->space))
+		isl_die(ctx, isl_error_invalid,
+			"unaligned unnamed parameters", goto error);
+	multi1 = FN(MULTI(BASE),align_params)(multi1,
+					    FN(MULTI(BASE),get_space)(multi2));
+	multi2 = FN(MULTI(BASE),align_params)(multi2,
+					    FN(MULTI(BASE),get_space)(multi1));
+	return fn(multi1, multi2);
+error:
+	FN(MULTI(BASE),free)(multi1);
+	FN(MULTI(BASE),free)(multi2);
+	return NULL;
+}
+
+/* Given two MULTI(BASE)s A -> B and C -> D,
+ * construct a MULTI(BASE) (A * C) -> [B -> D].
+ *
+ * The parameters are assumed to have been aligned.
+ */
+static __isl_give MULTI(BASE) *FN(MULTI(BASE),range_product_aligned)(
+	__isl_take MULTI(BASE) *multi1, __isl_take MULTI(BASE) *multi2)
+{
+	int i, n1, n2;
+	EL *el;
+	isl_space *space;
+	MULTI(BASE) *res;
+
+	if (!multi1 || !multi2)
+		goto error;
+
+	space = isl_space_range_product(FN(MULTI(BASE),get_space)(multi1),
+					FN(MULTI(BASE),get_space)(multi2));
+	res = FN(MULTI(BASE),alloc)(space);
+
+	n1 = FN(MULTI(BASE),dim)(multi1, isl_dim_out);
+	n2 = FN(MULTI(BASE),dim)(multi2, isl_dim_out);
+
+	for (i = 0; i < n1; ++i) {
+		el = FN(FN(MULTI(BASE),get),BASE)(multi1, i);
+		res = FN(FN(MULTI(BASE),set),BASE)(res, i, el);
+	}
+
+	for (i = 0; i < n2; ++i) {
+		el = FN(FN(MULTI(BASE),get),BASE)(multi2, i);
+		res = FN(FN(MULTI(BASE),set),BASE)(res, n1 + i, el);
+	}
+
+	FN(MULTI(BASE),free)(multi1);
+	FN(MULTI(BASE),free)(multi2);
+	return res;
+error:
+	FN(MULTI(BASE),free)(multi1);
+	FN(MULTI(BASE),free)(multi2);
+	return NULL;
+}
+
+/* Given two MULTI(BASE)s A -> B and C -> D,
+ * construct a MULTI(BASE) (A * C) -> [B -> D].
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),range_product)(
+	__isl_take MULTI(BASE) *multi1, __isl_take MULTI(BASE) *multi2)
+{
+	return FN(MULTI(BASE),align_params_multi_multi_and)(multi1, multi2,
+					&FN(MULTI(BASE),range_product_aligned));
+}
+
+/* Is the range of "multi" a wrapped relation?
+ */
+int FN(MULTI(BASE),range_is_wrapping)(__isl_keep MULTI(BASE) *multi)
+{
+	if (!multi)
+		return -1;
+	return isl_space_range_is_wrapping(multi->space);
+}
+
+/* Given a function A -> [B -> C], extract the function A -> B.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),range_factor_domain)(
+	__isl_take MULTI(BASE) *multi)
+{
+	isl_space *space;
+	int total, keep;
+
+	if (!multi)
+		return NULL;
+	if (!isl_space_range_is_wrapping(multi->space))
+		isl_die(FN(MULTI(BASE),get_ctx)(multi), isl_error_invalid,
+			"range is not a product",
+			return FN(MULTI(BASE),free)(multi));
+
+	space = FN(MULTI(BASE),get_space)(multi);
+	total = isl_space_dim(space, isl_dim_out);
+	space = isl_space_range_factor_domain(space);
+	keep = isl_space_dim(space, isl_dim_out);
+	multi = FN(MULTI(BASE),drop_dims)(multi,
+					isl_dim_out, keep, total - keep);
+	multi = FN(MULTI(BASE),reset_space)(multi, space);
+
+	return multi;
+}
+
+/* Given a function A -> [B -> C], extract the function A -> C.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),range_factor_range)(
+	__isl_take MULTI(BASE) *multi)
+{
+	isl_space *space;
+	int total, keep;
+
+	if (!multi)
+		return NULL;
+	if (!isl_space_range_is_wrapping(multi->space))
+		isl_die(FN(MULTI(BASE),get_ctx)(multi), isl_error_invalid,
+			"range is not a product",
+			return FN(MULTI(BASE),free)(multi));
+
+	space = FN(MULTI(BASE),get_space)(multi);
+	total = isl_space_dim(space, isl_dim_out);
+	space = isl_space_range_factor_range(space);
+	keep = isl_space_dim(space, isl_dim_out);
+	multi = FN(MULTI(BASE),drop_dims)(multi, isl_dim_out, 0, total - keep);
+	multi = FN(MULTI(BASE),reset_space)(multi, space);
+
+	return multi;
+}
+
+/* Given two MULTI(BASE)s A -> B and C -> D,
+ * construct a MULTI(BASE) [A -> C] -> [B -> D].
+ *
+ * The parameters are assumed to have been aligned.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),product_aligned)(
+	__isl_take MULTI(BASE) *multi1, __isl_take MULTI(BASE) *multi2)
+{
+	int i;
+	EL *el;
+	isl_space *space;
+	MULTI(BASE) *res;
+	int in1, in2, out1, out2;
+
+	in1 = FN(MULTI(BASE),dim)(multi1, isl_dim_in);
+	in2 = FN(MULTI(BASE),dim)(multi2, isl_dim_in);
+	out1 = FN(MULTI(BASE),dim)(multi1, isl_dim_out);
+	out2 = FN(MULTI(BASE),dim)(multi2, isl_dim_out);
+	space = isl_space_product(FN(MULTI(BASE),get_space)(multi1),
+				  FN(MULTI(BASE),get_space)(multi2));
+	res = FN(MULTI(BASE),alloc)(isl_space_copy(space));
+	space = isl_space_domain(space);
+
+	for (i = 0; i < out1; ++i) {
+		el = FN(FN(MULTI(BASE),get),BASE)(multi1, i);
+		el = FN(EL,insert_dims)(el, isl_dim_in, in1, in2);
+		el = FN(EL,reset_domain_space)(el, isl_space_copy(space));
+		res = FN(FN(MULTI(BASE),set),BASE)(res, i, el);
+	}
+
+	for (i = 0; i < out2; ++i) {
+		el = FN(FN(MULTI(BASE),get),BASE)(multi2, i);
+		el = FN(EL,insert_dims)(el, isl_dim_in, 0, in1);
+		el = FN(EL,reset_domain_space)(el, isl_space_copy(space));
+		res = FN(FN(MULTI(BASE),set),BASE)(res, out1 + i, el);
+	}
+
+	isl_space_free(space);
+	FN(MULTI(BASE),free)(multi1);
+	FN(MULTI(BASE),free)(multi2);
+	return res;
+}
+
+/* Given two MULTI(BASE)s A -> B and C -> D,
+ * construct a MULTI(BASE) [A -> C] -> [B -> D].
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),product)(
+	__isl_take MULTI(BASE) *multi1, __isl_take MULTI(BASE) *multi2)
+{
+	return FN(MULTI(BASE),align_params_multi_multi_and)(multi1, multi2,
+					&FN(MULTI(BASE),product_aligned));
+}
+
+__isl_give MULTI(BASE) *FN(MULTI(BASE),flatten_range)(
+	__isl_take MULTI(BASE) *multi)
+{
+	if (!multi)
+		return NULL;
+
+	if (!multi->space->nested[1])
+		return multi;
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi)
+		return NULL;
+
+	multi->space = isl_space_flatten_range(multi->space);
+	if (!multi->space)
+		return FN(MULTI(BASE),free)(multi);
+
+	return multi;
+}
+
+/* Given two MULTI(BASE)s A -> B and C -> D,
+ * construct a MULTI(BASE) (A * C) -> (B, D).
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),flat_range_product)(
+	__isl_take MULTI(BASE) *multi1, __isl_take MULTI(BASE) *multi2)
+{
+	MULTI(BASE) *multi;
+
+	multi = FN(MULTI(BASE),range_product)(multi1, multi2);
+	multi = FN(MULTI(BASE),flatten_range)(multi);
+	return multi;
+}
+
+/* Given two multi expressions, "multi1"
+ *
+ *	[A] -> [B1 B2]
+ *
+ * where B2 starts at position "pos", and "multi2"
+ *
+ *	[A] -> [D]
+ *
+ * return the multi expression
+ *
+ *	[A] -> [B1 D B2]
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),range_splice)(
+	__isl_take MULTI(BASE) *multi1, unsigned pos,
+	__isl_take MULTI(BASE) *multi2)
+{
+	MULTI(BASE) *res;
+	unsigned dim;
+
+	if (!multi1 || !multi2)
+		goto error;
+
+	dim = FN(MULTI(BASE),dim)(multi1, isl_dim_out);
+	if (pos > dim)
+		isl_die(FN(MULTI(BASE),get_ctx)(multi1), isl_error_invalid,
+			"index out of bounds", goto error);
+
+	res = FN(MULTI(BASE),copy)(multi1);
+	res = FN(MULTI(BASE),drop_dims)(res, isl_dim_out, pos, dim - pos);
+	multi1 = FN(MULTI(BASE),drop_dims)(multi1, isl_dim_out, 0, pos);
+
+	res = FN(MULTI(BASE),flat_range_product)(res, multi2);
+	res = FN(MULTI(BASE),flat_range_product)(res, multi1);
+
+	return res;
+error:
+	FN(MULTI(BASE),free)(multi1);
+	FN(MULTI(BASE),free)(multi2);
+	return NULL;
+}
+
+/* Given two multi expressions, "multi1"
+ *
+ *	[A1 A2] -> [B1 B2]
+ *
+ * where A2 starts at position "in_pos" and B2 starts at position "out_pos",
+ * and "multi2"
+ *
+ *	[C] -> [D]
+ *
+ * return the multi expression
+ *
+ *	[A1 C A2] -> [B1 D B2]
+ *
+ * We first insert input dimensions to obtain
+ *
+ *	[A1 C A2] -> [B1 B2]
+ *
+ * and
+ *
+ *	[A1 C A2] -> [D]
+ *
+ * and then apply range_splice.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),splice)(
+	__isl_take MULTI(BASE) *multi1, unsigned in_pos, unsigned out_pos,
+	__isl_take MULTI(BASE) *multi2)
+{
+	unsigned n_in1;
+	unsigned n_in2;
+
+	if (!multi1 || !multi2)
+		goto error;
+
+	n_in1 = FN(MULTI(BASE),dim)(multi1, isl_dim_in);
+	if (in_pos > n_in1)
+		isl_die(FN(MULTI(BASE),get_ctx)(multi1), isl_error_invalid,
+			"index out of bounds", goto error);
+
+	n_in2 = FN(MULTI(BASE),dim)(multi2, isl_dim_in);
+
+	multi1 = FN(MULTI(BASE),insert_dims)(multi1, isl_dim_in, in_pos, n_in2);
+	multi2 = FN(MULTI(BASE),insert_dims)(multi2, isl_dim_in, n_in2,
+						n_in1 - in_pos);
+	multi2 = FN(MULTI(BASE),insert_dims)(multi2, isl_dim_in, 0, in_pos);
+
+	return FN(MULTI(BASE),range_splice)(multi1, out_pos, multi2);
+error:
+	FN(MULTI(BASE),free)(multi1);
+	FN(MULTI(BASE),free)(multi2);
+	return NULL;
+}
+
+/* This function is currently only used from isl_aff.c
+ */
+static __isl_give MULTI(BASE) *FN(MULTI(BASE),bin_op)(
+	__isl_take MULTI(BASE) *multi1, __isl_take MULTI(BASE) *multi2,
+	__isl_give EL *(*fn)(__isl_take EL *, __isl_take EL *))
+	__attribute__ ((unused));
+
+/* Pairwise perform "fn" to the elements of "multi1" and "multi2" and
+ * return the result.
+ */
+static __isl_give MULTI(BASE) *FN(MULTI(BASE),bin_op)(
+	__isl_take MULTI(BASE) *multi1, __isl_take MULTI(BASE) *multi2,
+	__isl_give EL *(*fn)(__isl_take EL *, __isl_take EL *))
+{
+	int i;
+	isl_ctx *ctx;
+
+	multi1 = FN(MULTI(BASE),cow)(multi1);
+	if (!multi1 || !multi2)
+		goto error;
+
+	ctx = FN(MULTI(BASE),get_ctx)(multi1);
+	if (!isl_space_is_equal(multi1->space, multi2->space))
+		isl_die(ctx, isl_error_invalid,
+			"spaces don't match", goto error);
+
+	for (i = 0; i < multi1->n; ++i) {
+		multi1->p[i] = fn(multi1->p[i], FN(EL,copy)(multi2->p[i]));
+		if (!multi1->p[i])
+			goto error;
+	}
+
+	FN(MULTI(BASE),free)(multi2);
+	return multi1;
+error:
+	FN(MULTI(BASE),free)(multi1);
+	FN(MULTI(BASE),free)(multi2);
+	return NULL;
+}
+
+/* Multiply the elements of "multi" by "v" and return the result.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),scale_val)(__isl_take MULTI(BASE) *multi,
+	__isl_take isl_val *v)
+{
+	int i;
+
+	if (!multi || !v)
+		goto error;
+
+	if (isl_val_is_one(v)) {
+		isl_val_free(v);
+		return multi;
+	}
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting rational factor", goto error);
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi)
+		return NULL;
+
+	for (i = 0; i < multi->n; ++i) {
+		multi->p[i] = FN(EL,scale_val)(multi->p[i], isl_val_copy(v));
+		if (!multi->p[i])
+			goto error;
+	}
+
+	isl_val_free(v);
+	return multi;
+error:
+	isl_val_free(v);
+	return FN(MULTI(BASE),free)(multi);
+}
+
+/* Divide the elements of "multi" by "v" and return the result.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),scale_down_val)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_val *v)
+{
+	int i;
+
+	if (!multi || !v)
+		goto error;
+
+	if (isl_val_is_one(v)) {
+		isl_val_free(v);
+		return multi;
+	}
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting rational factor", goto error);
+	if (isl_val_is_zero(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"cannot scale down by zero", goto error);
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi)
+		return NULL;
+
+	for (i = 0; i < multi->n; ++i) {
+		multi->p[i] = FN(EL,scale_down_val)(multi->p[i],
+						    isl_val_copy(v));
+		if (!multi->p[i])
+			goto error;
+	}
+
+	isl_val_free(v);
+	return multi;
+error:
+	isl_val_free(v);
+	return FN(MULTI(BASE),free)(multi);
+}
+
+/* Multiply the elements of "multi" by the corresponding element of "mv"
+ * and return the result.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),scale_multi_val)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_multi_val *mv)
+{
+	int i;
+
+	if (!multi || !mv)
+		goto error;
+
+	if (!isl_space_tuple_is_equal(multi->space, isl_dim_out,
+					mv->space, isl_dim_set))
+		isl_die(isl_multi_val_get_ctx(mv), isl_error_invalid,
+			"spaces don't match", goto error);
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi)
+		return NULL;
+
+	for (i = 0; i < multi->n; ++i) {
+		isl_val *v;
+
+		v = isl_multi_val_get_val(mv, i);
+		multi->p[i] = FN(EL,scale_val)(multi->p[i], v);
+		if (!multi->p[i])
+			goto error;
+	}
+
+	isl_multi_val_free(mv);
+	return multi;
+error:
+	isl_multi_val_free(mv);
+	return FN(MULTI(BASE),free)(multi);
+}
+
+/* Divide the elements of "multi" by the corresponding element of "mv"
+ * and return the result.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),scale_down_multi_val)(
+	__isl_take MULTI(BASE) *multi, __isl_take isl_multi_val *mv)
+{
+	int i;
+
+	if (!multi || !mv)
+		goto error;
+
+	if (!isl_space_tuple_is_equal(multi->space, isl_dim_out,
+					mv->space, isl_dim_set))
+		isl_die(isl_multi_val_get_ctx(mv), isl_error_invalid,
+			"spaces don't match", goto error);
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi)
+		return NULL;
+
+	for (i = 0; i < multi->n; ++i) {
+		isl_val *v;
+
+		v = isl_multi_val_get_val(mv, i);
+		multi->p[i] = FN(EL,scale_down_val)(multi->p[i], v);
+		if (!multi->p[i])
+			goto error;
+	}
+
+	isl_multi_val_free(mv);
+	return multi;
+error:
+	isl_multi_val_free(mv);
+	return FN(MULTI(BASE),free)(multi);
+}
+
+#ifndef NO_MOVE_DIMS
+/* Move the "n" dimensions of "src_type" starting at "src_pos" of "multi"
+ * to dimensions of "dst_type" at "dst_pos".
+ *
+ * We only support moving input dimensions to parameters and vice versa.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),move_dims)(__isl_take MULTI(BASE) *multi,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n)
+{
+	int i;
+
+	if (!multi)
+		return NULL;
+
+	if (n == 0 &&
+	    !isl_space_is_named_or_nested(multi->space, src_type) &&
+	    !isl_space_is_named_or_nested(multi->space, dst_type))
+		return multi;
+
+	if (dst_type == isl_dim_out || src_type == isl_dim_out)
+		isl_die(FN(MULTI(BASE),get_ctx)(multi), isl_error_invalid,
+			"cannot move output/set dimension",
+			return FN(MULTI(BASE),free)(multi));
+	if (dst_type == isl_dim_div || src_type == isl_dim_div)
+		isl_die(FN(MULTI(BASE),get_ctx)(multi), isl_error_invalid,
+			"cannot move divs",
+			return FN(MULTI(BASE),free)(multi));
+	if (src_pos + n > isl_space_dim(multi->space, src_type))
+		isl_die(FN(MULTI(BASE),get_ctx)(multi), isl_error_invalid,
+			"range out of bounds",
+			return FN(MULTI(BASE),free)(multi));
+	if (dst_type == src_type)
+		isl_die(FN(MULTI(BASE),get_ctx)(multi), isl_error_unsupported,
+			"moving dims within the same type not supported",
+			return FN(MULTI(BASE),free)(multi));
+
+	multi = FN(MULTI(BASE),cow)(multi);
+	if (!multi)
+		return NULL;
+
+	multi->space = isl_space_move_dims(multi->space, dst_type, dst_pos,
+						src_type, src_pos, n);
+	if (!multi->space)
+		return FN(MULTI(BASE),free)(multi);
+
+	for (i = 0; i < multi->n; ++i) {
+		multi->p[i] = FN(EL,move_dims)(multi->p[i], dst_type, dst_pos,
+						src_type, src_pos, n);
+		if (!multi->p[i])
+			return FN(MULTI(BASE),free)(multi);
+	}
+
+	return multi;
+}
+#endif
+
+/* Convert a multiple expression defined over a parameter domain
+ * into one that is defined over a zero-dimensional set.
+ */
+__isl_give MULTI(BASE) *FN(MULTI(BASE),from_range)(
+	__isl_take MULTI(BASE) *multi)
+{
+	isl_space *space;
+
+	if (!multi)
+		return NULL;
+	if (!isl_space_is_set(multi->space))
+		isl_die(FN(MULTI(BASE),get_ctx)(multi), isl_error_invalid,
+			"not living in a set space",
+			return FN(MULTI(BASE),free)(multi));
+
+	space = FN(MULTI(BASE),get_space)(multi);
+	space = isl_space_from_range(space);
+	multi = FN(MULTI(BASE),reset_space)(multi, space);
+
+	return multi;
+}
+
+/* Are "multi1" and "multi2" obviously equal?
+ */
+int FN(MULTI(BASE),plain_is_equal)(__isl_keep MULTI(BASE) *multi1,
+	__isl_keep MULTI(BASE) *multi2)
+{
+	int i;
+	int equal;
+
+	if (!multi1 || !multi2)
+		return -1;
+	if (multi1->n != multi2->n)
+		return 0;
+	equal = isl_space_is_equal(multi1->space, multi2->space);
+	if (equal < 0 || !equal)
+		return equal;
+
+	for (i = 0; i < multi1->n; ++i) {
+		equal = FN(EL,plain_is_equal)(multi1->p[i], multi2->p[i]);
+		if (equal < 0 || !equal)
+			return equal;
+	}
+
+	return 1;
+}
+
+#ifndef NO_DOMAIN
+/* Return the shared domain of the elements of "multi".
+ */
+__isl_give isl_set *FN(MULTI(BASE),domain)(__isl_take MULTI(BASE) *multi)
+{
+	int i;
+	isl_set *dom;
+
+	if (!multi)
+		return NULL;
+
+	dom = isl_set_universe(FN(MULTI(BASE),get_domain_space)(multi));
+	for (i = 0; i < multi->n; ++i) {
+		isl_set *dom_i;
+
+		dom_i = FN(EL,domain)(FN(FN(MULTI(BASE),get),BASE)(multi, i));
+		dom = isl_set_intersect(dom, dom_i);
+	}
+
+	FN(MULTI(BASE),free)(multi);
+	return dom;
+}
+#endif
diff --git a/polly/lib/External/isl/isl_multi_templ.h b/polly/lib/External/isl/isl_multi_templ.h
new file mode 100644
index 00000000000..342d9a22315
--- /dev/null
+++ b/polly/lib/External/isl/isl_multi_templ.h
@@ -0,0 +1,18 @@
+#include <isl/space.h>
+
+#define xCAT(A,B) A ## B
+#define CAT(A,B) xCAT(A,B)
+#undef EL
+#define EL CAT(isl_,BASE)
+#define xMULTI(BASE) isl_multi_ ## BASE
+#define MULTI(BASE) xMULTI(BASE)
+
+struct MULTI(BASE) {
+	int ref;
+	isl_space *space;
+
+	int n;
+	EL *p[1];
+};
+
+__isl_give MULTI(BASE) *CAT(MULTI(BASE),_alloc)(__isl_take isl_space *space);
diff --git a/polly/lib/External/isl/isl_obj.c b/polly/lib/External/isl/isl_obj.c
new file mode 100644
index 00000000000..5bc26ce6cc7
--- /dev/null
+++ b/polly/lib/External/isl/isl_obj.c
@@ -0,0 +1,335 @@
+/*
+ * Copyright 2010      INRIA Saclay
+ * Copyright 2014      Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France 
+ * and Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France
+ */
+
+#include <isl/aff.h>
+#include <isl/set.h>
+#include <isl/map.h>
+#include <isl/polynomial.h>
+#include <isl/obj.h>
+
+static void *isl_obj_val_copy(void *v)
+{
+	return isl_val_copy((isl_val *)v);
+}
+
+static void isl_obj_val_free(void *v)
+{
+	isl_val_free((isl_val *)v);
+}
+
+static __isl_give isl_printer *isl_obj_val_print(__isl_take isl_printer *p,
+	void *v)
+{
+	return isl_printer_print_val(p, (isl_val *)v);
+}
+
+static void *isl_obj_val_add(void *v1, void *v2)
+{
+	return isl_val_add((isl_val *) v1, (isl_val *) v2);
+}
+
+struct isl_obj_vtable isl_obj_val_vtable = {
+	isl_obj_val_copy,
+	isl_obj_val_add,
+	isl_obj_val_print,
+	isl_obj_val_free
+};
+
+static void *isl_obj_map_copy(void *v)
+{
+	return isl_map_copy((struct isl_map *)v);
+}
+
+static void isl_obj_map_free(void *v)
+{
+	isl_map_free((struct isl_map *)v);
+}
+
+static __isl_give isl_printer *isl_obj_map_print(__isl_take isl_printer *p,
+	void *v)
+{
+	return isl_printer_print_map(p, (struct isl_map *)v);
+}
+
+static void *isl_obj_map_add(void *v1, void *v2)
+{
+	return isl_map_union((struct isl_map *)v1, (struct isl_map *)v2);
+}
+
+struct isl_obj_vtable isl_obj_map_vtable = {
+	isl_obj_map_copy,
+	isl_obj_map_add,
+	isl_obj_map_print,
+	isl_obj_map_free
+};
+
+static void *isl_obj_union_map_copy(void *v)
+{
+	return isl_union_map_copy((isl_union_map *)v);
+}
+
+static void isl_obj_union_map_free(void *v)
+{
+	isl_union_map_free((isl_union_map *)v);
+}
+
+static __isl_give isl_printer *isl_obj_union_map_print(__isl_take isl_printer *p,
+	void *v)
+{
+	return isl_printer_print_union_map(p, (isl_union_map *)v);
+}
+
+static void *isl_obj_union_map_add(void *v1, void *v2)
+{
+	return isl_union_map_union((isl_union_map *)v1, (isl_union_map *)v2);
+}
+
+struct isl_obj_vtable isl_obj_union_map_vtable = {
+	isl_obj_union_map_copy,
+	isl_obj_union_map_add,
+	isl_obj_union_map_print,
+	isl_obj_union_map_free
+};
+
+static void *isl_obj_set_copy(void *v)
+{
+	return isl_set_copy((struct isl_set *)v);
+}
+
+static void isl_obj_set_free(void *v)
+{
+	isl_set_free((struct isl_set *)v);
+}
+
+static __isl_give isl_printer *isl_obj_set_print(__isl_take isl_printer *p,
+	void *v)
+{
+	return isl_printer_print_set(p, (struct isl_set *)v);
+}
+
+static void *isl_obj_set_add(void *v1, void *v2)
+{
+	return isl_set_union((struct isl_set *)v1, (struct isl_set *)v2);
+}
+
+struct isl_obj_vtable isl_obj_set_vtable = {
+	isl_obj_set_copy,
+	isl_obj_set_add,
+	isl_obj_set_print,
+	isl_obj_set_free
+};
+
+static void *isl_obj_union_set_copy(void *v)
+{
+	return isl_union_set_copy((isl_union_set *)v);
+}
+
+static void isl_obj_union_set_free(void *v)
+{
+	isl_union_set_free((isl_union_set *)v);
+}
+
+static __isl_give isl_printer *isl_obj_union_set_print(__isl_take isl_printer *p,
+	void *v)
+{
+	return isl_printer_print_union_set(p, (isl_union_set *)v);
+}
+
+static void *isl_obj_union_set_add(void *v1, void *v2)
+{
+	return isl_union_set_union((isl_union_set *)v1, (isl_union_set *)v2);
+}
+
+struct isl_obj_vtable isl_obj_union_set_vtable = {
+	isl_obj_union_set_copy,
+	isl_obj_union_set_add,
+	isl_obj_union_set_print,
+	isl_obj_union_set_free
+};
+
+static void *isl_obj_pw_multi_aff_copy(void *v)
+{
+	return isl_pw_multi_aff_copy((isl_pw_multi_aff *) v);
+}
+
+static void isl_obj_pw_multi_aff_free(void *v)
+{
+	isl_pw_multi_aff_free((isl_pw_multi_aff *) v);
+}
+
+static __isl_give isl_printer *isl_obj_pw_multi_aff_print(
+	__isl_take isl_printer *p, void *v)
+{
+	return isl_printer_print_pw_multi_aff(p, (isl_pw_multi_aff *) v);
+}
+
+static void *isl_obj_pw_multi_aff_add(void *v1, void *v2)
+{
+	return isl_pw_multi_aff_add((isl_pw_multi_aff *) v1,
+				    (isl_pw_multi_aff *) v2);
+}
+
+struct isl_obj_vtable isl_obj_pw_multi_aff_vtable = {
+	isl_obj_pw_multi_aff_copy,
+	isl_obj_pw_multi_aff_add,
+	isl_obj_pw_multi_aff_print,
+	isl_obj_pw_multi_aff_free
+};
+
+static void *isl_obj_none_copy(void *v)
+{
+	return v;
+}
+
+static void isl_obj_none_free(void *v)
+{
+}
+
+static __isl_give isl_printer *isl_obj_none_print(__isl_take isl_printer *p,
+	void *v)
+{
+	return p;
+}
+
+static void *isl_obj_none_add(void *v1, void *v2)
+{
+	return NULL;
+}
+
+struct isl_obj_vtable isl_obj_none_vtable = {
+	isl_obj_none_copy,
+	isl_obj_none_add,
+	isl_obj_none_print,
+	isl_obj_none_free
+};
+
+static void *isl_obj_pw_qp_copy(void *v)
+{
+	return isl_pw_qpolynomial_copy((struct isl_pw_qpolynomial *)v);
+}
+
+static void isl_obj_pw_qp_free(void *v)
+{
+	isl_pw_qpolynomial_free((struct isl_pw_qpolynomial *)v);
+}
+
+static __isl_give isl_printer *isl_obj_pw_qp_print(__isl_take isl_printer *p,
+	void *v)
+{
+	return isl_printer_print_pw_qpolynomial(p,
+						(struct isl_pw_qpolynomial *)v);
+}
+
+static void *isl_obj_pw_qp_add(void *v1, void *v2)
+{
+	return isl_pw_qpolynomial_add((struct isl_pw_qpolynomial *)v1,
+					(struct isl_pw_qpolynomial *)v2);
+}
+
+struct isl_obj_vtable isl_obj_pw_qpolynomial_vtable = {
+	isl_obj_pw_qp_copy,
+	isl_obj_pw_qp_add,
+	isl_obj_pw_qp_print,
+	isl_obj_pw_qp_free
+};
+
+static void *isl_obj_union_pw_qp_copy(void *v)
+{
+	return isl_union_pw_qpolynomial_copy((struct isl_union_pw_qpolynomial *)v);
+}
+
+static void isl_obj_union_pw_qp_free(void *v)
+{
+	isl_union_pw_qpolynomial_free((struct isl_union_pw_qpolynomial *)v);
+}
+
+static __isl_give isl_printer *isl_obj_union_pw_qp_print(
+	__isl_take isl_printer *p, void *v)
+{
+	return isl_printer_print_union_pw_qpolynomial(p,
+					(struct isl_union_pw_qpolynomial *)v);
+}
+
+static void *isl_obj_union_pw_qp_add(void *v1, void *v2)
+{
+	return isl_union_pw_qpolynomial_add(
+					(struct isl_union_pw_qpolynomial *)v1,
+					(struct isl_union_pw_qpolynomial *)v2);
+}
+
+struct isl_obj_vtable isl_obj_union_pw_qpolynomial_vtable = {
+	isl_obj_union_pw_qp_copy,
+	isl_obj_union_pw_qp_add,
+	isl_obj_union_pw_qp_print,
+	isl_obj_union_pw_qp_free
+};
+
+static void *isl_obj_pw_qpf_copy(void *v)
+{
+	return isl_pw_qpolynomial_fold_copy((struct isl_pw_qpolynomial_fold *)v);
+}
+
+static void isl_obj_pw_qpf_free(void *v)
+{
+	isl_pw_qpolynomial_fold_free((struct isl_pw_qpolynomial_fold *)v);
+}
+
+static __isl_give isl_printer *isl_obj_pw_qpf_print(__isl_take isl_printer *p,
+	void *v)
+{
+	return isl_printer_print_pw_qpolynomial_fold(p,
+					(struct isl_pw_qpolynomial_fold *)v);
+}
+
+static void *isl_obj_pw_qpf_add(void *v1, void *v2)
+{
+	return isl_pw_qpolynomial_fold_fold((struct isl_pw_qpolynomial_fold *)v1,
+					    (struct isl_pw_qpolynomial_fold *)v2);
+}
+
+struct isl_obj_vtable isl_obj_pw_qpolynomial_fold_vtable = {
+	isl_obj_pw_qpf_copy,
+	isl_obj_pw_qpf_add,
+	isl_obj_pw_qpf_print,
+	isl_obj_pw_qpf_free
+};
+
+static void *isl_obj_union_pw_qpf_copy(void *v)
+{
+	return isl_union_pw_qpolynomial_fold_copy((struct isl_union_pw_qpolynomial_fold *)v);
+}
+
+static void isl_obj_union_pw_qpf_free(void *v)
+{
+	isl_union_pw_qpolynomial_fold_free((struct isl_union_pw_qpolynomial_fold *)v);
+}
+
+static __isl_give isl_printer *isl_obj_union_pw_qpf_print(
+	__isl_take isl_printer *p, void *v)
+{
+	return isl_printer_print_union_pw_qpolynomial_fold(p,
+				    (struct isl_union_pw_qpolynomial_fold *)v);
+}
+
+static void *isl_obj_union_pw_qpf_add(void *v1, void *v2)
+{
+	return isl_union_pw_qpolynomial_fold_fold(
+				    (struct isl_union_pw_qpolynomial_fold *)v1,
+				    (struct isl_union_pw_qpolynomial_fold *)v2);
+}
+
+struct isl_obj_vtable isl_obj_union_pw_qpolynomial_fold_vtable = {
+	isl_obj_union_pw_qpf_copy,
+	isl_obj_union_pw_qpf_add,
+	isl_obj_union_pw_qpf_print,
+	isl_obj_union_pw_qpf_free
+};
diff --git a/polly/lib/External/isl/isl_options.c b/polly/lib/External/isl/isl_options.c
new file mode 100644
index 00000000000..f41b6e0ec45
--- /dev/null
+++ b/polly/lib/External/isl/isl_options.c
@@ -0,0 +1,305 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+
+#include <isl/ctx.h>
+#include <isl_options_private.h>
+#include <isl/ast_build.h>
+#include <isl/schedule.h>
+#include <isl/version.h>
+
+struct isl_arg_choice isl_pip_context_choice[] = {
+	{"gbr",		ISL_CONTEXT_GBR},
+	{"lexmin",	ISL_CONTEXT_LEXMIN},
+	{0}
+};
+
+struct isl_arg_choice isl_gbr_choice[] = {
+	{"never",	ISL_GBR_NEVER},
+	{"once",	ISL_GBR_ONCE},
+	{"always",	ISL_GBR_ALWAYS},
+	{0}
+};
+
+struct isl_arg_choice isl_closure_choice[] = {
+	{"isl",		ISL_CLOSURE_ISL},
+	{"box",		ISL_CLOSURE_BOX},
+	{0}
+};
+
+static struct isl_arg_choice bound[] = {
+	{"bernstein",	ISL_BOUND_BERNSTEIN},
+	{"range",	ISL_BOUND_RANGE},
+	{0}
+};
+
+static struct isl_arg_choice on_error[] = {
+	{"warn",	ISL_ON_ERROR_WARN},
+	{"continue",	ISL_ON_ERROR_CONTINUE},
+	{"abort",	ISL_ON_ERROR_ABORT},
+	{0}
+};
+
+static struct isl_arg_choice isl_schedule_algorithm_choice[] = {
+	{"isl",		ISL_SCHEDULE_ALGORITHM_ISL},
+	{"feautrier",   ISL_SCHEDULE_ALGORITHM_FEAUTRIER},
+	{0}
+};
+
+static struct isl_arg_flags bernstein_recurse[] = {
+	{"none",	ISL_BERNSTEIN_FACTORS | ISL_BERNSTEIN_INTERVALS, 0},
+	{"factors",	ISL_BERNSTEIN_FACTORS | ISL_BERNSTEIN_INTERVALS,
+			ISL_BERNSTEIN_FACTORS},
+	{"intervals",	ISL_BERNSTEIN_FACTORS | ISL_BERNSTEIN_INTERVALS,
+			ISL_BERNSTEIN_INTERVALS},
+	{"full",	ISL_BERNSTEIN_FACTORS | ISL_BERNSTEIN_INTERVALS,
+			ISL_BERNSTEIN_FACTORS | ISL_BERNSTEIN_INTERVALS},
+	{0}
+};
+
+static struct isl_arg_choice convex[] = {
+	{"wrap",	ISL_CONVEX_HULL_WRAP},
+	{"fm",		ISL_CONVEX_HULL_FM},
+	{0}
+};
+
+static struct isl_arg_choice fuse[] = {
+	{"max",		ISL_SCHEDULE_FUSE_MAX},
+	{"min",		ISL_SCHEDULE_FUSE_MIN},
+	{0}
+};
+
+static struct isl_arg_choice separation_bounds[] = {
+	{"explicit",	ISL_AST_BUILD_SEPARATION_BOUNDS_EXPLICIT},
+	{"implicit",	ISL_AST_BUILD_SEPARATION_BOUNDS_IMPLICIT},
+	{0}
+};
+
+static void print_version(void)
+{
+	printf("%s", isl_version());
+}
+
+ISL_ARGS_START(struct isl_options, isl_options_args)
+ISL_ARG_CHOICE(struct isl_options, context, 0, "context", \
+	isl_pip_context_choice,	ISL_CONTEXT_GBR,
+	"how to handle the pip context tableau")
+ISL_ARG_CHOICE(struct isl_options, gbr, 0, "gbr", \
+	isl_gbr_choice,	ISL_GBR_ALWAYS,
+	"how often to use generalized basis reduction")
+ISL_ARG_CHOICE(struct isl_options, closure, 0, "closure", \
+	isl_closure_choice,	ISL_CLOSURE_ISL,
+	"closure operation to use")
+ISL_ARG_BOOL(struct isl_options, gbr_only_first, 0, "gbr-only-first", 0,
+	"only perform basis reduction in first direction")
+ISL_ARG_CHOICE(struct isl_options, bound, 0, "bound", bound,
+	ISL_BOUND_BERNSTEIN, "algorithm to use for computing bounds")
+ISL_ARG_CHOICE(struct isl_options, on_error, 0, "on-error", on_error,
+	ISL_ON_ERROR_WARN, "how to react if an error is detected")
+ISL_ARG_FLAGS(struct isl_options, bernstein_recurse, 0,
+	"bernstein-recurse", bernstein_recurse, ISL_BERNSTEIN_FACTORS, NULL)
+ISL_ARG_BOOL(struct isl_options, bernstein_triangulate, 0,
+	"bernstein-triangulate", 1,
+	"triangulate domains during Bernstein expansion")
+ISL_ARG_BOOL(struct isl_options, pip_symmetry, 0, "pip-symmetry", 1,
+	"detect simple symmetries in PIP input")
+ISL_ARG_CHOICE(struct isl_options, convex, 0, "convex-hull", \
+	convex,	ISL_CONVEX_HULL_WRAP, "convex hull algorithm to use")
+ISL_ARG_BOOL(struct isl_options, coalesce_bounded_wrapping, 0,
+	"coalesce-bounded-wrapping", 1, "bound wrapping during coalescing")
+ISL_ARG_INT(struct isl_options, schedule_max_coefficient, 0,
+	"schedule-max-coefficient", "limit", -1, "Only consider schedules "
+	"where the coefficients of the variable and parameter dimensions "
+        "do not exceed <limit>. A value of -1 allows arbitrary coefficients.")
+ISL_ARG_INT(struct isl_options, schedule_max_constant_term, 0,
+	"schedule-max-constant-term", "limit", -1, "Only consider schedules "
+	"where the coefficients of the constant dimension do not exceed "
+	"<limit>. A value of -1 allows arbitrary coefficients.")
+ISL_ARG_BOOL(struct isl_options, schedule_parametric, 0,
+	"schedule-parametric", 1, "construct possibly parametric schedules")
+ISL_ARG_BOOL(struct isl_options, schedule_outer_coincidence, 0,
+	"schedule-outer-coincidence", 0,
+	"try to construct schedules where the outer member of each band "
+	"satisfies the coincidence constraints")
+ISL_ARG_BOOL(struct isl_options, schedule_maximize_band_depth, 0,
+	"schedule-maximize-band-depth", 0,
+	"maximize the number of scheduling dimensions in a band")
+ISL_ARG_BOOL(struct isl_options, schedule_split_scaled, 0,
+	"schedule-split-scaled", 1,
+	"split non-tilable bands with scaled schedules")
+ISL_ARG_BOOL(struct isl_options, schedule_separate_components, 0,
+	"schedule-separate-components", 1,
+	"separate components in dependence graph")
+ISL_ARG_CHOICE(struct isl_options, schedule_algorithm, 0,
+	"schedule-algorithm", isl_schedule_algorithm_choice,
+	ISL_SCHEDULE_ALGORITHM_ISL, "scheduling algorithm to use")
+ISL_ARG_CHOICE(struct isl_options, schedule_fuse, 0, "schedule-fuse", fuse,
+	ISL_SCHEDULE_FUSE_MAX, "level of fusion during scheduling")
+ISL_ARG_BOOL(struct isl_options, tile_scale_tile_loops, 0,
+	"tile-scale-tile-loops", 1, "scale tile loops")
+ISL_ARG_BOOL(struct isl_options, tile_shift_point_loops, 0,
+	"tile-shift-point-loops", 1, "shift point loops to start at zero")
+ISL_ARG_STR(struct isl_options, ast_iterator_type, 0,
+	"ast-iterator-type", "type", "int",
+	"type used for iterators during printing of AST")
+ISL_ARG_BOOL(struct isl_options, ast_always_print_block, 0,
+	"ast-always-print-block", 0, "print for and if bodies as a block "
+	"regardless of the number of statements in the body")
+ISL_ARG_BOOL(struct isl_options, ast_build_atomic_upper_bound, 0,
+	"ast-build-atomic-upper-bound", 1, "generate atomic upper bounds")
+ISL_ARG_BOOL(struct isl_options, ast_build_prefer_pdiv, 0,
+	"ast-build-prefer-pdiv", 1, "prefer pdiv operation over fdiv")
+ISL_ARG_BOOL(struct isl_options, ast_build_exploit_nested_bounds, 0,
+	"ast-build-exploit-nested-bounds", 1,
+	"simplify conditions based on bounds of nested for loops")
+ISL_ARG_BOOL(struct isl_options, ast_build_group_coscheduled, 0,
+	"ast-build-group-coscheduled", 0,
+	"keep coscheduled domain elements together")
+ISL_ARG_CHOICE(struct isl_options, ast_build_separation_bounds, 0,
+	"ast-build-separation-bounds", separation_bounds,
+	ISL_AST_BUILD_SEPARATION_BOUNDS_EXPLICIT,
+	"bounds to use during separation")
+ISL_ARG_BOOL(struct isl_options, ast_build_scale_strides, 0,
+	"ast-build-scale-strides", 1,
+	"allow iterators of strided loops to be scaled down")
+ISL_ARG_BOOL(struct isl_options, ast_build_allow_else, 0,
+	"ast-build-allow-else", 1, "generate if statements with else branches")
+ISL_ARG_BOOL(struct isl_options, ast_build_allow_or, 0,
+	"ast-build-allow-or", 1, "generate if conditions with disjunctions")
+ISL_ARG_BOOL(struct isl_options, print_stats, 0, "print-stats", 0,
+	"print statistics for every isl_ctx")
+ISL_ARG_ULONG(struct isl_options, max_operations, 0,
+	"max-operations", 0, "default number of maximal operations per isl_ctx")
+ISL_ARG_VERSION(print_version)
+ISL_ARGS_END
+
+ISL_ARG_DEF(isl_options, struct isl_options, isl_options_args)
+
+ISL_ARG_CTX_DEF(isl_options, struct isl_options, isl_options_args)
+
+ISL_CTX_SET_CHOICE_DEF(isl_options, struct isl_options, isl_options_args, bound)
+ISL_CTX_GET_CHOICE_DEF(isl_options, struct isl_options, isl_options_args, bound)
+
+ISL_CTX_SET_CHOICE_DEF(isl_options, struct isl_options, isl_options_args,
+	on_error)
+ISL_CTX_GET_CHOICE_DEF(isl_options, struct isl_options, isl_options_args,
+	on_error)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	coalesce_bounded_wrapping)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	coalesce_bounded_wrapping)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	gbr_only_first)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	gbr_only_first)
+
+ISL_CTX_SET_INT_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_max_coefficient)
+ISL_CTX_GET_INT_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_max_coefficient)
+
+ISL_CTX_SET_INT_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_max_constant_term)
+ISL_CTX_GET_INT_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_max_constant_term)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_maximize_band_depth)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_maximize_band_depth)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_split_scaled)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_split_scaled)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_separate_components)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_separate_components)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_outer_coincidence)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_outer_coincidence)
+
+ISL_CTX_SET_CHOICE_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_algorithm)
+ISL_CTX_GET_CHOICE_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_algorithm)
+
+ISL_CTX_SET_CHOICE_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_fuse)
+ISL_CTX_GET_CHOICE_DEF(isl_options, struct isl_options, isl_options_args,
+	schedule_fuse)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	tile_scale_tile_loops)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	tile_scale_tile_loops)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	tile_shift_point_loops)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	tile_shift_point_loops)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_atomic_upper_bound)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_atomic_upper_bound)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_prefer_pdiv)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_prefer_pdiv)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_exploit_nested_bounds)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_exploit_nested_bounds)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_group_coscheduled)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_group_coscheduled)
+
+ISL_CTX_SET_STR_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_iterator_type)
+ISL_CTX_GET_STR_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_iterator_type)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_always_print_block)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_always_print_block)
+
+ISL_CTX_SET_CHOICE_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_separation_bounds)
+ISL_CTX_GET_CHOICE_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_separation_bounds)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_scale_strides)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_scale_strides)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_allow_else)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_allow_else)
+
+ISL_CTX_SET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_allow_or)
+ISL_CTX_GET_BOOL_DEF(isl_options, struct isl_options, isl_options_args,
+	ast_build_allow_or)
diff --git a/polly/lib/External/isl/isl_options_private.h b/polly/lib/External/isl/isl_options_private.h
new file mode 100644
index 00000000000..fbf81fa6ed0
--- /dev/null
+++ b/polly/lib/External/isl/isl_options_private.h
@@ -0,0 +1,67 @@
+#ifndef ISL_OPTIONS_PRIVATE_H
+#define ISL_OPTIONS_PRIVATE_H
+
+#include <isl/options.h>
+
+struct isl_options {
+	#define			ISL_CONTEXT_GBR		0
+	#define			ISL_CONTEXT_LEXMIN	1
+	unsigned		context;
+
+	#define			ISL_GBR_NEVER	0
+	#define			ISL_GBR_ONCE	1
+	#define			ISL_GBR_ALWAYS	2
+	unsigned		gbr;
+	unsigned		gbr_only_first;
+
+	#define			ISL_CLOSURE_ISL		0
+	#define			ISL_CLOSURE_BOX		1
+	unsigned		closure;
+
+	int			bound;
+	unsigned		on_error;
+
+	#define			ISL_BERNSTEIN_FACTORS	1
+	#define			ISL_BERNSTEIN_INTERVALS	2
+	int			bernstein_recurse;
+
+	int			bernstein_triangulate;
+
+	int			pip_symmetry;
+
+	#define			ISL_CONVEX_HULL_WRAP	0
+	#define			ISL_CONVEX_HULL_FM	1
+	int			convex;
+
+	int			coalesce_bounded_wrapping;
+
+	int			schedule_max_coefficient;
+	int			schedule_max_constant_term;
+	int			schedule_parametric;
+	int			schedule_outer_coincidence;
+	int			schedule_maximize_band_depth;
+	int			schedule_split_scaled;
+	int			schedule_separate_components;
+	unsigned		schedule_algorithm;
+	int			schedule_fuse;
+
+	int			tile_scale_tile_loops;
+	int			tile_shift_point_loops;
+
+	char			*ast_iterator_type;
+	int			ast_always_print_block;
+
+	int			ast_build_atomic_upper_bound;
+	int			ast_build_prefer_pdiv;
+	int			ast_build_exploit_nested_bounds;
+	int			ast_build_group_coscheduled;
+	int			ast_build_separation_bounds;
+	int			ast_build_scale_strides;
+	int			ast_build_allow_else;
+	int			ast_build_allow_or;
+
+	int			print_stats;
+	unsigned long		max_operations;
+};
+
+#endif
diff --git a/polly/lib/External/isl/isl_output.c b/polly/lib/External/isl/isl_output.c
new file mode 100644
index 00000000000..b5d6c923384
--- /dev/null
+++ b/polly/lib/External/isl/isl_output.c
@@ -0,0 +1,2698 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2010      INRIA Saclay
+ * Copyright 2012-2013 Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France 
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+#include <stdlib.h>
+#include <string.h>
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl/set.h>
+#include <isl_seq.h>
+#include <isl_polynomial_private.h>
+#include <isl_printer_private.h>
+#include <isl_space_private.h>
+#include <isl_mat_private.h>
+#include <isl_vec_private.h>
+#include <isl/union_map.h>
+#include <isl/constraint.h>
+#include <isl_local_space_private.h>
+#include <isl_aff_private.h>
+#include <isl_val_private.h>
+#include <isl/ast_build.h>
+#include <isl_sort.h>
+
+static const char *s_to[2] = { " -> ", " \\to " };
+static const char *s_and[2] = { " and ", " \\wedge " };
+static const char *s_or[2] = { " or ", " \\vee " };
+static const char *s_le[2] = { "<=", "\\le" };
+static const char *s_ge[2] = { ">=", "\\ge" };
+static const char *s_open_set[2] = { "{ ", "\\{\\, " };
+static const char *s_close_set[2] = { " }", " \\,\\}" };
+static const char *s_open_list[2] = { "[", "(" };
+static const char *s_close_list[2] = { "]", ")" };
+static const char *s_such_that[2] = { " : ", " \\mid " };
+static const char *s_open_exists[2] = { "exists (", "\\exists \\, " };
+static const char *s_close_exists[2] = { ")", "" };
+static const char *s_div_prefix[2] = { "e", "\\alpha_" };
+static const char *s_param_prefix[2] = { "p", "p_" };
+static const char *s_input_prefix[2] = { "i", "i_" };
+static const char *s_output_prefix[2] = { "o", "o_" };
+
+static __isl_give isl_printer *print_constraint_polylib(
+	struct isl_basic_map *bmap, int ineq, int n, __isl_take isl_printer *p)
+{
+	int i;
+	unsigned n_in = isl_basic_map_dim(bmap, isl_dim_in);
+	unsigned n_out = isl_basic_map_dim(bmap, isl_dim_out);
+	unsigned nparam = isl_basic_map_dim(bmap, isl_dim_param);
+	isl_int *c = ineq ? bmap->ineq[n] : bmap->eq[n];
+
+	p = isl_printer_start_line(p);
+	p = isl_printer_print_int(p, ineq);
+	for (i = 0; i < n_out; ++i) {
+		p = isl_printer_print_str(p, " ");
+		p = isl_printer_print_isl_int(p, c[1+nparam+n_in+i]);
+	}
+	for (i = 0; i < n_in; ++i) {
+		p = isl_printer_print_str(p, " ");
+		p = isl_printer_print_isl_int(p, c[1+nparam+i]);
+	}
+	for (i = 0; i < bmap->n_div; ++i) {
+		p = isl_printer_print_str(p, " ");
+		p = isl_printer_print_isl_int(p, c[1+nparam+n_in+n_out+i]);
+	}
+	for (i = 0; i < nparam; ++i) {
+		p = isl_printer_print_str(p, " ");
+		p = isl_printer_print_isl_int(p, c[1+i]);
+	}
+	p = isl_printer_print_str(p, " ");
+	p = isl_printer_print_isl_int(p, c[0]);
+	p = isl_printer_end_line(p);
+	return p;
+}
+
+static __isl_give isl_printer *print_constraints_polylib(
+	struct isl_basic_map *bmap, __isl_take isl_printer *p)
+{
+	int i;
+
+	p = isl_printer_set_isl_int_width(p, 5);
+
+	for (i = 0; i < bmap->n_eq; ++i)
+		p = print_constraint_polylib(bmap, 0, i, p);
+	for (i = 0; i < bmap->n_ineq; ++i)
+		p = print_constraint_polylib(bmap, 1, i, p);
+
+	return p;
+}
+
+static __isl_give isl_printer *bset_print_constraints_polylib(
+	struct isl_basic_set *bset, __isl_take isl_printer *p)
+{
+	return print_constraints_polylib((struct isl_basic_map *)bset, p);
+}
+
+static __isl_give isl_printer *isl_basic_map_print_polylib(
+	__isl_keep isl_basic_map *bmap, __isl_take isl_printer *p, int ext)
+{
+	unsigned total = isl_basic_map_total_dim(bmap);
+	p = isl_printer_start_line(p);
+	p = isl_printer_print_int(p, bmap->n_eq + bmap->n_ineq);
+	p = isl_printer_print_str(p, " ");
+	p = isl_printer_print_int(p, 1 + total + 1);
+	if (ext) {
+		p = isl_printer_print_str(p, " ");
+		p = isl_printer_print_int(p,
+				    isl_basic_map_dim(bmap, isl_dim_out));
+		p = isl_printer_print_str(p, " ");
+		p = isl_printer_print_int(p,
+				    isl_basic_map_dim(bmap, isl_dim_in));
+		p = isl_printer_print_str(p, " ");
+		p = isl_printer_print_int(p,
+				    isl_basic_map_dim(bmap, isl_dim_div));
+		p = isl_printer_print_str(p, " ");
+		p = isl_printer_print_int(p,
+				    isl_basic_map_dim(bmap, isl_dim_param));
+	}
+	p = isl_printer_end_line(p);
+	return print_constraints_polylib(bmap, p);
+}
+
+static __isl_give isl_printer *isl_basic_set_print_polylib(
+	__isl_keep isl_basic_set *bset, __isl_take isl_printer *p, int ext)
+{
+	return isl_basic_map_print_polylib((struct isl_basic_map *)bset, p, ext);
+}
+
+static __isl_give isl_printer *isl_map_print_polylib(__isl_keep isl_map *map,
+	__isl_take isl_printer *p, int ext)
+{
+	int i;
+
+	p = isl_printer_start_line(p);
+	p = isl_printer_print_int(p, map->n);
+	p = isl_printer_end_line(p);
+	for (i = 0; i < map->n; ++i) {
+		p = isl_printer_start_line(p);
+		p = isl_printer_end_line(p);
+		p = isl_basic_map_print_polylib(map->p[i], p, ext);
+	}
+	return p;
+}
+
+static __isl_give isl_printer *isl_set_print_polylib(__isl_keep isl_set *set,
+	__isl_take isl_printer *p, int ext)
+{
+	return isl_map_print_polylib((struct isl_map *)set, p, ext);
+}
+
+static int count_same_name(__isl_keep isl_space *dim,
+	enum isl_dim_type type, unsigned pos, const char *name)
+{
+	enum isl_dim_type t;
+	unsigned p, s;
+	int count = 0;
+
+	for (t = isl_dim_param; t <= type && t <= isl_dim_out; ++t) {
+		s = t == type ? pos : isl_space_dim(dim, t);
+		for (p = 0; p < s; ++p) {
+			const char *n = isl_space_get_dim_name(dim, t, p);
+			if (n && !strcmp(n, name))
+				count++;
+		}
+	}
+	return count;
+}
+
+static __isl_give isl_printer *print_name(__isl_keep isl_space *dim,
+	__isl_take isl_printer *p, enum isl_dim_type type, unsigned pos,
+	int latex)
+{
+	const char *name;
+	char buffer[20];
+	int primes;
+
+	name = type == isl_dim_div ? NULL : isl_space_get_dim_name(dim, type, pos);
+
+	if (!name) {
+		const char *prefix;
+		if (type == isl_dim_param)
+			prefix = s_param_prefix[latex];
+		else if (type == isl_dim_div)
+			prefix = s_div_prefix[latex];
+		else if (isl_space_is_set(dim) || type == isl_dim_in)
+			prefix = s_input_prefix[latex];
+		else
+			prefix = s_output_prefix[latex];
+		snprintf(buffer, sizeof(buffer), "%s%d", prefix, pos);
+		name = buffer;
+	}
+	primes = count_same_name(dim, name == buffer ? isl_dim_div : type,
+				 pos, name);
+	p = isl_printer_print_str(p, name);
+	while (primes-- > 0)
+		p = isl_printer_print_str(p, "'");
+	return p;
+}
+
+static enum isl_dim_type pos2type(__isl_keep isl_space *dim, unsigned *pos)
+{
+	enum isl_dim_type type;
+	unsigned n_in = isl_space_dim(dim, isl_dim_in);
+	unsigned n_out = isl_space_dim(dim, isl_dim_out);
+	unsigned nparam = isl_space_dim(dim, isl_dim_param);
+
+	if (*pos < 1 + nparam) {
+		type = isl_dim_param;
+		*pos -= 1;
+	} else if (*pos < 1 + nparam + n_in) {
+		type = isl_dim_in;
+		*pos -= 1 + nparam;
+	} else if (*pos < 1 + nparam + n_in + n_out) {
+		type = isl_dim_out;
+		*pos -= 1 + nparam + n_in;
+	} else {
+		type = isl_dim_div;
+		*pos -= 1 + nparam + n_in + n_out;
+	}
+
+	return type;
+}
+
+static __isl_give isl_printer *print_div(__isl_keep isl_space *dim,
+	__isl_keep isl_mat *div, int pos, __isl_take isl_printer *p);
+
+static __isl_give isl_printer *print_term(__isl_keep isl_space *dim,
+	__isl_keep isl_mat *div,
+	isl_int c, unsigned pos, __isl_take isl_printer *p, int latex)
+{
+	enum isl_dim_type type;
+	int print_div_def;
+
+	if (pos == 0)
+		return isl_printer_print_isl_int(p, c);
+
+	type = pos2type(dim, &pos);
+	print_div_def = type == isl_dim_div && div &&
+			!isl_int_is_zero(div->row[pos][0]);
+
+	if (isl_int_is_one(c))
+		;
+	else if (isl_int_is_negone(c))
+		p = isl_printer_print_str(p, "-");
+	else {
+		p = isl_printer_print_isl_int(p, c);
+		if (p->output_format == ISL_FORMAT_C || print_div_def)
+			p = isl_printer_print_str(p, "*");
+	}
+	if (print_div_def)
+		p = print_div(dim, div, pos, p);
+	else
+		p = print_name(dim, p, type, pos, latex);
+	return p;
+}
+
+static __isl_give isl_printer *print_affine_of_len(__isl_keep isl_space *dim,
+	__isl_keep isl_mat *div,
+	__isl_take isl_printer *p, isl_int *c, int len)
+{
+	int i;
+	int first;
+
+	for (i = 0, first = 1; i < len; ++i) {
+		int flip = 0;
+		if (isl_int_is_zero(c[i]))
+			continue;
+		if (!first) {
+			if (isl_int_is_neg(c[i])) {
+				flip = 1;
+				isl_int_neg(c[i], c[i]);
+				p = isl_printer_print_str(p, " - ");
+			} else 
+				p = isl_printer_print_str(p, " + ");
+		}
+		first = 0;
+		p = print_term(dim, div, c[i], i, p, 0);
+		if (flip)
+			isl_int_neg(c[i], c[i]);
+	}
+	if (first)
+		p = isl_printer_print_str(p, "0");
+	return p;
+}
+
+static __isl_give isl_printer *print_affine(__isl_keep isl_basic_map *bmap,
+	__isl_keep isl_space *dim, __isl_take isl_printer *p, isl_int *c)
+{
+	unsigned len = 1 + isl_basic_map_total_dim(bmap);
+	return print_affine_of_len(dim, NULL, p, c, len);
+}
+
+/* Internal data structure for print_space.
+ *
+ * latex is set if that is the output format.
+ * print_dim (if not NULL) is called on each dimension.
+ * user is set by the caller of print_space and may be used inside print_dim.
+ *
+ * space is the global space that is being printed.  This field is set by
+ *	print_space.
+ * type is the tuple of the global space that is currently being printed.
+ *	This field is set by print_space.
+ */
+struct isl_print_space_data {
+	int latex;
+	__isl_give isl_printer *(*print_dim)(__isl_take isl_printer *p,
+		struct isl_print_space_data *data, unsigned pos);
+	void *user;
+
+	isl_space *space;
+	enum isl_dim_type type;
+};
+
+/* offset is the offset of local_dim inside data->type of data->space.
+ */
+static __isl_give isl_printer *print_nested_var_list(__isl_take isl_printer *p,
+	__isl_keep isl_space *local_dim, enum isl_dim_type local_type,
+	struct isl_print_space_data *data, int offset)
+{
+	int i;
+
+	if (data->space != local_dim && local_type == isl_dim_out)
+		offset += local_dim->n_in;
+
+	for (i = 0; i < isl_space_dim(local_dim, local_type); ++i) {
+		if (i)
+			p = isl_printer_print_str(p, ", ");
+		if (data->print_dim)
+			p = data->print_dim(p, data, offset + i);
+		else
+			p = print_name(data->space, p, data->type, offset + i,
+					data->latex);
+	}
+	return p;
+}
+
+static __isl_give isl_printer *print_var_list(__isl_take isl_printer *p,
+	__isl_keep isl_space *space, enum isl_dim_type type)
+{
+	struct isl_print_space_data data = { .space = space, .type = type };
+
+	return print_nested_var_list(p, space, type, &data, 0);
+}
+
+static __isl_give isl_printer *print_nested_map_dim(__isl_take isl_printer *p,
+	__isl_keep isl_space *local_dim,
+	struct isl_print_space_data *data, int offset);
+
+static __isl_give isl_printer *print_nested_tuple(__isl_take isl_printer *p,
+	__isl_keep isl_space *local_dim, enum isl_dim_type local_type,
+	struct isl_print_space_data *data, int offset)
+{
+	const char *name = NULL;
+	unsigned n = isl_space_dim(local_dim, local_type);
+	if ((local_type == isl_dim_in || local_type == isl_dim_out)) {
+		name = isl_space_get_tuple_name(local_dim, local_type);
+		if (name) {
+			if (data->latex)
+				p = isl_printer_print_str(p, "\\mathrm{");
+			p = isl_printer_print_str(p, name);
+			if (data->latex)
+				p = isl_printer_print_str(p, "}");
+		}
+	}
+	if (!data->latex || n != 1 || name)
+		p = isl_printer_print_str(p, s_open_list[data->latex]);
+	if ((local_type == isl_dim_in || local_type == isl_dim_out) &&
+	    local_dim->nested[local_type - isl_dim_in]) {
+		if (data->space != local_dim && local_type == isl_dim_out)
+			offset += local_dim->n_in;
+		p = print_nested_map_dim(p,
+				local_dim->nested[local_type - isl_dim_in],
+				data, offset);
+	} else
+		p = print_nested_var_list(p, local_dim, local_type, data,
+					  offset);
+	if (!data->latex || n != 1 || name)
+		p = isl_printer_print_str(p, s_close_list[data->latex]);
+	return p;
+}
+
+static __isl_give isl_printer *print_tuple(__isl_keep isl_space *dim,
+	__isl_take isl_printer *p, enum isl_dim_type type,
+	struct isl_print_space_data *data)
+{
+	data->space = dim;
+	data->type = type;
+	return print_nested_tuple(p, dim, type, data, 0);
+}
+
+static __isl_give isl_printer *print_nested_map_dim(__isl_take isl_printer *p,
+	__isl_keep isl_space *local_dim,
+	struct isl_print_space_data *data, int offset)
+{
+	p = print_nested_tuple(p, local_dim, isl_dim_in, data, offset);
+	p = isl_printer_print_str(p, s_to[data->latex]);
+	p = print_nested_tuple(p, local_dim, isl_dim_out, data, offset);
+
+	return p;
+}
+
+static __isl_give isl_printer *print_space(__isl_keep isl_space *dim,
+	__isl_take isl_printer *p, int rational,
+	struct isl_print_space_data *data)
+{
+	if (rational && !data->latex)
+		p = isl_printer_print_str(p, "rat: ");
+	if (isl_space_is_params(dim))
+		;
+	else if (isl_space_is_set(dim))
+		p = print_tuple(dim, p, isl_dim_set, data);
+	else {
+		p = print_tuple(dim, p, isl_dim_in, data);
+		p = isl_printer_print_str(p, s_to[data->latex]);
+		p = print_tuple(dim, p, isl_dim_out, data);
+	}
+
+	return p;
+}
+
+static __isl_give isl_printer *print_omega_parameters(__isl_keep isl_space *dim,
+	__isl_take isl_printer *p)
+{
+	if (isl_space_dim(dim, isl_dim_param) == 0)
+		return p;
+
+	p = isl_printer_start_line(p);
+	p = isl_printer_print_str(p, "symbolic ");
+	p = print_var_list(p, dim, isl_dim_param);
+	p = isl_printer_print_str(p, ";");
+	p = isl_printer_end_line(p);
+	return p;
+}
+
+static __isl_give isl_printer *print_constraint(struct isl_basic_map *bmap,
+	__isl_keep isl_space *dim, __isl_take isl_printer *p,
+	isl_int *c, int last, const char *op, int first_constraint, int latex)
+{
+	if (!first_constraint)
+		p = isl_printer_print_str(p, s_and[latex]);
+
+	isl_int_abs(c[last], c[last]);
+
+	p = print_term(dim, NULL, c[last], last, p, latex);
+
+	p = isl_printer_print_str(p, " ");
+	p = isl_printer_print_str(p, op);
+	p = isl_printer_print_str(p, " ");
+
+	isl_int_set_si(c[last], 0);
+	p = print_affine(bmap, dim, p, c);
+
+	return p;
+}
+
+static __isl_give isl_printer *print_constraints(__isl_keep isl_basic_map *bmap,
+	__isl_keep isl_space *dim, __isl_take isl_printer *p, int latex)
+{
+	int i;
+	struct isl_vec *c;
+	unsigned total = isl_basic_map_total_dim(bmap);
+
+	c = isl_vec_alloc(bmap->ctx, 1 + total);
+	if (!c)
+		goto error;
+
+	for (i = bmap->n_eq - 1; i >= 0; --i) {
+		int l = isl_seq_last_non_zero(bmap->eq[i], 1 + total);
+		if (l < 0) {
+			if (i != bmap->n_eq - 1)
+				p = isl_printer_print_str(p, s_and[latex]);
+			p = isl_printer_print_str(p, "0 = 0");
+			continue;
+		}
+		if (isl_int_is_neg(bmap->eq[i][l]))
+			isl_seq_cpy(c->el, bmap->eq[i], 1 + total);
+		else
+			isl_seq_neg(c->el, bmap->eq[i], 1 + total);
+		p = print_constraint(bmap, dim, p, c->el, l,
+				    "=", i == bmap->n_eq - 1, latex);
+	}
+	for (i = 0; i < bmap->n_ineq; ++i) {
+		int l = isl_seq_last_non_zero(bmap->ineq[i], 1 + total);
+		int s;
+		const char *op;
+		if (l < 0)
+			continue;
+		s = isl_int_sgn(bmap->ineq[i][l]);
+		if (s < 0)
+			isl_seq_cpy(c->el, bmap->ineq[i], 1 + total);
+		else
+			isl_seq_neg(c->el, bmap->ineq[i], 1 + total);
+		op = s < 0 ? s_le[latex] : s_ge[latex];
+		p = print_constraint(bmap, dim, p, c->el, l,
+					op, !bmap->n_eq && !i, latex);
+	}
+
+	isl_vec_free(c);
+
+	return p;
+error:
+	isl_vec_free(c);
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *print_div(__isl_keep isl_space *dim,
+	__isl_keep isl_mat *div, int pos, __isl_take isl_printer *p)
+{
+	int c = p->output_format == ISL_FORMAT_C;
+	p = isl_printer_print_str(p, c ? "floord(" : "floor((");
+	p = print_affine_of_len(dim, div, p,
+				div->row[pos] + 1, div->n_col - 1);
+	p = isl_printer_print_str(p, c ? ", " : ")/");
+	p = isl_printer_print_isl_int(p, div->row[pos][0]);
+	p = isl_printer_print_str(p, ")");
+	return p;
+}
+
+/* Print a comma separated list of div names, with their definitions
+ * (provided that they have a definition and we are printing in isl format).
+ */
+static __isl_give isl_printer *print_div_list(__isl_take isl_printer *p,
+	__isl_keep isl_space *space, __isl_keep isl_mat *div, int latex)
+{
+	int i;
+	unsigned n_div;
+
+	if (!p || !space || !div)
+		return isl_printer_free(p);
+
+	n_div = isl_mat_rows(div);
+
+	for (i = 0; i < n_div; ++i) {
+		if (i)
+			p = isl_printer_print_str(p, ", ");
+		p = print_name(space, p, isl_dim_div, i, latex);
+		if (p->output_format != ISL_FORMAT_ISL ||
+		    isl_int_is_zero(div->row[i][0]))
+			continue;
+		p = isl_printer_print_str(p, " = ");
+		p = print_div(space, div, i, p);
+	}
+
+	return p;
+}
+
+static __isl_give isl_printer *print_disjunct(__isl_keep isl_basic_map *bmap,
+	__isl_keep isl_space *dim, __isl_take isl_printer *p, int latex)
+{
+	if (bmap->n_div > 0) {
+		isl_space *space;
+		isl_mat *div;
+
+		space = isl_basic_map_get_space(bmap);
+		div = isl_basic_map_get_divs(bmap);
+		p = isl_printer_print_str(p, s_open_exists[latex]);
+		p = print_div_list(p, space, div, latex);
+		isl_space_free(space);
+		isl_mat_free(div);
+		p = isl_printer_print_str(p, ": ");
+	}
+
+	p = print_constraints(bmap, dim, p, latex);
+
+	if (bmap->n_div > 0)
+		p = isl_printer_print_str(p, s_close_exists[latex]);
+	return p;
+}
+
+/* Print a colon followed by the constraints of "bmap"
+ * to "p", provided there are any constraints.
+ * The names of the variables are taken from "space".
+ * "latex" is set if the constraints should be printed in LaTeX format.
+ */
+static __isl_give isl_printer *print_optional_disjunct(
+	__isl_keep isl_basic_map *bmap, __isl_keep isl_space *space,
+	__isl_take isl_printer *p, int latex)
+{
+	if (isl_basic_map_is_universe(bmap))
+		return p;
+
+	p = isl_printer_print_str(p, ": ");
+	p = print_disjunct(bmap, space, p, latex);
+
+	return p;
+}
+
+static __isl_give isl_printer *basic_map_print_omega(
+	__isl_keep isl_basic_map *bmap, __isl_take isl_printer *p)
+{
+	p = isl_printer_print_str(p, "{ [");
+	p = print_var_list(p, bmap->dim, isl_dim_in);
+	p = isl_printer_print_str(p, "] -> [");
+	p = print_var_list(p, bmap->dim, isl_dim_out);
+	p = isl_printer_print_str(p, "] ");
+	p = print_optional_disjunct(bmap, bmap->dim, p, 0);
+	p = isl_printer_print_str(p, " }");
+	return p;
+}
+
+static __isl_give isl_printer *basic_set_print_omega(
+	__isl_keep isl_basic_set *bset, __isl_take isl_printer *p)
+{
+	p = isl_printer_print_str(p, "{ [");
+	p = print_var_list(p, bset->dim, isl_dim_set);
+	p = isl_printer_print_str(p, "] ");
+	p = print_optional_disjunct(bset, bset->dim, p, 0);
+	p = isl_printer_print_str(p, " }");
+	return p;
+}
+
+static __isl_give isl_printer *isl_map_print_omega(__isl_keep isl_map *map,
+	__isl_take isl_printer *p)
+{
+	int i;
+
+	for (i = 0; i < map->n; ++i) {
+		if (i)
+			p = isl_printer_print_str(p, " union ");
+		p = basic_map_print_omega(map->p[i], p);
+	}
+	return p;
+}
+
+static __isl_give isl_printer *isl_set_print_omega(__isl_keep isl_set *set,
+	__isl_take isl_printer *p)
+{
+	int i;
+
+	for (i = 0; i < set->n; ++i) {
+		if (i)
+			p = isl_printer_print_str(p, " union ");
+		p = basic_set_print_omega(set->p[i], p);
+	}
+	return p;
+}
+
+static __isl_give isl_printer *isl_basic_map_print_isl(
+	__isl_keep isl_basic_map *bmap, __isl_take isl_printer *p,
+	int latex)
+{
+	struct isl_print_space_data data = { .latex = latex };
+	int rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
+
+	if (isl_basic_map_dim(bmap, isl_dim_param) > 0) {
+		p = print_tuple(bmap->dim, p, isl_dim_param, &data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+	p = isl_printer_print_str(p, "{ ");
+	p = print_space(bmap->dim, p, rational, &data);
+	p = isl_printer_print_str(p, " : ");
+	p = print_disjunct(bmap, bmap->dim, p, latex);
+	p = isl_printer_print_str(p, " }");
+	return p;
+}
+
+static __isl_give isl_printer *print_disjuncts(__isl_keep isl_map *map,
+	__isl_take isl_printer *p, int latex)
+{
+	int i;
+
+	if (isl_map_plain_is_universe(map))
+		return p;
+
+	p = isl_printer_print_str(p, s_such_that[latex]);
+	if (map->n == 0)
+		p = isl_printer_print_str(p, "1 = 0");
+	for (i = 0; i < map->n; ++i) {
+		if (i)
+			p = isl_printer_print_str(p, s_or[latex]);
+		if (map->n > 1 && map->p[i]->n_eq + map->p[i]->n_ineq > 1)
+			p = isl_printer_print_str(p, "(");
+		p = print_disjunct(map->p[i], map->dim, p, latex);
+		if (map->n > 1 && map->p[i]->n_eq + map->p[i]->n_ineq > 1)
+			p = isl_printer_print_str(p, ")");
+	}
+	return p;
+}
+
+/* Print the disjuncts of a map (or set).
+ * If the map turns out to be a universal parameter domain, then
+ * we need to print the colon.  Otherwise, the output looks identical
+ * to the empty set.
+ */
+static __isl_give isl_printer *print_disjuncts_map(__isl_keep isl_map *map,
+	__isl_take isl_printer *p, int latex)
+{
+	if (isl_map_plain_is_universe(map) && isl_space_is_params(map->dim))
+		return isl_printer_print_str(p, s_such_that[latex]);
+	else
+		return print_disjuncts(map, p, latex);
+}
+
+struct isl_aff_split {
+	isl_basic_map *aff;
+	isl_map *map;
+};
+
+static void free_split(__isl_take struct isl_aff_split *split, int n)
+{
+	int i;
+
+	if (!split)
+		return;
+
+	for (i = 0; i < n; ++i) {
+		isl_basic_map_free(split[i].aff);
+		isl_map_free(split[i].map);
+	}
+
+	free(split);
+}
+
+static __isl_give isl_basic_map *get_aff(__isl_take isl_basic_map *bmap)
+{
+	int i, j;
+	unsigned nparam, n_in, n_out, total;
+
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		return NULL;
+	if (isl_basic_map_free_inequality(bmap, bmap->n_ineq) < 0)
+		goto error;
+
+	nparam = isl_basic_map_dim(bmap, isl_dim_param);
+	n_in = isl_basic_map_dim(bmap, isl_dim_in);
+	n_out = isl_basic_map_dim(bmap, isl_dim_out);
+	total = isl_basic_map_dim(bmap, isl_dim_all);
+	for (i = bmap->n_eq - 1; i >= 0; --i) {
+		j = isl_seq_last_non_zero(bmap->eq[i] + 1, total);
+		if (j >= nparam && j < nparam + n_in + n_out &&
+		    (isl_int_is_one(bmap->eq[i][1 + j]) ||
+		     isl_int_is_negone(bmap->eq[i][1 + j])))
+			continue;
+		if (isl_basic_map_drop_equality(bmap, i) < 0)
+			goto error;
+	}
+
+	bmap = isl_basic_map_finalize(bmap);
+
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+static int aff_split_cmp(const void *p1, const void *p2, void *user)
+{
+	const struct isl_aff_split *s1, *s2;
+	s1 = (const struct isl_aff_split *) p1;
+	s2 = (const struct isl_aff_split *) p2;
+
+	return isl_basic_map_plain_cmp(s1->aff, s2->aff);
+}
+
+static __isl_give isl_basic_map *drop_aff(__isl_take isl_basic_map *bmap,
+	__isl_keep isl_basic_map *aff)
+{
+	int i, j;
+	unsigned total;
+
+	if (!bmap || !aff)
+		goto error;
+
+	total = isl_space_dim(bmap->dim, isl_dim_all);
+
+	for (i = bmap->n_eq - 1; i >= 0; --i) {
+		if (isl_seq_first_non_zero(bmap->eq[i] + 1 + total,
+					    bmap->n_div) != -1)
+			continue;
+		for (j = 0; j < aff->n_eq; ++j) {
+			if (!isl_seq_eq(bmap->eq[i], aff->eq[j], 1 + total) &&
+			    !isl_seq_is_neg(bmap->eq[i], aff->eq[j], 1 + total))
+				continue;
+			if (isl_basic_map_drop_equality(bmap, i) < 0)
+				goto error;
+			break;
+		}
+	}
+
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+static __isl_give struct isl_aff_split *split_aff(__isl_keep isl_map *map)
+{
+	int i, n;
+	struct isl_aff_split *split;
+	isl_ctx *ctx;
+
+	ctx = isl_map_get_ctx(map);
+	split = isl_calloc_array(ctx, struct isl_aff_split, map->n);
+	if (!split)
+		return NULL;
+
+	for (i = 0; i < map->n; ++i) {
+		isl_basic_map *bmap;
+		split[i].aff = get_aff(isl_basic_map_copy(map->p[i]));
+		bmap = isl_basic_map_copy(map->p[i]);
+		bmap = isl_basic_map_cow(bmap);
+		bmap = drop_aff(bmap, split[i].aff);
+		split[i].map = isl_map_from_basic_map(bmap);
+		if (!split[i].aff || !split[i].map)
+			goto error;
+	}
+
+	if (isl_sort(split, map->n, sizeof(struct isl_aff_split),
+			&aff_split_cmp, NULL) < 0)
+		goto error;
+
+	n = map->n;
+	for (i = n - 1; i >= 1; --i) {
+		if (!isl_basic_map_plain_is_equal(split[i - 1].aff,
+						 split[i].aff))
+			continue;
+		isl_basic_map_free(split[i].aff);
+		split[i - 1].map = isl_map_union(split[i - 1].map,
+						 split[i].map);
+		if (i != n - 1)
+			split[i] = split[n - 1];
+		split[n - 1].aff = NULL;
+		split[n - 1].map = NULL;
+		--n;
+	}
+
+	return split;
+error:
+	free_split(split, map->n);
+	return NULL;
+}
+
+static int defining_equality(__isl_keep isl_basic_map *eq,
+	__isl_keep isl_space *dim, enum isl_dim_type type, int pos)
+{
+	int i;
+	unsigned total;
+
+	if (!eq)
+		return -1;
+
+	pos += isl_space_offset(dim, type);
+	total = isl_basic_map_total_dim(eq);
+
+	for (i = 0; i < eq->n_eq; ++i) {
+		if (isl_seq_last_non_zero(eq->eq[i] + 1, total) != pos)
+			continue;
+		if (isl_int_is_one(eq->eq[i][1 + pos]))
+			isl_seq_neg(eq->eq[i], eq->eq[i], 1 + total);
+		return i;
+	}
+
+	return -1;
+}
+
+/* Print dimension "pos" of data->space to "p".
+ *
+ * data->user is assumed to be an isl_basic_map keeping track of equalities.
+ *
+ * If the current dimension is defined by these equalities, then print
+ * the corresponding expression.  Otherwise, print the name of the dimension.
+ */
+static __isl_give isl_printer *print_dim_eq(__isl_take isl_printer *p,
+	struct isl_print_space_data *data, unsigned pos)
+{
+	isl_basic_map *eq = data->user;
+	int j;
+
+	j = defining_equality(eq, data->space, data->type, pos);
+	if (j >= 0) {
+		pos += 1 + isl_space_offset(data->space, data->type);
+		p = print_affine_of_len(eq->dim, NULL, p, eq->eq[j], pos);
+	} else {
+		p = print_name(data->space, p, data->type, pos, data->latex);
+	}
+
+	return p;
+}
+
+static __isl_give isl_printer *print_split_map(__isl_take isl_printer *p,
+	struct isl_aff_split *split, int n)
+{
+	struct isl_print_space_data data = { 0 };
+	int i;
+	int rational;
+
+	data.print_dim = &print_dim_eq;
+	for (i = 0; i < n; ++i) {
+		isl_space *dim;
+
+		if (!split[i].map)
+			break;
+		dim = split[i].map->dim;
+		rational = split[i].map->n > 0 &&
+		    ISL_F_ISSET(split[i].map->p[0], ISL_BASIC_MAP_RATIONAL);
+		if (i)
+			p = isl_printer_print_str(p, "; ");
+		data.user = split[i].aff;
+		p = print_space(dim, p, rational, &data);
+		p = print_disjuncts_map(split[i].map, p, 0);
+	}
+
+	return p;
+}
+
+static __isl_give isl_printer *isl_map_print_isl_body(__isl_keep isl_map *map,
+	__isl_take isl_printer *p)
+{
+	struct isl_print_space_data data = { 0 };
+	struct isl_aff_split *split = NULL;
+	int rational;
+
+	if (map->n > 0)
+		split = split_aff(map);
+	if (split) {
+		p = print_split_map(p, split, map->n);
+	} else {
+		rational = map->n > 0 &&
+		    ISL_F_ISSET(map->p[0], ISL_BASIC_MAP_RATIONAL);
+		p = print_space(map->dim, p, rational, &data);
+		p = print_disjuncts_map(map, p, 0);
+	}
+	free_split(split, map->n);
+	return p;
+}
+
+static __isl_give isl_printer *isl_map_print_isl(__isl_keep isl_map *map,
+	__isl_take isl_printer *p)
+{
+	struct isl_print_space_data data = { 0 };
+
+	if (isl_map_dim(map, isl_dim_param) > 0) {
+		p = print_tuple(map->dim, p, isl_dim_param, &data);
+		p = isl_printer_print_str(p, s_to[0]);
+	}
+	p = isl_printer_print_str(p, s_open_set[0]);
+	p = isl_map_print_isl_body(map, p);
+	p = isl_printer_print_str(p, s_close_set[0]);
+	return p;
+}
+
+static __isl_give isl_printer *print_latex_map(__isl_keep isl_map *map,
+	__isl_take isl_printer *p, __isl_keep isl_basic_map *aff)
+{
+	struct isl_print_space_data data = { 0 };
+
+	data.latex = 1;
+	if (isl_map_dim(map, isl_dim_param) > 0) {
+		p = print_tuple(map->dim, p, isl_dim_param, &data);
+		p = isl_printer_print_str(p, s_to[1]);
+	}
+	p = isl_printer_print_str(p, s_open_set[1]);
+	data.print_dim = &print_dim_eq;
+	data.user = aff;
+	p = print_space(map->dim, p, 0, &data);
+	p = print_disjuncts_map(map, p, 1);
+	p = isl_printer_print_str(p, s_close_set[1]);
+
+	return p;
+}
+
+static __isl_give isl_printer *isl_map_print_latex(__isl_keep isl_map *map,
+	__isl_take isl_printer *p)
+{
+	int i;
+	struct isl_aff_split *split = NULL;
+
+	if (map->n > 0)
+		split = split_aff(map);
+
+	if (!split)
+		return print_latex_map(map, p, NULL);
+
+	for (i = 0; i < map->n; ++i) {
+		if (!split[i].map)
+			break;
+		if (i)
+			p = isl_printer_print_str(p, " \\cup ");
+		p = print_latex_map(split[i].map, p, split[i].aff);
+	}
+
+	free_split(split, map->n);
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_print_basic_map(__isl_take isl_printer *p,
+	__isl_keep isl_basic_map *bmap)
+{
+	if (!p || !bmap)
+		goto error;
+	if (p->output_format == ISL_FORMAT_ISL)
+		return isl_basic_map_print_isl(bmap, p, 0);
+	else if (p->output_format == ISL_FORMAT_OMEGA)
+		return basic_map_print_omega(bmap, p);
+	isl_assert(bmap->ctx, 0, goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+void isl_basic_map_print(__isl_keep isl_basic_map *bmap, FILE *out, int indent,
+	const char *prefix, const char *suffix, unsigned output_format)
+{
+	isl_printer *printer;
+
+	if (!bmap)
+		return;
+
+	printer = isl_printer_to_file(bmap->ctx, out);
+	printer = isl_printer_set_indent(printer, indent);
+	printer = isl_printer_set_prefix(printer, prefix);
+	printer = isl_printer_set_suffix(printer, suffix);
+	printer = isl_printer_set_output_format(printer, output_format);
+	isl_printer_print_basic_map(printer, bmap);
+
+	isl_printer_free(printer);
+}
+
+__isl_give isl_printer *isl_printer_print_basic_set(__isl_take isl_printer *p,
+	__isl_keep isl_basic_set *bset)
+{
+	if (!p || !bset)
+		goto error;
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return isl_basic_map_print_isl(bset, p, 0);
+	else if (p->output_format == ISL_FORMAT_POLYLIB)
+		return isl_basic_set_print_polylib(bset, p, 0);
+	else if (p->output_format == ISL_FORMAT_EXT_POLYLIB)
+		return isl_basic_set_print_polylib(bset, p, 1);
+	else if (p->output_format == ISL_FORMAT_POLYLIB_CONSTRAINTS)
+		return bset_print_constraints_polylib(bset, p);
+	else if (p->output_format == ISL_FORMAT_OMEGA)
+		return basic_set_print_omega(bset, p);
+	isl_assert(p->ctx, 0, goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+void isl_basic_set_print(struct isl_basic_set *bset, FILE *out, int indent,
+	const char *prefix, const char *suffix, unsigned output_format)
+{
+	isl_printer *printer;
+
+	if (!bset)
+		return;
+
+	printer = isl_printer_to_file(bset->ctx, out);
+	printer = isl_printer_set_indent(printer, indent);
+	printer = isl_printer_set_prefix(printer, prefix);
+	printer = isl_printer_set_suffix(printer, suffix);
+	printer = isl_printer_set_output_format(printer, output_format);
+	isl_printer_print_basic_set(printer, bset);
+
+	isl_printer_free(printer);
+}
+
+__isl_give isl_printer *isl_printer_print_set(__isl_take isl_printer *p,
+	__isl_keep isl_set *set)
+{
+	if (!p || !set)
+		goto error;
+	if (p->output_format == ISL_FORMAT_ISL)
+		return isl_map_print_isl((isl_map *)set, p);
+	else if (p->output_format == ISL_FORMAT_POLYLIB)
+		return isl_set_print_polylib(set, p, 0);
+	else if (p->output_format == ISL_FORMAT_EXT_POLYLIB)
+		return isl_set_print_polylib(set, p, 1);
+	else if (p->output_format == ISL_FORMAT_OMEGA)
+		return isl_set_print_omega(set, p);
+	else if (p->output_format == ISL_FORMAT_LATEX)
+		return isl_map_print_latex((isl_map *)set, p);
+	isl_assert(set->ctx, 0, goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+void isl_set_print(struct isl_set *set, FILE *out, int indent,
+	unsigned output_format)
+{
+	isl_printer *printer;
+
+	if (!set)
+		return;
+
+	printer = isl_printer_to_file(set->ctx, out);
+	printer = isl_printer_set_indent(printer, indent);
+	printer = isl_printer_set_output_format(printer, output_format);
+	printer = isl_printer_print_set(printer, set);
+
+	isl_printer_free(printer);
+}
+
+__isl_give isl_printer *isl_printer_print_map(__isl_take isl_printer *p,
+	__isl_keep isl_map *map)
+{
+	if (!p || !map)
+		goto error;
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return isl_map_print_isl(map, p);
+	else if (p->output_format == ISL_FORMAT_POLYLIB)
+		return isl_map_print_polylib(map, p, 0);
+	else if (p->output_format == ISL_FORMAT_EXT_POLYLIB)
+		return isl_map_print_polylib(map, p, 1);
+	else if (p->output_format == ISL_FORMAT_OMEGA)
+		return isl_map_print_omega(map, p);
+	else if (p->output_format == ISL_FORMAT_LATEX)
+		return isl_map_print_latex(map, p);
+	isl_assert(map->ctx, 0, goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+struct isl_union_print_data {
+	isl_printer *p;
+	int first;
+};
+
+static int print_map_body(__isl_take isl_map *map, void *user)
+{
+	struct isl_union_print_data *data;
+	data = (struct isl_union_print_data *)user;
+
+	if (!data->first)
+		data->p = isl_printer_print_str(data->p, "; ");
+	data->first = 0;
+
+	data->p = isl_map_print_isl_body(map, data->p);
+	isl_map_free(map);
+
+	return 0;
+}
+
+static __isl_give isl_printer *isl_union_map_print_isl(
+	__isl_keep isl_union_map *umap, __isl_take isl_printer *p)
+{
+	struct isl_union_print_data data = { p, 1 };
+	struct isl_print_space_data space_data = { 0 };
+	isl_space *dim;
+
+	dim = isl_union_map_get_space(umap);
+	if (isl_space_dim(dim, isl_dim_param) > 0) {
+		p = print_tuple(dim, p, isl_dim_param, &space_data);
+		p = isl_printer_print_str(p, s_to[0]);
+	}
+	isl_space_free(dim);
+	p = isl_printer_print_str(p, s_open_set[0]);
+	isl_union_map_foreach_map(umap, &print_map_body, &data);
+	p = data.p;
+	p = isl_printer_print_str(p, s_close_set[0]);
+	return p;
+}
+
+static int print_latex_map_body(__isl_take isl_map *map, void *user)
+{
+	struct isl_union_print_data *data;
+	data = (struct isl_union_print_data *)user;
+
+	if (!data->first)
+		data->p = isl_printer_print_str(data->p, " \\cup ");
+	data->first = 0;
+
+	data->p = isl_map_print_latex(map, data->p);
+	isl_map_free(map);
+
+	return 0;
+}
+
+static __isl_give isl_printer *isl_union_map_print_latex(
+	__isl_keep isl_union_map *umap, __isl_take isl_printer *p)
+{
+	struct isl_union_print_data data = { p, 1 };
+	isl_union_map_foreach_map(umap, &print_latex_map_body, &data);
+	p = data.p;
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_print_union_map(__isl_take isl_printer *p,
+	__isl_keep isl_union_map *umap)
+{
+	if (!p || !umap)
+		goto error;
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return isl_union_map_print_isl(umap, p);
+	if (p->output_format == ISL_FORMAT_LATEX)
+		return isl_union_map_print_latex(umap, p);
+
+	isl_die(p->ctx, isl_error_invalid,
+		"invalid output format for isl_union_map", goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+__isl_give isl_printer *isl_printer_print_union_set(__isl_take isl_printer *p,
+	__isl_keep isl_union_set *uset)
+{
+	if (!p || !uset)
+		goto error;
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return isl_union_map_print_isl((isl_union_map *)uset, p);
+	if (p->output_format == ISL_FORMAT_LATEX)
+		return isl_union_map_print_latex((isl_union_map *)uset, p);
+
+	isl_die(p->ctx, isl_error_invalid,
+		"invalid output format for isl_union_set", goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+void isl_map_print(__isl_keep isl_map *map, FILE *out, int indent,
+	unsigned output_format)
+{
+	isl_printer *printer;
+
+	if (!map)
+		return;
+
+	printer = isl_printer_to_file(map->ctx, out);
+	printer = isl_printer_set_indent(printer, indent);
+	printer = isl_printer_set_output_format(printer, output_format);
+	printer = isl_printer_print_map(printer, map);
+
+	isl_printer_free(printer);
+}
+
+static int upoly_rec_n_non_zero(__isl_keep struct isl_upoly_rec *rec)
+{
+	int i;
+	int n;
+
+	for (i = 0, n = 0; i < rec->n; ++i)
+		if (!isl_upoly_is_zero(rec->p[i]))
+			++n;
+
+	return n;
+}
+
+static __isl_give isl_printer *upoly_print_cst(__isl_keep struct isl_upoly *up,
+	__isl_take isl_printer *p, int first)
+{
+	struct isl_upoly_cst *cst;
+	int neg;
+
+	cst = isl_upoly_as_cst(up);
+	if (!cst)
+		goto error;
+	neg = !first && isl_int_is_neg(cst->n);
+	if (!first)
+		p = isl_printer_print_str(p, neg ? " - " :  " + ");
+	if (neg)
+		isl_int_neg(cst->n, cst->n);
+	if (isl_int_is_zero(cst->d)) {
+		int sgn = isl_int_sgn(cst->n);
+		p = isl_printer_print_str(p, sgn < 0 ? "-infty" :
+					    sgn == 0 ? "NaN" : "infty");
+	} else
+		p = isl_printer_print_isl_int(p, cst->n);
+	if (neg)
+		isl_int_neg(cst->n, cst->n);
+	if (!isl_int_is_zero(cst->d) && !isl_int_is_one(cst->d)) {
+		p = isl_printer_print_str(p, "/");
+		p = isl_printer_print_isl_int(p, cst->d);
+	}
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *print_base(__isl_take isl_printer *p,
+	__isl_keep isl_space *dim, __isl_keep isl_mat *div, int var)
+{
+	unsigned total;
+
+	total = isl_space_dim(dim, isl_dim_all);
+	if (var < total)
+		p = print_term(dim, NULL, dim->ctx->one, 1 + var, p, 0);
+	else
+		p = print_div(dim, div, var - total, p);
+	return p;
+}
+
+static __isl_give isl_printer *print_pow(__isl_take isl_printer *p,
+	__isl_keep isl_space *dim, __isl_keep isl_mat *div, int var, int exp)
+{
+	p = print_base(p, dim, div, var);
+	if (exp == 1)
+		return p;
+	if (p->output_format == ISL_FORMAT_C) {
+		int i;
+		for (i = 1; i < exp; ++i) {
+			p = isl_printer_print_str(p, "*");
+			p = print_base(p, dim, div, var);
+		}
+	} else {
+		p = isl_printer_print_str(p, "^");
+		p = isl_printer_print_int(p, exp);
+	}
+	return p;
+}
+
+static __isl_give isl_printer *upoly_print(__isl_keep struct isl_upoly *up,
+	__isl_keep isl_space *dim, __isl_keep isl_mat *div,
+	__isl_take isl_printer *p, int outer)
+{
+	int i, n, first, print_parens;
+	struct isl_upoly_rec *rec;
+
+	if (!p || !up || !dim || !div)
+		goto error;
+
+	if (isl_upoly_is_cst(up))
+		return upoly_print_cst(up, p, 1);
+
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		goto error;
+	n = upoly_rec_n_non_zero(rec);
+	print_parens = n > 1 ||
+		    (outer && rec->up.var >= isl_space_dim(dim, isl_dim_all));
+	if (print_parens)
+		p = isl_printer_print_str(p, "(");
+	for (i = 0, first = 1; i < rec->n; ++i) {
+		if (isl_upoly_is_zero(rec->p[i]))
+			continue;
+		if (isl_upoly_is_negone(rec->p[i])) {
+			if (!i)
+				p = isl_printer_print_str(p, "-1");
+			else if (first)
+				p = isl_printer_print_str(p, "-");
+			else
+				p = isl_printer_print_str(p, " - ");
+		} else if (isl_upoly_is_cst(rec->p[i]) &&
+				!isl_upoly_is_one(rec->p[i]))
+			p = upoly_print_cst(rec->p[i], p, first);
+		else {
+			if (!first)
+				p = isl_printer_print_str(p, " + ");
+			if (i == 0 || !isl_upoly_is_one(rec->p[i]))
+				p = upoly_print(rec->p[i], dim, div, p, 0);
+		}
+		first = 0;
+		if (i == 0)
+			continue;
+		if (!isl_upoly_is_one(rec->p[i]) &&
+		    !isl_upoly_is_negone(rec->p[i]))
+			p = isl_printer_print_str(p, " * ");
+		p = print_pow(p, dim, div, rec->up.var, i);
+	}
+	if (print_parens)
+		p = isl_printer_print_str(p, ")");
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *print_qpolynomial(__isl_take isl_printer *p,
+	__isl_keep isl_qpolynomial *qp)
+{
+	if (!p || !qp)
+		goto error;
+	p = upoly_print(qp->upoly, qp->dim, qp->div, p, 1);
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *print_qpolynomial_isl(__isl_take isl_printer *p,
+	__isl_keep isl_qpolynomial *qp)
+{
+	struct isl_print_space_data data = { 0 };
+
+	if (!p || !qp)
+		goto error;
+
+	if (isl_space_dim(qp->dim, isl_dim_param) > 0) {
+		p = print_tuple(qp->dim, p, isl_dim_param, &data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+	p = isl_printer_print_str(p, "{ ");
+	if (!isl_space_is_params(qp->dim)) {
+		p = print_space(qp->dim, p, 0, &data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+	p = print_qpolynomial(p, qp);
+	p = isl_printer_print_str(p, " }");
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *print_qpolynomial_c(__isl_take isl_printer *p,
+	__isl_keep isl_space *dim, __isl_keep isl_qpolynomial *qp)
+{
+	isl_int den;
+
+	isl_int_init(den);
+	isl_qpolynomial_get_den(qp, &den);
+	if (!isl_int_is_one(den)) {
+		isl_qpolynomial *f;
+		p = isl_printer_print_str(p, "(");
+		qp = isl_qpolynomial_copy(qp);
+		f = isl_qpolynomial_rat_cst_on_domain(isl_space_copy(qp->dim),
+						den, qp->dim->ctx->one);
+		qp = isl_qpolynomial_mul(qp, f);
+	}
+	if (qp)
+		p = upoly_print(qp->upoly, dim, qp->div, p, 0);
+	else
+		p = isl_printer_free(p);
+	if (!isl_int_is_one(den)) {
+		p = isl_printer_print_str(p, ")/");
+		p = isl_printer_print_isl_int(p, den);
+		isl_qpolynomial_free(qp);
+	}
+	isl_int_clear(den);
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_print_qpolynomial(
+	__isl_take isl_printer *p, __isl_keep isl_qpolynomial *qp)
+{
+	if (!p || !qp)
+		goto error;
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return print_qpolynomial_isl(p, qp);
+	else if (p->output_format == ISL_FORMAT_C)
+		return print_qpolynomial_c(p, qp->dim, qp);
+	else
+		isl_die(qp->dim->ctx, isl_error_unsupported,
+			"output format not supported for isl_qpolynomials",
+			goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+void isl_qpolynomial_print(__isl_keep isl_qpolynomial *qp, FILE *out,
+	unsigned output_format)
+{
+	isl_printer *p;
+
+	if  (!qp)
+		return;
+
+	isl_assert(qp->dim->ctx, output_format == ISL_FORMAT_ISL, return);
+	p = isl_printer_to_file(qp->dim->ctx, out);
+	p = isl_printer_print_qpolynomial(p, qp);
+	isl_printer_free(p);
+}
+
+static __isl_give isl_printer *qpolynomial_fold_print(
+	__isl_keep isl_qpolynomial_fold *fold, __isl_take isl_printer *p)
+{
+	int i;
+
+	if (fold->type == isl_fold_min)
+		p = isl_printer_print_str(p, "min");
+	else if (fold->type == isl_fold_max)
+		p = isl_printer_print_str(p, "max");
+	p = isl_printer_print_str(p, "(");
+	for (i = 0; i < fold->n; ++i) {
+		if (i)
+			p = isl_printer_print_str(p, ", ");
+		p = print_qpolynomial(p, fold->qp[i]);
+	}
+	p = isl_printer_print_str(p, ")");
+	return p;
+}
+
+void isl_qpolynomial_fold_print(__isl_keep isl_qpolynomial_fold *fold,
+	FILE *out, unsigned output_format)
+{
+	isl_printer *p;
+
+	if (!fold)
+		return;
+
+	isl_assert(fold->dim->ctx, output_format == ISL_FORMAT_ISL, return);
+
+	p = isl_printer_to_file(fold->dim->ctx, out);
+	p = isl_printer_print_qpolynomial_fold(p, fold);
+
+	isl_printer_free(p);
+}
+
+static __isl_give isl_printer *isl_pwqp_print_isl_body(
+	__isl_take isl_printer *p, __isl_keep isl_pw_qpolynomial *pwqp)
+{
+	struct isl_print_space_data data = { 0 };
+	int i = 0;
+
+	for (i = 0; i < pwqp->n; ++i) {
+		if (i)
+			p = isl_printer_print_str(p, "; ");
+		if (!isl_space_is_params(pwqp->p[i].set->dim)) {
+			p = print_space(pwqp->p[i].set->dim, p, 0, &data);
+			p = isl_printer_print_str(p, " -> ");
+		}
+		p = print_qpolynomial(p, pwqp->p[i].qp);
+		p = print_disjuncts((isl_map *)pwqp->p[i].set, p, 0);
+	}
+
+	return p;
+}
+
+static __isl_give isl_printer *print_pw_qpolynomial_isl(
+	__isl_take isl_printer *p, __isl_keep isl_pw_qpolynomial *pwqp)
+{
+	struct isl_print_space_data data = { 0 };
+
+	if (!p || !pwqp)
+		goto error;
+
+	if (isl_space_dim(pwqp->dim, isl_dim_param) > 0) {
+		p = print_tuple(pwqp->dim, p, isl_dim_param, &data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+	p = isl_printer_print_str(p, "{ ");
+	if (pwqp->n == 0) {
+		if (!isl_space_is_set(pwqp->dim)) {
+			p = print_tuple(pwqp->dim, p, isl_dim_in, &data);
+			p = isl_printer_print_str(p, " -> ");
+		}
+		p = isl_printer_print_str(p, "0");
+	}
+	p = isl_pwqp_print_isl_body(p, pwqp);
+	p = isl_printer_print_str(p, " }");
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+void isl_pw_qpolynomial_print(__isl_keep isl_pw_qpolynomial *pwqp, FILE *out,
+	unsigned output_format)
+{
+	isl_printer *p;
+
+	if (!pwqp)
+		return;
+
+	p = isl_printer_to_file(pwqp->dim->ctx, out);
+	p = isl_printer_set_output_format(p, output_format);
+	p = isl_printer_print_pw_qpolynomial(p, pwqp);
+
+	isl_printer_free(p);
+}
+
+static __isl_give isl_printer *isl_pwf_print_isl_body(
+	__isl_take isl_printer *p, __isl_keep isl_pw_qpolynomial_fold *pwf)
+{
+	struct isl_print_space_data data = { 0 };
+	int i = 0;
+
+	for (i = 0; i < pwf->n; ++i) {
+		if (i)
+			p = isl_printer_print_str(p, "; ");
+		if (!isl_space_is_params(pwf->p[i].set->dim)) {
+			p = print_space(pwf->p[i].set->dim, p, 0, &data);
+			p = isl_printer_print_str(p, " -> ");
+		}
+		p = qpolynomial_fold_print(pwf->p[i].fold, p);
+		p = print_disjuncts((isl_map *)pwf->p[i].set, p, 0);
+	}
+
+	return p;
+}
+
+static __isl_give isl_printer *print_pw_qpolynomial_fold_isl(
+	__isl_take isl_printer *p, __isl_keep isl_pw_qpolynomial_fold *pwf)
+{
+	struct isl_print_space_data data = { 0 };
+
+	if (isl_space_dim(pwf->dim, isl_dim_param) > 0) {
+		p = print_tuple(pwf->dim, p, isl_dim_param, &data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+	p = isl_printer_print_str(p, "{ ");
+	if (pwf->n == 0) {
+		if (!isl_space_is_set(pwf->dim)) {
+			p = print_tuple(pwf->dim, p, isl_dim_in, &data);
+			p = isl_printer_print_str(p, " -> ");
+		}
+		p = isl_printer_print_str(p, "0");
+	}
+	p = isl_pwf_print_isl_body(p, pwf);
+	p = isl_printer_print_str(p, " }");
+	return p;
+}
+
+static __isl_give isl_printer *print_affine_c(__isl_take isl_printer *p,
+	__isl_keep isl_space *dim, __isl_keep isl_basic_set *bset, isl_int *c);
+
+static __isl_give isl_printer *print_name_c(__isl_take isl_printer *p,
+	__isl_keep isl_space *dim,
+	__isl_keep isl_basic_set *bset, enum isl_dim_type type, unsigned pos)
+{
+	if (type == isl_dim_div) {
+		p = isl_printer_print_str(p, "floord(");
+		p = print_affine_c(p, dim, bset, bset->div[pos] + 1);
+		p = isl_printer_print_str(p, ", ");
+		p = isl_printer_print_isl_int(p, bset->div[pos][0]);
+		p = isl_printer_print_str(p, ")");
+	} else {
+		const char *name;
+
+		name = isl_space_get_dim_name(dim, type, pos);
+		if (!name)
+			name = "UNNAMED";
+		p = isl_printer_print_str(p, name);
+	}
+	return p;
+}
+
+static __isl_give isl_printer *print_term_c(__isl_take isl_printer *p,
+	__isl_keep isl_space *dim,
+	__isl_keep isl_basic_set *bset, isl_int c, unsigned pos)
+{
+	enum isl_dim_type type;
+
+	if (pos == 0)
+		return isl_printer_print_isl_int(p, c);
+
+	if (isl_int_is_one(c))
+		;
+	else if (isl_int_is_negone(c))
+		p = isl_printer_print_str(p, "-");
+	else {
+		p = isl_printer_print_isl_int(p, c);
+		p = isl_printer_print_str(p, "*");
+	}
+	type = pos2type(dim, &pos);
+	p = print_name_c(p, dim, bset, type, pos);
+	return p;
+}
+
+static __isl_give isl_printer *print_partial_affine_c(__isl_take isl_printer *p,
+	__isl_keep isl_space *dim,
+	__isl_keep isl_basic_set *bset, isl_int *c, unsigned len)
+{
+	int i;
+	int first;
+
+	for (i = 0, first = 1; i < len; ++i) {
+		int flip = 0;
+		if (isl_int_is_zero(c[i]))
+			continue;
+		if (!first) {
+			if (isl_int_is_neg(c[i])) {
+				flip = 1;
+				isl_int_neg(c[i], c[i]);
+				p = isl_printer_print_str(p, " - ");
+			} else 
+				p = isl_printer_print_str(p, " + ");
+		}
+		first = 0;
+		p = print_term_c(p, dim, bset, c[i], i);
+		if (flip)
+			isl_int_neg(c[i], c[i]);
+	}
+	if (first)
+		p = isl_printer_print_str(p, "0");
+	return p;
+}
+
+static __isl_give isl_printer *print_affine_c(__isl_take isl_printer *p,
+	__isl_keep isl_space *dim, __isl_keep isl_basic_set *bset, isl_int *c)
+{
+	unsigned len = 1 + isl_basic_set_total_dim(bset);
+	return print_partial_affine_c(p, dim, bset, c, len);
+}
+
+/* We skip the constraint if it is implied by the div expression.
+ *
+ * *first indicates whether this is the first constraint in the conjunction and
+ * is updated if the constraint is actually printed.
+ */
+static __isl_give isl_printer *print_constraint_c(__isl_take isl_printer *p,
+	__isl_keep isl_space *dim,
+	__isl_keep isl_basic_set *bset, isl_int *c, const char *op, int *first)
+{
+	unsigned o_div;
+	unsigned n_div;
+	int div;
+
+	o_div = isl_basic_set_offset(bset, isl_dim_div);
+	n_div = isl_basic_set_dim(bset, isl_dim_div);
+	div = isl_seq_last_non_zero(c + o_div, n_div);
+	if (div >= 0 && isl_basic_set_is_div_constraint(bset, c, div))
+		return p;
+
+	if (!*first)
+		p = isl_printer_print_str(p, " && ");
+
+	p = print_affine_c(p, dim, bset, c);
+	p = isl_printer_print_str(p, " ");
+	p = isl_printer_print_str(p, op);
+	p = isl_printer_print_str(p, " 0");
+
+	*first = 0;
+
+	return p;
+}
+
+static __isl_give isl_printer *print_basic_set_c(__isl_take isl_printer *p,
+	__isl_keep isl_space *dim, __isl_keep isl_basic_set *bset)
+{
+	int i, j;
+	int first = 1;
+	unsigned n_div = isl_basic_set_dim(bset, isl_dim_div);
+	unsigned total = isl_basic_set_total_dim(bset) - n_div;
+
+	for (i = 0; i < bset->n_eq; ++i) {
+		j = isl_seq_last_non_zero(bset->eq[i] + 1 + total, n_div);
+		if (j < 0)
+			p = print_constraint_c(p, dim, bset,
+						bset->eq[i], "==", &first);
+		else {
+			if (i)
+				p = isl_printer_print_str(p, " && ");
+			p = isl_printer_print_str(p, "(");
+			p = print_partial_affine_c(p, dim, bset, bset->eq[i],
+						   1 + total + j);
+			p = isl_printer_print_str(p, ") % ");
+			p = isl_printer_print_isl_int(p,
+						bset->eq[i][1 + total + j]);
+			p = isl_printer_print_str(p, " == 0");
+			first = 0;
+		}
+	}
+	for (i = 0; i < bset->n_ineq; ++i)
+		p = print_constraint_c(p, dim, bset, bset->ineq[i], ">=",
+					&first);
+	return p;
+}
+
+static __isl_give isl_printer *print_set_c(__isl_take isl_printer *p,
+	__isl_keep isl_space *dim, __isl_keep isl_set *set)
+{
+	int i;
+
+	if (!set)
+		return isl_printer_free(p);
+
+	if (set->n == 0)
+		p = isl_printer_print_str(p, "0");
+
+	for (i = 0; i < set->n; ++i) {
+		if (i)
+			p = isl_printer_print_str(p, " || ");
+		if (set->n > 1)
+			p = isl_printer_print_str(p, "(");
+		p = print_basic_set_c(p, dim, set->p[i]);
+		if (set->n > 1)
+			p = isl_printer_print_str(p, ")");
+	}
+	return p;
+}
+
+static __isl_give isl_printer *print_pw_qpolynomial_c(
+	__isl_take isl_printer *p, __isl_keep isl_pw_qpolynomial *pwqp)
+{
+	int i;
+
+	if (pwqp->n == 1 && isl_set_plain_is_universe(pwqp->p[0].set))
+		return print_qpolynomial_c(p, pwqp->dim, pwqp->p[0].qp);
+
+	for (i = 0; i < pwqp->n; ++i) {
+		p = isl_printer_print_str(p, "(");
+		p = print_set_c(p, pwqp->dim, pwqp->p[i].set);
+		p = isl_printer_print_str(p, ") ? (");
+		p = print_qpolynomial_c(p, pwqp->dim, pwqp->p[i].qp);
+		p = isl_printer_print_str(p, ") : ");
+	}
+
+	p = isl_printer_print_str(p, "0");
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_print_pw_qpolynomial(
+	__isl_take isl_printer *p, __isl_keep isl_pw_qpolynomial *pwqp)
+{
+	if (!p || !pwqp)
+		goto error;
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return print_pw_qpolynomial_isl(p, pwqp);
+	else if (p->output_format == ISL_FORMAT_C)
+		return print_pw_qpolynomial_c(p, pwqp);
+	isl_assert(p->ctx, 0, goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static int print_pwqp_body(__isl_take isl_pw_qpolynomial *pwqp, void *user)
+{
+	struct isl_union_print_data *data;
+	data = (struct isl_union_print_data *)user;
+
+	if (!data->first)
+		data->p = isl_printer_print_str(data->p, "; ");
+	data->first = 0;
+
+	data->p = isl_pwqp_print_isl_body(data->p, pwqp);
+	isl_pw_qpolynomial_free(pwqp);
+
+	return 0;
+}
+
+static __isl_give isl_printer *print_union_pw_qpolynomial_isl(
+	__isl_take isl_printer *p, __isl_keep isl_union_pw_qpolynomial *upwqp)
+{
+	struct isl_union_print_data data = { p, 1 };
+	struct isl_print_space_data space_data = { 0 };
+	isl_space *dim;
+
+	dim = isl_union_pw_qpolynomial_get_space(upwqp);
+	if (isl_space_dim(dim, isl_dim_param) > 0) {
+		p = print_tuple(dim, p, isl_dim_param, &space_data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+	isl_space_free(dim);
+	p = isl_printer_print_str(p, "{ ");
+	isl_union_pw_qpolynomial_foreach_pw_qpolynomial(upwqp, &print_pwqp_body,
+							&data);
+	p = data.p;
+	p = isl_printer_print_str(p, " }");
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_print_union_pw_qpolynomial(
+	__isl_take isl_printer *p, __isl_keep isl_union_pw_qpolynomial *upwqp)
+{
+	if (!p || !upwqp)
+		goto error;
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return print_union_pw_qpolynomial_isl(p, upwqp);
+	isl_die(p->ctx, isl_error_invalid,
+		"invalid output format for isl_union_pw_qpolynomial",
+		goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *print_qpolynomial_fold_c(
+	__isl_take isl_printer *p, __isl_keep isl_space *dim,
+	__isl_keep isl_qpolynomial_fold *fold)
+{
+	int i;
+
+	for (i = 0; i < fold->n - 1; ++i)
+		if (fold->type == isl_fold_min)
+			p = isl_printer_print_str(p, "min(");
+		else if (fold->type == isl_fold_max)
+			p = isl_printer_print_str(p, "max(");
+
+	for (i = 0; i < fold->n; ++i) {
+		if (i)
+			p = isl_printer_print_str(p, ", ");
+		p = print_qpolynomial_c(p, dim, fold->qp[i]);
+		if (i)
+			p = isl_printer_print_str(p, ")");
+	}
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_print_qpolynomial_fold(
+	__isl_take isl_printer *p, __isl_keep isl_qpolynomial_fold *fold)
+{
+	if  (!p || !fold)
+		goto error;
+	if (p->output_format == ISL_FORMAT_ISL)
+		return qpolynomial_fold_print(fold, p);
+	else if (p->output_format == ISL_FORMAT_C)
+		return print_qpolynomial_fold_c(p, fold->dim, fold);
+	isl_die(p->ctx, isl_error_unsupported, "unsupported output format",
+		goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *print_pw_qpolynomial_fold_c(
+	__isl_take isl_printer *p, __isl_keep isl_pw_qpolynomial_fold *pwf)
+{
+	int i;
+
+	if (pwf->n == 1 && isl_set_plain_is_universe(pwf->p[0].set))
+		return print_qpolynomial_fold_c(p, pwf->dim, pwf->p[0].fold);
+
+	for (i = 0; i < pwf->n; ++i) {
+		p = isl_printer_print_str(p, "(");
+		p = print_set_c(p, pwf->dim, pwf->p[i].set);
+		p = isl_printer_print_str(p, ") ? (");
+		p = print_qpolynomial_fold_c(p, pwf->dim, pwf->p[i].fold);
+		p = isl_printer_print_str(p, ") : ");
+	}
+
+	p = isl_printer_print_str(p, "0");
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_print_pw_qpolynomial_fold(
+	__isl_take isl_printer *p, __isl_keep isl_pw_qpolynomial_fold *pwf)
+{
+	if (!p || !pwf)
+		goto error;
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return print_pw_qpolynomial_fold_isl(p, pwf);
+	else if (p->output_format == ISL_FORMAT_C)
+		return print_pw_qpolynomial_fold_c(p, pwf);
+	isl_assert(p->ctx, 0, goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+void isl_pw_qpolynomial_fold_print(__isl_keep isl_pw_qpolynomial_fold *pwf,
+	FILE *out, unsigned output_format)
+{
+	isl_printer *p;
+
+	if (!pwf)
+		return;
+
+	p = isl_printer_to_file(pwf->dim->ctx, out);
+	p = isl_printer_set_output_format(p, output_format);
+	p = isl_printer_print_pw_qpolynomial_fold(p, pwf);
+
+	isl_printer_free(p);
+}
+
+static int print_pwf_body(__isl_take isl_pw_qpolynomial_fold *pwf, void *user)
+{
+	struct isl_union_print_data *data;
+	data = (struct isl_union_print_data *)user;
+
+	if (!data->first)
+		data->p = isl_printer_print_str(data->p, "; ");
+	data->first = 0;
+
+	data->p = isl_pwf_print_isl_body(data->p, pwf);
+	isl_pw_qpolynomial_fold_free(pwf);
+
+	return 0;
+}
+
+static __isl_give isl_printer *print_union_pw_qpolynomial_fold_isl(
+	__isl_take isl_printer *p,
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf)
+{
+	struct isl_union_print_data data = { p, 1 };
+	struct isl_print_space_data space_data = { 0 };
+	isl_space *dim;
+
+	dim = isl_union_pw_qpolynomial_fold_get_space(upwf);
+	if (isl_space_dim(dim, isl_dim_param) > 0) {
+		p = print_tuple(dim, p, isl_dim_param, &space_data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+	isl_space_free(dim);
+	p = isl_printer_print_str(p, "{ ");
+	isl_union_pw_qpolynomial_fold_foreach_pw_qpolynomial_fold(upwf,
+							&print_pwf_body, &data);
+	p = data.p;
+	p = isl_printer_print_str(p, " }");
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_print_union_pw_qpolynomial_fold(
+	__isl_take isl_printer *p,
+	__isl_keep isl_union_pw_qpolynomial_fold *upwf)
+{
+	if (!p || !upwf)
+		goto error;
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return print_union_pw_qpolynomial_fold_isl(p, upwf);
+	isl_die(p->ctx, isl_error_invalid,
+		"invalid output format for isl_union_pw_qpolynomial_fold",
+		goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+__isl_give isl_printer *isl_printer_print_constraint(__isl_take isl_printer *p,
+	__isl_keep isl_constraint *c)
+{
+	isl_basic_map *bmap;
+
+	if (!p || !c)
+		goto error;
+
+	bmap = isl_basic_map_from_constraint(isl_constraint_copy(c));
+	p = isl_printer_print_basic_map(p, bmap);
+	isl_basic_map_free(bmap);
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *isl_printer_print_space_isl(
+	__isl_take isl_printer *p, __isl_keep isl_space *dim)
+{
+	struct isl_print_space_data data = { 0 };
+
+	if (!dim)
+		goto error;
+
+	if (isl_space_dim(dim, isl_dim_param) > 0) {
+		p = print_tuple(dim, p, isl_dim_param, &data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+
+	p = isl_printer_print_str(p, "{ ");
+	if (isl_space_is_params(dim))
+		p = isl_printer_print_str(p, s_such_that[0]);
+	else
+		p = print_space(dim, p, 0, &data);
+	p = isl_printer_print_str(p, " }");
+
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+__isl_give isl_printer *isl_printer_print_space(__isl_take isl_printer *p,
+	__isl_keep isl_space *space)
+{
+	if (!p || !space)
+		return isl_printer_free(p);
+	if (p->output_format == ISL_FORMAT_ISL)
+		return isl_printer_print_space_isl(p, space);
+	else if (p->output_format == ISL_FORMAT_OMEGA)
+		return print_omega_parameters(space, p);
+
+	isl_die(isl_space_get_ctx(space), isl_error_unsupported,
+		"output format not supported for space",
+		return isl_printer_free(p));
+}
+
+__isl_give isl_printer *isl_printer_print_local_space(__isl_take isl_printer *p,
+	__isl_keep isl_local_space *ls)
+{
+	struct isl_print_space_data data = { 0 };
+	unsigned total;
+	unsigned n_div;
+
+	if (!ls)
+		goto error;
+
+	total = isl_local_space_dim(ls, isl_dim_all);
+	if (isl_local_space_dim(ls, isl_dim_param) > 0) {
+		p = print_tuple(ls->dim, p, isl_dim_param, &data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+	p = isl_printer_print_str(p, "{ ");
+	p = print_space(ls->dim, p, 0, &data);
+	n_div = isl_local_space_dim(ls, isl_dim_div);
+	if (n_div > 0) {
+		p = isl_printer_print_str(p, " : ");
+		p = isl_printer_print_str(p, s_open_exists[0]);
+		p = print_div_list(p, ls->dim, ls->div, 0);
+	} else if (isl_space_is_params(ls->dim))
+		p = isl_printer_print_str(p, s_such_that[0]);
+	p = isl_printer_print_str(p, " }");
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *print_aff_body(__isl_take isl_printer *p,
+	__isl_keep isl_aff *aff)
+{
+	unsigned total;
+
+	if (isl_aff_is_nan(aff))
+		return isl_printer_print_str(p, "NaN");
+
+	total = isl_local_space_dim(aff->ls, isl_dim_all);
+	p = isl_printer_print_str(p, "(");
+	p = print_affine_of_len(aff->ls->dim, aff->ls->div, p,
+				aff->v->el + 1, 1 + total);
+	if (isl_int_is_one(aff->v->el[0]))
+		p = isl_printer_print_str(p, ")");
+	else {
+		p = isl_printer_print_str(p, ")/");
+		p = isl_printer_print_isl_int(p, aff->v->el[0]);
+	}
+
+	return p;
+}
+
+static __isl_give isl_printer *print_aff(__isl_take isl_printer *p,
+	__isl_keep isl_aff *aff)
+{
+	struct isl_print_space_data data = { 0 };
+
+	if (isl_space_is_params(aff->ls->dim))
+		;
+	else {
+		p = print_tuple(aff->ls->dim, p, isl_dim_set, &data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+	p = isl_printer_print_str(p, "[");
+	p = print_aff_body(p, aff);
+	p = isl_printer_print_str(p, "]");
+
+	return p;
+}
+
+static __isl_give isl_printer *print_aff_isl(__isl_take isl_printer *p,
+	__isl_keep isl_aff *aff)
+{
+	struct isl_print_space_data data = { 0 };
+
+	if (!aff)
+		goto error;
+
+	if (isl_local_space_dim(aff->ls, isl_dim_param) > 0) {
+		p = print_tuple(aff->ls->dim, p, isl_dim_param, &data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+	p = isl_printer_print_str(p, "{ ");
+	p = print_aff(p, aff);
+	p = isl_printer_print_str(p, " }");
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *print_pw_aff_isl(__isl_take isl_printer *p,
+	__isl_keep isl_pw_aff *pwaff)
+{
+	struct isl_print_space_data data = { 0 };
+	int i;
+
+	if (!pwaff)
+		goto error;
+
+	if (isl_space_dim(pwaff->dim, isl_dim_param) > 0) {
+		p = print_tuple(pwaff->dim, p, isl_dim_param, &data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+	p = isl_printer_print_str(p, "{ ");
+	for (i = 0; i < pwaff->n; ++i) {
+		if (i)
+			p = isl_printer_print_str(p, "; ");
+		p = print_aff(p, pwaff->p[i].aff);
+		p = print_disjuncts((isl_map *)pwaff->p[i].set, p, 0);
+	}
+	p = isl_printer_print_str(p, " }");
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *print_ls_affine_c(__isl_take isl_printer *p,
+	__isl_keep isl_local_space *ls, isl_int *c);
+
+static __isl_give isl_printer *print_ls_name_c(__isl_take isl_printer *p,
+	__isl_keep isl_local_space *ls, enum isl_dim_type type, unsigned pos)
+{
+	if (type == isl_dim_div) {
+		p = isl_printer_print_str(p, "floord(");
+		p = print_ls_affine_c(p, ls, ls->div->row[pos] + 1);
+		p = isl_printer_print_str(p, ", ");
+		p = isl_printer_print_isl_int(p, ls->div->row[pos][0]);
+		p = isl_printer_print_str(p, ")");
+	} else {
+		const char *name;
+
+		name = isl_space_get_dim_name(ls->dim, type, pos);
+		if (!name)
+			name = "UNNAMED";
+		p = isl_printer_print_str(p, name);
+	}
+	return p;
+}
+
+static __isl_give isl_printer *print_ls_term_c(__isl_take isl_printer *p,
+	__isl_keep isl_local_space *ls, isl_int c, unsigned pos)
+{
+	enum isl_dim_type type;
+
+	if (pos == 0)
+		return isl_printer_print_isl_int(p, c);
+
+	if (isl_int_is_one(c))
+		;
+	else if (isl_int_is_negone(c))
+		p = isl_printer_print_str(p, "-");
+	else {
+		p = isl_printer_print_isl_int(p, c);
+		p = isl_printer_print_str(p, "*");
+	}
+	type = pos2type(ls->dim, &pos);
+	p = print_ls_name_c(p, ls, type, pos);
+	return p;
+}
+
+static __isl_give isl_printer *print_ls_partial_affine_c(
+	__isl_take isl_printer *p, __isl_keep isl_local_space *ls,
+	isl_int *c, unsigned len)
+{
+	int i;
+	int first;
+
+	for (i = 0, first = 1; i < len; ++i) {
+		int flip = 0;
+		if (isl_int_is_zero(c[i]))
+			continue;
+		if (!first) {
+			if (isl_int_is_neg(c[i])) {
+				flip = 1;
+				isl_int_neg(c[i], c[i]);
+				p = isl_printer_print_str(p, " - ");
+			} else 
+				p = isl_printer_print_str(p, " + ");
+		}
+		first = 0;
+		p = print_ls_term_c(p, ls, c[i], i);
+		if (flip)
+			isl_int_neg(c[i], c[i]);
+	}
+	if (first)
+		p = isl_printer_print_str(p, "0");
+	return p;
+}
+
+static __isl_give isl_printer *print_ls_affine_c(__isl_take isl_printer *p,
+	__isl_keep isl_local_space *ls, isl_int *c)
+{
+	unsigned len = 1 + isl_local_space_dim(ls, isl_dim_all);
+	return print_ls_partial_affine_c(p, ls, c, len);
+}
+
+static __isl_give isl_printer *print_aff_c(__isl_take isl_printer *p,
+	__isl_keep isl_aff *aff)
+{
+	unsigned total;
+
+	total = isl_local_space_dim(aff->ls, isl_dim_all);
+	if (!isl_int_is_one(aff->v->el[0]))
+		p = isl_printer_print_str(p, "(");
+	p = print_ls_partial_affine_c(p, aff->ls, aff->v->el + 1, 1 + total);
+	if (!isl_int_is_one(aff->v->el[0])) {
+		p = isl_printer_print_str(p, ")/");
+		p = isl_printer_print_isl_int(p, aff->v->el[0]);
+	}
+	return p;
+}
+
+/* In the C format, we cannot express that "pwaff" may be undefined
+ * on parts of the domain space.  We therefore assume that the expression
+ * will only be evaluated on its definition domain and compute the gist
+ * of each cell with respect to this domain.
+ */
+static __isl_give isl_printer *print_pw_aff_c(__isl_take isl_printer *p,
+	__isl_keep isl_pw_aff *pwaff)
+{
+	isl_set *domain;
+	isl_ast_build *build;
+	isl_ast_expr *expr;
+
+	if (pwaff->n < 1)
+		isl_die(p->ctx, isl_error_unsupported,
+			"cannot print empty isl_pw_aff in C format",
+			return isl_printer_free(p));
+
+	domain = isl_pw_aff_domain(isl_pw_aff_copy(pwaff));
+	build = isl_ast_build_from_context(domain);
+	expr = isl_ast_build_expr_from_pw_aff(build, isl_pw_aff_copy(pwaff));
+	p = isl_printer_print_ast_expr(p, expr);
+	isl_ast_expr_free(expr);
+	isl_ast_build_free(build);
+
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_print_aff(__isl_take isl_printer *p,
+	__isl_keep isl_aff *aff)
+{
+	if (!p || !aff)
+		goto error;
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return print_aff_isl(p, aff);
+	else if (p->output_format == ISL_FORMAT_C)
+		return print_aff_c(p, aff);
+	isl_die(p->ctx, isl_error_unsupported, "unsupported output format",
+		goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+__isl_give isl_printer *isl_printer_print_pw_aff(__isl_take isl_printer *p,
+	__isl_keep isl_pw_aff *pwaff)
+{
+	if (!p || !pwaff)
+		goto error;
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return print_pw_aff_isl(p, pwaff);
+	else if (p->output_format == ISL_FORMAT_C)
+		return print_pw_aff_c(p, pwaff);
+	isl_die(p->ctx, isl_error_unsupported, "unsupported output format",
+		goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+/* Print dimension "pos" of data->space to "p".
+ *
+ * data->user is assumed to be an isl_multi_aff.
+ *
+ * If the current dimension is an output dimension, then print
+ * the corresponding expression.  Otherwise, print the name of the dimension.
+ */
+static __isl_give isl_printer *print_dim_ma(__isl_take isl_printer *p,
+	struct isl_print_space_data *data, unsigned pos)
+{
+	isl_multi_aff *ma = data->user;
+
+	if (data->type == isl_dim_out)
+		p = print_aff_body(p, ma->p[pos]);
+	else
+		p = print_name(data->space, p, data->type, pos, data->latex);
+
+	return p;
+}
+
+static __isl_give isl_printer *print_multi_aff(__isl_take isl_printer *p,
+	__isl_keep isl_multi_aff *maff)
+{
+	struct isl_print_space_data data = { 0 };
+
+	data.print_dim = &print_dim_ma;
+	data.user = maff;
+	return print_space(maff->space, p, 0, &data);
+}
+
+static __isl_give isl_printer *print_multi_aff_isl(__isl_take isl_printer *p,
+	__isl_keep isl_multi_aff *maff)
+{
+	struct isl_print_space_data data = { 0 };
+
+	if (!maff)
+		goto error;
+
+	if (isl_space_dim(maff->space, isl_dim_param) > 0) {
+		p = print_tuple(maff->space, p, isl_dim_param, &data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+	p = isl_printer_print_str(p, "{ ");
+	p = print_multi_aff(p, maff);
+	p = isl_printer_print_str(p, " }");
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+__isl_give isl_printer *isl_printer_print_multi_aff(__isl_take isl_printer *p,
+	__isl_keep isl_multi_aff *maff)
+{
+	if (!p || !maff)
+		goto error;
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return print_multi_aff_isl(p, maff);
+	isl_die(p->ctx, isl_error_unsupported, "unsupported output format",
+		goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *print_pw_multi_aff_body(
+	__isl_take isl_printer *p, __isl_keep isl_pw_multi_aff *pma)
+{
+	int i;
+
+	if (!pma)
+		goto error;
+
+	for (i = 0; i < pma->n; ++i) {
+		if (i)
+			p = isl_printer_print_str(p, "; ");
+		p = print_multi_aff(p, pma->p[i].maff);
+		p = print_disjuncts((isl_map *)pma->p[i].set, p, 0);
+	}
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *print_pw_multi_aff_isl(__isl_take isl_printer *p,
+	__isl_keep isl_pw_multi_aff *pma)
+{
+	struct isl_print_space_data data = { 0 };
+
+	if (!pma)
+		goto error;
+
+	if (isl_space_dim(pma->dim, isl_dim_param) > 0) {
+		p = print_tuple(pma->dim, p, isl_dim_param, &data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+	p = isl_printer_print_str(p, "{ ");
+	p = print_pw_multi_aff_body(p, pma);
+	p = isl_printer_print_str(p, " }");
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *print_unnamed_pw_multi_aff_c(
+	__isl_take isl_printer *p, __isl_keep isl_pw_multi_aff *pma)
+{
+	int i;
+
+	for (i = 0; i < pma->n - 1; ++i) {
+		p = isl_printer_print_str(p, "(");
+		p = print_set_c(p, pma->dim, pma->p[i].set);
+		p = isl_printer_print_str(p, ") ? (");
+		p = print_aff_c(p, pma->p[i].maff->p[0]);
+		p = isl_printer_print_str(p, ") : ");
+	}
+
+	return print_aff_c(p, pma->p[pma->n - 1].maff->p[0]);
+}
+
+static __isl_give isl_printer *print_pw_multi_aff_c(__isl_take isl_printer *p,
+	__isl_keep isl_pw_multi_aff *pma)
+{
+	int n;
+	const char *name;
+
+	if (!pma)
+		goto error;
+	if (pma->n < 1)
+		isl_die(p->ctx, isl_error_unsupported,
+			"cannot print empty isl_pw_multi_aff in C format",
+			goto error);
+	name = isl_pw_multi_aff_get_tuple_name(pma, isl_dim_out);
+	if (!name && isl_pw_multi_aff_dim(pma, isl_dim_out) == 1)
+		return print_unnamed_pw_multi_aff_c(p, pma);
+	if (!name)
+		isl_die(p->ctx, isl_error_unsupported,
+			"cannot print unnamed isl_pw_multi_aff in C format",
+			goto error);
+
+	p = isl_printer_print_str(p, name);
+	n = isl_pw_multi_aff_dim(pma, isl_dim_out);
+	if (n != 0)
+		isl_die(p->ctx, isl_error_unsupported,
+			"not supported yet", goto error);
+
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+__isl_give isl_printer *isl_printer_print_pw_multi_aff(
+	__isl_take isl_printer *p, __isl_keep isl_pw_multi_aff *pma)
+{
+	if (!p || !pma)
+		goto error;
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return print_pw_multi_aff_isl(p, pma);
+	if (p->output_format == ISL_FORMAT_C)
+		return print_pw_multi_aff_c(p, pma);
+	isl_die(p->ctx, isl_error_unsupported, "unsupported output format",
+		goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static int print_pw_multi_aff_body_wrap(__isl_take isl_pw_multi_aff *pma,
+	void *user)
+{
+	struct isl_union_print_data *data;
+	data = (struct isl_union_print_data *) user;
+
+	if (!data->first)
+		data->p = isl_printer_print_str(data->p, "; ");
+	data->first = 0;
+
+	data->p = print_pw_multi_aff_body(data->p, pma);
+	isl_pw_multi_aff_free(pma);
+
+	return 0;
+}
+
+static __isl_give isl_printer *print_union_pw_multi_aff_isl(
+	__isl_take isl_printer *p, __isl_keep isl_union_pw_multi_aff *upma)
+{
+	struct isl_union_print_data data = { p, 1 };
+	struct isl_print_space_data space_data = { 0 };
+	isl_space *space;
+
+	space = isl_union_pw_multi_aff_get_space(upma);
+	if (isl_space_dim(space, isl_dim_param) > 0) {
+		p = print_tuple(space, p, isl_dim_param, &space_data);
+		p = isl_printer_print_str(p, s_to[0]);
+	}
+	isl_space_free(space);
+	p = isl_printer_print_str(p, s_open_set[0]);
+	isl_union_pw_multi_aff_foreach_pw_multi_aff(upma,
+					&print_pw_multi_aff_body_wrap, &data);
+	p = data.p;
+	p = isl_printer_print_str(p, s_close_set[0]);
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_print_union_pw_multi_aff(
+	__isl_take isl_printer *p, __isl_keep isl_union_pw_multi_aff *upma)
+{
+	if (!p || !upma)
+		goto error;
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return print_union_pw_multi_aff_isl(p, upma);
+	isl_die(p->ctx, isl_error_unsupported, "unsupported output format",
+		goto error);
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+/* Print dimension "pos" of data->space to "p".
+ *
+ * data->user is assumed to be an isl_multi_pw_aff.
+ *
+ * If the current dimension is an output dimension, then print
+ * the corresponding piecewise affine expression.
+ * Otherwise, print the name of the dimension.
+ */
+static __isl_give isl_printer *print_dim_mpa(__isl_take isl_printer *p,
+	struct isl_print_space_data *data, unsigned pos)
+{
+	int i;
+	int need_parens;
+	isl_multi_pw_aff *mpa = data->user;
+	isl_pw_aff *pa;
+
+	if (data->type != isl_dim_out)
+		return print_name(data->space, p, data->type, pos, data->latex);
+
+	pa = mpa->p[pos];
+	if (pa->n == 0)
+		return isl_printer_print_str(p, "(0 : 1 = 0)");
+
+	need_parens = pa->n != 1 || !isl_set_plain_is_universe(pa->p[0].set);
+	if (need_parens)
+		p = isl_printer_print_str(p, "(");
+	for (i = 0; i < pa->n; ++i) {
+		if (i)
+			p = isl_printer_print_str(p, "; ");
+		p = print_aff_body(p, pa->p[i].aff);
+		p = print_disjuncts(pa->p[i].set, p, 0);
+	}
+	if (need_parens)
+		p = isl_printer_print_str(p, ")");
+
+	return p;
+}
+
+/* Print "mpa" to "p" in isl format.
+ */
+static __isl_give isl_printer *print_multi_pw_aff_isl(__isl_take isl_printer *p,
+	__isl_keep isl_multi_pw_aff *mpa)
+{
+	struct isl_print_space_data data = { 0 };
+
+	if (!mpa)
+		return isl_printer_free(p);
+
+	if (isl_space_dim(mpa->space, isl_dim_param) > 0) {
+		p = print_tuple(mpa->space, p, isl_dim_param, &data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+	p = isl_printer_print_str(p, "{ ");
+	data.print_dim = &print_dim_mpa;
+	data.user = mpa;
+	p = print_space(mpa->space, p, 0, &data);
+	p = isl_printer_print_str(p, " }");
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_print_multi_pw_aff(
+	__isl_take isl_printer *p, __isl_keep isl_multi_pw_aff *mpa)
+{
+	if (!p || !mpa)
+		return isl_printer_free(p);
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return print_multi_pw_aff_isl(p, mpa);
+	isl_die(p->ctx, isl_error_unsupported, "unsupported output format",
+		return isl_printer_free(p));
+}
+
+/* Print dimension "pos" of data->space to "p".
+ *
+ * data->user is assumed to be an isl_multi_val.
+ *
+ * If the current dimension is an output dimension, then print
+ * the corresponding value.  Otherwise, print the name of the dimension.
+ */
+static __isl_give isl_printer *print_dim_mv(__isl_take isl_printer *p,
+	struct isl_print_space_data *data, unsigned pos)
+{
+	isl_multi_val *mv = data->user;
+
+	if (data->type == isl_dim_out)
+		return isl_printer_print_val(p, mv->p[pos]);
+	else
+		return print_name(data->space, p, data->type, pos, data->latex);
+}
+
+/* Print the isl_multi_val "mv" to "p" in isl format.
+ */
+static __isl_give isl_printer *print_multi_val_isl(__isl_take isl_printer *p,
+	__isl_keep isl_multi_val *mv)
+{
+	struct isl_print_space_data data = { 0 };
+
+	if (!mv)
+		return isl_printer_free(p);
+
+	if (isl_space_dim(mv->space, isl_dim_param) > 0) {
+		p = print_tuple(mv->space, p, isl_dim_param, &data);
+		p = isl_printer_print_str(p, " -> ");
+	}
+	p = isl_printer_print_str(p, "{ ");
+	data.print_dim = &print_dim_mv;
+	data.user = mv;
+	p = print_space(mv->space, p, 0, &data);
+	p = isl_printer_print_str(p, " }");
+	return p;
+}
+
+/* Print the isl_multi_val "mv" to "p".
+ *
+ * Currently only supported in isl format.
+ */
+__isl_give isl_printer *isl_printer_print_multi_val(
+	__isl_take isl_printer *p, __isl_keep isl_multi_val *mv)
+{
+	if (!p || !mv)
+		return isl_printer_free(p);
+
+	if (p->output_format == ISL_FORMAT_ISL)
+		return print_multi_val_isl(p, mv);
+	isl_die(p->ctx, isl_error_unsupported, "unsupported output format",
+		return isl_printer_free(p));
+}
diff --git a/polly/lib/External/isl/isl_point.c b/polly/lib/External/isl/isl_point.c
new file mode 100644
index 00000000000..f5320f2ede8
--- /dev/null
+++ b/polly/lib/External/isl/isl_point.c
@@ -0,0 +1,585 @@
+#include <isl_map_private.h>
+#include <isl_point_private.h>
+#include <isl/set.h>
+#include <isl_sample.h>
+#include <isl_scan.h>
+#include <isl_seq.h>
+#include <isl_space_private.h>
+#include <isl_val_private.h>
+#include <isl_vec_private.h>
+#include <isl/deprecated/point_int.h>
+
+isl_ctx *isl_point_get_ctx(__isl_keep isl_point *pnt)
+{
+	return pnt ? isl_space_get_ctx(pnt->dim) : NULL;
+}
+
+__isl_give isl_space *isl_point_get_space(__isl_keep isl_point *pnt)
+{
+	return pnt ? isl_space_copy(pnt->dim) : NULL;
+}
+
+__isl_give isl_point *isl_point_alloc(__isl_take isl_space *dim,
+	__isl_take isl_vec *vec)
+{
+	struct isl_point *pnt;
+
+	if (!dim || !vec)
+		goto error;
+
+	if (vec->size > 1 + isl_space_dim(dim, isl_dim_all)) {
+		vec = isl_vec_cow(vec);
+		if (!vec)
+			goto error;
+		vec->size = 1 + isl_space_dim(dim, isl_dim_all);
+	}
+
+	pnt = isl_alloc_type(dim->ctx, struct isl_point);
+	if (!pnt)
+		goto error;
+
+	pnt->ref = 1;
+	pnt->dim = dim;
+	pnt->vec = vec;
+
+	return pnt;
+error:
+	isl_space_free(dim);
+	isl_vec_free(vec);
+	return NULL;
+}
+
+__isl_give isl_point *isl_point_zero(__isl_take isl_space *dim)
+{
+	isl_vec *vec;
+
+	if (!dim)
+		return NULL;
+	vec = isl_vec_alloc(dim->ctx, 1 + isl_space_dim(dim, isl_dim_all));
+	if (!vec)
+		goto error;
+	isl_int_set_si(vec->el[0], 1);
+	isl_seq_clr(vec->el + 1, vec->size - 1);
+	return isl_point_alloc(dim, vec);
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+__isl_give isl_point *isl_point_dup(__isl_keep isl_point *pnt)
+{
+	struct isl_point *pnt2;
+
+	if (!pnt)
+		return NULL;
+	pnt2 = isl_point_alloc(isl_space_copy(pnt->dim), isl_vec_copy(pnt->vec));
+	return pnt2;
+}
+
+__isl_give isl_point *isl_point_cow(__isl_take isl_point *pnt)
+{
+	struct isl_point *pnt2;
+	if (!pnt)
+		return NULL;
+
+	if (pnt->ref == 1)
+		return pnt;
+
+	pnt2 = isl_point_dup(pnt);
+	isl_point_free(pnt);
+	return pnt2;
+}
+
+__isl_give isl_point *isl_point_copy(__isl_keep isl_point *pnt)
+{
+	if (!pnt)
+		return NULL;
+
+	pnt->ref++;
+	return pnt;
+}
+
+void isl_point_free(__isl_take isl_point *pnt)
+{
+	if (!pnt)
+		return;
+
+	if (--pnt->ref > 0)
+		return;
+
+	isl_space_free(pnt->dim);
+	isl_vec_free(pnt->vec);
+	free(pnt);
+}
+
+__isl_give isl_point *isl_point_void(__isl_take isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+
+	return isl_point_alloc(dim, isl_vec_alloc(dim->ctx, 0));
+}
+
+int isl_point_is_void(__isl_keep isl_point *pnt)
+{
+	if (!pnt)
+		return -1;
+
+	return pnt->vec->size == 0;
+}
+
+int isl_point_get_coordinate(__isl_keep isl_point *pnt,
+	enum isl_dim_type type, int pos, isl_int *v)
+{
+	if (!pnt || isl_point_is_void(pnt))
+		return -1;
+
+	if (pos < 0 || pos >= isl_space_dim(pnt->dim, type))
+		isl_die(isl_point_get_ctx(pnt), isl_error_invalid,
+			"position out of bounds", return -1);
+
+	if (type == isl_dim_set)
+		pos += isl_space_dim(pnt->dim, isl_dim_param);
+	isl_int_set(*v, pnt->vec->el[1 + pos]);
+
+	return 0;
+}
+
+/* Return the value of coordinate "pos" of type "type" of "pnt".
+ */
+__isl_give isl_val *isl_point_get_coordinate_val(__isl_keep isl_point *pnt,
+	enum isl_dim_type type, int pos)
+{
+	isl_ctx *ctx;
+	isl_val *v;
+
+	if (!pnt)
+		return NULL;
+
+	ctx = isl_point_get_ctx(pnt);
+	if (isl_point_is_void(pnt))
+		isl_die(ctx, isl_error_invalid,
+			"void point does not have coordinates", return NULL);
+	if (pos < 0 || pos >= isl_space_dim(pnt->dim, type))
+		isl_die(ctx, isl_error_invalid,
+			"position out of bounds", return NULL);
+
+	if (type == isl_dim_set)
+		pos += isl_space_dim(pnt->dim, isl_dim_param);
+
+	v = isl_val_rat_from_isl_int(ctx, pnt->vec->el[1 + pos],
+						pnt->vec->el[0]);
+	return isl_val_normalize(v);
+}
+
+__isl_give isl_point *isl_point_set_coordinate(__isl_take isl_point *pnt,
+	enum isl_dim_type type, int pos, isl_int v)
+{
+	if (!pnt || isl_point_is_void(pnt))
+		return pnt;
+
+	pnt = isl_point_cow(pnt);
+	if (!pnt)
+		return NULL;
+	pnt->vec = isl_vec_cow(pnt->vec);
+	if (!pnt->vec)
+		goto error;
+
+	if (type == isl_dim_set)
+		pos += isl_space_dim(pnt->dim, isl_dim_param);
+
+	isl_int_set(pnt->vec->el[1 + pos], v);
+
+	return pnt;
+error:
+	isl_point_free(pnt);
+	return NULL;
+}
+
+/* Replace coordinate "pos" of type "type" of "pnt" by "v".
+ */
+__isl_give isl_point *isl_point_set_coordinate_val(__isl_take isl_point *pnt,
+	enum isl_dim_type type, int pos, __isl_take isl_val *v)
+{
+	if (!pnt || !v)
+		goto error;
+	if (isl_point_is_void(pnt))
+		isl_die(isl_point_get_ctx(pnt), isl_error_invalid,
+			"void point does not have coordinates", goto error);
+	if (pos < 0 || pos >= isl_space_dim(pnt->dim, type))
+		isl_die(isl_point_get_ctx(pnt), isl_error_invalid,
+			"position out of bounds", goto error);
+	if (!isl_val_is_rat(v))
+		isl_die(isl_point_get_ctx(pnt), isl_error_invalid,
+			"expecting rational value", goto error);
+
+	if (isl_int_eq(pnt->vec->el[1 + pos], v->n) &&
+	    isl_int_eq(pnt->vec->el[0], v->d)) {
+		isl_val_free(v);
+		return pnt;
+	}
+
+	pnt = isl_point_cow(pnt);
+	if (!pnt)
+		goto error;
+	pnt->vec = isl_vec_cow(pnt->vec);
+	if (!pnt->vec)
+		goto error;
+
+	if (isl_int_eq(pnt->vec->el[0], v->d)) {
+		isl_int_set(pnt->vec->el[1 + pos], v->n);
+	} else if (isl_int_is_one(v->d)) {
+		isl_int_mul(pnt->vec->el[1 + pos], pnt->vec->el[0], v->n);
+	} else {
+		isl_seq_scale(pnt->vec->el + 1,
+				pnt->vec->el + 1, v->d, pnt->vec->size - 1);
+		isl_int_mul(pnt->vec->el[1 + pos], pnt->vec->el[0], v->n);
+		isl_int_mul(pnt->vec->el[0], pnt->vec->el[0], v->d);
+		pnt->vec = isl_vec_normalize(pnt->vec);
+		if (!pnt->vec)
+			goto error;
+	}
+
+	isl_val_free(v);
+	return pnt;
+error:
+	isl_val_free(v);
+	isl_point_free(pnt);
+	return NULL;
+}
+
+__isl_give isl_point *isl_point_add_ui(__isl_take isl_point *pnt,
+	enum isl_dim_type type, int pos, unsigned val)
+{
+	if (!pnt || isl_point_is_void(pnt))
+		return pnt;
+
+	pnt = isl_point_cow(pnt);
+	if (!pnt)
+		return NULL;
+	pnt->vec = isl_vec_cow(pnt->vec);
+	if (!pnt->vec)
+		goto error;
+
+	if (type == isl_dim_set)
+		pos += isl_space_dim(pnt->dim, isl_dim_param);
+
+	isl_int_add_ui(pnt->vec->el[1 + pos], pnt->vec->el[1 + pos], val);
+
+	return pnt;
+error:
+	isl_point_free(pnt);
+	return NULL;
+}
+
+__isl_give isl_point *isl_point_sub_ui(__isl_take isl_point *pnt,
+	enum isl_dim_type type, int pos, unsigned val)
+{
+	if (!pnt || isl_point_is_void(pnt))
+		return pnt;
+
+	pnt = isl_point_cow(pnt);
+	if (!pnt)
+		return NULL;
+	pnt->vec = isl_vec_cow(pnt->vec);
+	if (!pnt->vec)
+		goto error;
+
+	if (type == isl_dim_set)
+		pos += isl_space_dim(pnt->dim, isl_dim_param);
+
+	isl_int_sub_ui(pnt->vec->el[1 + pos], pnt->vec->el[1 + pos], val);
+
+	return pnt;
+error:
+	isl_point_free(pnt);
+	return NULL;
+}
+
+struct isl_foreach_point {
+	struct isl_scan_callback callback;
+	int (*fn)(__isl_take isl_point *pnt, void *user);
+	void *user;
+	isl_space *dim;
+};
+
+static int foreach_point(struct isl_scan_callback *cb, __isl_take isl_vec *sample)
+{
+	struct isl_foreach_point *fp = (struct isl_foreach_point *)cb;
+	isl_point *pnt;
+
+	pnt = isl_point_alloc(isl_space_copy(fp->dim), sample);
+
+	return fp->fn(pnt, fp->user);
+}
+
+int isl_set_foreach_point(__isl_keep isl_set *set,
+	int (*fn)(__isl_take isl_point *pnt, void *user), void *user)
+{
+	struct isl_foreach_point fp = { { &foreach_point }, fn, user };
+	int i;
+
+	if (!set)
+		return -1;
+
+	fp.dim = isl_set_get_space(set);
+	if (!fp.dim)
+		return -1;
+
+	set = isl_set_copy(set);
+	set = isl_set_cow(set);
+	set = isl_set_make_disjoint(set);
+	set = isl_set_compute_divs(set);
+	if (!set)
+		goto error;
+
+	for (i = 0; i < set->n; ++i)
+		if (isl_basic_set_scan(isl_basic_set_copy(set->p[i]),
+					&fp.callback) < 0)
+			goto error;
+
+	isl_set_free(set);
+	isl_space_free(fp.dim);
+
+	return 0;
+error:
+	isl_set_free(set);
+	isl_space_free(fp.dim);
+	return -1;
+}
+
+/* Return 1 if "bmap" contains the point "point".
+ * "bmap" is assumed to have known divs.
+ * The point is first extended with the divs and then passed
+ * to basic_map_contains.
+ */
+int isl_basic_map_contains_point(__isl_keep isl_basic_map *bmap,
+	__isl_keep isl_point *point)
+{
+	int i;
+	struct isl_vec *vec;
+	unsigned dim;
+	int contains;
+
+	if (!bmap || !point)
+		return -1;
+	isl_assert(bmap->ctx, isl_space_is_equal(bmap->dim, point->dim), return -1);
+	if (bmap->n_div == 0)
+		return isl_basic_map_contains(bmap, point->vec);
+
+	dim = isl_basic_map_total_dim(bmap) - bmap->n_div;
+	vec = isl_vec_alloc(bmap->ctx, 1 + dim + bmap->n_div);
+	if (!vec)
+		return -1;
+
+	isl_seq_cpy(vec->el, point->vec->el, point->vec->size);
+	for (i = 0; i < bmap->n_div; ++i) {
+		isl_seq_inner_product(bmap->div[i] + 1, vec->el,
+					1 + dim + i, &vec->el[1+dim+i]);
+		isl_int_fdiv_q(vec->el[1+dim+i], vec->el[1+dim+i],
+				bmap->div[i][0]);
+	}
+
+	contains = isl_basic_map_contains(bmap, vec);
+
+	isl_vec_free(vec);
+	return contains;
+}
+
+int isl_map_contains_point(__isl_keep isl_map *map, __isl_keep isl_point *point)
+{
+	int i;
+	int found = 0;
+
+	if (!map || !point)
+		return -1;
+
+	map = isl_map_copy(map);
+	map = isl_map_compute_divs(map);
+	if (!map)
+		return -1;
+
+	for (i = 0; i < map->n; ++i) {
+		found = isl_basic_map_contains_point(map->p[i], point);
+		if (found < 0)
+			goto error;
+		if (found)
+			break;
+	}
+	isl_map_free(map);
+
+	return found;
+error:
+	isl_map_free(map);
+	return -1;
+}
+
+int isl_set_contains_point(__isl_keep isl_set *set, __isl_keep isl_point *point)
+{
+	return isl_map_contains_point((isl_map *)set, point);
+}
+
+__isl_give isl_basic_set *isl_basic_set_from_point(__isl_take isl_point *pnt)
+{
+	isl_basic_set *bset;
+	isl_basic_set *model;
+
+	if (!pnt)
+		return NULL;
+
+	model = isl_basic_set_empty(isl_space_copy(pnt->dim));
+	bset = isl_basic_set_from_vec(isl_vec_copy(pnt->vec));
+	bset = isl_basic_set_from_underlying_set(bset, model);
+	isl_point_free(pnt);
+
+	return bset;
+}
+
+__isl_give isl_set *isl_set_from_point(__isl_take isl_point *pnt)
+{
+	isl_basic_set *bset;
+	bset = isl_basic_set_from_point(pnt);
+	return isl_set_from_basic_set(bset);
+}
+
+__isl_give isl_basic_set *isl_basic_set_box_from_points(
+	__isl_take isl_point *pnt1, __isl_take isl_point *pnt2)
+{
+	isl_basic_set *bset;
+	unsigned total;
+	int i;
+	int k;
+	isl_int t;
+
+	isl_int_init(t);
+
+	if (!pnt1 || !pnt2)
+		goto error;
+
+	isl_assert(pnt1->dim->ctx,
+			isl_space_is_equal(pnt1->dim, pnt2->dim), goto error);
+
+	if (isl_point_is_void(pnt1) && isl_point_is_void(pnt2)) {
+		isl_space *dim = isl_space_copy(pnt1->dim);
+		isl_point_free(pnt1);
+		isl_point_free(pnt2);
+		isl_int_clear(t);
+		return isl_basic_set_empty(dim);
+	}
+	if (isl_point_is_void(pnt1)) {
+		isl_point_free(pnt1);
+		isl_int_clear(t);
+		return isl_basic_set_from_point(pnt2);
+	}
+	if (isl_point_is_void(pnt2)) {
+		isl_point_free(pnt2);
+		isl_int_clear(t);
+		return isl_basic_set_from_point(pnt1);
+	}
+
+	total = isl_space_dim(pnt1->dim, isl_dim_all);
+	bset = isl_basic_set_alloc_space(isl_space_copy(pnt1->dim), 0, 0, 2 * total);
+
+	for (i = 0; i < total; ++i) {
+		isl_int_mul(t, pnt1->vec->el[1 + i], pnt2->vec->el[0]);
+		isl_int_submul(t, pnt2->vec->el[1 + i], pnt1->vec->el[0]);
+
+		k = isl_basic_set_alloc_inequality(bset);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(bset->ineq[k] + 1, total);
+		if (isl_int_is_pos(t)) {
+			isl_int_set_si(bset->ineq[k][1 + i], -1);
+			isl_int_set(bset->ineq[k][0], pnt1->vec->el[1 + i]);
+		} else {
+			isl_int_set_si(bset->ineq[k][1 + i], 1);
+			isl_int_neg(bset->ineq[k][0], pnt1->vec->el[1 + i]);
+		}
+		isl_int_fdiv_q(bset->ineq[k][0], bset->ineq[k][0], pnt1->vec->el[0]);
+
+		k = isl_basic_set_alloc_inequality(bset);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(bset->ineq[k] + 1, total);
+		if (isl_int_is_pos(t)) {
+			isl_int_set_si(bset->ineq[k][1 + i], 1);
+			isl_int_neg(bset->ineq[k][0], pnt2->vec->el[1 + i]);
+		} else {
+			isl_int_set_si(bset->ineq[k][1 + i], -1);
+			isl_int_set(bset->ineq[k][0], pnt2->vec->el[1 + i]);
+		}
+		isl_int_fdiv_q(bset->ineq[k][0], bset->ineq[k][0], pnt2->vec->el[0]);
+	}
+
+	bset = isl_basic_set_finalize(bset);
+
+	isl_point_free(pnt1);
+	isl_point_free(pnt2);
+
+	isl_int_clear(t);
+
+	return bset;
+error:
+	isl_point_free(pnt1);
+	isl_point_free(pnt2);
+	isl_int_clear(t);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_box_from_points(__isl_take isl_point *pnt1,
+	__isl_take isl_point *pnt2)
+{
+	isl_basic_set *bset;
+	bset = isl_basic_set_box_from_points(pnt1, pnt2);
+	return isl_set_from_basic_set(bset);
+}
+
+__isl_give isl_printer *isl_printer_print_point(
+	__isl_take isl_printer *p, __isl_keep isl_point *pnt)
+{
+	int i;
+	unsigned nparam;
+	unsigned dim;
+
+	if (!pnt)
+		return p;
+	if (isl_point_is_void(pnt)) {
+		p = isl_printer_print_str(p, "void");
+		return p;
+	}
+
+	nparam = isl_space_dim(pnt->dim, isl_dim_param);
+	dim = isl_space_dim(pnt->dim, isl_dim_set);
+	if (nparam > 0) {
+		p = isl_printer_print_str(p, "[");
+		for (i = 0; i < nparam; ++i) {
+			const char *name;
+			if (i)
+				p = isl_printer_print_str(p, ", ");
+			name = isl_space_get_dim_name(pnt->dim, isl_dim_param, i);
+			if (name) {
+				p = isl_printer_print_str(p, name);
+				p = isl_printer_print_str(p, " = ");
+			}
+			p = isl_printer_print_isl_int(p, pnt->vec->el[1 + i]);
+			if (!isl_int_is_one(pnt->vec->el[0])) {
+				p = isl_printer_print_str(p, "/");
+				p = isl_printer_print_isl_int(p, pnt->vec->el[0]);
+			}
+		}
+		p = isl_printer_print_str(p, "]");
+		p = isl_printer_print_str(p, " -> ");
+	}
+	p = isl_printer_print_str(p, "[");
+	for (i = 0; i < dim; ++i) {
+		if (i)
+			p = isl_printer_print_str(p, ", ");
+		p = isl_printer_print_isl_int(p, pnt->vec->el[1 + nparam + i]);
+		if (!isl_int_is_one(pnt->vec->el[0])) {
+			p = isl_printer_print_str(p, "/");
+			p = isl_printer_print_isl_int(p, pnt->vec->el[0]);
+		}
+	}
+	p = isl_printer_print_str(p, "]");
+	return p;
+}
diff --git a/polly/lib/External/isl/isl_point_private.h b/polly/lib/External/isl/isl_point_private.h
new file mode 100644
index 00000000000..305b81e381a
--- /dev/null
+++ b/polly/lib/External/isl/isl_point_private.h
@@ -0,0 +1,12 @@
+#include <isl/space.h>
+#include <isl/point.h>
+#include <isl/vec.h>
+
+struct isl_point {
+	int		ref;
+	isl_space	*dim;
+	struct isl_vec	*vec;
+};
+
+__isl_give isl_point *isl_point_alloc(__isl_take isl_space *dim,
+	__isl_take isl_vec *vec);
diff --git a/polly/lib/External/isl/isl_polynomial.c b/polly/lib/External/isl/isl_polynomial.c
new file mode 100644
index 00000000000..6b73bb397d0
--- /dev/null
+++ b/polly/lib/External/isl/isl_polynomial.c
@@ -0,0 +1,4866 @@
+/*
+ * Copyright 2010      INRIA Saclay
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France 
+ */
+
+#include <stdlib.h>
+#define ISL_DIM_H
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl_factorization.h>
+#include <isl_lp_private.h>
+#include <isl_seq.h>
+#include <isl_union_map_private.h>
+#include <isl_constraint_private.h>
+#include <isl_polynomial_private.h>
+#include <isl_point_private.h>
+#include <isl_space_private.h>
+#include <isl_mat_private.h>
+#include <isl_vec_private.h>
+#include <isl_range.h>
+#include <isl_local_space_private.h>
+#include <isl_aff_private.h>
+#include <isl_val_private.h>
+#include <isl_config.h>
+#include <isl/deprecated/polynomial_int.h>
+
+static unsigned pos(__isl_keep isl_space *dim, enum isl_dim_type type)
+{
+	switch (type) {
+	case isl_dim_param:	return 0;
+	case isl_dim_in:	return dim->nparam;
+	case isl_dim_out:	return dim->nparam + dim->n_in;
+	default:		return 0;
+	}
+}
+
+int isl_upoly_is_cst(__isl_keep struct isl_upoly *up)
+{
+	if (!up)
+		return -1;
+
+	return up->var < 0;
+}
+
+__isl_keep struct isl_upoly_cst *isl_upoly_as_cst(__isl_keep struct isl_upoly *up)
+{
+	if (!up)
+		return NULL;
+
+	isl_assert(up->ctx, up->var < 0, return NULL);
+
+	return (struct isl_upoly_cst *)up;
+}
+
+__isl_keep struct isl_upoly_rec *isl_upoly_as_rec(__isl_keep struct isl_upoly *up)
+{
+	if (!up)
+		return NULL;
+
+	isl_assert(up->ctx, up->var >= 0, return NULL);
+
+	return (struct isl_upoly_rec *)up;
+}
+
+int isl_upoly_is_equal(__isl_keep struct isl_upoly *up1,
+	__isl_keep struct isl_upoly *up2)
+{
+	int i;
+	struct isl_upoly_rec *rec1, *rec2;
+
+	if (!up1 || !up2)
+		return -1;
+	if (up1 == up2)
+		return 1;
+	if (up1->var != up2->var)
+		return 0;
+	if (isl_upoly_is_cst(up1)) {
+		struct isl_upoly_cst *cst1, *cst2;
+		cst1 = isl_upoly_as_cst(up1);
+		cst2 = isl_upoly_as_cst(up2);
+		if (!cst1 || !cst2)
+			return -1;
+		return isl_int_eq(cst1->n, cst2->n) &&
+		       isl_int_eq(cst1->d, cst2->d);
+	}
+
+	rec1 = isl_upoly_as_rec(up1);
+	rec2 = isl_upoly_as_rec(up2);
+	if (!rec1 || !rec2)
+		return -1;
+
+	if (rec1->n != rec2->n)
+		return 0;
+
+	for (i = 0; i < rec1->n; ++i) {
+		int eq = isl_upoly_is_equal(rec1->p[i], rec2->p[i]);
+		if (eq < 0 || !eq)
+			return eq;
+	}
+
+	return 1;
+}
+
+int isl_upoly_is_zero(__isl_keep struct isl_upoly *up)
+{
+	struct isl_upoly_cst *cst;
+
+	if (!up)
+		return -1;
+	if (!isl_upoly_is_cst(up))
+		return 0;
+
+	cst = isl_upoly_as_cst(up);
+	if (!cst)
+		return -1;
+
+	return isl_int_is_zero(cst->n) && isl_int_is_pos(cst->d);
+}
+
+int isl_upoly_sgn(__isl_keep struct isl_upoly *up)
+{
+	struct isl_upoly_cst *cst;
+
+	if (!up)
+		return 0;
+	if (!isl_upoly_is_cst(up))
+		return 0;
+
+	cst = isl_upoly_as_cst(up);
+	if (!cst)
+		return 0;
+
+	return isl_int_sgn(cst->n);
+}
+
+int isl_upoly_is_nan(__isl_keep struct isl_upoly *up)
+{
+	struct isl_upoly_cst *cst;
+
+	if (!up)
+		return -1;
+	if (!isl_upoly_is_cst(up))
+		return 0;
+
+	cst = isl_upoly_as_cst(up);
+	if (!cst)
+		return -1;
+
+	return isl_int_is_zero(cst->n) && isl_int_is_zero(cst->d);
+}
+
+int isl_upoly_is_infty(__isl_keep struct isl_upoly *up)
+{
+	struct isl_upoly_cst *cst;
+
+	if (!up)
+		return -1;
+	if (!isl_upoly_is_cst(up))
+		return 0;
+
+	cst = isl_upoly_as_cst(up);
+	if (!cst)
+		return -1;
+
+	return isl_int_is_pos(cst->n) && isl_int_is_zero(cst->d);
+}
+
+int isl_upoly_is_neginfty(__isl_keep struct isl_upoly *up)
+{
+	struct isl_upoly_cst *cst;
+
+	if (!up)
+		return -1;
+	if (!isl_upoly_is_cst(up))
+		return 0;
+
+	cst = isl_upoly_as_cst(up);
+	if (!cst)
+		return -1;
+
+	return isl_int_is_neg(cst->n) && isl_int_is_zero(cst->d);
+}
+
+int isl_upoly_is_one(__isl_keep struct isl_upoly *up)
+{
+	struct isl_upoly_cst *cst;
+
+	if (!up)
+		return -1;
+	if (!isl_upoly_is_cst(up))
+		return 0;
+
+	cst = isl_upoly_as_cst(up);
+	if (!cst)
+		return -1;
+
+	return isl_int_eq(cst->n, cst->d) && isl_int_is_pos(cst->d);
+}
+
+int isl_upoly_is_negone(__isl_keep struct isl_upoly *up)
+{
+	struct isl_upoly_cst *cst;
+
+	if (!up)
+		return -1;
+	if (!isl_upoly_is_cst(up))
+		return 0;
+
+	cst = isl_upoly_as_cst(up);
+	if (!cst)
+		return -1;
+
+	return isl_int_is_negone(cst->n) && isl_int_is_one(cst->d);
+}
+
+__isl_give struct isl_upoly_cst *isl_upoly_cst_alloc(struct isl_ctx *ctx)
+{
+	struct isl_upoly_cst *cst;
+
+	cst = isl_alloc_type(ctx, struct isl_upoly_cst);
+	if (!cst)
+		return NULL;
+
+	cst->up.ref = 1;
+	cst->up.ctx = ctx;
+	isl_ctx_ref(ctx);
+	cst->up.var = -1;
+
+	isl_int_init(cst->n);
+	isl_int_init(cst->d);
+
+	return cst;
+}
+
+__isl_give struct isl_upoly *isl_upoly_zero(struct isl_ctx *ctx)
+{
+	struct isl_upoly_cst *cst;
+
+	cst = isl_upoly_cst_alloc(ctx);
+	if (!cst)
+		return NULL;
+
+	isl_int_set_si(cst->n, 0);
+	isl_int_set_si(cst->d, 1);
+
+	return &cst->up;
+}
+
+__isl_give struct isl_upoly *isl_upoly_one(struct isl_ctx *ctx)
+{
+	struct isl_upoly_cst *cst;
+
+	cst = isl_upoly_cst_alloc(ctx);
+	if (!cst)
+		return NULL;
+
+	isl_int_set_si(cst->n, 1);
+	isl_int_set_si(cst->d, 1);
+
+	return &cst->up;
+}
+
+__isl_give struct isl_upoly *isl_upoly_infty(struct isl_ctx *ctx)
+{
+	struct isl_upoly_cst *cst;
+
+	cst = isl_upoly_cst_alloc(ctx);
+	if (!cst)
+		return NULL;
+
+	isl_int_set_si(cst->n, 1);
+	isl_int_set_si(cst->d, 0);
+
+	return &cst->up;
+}
+
+__isl_give struct isl_upoly *isl_upoly_neginfty(struct isl_ctx *ctx)
+{
+	struct isl_upoly_cst *cst;
+
+	cst = isl_upoly_cst_alloc(ctx);
+	if (!cst)
+		return NULL;
+
+	isl_int_set_si(cst->n, -1);
+	isl_int_set_si(cst->d, 0);
+
+	return &cst->up;
+}
+
+__isl_give struct isl_upoly *isl_upoly_nan(struct isl_ctx *ctx)
+{
+	struct isl_upoly_cst *cst;
+
+	cst = isl_upoly_cst_alloc(ctx);
+	if (!cst)
+		return NULL;
+
+	isl_int_set_si(cst->n, 0);
+	isl_int_set_si(cst->d, 0);
+
+	return &cst->up;
+}
+
+__isl_give struct isl_upoly *isl_upoly_rat_cst(struct isl_ctx *ctx,
+	isl_int n, isl_int d)
+{
+	struct isl_upoly_cst *cst;
+
+	cst = isl_upoly_cst_alloc(ctx);
+	if (!cst)
+		return NULL;
+
+	isl_int_set(cst->n, n);
+	isl_int_set(cst->d, d);
+
+	return &cst->up;
+}
+
+__isl_give struct isl_upoly_rec *isl_upoly_alloc_rec(struct isl_ctx *ctx,
+	int var, int size)
+{
+	struct isl_upoly_rec *rec;
+
+	isl_assert(ctx, var >= 0, return NULL);
+	isl_assert(ctx, size >= 0, return NULL);
+	rec = isl_calloc(ctx, struct isl_upoly_rec,
+			sizeof(struct isl_upoly_rec) +
+			size * sizeof(struct isl_upoly *));
+	if (!rec)
+		return NULL;
+
+	rec->up.ref = 1;
+	rec->up.ctx = ctx;
+	isl_ctx_ref(ctx);
+	rec->up.var = var;
+
+	rec->n = 0;
+	rec->size = size;
+
+	return rec;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_reset_domain_space(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_space *dim)
+{
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp || !dim)
+		goto error;
+
+	isl_space_free(qp->dim);
+	qp->dim = dim;
+
+	return qp;
+error:
+	isl_qpolynomial_free(qp);
+	isl_space_free(dim);
+	return NULL;
+}
+
+/* Reset the space of "qp".  This function is called from isl_pw_templ.c
+ * and doesn't know if the space of an element object is represented
+ * directly or through its domain.  It therefore passes along both.
+ */
+__isl_give isl_qpolynomial *isl_qpolynomial_reset_space_and_domain(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_space *space,
+	__isl_take isl_space *domain)
+{
+	isl_space_free(space);
+	return isl_qpolynomial_reset_domain_space(qp, domain);
+}
+
+isl_ctx *isl_qpolynomial_get_ctx(__isl_keep isl_qpolynomial *qp)
+{
+	return qp ? qp->dim->ctx : NULL;
+}
+
+__isl_give isl_space *isl_qpolynomial_get_domain_space(
+	__isl_keep isl_qpolynomial *qp)
+{
+	return qp ? isl_space_copy(qp->dim) : NULL;
+}
+
+__isl_give isl_space *isl_qpolynomial_get_space(__isl_keep isl_qpolynomial *qp)
+{
+	isl_space *space;
+	if (!qp)
+		return NULL;
+	space = isl_space_copy(qp->dim);
+	space = isl_space_from_domain(space);
+	space = isl_space_add_dims(space, isl_dim_out, 1);
+	return space;
+}
+
+/* Externally, an isl_qpolynomial has a map space, but internally, the
+ * ls field corresponds to the domain of that space.
+ */
+unsigned isl_qpolynomial_dim(__isl_keep isl_qpolynomial *qp,
+	enum isl_dim_type type)
+{
+	if (!qp)
+		return 0;
+	if (type == isl_dim_out)
+		return 1;
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+	return isl_space_dim(qp->dim, type);
+}
+
+int isl_qpolynomial_is_zero(__isl_keep isl_qpolynomial *qp)
+{
+	return qp ? isl_upoly_is_zero(qp->upoly) : -1;
+}
+
+int isl_qpolynomial_is_one(__isl_keep isl_qpolynomial *qp)
+{
+	return qp ? isl_upoly_is_one(qp->upoly) : -1;
+}
+
+int isl_qpolynomial_is_nan(__isl_keep isl_qpolynomial *qp)
+{
+	return qp ? isl_upoly_is_nan(qp->upoly) : -1;
+}
+
+int isl_qpolynomial_is_infty(__isl_keep isl_qpolynomial *qp)
+{
+	return qp ? isl_upoly_is_infty(qp->upoly) : -1;
+}
+
+int isl_qpolynomial_is_neginfty(__isl_keep isl_qpolynomial *qp)
+{
+	return qp ? isl_upoly_is_neginfty(qp->upoly) : -1;
+}
+
+int isl_qpolynomial_sgn(__isl_keep isl_qpolynomial *qp)
+{
+	return qp ? isl_upoly_sgn(qp->upoly) : 0;
+}
+
+static void upoly_free_cst(__isl_take struct isl_upoly_cst *cst)
+{
+	isl_int_clear(cst->n);
+	isl_int_clear(cst->d);
+}
+
+static void upoly_free_rec(__isl_take struct isl_upoly_rec *rec)
+{
+	int i;
+
+	for (i = 0; i < rec->n; ++i)
+		isl_upoly_free(rec->p[i]);
+}
+
+__isl_give struct isl_upoly *isl_upoly_copy(__isl_keep struct isl_upoly *up)
+{
+	if (!up)
+		return NULL;
+
+	up->ref++;
+	return up;
+}
+
+__isl_give struct isl_upoly *isl_upoly_dup_cst(__isl_keep struct isl_upoly *up)
+{
+	struct isl_upoly_cst *cst;
+	struct isl_upoly_cst *dup;
+
+	cst = isl_upoly_as_cst(up);
+	if (!cst)
+		return NULL;
+
+	dup = isl_upoly_as_cst(isl_upoly_zero(up->ctx));
+	if (!dup)
+		return NULL;
+	isl_int_set(dup->n, cst->n);
+	isl_int_set(dup->d, cst->d);
+
+	return &dup->up;
+}
+
+__isl_give struct isl_upoly *isl_upoly_dup_rec(__isl_keep struct isl_upoly *up)
+{
+	int i;
+	struct isl_upoly_rec *rec;
+	struct isl_upoly_rec *dup;
+
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		return NULL;
+
+	dup = isl_upoly_alloc_rec(up->ctx, up->var, rec->n);
+	if (!dup)
+		return NULL;
+
+	for (i = 0; i < rec->n; ++i) {
+		dup->p[i] = isl_upoly_copy(rec->p[i]);
+		if (!dup->p[i])
+			goto error;
+		dup->n++;
+	}
+
+	return &dup->up;
+error:
+	isl_upoly_free(&dup->up);
+	return NULL;
+}
+
+__isl_give struct isl_upoly *isl_upoly_dup(__isl_keep struct isl_upoly *up)
+{
+	if (!up)
+		return NULL;
+
+	if (isl_upoly_is_cst(up))
+		return isl_upoly_dup_cst(up);
+	else
+		return isl_upoly_dup_rec(up);
+}
+
+__isl_give struct isl_upoly *isl_upoly_cow(__isl_take struct isl_upoly *up)
+{
+	if (!up)
+		return NULL;
+
+	if (up->ref == 1)
+		return up;
+	up->ref--;
+	return isl_upoly_dup(up);
+}
+
+void isl_upoly_free(__isl_take struct isl_upoly *up)
+{
+	if (!up)
+		return;
+
+	if (--up->ref > 0)
+		return;
+
+	if (up->var < 0)
+		upoly_free_cst((struct isl_upoly_cst *)up);
+	else
+		upoly_free_rec((struct isl_upoly_rec *)up);
+
+	isl_ctx_deref(up->ctx);
+	free(up);
+}
+
+static void isl_upoly_cst_reduce(__isl_keep struct isl_upoly_cst *cst)
+{
+	isl_int gcd;
+
+	isl_int_init(gcd);
+	isl_int_gcd(gcd, cst->n, cst->d);
+	if (!isl_int_is_zero(gcd) && !isl_int_is_one(gcd)) {
+		isl_int_divexact(cst->n, cst->n, gcd);
+		isl_int_divexact(cst->d, cst->d, gcd);
+	}
+	isl_int_clear(gcd);
+}
+
+__isl_give struct isl_upoly *isl_upoly_sum_cst(__isl_take struct isl_upoly *up1,
+	__isl_take struct isl_upoly *up2)
+{
+	struct isl_upoly_cst *cst1;
+	struct isl_upoly_cst *cst2;
+
+	up1 = isl_upoly_cow(up1);
+	if (!up1 || !up2)
+		goto error;
+
+	cst1 = isl_upoly_as_cst(up1);
+	cst2 = isl_upoly_as_cst(up2);
+
+	if (isl_int_eq(cst1->d, cst2->d))
+		isl_int_add(cst1->n, cst1->n, cst2->n);
+	else {
+		isl_int_mul(cst1->n, cst1->n, cst2->d);
+		isl_int_addmul(cst1->n, cst2->n, cst1->d);
+		isl_int_mul(cst1->d, cst1->d, cst2->d);
+	}
+
+	isl_upoly_cst_reduce(cst1);
+
+	isl_upoly_free(up2);
+	return up1;
+error:
+	isl_upoly_free(up1);
+	isl_upoly_free(up2);
+	return NULL;
+}
+
+static __isl_give struct isl_upoly *replace_by_zero(
+	__isl_take struct isl_upoly *up)
+{
+	struct isl_ctx *ctx;
+
+	if (!up)
+		return NULL;
+	ctx = up->ctx;
+	isl_upoly_free(up);
+	return isl_upoly_zero(ctx);
+}
+
+static __isl_give struct isl_upoly *replace_by_constant_term(
+	__isl_take struct isl_upoly *up)
+{
+	struct isl_upoly_rec *rec;
+	struct isl_upoly *cst;
+
+	if (!up)
+		return NULL;
+
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		goto error;
+	cst = isl_upoly_copy(rec->p[0]);
+	isl_upoly_free(up);
+	return cst;
+error:
+	isl_upoly_free(up);
+	return NULL;
+}
+
+__isl_give struct isl_upoly *isl_upoly_sum(__isl_take struct isl_upoly *up1,
+	__isl_take struct isl_upoly *up2)
+{
+	int i;
+	struct isl_upoly_rec *rec1, *rec2;
+
+	if (!up1 || !up2)
+		goto error;
+
+	if (isl_upoly_is_nan(up1)) {
+		isl_upoly_free(up2);
+		return up1;
+	}
+
+	if (isl_upoly_is_nan(up2)) {
+		isl_upoly_free(up1);
+		return up2;
+	}
+
+	if (isl_upoly_is_zero(up1)) {
+		isl_upoly_free(up1);
+		return up2;
+	}
+
+	if (isl_upoly_is_zero(up2)) {
+		isl_upoly_free(up2);
+		return up1;
+	}
+
+	if (up1->var < up2->var)
+		return isl_upoly_sum(up2, up1);
+
+	if (up2->var < up1->var) {
+		struct isl_upoly_rec *rec;
+		if (isl_upoly_is_infty(up2) || isl_upoly_is_neginfty(up2)) {
+			isl_upoly_free(up1);
+			return up2;
+		}
+		up1 = isl_upoly_cow(up1);
+		rec = isl_upoly_as_rec(up1);
+		if (!rec)
+			goto error;
+		rec->p[0] = isl_upoly_sum(rec->p[0], up2);
+		if (rec->n == 1)
+			up1 = replace_by_constant_term(up1);
+		return up1;
+	}
+
+	if (isl_upoly_is_cst(up1))
+		return isl_upoly_sum_cst(up1, up2);
+
+	rec1 = isl_upoly_as_rec(up1);
+	rec2 = isl_upoly_as_rec(up2);
+	if (!rec1 || !rec2)
+		goto error;
+
+	if (rec1->n < rec2->n)
+		return isl_upoly_sum(up2, up1);
+
+	up1 = isl_upoly_cow(up1);
+	rec1 = isl_upoly_as_rec(up1);
+	if (!rec1)
+		goto error;
+
+	for (i = rec2->n - 1; i >= 0; --i) {
+		rec1->p[i] = isl_upoly_sum(rec1->p[i],
+					    isl_upoly_copy(rec2->p[i]));
+		if (!rec1->p[i])
+			goto error;
+		if (i == rec1->n - 1 && isl_upoly_is_zero(rec1->p[i])) {
+			isl_upoly_free(rec1->p[i]);
+			rec1->n--;
+		}
+	}
+
+	if (rec1->n == 0)
+		up1 = replace_by_zero(up1);
+	else if (rec1->n == 1)
+		up1 = replace_by_constant_term(up1);
+
+	isl_upoly_free(up2);
+
+	return up1;
+error:
+	isl_upoly_free(up1);
+	isl_upoly_free(up2);
+	return NULL;
+}
+
+__isl_give struct isl_upoly *isl_upoly_cst_add_isl_int(
+	__isl_take struct isl_upoly *up, isl_int v)
+{
+	struct isl_upoly_cst *cst;
+
+	up = isl_upoly_cow(up);
+	if (!up)
+		return NULL;
+
+	cst = isl_upoly_as_cst(up);
+
+	isl_int_addmul(cst->n, cst->d, v);
+
+	return up;
+}
+
+__isl_give struct isl_upoly *isl_upoly_add_isl_int(
+	__isl_take struct isl_upoly *up, isl_int v)
+{
+	struct isl_upoly_rec *rec;
+
+	if (!up)
+		return NULL;
+
+	if (isl_upoly_is_cst(up))
+		return isl_upoly_cst_add_isl_int(up, v);
+
+	up = isl_upoly_cow(up);
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		goto error;
+
+	rec->p[0] = isl_upoly_add_isl_int(rec->p[0], v);
+	if (!rec->p[0])
+		goto error;
+
+	return up;
+error:
+	isl_upoly_free(up);
+	return NULL;
+}
+
+__isl_give struct isl_upoly *isl_upoly_cst_mul_isl_int(
+	__isl_take struct isl_upoly *up, isl_int v)
+{
+	struct isl_upoly_cst *cst;
+
+	if (isl_upoly_is_zero(up))
+		return up;
+
+	up = isl_upoly_cow(up);
+	if (!up)
+		return NULL;
+
+	cst = isl_upoly_as_cst(up);
+
+	isl_int_mul(cst->n, cst->n, v);
+
+	return up;
+}
+
+__isl_give struct isl_upoly *isl_upoly_mul_isl_int(
+	__isl_take struct isl_upoly *up, isl_int v)
+{
+	int i;
+	struct isl_upoly_rec *rec;
+
+	if (!up)
+		return NULL;
+
+	if (isl_upoly_is_cst(up))
+		return isl_upoly_cst_mul_isl_int(up, v);
+
+	up = isl_upoly_cow(up);
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		goto error;
+
+	for (i = 0; i < rec->n; ++i) {
+		rec->p[i] = isl_upoly_mul_isl_int(rec->p[i], v);
+		if (!rec->p[i])
+			goto error;
+	}
+
+	return up;
+error:
+	isl_upoly_free(up);
+	return NULL;
+}
+
+/* Multiply the constant polynomial "up" by "v".
+ */
+static __isl_give struct isl_upoly *isl_upoly_cst_scale_val(
+	__isl_take struct isl_upoly *up, __isl_keep isl_val *v)
+{
+	struct isl_upoly_cst *cst;
+
+	if (isl_upoly_is_zero(up))
+		return up;
+
+	up = isl_upoly_cow(up);
+	if (!up)
+		return NULL;
+
+	cst = isl_upoly_as_cst(up);
+
+	isl_int_mul(cst->n, cst->n, v->n);
+	isl_int_mul(cst->d, cst->d, v->d);
+	isl_upoly_cst_reduce(cst);
+
+	return up;
+}
+
+/* Multiply the polynomial "up" by "v".
+ */
+static __isl_give struct isl_upoly *isl_upoly_scale_val(
+	__isl_take struct isl_upoly *up, __isl_keep isl_val *v)
+{
+	int i;
+	struct isl_upoly_rec *rec;
+
+	if (!up)
+		return NULL;
+
+	if (isl_upoly_is_cst(up))
+		return isl_upoly_cst_scale_val(up, v);
+
+	up = isl_upoly_cow(up);
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		goto error;
+
+	for (i = 0; i < rec->n; ++i) {
+		rec->p[i] = isl_upoly_scale_val(rec->p[i], v);
+		if (!rec->p[i])
+			goto error;
+	}
+
+	return up;
+error:
+	isl_upoly_free(up);
+	return NULL;
+}
+
+__isl_give struct isl_upoly *isl_upoly_mul_cst(__isl_take struct isl_upoly *up1,
+	__isl_take struct isl_upoly *up2)
+{
+	struct isl_upoly_cst *cst1;
+	struct isl_upoly_cst *cst2;
+
+	up1 = isl_upoly_cow(up1);
+	if (!up1 || !up2)
+		goto error;
+
+	cst1 = isl_upoly_as_cst(up1);
+	cst2 = isl_upoly_as_cst(up2);
+
+	isl_int_mul(cst1->n, cst1->n, cst2->n);
+	isl_int_mul(cst1->d, cst1->d, cst2->d);
+
+	isl_upoly_cst_reduce(cst1);
+
+	isl_upoly_free(up2);
+	return up1;
+error:
+	isl_upoly_free(up1);
+	isl_upoly_free(up2);
+	return NULL;
+}
+
+__isl_give struct isl_upoly *isl_upoly_mul_rec(__isl_take struct isl_upoly *up1,
+	__isl_take struct isl_upoly *up2)
+{
+	struct isl_upoly_rec *rec1;
+	struct isl_upoly_rec *rec2;
+	struct isl_upoly_rec *res = NULL;
+	int i, j;
+	int size;
+
+	rec1 = isl_upoly_as_rec(up1);
+	rec2 = isl_upoly_as_rec(up2);
+	if (!rec1 || !rec2)
+		goto error;
+	size = rec1->n + rec2->n - 1;
+	res = isl_upoly_alloc_rec(up1->ctx, up1->var, size);
+	if (!res)
+		goto error;
+
+	for (i = 0; i < rec1->n; ++i) {
+		res->p[i] = isl_upoly_mul(isl_upoly_copy(rec2->p[0]),
+					    isl_upoly_copy(rec1->p[i]));
+		if (!res->p[i])
+			goto error;
+		res->n++;
+	}
+	for (; i < size; ++i) {
+		res->p[i] = isl_upoly_zero(up1->ctx);
+		if (!res->p[i])
+			goto error;
+		res->n++;
+	}
+	for (i = 0; i < rec1->n; ++i) {
+		for (j = 1; j < rec2->n; ++j) {
+			struct isl_upoly *up;
+			up = isl_upoly_mul(isl_upoly_copy(rec2->p[j]),
+					    isl_upoly_copy(rec1->p[i]));
+			res->p[i + j] = isl_upoly_sum(res->p[i + j], up);
+			if (!res->p[i + j])
+				goto error;
+		}
+	}
+
+	isl_upoly_free(up1);
+	isl_upoly_free(up2);
+
+	return &res->up;
+error:
+	isl_upoly_free(up1);
+	isl_upoly_free(up2);
+	isl_upoly_free(&res->up);
+	return NULL;
+}
+
+__isl_give struct isl_upoly *isl_upoly_mul(__isl_take struct isl_upoly *up1,
+	__isl_take struct isl_upoly *up2)
+{
+	if (!up1 || !up2)
+		goto error;
+
+	if (isl_upoly_is_nan(up1)) {
+		isl_upoly_free(up2);
+		return up1;
+	}
+
+	if (isl_upoly_is_nan(up2)) {
+		isl_upoly_free(up1);
+		return up2;
+	}
+
+	if (isl_upoly_is_zero(up1)) {
+		isl_upoly_free(up2);
+		return up1;
+	}
+
+	if (isl_upoly_is_zero(up2)) {
+		isl_upoly_free(up1);
+		return up2;
+	}
+
+	if (isl_upoly_is_one(up1)) {
+		isl_upoly_free(up1);
+		return up2;
+	}
+
+	if (isl_upoly_is_one(up2)) {
+		isl_upoly_free(up2);
+		return up1;
+	}
+
+	if (up1->var < up2->var)
+		return isl_upoly_mul(up2, up1);
+
+	if (up2->var < up1->var) {
+		int i;
+		struct isl_upoly_rec *rec;
+		if (isl_upoly_is_infty(up2) || isl_upoly_is_neginfty(up2)) {
+			isl_ctx *ctx = up1->ctx;
+			isl_upoly_free(up1);
+			isl_upoly_free(up2);
+			return isl_upoly_nan(ctx);
+		}
+		up1 = isl_upoly_cow(up1);
+		rec = isl_upoly_as_rec(up1);
+		if (!rec)
+			goto error;
+
+		for (i = 0; i < rec->n; ++i) {
+			rec->p[i] = isl_upoly_mul(rec->p[i],
+						    isl_upoly_copy(up2));
+			if (!rec->p[i])
+				goto error;
+		}
+		isl_upoly_free(up2);
+		return up1;
+	}
+
+	if (isl_upoly_is_cst(up1))
+		return isl_upoly_mul_cst(up1, up2);
+
+	return isl_upoly_mul_rec(up1, up2);
+error:
+	isl_upoly_free(up1);
+	isl_upoly_free(up2);
+	return NULL;
+}
+
+__isl_give struct isl_upoly *isl_upoly_pow(__isl_take struct isl_upoly *up,
+	unsigned power)
+{
+	struct isl_upoly *res;
+
+	if (!up)
+		return NULL;
+	if (power == 1)
+		return up;
+
+	if (power % 2)
+		res = isl_upoly_copy(up);
+	else
+		res = isl_upoly_one(up->ctx);
+
+	while (power >>= 1) {
+		up = isl_upoly_mul(up, isl_upoly_copy(up));
+		if (power % 2)
+			res = isl_upoly_mul(res, isl_upoly_copy(up));
+	}
+
+	isl_upoly_free(up);
+	return res;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_alloc(__isl_take isl_space *dim,
+	unsigned n_div, __isl_take struct isl_upoly *up)
+{
+	struct isl_qpolynomial *qp = NULL;
+	unsigned total;
+
+	if (!dim || !up)
+		goto error;
+
+	if (!isl_space_is_set(dim))
+		isl_die(isl_space_get_ctx(dim), isl_error_invalid,
+			"domain of polynomial should be a set", goto error);
+
+	total = isl_space_dim(dim, isl_dim_all);
+
+	qp = isl_calloc_type(dim->ctx, struct isl_qpolynomial);
+	if (!qp)
+		goto error;
+
+	qp->ref = 1;
+	qp->div = isl_mat_alloc(dim->ctx, n_div, 1 + 1 + total + n_div);
+	if (!qp->div)
+		goto error;
+
+	qp->dim = dim;
+	qp->upoly = up;
+
+	return qp;
+error:
+	isl_space_free(dim);
+	isl_upoly_free(up);
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_copy(__isl_keep isl_qpolynomial *qp)
+{
+	if (!qp)
+		return NULL;
+
+	qp->ref++;
+	return qp;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_dup(__isl_keep isl_qpolynomial *qp)
+{
+	struct isl_qpolynomial *dup;
+
+	if (!qp)
+		return NULL;
+
+	dup = isl_qpolynomial_alloc(isl_space_copy(qp->dim), qp->div->n_row,
+				    isl_upoly_copy(qp->upoly));
+	if (!dup)
+		return NULL;
+	isl_mat_free(dup->div);
+	dup->div = isl_mat_copy(qp->div);
+	if (!dup->div)
+		goto error;
+
+	return dup;
+error:
+	isl_qpolynomial_free(dup);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_cow(__isl_take isl_qpolynomial *qp)
+{
+	if (!qp)
+		return NULL;
+
+	if (qp->ref == 1)
+		return qp;
+	qp->ref--;
+	return isl_qpolynomial_dup(qp);
+}
+
+__isl_null isl_qpolynomial *isl_qpolynomial_free(
+	__isl_take isl_qpolynomial *qp)
+{
+	if (!qp)
+		return NULL;
+
+	if (--qp->ref > 0)
+		return NULL;
+
+	isl_space_free(qp->dim);
+	isl_mat_free(qp->div);
+	isl_upoly_free(qp->upoly);
+
+	free(qp);
+	return NULL;
+}
+
+__isl_give struct isl_upoly *isl_upoly_var_pow(isl_ctx *ctx, int pos, int power)
+{
+	int i;
+	struct isl_upoly_rec *rec;
+	struct isl_upoly_cst *cst;
+
+	rec = isl_upoly_alloc_rec(ctx, pos, 1 + power);
+	if (!rec)
+		return NULL;
+	for (i = 0; i < 1 + power; ++i) {
+		rec->p[i] = isl_upoly_zero(ctx);
+		if (!rec->p[i])
+			goto error;
+		rec->n++;
+	}
+	cst = isl_upoly_as_cst(rec->p[power]);
+	isl_int_set_si(cst->n, 1);
+
+	return &rec->up;
+error:
+	isl_upoly_free(&rec->up);
+	return NULL;
+}
+
+/* r array maps original positions to new positions.
+ */
+static __isl_give struct isl_upoly *reorder(__isl_take struct isl_upoly *up,
+	int *r)
+{
+	int i;
+	struct isl_upoly_rec *rec;
+	struct isl_upoly *base;
+	struct isl_upoly *res;
+
+	if (isl_upoly_is_cst(up))
+		return up;
+
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		goto error;
+
+	isl_assert(up->ctx, rec->n >= 1, goto error);
+
+	base = isl_upoly_var_pow(up->ctx, r[up->var], 1);
+	res = reorder(isl_upoly_copy(rec->p[rec->n - 1]), r);
+
+	for (i = rec->n - 2; i >= 0; --i) {
+		res = isl_upoly_mul(res, isl_upoly_copy(base));
+		res = isl_upoly_sum(res, reorder(isl_upoly_copy(rec->p[i]), r));
+	}
+
+	isl_upoly_free(base);
+	isl_upoly_free(up);
+
+	return res;
+error:
+	isl_upoly_free(up);
+	return NULL;
+}
+
+static int compatible_divs(__isl_keep isl_mat *div1, __isl_keep isl_mat *div2)
+{
+	int n_row, n_col;
+	int equal;
+
+	isl_assert(div1->ctx, div1->n_row >= div2->n_row &&
+				div1->n_col >= div2->n_col, return -1);
+
+	if (div1->n_row == div2->n_row)
+		return isl_mat_is_equal(div1, div2);
+
+	n_row = div1->n_row;
+	n_col = div1->n_col;
+	div1->n_row = div2->n_row;
+	div1->n_col = div2->n_col;
+
+	equal = isl_mat_is_equal(div1, div2);
+
+	div1->n_row = n_row;
+	div1->n_col = n_col;
+
+	return equal;
+}
+
+static int cmp_row(__isl_keep isl_mat *div, int i, int j)
+{
+	int li, lj;
+
+	li = isl_seq_last_non_zero(div->row[i], div->n_col);
+	lj = isl_seq_last_non_zero(div->row[j], div->n_col);
+
+	if (li != lj)
+		return li - lj;
+
+	return isl_seq_cmp(div->row[i], div->row[j], div->n_col);
+}
+
+struct isl_div_sort_info {
+	isl_mat	*div;
+	int	 row;
+};
+
+static int div_sort_cmp(const void *p1, const void *p2)
+{
+	const struct isl_div_sort_info *i1, *i2;
+	i1 = (const struct isl_div_sort_info *) p1;
+	i2 = (const struct isl_div_sort_info *) p2;
+
+	return cmp_row(i1->div, i1->row, i2->row);
+}
+
+/* Sort divs and remove duplicates.
+ */
+static __isl_give isl_qpolynomial *sort_divs(__isl_take isl_qpolynomial *qp)
+{
+	int i;
+	int skip;
+	int len;
+	struct isl_div_sort_info *array = NULL;
+	int *pos = NULL, *at = NULL;
+	int *reordering = NULL;
+	unsigned div_pos;
+
+	if (!qp)
+		return NULL;
+	if (qp->div->n_row <= 1)
+		return qp;
+
+	div_pos = isl_space_dim(qp->dim, isl_dim_all);
+
+	array = isl_alloc_array(qp->div->ctx, struct isl_div_sort_info,
+				qp->div->n_row);
+	pos = isl_alloc_array(qp->div->ctx, int, qp->div->n_row);
+	at = isl_alloc_array(qp->div->ctx, int, qp->div->n_row);
+	len = qp->div->n_col - 2;
+	reordering = isl_alloc_array(qp->div->ctx, int, len);
+	if (!array || !pos || !at || !reordering)
+		goto error;
+
+	for (i = 0; i < qp->div->n_row; ++i) {
+		array[i].div = qp->div;
+		array[i].row = i;
+		pos[i] = i;
+		at[i] = i;
+	}
+
+	qsort(array, qp->div->n_row, sizeof(struct isl_div_sort_info),
+		div_sort_cmp);
+
+	for (i = 0; i < div_pos; ++i)
+		reordering[i] = i;
+
+	for (i = 0; i < qp->div->n_row; ++i) {
+		if (pos[array[i].row] == i)
+			continue;
+		qp->div = isl_mat_swap_rows(qp->div, i, pos[array[i].row]);
+		pos[at[i]] = pos[array[i].row];
+		at[pos[array[i].row]] = at[i];
+		at[i] = array[i].row;
+		pos[array[i].row] = i;
+	}
+
+	skip = 0;
+	for (i = 0; i < len - div_pos; ++i) {
+		if (i > 0 &&
+		    isl_seq_eq(qp->div->row[i - skip - 1],
+			       qp->div->row[i - skip], qp->div->n_col)) {
+			qp->div = isl_mat_drop_rows(qp->div, i - skip, 1);
+			isl_mat_col_add(qp->div, 2 + div_pos + i - skip - 1,
+						 2 + div_pos + i - skip);
+			qp->div = isl_mat_drop_cols(qp->div,
+						    2 + div_pos + i - skip, 1);
+			skip++;
+		}
+		reordering[div_pos + array[i].row] = div_pos + i - skip;
+	}
+
+	qp->upoly = reorder(qp->upoly, reordering);
+
+	if (!qp->upoly || !qp->div)
+		goto error;
+
+	free(at);
+	free(pos);
+	free(array);
+	free(reordering);
+
+	return qp;
+error:
+	free(at);
+	free(pos);
+	free(array);
+	free(reordering);
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+static __isl_give struct isl_upoly *expand(__isl_take struct isl_upoly *up,
+	int *exp, int first)
+{
+	int i;
+	struct isl_upoly_rec *rec;
+
+	if (isl_upoly_is_cst(up))
+		return up;
+
+	if (up->var < first)
+		return up;
+
+	if (exp[up->var - first] == up->var - first)
+		return up;
+
+	up = isl_upoly_cow(up);
+	if (!up)
+		goto error;
+
+	up->var = exp[up->var - first] + first;
+
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		goto error;
+
+	for (i = 0; i < rec->n; ++i) {
+		rec->p[i] = expand(rec->p[i], exp, first);
+		if (!rec->p[i])
+			goto error;
+	}
+
+	return up;
+error:
+	isl_upoly_free(up);
+	return NULL;
+}
+
+static __isl_give isl_qpolynomial *with_merged_divs(
+	__isl_give isl_qpolynomial *(*fn)(__isl_take isl_qpolynomial *qp1,
+					  __isl_take isl_qpolynomial *qp2),
+	__isl_take isl_qpolynomial *qp1, __isl_take isl_qpolynomial *qp2)
+{
+	int *exp1 = NULL;
+	int *exp2 = NULL;
+	isl_mat *div = NULL;
+	int n_div1, n_div2;
+
+	qp1 = isl_qpolynomial_cow(qp1);
+	qp2 = isl_qpolynomial_cow(qp2);
+
+	if (!qp1 || !qp2)
+		goto error;
+
+	isl_assert(qp1->div->ctx, qp1->div->n_row >= qp2->div->n_row &&
+				qp1->div->n_col >= qp2->div->n_col, goto error);
+
+	n_div1 = qp1->div->n_row;
+	n_div2 = qp2->div->n_row;
+	exp1 = isl_alloc_array(qp1->div->ctx, int, n_div1);
+	exp2 = isl_alloc_array(qp2->div->ctx, int, n_div2);
+	if ((n_div1 && !exp1) || (n_div2 && !exp2))
+		goto error;
+
+	div = isl_merge_divs(qp1->div, qp2->div, exp1, exp2);
+	if (!div)
+		goto error;
+
+	isl_mat_free(qp1->div);
+	qp1->div = isl_mat_copy(div);
+	isl_mat_free(qp2->div);
+	qp2->div = isl_mat_copy(div);
+
+	qp1->upoly = expand(qp1->upoly, exp1, div->n_col - div->n_row - 2);
+	qp2->upoly = expand(qp2->upoly, exp2, div->n_col - div->n_row - 2);
+
+	if (!qp1->upoly || !qp2->upoly)
+		goto error;
+
+	isl_mat_free(div);
+	free(exp1);
+	free(exp2);
+
+	return fn(qp1, qp2);
+error:
+	isl_mat_free(div);
+	free(exp1);
+	free(exp2);
+	isl_qpolynomial_free(qp1);
+	isl_qpolynomial_free(qp2);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_add(__isl_take isl_qpolynomial *qp1,
+	__isl_take isl_qpolynomial *qp2)
+{
+	qp1 = isl_qpolynomial_cow(qp1);
+
+	if (!qp1 || !qp2)
+		goto error;
+
+	if (qp1->div->n_row < qp2->div->n_row)
+		return isl_qpolynomial_add(qp2, qp1);
+
+	isl_assert(qp1->dim->ctx, isl_space_is_equal(qp1->dim, qp2->dim), goto error);
+	if (!compatible_divs(qp1->div, qp2->div))
+		return with_merged_divs(isl_qpolynomial_add, qp1, qp2);
+
+	qp1->upoly = isl_upoly_sum(qp1->upoly, isl_upoly_copy(qp2->upoly));
+	if (!qp1->upoly)
+		goto error;
+
+	isl_qpolynomial_free(qp2);
+
+	return qp1;
+error:
+	isl_qpolynomial_free(qp1);
+	isl_qpolynomial_free(qp2);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_add_on_domain(
+	__isl_keep isl_set *dom,
+	__isl_take isl_qpolynomial *qp1,
+	__isl_take isl_qpolynomial *qp2)
+{
+	qp1 = isl_qpolynomial_add(qp1, qp2);
+	qp1 = isl_qpolynomial_gist(qp1, isl_set_copy(dom));
+	return qp1;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_sub(__isl_take isl_qpolynomial *qp1,
+	__isl_take isl_qpolynomial *qp2)
+{
+	return isl_qpolynomial_add(qp1, isl_qpolynomial_neg(qp2));
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_add_isl_int(
+	__isl_take isl_qpolynomial *qp, isl_int v)
+{
+	if (isl_int_is_zero(v))
+		return qp;
+
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp)
+		return NULL;
+
+	qp->upoly = isl_upoly_add_isl_int(qp->upoly, v);
+	if (!qp->upoly)
+		goto error;
+
+	return qp;
+error:
+	isl_qpolynomial_free(qp);
+	return NULL;
+
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_neg(__isl_take isl_qpolynomial *qp)
+{
+	if (!qp)
+		return NULL;
+
+	return isl_qpolynomial_mul_isl_int(qp, qp->dim->ctx->negone);
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_mul_isl_int(
+	__isl_take isl_qpolynomial *qp, isl_int v)
+{
+	if (isl_int_is_one(v))
+		return qp;
+
+	if (qp && isl_int_is_zero(v)) {
+		isl_qpolynomial *zero;
+		zero = isl_qpolynomial_zero_on_domain(isl_space_copy(qp->dim));
+		isl_qpolynomial_free(qp);
+		return zero;
+	}
+	
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp)
+		return NULL;
+
+	qp->upoly = isl_upoly_mul_isl_int(qp->upoly, v);
+	if (!qp->upoly)
+		goto error;
+
+	return qp;
+error:
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_scale(
+	__isl_take isl_qpolynomial *qp, isl_int v)
+{
+	return isl_qpolynomial_mul_isl_int(qp, v);
+}
+
+/* Multiply "qp" by "v".
+ */
+__isl_give isl_qpolynomial *isl_qpolynomial_scale_val(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_val *v)
+{
+	if (!qp || !v)
+		goto error;
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_qpolynomial_get_ctx(qp), isl_error_invalid,
+			"expecting rational factor", goto error);
+
+	if (isl_val_is_one(v)) {
+		isl_val_free(v);
+		return qp;
+	}
+
+	if (isl_val_is_zero(v)) {
+		isl_space *space;
+
+		space = isl_qpolynomial_get_domain_space(qp);
+		isl_qpolynomial_free(qp);
+		isl_val_free(v);
+		return isl_qpolynomial_zero_on_domain(space);
+	}
+
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp)
+		goto error;
+
+	qp->upoly = isl_upoly_scale_val(qp->upoly, v);
+	if (!qp->upoly)
+		qp = isl_qpolynomial_free(qp);
+
+	isl_val_free(v);
+	return qp;
+error:
+	isl_val_free(v);
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+/* Divide "qp" by "v".
+ */
+__isl_give isl_qpolynomial *isl_qpolynomial_scale_down_val(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_val *v)
+{
+	if (!qp || !v)
+		goto error;
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_qpolynomial_get_ctx(qp), isl_error_invalid,
+			"expecting rational factor", goto error);
+	if (isl_val_is_zero(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"cannot scale down by zero", goto error);
+
+	return isl_qpolynomial_scale_val(qp, isl_val_inv(v));
+error:
+	isl_val_free(v);
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_mul(__isl_take isl_qpolynomial *qp1,
+	__isl_take isl_qpolynomial *qp2)
+{
+	qp1 = isl_qpolynomial_cow(qp1);
+
+	if (!qp1 || !qp2)
+		goto error;
+
+	if (qp1->div->n_row < qp2->div->n_row)
+		return isl_qpolynomial_mul(qp2, qp1);
+
+	isl_assert(qp1->dim->ctx, isl_space_is_equal(qp1->dim, qp2->dim), goto error);
+	if (!compatible_divs(qp1->div, qp2->div))
+		return with_merged_divs(isl_qpolynomial_mul, qp1, qp2);
+
+	qp1->upoly = isl_upoly_mul(qp1->upoly, isl_upoly_copy(qp2->upoly));
+	if (!qp1->upoly)
+		goto error;
+
+	isl_qpolynomial_free(qp2);
+
+	return qp1;
+error:
+	isl_qpolynomial_free(qp1);
+	isl_qpolynomial_free(qp2);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_pow(__isl_take isl_qpolynomial *qp,
+	unsigned power)
+{
+	qp = isl_qpolynomial_cow(qp);
+
+	if (!qp)
+		return NULL;
+
+	qp->upoly = isl_upoly_pow(qp->upoly, power);
+	if (!qp->upoly)
+		goto error;
+
+	return qp;
+error:
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_pow(
+	__isl_take isl_pw_qpolynomial *pwqp, unsigned power)
+{
+	int i;
+
+	if (power == 1)
+		return pwqp;
+
+	pwqp = isl_pw_qpolynomial_cow(pwqp);
+	if (!pwqp)
+		return NULL;
+
+	for (i = 0; i < pwqp->n; ++i) {
+		pwqp->p[i].qp = isl_qpolynomial_pow(pwqp->p[i].qp, power);
+		if (!pwqp->p[i].qp)
+			return isl_pw_qpolynomial_free(pwqp);
+	}
+
+	return pwqp;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_zero_on_domain(
+	__isl_take isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+	return isl_qpolynomial_alloc(dim, 0, isl_upoly_zero(dim->ctx));
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_one_on_domain(
+	__isl_take isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+	return isl_qpolynomial_alloc(dim, 0, isl_upoly_one(dim->ctx));
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_infty_on_domain(
+	__isl_take isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+	return isl_qpolynomial_alloc(dim, 0, isl_upoly_infty(dim->ctx));
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_neginfty_on_domain(
+	__isl_take isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+	return isl_qpolynomial_alloc(dim, 0, isl_upoly_neginfty(dim->ctx));
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_nan_on_domain(
+	__isl_take isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+	return isl_qpolynomial_alloc(dim, 0, isl_upoly_nan(dim->ctx));
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_cst_on_domain(
+	__isl_take isl_space *dim,
+	isl_int v)
+{
+	struct isl_qpolynomial *qp;
+	struct isl_upoly_cst *cst;
+
+	if (!dim)
+		return NULL;
+
+	qp = isl_qpolynomial_alloc(dim, 0, isl_upoly_zero(dim->ctx));
+	if (!qp)
+		return NULL;
+
+	cst = isl_upoly_as_cst(qp->upoly);
+	isl_int_set(cst->n, v);
+
+	return qp;
+}
+
+int isl_qpolynomial_is_cst(__isl_keep isl_qpolynomial *qp,
+	isl_int *n, isl_int *d)
+{
+	struct isl_upoly_cst *cst;
+
+	if (!qp)
+		return -1;
+
+	if (!isl_upoly_is_cst(qp->upoly))
+		return 0;
+
+	cst = isl_upoly_as_cst(qp->upoly);
+	if (!cst)
+		return -1;
+
+	if (n)
+		isl_int_set(*n, cst->n);
+	if (d)
+		isl_int_set(*d, cst->d);
+
+	return 1;
+}
+
+/* Return the constant term of "up".
+ */
+static __isl_give isl_val *isl_upoly_get_constant_val(
+	__isl_keep struct isl_upoly *up)
+{
+	struct isl_upoly_cst *cst;
+
+	if (!up)
+		return NULL;
+
+	while (!isl_upoly_is_cst(up)) {
+		struct isl_upoly_rec *rec;
+
+		rec = isl_upoly_as_rec(up);
+		if (!rec)
+			return NULL;
+		up = rec->p[0];
+	}
+
+	cst = isl_upoly_as_cst(up);
+	if (!cst)
+		return NULL;
+	return isl_val_rat_from_isl_int(cst->up.ctx, cst->n, cst->d);
+}
+
+/* Return the constant term of "qp".
+ */
+__isl_give isl_val *isl_qpolynomial_get_constant_val(
+	__isl_keep isl_qpolynomial *qp)
+{
+	if (!qp)
+		return NULL;
+
+	return isl_upoly_get_constant_val(qp->upoly);
+}
+
+int isl_upoly_is_affine(__isl_keep struct isl_upoly *up)
+{
+	int is_cst;
+	struct isl_upoly_rec *rec;
+
+	if (!up)
+		return -1;
+
+	if (up->var < 0)
+		return 1;
+
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		return -1;
+
+	if (rec->n > 2)
+		return 0;
+
+	isl_assert(up->ctx, rec->n > 1, return -1);
+
+	is_cst = isl_upoly_is_cst(rec->p[1]);
+	if (is_cst < 0)
+		return -1;
+	if (!is_cst)
+		return 0;
+
+	return isl_upoly_is_affine(rec->p[0]);
+}
+
+int isl_qpolynomial_is_affine(__isl_keep isl_qpolynomial *qp)
+{
+	if (!qp)
+		return -1;
+
+	if (qp->div->n_row > 0)
+		return 0;
+
+	return isl_upoly_is_affine(qp->upoly);
+}
+
+static void update_coeff(__isl_keep isl_vec *aff,
+	__isl_keep struct isl_upoly_cst *cst, int pos)
+{
+	isl_int gcd;
+	isl_int f;
+
+	if (isl_int_is_zero(cst->n))
+		return;
+
+	isl_int_init(gcd);
+	isl_int_init(f);
+	isl_int_gcd(gcd, cst->d, aff->el[0]);
+	isl_int_divexact(f, cst->d, gcd);
+	isl_int_divexact(gcd, aff->el[0], gcd);
+	isl_seq_scale(aff->el, aff->el, f, aff->size);
+	isl_int_mul(aff->el[1 + pos], gcd, cst->n);
+	isl_int_clear(gcd);
+	isl_int_clear(f);
+}
+
+int isl_upoly_update_affine(__isl_keep struct isl_upoly *up,
+	__isl_keep isl_vec *aff)
+{
+	struct isl_upoly_cst *cst;
+	struct isl_upoly_rec *rec;
+
+	if (!up || !aff)
+		return -1;
+
+	if (up->var < 0) {
+		struct isl_upoly_cst *cst;
+
+		cst = isl_upoly_as_cst(up);
+		if (!cst)
+			return -1;
+		update_coeff(aff, cst, 0);
+		return 0;
+	}
+
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		return -1;
+	isl_assert(up->ctx, rec->n == 2, return -1);
+
+	cst = isl_upoly_as_cst(rec->p[1]);
+	if (!cst)
+		return -1;
+	update_coeff(aff, cst, 1 + up->var);
+
+	return isl_upoly_update_affine(rec->p[0], aff);
+}
+
+__isl_give isl_vec *isl_qpolynomial_extract_affine(
+	__isl_keep isl_qpolynomial *qp)
+{
+	isl_vec *aff;
+	unsigned d;
+
+	if (!qp)
+		return NULL;
+
+	d = isl_space_dim(qp->dim, isl_dim_all);
+	aff = isl_vec_alloc(qp->div->ctx, 2 + d + qp->div->n_row);
+	if (!aff)
+		return NULL;
+
+	isl_seq_clr(aff->el + 1, 1 + d + qp->div->n_row);
+	isl_int_set_si(aff->el[0], 1);
+
+	if (isl_upoly_update_affine(qp->upoly, aff) < 0)
+		goto error;
+
+	return aff;
+error:
+	isl_vec_free(aff);
+	return NULL;
+}
+
+int isl_qpolynomial_plain_is_equal(__isl_keep isl_qpolynomial *qp1,
+	__isl_keep isl_qpolynomial *qp2)
+{
+	int equal;
+
+	if (!qp1 || !qp2)
+		return -1;
+
+	equal = isl_space_is_equal(qp1->dim, qp2->dim);
+	if (equal < 0 || !equal)
+		return equal;
+
+	equal = isl_mat_is_equal(qp1->div, qp2->div);
+	if (equal < 0 || !equal)
+		return equal;
+
+	return isl_upoly_is_equal(qp1->upoly, qp2->upoly);
+}
+
+static void upoly_update_den(__isl_keep struct isl_upoly *up, isl_int *d)
+{
+	int i;
+	struct isl_upoly_rec *rec;
+
+	if (isl_upoly_is_cst(up)) {
+		struct isl_upoly_cst *cst;
+		cst = isl_upoly_as_cst(up);
+		if (!cst)
+			return;
+		isl_int_lcm(*d, *d, cst->d);
+		return;
+	}
+
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		return;
+
+	for (i = 0; i < rec->n; ++i)
+		upoly_update_den(rec->p[i], d);
+}
+
+void isl_qpolynomial_get_den(__isl_keep isl_qpolynomial *qp, isl_int *d)
+{
+	isl_int_set_si(*d, 1);
+	if (!qp)
+		return;
+	upoly_update_den(qp->upoly, d);
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_var_pow_on_domain(
+	__isl_take isl_space *dim, int pos, int power)
+{
+	struct isl_ctx *ctx;
+
+	if (!dim)
+		return NULL;
+
+	ctx = dim->ctx;
+
+	return isl_qpolynomial_alloc(dim, 0, isl_upoly_var_pow(ctx, pos, power));
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_var_on_domain(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned pos)
+{
+	if (!dim)
+		return NULL;
+
+	isl_assert(dim->ctx, isl_space_dim(dim, isl_dim_in) == 0, goto error);
+	isl_assert(dim->ctx, pos < isl_space_dim(dim, type), goto error);
+
+	if (type == isl_dim_set)
+		pos += isl_space_dim(dim, isl_dim_param);
+
+	return isl_qpolynomial_var_pow_on_domain(dim, pos, 1);
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+__isl_give struct isl_upoly *isl_upoly_subs(__isl_take struct isl_upoly *up,
+	unsigned first, unsigned n, __isl_keep struct isl_upoly **subs)
+{
+	int i;
+	struct isl_upoly_rec *rec;
+	struct isl_upoly *base, *res;
+
+	if (!up)
+		return NULL;
+
+	if (isl_upoly_is_cst(up))
+		return up;
+
+	if (up->var < first)
+		return up;
+
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		goto error;
+
+	isl_assert(up->ctx, rec->n >= 1, goto error);
+
+	if (up->var >= first + n)
+		base = isl_upoly_var_pow(up->ctx, up->var, 1);
+	else
+		base = isl_upoly_copy(subs[up->var - first]);
+
+	res = isl_upoly_subs(isl_upoly_copy(rec->p[rec->n - 1]), first, n, subs);
+	for (i = rec->n - 2; i >= 0; --i) {
+		struct isl_upoly *t;
+		t = isl_upoly_subs(isl_upoly_copy(rec->p[i]), first, n, subs);
+		res = isl_upoly_mul(res, isl_upoly_copy(base));
+		res = isl_upoly_sum(res, t);
+	}
+
+	isl_upoly_free(base);
+	isl_upoly_free(up);
+				
+	return res;
+error:
+	isl_upoly_free(up);
+	return NULL;
+}	
+
+__isl_give struct isl_upoly *isl_upoly_from_affine(isl_ctx *ctx, isl_int *f,
+	isl_int denom, unsigned len)
+{
+	int i;
+	struct isl_upoly *up;
+
+	isl_assert(ctx, len >= 1, return NULL);
+
+	up = isl_upoly_rat_cst(ctx, f[0], denom);
+	for (i = 0; i < len - 1; ++i) {
+		struct isl_upoly *t;
+		struct isl_upoly *c;
+
+		if (isl_int_is_zero(f[1 + i]))
+			continue;
+
+		c = isl_upoly_rat_cst(ctx, f[1 + i], denom);
+		t = isl_upoly_var_pow(ctx, i, 1);
+		t = isl_upoly_mul(c, t);
+		up = isl_upoly_sum(up, t);
+	}
+
+	return up;
+}
+
+/* Remove common factor of non-constant terms and denominator.
+ */
+static void normalize_div(__isl_keep isl_qpolynomial *qp, int div)
+{
+	isl_ctx *ctx = qp->div->ctx;
+	unsigned total = qp->div->n_col - 2;
+
+	isl_seq_gcd(qp->div->row[div] + 2, total, &ctx->normalize_gcd);
+	isl_int_gcd(ctx->normalize_gcd,
+		    ctx->normalize_gcd, qp->div->row[div][0]);
+	if (isl_int_is_one(ctx->normalize_gcd))
+		return;
+
+	isl_seq_scale_down(qp->div->row[div] + 2, qp->div->row[div] + 2,
+			    ctx->normalize_gcd, total);
+	isl_int_divexact(qp->div->row[div][0], qp->div->row[div][0],
+			    ctx->normalize_gcd);
+	isl_int_fdiv_q(qp->div->row[div][1], qp->div->row[div][1],
+			    ctx->normalize_gcd);
+}
+
+/* Replace the integer division identified by "div" by the polynomial "s".
+ * The integer division is assumed not to appear in the definition
+ * of any other integer divisions.
+ */
+static __isl_give isl_qpolynomial *substitute_div(
+	__isl_take isl_qpolynomial *qp,
+	int div, __isl_take struct isl_upoly *s)
+{
+	int i;
+	int total;
+	int *reordering;
+
+	if (!qp || !s)
+		goto error;
+
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp)
+		goto error;
+
+	total = isl_space_dim(qp->dim, isl_dim_all);
+	qp->upoly = isl_upoly_subs(qp->upoly, total + div, 1, &s);
+	if (!qp->upoly)
+		goto error;
+
+	reordering = isl_alloc_array(qp->dim->ctx, int, total + qp->div->n_row);
+	if (!reordering)
+		goto error;
+	for (i = 0; i < total + div; ++i)
+		reordering[i] = i;
+	for (i = total + div + 1; i < total + qp->div->n_row; ++i)
+		reordering[i] = i - 1;
+	qp->div = isl_mat_drop_rows(qp->div, div, 1);
+	qp->div = isl_mat_drop_cols(qp->div, 2 + total + div, 1);
+	qp->upoly = reorder(qp->upoly, reordering);
+	free(reordering);
+
+	if (!qp->upoly || !qp->div)
+		goto error;
+
+	isl_upoly_free(s);
+	return qp;
+error:
+	isl_qpolynomial_free(qp);
+	isl_upoly_free(s);
+	return NULL;
+}
+
+/* Replace all integer divisions [e/d] that turn out to not actually be integer
+ * divisions because d is equal to 1 by their definition, i.e., e.
+ */
+static __isl_give isl_qpolynomial *substitute_non_divs(
+	__isl_take isl_qpolynomial *qp)
+{
+	int i, j;
+	int total;
+	struct isl_upoly *s;
+
+	if (!qp)
+		return NULL;
+
+	total = isl_space_dim(qp->dim, isl_dim_all);
+	for (i = 0; qp && i < qp->div->n_row; ++i) {
+		if (!isl_int_is_one(qp->div->row[i][0]))
+			continue;
+		for (j = i + 1; j < qp->div->n_row; ++j) {
+			if (isl_int_is_zero(qp->div->row[j][2 + total + i]))
+				continue;
+			isl_seq_combine(qp->div->row[j] + 1,
+				qp->div->ctx->one, qp->div->row[j] + 1,
+				qp->div->row[j][2 + total + i],
+				qp->div->row[i] + 1, 1 + total + i);
+			isl_int_set_si(qp->div->row[j][2 + total + i], 0);
+			normalize_div(qp, j);
+		}
+		s = isl_upoly_from_affine(qp->dim->ctx, qp->div->row[i] + 1,
+					qp->div->row[i][0], qp->div->n_col - 1);
+		qp = substitute_div(qp, i, s);
+		--i;
+	}
+
+	return qp;
+}
+
+/* Reduce the coefficients of div "div" to lie in the interval [0, d-1],
+ * with d the denominator.  When replacing the coefficient e of x by
+ * d * frac(e/d) = e - d * floor(e/d), we are subtracting d * floor(e/d) * x
+ * inside the division, so we need to add floor(e/d) * x outside.
+ * That is, we replace q by q' + floor(e/d) * x and we therefore need
+ * to adjust the coefficient of x in each later div that depends on the
+ * current div "div" and also in the affine expression "aff"
+ * (if it too depends on "div").
+ */
+static void reduce_div(__isl_keep isl_qpolynomial *qp, int div,
+	__isl_keep isl_vec *aff)
+{
+	int i, j;
+	isl_int v;
+	unsigned total = qp->div->n_col - qp->div->n_row - 2;
+
+	isl_int_init(v);
+	for (i = 0; i < 1 + total + div; ++i) {
+		if (isl_int_is_nonneg(qp->div->row[div][1 + i]) &&
+		    isl_int_lt(qp->div->row[div][1 + i], qp->div->row[div][0]))
+			continue;
+		isl_int_fdiv_q(v, qp->div->row[div][1 + i], qp->div->row[div][0]);
+		isl_int_fdiv_r(qp->div->row[div][1 + i],
+				qp->div->row[div][1 + i], qp->div->row[div][0]);
+		if (!isl_int_is_zero(aff->el[1 + total + div]))
+			isl_int_addmul(aff->el[i], v, aff->el[1 + total + div]);
+		for (j = div + 1; j < qp->div->n_row; ++j) {
+			if (isl_int_is_zero(qp->div->row[j][2 + total + div]))
+				continue;
+			isl_int_addmul(qp->div->row[j][1 + i],
+					v, qp->div->row[j][2 + total + div]);
+		}
+	}
+	isl_int_clear(v);
+}
+
+/* Check if the last non-zero coefficient is bigger that half of the
+ * denominator.  If so, we will invert the div to further reduce the number
+ * of distinct divs that may appear.
+ * If the last non-zero coefficient is exactly half the denominator,
+ * then we continue looking for earlier coefficients that are bigger
+ * than half the denominator.
+ */
+static int needs_invert(__isl_keep isl_mat *div, int row)
+{
+	int i;
+	int cmp;
+
+	for (i = div->n_col - 1; i >= 1; --i) {
+		if (isl_int_is_zero(div->row[row][i]))
+			continue;
+		isl_int_mul_ui(div->row[row][i], div->row[row][i], 2);
+		cmp = isl_int_cmp(div->row[row][i], div->row[row][0]);
+		isl_int_divexact_ui(div->row[row][i], div->row[row][i], 2);
+		if (cmp)
+			return cmp > 0;
+		if (i == 1)
+			return 1;
+	}
+
+	return 0;
+}
+
+/* Replace div "div" q = [e/d] by -[(-e+(d-1))/d].
+ * We only invert the coefficients of e (and the coefficient of q in
+ * later divs and in "aff").  After calling this function, the
+ * coefficients of e should be reduced again.
+ */
+static void invert_div(__isl_keep isl_qpolynomial *qp, int div,
+	__isl_keep isl_vec *aff)
+{
+	unsigned total = qp->div->n_col - qp->div->n_row - 2;
+
+	isl_seq_neg(qp->div->row[div] + 1,
+		    qp->div->row[div] + 1, qp->div->n_col - 1);
+	isl_int_sub_ui(qp->div->row[div][1], qp->div->row[div][1], 1);
+	isl_int_add(qp->div->row[div][1],
+		    qp->div->row[div][1], qp->div->row[div][0]);
+	if (!isl_int_is_zero(aff->el[1 + total + div]))
+		isl_int_neg(aff->el[1 + total + div], aff->el[1 + total + div]);
+	isl_mat_col_mul(qp->div, 2 + total + div,
+			qp->div->ctx->negone, 2 + total + div);
+}
+
+/* Assuming "qp" is a monomial, reduce all its divs to have coefficients
+ * in the interval [0, d-1], with d the denominator and such that the
+ * last non-zero coefficient that is not equal to d/2 is smaller than d/2.
+ *
+ * After the reduction, some divs may have become redundant or identical,
+ * so we call substitute_non_divs and sort_divs.  If these functions
+ * eliminate divs or merge two or more divs into one, the coefficients
+ * of the enclosing divs may have to be reduced again, so we call
+ * ourselves recursively if the number of divs decreases.
+ */
+static __isl_give isl_qpolynomial *reduce_divs(__isl_take isl_qpolynomial *qp)
+{
+	int i;
+	isl_vec *aff = NULL;
+	struct isl_upoly *s;
+	unsigned n_div;
+
+	if (!qp)
+		return NULL;
+
+	aff = isl_vec_alloc(qp->div->ctx, qp->div->n_col - 1);
+	aff = isl_vec_clr(aff);
+	if (!aff)
+		goto error;
+
+	isl_int_set_si(aff->el[1 + qp->upoly->var], 1);
+
+	for (i = 0; i < qp->div->n_row; ++i) {
+		normalize_div(qp, i);
+		reduce_div(qp, i, aff);
+		if (needs_invert(qp->div, i)) {
+			invert_div(qp, i, aff);
+			reduce_div(qp, i, aff);
+		}
+	}
+
+	s = isl_upoly_from_affine(qp->div->ctx, aff->el,
+				  qp->div->ctx->one, aff->size);
+	qp->upoly = isl_upoly_subs(qp->upoly, qp->upoly->var, 1, &s);
+	isl_upoly_free(s);
+	if (!qp->upoly)
+		goto error;
+
+	isl_vec_free(aff);
+
+	n_div = qp->div->n_row;
+	qp = substitute_non_divs(qp);
+	qp = sort_divs(qp);
+	if (qp && qp->div->n_row < n_div)
+		return reduce_divs(qp);
+
+	return qp;
+error:
+	isl_qpolynomial_free(qp);
+	isl_vec_free(aff);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_rat_cst_on_domain(
+	__isl_take isl_space *dim, const isl_int n, const isl_int d)
+{
+	struct isl_qpolynomial *qp;
+	struct isl_upoly_cst *cst;
+
+	if (!dim)
+		return NULL;
+
+	qp = isl_qpolynomial_alloc(dim, 0, isl_upoly_zero(dim->ctx));
+	if (!qp)
+		return NULL;
+
+	cst = isl_upoly_as_cst(qp->upoly);
+	isl_int_set(cst->n, n);
+	isl_int_set(cst->d, d);
+
+	return qp;
+}
+
+/* Return an isl_qpolynomial that is equal to "val" on domain space "domain".
+ */
+__isl_give isl_qpolynomial *isl_qpolynomial_val_on_domain(
+	__isl_take isl_space *domain, __isl_take isl_val *val)
+{
+	isl_qpolynomial *qp;
+	struct isl_upoly_cst *cst;
+
+	if (!domain || !val)
+		goto error;
+
+	qp = isl_qpolynomial_alloc(isl_space_copy(domain), 0,
+					isl_upoly_zero(domain->ctx));
+	if (!qp)
+		goto error;
+
+	cst = isl_upoly_as_cst(qp->upoly);
+	isl_int_set(cst->n, val->n);
+	isl_int_set(cst->d, val->d);
+
+	isl_space_free(domain);
+	isl_val_free(val);
+	return qp;
+error:
+	isl_space_free(domain);
+	isl_val_free(val);
+	return NULL;
+}
+
+static int up_set_active(__isl_keep struct isl_upoly *up, int *active, int d)
+{
+	struct isl_upoly_rec *rec;
+	int i;
+
+	if (!up)
+		return -1;
+
+	if (isl_upoly_is_cst(up))
+		return 0;
+
+	if (up->var < d)
+		active[up->var] = 1;
+
+	rec = isl_upoly_as_rec(up);
+	for (i = 0; i < rec->n; ++i)
+		if (up_set_active(rec->p[i], active, d) < 0)
+			return -1;
+
+	return 0;
+}
+
+static int set_active(__isl_keep isl_qpolynomial *qp, int *active)
+{
+	int i, j;
+	int d = isl_space_dim(qp->dim, isl_dim_all);
+
+	if (!qp || !active)
+		return -1;
+
+	for (i = 0; i < d; ++i)
+		for (j = 0; j < qp->div->n_row; ++j) {
+			if (isl_int_is_zero(qp->div->row[j][2 + i]))
+				continue;
+			active[i] = 1;
+			break;
+		}
+
+	return up_set_active(qp->upoly, active, d);
+}
+
+int isl_qpolynomial_involves_dims(__isl_keep isl_qpolynomial *qp,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+	int *active = NULL;
+	int involves = 0;
+
+	if (!qp)
+		return -1;
+	if (n == 0)
+		return 0;
+
+	isl_assert(qp->dim->ctx,
+		    first + n <= isl_qpolynomial_dim(qp, type), return -1);
+	isl_assert(qp->dim->ctx, type == isl_dim_param ||
+				 type == isl_dim_in, return -1);
+
+	active = isl_calloc_array(qp->dim->ctx, int,
+					isl_space_dim(qp->dim, isl_dim_all));
+	if (set_active(qp, active) < 0)
+		goto error;
+
+	if (type == isl_dim_in)
+		first += isl_space_dim(qp->dim, isl_dim_param);
+	for (i = 0; i < n; ++i)
+		if (active[first + i]) {
+			involves = 1;
+			break;
+		}
+
+	free(active);
+
+	return involves;
+error:
+	free(active);
+	return -1;
+}
+
+/* Remove divs that do not appear in the quasi-polynomial, nor in any
+ * of the divs that do appear in the quasi-polynomial.
+ */
+static __isl_give isl_qpolynomial *remove_redundant_divs(
+	__isl_take isl_qpolynomial *qp)
+{
+	int i, j;
+	int d;
+	int len;
+	int skip;
+	int *active = NULL;
+	int *reordering = NULL;
+	int redundant = 0;
+	int n_div;
+	isl_ctx *ctx;
+
+	if (!qp)
+		return NULL;
+	if (qp->div->n_row == 0)
+		return qp;
+
+	d = isl_space_dim(qp->dim, isl_dim_all);
+	len = qp->div->n_col - 2;
+	ctx = isl_qpolynomial_get_ctx(qp);
+	active = isl_calloc_array(ctx, int, len);
+	if (!active)
+		goto error;
+
+	if (up_set_active(qp->upoly, active, len) < 0)
+		goto error;
+
+	for (i = qp->div->n_row - 1; i >= 0; --i) {
+		if (!active[d + i]) {
+			redundant = 1;
+			continue;
+		}
+		for (j = 0; j < i; ++j) {
+			if (isl_int_is_zero(qp->div->row[i][2 + d + j]))
+				continue;
+			active[d + j] = 1;
+			break;
+		}
+	}
+
+	if (!redundant) {
+		free(active);
+		return qp;
+	}
+
+	reordering = isl_alloc_array(qp->div->ctx, int, len);
+	if (!reordering)
+		goto error;
+
+	for (i = 0; i < d; ++i)
+		reordering[i] = i;
+
+	skip = 0;
+	n_div = qp->div->n_row;
+	for (i = 0; i < n_div; ++i) {
+		if (!active[d + i]) {
+			qp->div = isl_mat_drop_rows(qp->div, i - skip, 1);
+			qp->div = isl_mat_drop_cols(qp->div,
+						    2 + d + i - skip, 1);
+			skip++;
+		}
+		reordering[d + i] = d + i - skip;
+	}
+
+	qp->upoly = reorder(qp->upoly, reordering);
+
+	if (!qp->upoly || !qp->div)
+		goto error;
+
+	free(active);
+	free(reordering);
+
+	return qp;
+error:
+	free(active);
+	free(reordering);
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+__isl_give struct isl_upoly *isl_upoly_drop(__isl_take struct isl_upoly *up,
+	unsigned first, unsigned n)
+{
+	int i;
+	struct isl_upoly_rec *rec;
+
+	if (!up)
+		return NULL;
+	if (n == 0 || up->var < 0 || up->var < first)
+		return up;
+	if (up->var < first + n) {
+		up = replace_by_constant_term(up);
+		return isl_upoly_drop(up, first, n);
+	}
+	up = isl_upoly_cow(up);
+	if (!up)
+		return NULL;
+	up->var -= n;
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		goto error;
+
+	for (i = 0; i < rec->n; ++i) {
+		rec->p[i] = isl_upoly_drop(rec->p[i], first, n);
+		if (!rec->p[i])
+			goto error;
+	}
+
+	return up;
+error:
+	isl_upoly_free(up);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_set_dim_name(
+	__isl_take isl_qpolynomial *qp,
+	enum isl_dim_type type, unsigned pos, const char *s)
+{
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp)
+		return NULL;
+	qp->dim = isl_space_set_dim_name(qp->dim, type, pos, s);
+	if (!qp->dim)
+		goto error;
+	return qp;
+error:
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_drop_dims(
+	__isl_take isl_qpolynomial *qp,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	if (!qp)
+		return NULL;
+	if (type == isl_dim_out)
+		isl_die(qp->dim->ctx, isl_error_invalid,
+			"cannot drop output/set dimension",
+			goto error);
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+	if (n == 0 && !isl_space_is_named_or_nested(qp->dim, type))
+		return qp;
+
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp)
+		return NULL;
+
+	isl_assert(qp->dim->ctx, first + n <= isl_space_dim(qp->dim, type),
+			goto error);
+	isl_assert(qp->dim->ctx, type == isl_dim_param ||
+				 type == isl_dim_set, goto error);
+
+	qp->dim = isl_space_drop_dims(qp->dim, type, first, n);
+	if (!qp->dim)
+		goto error;
+
+	if (type == isl_dim_set)
+		first += isl_space_dim(qp->dim, isl_dim_param);
+
+	qp->div = isl_mat_drop_cols(qp->div, 2 + first, n);
+	if (!qp->div)
+		goto error;
+
+	qp->upoly = isl_upoly_drop(qp->upoly, first, n);
+	if (!qp->upoly)
+		goto error;
+
+	return qp;
+error:
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+/* Project the domain of the quasi-polynomial onto its parameter space.
+ * The quasi-polynomial may not involve any of the domain dimensions.
+ */
+__isl_give isl_qpolynomial *isl_qpolynomial_project_domain_on_params(
+	__isl_take isl_qpolynomial *qp)
+{
+	isl_space *space;
+	unsigned n;
+	int involves;
+
+	n = isl_qpolynomial_dim(qp, isl_dim_in);
+	involves = isl_qpolynomial_involves_dims(qp, isl_dim_in, 0, n);
+	if (involves < 0)
+		return isl_qpolynomial_free(qp);
+	if (involves)
+		isl_die(isl_qpolynomial_get_ctx(qp), isl_error_invalid,
+			"polynomial involves some of the domain dimensions",
+			return isl_qpolynomial_free(qp));
+	qp = isl_qpolynomial_drop_dims(qp, isl_dim_in, 0, n);
+	space = isl_qpolynomial_get_domain_space(qp);
+	space = isl_space_params(space);
+	qp = isl_qpolynomial_reset_domain_space(qp, space);
+	return qp;
+}
+
+static __isl_give isl_qpolynomial *isl_qpolynomial_substitute_equalities_lifted(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_basic_set *eq)
+{
+	int i, j, k;
+	isl_int denom;
+	unsigned total;
+	unsigned n_div;
+	struct isl_upoly *up;
+
+	if (!eq)
+		goto error;
+	if (eq->n_eq == 0) {
+		isl_basic_set_free(eq);
+		return qp;
+	}
+
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp)
+		goto error;
+	qp->div = isl_mat_cow(qp->div);
+	if (!qp->div)
+		goto error;
+
+	total = 1 + isl_space_dim(eq->dim, isl_dim_all);
+	n_div = eq->n_div;
+	isl_int_init(denom);
+	for (i = 0; i < eq->n_eq; ++i) {
+		j = isl_seq_last_non_zero(eq->eq[i], total + n_div);
+		if (j < 0 || j == 0 || j >= total)
+			continue;
+
+		for (k = 0; k < qp->div->n_row; ++k) {
+			if (isl_int_is_zero(qp->div->row[k][1 + j]))
+				continue;
+			isl_seq_elim(qp->div->row[k] + 1, eq->eq[i], j, total,
+					&qp->div->row[k][0]);
+			normalize_div(qp, k);
+		}
+
+		if (isl_int_is_pos(eq->eq[i][j]))
+			isl_seq_neg(eq->eq[i], eq->eq[i], total);
+		isl_int_abs(denom, eq->eq[i][j]);
+		isl_int_set_si(eq->eq[i][j], 0);
+
+		up = isl_upoly_from_affine(qp->dim->ctx,
+						   eq->eq[i], denom, total);
+		qp->upoly = isl_upoly_subs(qp->upoly, j - 1, 1, &up);
+		isl_upoly_free(up);
+	}
+	isl_int_clear(denom);
+
+	if (!qp->upoly)
+		goto error;
+
+	isl_basic_set_free(eq);
+
+	qp = substitute_non_divs(qp);
+	qp = sort_divs(qp);
+
+	return qp;
+error:
+	isl_basic_set_free(eq);
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+/* Exploit the equalities in "eq" to simplify the quasi-polynomial.
+ */
+__isl_give isl_qpolynomial *isl_qpolynomial_substitute_equalities(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_basic_set *eq)
+{
+	if (!qp || !eq)
+		goto error;
+	if (qp->div->n_row > 0)
+		eq = isl_basic_set_add_dims(eq, isl_dim_set, qp->div->n_row);
+	return isl_qpolynomial_substitute_equalities_lifted(qp, eq);
+error:
+	isl_basic_set_free(eq);
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+static __isl_give isl_basic_set *add_div_constraints(
+	__isl_take isl_basic_set *bset, __isl_take isl_mat *div)
+{
+	int i;
+	unsigned total;
+
+	if (!bset || !div)
+		goto error;
+
+	bset = isl_basic_set_extend_constraints(bset, 0, 2 * div->n_row);
+	if (!bset)
+		goto error;
+	total = isl_basic_set_total_dim(bset);
+	for (i = 0; i < div->n_row; ++i)
+		if (isl_basic_set_add_div_constraints_var(bset,
+				    total - div->n_row + i, div->row[i]) < 0)
+			goto error;
+
+	isl_mat_free(div);
+	return bset;
+error:
+	isl_mat_free(div);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Look for equalities among the variables shared by context and qp
+ * and the integer divisions of qp, if any.
+ * The equalities are then used to eliminate variables and/or integer
+ * divisions from qp.
+ */
+__isl_give isl_qpolynomial *isl_qpolynomial_gist(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_set *context)
+{
+	isl_basic_set *aff;
+
+	if (!qp)
+		goto error;
+	if (qp->div->n_row > 0) {
+		isl_basic_set *bset;
+		context = isl_set_add_dims(context, isl_dim_set,
+					    qp->div->n_row);
+		bset = isl_basic_set_universe(isl_set_get_space(context));
+		bset = add_div_constraints(bset, isl_mat_copy(qp->div));
+		context = isl_set_intersect(context,
+					    isl_set_from_basic_set(bset));
+	}
+
+	aff = isl_set_affine_hull(context);
+	return isl_qpolynomial_substitute_equalities_lifted(qp, aff);
+error:
+	isl_qpolynomial_free(qp);
+	isl_set_free(context);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_gist_params(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_set *context)
+{
+	isl_space *space = isl_qpolynomial_get_domain_space(qp);
+	isl_set *dom_context = isl_set_universe(space);
+	dom_context = isl_set_intersect_params(dom_context, context);
+	return isl_qpolynomial_gist(qp, dom_context);
+}
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_from_qpolynomial(
+	__isl_take isl_qpolynomial *qp)
+{
+	isl_set *dom;
+
+	if (!qp)
+		return NULL;
+	if (isl_qpolynomial_is_zero(qp)) {
+		isl_space *dim = isl_qpolynomial_get_space(qp);
+		isl_qpolynomial_free(qp);
+		return isl_pw_qpolynomial_zero(dim);
+	}
+
+	dom = isl_set_universe(isl_qpolynomial_get_domain_space(qp));
+	return isl_pw_qpolynomial_alloc(dom, qp);
+}
+
+#undef PW
+#define PW isl_pw_qpolynomial
+#undef EL
+#define EL isl_qpolynomial
+#undef EL_IS_ZERO
+#define EL_IS_ZERO is_zero
+#undef ZERO
+#define ZERO zero
+#undef IS_ZERO
+#define IS_ZERO is_zero
+#undef FIELD
+#define FIELD qp
+#undef DEFAULT_IS_ZERO
+#define DEFAULT_IS_ZERO 1
+
+#define NO_PULLBACK
+
+#include <isl_pw_templ.c>
+
+#undef UNION
+#define UNION isl_union_pw_qpolynomial
+#undef PART
+#define PART isl_pw_qpolynomial
+#undef PARTS
+#define PARTS pw_qpolynomial
+#define ALIGN_DOMAIN
+
+#include <isl_union_templ.c>
+
+int isl_pw_qpolynomial_is_one(__isl_keep isl_pw_qpolynomial *pwqp)
+{
+	if (!pwqp)
+		return -1;
+
+	if (pwqp->n != -1)
+		return 0;
+
+	if (!isl_set_plain_is_universe(pwqp->p[0].set))
+		return 0;
+
+	return isl_qpolynomial_is_one(pwqp->p[0].qp);
+}
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add(
+	__isl_take isl_pw_qpolynomial *pwqp1,
+	__isl_take isl_pw_qpolynomial *pwqp2)
+{
+	return isl_pw_qpolynomial_union_add_(pwqp1, pwqp2);
+}
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_mul(
+	__isl_take isl_pw_qpolynomial *pwqp1,
+	__isl_take isl_pw_qpolynomial *pwqp2)
+{
+	int i, j, n;
+	struct isl_pw_qpolynomial *res;
+
+	if (!pwqp1 || !pwqp2)
+		goto error;
+
+	isl_assert(pwqp1->dim->ctx, isl_space_is_equal(pwqp1->dim, pwqp2->dim),
+			goto error);
+
+	if (isl_pw_qpolynomial_is_zero(pwqp1)) {
+		isl_pw_qpolynomial_free(pwqp2);
+		return pwqp1;
+	}
+
+	if (isl_pw_qpolynomial_is_zero(pwqp2)) {
+		isl_pw_qpolynomial_free(pwqp1);
+		return pwqp2;
+	}
+
+	if (isl_pw_qpolynomial_is_one(pwqp1)) {
+		isl_pw_qpolynomial_free(pwqp1);
+		return pwqp2;
+	}
+
+	if (isl_pw_qpolynomial_is_one(pwqp2)) {
+		isl_pw_qpolynomial_free(pwqp2);
+		return pwqp1;
+	}
+
+	n = pwqp1->n * pwqp2->n;
+	res = isl_pw_qpolynomial_alloc_size(isl_space_copy(pwqp1->dim), n);
+
+	for (i = 0; i < pwqp1->n; ++i) {
+		for (j = 0; j < pwqp2->n; ++j) {
+			struct isl_set *common;
+			struct isl_qpolynomial *prod;
+			common = isl_set_intersect(isl_set_copy(pwqp1->p[i].set),
+						isl_set_copy(pwqp2->p[j].set));
+			if (isl_set_plain_is_empty(common)) {
+				isl_set_free(common);
+				continue;
+			}
+
+			prod = isl_qpolynomial_mul(
+				isl_qpolynomial_copy(pwqp1->p[i].qp),
+				isl_qpolynomial_copy(pwqp2->p[j].qp));
+
+			res = isl_pw_qpolynomial_add_piece(res, common, prod);
+		}
+	}
+
+	isl_pw_qpolynomial_free(pwqp1);
+	isl_pw_qpolynomial_free(pwqp2);
+
+	return res;
+error:
+	isl_pw_qpolynomial_free(pwqp1);
+	isl_pw_qpolynomial_free(pwqp2);
+	return NULL;
+}
+
+__isl_give isl_val *isl_upoly_eval(__isl_take struct isl_upoly *up,
+	__isl_take isl_vec *vec)
+{
+	int i;
+	struct isl_upoly_rec *rec;
+	isl_val *res;
+	isl_val *base;
+
+	if (isl_upoly_is_cst(up)) {
+		isl_vec_free(vec);
+		res = isl_upoly_get_constant_val(up);
+		isl_upoly_free(up);
+		return res;
+	}
+
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		goto error;
+
+	isl_assert(up->ctx, rec->n >= 1, goto error);
+
+	base = isl_val_rat_from_isl_int(up->ctx,
+					vec->el[1 + up->var], vec->el[0]);
+
+	res = isl_upoly_eval(isl_upoly_copy(rec->p[rec->n - 1]),
+				isl_vec_copy(vec));
+
+	for (i = rec->n - 2; i >= 0; --i) {
+		res = isl_val_mul(res, isl_val_copy(base));
+		res = isl_val_add(res,
+			    isl_upoly_eval(isl_upoly_copy(rec->p[i]),
+							    isl_vec_copy(vec)));
+	}
+
+	isl_val_free(base);
+	isl_upoly_free(up);
+	isl_vec_free(vec);
+	return res;
+error:
+	isl_upoly_free(up);
+	isl_vec_free(vec);
+	return NULL;
+}
+
+__isl_give isl_val *isl_qpolynomial_eval(__isl_take isl_qpolynomial *qp,
+	__isl_take isl_point *pnt)
+{
+	isl_vec *ext;
+	isl_val *v;
+
+	if (!qp || !pnt)
+		goto error;
+	isl_assert(pnt->dim->ctx, isl_space_is_equal(pnt->dim, qp->dim), goto error);
+
+	if (qp->div->n_row == 0)
+		ext = isl_vec_copy(pnt->vec);
+	else {
+		int i;
+		unsigned dim = isl_space_dim(qp->dim, isl_dim_all);
+		ext = isl_vec_alloc(qp->dim->ctx, 1 + dim + qp->div->n_row);
+		if (!ext)
+			goto error;
+
+		isl_seq_cpy(ext->el, pnt->vec->el, pnt->vec->size);
+		for (i = 0; i < qp->div->n_row; ++i) {
+			isl_seq_inner_product(qp->div->row[i] + 1, ext->el,
+						1 + dim + i, &ext->el[1+dim+i]);
+			isl_int_fdiv_q(ext->el[1+dim+i], ext->el[1+dim+i],
+					qp->div->row[i][0]);
+		}
+	}
+
+	v = isl_upoly_eval(isl_upoly_copy(qp->upoly), ext);
+
+	isl_qpolynomial_free(qp);
+	isl_point_free(pnt);
+
+	return v;
+error:
+	isl_qpolynomial_free(qp);
+	isl_point_free(pnt);
+	return NULL;
+}
+
+int isl_upoly_cmp(__isl_keep struct isl_upoly_cst *cst1,
+	__isl_keep struct isl_upoly_cst *cst2)
+{
+	int cmp;
+	isl_int t;
+	isl_int_init(t);
+	isl_int_mul(t, cst1->n, cst2->d);
+	isl_int_submul(t, cst2->n, cst1->d);
+	cmp = isl_int_sgn(t);
+	isl_int_clear(t);
+	return cmp;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_insert_dims(
+	__isl_take isl_qpolynomial *qp, enum isl_dim_type type,
+	unsigned first, unsigned n)
+{
+	unsigned total;
+	unsigned g_pos;
+	int *exp;
+
+	if (!qp)
+		return NULL;
+	if (type == isl_dim_out)
+		isl_die(qp->div->ctx, isl_error_invalid,
+			"cannot insert output/set dimensions",
+			goto error);
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+	if (n == 0 && !isl_space_is_named_or_nested(qp->dim, type))
+		return qp;
+
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp)
+		return NULL;
+
+	isl_assert(qp->div->ctx, first <= isl_space_dim(qp->dim, type),
+		    goto error);
+
+	g_pos = pos(qp->dim, type) + first;
+
+	qp->div = isl_mat_insert_zero_cols(qp->div, 2 + g_pos, n);
+	if (!qp->div)
+		goto error;
+
+	total = qp->div->n_col - 2;
+	if (total > g_pos) {
+		int i;
+		exp = isl_alloc_array(qp->div->ctx, int, total - g_pos);
+		if (!exp)
+			goto error;
+		for (i = 0; i < total - g_pos; ++i)
+			exp[i] = i + n;
+		qp->upoly = expand(qp->upoly, exp, g_pos);
+		free(exp);
+		if (!qp->upoly)
+			goto error;
+	}
+
+	qp->dim = isl_space_insert_dims(qp->dim, type, first, n);
+	if (!qp->dim)
+		goto error;
+
+	return qp;
+error:
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_add_dims(
+	__isl_take isl_qpolynomial *qp, enum isl_dim_type type, unsigned n)
+{
+	unsigned pos;
+
+	pos = isl_qpolynomial_dim(qp, type);
+
+	return isl_qpolynomial_insert_dims(qp, type, pos, n);
+}
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add_dims(
+	__isl_take isl_pw_qpolynomial *pwqp,
+	enum isl_dim_type type, unsigned n)
+{
+	unsigned pos;
+
+	pos = isl_pw_qpolynomial_dim(pwqp, type);
+
+	return isl_pw_qpolynomial_insert_dims(pwqp, type, pos, n);
+}
+
+static int *reordering_move(isl_ctx *ctx,
+	unsigned len, unsigned dst, unsigned src, unsigned n)
+{
+	int i;
+	int *reordering;
+
+	reordering = isl_alloc_array(ctx, int, len);
+	if (!reordering)
+		return NULL;
+
+	if (dst <= src) {
+		for (i = 0; i < dst; ++i)
+			reordering[i] = i;
+		for (i = 0; i < n; ++i)
+			reordering[src + i] = dst + i;
+		for (i = 0; i < src - dst; ++i)
+			reordering[dst + i] = dst + n + i;
+		for (i = 0; i < len - src - n; ++i)
+			reordering[src + n + i] = src + n + i;
+	} else {
+		for (i = 0; i < src; ++i)
+			reordering[i] = i;
+		for (i = 0; i < n; ++i)
+			reordering[src + i] = dst + i;
+		for (i = 0; i < dst - src; ++i)
+			reordering[src + n + i] = src + i;
+		for (i = 0; i < len - dst - n; ++i)
+			reordering[dst + n + i] = dst + n + i;
+	}
+
+	return reordering;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_move_dims(
+	__isl_take isl_qpolynomial *qp,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n)
+{
+	unsigned g_dst_pos;
+	unsigned g_src_pos;
+	int *reordering;
+
+	if (n == 0)
+		return qp;
+
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp)
+		return NULL;
+
+	if (dst_type == isl_dim_out || src_type == isl_dim_out)
+		isl_die(qp->dim->ctx, isl_error_invalid,
+			"cannot move output/set dimension",
+			goto error);
+	if (dst_type == isl_dim_in)
+		dst_type = isl_dim_set;
+	if (src_type == isl_dim_in)
+		src_type = isl_dim_set;
+
+	isl_assert(qp->dim->ctx, src_pos + n <= isl_space_dim(qp->dim, src_type),
+		goto error);
+
+	g_dst_pos = pos(qp->dim, dst_type) + dst_pos;
+	g_src_pos = pos(qp->dim, src_type) + src_pos;
+	if (dst_type > src_type)
+		g_dst_pos -= n;
+
+	qp->div = isl_mat_move_cols(qp->div, 2 + g_dst_pos, 2 + g_src_pos, n);
+	if (!qp->div)
+		goto error;
+	qp = sort_divs(qp);
+	if (!qp)
+		goto error;
+
+	reordering = reordering_move(qp->dim->ctx,
+				qp->div->n_col - 2, g_dst_pos, g_src_pos, n);
+	if (!reordering)
+		goto error;
+
+	qp->upoly = reorder(qp->upoly, reordering);
+	free(reordering);
+	if (!qp->upoly)
+		goto error;
+
+	qp->dim = isl_space_move_dims(qp->dim, dst_type, dst_pos, src_type, src_pos, n);
+	if (!qp->dim)
+		goto error;
+
+	return qp;
+error:
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_from_affine(__isl_take isl_space *dim,
+	isl_int *f, isl_int denom)
+{
+	struct isl_upoly *up;
+
+	dim = isl_space_domain(dim);
+	if (!dim)
+		return NULL;
+
+	up = isl_upoly_from_affine(dim->ctx, f, denom,
+					1 + isl_space_dim(dim, isl_dim_all));
+
+	return isl_qpolynomial_alloc(dim, 0, up);
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_from_aff(__isl_take isl_aff *aff)
+{
+	isl_ctx *ctx;
+	struct isl_upoly *up;
+	isl_qpolynomial *qp;
+
+	if (!aff)
+		return NULL;
+
+	ctx = isl_aff_get_ctx(aff);
+	up = isl_upoly_from_affine(ctx, aff->v->el + 1, aff->v->el[0],
+				    aff->v->size - 1);
+
+	qp = isl_qpolynomial_alloc(isl_aff_get_domain_space(aff),
+				    aff->ls->div->n_row, up);
+	if (!qp)
+		goto error;
+
+	isl_mat_free(qp->div);
+	qp->div = isl_mat_copy(aff->ls->div);
+	qp->div = isl_mat_cow(qp->div);
+	if (!qp->div)
+		goto error;
+
+	isl_aff_free(aff);
+	qp = reduce_divs(qp);
+	qp = remove_redundant_divs(qp);
+	return qp;
+error:
+	isl_aff_free(aff);
+	return isl_qpolynomial_free(qp);
+}
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_from_pw_aff(
+	__isl_take isl_pw_aff *pwaff)
+{
+	int i;
+	isl_pw_qpolynomial *pwqp;
+
+	if (!pwaff)
+		return NULL;
+
+	pwqp = isl_pw_qpolynomial_alloc_size(isl_pw_aff_get_space(pwaff),
+						pwaff->n);
+
+	for (i = 0; i < pwaff->n; ++i) {
+		isl_set *dom;
+		isl_qpolynomial *qp;
+
+		dom = isl_set_copy(pwaff->p[i].set);
+		qp = isl_qpolynomial_from_aff(isl_aff_copy(pwaff->p[i].aff));
+		pwqp = isl_pw_qpolynomial_add_piece(pwqp,  dom, qp);
+	}
+
+	isl_pw_aff_free(pwaff);
+	return pwqp;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_from_constraint(
+	__isl_take isl_constraint *c, enum isl_dim_type type, unsigned pos)
+{
+	isl_aff *aff;
+
+	aff = isl_constraint_get_bound(c, type, pos);
+	isl_constraint_free(c);
+	return isl_qpolynomial_from_aff(aff);
+}
+
+/* For each 0 <= i < "n", replace variable "first" + i of type "type"
+ * in "qp" by subs[i].
+ */
+__isl_give isl_qpolynomial *isl_qpolynomial_substitute(
+	__isl_take isl_qpolynomial *qp,
+	enum isl_dim_type type, unsigned first, unsigned n,
+	__isl_keep isl_qpolynomial **subs)
+{
+	int i;
+	struct isl_upoly **ups;
+
+	if (n == 0)
+		return qp;
+
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp)
+		return NULL;
+
+	if (type == isl_dim_out)
+		isl_die(qp->dim->ctx, isl_error_invalid,
+			"cannot substitute output/set dimension",
+			goto error);
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	for (i = 0; i < n; ++i)
+		if (!subs[i])
+			goto error;
+
+	isl_assert(qp->dim->ctx, first + n <= isl_space_dim(qp->dim, type),
+			goto error);
+
+	for (i = 0; i < n; ++i)
+		isl_assert(qp->dim->ctx, isl_space_is_equal(qp->dim, subs[i]->dim),
+				goto error);
+
+	isl_assert(qp->dim->ctx, qp->div->n_row == 0, goto error);
+	for (i = 0; i < n; ++i)
+		isl_assert(qp->dim->ctx, subs[i]->div->n_row == 0, goto error);
+
+	first += pos(qp->dim, type);
+
+	ups = isl_alloc_array(qp->dim->ctx, struct isl_upoly *, n);
+	if (!ups)
+		goto error;
+	for (i = 0; i < n; ++i)
+		ups[i] = subs[i]->upoly;
+
+	qp->upoly = isl_upoly_subs(qp->upoly, first, n, ups);
+
+	free(ups);
+
+	if (!qp->upoly)
+		goto error;
+
+	return qp;
+error:
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+/* Extend "bset" with extra set dimensions for each integer division
+ * in "qp" and then call "fn" with the extended bset and the polynomial
+ * that results from replacing each of the integer divisions by the
+ * corresponding extra set dimension.
+ */
+int isl_qpolynomial_as_polynomial_on_domain(__isl_keep isl_qpolynomial *qp,
+	__isl_keep isl_basic_set *bset,
+	int (*fn)(__isl_take isl_basic_set *bset,
+		  __isl_take isl_qpolynomial *poly, void *user), void *user)
+{
+	isl_space *dim;
+	isl_mat *div;
+	isl_qpolynomial *poly;
+
+	if (!qp || !bset)
+		goto error;
+	if (qp->div->n_row == 0)
+		return fn(isl_basic_set_copy(bset), isl_qpolynomial_copy(qp),
+			  user);
+
+	div = isl_mat_copy(qp->div);
+	dim = isl_space_copy(qp->dim);
+	dim = isl_space_add_dims(dim, isl_dim_set, qp->div->n_row);
+	poly = isl_qpolynomial_alloc(dim, 0, isl_upoly_copy(qp->upoly));
+	bset = isl_basic_set_copy(bset);
+	bset = isl_basic_set_add_dims(bset, isl_dim_set, qp->div->n_row);
+	bset = add_div_constraints(bset, div);
+
+	return fn(bset, poly, user);
+error:
+	return -1;
+}
+
+/* Return total degree in variables first (inclusive) up to last (exclusive).
+ */
+int isl_upoly_degree(__isl_keep struct isl_upoly *up, int first, int last)
+{
+	int deg = -1;
+	int i;
+	struct isl_upoly_rec *rec;
+
+	if (!up)
+		return -2;
+	if (isl_upoly_is_zero(up))
+		return -1;
+	if (isl_upoly_is_cst(up) || up->var < first)
+		return 0;
+
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		return -2;
+
+	for (i = 0; i < rec->n; ++i) {
+		int d;
+
+		if (isl_upoly_is_zero(rec->p[i]))
+			continue;
+		d = isl_upoly_degree(rec->p[i], first, last);
+		if (up->var < last)
+			d += i;
+		if (d > deg)
+			deg = d;
+	}
+
+	return deg;
+}
+
+/* Return total degree in set variables.
+ */
+int isl_qpolynomial_degree(__isl_keep isl_qpolynomial *poly)
+{
+	unsigned ovar;
+	unsigned nvar;
+
+	if (!poly)
+		return -2;
+
+	ovar = isl_space_offset(poly->dim, isl_dim_set);
+	nvar = isl_space_dim(poly->dim, isl_dim_set);
+	return isl_upoly_degree(poly->upoly, ovar, ovar + nvar);
+}
+
+__isl_give struct isl_upoly *isl_upoly_coeff(__isl_keep struct isl_upoly *up,
+	unsigned pos, int deg)
+{
+	int i;
+	struct isl_upoly_rec *rec;
+
+	if (!up)
+		return NULL;
+
+	if (isl_upoly_is_cst(up) || up->var < pos) {
+		if (deg == 0)
+			return isl_upoly_copy(up);
+		else
+			return isl_upoly_zero(up->ctx);
+	}
+
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		return NULL;
+
+	if (up->var == pos) {
+		if (deg < rec->n)
+			return isl_upoly_copy(rec->p[deg]);
+		else
+			return isl_upoly_zero(up->ctx);
+	}
+
+	up = isl_upoly_copy(up);
+	up = isl_upoly_cow(up);
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		goto error;
+
+	for (i = 0; i < rec->n; ++i) {
+		struct isl_upoly *t;
+		t = isl_upoly_coeff(rec->p[i], pos, deg);
+		if (!t)
+			goto error;
+		isl_upoly_free(rec->p[i]);
+		rec->p[i] = t;
+	}
+
+	return up;
+error:
+	isl_upoly_free(up);
+	return NULL;
+}
+
+/* Return coefficient of power "deg" of variable "t_pos" of type "type".
+ */
+__isl_give isl_qpolynomial *isl_qpolynomial_coeff(
+	__isl_keep isl_qpolynomial *qp,
+	enum isl_dim_type type, unsigned t_pos, int deg)
+{
+	unsigned g_pos;
+	struct isl_upoly *up;
+	isl_qpolynomial *c;
+
+	if (!qp)
+		return NULL;
+
+	if (type == isl_dim_out)
+		isl_die(qp->div->ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			return NULL);
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	isl_assert(qp->div->ctx, t_pos < isl_space_dim(qp->dim, type),
+			return NULL);
+
+	g_pos = pos(qp->dim, type) + t_pos;
+	up = isl_upoly_coeff(qp->upoly, g_pos, deg);
+
+	c = isl_qpolynomial_alloc(isl_space_copy(qp->dim), qp->div->n_row, up);
+	if (!c)
+		return NULL;
+	isl_mat_free(c->div);
+	c->div = isl_mat_copy(qp->div);
+	if (!c->div)
+		goto error;
+	return c;
+error:
+	isl_qpolynomial_free(c);
+	return NULL;
+}
+
+/* Homogenize the polynomial in the variables first (inclusive) up to
+ * last (exclusive) by inserting powers of variable first.
+ * Variable first is assumed not to appear in the input.
+ */
+__isl_give struct isl_upoly *isl_upoly_homogenize(
+	__isl_take struct isl_upoly *up, int deg, int target,
+	int first, int last)
+{
+	int i;
+	struct isl_upoly_rec *rec;
+
+	if (!up)
+		return NULL;
+	if (isl_upoly_is_zero(up))
+		return up;
+	if (deg == target)
+		return up;
+	if (isl_upoly_is_cst(up) || up->var < first) {
+		struct isl_upoly *hom;
+
+		hom = isl_upoly_var_pow(up->ctx, first, target - deg);
+		if (!hom)
+			goto error;
+		rec = isl_upoly_as_rec(hom);
+		rec->p[target - deg] = isl_upoly_mul(rec->p[target - deg], up);
+
+		return hom;
+	}
+
+	up = isl_upoly_cow(up);
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		goto error;
+
+	for (i = 0; i < rec->n; ++i) {
+		if (isl_upoly_is_zero(rec->p[i]))
+			continue;
+		rec->p[i] = isl_upoly_homogenize(rec->p[i],
+				up->var < last ? deg + i : i, target,
+				first, last);
+		if (!rec->p[i])
+			goto error;
+	}
+
+	return up;
+error:
+	isl_upoly_free(up);
+	return NULL;
+}
+
+/* Homogenize the polynomial in the set variables by introducing
+ * powers of an extra set variable at position 0.
+ */
+__isl_give isl_qpolynomial *isl_qpolynomial_homogenize(
+	__isl_take isl_qpolynomial *poly)
+{
+	unsigned ovar;
+	unsigned nvar;
+	int deg = isl_qpolynomial_degree(poly);
+
+	if (deg < -1)
+		goto error;
+
+	poly = isl_qpolynomial_insert_dims(poly, isl_dim_in, 0, 1);
+	poly = isl_qpolynomial_cow(poly);
+	if (!poly)
+		goto error;
+
+	ovar = isl_space_offset(poly->dim, isl_dim_set);
+	nvar = isl_space_dim(poly->dim, isl_dim_set);
+	poly->upoly = isl_upoly_homogenize(poly->upoly, 0, deg,
+						ovar, ovar + nvar);
+	if (!poly->upoly)
+		goto error;
+
+	return poly;
+error:
+	isl_qpolynomial_free(poly);
+	return NULL;
+}
+
+__isl_give isl_term *isl_term_alloc(__isl_take isl_space *dim,
+	__isl_take isl_mat *div)
+{
+	isl_term *term;
+	int n;
+
+	if (!dim || !div)
+		goto error;
+
+	n = isl_space_dim(dim, isl_dim_all) + div->n_row;
+
+	term = isl_calloc(dim->ctx, struct isl_term,
+			sizeof(struct isl_term) + (n - 1) * sizeof(int));
+	if (!term)
+		goto error;
+
+	term->ref = 1;
+	term->dim = dim;
+	term->div = div;
+	isl_int_init(term->n);
+	isl_int_init(term->d);
+	
+	return term;
+error:
+	isl_space_free(dim);
+	isl_mat_free(div);
+	return NULL;
+}
+
+__isl_give isl_term *isl_term_copy(__isl_keep isl_term *term)
+{
+	if (!term)
+		return NULL;
+
+	term->ref++;
+	return term;
+}
+
+__isl_give isl_term *isl_term_dup(__isl_keep isl_term *term)
+{
+	int i;
+	isl_term *dup;
+	unsigned total;
+
+	if (!term)
+		return NULL;
+
+	total = isl_space_dim(term->dim, isl_dim_all) + term->div->n_row;
+
+	dup = isl_term_alloc(isl_space_copy(term->dim), isl_mat_copy(term->div));
+	if (!dup)
+		return NULL;
+
+	isl_int_set(dup->n, term->n);
+	isl_int_set(dup->d, term->d);
+
+	for (i = 0; i < total; ++i)
+		dup->pow[i] = term->pow[i];
+
+	return dup;
+}
+
+__isl_give isl_term *isl_term_cow(__isl_take isl_term *term)
+{
+	if (!term)
+		return NULL;
+
+	if (term->ref == 1)
+		return term;
+	term->ref--;
+	return isl_term_dup(term);
+}
+
+void isl_term_free(__isl_take isl_term *term)
+{
+	if (!term)
+		return;
+
+	if (--term->ref > 0)
+		return;
+
+	isl_space_free(term->dim);
+	isl_mat_free(term->div);
+	isl_int_clear(term->n);
+	isl_int_clear(term->d);
+	free(term);
+}
+
+unsigned isl_term_dim(__isl_keep isl_term *term, enum isl_dim_type type)
+{
+	if (!term)
+		return 0;
+
+	switch (type) {
+	case isl_dim_param:
+	case isl_dim_in:
+	case isl_dim_out:	return isl_space_dim(term->dim, type);
+	case isl_dim_div:	return term->div->n_row;
+	case isl_dim_all:	return isl_space_dim(term->dim, isl_dim_all) +
+								term->div->n_row;
+	default:		return 0;
+	}
+}
+
+isl_ctx *isl_term_get_ctx(__isl_keep isl_term *term)
+{
+	return term ? term->dim->ctx : NULL;
+}
+
+void isl_term_get_num(__isl_keep isl_term *term, isl_int *n)
+{
+	if (!term)
+		return;
+	isl_int_set(*n, term->n);
+}
+
+void isl_term_get_den(__isl_keep isl_term *term, isl_int *d)
+{
+	if (!term)
+		return;
+	isl_int_set(*d, term->d);
+}
+
+/* Return the coefficient of the term "term".
+ */
+__isl_give isl_val *isl_term_get_coefficient_val(__isl_keep isl_term *term)
+{
+	if (!term)
+		return NULL;
+
+	return isl_val_rat_from_isl_int(isl_term_get_ctx(term),
+					term->n, term->d);
+}
+
+int isl_term_get_exp(__isl_keep isl_term *term,
+	enum isl_dim_type type, unsigned pos)
+{
+	if (!term)
+		return -1;
+
+	isl_assert(term->dim->ctx, pos < isl_term_dim(term, type), return -1);
+
+	if (type >= isl_dim_set)
+		pos += isl_space_dim(term->dim, isl_dim_param);
+	if (type >= isl_dim_div)
+		pos += isl_space_dim(term->dim, isl_dim_set);
+
+	return term->pow[pos];
+}
+
+__isl_give isl_aff *isl_term_get_div(__isl_keep isl_term *term, unsigned pos)
+{
+	isl_local_space *ls;
+	isl_aff *aff;
+
+	if (!term)
+		return NULL;
+
+	isl_assert(term->dim->ctx, pos < isl_term_dim(term, isl_dim_div),
+			return NULL);
+
+	ls = isl_local_space_alloc_div(isl_space_copy(term->dim),
+					isl_mat_copy(term->div));
+	aff = isl_aff_alloc(ls);
+	if (!aff)
+		return NULL;
+
+	isl_seq_cpy(aff->v->el, term->div->row[pos], aff->v->size);
+
+	aff = isl_aff_normalize(aff);
+
+	return aff;
+}
+
+__isl_give isl_term *isl_upoly_foreach_term(__isl_keep struct isl_upoly *up,
+	int (*fn)(__isl_take isl_term *term, void *user),
+	__isl_take isl_term *term, void *user)
+{
+	int i;
+	struct isl_upoly_rec *rec;
+
+	if (!up || !term)
+		goto error;
+
+	if (isl_upoly_is_zero(up))
+		return term;
+
+	isl_assert(up->ctx, !isl_upoly_is_nan(up), goto error);
+	isl_assert(up->ctx, !isl_upoly_is_infty(up), goto error);
+	isl_assert(up->ctx, !isl_upoly_is_neginfty(up), goto error);
+
+	if (isl_upoly_is_cst(up)) {
+		struct isl_upoly_cst *cst;
+		cst = isl_upoly_as_cst(up);
+		if (!cst)
+			goto error;
+		term = isl_term_cow(term);
+		if (!term)
+			goto error;
+		isl_int_set(term->n, cst->n);
+		isl_int_set(term->d, cst->d);
+		if (fn(isl_term_copy(term), user) < 0)
+			goto error;
+		return term;
+	}
+
+	rec = isl_upoly_as_rec(up);
+	if (!rec)
+		goto error;
+
+	for (i = 0; i < rec->n; ++i) {
+		term = isl_term_cow(term);
+		if (!term)
+			goto error;
+		term->pow[up->var] = i;
+		term = isl_upoly_foreach_term(rec->p[i], fn, term, user);
+		if (!term)
+			goto error;
+	}
+	term->pow[up->var] = 0;
+
+	return term;
+error:
+	isl_term_free(term);
+	return NULL;
+}
+
+int isl_qpolynomial_foreach_term(__isl_keep isl_qpolynomial *qp,
+	int (*fn)(__isl_take isl_term *term, void *user), void *user)
+{
+	isl_term *term;
+
+	if (!qp)
+		return -1;
+
+	term = isl_term_alloc(isl_space_copy(qp->dim), isl_mat_copy(qp->div));
+	if (!term)
+		return -1;
+
+	term = isl_upoly_foreach_term(qp->upoly, fn, term, user);
+
+	isl_term_free(term);
+
+	return term ? 0 : -1;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_from_term(__isl_take isl_term *term)
+{
+	struct isl_upoly *up;
+	isl_qpolynomial *qp;
+	int i, n;
+
+	if (!term)
+		return NULL;
+
+	n = isl_space_dim(term->dim, isl_dim_all) + term->div->n_row;
+
+	up = isl_upoly_rat_cst(term->dim->ctx, term->n, term->d);
+	for (i = 0; i < n; ++i) {
+		if (!term->pow[i])
+			continue;
+		up = isl_upoly_mul(up,
+			isl_upoly_var_pow(term->dim->ctx, i, term->pow[i]));
+	}
+
+	qp = isl_qpolynomial_alloc(isl_space_copy(term->dim), term->div->n_row, up);
+	if (!qp)
+		goto error;
+	isl_mat_free(qp->div);
+	qp->div = isl_mat_copy(term->div);
+	if (!qp->div)
+		goto error;
+
+	isl_term_free(term);
+	return qp;
+error:
+	isl_qpolynomial_free(qp);
+	isl_term_free(term);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_lift(__isl_take isl_qpolynomial *qp,
+	__isl_take isl_space *dim)
+{
+	int i;
+	int extra;
+	unsigned total;
+
+	if (!qp || !dim)
+		goto error;
+
+	if (isl_space_is_equal(qp->dim, dim)) {
+		isl_space_free(dim);
+		return qp;
+	}
+
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp)
+		goto error;
+
+	extra = isl_space_dim(dim, isl_dim_set) -
+			isl_space_dim(qp->dim, isl_dim_set);
+	total = isl_space_dim(qp->dim, isl_dim_all);
+	if (qp->div->n_row) {
+		int *exp;
+
+		exp = isl_alloc_array(qp->div->ctx, int, qp->div->n_row);
+		if (!exp)
+			goto error;
+		for (i = 0; i < qp->div->n_row; ++i)
+			exp[i] = extra + i;
+		qp->upoly = expand(qp->upoly, exp, total);
+		free(exp);
+		if (!qp->upoly)
+			goto error;
+	}
+	qp->div = isl_mat_insert_cols(qp->div, 2 + total, extra);
+	if (!qp->div)
+		goto error;
+	for (i = 0; i < qp->div->n_row; ++i)
+		isl_seq_clr(qp->div->row[i] + 2 + total, extra);
+
+	isl_space_free(qp->dim);
+	qp->dim = dim;
+
+	return qp;
+error:
+	isl_space_free(dim);
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+/* For each parameter or variable that does not appear in qp,
+ * first eliminate the variable from all constraints and then set it to zero.
+ */
+static __isl_give isl_set *fix_inactive(__isl_take isl_set *set,
+	__isl_keep isl_qpolynomial *qp)
+{
+	int *active = NULL;
+	int i;
+	int d;
+	unsigned nparam;
+	unsigned nvar;
+
+	if (!set || !qp)
+		goto error;
+
+	d = isl_space_dim(set->dim, isl_dim_all);
+	active = isl_calloc_array(set->ctx, int, d);
+	if (set_active(qp, active) < 0)
+		goto error;
+
+	for (i = 0; i < d; ++i)
+		if (!active[i])
+			break;
+
+	if (i == d) {
+		free(active);
+		return set;
+	}
+
+	nparam = isl_space_dim(set->dim, isl_dim_param);
+	nvar = isl_space_dim(set->dim, isl_dim_set);
+	for (i = 0; i < nparam; ++i) {
+		if (active[i])
+			continue;
+		set = isl_set_eliminate(set, isl_dim_param, i, 1);
+		set = isl_set_fix_si(set, isl_dim_param, i, 0);
+	}
+	for (i = 0; i < nvar; ++i) {
+		if (active[nparam + i])
+			continue;
+		set = isl_set_eliminate(set, isl_dim_set, i, 1);
+		set = isl_set_fix_si(set, isl_dim_set, i, 0);
+	}
+
+	free(active);
+
+	return set;
+error:
+	free(active);
+	isl_set_free(set);
+	return NULL;
+}
+
+struct isl_opt_data {
+	isl_qpolynomial *qp;
+	int first;
+	isl_val *opt;
+	int max;
+};
+
+static int opt_fn(__isl_take isl_point *pnt, void *user)
+{
+	struct isl_opt_data *data = (struct isl_opt_data *)user;
+	isl_val *val;
+
+	val = isl_qpolynomial_eval(isl_qpolynomial_copy(data->qp), pnt);
+	if (data->first) {
+		data->first = 0;
+		data->opt = val;
+	} else if (data->max) {
+		data->opt = isl_val_max(data->opt, val);
+	} else {
+		data->opt = isl_val_min(data->opt, val);
+	}
+
+	return 0;
+}
+
+__isl_give isl_val *isl_qpolynomial_opt_on_domain(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_set *set, int max)
+{
+	struct isl_opt_data data = { NULL, 1, NULL, max };
+
+	if (!set || !qp)
+		goto error;
+
+	if (isl_upoly_is_cst(qp->upoly)) {
+		isl_set_free(set);
+		data.opt = isl_qpolynomial_get_constant_val(qp);
+		isl_qpolynomial_free(qp);
+		return data.opt;
+	}
+
+	set = fix_inactive(set, qp);
+
+	data.qp = qp;
+	if (isl_set_foreach_point(set, opt_fn, &data) < 0)
+		goto error;
+
+	if (data.first)
+		data.opt = isl_val_zero(isl_set_get_ctx(set));
+
+	isl_set_free(set);
+	isl_qpolynomial_free(qp);
+	return data.opt;
+error:
+	isl_set_free(set);
+	isl_qpolynomial_free(qp);
+	isl_val_free(data.opt);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_morph_domain(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_morph *morph)
+{
+	int i;
+	int n_sub;
+	isl_ctx *ctx;
+	struct isl_upoly **subs;
+	isl_mat *mat, *diag;
+
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp || !morph)
+		goto error;
+
+	ctx = qp->dim->ctx;
+	isl_assert(ctx, isl_space_is_equal(qp->dim, morph->dom->dim), goto error);
+
+	n_sub = morph->inv->n_row - 1;
+	if (morph->inv->n_row != morph->inv->n_col)
+		n_sub += qp->div->n_row;
+	subs = isl_calloc_array(ctx, struct isl_upoly *, n_sub);
+	if (n_sub && !subs)
+		goto error;
+
+	for (i = 0; 1 + i < morph->inv->n_row; ++i)
+		subs[i] = isl_upoly_from_affine(ctx, morph->inv->row[1 + i],
+					morph->inv->row[0][0], morph->inv->n_col);
+	if (morph->inv->n_row != morph->inv->n_col)
+		for (i = 0; i < qp->div->n_row; ++i)
+			subs[morph->inv->n_row - 1 + i] =
+			    isl_upoly_var_pow(ctx, morph->inv->n_col - 1 + i, 1);
+
+	qp->upoly = isl_upoly_subs(qp->upoly, 0, n_sub, subs);
+
+	for (i = 0; i < n_sub; ++i)
+		isl_upoly_free(subs[i]);
+	free(subs);
+
+	diag = isl_mat_diag(ctx, 1, morph->inv->row[0][0]);
+	mat = isl_mat_diagonal(diag, isl_mat_copy(morph->inv));
+	diag = isl_mat_diag(ctx, qp->div->n_row, morph->inv->row[0][0]);
+	mat = isl_mat_diagonal(mat, diag);
+	qp->div = isl_mat_product(qp->div, mat);
+	isl_space_free(qp->dim);
+	qp->dim = isl_space_copy(morph->ran->dim);
+
+	if (!qp->upoly || !qp->div || !qp->dim)
+		goto error;
+
+	isl_morph_free(morph);
+
+	return qp;
+error:
+	isl_qpolynomial_free(qp);
+	isl_morph_free(morph);
+	return NULL;
+}
+
+static int neg_entry(void **entry, void *user)
+{
+	isl_pw_qpolynomial **pwqp = (isl_pw_qpolynomial **)entry;
+
+	*pwqp = isl_pw_qpolynomial_neg(*pwqp);
+
+	return *pwqp ? 0 : -1;
+}
+
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_neg(
+	__isl_take isl_union_pw_qpolynomial *upwqp)
+{
+	upwqp = isl_union_pw_qpolynomial_cow(upwqp);
+	if (!upwqp)
+		return NULL;
+
+	if (isl_hash_table_foreach(upwqp->space->ctx, &upwqp->table,
+				   &neg_entry, NULL) < 0)
+		goto error;
+
+	return upwqp;
+error:
+	isl_union_pw_qpolynomial_free(upwqp);
+	return NULL;
+}
+
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_mul(
+	__isl_take isl_union_pw_qpolynomial *upwqp1,
+	__isl_take isl_union_pw_qpolynomial *upwqp2)
+{
+	return match_bin_op(upwqp1, upwqp2, &isl_pw_qpolynomial_mul);
+}
+
+/* Reorder the columns of the given div definitions according to the
+ * given reordering.
+ */
+static __isl_give isl_mat *reorder_divs(__isl_take isl_mat *div,
+	__isl_take isl_reordering *r)
+{
+	int i, j;
+	isl_mat *mat;
+	int extra;
+
+	if (!div || !r)
+		goto error;
+
+	extra = isl_space_dim(r->dim, isl_dim_all) + div->n_row - r->len;
+	mat = isl_mat_alloc(div->ctx, div->n_row, div->n_col + extra);
+	if (!mat)
+		goto error;
+
+	for (i = 0; i < div->n_row; ++i) {
+		isl_seq_cpy(mat->row[i], div->row[i], 2);
+		isl_seq_clr(mat->row[i] + 2, mat->n_col - 2);
+		for (j = 0; j < r->len; ++j)
+			isl_int_set(mat->row[i][2 + r->pos[j]],
+				    div->row[i][2 + j]);
+	}
+
+	isl_reordering_free(r);
+	isl_mat_free(div);
+	return mat;
+error:
+	isl_reordering_free(r);
+	isl_mat_free(div);
+	return NULL;
+}
+
+/* Reorder the dimension of "qp" according to the given reordering.
+ */
+__isl_give isl_qpolynomial *isl_qpolynomial_realign_domain(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_reordering *r)
+{
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp)
+		goto error;
+
+	r = isl_reordering_extend(r, qp->div->n_row);
+	if (!r)
+		goto error;
+
+	qp->div = reorder_divs(qp->div, isl_reordering_copy(r));
+	if (!qp->div)
+		goto error;
+
+	qp->upoly = reorder(qp->upoly, r->pos);
+	if (!qp->upoly)
+		goto error;
+
+	qp = isl_qpolynomial_reset_domain_space(qp, isl_space_copy(r->dim));
+
+	isl_reordering_free(r);
+	return qp;
+error:
+	isl_qpolynomial_free(qp);
+	isl_reordering_free(r);
+	return NULL;
+}
+
+__isl_give isl_qpolynomial *isl_qpolynomial_align_params(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_space *model)
+{
+	if (!qp || !model)
+		goto error;
+
+	if (!isl_space_match(qp->dim, isl_dim_param, model, isl_dim_param)) {
+		isl_reordering *exp;
+
+		model = isl_space_drop_dims(model, isl_dim_in,
+					0, isl_space_dim(model, isl_dim_in));
+		model = isl_space_drop_dims(model, isl_dim_out,
+					0, isl_space_dim(model, isl_dim_out));
+		exp = isl_parameter_alignment_reordering(qp->dim, model);
+		exp = isl_reordering_extend_space(exp,
+					isl_qpolynomial_get_domain_space(qp));
+		qp = isl_qpolynomial_realign_domain(qp, exp);
+	}
+
+	isl_space_free(model);
+	return qp;
+error:
+	isl_space_free(model);
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+struct isl_split_periods_data {
+	int max_periods;
+	isl_pw_qpolynomial *res;
+};
+
+/* Create a slice where the integer division "div" has the fixed value "v".
+ * In particular, if "div" refers to floor(f/m), then create a slice
+ *
+ *	m v <= f <= m v + (m - 1)
+ *
+ * or
+ *
+ *	f - m v >= 0
+ *	-f + m v + (m - 1) >= 0
+ */
+static __isl_give isl_set *set_div_slice(__isl_take isl_space *dim,
+	__isl_keep isl_qpolynomial *qp, int div, isl_int v)
+{
+	int total;
+	isl_basic_set *bset = NULL;
+	int k;
+
+	if (!dim || !qp)
+		goto error;
+
+	total = isl_space_dim(dim, isl_dim_all);
+	bset = isl_basic_set_alloc_space(isl_space_copy(dim), 0, 0, 2);
+
+	k = isl_basic_set_alloc_inequality(bset);
+	if (k < 0)
+		goto error;
+	isl_seq_cpy(bset->ineq[k], qp->div->row[div] + 1, 1 + total);
+	isl_int_submul(bset->ineq[k][0], v, qp->div->row[div][0]);
+
+	k = isl_basic_set_alloc_inequality(bset);
+	if (k < 0)
+		goto error;
+	isl_seq_neg(bset->ineq[k], qp->div->row[div] + 1, 1 + total);
+	isl_int_addmul(bset->ineq[k][0], v, qp->div->row[div][0]);
+	isl_int_add(bset->ineq[k][0], bset->ineq[k][0], qp->div->row[div][0]);
+	isl_int_sub_ui(bset->ineq[k][0], bset->ineq[k][0], 1);
+
+	isl_space_free(dim);
+	return isl_set_from_basic_set(bset);
+error:
+	isl_basic_set_free(bset);
+	isl_space_free(dim);
+	return NULL;
+}
+
+static int split_periods(__isl_take isl_set *set,
+	__isl_take isl_qpolynomial *qp, void *user);
+
+/* Create a slice of the domain "set" such that integer division "div"
+ * has the fixed value "v" and add the results to data->res,
+ * replacing the integer division by "v" in "qp".
+ */
+static int set_div(__isl_take isl_set *set,
+	__isl_take isl_qpolynomial *qp, int div, isl_int v,
+	struct isl_split_periods_data *data)
+{
+	int i;
+	int total;
+	isl_set *slice;
+	struct isl_upoly *cst;
+
+	slice = set_div_slice(isl_set_get_space(set), qp, div, v);
+	set = isl_set_intersect(set, slice);
+
+	if (!qp)
+		goto error;
+
+	total = isl_space_dim(qp->dim, isl_dim_all);
+
+	for (i = div + 1; i < qp->div->n_row; ++i) {
+		if (isl_int_is_zero(qp->div->row[i][2 + total + div]))
+			continue;
+		isl_int_addmul(qp->div->row[i][1],
+				qp->div->row[i][2 + total + div], v);
+		isl_int_set_si(qp->div->row[i][2 + total + div], 0);
+	}
+
+	cst = isl_upoly_rat_cst(qp->dim->ctx, v, qp->dim->ctx->one);
+	qp = substitute_div(qp, div, cst);
+
+	return split_periods(set, qp, data);
+error:
+	isl_set_free(set);
+	isl_qpolynomial_free(qp);
+	return -1;
+}
+
+/* Split the domain "set" such that integer division "div"
+ * has a fixed value (ranging from "min" to "max") on each slice
+ * and add the results to data->res.
+ */
+static int split_div(__isl_take isl_set *set,
+	__isl_take isl_qpolynomial *qp, int div, isl_int min, isl_int max,
+	struct isl_split_periods_data *data)
+{
+	for (; isl_int_le(min, max); isl_int_add_ui(min, min, 1)) {
+		isl_set *set_i = isl_set_copy(set);
+		isl_qpolynomial *qp_i = isl_qpolynomial_copy(qp);
+
+		if (set_div(set_i, qp_i, div, min, data) < 0)
+			goto error;
+	}
+	isl_set_free(set);
+	isl_qpolynomial_free(qp);
+	return 0;
+error:
+	isl_set_free(set);
+	isl_qpolynomial_free(qp);
+	return -1;
+}
+
+/* If "qp" refers to any integer division
+ * that can only attain "max_periods" distinct values on "set"
+ * then split the domain along those distinct values.
+ * Add the results (or the original if no splitting occurs)
+ * to data->res.
+ */
+static int split_periods(__isl_take isl_set *set,
+	__isl_take isl_qpolynomial *qp, void *user)
+{
+	int i;
+	isl_pw_qpolynomial *pwqp;
+	struct isl_split_periods_data *data;
+	isl_int min, max;
+	int total;
+	int r = 0;
+
+	data = (struct isl_split_periods_data *)user;
+
+	if (!set || !qp)
+		goto error;
+
+	if (qp->div->n_row == 0) {
+		pwqp = isl_pw_qpolynomial_alloc(set, qp);
+		data->res = isl_pw_qpolynomial_add_disjoint(data->res, pwqp);
+		return 0;
+	}
+
+	isl_int_init(min);
+	isl_int_init(max);
+	total = isl_space_dim(qp->dim, isl_dim_all);
+	for (i = 0; i < qp->div->n_row; ++i) {
+		enum isl_lp_result lp_res;
+
+		if (isl_seq_first_non_zero(qp->div->row[i] + 2 + total,
+						qp->div->n_row) != -1)
+			continue;
+
+		lp_res = isl_set_solve_lp(set, 0, qp->div->row[i] + 1,
+					  set->ctx->one, &min, NULL, NULL);
+		if (lp_res == isl_lp_error)
+			goto error2;
+		if (lp_res == isl_lp_unbounded || lp_res == isl_lp_empty)
+			continue;
+		isl_int_fdiv_q(min, min, qp->div->row[i][0]);
+
+		lp_res = isl_set_solve_lp(set, 1, qp->div->row[i] + 1,
+					  set->ctx->one, &max, NULL, NULL);
+		if (lp_res == isl_lp_error)
+			goto error2;
+		if (lp_res == isl_lp_unbounded || lp_res == isl_lp_empty)
+			continue;
+		isl_int_fdiv_q(max, max, qp->div->row[i][0]);
+
+		isl_int_sub(max, max, min);
+		if (isl_int_cmp_si(max, data->max_periods) < 0) {
+			isl_int_add(max, max, min);
+			break;
+		}
+	}
+
+	if (i < qp->div->n_row) {
+		r = split_div(set, qp, i, min, max, data);
+	} else {
+		pwqp = isl_pw_qpolynomial_alloc(set, qp);
+		data->res = isl_pw_qpolynomial_add_disjoint(data->res, pwqp);
+	}
+
+	isl_int_clear(max);
+	isl_int_clear(min);
+
+	return r;
+error2:
+	isl_int_clear(max);
+	isl_int_clear(min);
+error:
+	isl_set_free(set);
+	isl_qpolynomial_free(qp);
+	return -1;
+}
+
+/* If any quasi-polynomial in pwqp refers to any integer division
+ * that can only attain "max_periods" distinct values on its domain
+ * then split the domain along those distinct values.
+ */
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_split_periods(
+	__isl_take isl_pw_qpolynomial *pwqp, int max_periods)
+{
+	struct isl_split_periods_data data;
+
+	data.max_periods = max_periods;
+	data.res = isl_pw_qpolynomial_zero(isl_pw_qpolynomial_get_space(pwqp));
+
+	if (isl_pw_qpolynomial_foreach_piece(pwqp, &split_periods, &data) < 0)
+		goto error;
+
+	isl_pw_qpolynomial_free(pwqp);
+
+	return data.res;
+error:
+	isl_pw_qpolynomial_free(data.res);
+	isl_pw_qpolynomial_free(pwqp);
+	return NULL;
+}
+
+/* Construct a piecewise quasipolynomial that is constant on the given
+ * domain.  In particular, it is
+ *	0	if cst == 0
+ *	1	if cst == 1
+ *  infinity	if cst == -1
+ */
+static __isl_give isl_pw_qpolynomial *constant_on_domain(
+	__isl_take isl_basic_set *bset, int cst)
+{
+	isl_space *dim;
+	isl_qpolynomial *qp;
+
+	if (!bset)
+		return NULL;
+
+	bset = isl_basic_set_params(bset);
+	dim = isl_basic_set_get_space(bset);
+	if (cst < 0)
+		qp = isl_qpolynomial_infty_on_domain(dim);
+	else if (cst == 0)
+		qp = isl_qpolynomial_zero_on_domain(dim);
+	else
+		qp = isl_qpolynomial_one_on_domain(dim);
+	return isl_pw_qpolynomial_alloc(isl_set_from_basic_set(bset), qp);
+}
+
+/* Factor bset, call fn on each of the factors and return the product.
+ *
+ * If no factors can be found, simply call fn on the input.
+ * Otherwise, construct the factors based on the factorizer,
+ * call fn on each factor and compute the product.
+ */
+static __isl_give isl_pw_qpolynomial *compressed_multiplicative_call(
+	__isl_take isl_basic_set *bset,
+	__isl_give isl_pw_qpolynomial *(*fn)(__isl_take isl_basic_set *bset))
+{
+	int i, n;
+	isl_space *dim;
+	isl_set *set;
+	isl_factorizer *f;
+	isl_qpolynomial *qp;
+	isl_pw_qpolynomial *pwqp;
+	unsigned nparam;
+	unsigned nvar;
+
+	f = isl_basic_set_factorizer(bset);
+	if (!f)
+		goto error;
+	if (f->n_group == 0) {
+		isl_factorizer_free(f);
+		return fn(bset);
+	}
+
+	nparam = isl_basic_set_dim(bset, isl_dim_param);
+	nvar = isl_basic_set_dim(bset, isl_dim_set);
+
+	dim = isl_basic_set_get_space(bset);
+	dim = isl_space_domain(dim);
+	set = isl_set_universe(isl_space_copy(dim));
+	qp = isl_qpolynomial_one_on_domain(dim);
+	pwqp = isl_pw_qpolynomial_alloc(set, qp);
+
+	bset = isl_morph_basic_set(isl_morph_copy(f->morph), bset);
+
+	for (i = 0, n = 0; i < f->n_group; ++i) {
+		isl_basic_set *bset_i;
+		isl_pw_qpolynomial *pwqp_i;
+
+		bset_i = isl_basic_set_copy(bset);
+		bset_i = isl_basic_set_drop_constraints_involving(bset_i,
+			    nparam + n + f->len[i], nvar - n - f->len[i]);
+		bset_i = isl_basic_set_drop_constraints_involving(bset_i,
+			    nparam, n);
+		bset_i = isl_basic_set_drop(bset_i, isl_dim_set,
+			    n + f->len[i], nvar - n - f->len[i]);
+		bset_i = isl_basic_set_drop(bset_i, isl_dim_set, 0, n);
+
+		pwqp_i = fn(bset_i);
+		pwqp = isl_pw_qpolynomial_mul(pwqp, pwqp_i);
+
+		n += f->len[i];
+	}
+
+	isl_basic_set_free(bset);
+	isl_factorizer_free(f);
+
+	return pwqp;
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Factor bset, call fn on each of the factors and return the product.
+ * The function is assumed to evaluate to zero on empty domains,
+ * to one on zero-dimensional domains and to infinity on unbounded domains
+ * and will not be called explicitly on zero-dimensional or unbounded domains.
+ *
+ * We first check for some special cases and remove all equalities.
+ * Then we hand over control to compressed_multiplicative_call.
+ */
+__isl_give isl_pw_qpolynomial *isl_basic_set_multiplicative_call(
+	__isl_take isl_basic_set *bset,
+	__isl_give isl_pw_qpolynomial *(*fn)(__isl_take isl_basic_set *bset))
+{
+	int bounded;
+	isl_morph *morph;
+	isl_pw_qpolynomial *pwqp;
+
+	if (!bset)
+		return NULL;
+
+	if (isl_basic_set_plain_is_empty(bset))
+		return constant_on_domain(bset, 0);
+
+	if (isl_basic_set_dim(bset, isl_dim_set) == 0)
+		return constant_on_domain(bset, 1);
+
+	bounded = isl_basic_set_is_bounded(bset);
+	if (bounded < 0)
+		goto error;
+	if (!bounded)
+		return constant_on_domain(bset, -1);
+
+	if (bset->n_eq == 0)
+		return compressed_multiplicative_call(bset, fn);
+
+	morph = isl_basic_set_full_compression(bset);
+	bset = isl_morph_basic_set(isl_morph_copy(morph), bset);
+
+	pwqp = compressed_multiplicative_call(bset, fn);
+
+	morph = isl_morph_dom_params(morph);
+	morph = isl_morph_ran_params(morph);
+	morph = isl_morph_inverse(morph);
+
+	pwqp = isl_pw_qpolynomial_morph_domain(pwqp, morph);
+
+	return pwqp;
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Drop all floors in "qp", turning each integer division [a/m] into
+ * a rational division a/m.  If "down" is set, then the integer division
+ * is replaced by (a-(m-1))/m instead.
+ */
+static __isl_give isl_qpolynomial *qp_drop_floors(
+	__isl_take isl_qpolynomial *qp, int down)
+{
+	int i;
+	struct isl_upoly *s;
+
+	if (!qp)
+		return NULL;
+	if (qp->div->n_row == 0)
+		return qp;
+
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp)
+		return NULL;
+
+	for (i = qp->div->n_row - 1; i >= 0; --i) {
+		if (down) {
+			isl_int_sub(qp->div->row[i][1],
+				    qp->div->row[i][1], qp->div->row[i][0]);
+			isl_int_add_ui(qp->div->row[i][1],
+				       qp->div->row[i][1], 1);
+		}
+		s = isl_upoly_from_affine(qp->dim->ctx, qp->div->row[i] + 1,
+					qp->div->row[i][0], qp->div->n_col - 1);
+		qp = substitute_div(qp, i, s);
+		if (!qp)
+			return NULL;
+	}
+
+	return qp;
+}
+
+/* Drop all floors in "pwqp", turning each integer division [a/m] into
+ * a rational division a/m.
+ */
+static __isl_give isl_pw_qpolynomial *pwqp_drop_floors(
+	__isl_take isl_pw_qpolynomial *pwqp)
+{
+	int i;
+
+	if (!pwqp)
+		return NULL;
+
+	if (isl_pw_qpolynomial_is_zero(pwqp))
+		return pwqp;
+
+	pwqp = isl_pw_qpolynomial_cow(pwqp);
+	if (!pwqp)
+		return NULL;
+
+	for (i = 0; i < pwqp->n; ++i) {
+		pwqp->p[i].qp = qp_drop_floors(pwqp->p[i].qp, 0);
+		if (!pwqp->p[i].qp)
+			goto error;
+	}
+
+	return pwqp;
+error:
+	isl_pw_qpolynomial_free(pwqp);
+	return NULL;
+}
+
+/* Adjust all the integer divisions in "qp" such that they are at least
+ * one over the given orthant (identified by "signs").  This ensures
+ * that they will still be non-negative even after subtracting (m-1)/m.
+ *
+ * In particular, f is replaced by f' + v, changing f = [a/m]
+ * to f' = [(a - m v)/m].
+ * If the constant term k in a is smaller than m,
+ * the constant term of v is set to floor(k/m) - 1.
+ * For any other term, if the coefficient c and the variable x have
+ * the same sign, then no changes are needed.
+ * Otherwise, if the variable is positive (and c is negative),
+ * then the coefficient of x in v is set to floor(c/m).
+ * If the variable is negative (and c is positive),
+ * then the coefficient of x in v is set to ceil(c/m).
+ */
+static __isl_give isl_qpolynomial *make_divs_pos(__isl_take isl_qpolynomial *qp,
+	int *signs)
+{
+	int i, j;
+	int total;
+	isl_vec *v = NULL;
+	struct isl_upoly *s;
+
+	qp = isl_qpolynomial_cow(qp);
+	if (!qp)
+		return NULL;
+	qp->div = isl_mat_cow(qp->div);
+	if (!qp->div)
+		goto error;
+
+	total = isl_space_dim(qp->dim, isl_dim_all);
+	v = isl_vec_alloc(qp->div->ctx, qp->div->n_col - 1);
+
+	for (i = 0; i < qp->div->n_row; ++i) {
+		isl_int *row = qp->div->row[i];
+		v = isl_vec_clr(v);
+		if (!v)
+			goto error;
+		if (isl_int_lt(row[1], row[0])) {
+			isl_int_fdiv_q(v->el[0], row[1], row[0]);
+			isl_int_sub_ui(v->el[0], v->el[0], 1);
+			isl_int_submul(row[1], row[0], v->el[0]);
+		}
+		for (j = 0; j < total; ++j) {
+			if (isl_int_sgn(row[2 + j]) * signs[j] >= 0)
+				continue;
+			if (signs[j] < 0)
+				isl_int_cdiv_q(v->el[1 + j], row[2 + j], row[0]);
+			else
+				isl_int_fdiv_q(v->el[1 + j], row[2 + j], row[0]);
+			isl_int_submul(row[2 + j], row[0], v->el[1 + j]);
+		}
+		for (j = 0; j < i; ++j) {
+			if (isl_int_sgn(row[2 + total + j]) >= 0)
+				continue;
+			isl_int_fdiv_q(v->el[1 + total + j],
+					row[2 + total + j], row[0]);
+			isl_int_submul(row[2 + total + j],
+					row[0], v->el[1 + total + j]);
+		}
+		for (j = i + 1; j < qp->div->n_row; ++j) {
+			if (isl_int_is_zero(qp->div->row[j][2 + total + i]))
+				continue;
+			isl_seq_combine(qp->div->row[j] + 1,
+				qp->div->ctx->one, qp->div->row[j] + 1,
+				qp->div->row[j][2 + total + i], v->el, v->size);
+		}
+		isl_int_set_si(v->el[1 + total + i], 1);
+		s = isl_upoly_from_affine(qp->dim->ctx, v->el,
+					qp->div->ctx->one, v->size);
+		qp->upoly = isl_upoly_subs(qp->upoly, total + i, 1, &s);
+		isl_upoly_free(s);
+		if (!qp->upoly)
+			goto error;
+	}
+
+	isl_vec_free(v);
+	return qp;
+error:
+	isl_vec_free(v);
+	isl_qpolynomial_free(qp);
+	return NULL;
+}
+
+struct isl_to_poly_data {
+	int sign;
+	isl_pw_qpolynomial *res;
+	isl_qpolynomial *qp;
+};
+
+/* Appoximate data->qp by a polynomial on the orthant identified by "signs".
+ * We first make all integer divisions positive and then split the
+ * quasipolynomials into terms with sign data->sign (the direction
+ * of the requested approximation) and terms with the opposite sign.
+ * In the first set of terms, each integer division [a/m] is
+ * overapproximated by a/m, while in the second it is underapproximated
+ * by (a-(m-1))/m.
+ */
+static int to_polynomial_on_orthant(__isl_take isl_set *orthant, int *signs,
+	void *user)
+{
+	struct isl_to_poly_data *data = user;
+	isl_pw_qpolynomial *t;
+	isl_qpolynomial *qp, *up, *down;
+
+	qp = isl_qpolynomial_copy(data->qp);
+	qp = make_divs_pos(qp, signs);
+
+	up = isl_qpolynomial_terms_of_sign(qp, signs, data->sign);
+	up = qp_drop_floors(up, 0);
+	down = isl_qpolynomial_terms_of_sign(qp, signs, -data->sign);
+	down = qp_drop_floors(down, 1);
+
+	isl_qpolynomial_free(qp);
+	qp = isl_qpolynomial_add(up, down);
+
+	t = isl_pw_qpolynomial_alloc(orthant, qp);
+	data->res = isl_pw_qpolynomial_add_disjoint(data->res, t);
+
+	return 0;
+}
+
+/* Approximate each quasipolynomial by a polynomial.  If "sign" is positive,
+ * the polynomial will be an overapproximation.  If "sign" is negative,
+ * it will be an underapproximation.  If "sign" is zero, the approximation
+ * will lie somewhere in between.
+ *
+ * In particular, is sign == 0, we simply drop the floors, turning
+ * the integer divisions into rational divisions.
+ * Otherwise, we split the domains into orthants, make all integer divisions
+ * positive and then approximate each [a/m] by either a/m or (a-(m-1))/m,
+ * depending on the requested sign and the sign of the term in which
+ * the integer division appears.
+ */
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_to_polynomial(
+	__isl_take isl_pw_qpolynomial *pwqp, int sign)
+{
+	int i;
+	struct isl_to_poly_data data;
+
+	if (sign == 0)
+		return pwqp_drop_floors(pwqp);
+
+	if (!pwqp)
+		return NULL;
+
+	data.sign = sign;
+	data.res = isl_pw_qpolynomial_zero(isl_pw_qpolynomial_get_space(pwqp));
+
+	for (i = 0; i < pwqp->n; ++i) {
+		if (pwqp->p[i].qp->div->n_row == 0) {
+			isl_pw_qpolynomial *t;
+			t = isl_pw_qpolynomial_alloc(
+					isl_set_copy(pwqp->p[i].set),
+					isl_qpolynomial_copy(pwqp->p[i].qp));
+			data.res = isl_pw_qpolynomial_add_disjoint(data.res, t);
+			continue;
+		}
+		data.qp = pwqp->p[i].qp;
+		if (isl_set_foreach_orthant(pwqp->p[i].set,
+					&to_polynomial_on_orthant, &data) < 0)
+			goto error;
+	}
+
+	isl_pw_qpolynomial_free(pwqp);
+
+	return data.res;
+error:
+	isl_pw_qpolynomial_free(pwqp);
+	isl_pw_qpolynomial_free(data.res);
+	return NULL;
+}
+
+static int poly_entry(void **entry, void *user)
+{
+	int *sign = user;
+	isl_pw_qpolynomial **pwqp = (isl_pw_qpolynomial **)entry;
+
+	*pwqp = isl_pw_qpolynomial_to_polynomial(*pwqp, *sign);
+
+	return *pwqp ? 0 : -1;
+}
+
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_to_polynomial(
+	__isl_take isl_union_pw_qpolynomial *upwqp, int sign)
+{
+	upwqp = isl_union_pw_qpolynomial_cow(upwqp);
+	if (!upwqp)
+		return NULL;
+
+	if (isl_hash_table_foreach(upwqp->space->ctx, &upwqp->table,
+				   &poly_entry, &sign) < 0)
+		goto error;
+
+	return upwqp;
+error:
+	isl_union_pw_qpolynomial_free(upwqp);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_from_qpolynomial(
+	__isl_take isl_qpolynomial *qp)
+{
+	int i, k;
+	isl_space *dim;
+	isl_vec *aff = NULL;
+	isl_basic_map *bmap = NULL;
+	unsigned pos;
+	unsigned n_div;
+
+	if (!qp)
+		return NULL;
+	if (!isl_upoly_is_affine(qp->upoly))
+		isl_die(qp->dim->ctx, isl_error_invalid,
+			"input quasi-polynomial not affine", goto error);
+	aff = isl_qpolynomial_extract_affine(qp);
+	if (!aff)
+		goto error;
+	dim = isl_qpolynomial_get_space(qp);
+	pos = 1 + isl_space_offset(dim, isl_dim_out);
+	n_div = qp->div->n_row;
+	bmap = isl_basic_map_alloc_space(dim, n_div, 1, 2 * n_div);
+
+	for (i = 0; i < n_div; ++i) {
+		k = isl_basic_map_alloc_div(bmap);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(bmap->div[k], qp->div->row[i], qp->div->n_col);
+		isl_int_set_si(bmap->div[k][qp->div->n_col], 0);
+		if (isl_basic_map_add_div_constraints(bmap, k) < 0)
+			goto error;
+	}
+	k = isl_basic_map_alloc_equality(bmap);
+	if (k < 0)
+		goto error;
+	isl_int_neg(bmap->eq[k][pos], aff->el[0]);
+	isl_seq_cpy(bmap->eq[k], aff->el + 1, pos);
+	isl_seq_cpy(bmap->eq[k] + pos + 1, aff->el + 1 + pos, n_div);
+
+	isl_vec_free(aff);
+	isl_qpolynomial_free(qp);
+	bmap = isl_basic_map_finalize(bmap);
+	return bmap;
+error:
+	isl_vec_free(aff);
+	isl_qpolynomial_free(qp);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
diff --git a/polly/lib/External/isl/isl_polynomial_private.h b/polly/lib/External/isl/isl_polynomial_private.h
new file mode 100644
index 00000000000..a668469e124
--- /dev/null
+++ b/polly/lib/External/isl/isl_polynomial_private.h
@@ -0,0 +1,253 @@
+#include <stdio.h>
+#include <isl_int.h>
+#include <isl/map.h>
+#include <isl/mat.h>
+#include <isl_morph.h>
+#include <isl/polynomial.h>
+#include <isl_reordering.h>
+
+struct isl_upoly {
+	int ref;
+	struct isl_ctx *ctx;
+
+	int var;
+};
+
+struct isl_upoly_cst {
+	struct isl_upoly up;
+	isl_int n;
+	isl_int d;
+};
+
+struct isl_upoly_rec {
+	struct isl_upoly up;
+	int n;
+
+	size_t size;
+	struct isl_upoly *p[];
+};
+
+/* dim represents the domain space.
+ */
+struct isl_qpolynomial {
+	int ref;
+
+	isl_space *dim;
+	struct isl_mat *div;
+	struct isl_upoly *upoly;
+};
+
+struct isl_term {
+	int ref;
+
+	isl_int n;
+	isl_int d;
+
+	isl_space *dim;
+	struct isl_mat *div;
+
+	int pow[1];
+};
+
+struct isl_pw_qpolynomial_piece {
+	struct isl_set *set;
+	struct isl_qpolynomial *qp;
+};
+
+struct isl_pw_qpolynomial {
+	int ref;
+
+	isl_space *dim;
+
+	int n;
+
+	size_t size;
+	struct isl_pw_qpolynomial_piece p[1];
+};
+
+/* dim represents the domain space.
+ */
+struct isl_qpolynomial_fold {
+	int ref;
+
+	enum isl_fold type;
+	isl_space *dim;
+
+	int n;
+
+	size_t size;
+	struct isl_qpolynomial *qp[1];
+};
+
+struct isl_pw_qpolynomial_fold_piece {
+	struct isl_set *set;
+	struct isl_qpolynomial_fold *fold;
+};
+
+struct isl_pw_qpolynomial_fold {
+	int ref;
+
+	enum isl_fold type;
+	isl_space *dim;
+
+	int n;
+
+	size_t size;
+	struct isl_pw_qpolynomial_fold_piece p[1];
+};
+
+void isl_term_get_num(__isl_keep isl_term *term, isl_int *n);
+
+__isl_give struct isl_upoly *isl_upoly_zero(struct isl_ctx *ctx);
+__isl_give struct isl_upoly *isl_upoly_copy(__isl_keep struct isl_upoly *up);
+__isl_give struct isl_upoly *isl_upoly_cow(__isl_take struct isl_upoly *up);
+__isl_give struct isl_upoly *isl_upoly_dup(__isl_keep struct isl_upoly *up);
+void isl_upoly_free(__isl_take struct isl_upoly *up);
+__isl_give struct isl_upoly *isl_upoly_mul(__isl_take struct isl_upoly *up1,
+	__isl_take struct isl_upoly *up2);
+
+int isl_upoly_is_cst(__isl_keep struct isl_upoly *up);
+int isl_upoly_is_zero(__isl_keep struct isl_upoly *up);
+int isl_upoly_is_one(__isl_keep struct isl_upoly *up);
+int isl_upoly_is_negone(__isl_keep struct isl_upoly *up);
+__isl_keep struct isl_upoly_cst *isl_upoly_as_cst(__isl_keep struct isl_upoly *up);
+__isl_keep struct isl_upoly_rec *isl_upoly_as_rec(__isl_keep struct isl_upoly *up);
+
+__isl_give struct isl_upoly *isl_upoly_sum(__isl_take struct isl_upoly *up1,
+	__isl_take struct isl_upoly *up2);
+__isl_give struct isl_upoly *isl_upoly_mul_isl_int(
+	__isl_take struct isl_upoly *up, isl_int v);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_alloc(__isl_take isl_space *dim,
+	unsigned n_div, __isl_take struct isl_upoly *up);
+__isl_give isl_qpolynomial *isl_qpolynomial_cow(__isl_take isl_qpolynomial *qp);
+__isl_give isl_qpolynomial *isl_qpolynomial_dup(__isl_keep isl_qpolynomial *qp);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_cst_on_domain(__isl_take isl_space *dim,
+	isl_int v);
+__isl_give isl_qpolynomial *isl_qpolynomial_rat_cst_on_domain(
+	__isl_take isl_space *space, const isl_int n, const isl_int d);
+__isl_give isl_qpolynomial *isl_qpolynomial_var_pow_on_domain(__isl_take isl_space *dim,
+	int pos, int power);
+int isl_qpolynomial_is_one(__isl_keep isl_qpolynomial *qp);
+int isl_qpolynomial_is_affine(__isl_keep isl_qpolynomial *qp);
+int isl_qpolynomial_is_cst(__isl_keep isl_qpolynomial *qp,
+	isl_int *n, isl_int *d);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_add_on_domain(
+	__isl_keep isl_set *dom,
+	__isl_take isl_qpolynomial *qp1,
+	__isl_take isl_qpolynomial *qp2);
+
+int isl_qpolynomial_degree(__isl_keep isl_qpolynomial *poly);
+__isl_give isl_qpolynomial *isl_qpolynomial_coeff(
+	__isl_keep isl_qpolynomial *poly,
+	enum isl_dim_type type, unsigned pos, int deg);
+
+__isl_give isl_vec *isl_qpolynomial_extract_affine(
+	__isl_keep isl_qpolynomial *qp);
+__isl_give isl_qpolynomial *isl_qpolynomial_from_affine(__isl_take isl_space *dim,
+	isl_int *f, isl_int denom);
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_cow(
+	__isl_take isl_pw_qpolynomial *pwqp);
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add_piece(
+	__isl_take isl_pw_qpolynomial *pwqp,
+	__isl_take isl_set *set, __isl_take isl_qpolynomial *qp);
+int isl_pw_qpolynomial_is_one(__isl_keep isl_pw_qpolynomial *pwqp);
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_project_out(
+	__isl_take isl_pw_qpolynomial *pwqp,
+	enum isl_dim_type type, unsigned first, unsigned n);
+
+__isl_give isl_val *isl_qpolynomial_opt_on_domain(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_set *set, int max);
+
+enum isl_fold isl_fold_type_negate(enum isl_fold type);
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_cow(
+	__isl_take isl_qpolynomial_fold *fold);
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_dup(
+	__isl_keep isl_qpolynomial_fold *fold);
+
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_cow(
+	__isl_take isl_pw_qpolynomial_fold *pwf);
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_add_on_domain(
+	__isl_keep isl_set *set,
+	__isl_take isl_qpolynomial_fold *fold1,
+	__isl_take isl_qpolynomial_fold *fold2);
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_fold_on_domain(
+	__isl_keep isl_set *set,
+	__isl_take isl_qpolynomial_fold *fold1,
+	__isl_take isl_qpolynomial_fold *fold2);
+
+__isl_give isl_val *isl_qpolynomial_fold_opt_on_domain(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_set *set, int max);
+
+int isl_pw_qpolynomial_fold_covers(__isl_keep isl_pw_qpolynomial_fold *pwf1,
+	__isl_keep isl_pw_qpolynomial_fold *pwf2);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_morph_domain(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_morph *morph);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_morph_domain(
+	__isl_take isl_pw_qpolynomial *pwqp, __isl_take isl_morph *morph);
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_morph_domain(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_morph *morph);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_morph_domain(
+	__isl_take isl_pw_qpolynomial_fold *pwf, __isl_take isl_morph *morph);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_lift(__isl_take isl_qpolynomial *qp,
+	__isl_take isl_space *dim);
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_lift(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_space *dim);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_substitute_equalities(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_basic_set *eq);
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_substitute_equalities(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_basic_set *eq);
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_gist(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_set *context);
+
+__isl_give isl_qpolynomial *isl_qpolynomial_realign_domain(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_reordering *r);
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_realign_domain(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_reordering *r);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_realign_domain(
+	__isl_take isl_pw_qpolynomial *pwqp, __isl_take isl_reordering *r);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_realign_domain(
+	__isl_take isl_pw_qpolynomial_fold *pwf, __isl_take isl_reordering *r);
+
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_reset_space(
+	__isl_take isl_pw_qpolynomial *pwqp, __isl_take isl_space *space);
+__isl_give isl_qpolynomial *isl_qpolynomial_reset_domain_space(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_space *dim);
+__isl_give isl_qpolynomial *isl_qpolynomial_reset_space_and_domain(
+	__isl_take isl_qpolynomial *qp, __isl_take isl_space *space,
+	__isl_take isl_space *domain);
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_reset_domain_space(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_space *dim);
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_reset_space_and_domain(
+	__isl_take isl_qpolynomial_fold *fold, __isl_take isl_space *space,
+	__isl_take isl_space *domain);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_reset_domain_space(
+	__isl_take isl_pw_qpolynomial_fold *pwf, __isl_take isl_space *dim);
+
+void isl_qpolynomial_get_den(__isl_keep isl_qpolynomial *qp, isl_int *d);
+__isl_give isl_qpolynomial *isl_qpolynomial_add_isl_int(
+	__isl_take isl_qpolynomial *qp, isl_int v);
+__isl_give isl_qpolynomial *isl_qpolynomial_mul_isl_int(
+	__isl_take isl_qpolynomial *qp, isl_int v);
+__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_mul_isl_int(
+	__isl_take isl_pw_qpolynomial *pwqp, isl_int v);
+
+__isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_mul_isl_int(
+	__isl_take isl_qpolynomial_fold *fold, isl_int v);
+__isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_mul_isl_int(
+	__isl_take isl_pw_qpolynomial_fold *pwf, isl_int v);
+__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_mul_isl_int(
+	__isl_take isl_union_pw_qpolynomial *upwqp, isl_int v);
+__isl_give isl_union_pw_qpolynomial_fold *
+isl_union_pw_qpolynomial_fold_mul_isl_int(
+	__isl_take isl_union_pw_qpolynomial_fold *upwf, isl_int v);
diff --git a/polly/lib/External/isl/isl_power_templ.c b/polly/lib/External/isl/isl_power_templ.c
new file mode 100644
index 00000000000..65253bdcb96
--- /dev/null
+++ b/polly/lib/External/isl/isl_power_templ.c
@@ -0,0 +1,81 @@
+#include <isl_val_private.h>
+
+#define xFN(TYPE,NAME) TYPE ## _ ## NAME
+#define FN(TYPE,NAME) xFN(TYPE,NAME)
+
+/* Compute the given non-zero power of "map" and return the result.
+ * If the exponent "exp" is negative, then the -exp th power of the inverse
+ * relation is computed.
+ */
+__isl_give TYPE *FN(TYPE,fixed_power)(__isl_take TYPE *map, isl_int exp)
+{
+	isl_ctx *ctx;
+	TYPE *res = NULL;
+	isl_int r;
+
+	if (!map)
+		return NULL;
+
+	ctx = FN(TYPE,get_ctx)(map);
+	if (isl_int_is_zero(exp))
+		isl_die(ctx, isl_error_invalid,
+			"expecting non-zero exponent", goto error);
+
+	if (isl_int_is_neg(exp)) {
+		isl_int_neg(exp, exp);
+		map = FN(TYPE,reverse)(map);
+		return FN(TYPE,fixed_power)(map, exp);
+	}
+
+	isl_int_init(r);
+	for (;;) {
+		isl_int_fdiv_r(r, exp, ctx->two);
+
+		if (!isl_int_is_zero(r)) {
+			if (!res)
+				res = FN(TYPE,copy)(map);
+			else {
+				res = FN(TYPE,apply_range)(res,
+							  FN(TYPE,copy)(map));
+				res = FN(TYPE,coalesce)(res);
+			}
+			if (!res)
+				break;
+		}
+
+		isl_int_fdiv_q(exp, exp, ctx->two);
+		if (isl_int_is_zero(exp))
+			break;
+
+		map = FN(TYPE,apply_range)(map, FN(TYPE,copy)(map));
+		map = FN(TYPE,coalesce)(map);
+	}
+	isl_int_clear(r);
+
+	FN(TYPE,free)(map);
+	return res;
+error:
+	FN(TYPE,free)(map);
+	return NULL;
+}
+
+/* Compute the given non-zero power of "map" and return the result.
+ * If the exponent "exp" is negative, then the -exp th power of the inverse
+ * relation is computed.
+ */
+__isl_give TYPE *FN(TYPE,fixed_power_val)(__isl_take TYPE *map,
+	__isl_take isl_val *exp)
+{
+	if (!map || !exp)
+		goto error;
+	if (!isl_val_is_int(exp))
+		isl_die(FN(TYPE,get_ctx)(map), isl_error_invalid,
+			"expecting integer exponent", goto error);
+	map = FN(TYPE,fixed_power)(map, exp->n);
+	isl_val_free(exp);
+	return map;
+error:
+	FN(TYPE,free)(map);
+	isl_val_free(exp);
+	return NULL;
+}
diff --git a/polly/lib/External/isl/isl_printer.c b/polly/lib/External/isl/isl_printer.c
new file mode 100644
index 00000000000..d5407fce6ad
--- /dev/null
+++ b/polly/lib/External/isl/isl_printer.c
@@ -0,0 +1,449 @@
+#include <string.h>
+#include <isl_int.h>
+#include <isl_printer_private.h>
+
+static __isl_give isl_printer *file_start_line(__isl_take isl_printer *p)
+{
+	fprintf(p->file, "%s%*s%s", p->indent_prefix ? p->indent_prefix : "",
+				    p->indent, "", p->prefix ? p->prefix : "");
+	return p;
+}
+
+static __isl_give isl_printer *file_end_line(__isl_take isl_printer *p)
+{
+	fprintf(p->file, "%s\n", p->suffix ? p->suffix : "");
+	return p;
+}
+
+static __isl_give isl_printer *file_flush(__isl_take isl_printer *p)
+{
+	fflush(p->file);
+	return p;
+}
+
+static __isl_give isl_printer *file_print_str(__isl_take isl_printer *p,
+	const char *s)
+{
+	fprintf(p->file, "%s", s);
+	return p;
+}
+
+static __isl_give isl_printer *file_print_double(__isl_take isl_printer *p,
+	double d)
+{
+	fprintf(p->file, "%g", d);
+	return p;
+}
+
+static __isl_give isl_printer *file_print_int(__isl_take isl_printer *p, int i)
+{
+	fprintf(p->file, "%d", i);
+	return p;
+}
+
+static __isl_give isl_printer *file_print_isl_int(__isl_take isl_printer *p, isl_int i)
+{
+	isl_int_print(p->file, i, p->width);
+	return p;
+}
+
+static int grow_buf(__isl_keep isl_printer *p, int extra)
+{
+	int new_size;
+	char *new_buf;
+
+	if (p->buf_size == 0)
+		return -1;
+
+	new_size = ((p->buf_n + extra + 1) * 3) / 2;
+	new_buf = isl_realloc_array(p->ctx, p->buf, char, new_size);
+	if (!new_buf) {
+		p->buf_size = 0;
+		return -1;
+	}
+	p->buf = new_buf;
+	p->buf_size = new_size;
+
+	return 0;
+}
+
+static __isl_give isl_printer *str_print(__isl_take isl_printer *p,
+	const char *s, int len)
+{
+	if (p->buf_n + len + 1 >= p->buf_size && grow_buf(p, len))
+		goto error;
+	memcpy(p->buf + p->buf_n, s, len);
+	p->buf_n += len;
+
+	p->buf[p->buf_n] = '\0';
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *str_print_indent(__isl_take isl_printer *p,
+	int indent)
+{
+	int i;
+
+	if (p->buf_n + indent + 1 >= p->buf_size && grow_buf(p, indent))
+		goto error;
+	for (i = 0; i < indent; ++i)
+		p->buf[p->buf_n++] = ' ';
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *str_start_line(__isl_take isl_printer *p)
+{
+	if (p->indent_prefix)
+		p = str_print(p, p->indent_prefix, strlen(p->indent_prefix));
+	p = str_print_indent(p, p->indent);
+	if (p->prefix)
+		p = str_print(p, p->prefix, strlen(p->prefix));
+	return p;
+}
+
+static __isl_give isl_printer *str_end_line(__isl_take isl_printer *p)
+{
+	if (p->suffix)
+		p = str_print(p, p->suffix, strlen(p->suffix));
+	p = str_print(p, "\n", strlen("\n"));
+	return p;
+}
+
+static __isl_give isl_printer *str_flush(__isl_take isl_printer *p)
+{
+	p->buf_n = 0;
+	return p;
+}
+
+static __isl_give isl_printer *str_print_str(__isl_take isl_printer *p,
+	const char *s)
+{
+	return str_print(p, s, strlen(s));
+}
+
+static __isl_give isl_printer *str_print_double(__isl_take isl_printer *p,
+	double d)
+{
+	int left = p->buf_size - p->buf_n;
+	int need = snprintf(p->buf + p->buf_n, left, "%g", d);
+	if (need >= left) {
+		if (grow_buf(p, need))
+			goto error;
+		left = p->buf_size - p->buf_n;
+		need = snprintf(p->buf + p->buf_n, left, "%g", d);
+	}
+	p->buf_n += need;
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *str_print_int(__isl_take isl_printer *p, int i)
+{
+	int left = p->buf_size - p->buf_n;
+	int need = snprintf(p->buf + p->buf_n, left, "%d", i);
+	if (need >= left) {
+		if (grow_buf(p, need))
+			goto error;
+		left = p->buf_size - p->buf_n;
+		need = snprintf(p->buf + p->buf_n, left, "%d", i);
+	}
+	p->buf_n += need;
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+static __isl_give isl_printer *str_print_isl_int(__isl_take isl_printer *p,
+	isl_int i)
+{
+	char *s;
+	int len;
+
+	s = isl_int_get_str(i);
+	len = strlen(s);
+	if (len < p->width)
+		p = str_print_indent(p, p->width - len);
+	p = str_print(p, s, len);
+	isl_int_free_str(s);
+	return p;
+}
+
+struct isl_printer_ops {
+	__isl_give isl_printer *(*start_line)(__isl_take isl_printer *p);
+	__isl_give isl_printer *(*end_line)(__isl_take isl_printer *p);
+	__isl_give isl_printer *(*print_double)(__isl_take isl_printer *p,
+		double d);
+	__isl_give isl_printer *(*print_int)(__isl_take isl_printer *p, int i);
+	__isl_give isl_printer *(*print_isl_int)(__isl_take isl_printer *p,
+						isl_int i);
+	__isl_give isl_printer *(*print_str)(__isl_take isl_printer *p,
+						const char *s);
+	__isl_give isl_printer *(*flush)(__isl_take isl_printer *p);
+};
+
+static struct isl_printer_ops file_ops = {
+	file_start_line,
+	file_end_line,
+	file_print_double,
+	file_print_int,
+	file_print_isl_int,
+	file_print_str,
+	file_flush
+};
+
+static struct isl_printer_ops str_ops = {
+	str_start_line,
+	str_end_line,
+	str_print_double,
+	str_print_int,
+	str_print_isl_int,
+	str_print_str,
+	str_flush
+};
+
+__isl_give isl_printer *isl_printer_to_file(isl_ctx *ctx, FILE *file)
+{
+	struct isl_printer *p = isl_alloc_type(ctx, struct isl_printer);
+	if (!p)
+		return NULL;
+	p->ctx = ctx;
+	isl_ctx_ref(p->ctx);
+	p->ops = &file_ops;
+	p->file = file;
+	p->buf = NULL;
+	p->buf_n = 0;
+	p->buf_size = 0;
+	p->indent = 0;
+	p->output_format = ISL_FORMAT_ISL;
+	p->indent_prefix = NULL;
+	p->prefix = NULL;
+	p->suffix = NULL;
+	p->width = 0;
+
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_to_str(isl_ctx *ctx)
+{
+	struct isl_printer *p = isl_calloc_type(ctx, struct isl_printer);
+	if (!p)
+		return NULL;
+	p->ctx = ctx;
+	isl_ctx_ref(p->ctx);
+	p->ops = &str_ops;
+	p->file = NULL;
+	p->buf = isl_alloc_array(ctx, char, 256);
+	if (!p->buf)
+		goto error;
+	p->buf_n = 0;
+	p->buf[0] = '\0';
+	p->buf_size = 256;
+	p->indent = 0;
+	p->output_format = ISL_FORMAT_ISL;
+	p->indent_prefix = NULL;
+	p->prefix = NULL;
+	p->suffix = NULL;
+	p->width = 0;
+
+	return p;
+error:
+	isl_printer_free(p);
+	return NULL;
+}
+
+__isl_null isl_printer *isl_printer_free(__isl_take isl_printer *p)
+{
+	if (!p)
+		return NULL;
+	free(p->buf);
+	free(p->indent_prefix);
+	free(p->prefix);
+	free(p->suffix);
+	isl_ctx_deref(p->ctx);
+	free(p);
+
+	return NULL;
+}
+
+isl_ctx *isl_printer_get_ctx(__isl_keep isl_printer *printer)
+{
+	return printer ? printer->ctx : NULL;
+}
+
+FILE *isl_printer_get_file(__isl_keep isl_printer *printer)
+{
+	if (!printer)
+		return NULL;
+	if (!printer->file)
+		isl_die(isl_printer_get_ctx(printer), isl_error_invalid,
+			"not a file printer", return NULL);
+	return printer->file;
+}
+
+__isl_give isl_printer *isl_printer_set_isl_int_width(__isl_take isl_printer *p,
+	int width)
+{
+	if (!p)
+		return NULL;
+
+	p->width = width;
+
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_set_indent(__isl_take isl_printer *p,
+	int indent)
+{
+	if (!p)
+		return NULL;
+
+	p->indent = indent;
+
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_indent(__isl_take isl_printer *p,
+	int indent)
+{
+	if (!p)
+		return NULL;
+
+	p->indent += indent;
+	if (p->indent < 0)
+		p->indent = 0;
+
+	return p;
+}
+
+/* Replace the indent prefix of "p" by "prefix".
+ */
+__isl_give isl_printer *isl_printer_set_indent_prefix(__isl_take isl_printer *p,
+	const char *prefix)
+{
+	if (!p)
+		return NULL;
+
+	free(p->indent_prefix);
+	p->indent_prefix = prefix ? strdup(prefix) : NULL;
+
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_set_prefix(__isl_take isl_printer *p,
+	const char *prefix)
+{
+	if (!p)
+		return NULL;
+
+	free(p->prefix);
+	p->prefix = prefix ? strdup(prefix) : NULL;
+
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_set_suffix(__isl_take isl_printer *p,
+	const char *suffix)
+{
+	if (!p)
+		return NULL;
+
+	free(p->suffix);
+	p->suffix = suffix ? strdup(suffix) : NULL;
+
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_set_output_format(__isl_take isl_printer *p,
+	int output_format)
+{
+	if (!p)
+		return NULL;
+
+	p->output_format = output_format;
+
+	return p;
+}
+
+int isl_printer_get_output_format(__isl_keep isl_printer *p)
+{
+	if (!p)
+		return -1;
+	return p->output_format;
+}
+
+__isl_give isl_printer *isl_printer_print_str(__isl_take isl_printer *p,
+	const char *s)
+{
+	if (!p)
+		return NULL;
+	if (!s)
+		return isl_printer_free(p);
+
+	return p->ops->print_str(p, s);
+}
+
+__isl_give isl_printer *isl_printer_print_double(__isl_take isl_printer *p,
+	double d)
+{
+	if (!p)
+		return NULL;
+
+	return p->ops->print_double(p, d);
+}
+
+__isl_give isl_printer *isl_printer_print_int(__isl_take isl_printer *p, int i)
+{
+	if (!p)
+		return NULL;
+
+	return p->ops->print_int(p, i);
+}
+
+__isl_give isl_printer *isl_printer_print_isl_int(__isl_take isl_printer *p,
+	isl_int i)
+{
+	if (!p)
+		return NULL;
+
+	return p->ops->print_isl_int(p, i);
+}
+
+__isl_give isl_printer *isl_printer_start_line(__isl_take isl_printer *p)
+{
+	if (!p)
+		return NULL;
+
+	return p->ops->start_line(p);
+}
+
+__isl_give isl_printer *isl_printer_end_line(__isl_take isl_printer *p)
+{
+	if (!p)
+		return NULL;
+
+	return p->ops->end_line(p);
+}
+
+char *isl_printer_get_str(__isl_keep isl_printer *printer)
+{
+	if (!printer || !printer->buf)
+		return NULL;
+	return strdup(printer->buf);
+}
+
+__isl_give isl_printer *isl_printer_flush(__isl_take isl_printer *p)
+{
+	if (!p)
+		return NULL;
+
+	return p->ops->flush(p);
+}
diff --git a/polly/lib/External/isl/isl_printer_private.h b/polly/lib/External/isl/isl_printer_private.h
new file mode 100644
index 00000000000..c977214ce49
--- /dev/null
+++ b/polly/lib/External/isl/isl_printer_private.h
@@ -0,0 +1,18 @@
+#include <isl/printer.h>
+
+struct isl_printer_ops;
+
+struct isl_printer {
+	struct isl_ctx	*ctx;
+	struct isl_printer_ops *ops;
+	FILE        	*file;
+	int		buf_n;
+	int		buf_size;
+	char		*buf;
+	int		indent;
+	int		output_format;
+	char		*indent_prefix;
+	char		*prefix;
+	char		*suffix;
+	int		width;
+};
diff --git a/polly/lib/External/isl/isl_pw_templ.c b/polly/lib/External/isl/isl_pw_templ.c
new file mode 100644
index 00000000000..594f6b73cc6
--- /dev/null
+++ b/polly/lib/External/isl/isl_pw_templ.c
@@ -0,0 +1,2100 @@
+/*
+ * Copyright 2010-2011 INRIA Saclay
+ * Copyright 2011      Sven Verdoolaege
+ * Copyright 2012-2014 Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+#include <isl/aff.h>
+#include <isl_val_private.h>
+
+#define xFN(TYPE,NAME) TYPE ## _ ## NAME
+#define FN(TYPE,NAME) xFN(TYPE,NAME)
+#define xS(TYPE,NAME) struct TYPE ## _ ## NAME
+#define S(TYPE,NAME) xS(TYPE,NAME)
+
+#ifdef HAS_TYPE
+__isl_give PW *FN(PW,alloc_size)(__isl_take isl_space *dim,
+	enum isl_fold type, int n)
+#else
+__isl_give PW *FN(PW,alloc_size)(__isl_take isl_space *dim, int n)
+#endif
+{
+	isl_ctx *ctx;
+	struct PW *pw;
+
+	if (!dim)
+		return NULL;
+	ctx = isl_space_get_ctx(dim);
+	isl_assert(ctx, n >= 0, goto error);
+	pw = isl_alloc(ctx, struct PW,
+			sizeof(struct PW) + (n - 1) * sizeof(S(PW,piece)));
+	if (!pw)
+		goto error;
+
+	pw->ref = 1;
+#ifdef HAS_TYPE
+	pw->type = type;
+#endif
+	pw->size = n;
+	pw->n = 0;
+	pw->dim = dim;
+	return pw;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+#ifdef HAS_TYPE
+__isl_give PW *FN(PW,ZERO)(__isl_take isl_space *dim, enum isl_fold type)
+{
+	return FN(PW,alloc_size)(dim, type, 0);
+}
+#else
+__isl_give PW *FN(PW,ZERO)(__isl_take isl_space *dim)
+{
+	return FN(PW,alloc_size)(dim, 0);
+}
+#endif
+
+__isl_give PW *FN(PW,add_piece)(__isl_take PW *pw,
+	__isl_take isl_set *set, __isl_take EL *el)
+{
+	isl_ctx *ctx;
+	isl_space *el_dim = NULL;
+
+	if (!pw || !set || !el)
+		goto error;
+
+	if (isl_set_plain_is_empty(set) || FN(EL,EL_IS_ZERO)(el)) {
+		isl_set_free(set);
+		FN(EL,free)(el);
+		return pw;
+	}
+
+	ctx = isl_set_get_ctx(set);
+#ifdef HAS_TYPE
+	if (pw->type != el->type)
+		isl_die(ctx, isl_error_invalid, "fold types don't match",
+			goto error);
+#endif
+	el_dim = FN(EL,get_space(el));
+	isl_assert(ctx, isl_space_is_equal(pw->dim, el_dim), goto error);
+	isl_assert(ctx, pw->n < pw->size, goto error);
+
+	pw->p[pw->n].set = set;
+	pw->p[pw->n].FIELD = el;
+	pw->n++;
+	
+	isl_space_free(el_dim);
+	return pw;
+error:
+	isl_space_free(el_dim);
+	FN(PW,free)(pw);
+	isl_set_free(set);
+	FN(EL,free)(el);
+	return NULL;
+}
+
+#ifdef HAS_TYPE
+__isl_give PW *FN(PW,alloc)(enum isl_fold type,
+	__isl_take isl_set *set, __isl_take EL *el)
+#else
+__isl_give PW *FN(PW,alloc)(__isl_take isl_set *set, __isl_take EL *el)
+#endif
+{
+	PW *pw;
+
+	if (!set || !el)
+		goto error;
+
+#ifdef HAS_TYPE
+	pw = FN(PW,alloc_size)(FN(EL,get_space)(el), type, 1);
+#else
+	pw = FN(PW,alloc_size)(FN(EL,get_space)(el), 1);
+#endif
+
+	return FN(PW,add_piece)(pw, set, el);
+error:
+	isl_set_free(set);
+	FN(EL,free)(el);
+	return NULL;
+}
+
+__isl_give PW *FN(PW,dup)(__isl_keep PW *pw)
+{
+	int i;
+	PW *dup;
+
+	if (!pw)
+		return NULL;
+
+#ifdef HAS_TYPE
+	dup = FN(PW,alloc_size)(isl_space_copy(pw->dim), pw->type, pw->n);
+#else
+	dup = FN(PW,alloc_size)(isl_space_copy(pw->dim), pw->n);
+#endif
+	if (!dup)
+		return NULL;
+
+	for (i = 0; i < pw->n; ++i)
+		dup = FN(PW,add_piece)(dup, isl_set_copy(pw->p[i].set),
+					    FN(EL,copy)(pw->p[i].FIELD));
+
+	return dup;
+}
+
+__isl_give PW *FN(PW,cow)(__isl_take PW *pw)
+{
+	if (!pw)
+		return NULL;
+
+	if (pw->ref == 1)
+		return pw;
+	pw->ref--;
+	return FN(PW,dup)(pw);
+}
+
+__isl_give PW *FN(PW,copy)(__isl_keep PW *pw)
+{
+	if (!pw)
+		return NULL;
+
+	pw->ref++;
+	return pw;
+}
+
+__isl_null PW *FN(PW,free)(__isl_take PW *pw)
+{
+	int i;
+
+	if (!pw)
+		return NULL;
+	if (--pw->ref > 0)
+		return NULL;
+
+	for (i = 0; i < pw->n; ++i) {
+		isl_set_free(pw->p[i].set);
+		FN(EL,free)(pw->p[i].FIELD);
+	}
+	isl_space_free(pw->dim);
+	free(pw);
+
+	return NULL;
+}
+
+const char *FN(PW,get_dim_name)(__isl_keep PW *pw, enum isl_dim_type type,
+	unsigned pos)
+{
+	return pw ? isl_space_get_dim_name(pw->dim, type, pos) : NULL;
+}
+
+int FN(PW,has_dim_id)(__isl_keep PW *pw, enum isl_dim_type type, unsigned pos)
+{
+	return pw ? isl_space_has_dim_id(pw->dim, type, pos) : -1;
+}
+
+__isl_give isl_id *FN(PW,get_dim_id)(__isl_keep PW *pw, enum isl_dim_type type,
+	unsigned pos)
+{
+	return pw ? isl_space_get_dim_id(pw->dim, type, pos) : NULL;
+}
+
+int FN(PW,has_tuple_name)(__isl_keep PW *pw, enum isl_dim_type type)
+{
+	return pw ? isl_space_has_tuple_name(pw->dim, type) : -1;
+}
+
+const char *FN(PW,get_tuple_name)(__isl_keep PW *pw, enum isl_dim_type type)
+{
+	return pw ? isl_space_get_tuple_name(pw->dim, type) : NULL;
+}
+
+int FN(PW,has_tuple_id)(__isl_keep PW *pw, enum isl_dim_type type)
+{
+	return pw ? isl_space_has_tuple_id(pw->dim, type) : -1;
+}
+
+__isl_give isl_id *FN(PW,get_tuple_id)(__isl_keep PW *pw, enum isl_dim_type type)
+{
+	return pw ? isl_space_get_tuple_id(pw->dim, type) : NULL;
+}
+
+int FN(PW,IS_ZERO)(__isl_keep PW *pw)
+{
+	if (!pw)
+		return -1;
+
+	return pw->n == 0;
+}
+
+#ifndef NO_REALIGN
+__isl_give PW *FN(PW,realign_domain)(__isl_take PW *pw,
+	__isl_take isl_reordering *exp)
+{
+	int i;
+
+	pw = FN(PW,cow)(pw);
+	if (!pw || !exp)
+		goto error;
+
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].set = isl_set_realign(pw->p[i].set,
+						    isl_reordering_copy(exp));
+		if (!pw->p[i].set)
+			goto error;
+		pw->p[i].FIELD = FN(EL,realign_domain)(pw->p[i].FIELD,
+						    isl_reordering_copy(exp));
+		if (!pw->p[i].FIELD)
+			goto error;
+	}
+
+	pw = FN(PW,reset_domain_space)(pw, isl_space_copy(exp->dim));
+
+	isl_reordering_free(exp);
+	return pw;
+error:
+	isl_reordering_free(exp);
+	FN(PW,free)(pw);
+	return NULL;
+}
+
+/* Align the parameters of "pw" to those of "model".
+ */
+__isl_give PW *FN(PW,align_params)(__isl_take PW *pw, __isl_take isl_space *model)
+{
+	isl_ctx *ctx;
+
+	if (!pw || !model)
+		goto error;
+
+	ctx = isl_space_get_ctx(model);
+	if (!isl_space_has_named_params(model))
+		isl_die(ctx, isl_error_invalid,
+			"model has unnamed parameters", goto error);
+	if (!isl_space_has_named_params(pw->dim))
+		isl_die(ctx, isl_error_invalid,
+			"input has unnamed parameters", goto error);
+	if (!isl_space_match(pw->dim, isl_dim_param, model, isl_dim_param)) {
+		isl_reordering *exp;
+
+		model = isl_space_drop_dims(model, isl_dim_in,
+					0, isl_space_dim(model, isl_dim_in));
+		model = isl_space_drop_dims(model, isl_dim_out,
+					0, isl_space_dim(model, isl_dim_out));
+		exp = isl_parameter_alignment_reordering(pw->dim, model);
+		exp = isl_reordering_extend_space(exp,
+					FN(PW,get_domain_space)(pw));
+		pw = FN(PW,realign_domain)(pw, exp);
+	}
+
+	isl_space_free(model);
+	return pw;
+error:
+	isl_space_free(model);
+	FN(PW,free)(pw);
+	return NULL;
+}
+
+static __isl_give PW *FN(PW,align_params_pw_pw_and)(__isl_take PW *pw1,
+	__isl_take PW *pw2,
+	__isl_give PW *(*fn)(__isl_take PW *pw1, __isl_take PW *pw2))
+{
+	isl_ctx *ctx;
+
+	if (!pw1 || !pw2)
+		goto error;
+	if (isl_space_match(pw1->dim, isl_dim_param, pw2->dim, isl_dim_param))
+		return fn(pw1, pw2);
+	ctx = FN(PW,get_ctx)(pw1);
+	if (!isl_space_has_named_params(pw1->dim) ||
+	    !isl_space_has_named_params(pw2->dim))
+		isl_die(ctx, isl_error_invalid,
+			"unaligned unnamed parameters", goto error);
+	pw1 = FN(PW,align_params)(pw1, FN(PW,get_space)(pw2));
+	pw2 = FN(PW,align_params)(pw2, FN(PW,get_space)(pw1));
+	return fn(pw1, pw2);
+error:
+	FN(PW,free)(pw1);
+	FN(PW,free)(pw2);
+	return NULL;
+}
+
+static __isl_give PW *FN(PW,align_params_pw_set_and)(__isl_take PW *pw,
+	__isl_take isl_set *set,
+	__isl_give PW *(*fn)(__isl_take PW *pw, __isl_take isl_set *set))
+{
+	isl_ctx *ctx;
+
+	if (!pw || !set)
+		goto error;
+	if (isl_space_match(pw->dim, isl_dim_param, set->dim, isl_dim_param))
+		return fn(pw, set);
+	ctx = FN(PW,get_ctx)(pw);
+	if (!isl_space_has_named_params(pw->dim) ||
+	    !isl_space_has_named_params(set->dim))
+		isl_die(ctx, isl_error_invalid,
+			"unaligned unnamed parameters", goto error);
+	pw = FN(PW,align_params)(pw, isl_set_get_space(set));
+	set = isl_set_align_params(set, FN(PW,get_space)(pw));
+	return fn(pw, set);
+error:
+	FN(PW,free)(pw);
+	isl_set_free(set);
+	return NULL;
+}
+#endif
+
+static __isl_give PW *FN(PW,union_add_aligned)(__isl_take PW *pw1,
+	__isl_take PW *pw2)
+{
+	int i, j, n;
+	struct PW *res;
+	isl_ctx *ctx;
+	isl_set *set;
+
+	if (!pw1 || !pw2)
+		goto error;
+
+	ctx = isl_space_get_ctx(pw1->dim);
+#ifdef HAS_TYPE
+	if (pw1->type != pw2->type)
+		isl_die(ctx, isl_error_invalid,
+			"fold types don't match", goto error);
+#endif
+	isl_assert(ctx, isl_space_is_equal(pw1->dim, pw2->dim), goto error);
+
+	if (FN(PW,IS_ZERO)(pw1)) {
+		FN(PW,free)(pw1);
+		return pw2;
+	}
+
+	if (FN(PW,IS_ZERO)(pw2)) {
+		FN(PW,free)(pw2);
+		return pw1;
+	}
+
+	n = (pw1->n + 1) * (pw2->n + 1);
+#ifdef HAS_TYPE
+	res = FN(PW,alloc_size)(isl_space_copy(pw1->dim), pw1->type, n);
+#else
+	res = FN(PW,alloc_size)(isl_space_copy(pw1->dim), n);
+#endif
+
+	for (i = 0; i < pw1->n; ++i) {
+		set = isl_set_copy(pw1->p[i].set);
+		for (j = 0; j < pw2->n; ++j) {
+			struct isl_set *common;
+			EL *sum;
+			common = isl_set_intersect(isl_set_copy(pw1->p[i].set),
+						isl_set_copy(pw2->p[j].set));
+			if (isl_set_plain_is_empty(common)) {
+				isl_set_free(common);
+				continue;
+			}
+			set = isl_set_subtract(set,
+					isl_set_copy(pw2->p[j].set));
+
+			sum = FN(EL,add_on_domain)(common,
+						   FN(EL,copy)(pw1->p[i].FIELD),
+						   FN(EL,copy)(pw2->p[j].FIELD));
+
+			res = FN(PW,add_piece)(res, common, sum);
+		}
+		res = FN(PW,add_piece)(res, set, FN(EL,copy)(pw1->p[i].FIELD));
+	}
+
+	for (j = 0; j < pw2->n; ++j) {
+		set = isl_set_copy(pw2->p[j].set);
+		for (i = 0; i < pw1->n; ++i)
+			set = isl_set_subtract(set,
+					isl_set_copy(pw1->p[i].set));
+		res = FN(PW,add_piece)(res, set, FN(EL,copy)(pw2->p[j].FIELD));
+	}
+
+	FN(PW,free)(pw1);
+	FN(PW,free)(pw2);
+
+	return res;
+error:
+	FN(PW,free)(pw1);
+	FN(PW,free)(pw2);
+	return NULL;
+}
+
+/* Private version of "union_add".  For isl_pw_qpolynomial and
+ * isl_pw_qpolynomial_fold, we prefer to simply call it "add".
+ */
+static __isl_give PW *FN(PW,union_add_)(__isl_take PW *pw1, __isl_take PW *pw2)
+{
+	return FN(PW,align_params_pw_pw_and)(pw1, pw2,
+						&FN(PW,union_add_aligned));
+}
+
+/* Make sure "pw" has room for at least "n" more pieces.
+ *
+ * If there is only one reference to pw, we extend it in place.
+ * Otherwise, we create a new PW and copy the pieces.
+ */
+static __isl_give PW *FN(PW,grow)(__isl_take PW *pw, int n)
+{
+	int i;
+	isl_ctx *ctx;
+	PW *res;
+
+	if (!pw)
+		return NULL;
+	if (pw->n + n <= pw->size)
+		return pw;
+	ctx = FN(PW,get_ctx)(pw);
+	n += pw->n;
+	if (pw->ref == 1) {
+		res = isl_realloc(ctx, pw, struct PW,
+			    sizeof(struct PW) + (n - 1) * sizeof(S(PW,piece)));
+		if (!res)
+			return FN(PW,free)(pw);
+		res->size = n;
+		return res;
+	}
+#ifdef HAS_TYPE
+	res = FN(PW,alloc_size)(isl_space_copy(pw->dim), pw->type, n);
+#else
+	res = FN(PW,alloc_size)(isl_space_copy(pw->dim), n);
+#endif
+	if (!res)
+		return FN(PW,free)(pw);
+	for (i = 0; i < pw->n; ++i)
+		res = FN(PW,add_piece)(res, isl_set_copy(pw->p[i].set),
+					    FN(EL,copy)(pw->p[i].FIELD));
+	FN(PW,free)(pw);
+	return res;
+}
+
+static __isl_give PW *FN(PW,add_disjoint_aligned)(__isl_take PW *pw1,
+	__isl_take PW *pw2)
+{
+	int i;
+	isl_ctx *ctx;
+
+	if (!pw1 || !pw2)
+		goto error;
+
+	if (pw1->size < pw1->n + pw2->n && pw1->n < pw2->n)
+		return FN(PW,add_disjoint_aligned)(pw2, pw1);
+
+	ctx = isl_space_get_ctx(pw1->dim);
+#ifdef HAS_TYPE
+	if (pw1->type != pw2->type)
+		isl_die(ctx, isl_error_invalid,
+			"fold types don't match", goto error);
+#endif
+	isl_assert(ctx, isl_space_is_equal(pw1->dim, pw2->dim), goto error);
+
+	if (FN(PW,IS_ZERO)(pw1)) {
+		FN(PW,free)(pw1);
+		return pw2;
+	}
+
+	if (FN(PW,IS_ZERO)(pw2)) {
+		FN(PW,free)(pw2);
+		return pw1;
+	}
+
+	pw1 = FN(PW,grow)(pw1, pw2->n);
+	if (!pw1)
+		goto error;
+
+	for (i = 0; i < pw2->n; ++i)
+		pw1 = FN(PW,add_piece)(pw1,
+				isl_set_copy(pw2->p[i].set),
+				FN(EL,copy)(pw2->p[i].FIELD));
+
+	FN(PW,free)(pw2);
+
+	return pw1;
+error:
+	FN(PW,free)(pw1);
+	FN(PW,free)(pw2);
+	return NULL;
+}
+
+__isl_give PW *FN(PW,add_disjoint)(__isl_take PW *pw1, __isl_take PW *pw2)
+{
+	return FN(PW,align_params_pw_pw_and)(pw1, pw2,
+						&FN(PW,add_disjoint_aligned));
+}
+
+/* This function is currently only used from isl_aff.c
+ */
+static __isl_give PW *FN(PW,on_shared_domain_in)(__isl_take PW *pw1,
+	__isl_take PW *pw2, __isl_take isl_space *space,
+	__isl_give EL *(*fn)(__isl_take EL *el1, __isl_take EL *el2))
+	__attribute__ ((unused));
+
+/* Apply "fn" to pairs of elements from pw1 and pw2 on shared domains.
+ * The result of "fn" (and therefore also of this function) lives in "space".
+ */
+static __isl_give PW *FN(PW,on_shared_domain_in)(__isl_take PW *pw1,
+	__isl_take PW *pw2, __isl_take isl_space *space,
+	__isl_give EL *(*fn)(__isl_take EL *el1, __isl_take EL *el2))
+{
+	int i, j, n;
+	PW *res = NULL;
+
+	if (!pw1 || !pw2)
+		goto error;
+
+	n = pw1->n * pw2->n;
+#ifdef HAS_TYPE
+	res = FN(PW,alloc_size)(isl_space_copy(space), pw1->type, n);
+#else
+	res = FN(PW,alloc_size)(isl_space_copy(space), n);
+#endif
+
+	for (i = 0; i < pw1->n; ++i) {
+		for (j = 0; j < pw2->n; ++j) {
+			isl_set *common;
+			EL *res_ij;
+			int empty;
+
+			common = isl_set_intersect(
+					isl_set_copy(pw1->p[i].set),
+					isl_set_copy(pw2->p[j].set));
+			empty = isl_set_plain_is_empty(common);
+			if (empty < 0 || empty) {
+				isl_set_free(common);
+				if (empty < 0)
+					goto error;
+				continue;
+			}
+
+			res_ij = fn(FN(EL,copy)(pw1->p[i].FIELD),
+				    FN(EL,copy)(pw2->p[j].FIELD));
+			res_ij = FN(EL,gist)(res_ij, isl_set_copy(common));
+
+			res = FN(PW,add_piece)(res, common, res_ij);
+		}
+	}
+
+	isl_space_free(space);
+	FN(PW,free)(pw1);
+	FN(PW,free)(pw2);
+	return res;
+error:
+	isl_space_free(space);
+	FN(PW,free)(pw1);
+	FN(PW,free)(pw2);
+	FN(PW,free)(res);
+	return NULL;
+}
+
+/* This function is currently only used from isl_aff.c
+ */
+static __isl_give PW *FN(PW,on_shared_domain)(__isl_take PW *pw1,
+	__isl_take PW *pw2,
+	__isl_give EL *(*fn)(__isl_take EL *el1, __isl_take EL *el2))
+	__attribute__ ((unused));
+
+/* Apply "fn" to pairs of elements from pw1 and pw2 on shared domains.
+ * The result of "fn" is assumed to live in the same space as "pw1" and "pw2".
+ */
+static __isl_give PW *FN(PW,on_shared_domain)(__isl_take PW *pw1,
+	__isl_take PW *pw2,
+	__isl_give EL *(*fn)(__isl_take EL *el1, __isl_take EL *el2))
+{
+	isl_space *space;
+
+	if (!pw1 || !pw2)
+		goto error;
+
+	space = isl_space_copy(pw1->dim);
+	return FN(PW,on_shared_domain_in)(pw1, pw2, space, fn);
+error:
+	FN(PW,free)(pw1);
+	FN(PW,free)(pw2);
+	return NULL;
+}
+
+#ifndef NO_NEG
+__isl_give PW *FN(PW,neg)(__isl_take PW *pw)
+{
+	int i;
+
+	if (!pw)
+		return NULL;
+
+	if (FN(PW,IS_ZERO)(pw))
+		return pw;
+
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		return NULL;
+
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].FIELD = FN(EL,neg)(pw->p[i].FIELD);
+		if (!pw->p[i].FIELD)
+			return FN(PW,free)(pw);
+	}
+
+	return pw;
+}
+
+__isl_give PW *FN(PW,sub)(__isl_take PW *pw1, __isl_take PW *pw2)
+{
+	return FN(PW,add)(pw1, FN(PW,neg)(pw2));
+}
+#endif
+
+#ifndef NO_EVAL
+__isl_give isl_val *FN(PW,eval)(__isl_take PW *pw, __isl_take isl_point *pnt)
+{
+	int i;
+	int found = 0;
+	isl_ctx *ctx;
+	isl_space *pnt_dim = NULL;
+	isl_val *v;
+
+	if (!pw || !pnt)
+		goto error;
+	ctx = isl_point_get_ctx(pnt);
+	pnt_dim = isl_point_get_space(pnt);
+	isl_assert(ctx, isl_space_is_domain_internal(pnt_dim, pw->dim),
+		    goto error);
+
+	for (i = 0; i < pw->n; ++i) {
+		found = isl_set_contains_point(pw->p[i].set, pnt);
+		if (found < 0)
+			goto error;
+		if (found)
+			break;
+	}
+	if (found)
+		v = FN(EL,eval)(FN(EL,copy)(pw->p[i].FIELD),
+					    isl_point_copy(pnt));
+	else
+		v = isl_val_zero(ctx);
+	FN(PW,free)(pw);
+	isl_space_free(pnt_dim);
+	isl_point_free(pnt);
+	return v;
+error:
+	FN(PW,free)(pw);
+	isl_space_free(pnt_dim);
+	isl_point_free(pnt);
+	return NULL;
+}
+#endif
+
+/* Return the parameter domain of "pw".
+ */
+__isl_give isl_set *FN(PW,params)(__isl_take PW *pw)
+{
+	return isl_set_params(FN(PW,domain)(pw));
+}
+
+__isl_give isl_set *FN(PW,domain)(__isl_take PW *pw)
+{
+	int i;
+	isl_set *dom;
+
+	if (!pw)
+		return NULL;
+
+	dom = isl_set_empty(FN(PW,get_domain_space)(pw));
+	for (i = 0; i < pw->n; ++i)
+		dom = isl_set_union_disjoint(dom, isl_set_copy(pw->p[i].set));
+
+	FN(PW,free)(pw);
+
+	return dom;
+}
+
+/* Exploit the equalities in the domain of piece "i" of "pw"
+ * to simplify the associated function.
+ * If the domain of piece "i" is empty, then remove it entirely,
+ * replacing it with the final piece.
+ */
+static int FN(PW,exploit_equalities_and_remove_if_empty)(__isl_keep PW *pw,
+	int i)
+{
+	isl_basic_set *aff;
+	int empty = isl_set_plain_is_empty(pw->p[i].set);
+
+	if (empty < 0)
+		return -1;
+	if (empty) {
+		isl_set_free(pw->p[i].set);
+		FN(EL,free)(pw->p[i].FIELD);
+		if (i != pw->n - 1)
+			pw->p[i] = pw->p[pw->n - 1];
+		pw->n--;
+
+		return 0;
+	}
+
+	aff = isl_set_affine_hull(isl_set_copy(pw->p[i].set));
+	pw->p[i].FIELD = FN(EL,substitute_equalities)(pw->p[i].FIELD, aff);
+	if (!pw->p[i].FIELD)
+		return -1;
+
+	return 0;
+}
+
+/* Convert a piecewise expression defined over a parameter domain
+ * into one that is defined over a zero-dimensional set.
+ */
+__isl_give PW *FN(PW,from_range)(__isl_take PW *pw)
+{
+	isl_space *space;
+
+	if (!pw)
+		return NULL;
+	if (!isl_space_is_set(pw->dim))
+		isl_die(FN(PW,get_ctx)(pw), isl_error_invalid,
+			"not living in a set space", return FN(PW,free)(pw));
+
+	space = FN(PW,get_space)(pw);
+	space = isl_space_from_range(space);
+	pw = FN(PW,reset_space)(pw, space);
+
+	return pw;
+}
+
+/* Fix the value of the given parameter or domain dimension of "pw"
+ * to be equal to "value".
+ */
+__isl_give PW *FN(PW,fix_si)(__isl_take PW *pw, enum isl_dim_type type,
+	unsigned pos, int value)
+{
+	int i;
+
+	if (!pw)
+		return NULL;
+
+	if (type == isl_dim_out)
+		isl_die(FN(PW,get_ctx)(pw), isl_error_invalid,
+			"cannot fix output dimension", return FN(PW,free)(pw));
+
+	if (pw->n == 0)
+		return pw;
+
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		return FN(PW,free)(pw);
+
+	for (i = pw->n - 1; i >= 0; --i) {
+		pw->p[i].set = isl_set_fix_si(pw->p[i].set, type, pos, value);
+		if (FN(PW,exploit_equalities_and_remove_if_empty)(pw, i) < 0)
+			return FN(PW,free)(pw);
+	}
+
+	return pw;
+}
+
+/* Restrict the domain of "pw" by combining each cell
+ * with "set" through a call to "fn", where "fn" may be
+ * isl_set_intersect or isl_set_intersect_params.
+ */
+static __isl_give PW *FN(PW,intersect_aligned)(__isl_take PW *pw,
+	__isl_take isl_set *set,
+	__isl_give isl_set *(*fn)(__isl_take isl_set *set1,
+				    __isl_take isl_set *set2))
+{
+	int i;
+
+	if (!pw || !set)
+		goto error;
+
+	if (pw->n == 0) {
+		isl_set_free(set);
+		return pw;
+	}
+
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		goto error;
+
+	for (i = pw->n - 1; i >= 0; --i) {
+		pw->p[i].set = fn(pw->p[i].set, isl_set_copy(set));
+		if (FN(PW,exploit_equalities_and_remove_if_empty)(pw, i) < 0)
+			goto error;
+	}
+	
+	isl_set_free(set);
+	return pw;
+error:
+	isl_set_free(set);
+	FN(PW,free)(pw);
+	return NULL;
+}
+
+static __isl_give PW *FN(PW,intersect_domain_aligned)(__isl_take PW *pw,
+	__isl_take isl_set *set)
+{
+	return FN(PW,intersect_aligned)(pw, set, &isl_set_intersect);
+}
+
+__isl_give PW *FN(PW,intersect_domain)(__isl_take PW *pw,
+	__isl_take isl_set *context)
+{
+	return FN(PW,align_params_pw_set_and)(pw, context,
+					&FN(PW,intersect_domain_aligned));
+}
+
+static __isl_give PW *FN(PW,intersect_params_aligned)(__isl_take PW *pw,
+	__isl_take isl_set *set)
+{
+	return FN(PW,intersect_aligned)(pw, set, &isl_set_intersect_params);
+}
+
+/* Intersect the domain of "pw" with the parameter domain "context".
+ */
+__isl_give PW *FN(PW,intersect_params)(__isl_take PW *pw,
+	__isl_take isl_set *context)
+{
+	return FN(PW,align_params_pw_set_and)(pw, context,
+					&FN(PW,intersect_params_aligned));
+}
+
+/* Compute the gist of "pw" with respect to the domain constraints
+ * of "context" for the case where the domain of the last element
+ * of "pw" is equal to "context".
+ * Call "fn_el" to compute the gist of this element, replace
+ * its domain by the universe and drop all other elements
+ * as their domains are necessarily disjoint from "context".
+ */
+static __isl_give PW *FN(PW,gist_last)(__isl_take PW *pw,
+	__isl_take isl_set *context,
+	__isl_give EL *(*fn_el)(__isl_take EL *el, __isl_take isl_set *set))
+{
+	int i;
+	isl_space *space;
+
+	for (i = 0; i < pw->n - 1; ++i) {
+		isl_set_free(pw->p[i].set);
+		FN(EL,free)(pw->p[i].FIELD);
+	}
+	pw->p[0].FIELD = pw->p[pw->n - 1].FIELD;
+	pw->p[0].set = pw->p[pw->n - 1].set;
+	pw->n = 1;
+
+	space = isl_set_get_space(context);
+	pw->p[0].FIELD = fn_el(pw->p[0].FIELD, context);
+	context = isl_set_universe(space);
+	isl_set_free(pw->p[0].set);
+	pw->p[0].set = context;
+
+	if (!pw->p[0].FIELD || !pw->p[0].set)
+		return FN(PW,free)(pw);
+
+	return pw;
+}
+
+/* Compute the gist of "pw" with respect to the domain constraints
+ * of "context".  Call "fn_el" to compute the gist of the elements
+ * and "fn_dom" to compute the gist of the domains.
+ *
+ * If the piecewise expression is empty or the context is the universe,
+ * then nothing can be simplified.
+ */
+static __isl_give PW *FN(PW,gist_aligned)(__isl_take PW *pw,
+	__isl_take isl_set *context,
+	__isl_give EL *(*fn_el)(__isl_take EL *el,
+				    __isl_take isl_set *set),
+	__isl_give isl_set *(*fn_dom)(__isl_take isl_set *set,
+				    __isl_take isl_basic_set *bset))
+{
+	int i;
+	int is_universe;
+	isl_basic_set *hull = NULL;
+
+	if (!pw || !context)
+		goto error;
+
+	if (pw->n == 0) {
+		isl_set_free(context);
+		return pw;
+	}
+
+	is_universe = isl_set_plain_is_universe(context);
+	if (is_universe < 0)
+		goto error;
+	if (is_universe) {
+		isl_set_free(context);
+		return pw;
+	}
+
+	if (!isl_space_match(pw->dim, isl_dim_param,
+				context->dim, isl_dim_param)) {
+		pw = FN(PW,align_params)(pw, isl_set_get_space(context));
+		context = isl_set_align_params(context, FN(PW,get_space)(pw));
+	}
+
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		goto error;
+
+	if (pw->n == 1) {
+		int equal;
+
+		equal = isl_set_plain_is_equal(pw->p[0].set, context);
+		if (equal < 0)
+			goto error;
+		if (equal)
+			return FN(PW,gist_last)(pw, context, fn_el);
+	}
+
+	context = isl_set_compute_divs(context);
+	hull = isl_set_simple_hull(isl_set_copy(context));
+
+	for (i = pw->n - 1; i >= 0; --i) {
+		isl_set *set_i;
+		int empty;
+
+		if (i == pw->n - 1) {
+			int equal;
+			equal = isl_set_plain_is_equal(pw->p[i].set, context);
+			if (equal < 0)
+				goto error;
+			if (equal) {
+				isl_basic_set_free(hull);
+				return FN(PW,gist_last)(pw, context, fn_el);
+			}
+		}
+		set_i = isl_set_intersect(isl_set_copy(pw->p[i].set),
+						 isl_set_copy(context));
+		empty = isl_set_plain_is_empty(set_i);
+		pw->p[i].FIELD = fn_el(pw->p[i].FIELD, set_i);
+		pw->p[i].set = fn_dom(pw->p[i].set, isl_basic_set_copy(hull));
+		if (empty < 0 || !pw->p[i].FIELD || !pw->p[i].set)
+			goto error;
+		if (empty) {
+			isl_set_free(pw->p[i].set);
+			FN(EL,free)(pw->p[i].FIELD);
+			if (i != pw->n - 1)
+				pw->p[i] = pw->p[pw->n - 1];
+			pw->n--;
+		}
+	}
+
+	isl_basic_set_free(hull);
+	isl_set_free(context);
+
+	return pw;
+error:
+	FN(PW,free)(pw);
+	isl_basic_set_free(hull);
+	isl_set_free(context);
+	return NULL;
+}
+
+static __isl_give PW *FN(PW,gist_domain_aligned)(__isl_take PW *pw,
+	__isl_take isl_set *set)
+{
+	return FN(PW,gist_aligned)(pw, set, &FN(EL,gist),
+					&isl_set_gist_basic_set);
+}
+
+__isl_give PW *FN(PW,gist)(__isl_take PW *pw, __isl_take isl_set *context)
+{
+	return FN(PW,align_params_pw_set_and)(pw, context,
+						&FN(PW,gist_domain_aligned));
+}
+
+static __isl_give PW *FN(PW,gist_params_aligned)(__isl_take PW *pw,
+	__isl_take isl_set *set)
+{
+	return FN(PW,gist_aligned)(pw, set, &FN(EL,gist_params),
+					&isl_set_gist_params_basic_set);
+}
+
+__isl_give PW *FN(PW,gist_params)(__isl_take PW *pw,
+	__isl_take isl_set *context)
+{
+	return FN(PW,align_params_pw_set_and)(pw, context,
+						&FN(PW,gist_params_aligned));
+}
+
+__isl_give PW *FN(PW,coalesce)(__isl_take PW *pw)
+{
+	int i, j;
+
+	if (!pw)
+		return NULL;
+	if (pw->n == 0)
+		return pw;
+
+	for (i = pw->n - 1; i >= 0; --i) {
+		for (j = i - 1; j >= 0; --j) {
+			if (!FN(EL,plain_is_equal)(pw->p[i].FIELD,
+							pw->p[j].FIELD))
+				continue;
+			pw->p[j].set = isl_set_union(pw->p[j].set,
+							pw->p[i].set);
+			FN(EL,free)(pw->p[i].FIELD);
+			if (i != pw->n - 1)
+				pw->p[i] = pw->p[pw->n - 1];
+			pw->n--;
+			break;
+		}
+		if (j >= 0)
+			continue;
+		pw->p[i].set = isl_set_coalesce(pw->p[i].set);
+		if (!pw->p[i].set)
+			goto error;
+	}
+
+	return pw;
+error:
+	FN(PW,free)(pw);
+	return NULL;
+}
+
+isl_ctx *FN(PW,get_ctx)(__isl_keep PW *pw)
+{
+	return pw ? isl_space_get_ctx(pw->dim) : NULL;
+}
+
+#ifndef NO_INVOLVES_DIMS
+int FN(PW,involves_dims)(__isl_keep PW *pw, enum isl_dim_type type,
+	unsigned first, unsigned n)
+{
+	int i;
+	enum isl_dim_type set_type;
+
+	if (!pw)
+		return -1;
+	if (pw->n == 0 || n == 0)
+		return 0;
+
+	set_type = type == isl_dim_in ? isl_dim_set : type;
+
+	for (i = 0; i < pw->n; ++i) {
+		int involves = FN(EL,involves_dims)(pw->p[i].FIELD,
+							type, first, n);
+		if (involves < 0 || involves)
+			return involves;
+		involves = isl_set_involves_dims(pw->p[i].set,
+							set_type, first, n);
+		if (involves < 0 || involves)
+			return involves;
+	}
+	return 0;
+}
+#endif
+
+__isl_give PW *FN(PW,set_dim_name)(__isl_take PW *pw,
+	enum isl_dim_type type, unsigned pos, const char *s)
+{
+	int i;
+	enum isl_dim_type set_type;
+
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		return NULL;
+
+	set_type = type == isl_dim_in ? isl_dim_set : type;
+
+	pw->dim = isl_space_set_dim_name(pw->dim, type, pos, s);
+	if (!pw->dim)
+		goto error;
+
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].set = isl_set_set_dim_name(pw->p[i].set,
+							set_type, pos, s);
+		if (!pw->p[i].set)
+			goto error;
+		pw->p[i].FIELD = FN(EL,set_dim_name)(pw->p[i].FIELD, type, pos, s);
+		if (!pw->p[i].FIELD)
+			goto error;
+	}
+
+	return pw;
+error:
+	FN(PW,free)(pw);
+	return NULL;
+}
+
+#ifndef NO_DROP_DIMS
+__isl_give PW *FN(PW,drop_dims)(__isl_take PW *pw,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+	enum isl_dim_type set_type;
+
+	if (!pw)
+		return NULL;
+	if (n == 0 && !isl_space_get_tuple_name(pw->dim, type))
+		return pw;
+
+	set_type = type == isl_dim_in ? isl_dim_set : type;
+
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		return NULL;
+	pw->dim = isl_space_drop_dims(pw->dim, type, first, n);
+	if (!pw->dim)
+		goto error;
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].FIELD = FN(EL,drop_dims)(pw->p[i].FIELD, type, first, n);
+		if (!pw->p[i].FIELD)
+			goto error;
+		if (type == isl_dim_out)
+			continue;
+		pw->p[i].set = isl_set_drop(pw->p[i].set, set_type, first, n);
+		if (!pw->p[i].set)
+			goto error;
+	}
+
+	return pw;
+error:
+	FN(PW,free)(pw);
+	return NULL;
+}
+
+/* This function is very similar to drop_dims.
+ * The only difference is that the cells may still involve
+ * the specified dimensions.  They are removed using
+ * isl_set_project_out instead of isl_set_drop.
+ */
+__isl_give PW *FN(PW,project_out)(__isl_take PW *pw,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+	enum isl_dim_type set_type;
+
+	if (!pw)
+		return NULL;
+	if (n == 0 && !isl_space_get_tuple_name(pw->dim, type))
+		return pw;
+
+	set_type = type == isl_dim_in ? isl_dim_set : type;
+
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		return NULL;
+	pw->dim = isl_space_drop_dims(pw->dim, type, first, n);
+	if (!pw->dim)
+		goto error;
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].set = isl_set_project_out(pw->p[i].set,
+							set_type, first, n);
+		if (!pw->p[i].set)
+			goto error;
+		pw->p[i].FIELD = FN(EL,drop_dims)(pw->p[i].FIELD, type, first, n);
+		if (!pw->p[i].FIELD)
+			goto error;
+	}
+
+	return pw;
+error:
+	FN(PW,free)(pw);
+	return NULL;
+}
+
+/* Project the domain of pw onto its parameter space.
+ */
+__isl_give PW *FN(PW,project_domain_on_params)(__isl_take PW *pw)
+{
+	isl_space *space;
+	unsigned n;
+
+	n = FN(PW,dim)(pw, isl_dim_in);
+	pw = FN(PW,project_out)(pw, isl_dim_in, 0, n);
+	space = FN(PW,get_domain_space)(pw);
+	space = isl_space_params(space);
+	pw = FN(PW,reset_domain_space)(pw, space);
+	return pw;
+}
+#endif
+
+#ifndef NO_INSERT_DIMS
+__isl_give PW *FN(PW,insert_dims)(__isl_take PW *pw, enum isl_dim_type type,
+	unsigned first, unsigned n)
+{
+	int i;
+	enum isl_dim_type set_type;
+
+	if (!pw)
+		return NULL;
+	if (n == 0 && !isl_space_is_named_or_nested(pw->dim, type))
+		return pw;
+
+	set_type = type == isl_dim_in ? isl_dim_set : type;
+
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		return NULL;
+
+	pw->dim = isl_space_insert_dims(pw->dim, type, first, n);
+	if (!pw->dim)
+		goto error;
+
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].set = isl_set_insert_dims(pw->p[i].set,
+							    set_type, first, n);
+		if (!pw->p[i].set)
+			goto error;
+		pw->p[i].FIELD = FN(EL,insert_dims)(pw->p[i].FIELD,
+								type, first, n);
+		if (!pw->p[i].FIELD)
+			goto error;
+	}
+
+	return pw;
+error:
+	FN(PW,free)(pw);
+	return NULL;
+}
+#endif
+
+__isl_give PW *FN(PW,fix_dim)(__isl_take PW *pw,
+	enum isl_dim_type type, unsigned pos, isl_int v)
+{
+	int i;
+
+	if (!pw)
+		return NULL;
+
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		return NULL;
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].set = isl_set_fix(pw->p[i].set, type, pos, v);
+		if (FN(PW,exploit_equalities_and_remove_if_empty)(pw, i) < 0)
+			return FN(PW,free)(pw);
+	}
+
+	return pw;
+}
+
+/* Fix the value of the variable at position "pos" of type "type" of "pw"
+ * to be equal to "v".
+ */
+__isl_give PW *FN(PW,fix_val)(__isl_take PW *pw,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_val *v)
+{
+	if (!v)
+		return FN(PW,free)(pw);
+	if (!isl_val_is_int(v))
+		isl_die(FN(PW,get_ctx)(pw), isl_error_invalid,
+			"expecting integer value", goto error);
+
+	pw = FN(PW,fix_dim)(pw, type, pos, v->n);
+	isl_val_free(v);
+
+	return pw;
+error:
+	isl_val_free(v);
+	return FN(PW,free)(pw);
+}
+
+unsigned FN(PW,dim)(__isl_keep PW *pw, enum isl_dim_type type)
+{
+	return pw ? isl_space_dim(pw->dim, type) : 0;
+}
+
+__isl_give PW *FN(PW,split_dims)(__isl_take PW *pw,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+
+	if (!pw)
+		return NULL;
+	if (n == 0)
+		return pw;
+
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		return NULL;
+	if (!pw->dim)
+		goto error;
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].set = isl_set_split_dims(pw->p[i].set, type, first, n);
+		if (!pw->p[i].set)
+			goto error;
+	}
+
+	return pw;
+error:
+	FN(PW,free)(pw);
+	return NULL;
+}
+
+#ifndef NO_OPT
+/* Compute the maximal value attained by the piecewise quasipolynomial
+ * on its domain or zero if the domain is empty.
+ * In the worst case, the domain is scanned completely,
+ * so the domain is assumed to be bounded.
+ */
+__isl_give isl_val *FN(PW,opt)(__isl_take PW *pw, int max)
+{
+	int i;
+	isl_val *opt;
+
+	if (!pw)
+		return NULL;
+
+	if (pw->n == 0) {
+		opt = isl_val_zero(FN(PW,get_ctx)(pw));
+		FN(PW,free)(pw);
+		return opt;
+	}
+
+	opt = FN(EL,opt_on_domain)(FN(EL,copy)(pw->p[0].FIELD),
+					isl_set_copy(pw->p[0].set), max);
+	for (i = 1; i < pw->n; ++i) {
+		isl_val *opt_i;
+		opt_i = FN(EL,opt_on_domain)(FN(EL,copy)(pw->p[i].FIELD),
+						isl_set_copy(pw->p[i].set), max);
+		if (max)
+			opt = isl_val_max(opt, opt_i);
+		else
+			opt = isl_val_min(opt, opt_i);
+	}
+
+	FN(PW,free)(pw);
+	return opt;
+}
+
+__isl_give isl_val *FN(PW,max)(__isl_take PW *pw)
+{
+	return FN(PW,opt)(pw, 1);
+}
+
+__isl_give isl_val *FN(PW,min)(__isl_take PW *pw)
+{
+	return FN(PW,opt)(pw, 0);
+}
+#endif
+
+__isl_give isl_space *FN(PW,get_space)(__isl_keep PW *pw)
+{
+	return pw ? isl_space_copy(pw->dim) : NULL;
+}
+
+__isl_give isl_space *FN(PW,get_domain_space)(__isl_keep PW *pw)
+{
+	return pw ? isl_space_domain(isl_space_copy(pw->dim)) : NULL;
+}
+
+/* Return the position of the dimension of the given type and name
+ * in "pw".
+ * Return -1 if no such dimension can be found.
+ */
+int FN(PW,find_dim_by_name)(__isl_keep PW *pw,
+	enum isl_dim_type type, const char *name)
+{
+	if (!pw)
+		return -1;
+	return isl_space_find_dim_by_name(pw->dim, type, name);
+}
+
+#ifndef NO_RESET_DIM
+/* Reset the space of "pw".  Since we don't know if the elements
+ * represent the spaces themselves or their domains, we pass along
+ * both when we call their reset_space_and_domain.
+ */
+static __isl_give PW *FN(PW,reset_space_and_domain)(__isl_take PW *pw,
+	__isl_take isl_space *space, __isl_take isl_space *domain)
+{
+	int i;
+
+	pw = FN(PW,cow)(pw);
+	if (!pw || !space || !domain)
+		goto error;
+
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].set = isl_set_reset_space(pw->p[i].set,
+						 isl_space_copy(domain));
+		if (!pw->p[i].set)
+			goto error;
+		pw->p[i].FIELD = FN(EL,reset_space_and_domain)(pw->p[i].FIELD,
+			      isl_space_copy(space), isl_space_copy(domain));
+		if (!pw->p[i].FIELD)
+			goto error;
+	}
+
+	isl_space_free(domain);
+
+	isl_space_free(pw->dim);
+	pw->dim = space;
+
+	return pw;
+error:
+	isl_space_free(domain);
+	isl_space_free(space);
+	FN(PW,free)(pw);
+	return NULL;
+}
+
+__isl_give PW *FN(PW,reset_domain_space)(__isl_take PW *pw,
+	__isl_take isl_space *domain)
+{
+	isl_space *space;
+
+	space = isl_space_extend_domain_with_range(isl_space_copy(domain),
+						   FN(PW,get_space)(pw));
+	return FN(PW,reset_space_and_domain)(pw, space, domain);
+}
+
+__isl_give PW *FN(PW,reset_space)(__isl_take PW *pw, __isl_take isl_space *dim)
+{
+	isl_space *domain;
+
+	domain = isl_space_domain(isl_space_copy(dim));
+	return FN(PW,reset_space_and_domain)(pw, dim, domain);
+}
+
+__isl_give PW *FN(PW,set_tuple_id)(__isl_take PW *pw, enum isl_dim_type type,
+	__isl_take isl_id *id)
+{
+	isl_space *space;
+
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		goto error;
+
+	space = FN(PW,get_space)(pw);
+	space = isl_space_set_tuple_id(space, type, id);
+
+	return FN(PW,reset_space)(pw, space);
+error:
+	isl_id_free(id);
+	return FN(PW,free)(pw);
+}
+
+/* Drop the id on the specified tuple.
+ */
+__isl_give PW *FN(PW,reset_tuple_id)(__isl_take PW *pw, enum isl_dim_type type)
+{
+	isl_space *space;
+
+	if (!pw)
+		return NULL;
+	if (!FN(PW,has_tuple_id)(pw, type))
+		return pw;
+
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		return NULL;
+
+	space = FN(PW,get_space)(pw);
+	space = isl_space_reset_tuple_id(space, type);
+
+	return FN(PW,reset_space)(pw, space);
+}
+
+__isl_give PW *FN(PW,set_dim_id)(__isl_take PW *pw,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id)
+{
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		goto error;
+	pw->dim = isl_space_set_dim_id(pw->dim, type, pos, id);
+	return FN(PW,reset_space)(pw, isl_space_copy(pw->dim));
+error:
+	isl_id_free(id);
+	return FN(PW,free)(pw);
+}
+#endif
+
+int FN(PW,has_equal_space)(__isl_keep PW *pw1, __isl_keep PW *pw2)
+{
+	if (!pw1 || !pw2)
+		return -1;
+
+	return isl_space_is_equal(pw1->dim, pw2->dim);
+}
+
+#ifndef NO_MORPH
+__isl_give PW *FN(PW,morph_domain)(__isl_take PW *pw,
+	__isl_take isl_morph *morph)
+{
+	int i;
+	isl_ctx *ctx;
+
+	if (!pw || !morph)
+		goto error;
+
+	ctx = isl_space_get_ctx(pw->dim);
+	isl_assert(ctx, isl_space_is_domain_internal(morph->dom->dim, pw->dim),
+		goto error);
+
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		goto error;
+	pw->dim = isl_space_extend_domain_with_range(
+			isl_space_copy(morph->ran->dim), pw->dim);
+	if (!pw->dim)
+		goto error;
+
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].set = isl_morph_set(isl_morph_copy(morph), pw->p[i].set);
+		if (!pw->p[i].set)
+			goto error;
+		pw->p[i].FIELD = FN(EL,morph_domain)(pw->p[i].FIELD,
+						isl_morph_copy(morph));
+		if (!pw->p[i].FIELD)
+			goto error;
+	}
+
+	isl_morph_free(morph);
+
+	return pw;
+error:
+	FN(PW,free)(pw);
+	isl_morph_free(morph);
+	return NULL;
+}
+#endif
+
+int FN(PW,n_piece)(__isl_keep PW *pw)
+{
+	return pw ? pw->n : 0;
+}
+
+int FN(PW,foreach_piece)(__isl_keep PW *pw,
+	int (*fn)(__isl_take isl_set *set, __isl_take EL *el, void *user),
+	void *user)
+{
+	int i;
+
+	if (!pw)
+		return -1;
+
+	for (i = 0; i < pw->n; ++i)
+		if (fn(isl_set_copy(pw->p[i].set),
+				FN(EL,copy)(pw->p[i].FIELD), user) < 0)
+			return -1;
+
+	return 0;
+}
+
+#ifndef NO_LIFT
+static int any_divs(__isl_keep isl_set *set)
+{
+	int i;
+
+	if (!set)
+		return -1;
+
+	for (i = 0; i < set->n; ++i)
+		if (set->p[i]->n_div > 0)
+			return 1;
+
+	return 0;
+}
+
+static int foreach_lifted_subset(__isl_take isl_set *set, __isl_take EL *el,
+	int (*fn)(__isl_take isl_set *set, __isl_take EL *el,
+		    void *user), void *user)
+{
+	int i;
+
+	if (!set || !el)
+		goto error;
+
+	for (i = 0; i < set->n; ++i) {
+		isl_set *lift;
+		EL *copy;
+
+		lift = isl_set_from_basic_set(isl_basic_set_copy(set->p[i]));
+		lift = isl_set_lift(lift);
+
+		copy = FN(EL,copy)(el);
+		copy = FN(EL,lift)(copy, isl_set_get_space(lift));
+
+		if (fn(lift, copy, user) < 0)
+			goto error;
+	}
+
+	isl_set_free(set);
+	FN(EL,free)(el);
+
+	return 0;
+error:
+	isl_set_free(set);
+	FN(EL,free)(el);
+	return -1;
+}
+
+int FN(PW,foreach_lifted_piece)(__isl_keep PW *pw,
+	int (*fn)(__isl_take isl_set *set, __isl_take EL *el,
+		    void *user), void *user)
+{
+	int i;
+
+	if (!pw)
+		return -1;
+
+	for (i = 0; i < pw->n; ++i) {
+		isl_set *set;
+		EL *el;
+
+		set = isl_set_copy(pw->p[i].set);
+		el = FN(EL,copy)(pw->p[i].FIELD);
+		if (!any_divs(set)) {
+			if (fn(set, el, user) < 0)
+				return -1;
+			continue;
+		}
+		if (foreach_lifted_subset(set, el, fn, user) < 0)
+			return -1;
+	}
+
+	return 0;
+}
+#endif
+
+#ifndef NO_MOVE_DIMS
+__isl_give PW *FN(PW,move_dims)(__isl_take PW *pw,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n)
+{
+	int i;
+
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		return NULL;
+
+	pw->dim = isl_space_move_dims(pw->dim, dst_type, dst_pos, src_type, src_pos, n);
+	if (!pw->dim)
+		goto error;
+
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].FIELD = FN(EL,move_dims)(pw->p[i].FIELD,
+					dst_type, dst_pos, src_type, src_pos, n);
+		if (!pw->p[i].FIELD)
+			goto error;
+	}
+
+	if (dst_type == isl_dim_in)
+		dst_type = isl_dim_set;
+	if (src_type == isl_dim_in)
+		src_type = isl_dim_set;
+
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].set = isl_set_move_dims(pw->p[i].set,
+						dst_type, dst_pos,
+						src_type, src_pos, n);
+		if (!pw->p[i].set)
+			goto error;
+	}
+
+	return pw;
+error:
+	FN(PW,free)(pw);
+	return NULL;
+}
+#endif
+
+__isl_give PW *FN(PW,mul_isl_int)(__isl_take PW *pw, isl_int v)
+{
+	int i;
+
+	if (isl_int_is_one(v))
+		return pw;
+	if (pw && DEFAULT_IS_ZERO && isl_int_is_zero(v)) {
+		PW *zero;
+		isl_space *dim = FN(PW,get_space)(pw);
+#ifdef HAS_TYPE
+		zero = FN(PW,ZERO)(dim, pw->type);
+#else
+		zero = FN(PW,ZERO)(dim);
+#endif
+		FN(PW,free)(pw);
+		return zero;
+	}
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		return NULL;
+	if (pw->n == 0)
+		return pw;
+
+#ifdef HAS_TYPE
+	if (isl_int_is_neg(v))
+		pw->type = isl_fold_type_negate(pw->type);
+#endif
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].FIELD = FN(EL,scale)(pw->p[i].FIELD, v);
+		if (!pw->p[i].FIELD)
+			goto error;
+	}
+
+	return pw;
+error:
+	FN(PW,free)(pw);
+	return NULL;
+}
+
+/* Multiply the pieces of "pw" by "v" and return the result.
+ */
+__isl_give PW *FN(PW,scale_val)(__isl_take PW *pw, __isl_take isl_val *v)
+{
+	int i;
+
+	if (!pw || !v)
+		goto error;
+
+	if (isl_val_is_one(v)) {
+		isl_val_free(v);
+		return pw;
+	}
+	if (pw && DEFAULT_IS_ZERO && isl_val_is_zero(v)) {
+		PW *zero;
+		isl_space *space = FN(PW,get_space)(pw);
+#ifdef HAS_TYPE
+		zero = FN(PW,ZERO)(space, pw->type);
+#else
+		zero = FN(PW,ZERO)(space);
+#endif
+		FN(PW,free)(pw);
+		isl_val_free(v);
+		return zero;
+	}
+	if (pw->n == 0) {
+		isl_val_free(v);
+		return pw;
+	}
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		goto error;
+
+#ifdef HAS_TYPE
+	if (isl_val_is_neg(v))
+		pw->type = isl_fold_type_negate(pw->type);
+#endif
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].FIELD = FN(EL,scale_val)(pw->p[i].FIELD,
+						    isl_val_copy(v));
+		if (!pw->p[i].FIELD)
+			goto error;
+	}
+
+	isl_val_free(v);
+	return pw;
+error:
+	isl_val_free(v);
+	FN(PW,free)(pw);
+	return NULL;
+}
+
+/* Divide the pieces of "pw" by "v" and return the result.
+ */
+__isl_give PW *FN(PW,scale_down_val)(__isl_take PW *pw, __isl_take isl_val *v)
+{
+	int i;
+
+	if (!pw || !v)
+		goto error;
+
+	if (isl_val_is_one(v)) {
+		isl_val_free(v);
+		return pw;
+	}
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting rational factor", goto error);
+	if (isl_val_is_zero(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"cannot scale down by zero", goto error);
+
+	if (pw->n == 0) {
+		isl_val_free(v);
+		return pw;
+	}
+	pw = FN(PW,cow)(pw);
+	if (!pw)
+		goto error;
+
+#ifdef HAS_TYPE
+	if (isl_val_is_neg(v))
+		pw->type = isl_fold_type_negate(pw->type);
+#endif
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].FIELD = FN(EL,scale_down_val)(pw->p[i].FIELD,
+						    isl_val_copy(v));
+		if (!pw->p[i].FIELD)
+			goto error;
+	}
+
+	isl_val_free(v);
+	return pw;
+error:
+	isl_val_free(v);
+	FN(PW,free)(pw);
+	return NULL;
+}
+
+__isl_give PW *FN(PW,scale)(__isl_take PW *pw, isl_int v)
+{
+	return FN(PW,mul_isl_int)(pw, v);
+}
+
+static int FN(PW,qsort_set_cmp)(const void *p1, const void *p2)
+{
+	isl_set *set1 = *(isl_set * const *)p1;
+	isl_set *set2 = *(isl_set * const *)p2;
+
+	return isl_set_plain_cmp(set1, set2);
+}
+
+/* We normalize in place, but if anything goes wrong we need
+ * to return NULL, so we need to make sure we don't change the
+ * meaning of any possible other copies of map.
+ */
+__isl_give PW *FN(PW,normalize)(__isl_take PW *pw)
+{
+	int i, j;
+	isl_set *set;
+
+	if (!pw)
+		return NULL;
+	for (i = 0; i < pw->n; ++i) {
+		set = isl_set_normalize(isl_set_copy(pw->p[i].set));
+		if (!set)
+			return FN(PW,free)(pw);
+		isl_set_free(pw->p[i].set);
+		pw->p[i].set = set;
+	}
+	qsort(pw->p, pw->n, sizeof(pw->p[0]), &FN(PW,qsort_set_cmp));
+	for (i = pw->n - 1; i >= 1; --i) {
+		if (!isl_set_plain_is_equal(pw->p[i - 1].set, pw->p[i].set))
+			continue;
+		if (!FN(EL,plain_is_equal)(pw->p[i - 1].FIELD, pw->p[i].FIELD))
+			continue;
+		set = isl_set_union(isl_set_copy(pw->p[i - 1].set),
+				    isl_set_copy(pw->p[i].set));
+		if (!set)
+			return FN(PW,free)(pw);
+		isl_set_free(pw->p[i].set);
+		FN(EL,free)(pw->p[i].FIELD);
+		isl_set_free(pw->p[i - 1].set);
+		pw->p[i - 1].set = set;
+		for (j = i + 1; j < pw->n; ++j)
+			pw->p[j - 1] = pw->p[j];
+		pw->n--;
+	}
+
+	return pw;
+}
+
+/* Is pw1 obviously equal to pw2?
+ * That is, do they have obviously identical cells and obviously identical
+ * elements on each cell?
+ */
+int FN(PW,plain_is_equal)(__isl_keep PW *pw1, __isl_keep PW *pw2)
+{
+	int i;
+	int equal;
+
+	if (!pw1 || !pw2)
+		return -1;
+
+	if (pw1 == pw2)
+		return 1;
+	if (!isl_space_is_equal(pw1->dim, pw2->dim))
+		return 0;
+
+	pw1 = FN(PW,copy)(pw1);
+	pw2 = FN(PW,copy)(pw2);
+	pw1 = FN(PW,normalize)(pw1);
+	pw2 = FN(PW,normalize)(pw2);
+	if (!pw1 || !pw2)
+		goto error;
+
+	equal = pw1->n == pw2->n;
+	for (i = 0; equal && i < pw1->n; ++i) {
+		equal = isl_set_plain_is_equal(pw1->p[i].set, pw2->p[i].set);
+		if (equal < 0)
+			goto error;
+		if (!equal)
+			break;
+		equal = FN(EL,plain_is_equal)(pw1->p[i].FIELD, pw2->p[i].FIELD);
+		if (equal < 0)
+			goto error;
+	}
+
+	FN(PW,free)(pw1);
+	FN(PW,free)(pw2);
+	return equal;
+error:
+	FN(PW,free)(pw1);
+	FN(PW,free)(pw2);
+	return -1;
+}
+
+#ifndef NO_PULLBACK
+static __isl_give PW *FN(PW,align_params_pw_multi_aff_and)(__isl_take PW *pw,
+	__isl_take isl_multi_aff *ma,
+	__isl_give PW *(*fn)(__isl_take PW *pw, __isl_take isl_multi_aff *ma))
+{
+	isl_ctx *ctx;
+	isl_space *ma_space;
+
+	ma_space = isl_multi_aff_get_space(ma);
+	if (!pw || !ma || !ma_space)
+		goto error;
+	if (isl_space_match(pw->dim, isl_dim_param, ma_space, isl_dim_param)) {
+		isl_space_free(ma_space);
+		return fn(pw, ma);
+	}
+	ctx = FN(PW,get_ctx)(pw);
+	if (!isl_space_has_named_params(pw->dim) ||
+	    !isl_space_has_named_params(ma_space))
+		isl_die(ctx, isl_error_invalid,
+			"unaligned unnamed parameters", goto error);
+	pw = FN(PW,align_params)(pw, ma_space);
+	ma = isl_multi_aff_align_params(ma, FN(PW,get_space)(pw));
+	return fn(pw, ma);
+error:
+	isl_space_free(ma_space);
+	FN(PW,free)(pw);
+	isl_multi_aff_free(ma);
+	return NULL;
+}
+
+static __isl_give PW *FN(PW,align_params_pw_pw_multi_aff_and)(__isl_take PW *pw,
+	__isl_take isl_pw_multi_aff *pma,
+	__isl_give PW *(*fn)(__isl_take PW *pw,
+		__isl_take isl_pw_multi_aff *ma))
+{
+	isl_ctx *ctx;
+	isl_space *pma_space;
+
+	pma_space = isl_pw_multi_aff_get_space(pma);
+	if (!pw || !pma || !pma_space)
+		goto error;
+	if (isl_space_match(pw->dim, isl_dim_param, pma_space, isl_dim_param)) {
+		isl_space_free(pma_space);
+		return fn(pw, pma);
+	}
+	ctx = FN(PW,get_ctx)(pw);
+	if (!isl_space_has_named_params(pw->dim) ||
+	    !isl_space_has_named_params(pma_space))
+		isl_die(ctx, isl_error_invalid,
+			"unaligned unnamed parameters", goto error);
+	pw = FN(PW,align_params)(pw, pma_space);
+	pma = isl_pw_multi_aff_align_params(pma, FN(PW,get_space)(pw));
+	return fn(pw, pma);
+error:
+	isl_space_free(pma_space);
+	FN(PW,free)(pw);
+	isl_pw_multi_aff_free(pma);
+	return NULL;
+}
+
+/* Compute the pullback of "pw" by the function represented by "ma".
+ * In other words, plug in "ma" in "pw".
+ */
+static __isl_give PW *FN(PW,pullback_multi_aff_aligned)(__isl_take PW *pw,
+	__isl_take isl_multi_aff *ma)
+{
+	int i;
+	isl_space *space = NULL;
+
+	ma = isl_multi_aff_align_divs(ma);
+	pw = FN(PW,cow)(pw);
+	if (!pw || !ma)
+		goto error;
+
+	space = isl_space_join(isl_multi_aff_get_space(ma),
+				FN(PW,get_space)(pw));
+
+	for (i = 0; i < pw->n; ++i) {
+		pw->p[i].set = isl_set_preimage_multi_aff(pw->p[i].set,
+						    isl_multi_aff_copy(ma));
+		if (!pw->p[i].set)
+			goto error;
+		pw->p[i].FIELD = FN(EL,pullback_multi_aff)(pw->p[i].FIELD,
+						    isl_multi_aff_copy(ma));
+		if (!pw->p[i].FIELD)
+			goto error;
+	}
+
+	pw = FN(PW,reset_space)(pw, space);
+	isl_multi_aff_free(ma);
+	return pw;
+error:
+	isl_space_free(space);
+	isl_multi_aff_free(ma);
+	FN(PW,free)(pw);
+	return NULL;
+}
+
+__isl_give PW *FN(PW,pullback_multi_aff)(__isl_take PW *pw,
+	__isl_take isl_multi_aff *ma)
+{
+	return FN(PW,align_params_pw_multi_aff_and)(pw, ma,
+					&FN(PW,pullback_multi_aff_aligned));
+}
+
+/* Compute the pullback of "pw" by the function represented by "pma".
+ * In other words, plug in "pma" in "pw".
+ */
+static __isl_give PW *FN(PW,pullback_pw_multi_aff_aligned)(__isl_take PW *pw,
+	__isl_take isl_pw_multi_aff *pma)
+{
+	int i;
+	PW *res;
+
+	if (!pma)
+		goto error;
+
+	if (pma->n == 0) {
+		isl_space *space;
+		space = isl_space_join(isl_pw_multi_aff_get_space(pma),
+					FN(PW,get_space)(pw));
+		isl_pw_multi_aff_free(pma);
+		res = FN(PW,empty)(space);
+		FN(PW,free)(pw);
+		return res;
+	}
+
+	res = FN(PW,pullback_multi_aff)(FN(PW,copy)(pw),
+					isl_multi_aff_copy(pma->p[0].maff));
+	res = FN(PW,intersect_domain)(res, isl_set_copy(pma->p[0].set));
+
+	for (i = 1; i < pma->n; ++i) {
+		PW *res_i;
+
+		res_i = FN(PW,pullback_multi_aff)(FN(PW,copy)(pw),
+					isl_multi_aff_copy(pma->p[i].maff));
+		res_i = FN(PW,intersect_domain)(res_i,
+					isl_set_copy(pma->p[i].set));
+		res = FN(PW,add_disjoint)(res, res_i);
+	}
+
+	isl_pw_multi_aff_free(pma);
+	FN(PW,free)(pw);
+	return res;
+error:
+	isl_pw_multi_aff_free(pma);
+	FN(PW,free)(pw);
+	return NULL;
+}
+
+__isl_give PW *FN(PW,pullback_pw_multi_aff)(__isl_take PW *pw,
+	__isl_take isl_pw_multi_aff *pma)
+{
+	return FN(PW,align_params_pw_pw_multi_aff_and)(pw, pma,
+					&FN(PW,pullback_pw_multi_aff_aligned));
+}
+#endif
diff --git a/polly/lib/External/isl/isl_range.c b/polly/lib/External/isl/isl_range.c
new file mode 100644
index 00000000000..5599582cb83
--- /dev/null
+++ b/polly/lib/External/isl/isl_range.c
@@ -0,0 +1,491 @@
+#include <isl_ctx_private.h>
+#include <isl_constraint_private.h>
+#include <isl/set.h>
+#include <isl_polynomial_private.h>
+#include <isl_morph.h>
+#include <isl_range.h>
+
+struct range_data {
+	struct isl_bound	*bound;
+	int 		    	*signs;
+	int			sign;
+	int			test_monotonicity;
+	int		    	monotonicity;
+	int			tight;
+	isl_qpolynomial	    	*poly;
+	isl_pw_qpolynomial_fold *pwf;
+	isl_pw_qpolynomial_fold *pwf_tight;
+};
+
+static int propagate_on_domain(__isl_take isl_basic_set *bset,
+	__isl_take isl_qpolynomial *poly, struct range_data *data);
+
+/* Check whether the polynomial "poly" has sign "sign" over "bset",
+ * i.e., if sign == 1, check that the lower bound on the polynomial
+ * is non-negative and if sign == -1, check that the upper bound on
+ * the polynomial is non-positive.
+ */
+static int has_sign(__isl_keep isl_basic_set *bset,
+	__isl_keep isl_qpolynomial *poly, int sign, int *signs)
+{
+	struct range_data data_m;
+	unsigned nvar;
+	unsigned nparam;
+	isl_space *dim;
+	isl_val *opt;
+	int r;
+	enum isl_fold type;
+
+	nparam = isl_basic_set_dim(bset, isl_dim_param);
+	nvar = isl_basic_set_dim(bset, isl_dim_set);
+
+	bset = isl_basic_set_copy(bset);
+	poly = isl_qpolynomial_copy(poly);
+
+	bset = isl_basic_set_move_dims(bset, isl_dim_set, 0,
+					isl_dim_param, 0, nparam);
+	poly = isl_qpolynomial_move_dims(poly, isl_dim_in, 0,
+					isl_dim_param, 0, nparam);
+
+	dim = isl_qpolynomial_get_space(poly);
+	dim = isl_space_params(dim);
+	dim = isl_space_from_domain(dim);
+	dim = isl_space_add_dims(dim, isl_dim_out, 1);
+
+	data_m.test_monotonicity = 0;
+	data_m.signs = signs;
+	data_m.sign = -sign;
+	type = data_m.sign < 0 ? isl_fold_min : isl_fold_max;
+	data_m.pwf = isl_pw_qpolynomial_fold_zero(dim, type);
+	data_m.tight = 0;
+	data_m.pwf_tight = NULL;
+
+	if (propagate_on_domain(bset, poly, &data_m) < 0)
+		goto error;
+
+	if (sign > 0)
+		opt = isl_pw_qpolynomial_fold_min(data_m.pwf);
+	else
+		opt = isl_pw_qpolynomial_fold_max(data_m.pwf);
+
+	if (!opt)
+		r = -1;
+	else if (isl_val_is_nan(opt) ||
+		 isl_val_is_infty(opt) ||
+		 isl_val_is_neginfty(opt))
+		r = 0;
+	else
+		r = sign * isl_val_sgn(opt) >= 0;
+
+	isl_val_free(opt);
+
+	return r;
+error:
+	isl_pw_qpolynomial_fold_free(data_m.pwf);
+	return -1;
+}
+
+/* Return  1 if poly is monotonically increasing in the last set variable,
+ *        -1 if poly is monotonically decreasing in the last set variable,
+ *	   0 if no conclusion,
+ *	  -2 on error.
+ *
+ * We simply check the sign of p(x+1)-p(x)
+ */
+static int monotonicity(__isl_keep isl_basic_set *bset,
+	__isl_keep isl_qpolynomial *poly, struct range_data *data)
+{
+	isl_ctx *ctx;
+	isl_space *dim;
+	isl_qpolynomial *sub = NULL;
+	isl_qpolynomial *diff = NULL;
+	int result = 0;
+	int s;
+	unsigned nvar;
+
+	ctx = isl_qpolynomial_get_ctx(poly);
+	dim = isl_qpolynomial_get_domain_space(poly);
+
+	nvar = isl_basic_set_dim(bset, isl_dim_set);
+
+	sub = isl_qpolynomial_var_on_domain(isl_space_copy(dim), isl_dim_set, nvar - 1);
+	sub = isl_qpolynomial_add(sub,
+		isl_qpolynomial_rat_cst_on_domain(dim, ctx->one, ctx->one));
+
+	diff = isl_qpolynomial_substitute(isl_qpolynomial_copy(poly),
+			isl_dim_in, nvar - 1, 1, &sub);
+	diff = isl_qpolynomial_sub(diff, isl_qpolynomial_copy(poly));
+
+	s = has_sign(bset, diff, 1, data->signs);
+	if (s < 0)
+		goto error;
+	if (s)
+		result = 1;
+	else {
+		s = has_sign(bset, diff, -1, data->signs);
+		if (s < 0)
+			goto error;
+		if (s)
+			result = -1;
+	}
+
+	isl_qpolynomial_free(diff);
+	isl_qpolynomial_free(sub);
+
+	return result;
+error:
+	isl_qpolynomial_free(diff);
+	isl_qpolynomial_free(sub);
+	return -2;
+}
+
+static __isl_give isl_qpolynomial *bound2poly(__isl_take isl_constraint *bound,
+	__isl_take isl_space *dim, unsigned pos, int sign)
+{
+	if (!bound) {
+		if (sign > 0)
+			return isl_qpolynomial_infty_on_domain(dim);
+		else
+			return isl_qpolynomial_neginfty_on_domain(dim);
+	}
+	isl_space_free(dim);
+	return isl_qpolynomial_from_constraint(bound, isl_dim_set, pos);
+}
+
+static int bound_is_integer(__isl_take isl_constraint *bound, unsigned pos)
+{
+	isl_int c;
+	int is_int;
+
+	if (!bound)
+		return 1;
+
+	isl_int_init(c);
+	isl_constraint_get_coefficient(bound, isl_dim_set, pos, &c);
+	is_int = isl_int_is_one(c) || isl_int_is_negone(c);
+	isl_int_clear(c);
+
+	return is_int;
+}
+
+struct isl_fixed_sign_data {
+	int		*signs;
+	int		sign;
+	isl_qpolynomial	*poly;
+};
+
+/* Add term "term" to data->poly if it has sign data->sign.
+ * The sign is determined based on the signs of the parameters
+ * and variables in data->signs.  The integer divisions, if
+ * any, are assumed to be non-negative.
+ */
+static int collect_fixed_sign_terms(__isl_take isl_term *term, void *user)
+{
+	struct isl_fixed_sign_data *data = (struct isl_fixed_sign_data *)user;
+	isl_int n;
+	int i;
+	int sign;
+	unsigned nparam;
+	unsigned nvar;
+
+	if (!term)
+		return -1;
+
+	nparam = isl_term_dim(term, isl_dim_param);
+	nvar = isl_term_dim(term, isl_dim_set);
+
+	isl_int_init(n);
+
+	isl_term_get_num(term, &n);
+
+	sign = isl_int_sgn(n);
+	for (i = 0; i < nparam; ++i) {
+		if (data->signs[i] > 0)
+			continue;
+		if (isl_term_get_exp(term, isl_dim_param, i) % 2)
+			sign = -sign;
+	}
+	for (i = 0; i < nvar; ++i) {
+		if (data->signs[nparam + i] > 0)
+			continue;
+		if (isl_term_get_exp(term, isl_dim_set, i) % 2)
+			sign = -sign;
+	}
+
+	if (sign == data->sign) {
+		isl_qpolynomial *t = isl_qpolynomial_from_term(term);
+
+		data->poly = isl_qpolynomial_add(data->poly, t);
+	} else
+		isl_term_free(term);
+
+	isl_int_clear(n);
+
+	return 0;
+}
+
+/* Construct and return a polynomial that consists of the terms
+ * in "poly" that have sign "sign".  The integer divisions, if
+ * any, are assumed to be non-negative.
+ */
+__isl_give isl_qpolynomial *isl_qpolynomial_terms_of_sign(
+	__isl_keep isl_qpolynomial *poly, int *signs, int sign)
+{
+	isl_space *space;
+	struct isl_fixed_sign_data data = { signs, sign };
+
+	space = isl_qpolynomial_get_domain_space(poly);
+	data.poly = isl_qpolynomial_zero_on_domain(space);
+
+	if (isl_qpolynomial_foreach_term(poly, collect_fixed_sign_terms, &data) < 0)
+		goto error;
+
+	return data.poly;
+error:
+	isl_qpolynomial_free(data.poly);
+	return NULL;
+}
+
+/* Helper function to add a guarded polynomial to either pwf_tight or pwf,
+ * depending on whether the result has been determined to be tight.
+ */
+static int add_guarded_poly(__isl_take isl_basic_set *bset,
+	__isl_take isl_qpolynomial *poly, struct range_data *data)
+{
+	enum isl_fold type = data->sign < 0 ? isl_fold_min : isl_fold_max;
+	isl_set *set;
+	isl_qpolynomial_fold *fold;
+	isl_pw_qpolynomial_fold *pwf;
+
+	bset = isl_basic_set_params(bset);
+	poly = isl_qpolynomial_project_domain_on_params(poly);
+
+	fold = isl_qpolynomial_fold_alloc(type, poly);
+	set = isl_set_from_basic_set(bset);
+	pwf = isl_pw_qpolynomial_fold_alloc(type, set, fold);
+	if (data->tight)
+		data->pwf_tight = isl_pw_qpolynomial_fold_fold(
+						data->pwf_tight, pwf);
+	else
+		data->pwf = isl_pw_qpolynomial_fold_fold(data->pwf, pwf);
+
+	return 0;
+}
+
+/* Given a lower and upper bound on the final variable and constraints
+ * on the remaining variables where these bounds are active,
+ * eliminate the variable from data->poly based on these bounds.
+ * If the polynomial has been determined to be monotonic
+ * in the variable, then simply plug in the appropriate bound.
+ * If the current polynomial is tight and if this bound is integer,
+ * then the result is still tight.  In all other cases, the results
+ * may not be tight.
+ * Otherwise, plug in the largest bound (in absolute value) in
+ * the positive terms (if an upper bound is wanted) or the negative terms
+ * (if a lower bounded is wanted) and the other bound in the other terms.
+ *
+ * If all variables have been eliminated, then record the result.
+ * Ohterwise, recurse on the next variable.
+ */
+static int propagate_on_bound_pair(__isl_take isl_constraint *lower,
+	__isl_take isl_constraint *upper, __isl_take isl_basic_set *bset,
+	void *user)
+{
+	struct range_data *data = (struct range_data *)user;
+	int save_tight = data->tight;
+	isl_qpolynomial *poly;
+	int r;
+	unsigned nvar;
+
+	nvar = isl_basic_set_dim(bset, isl_dim_set);
+
+	if (data->monotonicity) {
+		isl_qpolynomial *sub;
+		isl_space *dim = isl_qpolynomial_get_domain_space(data->poly);
+		if (data->monotonicity * data->sign > 0) {
+			if (data->tight)
+				data->tight = bound_is_integer(upper, nvar);
+			sub = bound2poly(upper, dim, nvar, 1);
+			isl_constraint_free(lower);
+		} else {
+			if (data->tight)
+				data->tight = bound_is_integer(lower, nvar);
+			sub = bound2poly(lower, dim, nvar, -1);
+			isl_constraint_free(upper);
+		}
+		poly = isl_qpolynomial_copy(data->poly);
+		poly = isl_qpolynomial_substitute(poly, isl_dim_in, nvar, 1, &sub);
+		poly = isl_qpolynomial_drop_dims(poly, isl_dim_in, nvar, 1);
+
+		isl_qpolynomial_free(sub);
+	} else {
+		isl_qpolynomial *l, *u;
+		isl_qpolynomial *pos, *neg;
+		isl_space *dim = isl_qpolynomial_get_domain_space(data->poly);
+		unsigned nparam = isl_basic_set_dim(bset, isl_dim_param);
+		int sign = data->sign * data->signs[nparam + nvar];
+
+		data->tight = 0;
+
+		u = bound2poly(upper, isl_space_copy(dim), nvar, 1);
+		l = bound2poly(lower, dim, nvar, -1);
+
+		pos = isl_qpolynomial_terms_of_sign(data->poly, data->signs, sign);
+		neg = isl_qpolynomial_terms_of_sign(data->poly, data->signs, -sign);
+
+		pos = isl_qpolynomial_substitute(pos, isl_dim_in, nvar, 1, &u);
+		neg = isl_qpolynomial_substitute(neg, isl_dim_in, nvar, 1, &l);
+
+		poly = isl_qpolynomial_add(pos, neg);
+		poly = isl_qpolynomial_drop_dims(poly, isl_dim_in, nvar, 1);
+
+		isl_qpolynomial_free(u);
+		isl_qpolynomial_free(l);
+	}
+
+	if (isl_basic_set_dim(bset, isl_dim_set) == 0)
+		r = add_guarded_poly(bset, poly, data);
+	else
+		r = propagate_on_domain(bset, poly, data);
+
+	data->tight = save_tight;
+
+	return r;
+}
+
+/* Recursively perform range propagation on the polynomial "poly"
+ * defined over the basic set "bset" and collect the results in "data".
+ */
+static int propagate_on_domain(__isl_take isl_basic_set *bset,
+	__isl_take isl_qpolynomial *poly, struct range_data *data)
+{
+	isl_ctx *ctx;
+	isl_qpolynomial *save_poly = data->poly;
+	int save_monotonicity = data->monotonicity;
+	unsigned d;
+
+	if (!bset || !poly)
+		goto error;
+
+	ctx = isl_basic_set_get_ctx(bset);
+	d = isl_basic_set_dim(bset, isl_dim_set);
+	isl_assert(ctx, d >= 1, goto error);
+
+	if (isl_qpolynomial_is_cst(poly, NULL, NULL)) {
+		bset = isl_basic_set_project_out(bset, isl_dim_set, 0, d);
+		poly = isl_qpolynomial_drop_dims(poly, isl_dim_in, 0, d);
+		return add_guarded_poly(bset, poly, data);
+	}
+
+	if (data->test_monotonicity)
+		data->monotonicity = monotonicity(bset, poly, data);
+	else
+		data->monotonicity = 0;
+	if (data->monotonicity < -1)
+		goto error;
+
+	data->poly = poly;
+	if (isl_basic_set_foreach_bound_pair(bset, isl_dim_set, d - 1,
+					    &propagate_on_bound_pair, data) < 0)
+		goto error;
+
+	isl_basic_set_free(bset);
+	isl_qpolynomial_free(poly);
+	data->monotonicity = save_monotonicity;
+	data->poly = save_poly;
+
+	return 0;
+error:
+	isl_basic_set_free(bset);
+	isl_qpolynomial_free(poly);
+	data->monotonicity = save_monotonicity;
+	data->poly = save_poly;
+	return -1;
+}
+
+static int basic_guarded_poly_bound(__isl_take isl_basic_set *bset, void *user)
+{
+	struct range_data *data = (struct range_data *)user;
+	isl_ctx *ctx;
+	unsigned nparam = isl_basic_set_dim(bset, isl_dim_param);
+	unsigned dim = isl_basic_set_dim(bset, isl_dim_set);
+	int r;
+
+	data->signs = NULL;
+
+	ctx = isl_basic_set_get_ctx(bset);
+	data->signs = isl_alloc_array(ctx, int,
+					isl_basic_set_dim(bset, isl_dim_all));
+
+	if (isl_basic_set_dims_get_sign(bset, isl_dim_set, 0, dim,
+					data->signs + nparam) < 0)
+		goto error;
+	if (isl_basic_set_dims_get_sign(bset, isl_dim_param, 0, nparam,
+					data->signs) < 0)
+		goto error;
+
+	r = propagate_on_domain(bset, isl_qpolynomial_copy(data->poly), data);
+
+	free(data->signs);
+
+	return r;
+error:
+	free(data->signs);
+	isl_basic_set_free(bset);
+	return -1;
+}
+
+static int qpolynomial_bound_on_domain_range(__isl_take isl_basic_set *bset,
+	__isl_take isl_qpolynomial *poly, struct range_data *data)
+{
+	unsigned nparam = isl_basic_set_dim(bset, isl_dim_param);
+	unsigned nvar = isl_basic_set_dim(bset, isl_dim_set);
+	isl_set *set = NULL;
+
+	if (!bset)
+		goto error;
+
+	if (nvar == 0)
+		return add_guarded_poly(bset, poly, data);
+
+	set = isl_set_from_basic_set(bset);
+	set = isl_set_split_dims(set, isl_dim_param, 0, nparam);
+	set = isl_set_split_dims(set, isl_dim_set, 0, nvar);
+
+	data->poly = poly;
+
+	data->test_monotonicity = 1;
+	if (isl_set_foreach_basic_set(set, &basic_guarded_poly_bound, data) < 0)
+		goto error;
+
+	isl_set_free(set);
+	isl_qpolynomial_free(poly);
+
+	return 0;
+error:
+	isl_set_free(set);
+	isl_qpolynomial_free(poly);
+	return -1;
+}
+
+int isl_qpolynomial_bound_on_domain_range(__isl_take isl_basic_set *bset,
+	__isl_take isl_qpolynomial *poly, struct isl_bound *bound)
+{
+	struct range_data data;
+	int r;
+
+	data.pwf = bound->pwf;
+	data.pwf_tight = bound->pwf_tight;
+	data.tight = bound->check_tight;
+	if (bound->type == isl_fold_min)
+		data.sign = -1;
+	else
+		data.sign = 1;
+
+	r = qpolynomial_bound_on_domain_range(bset, poly, &data);
+
+	bound->pwf = data.pwf;
+	bound->pwf_tight = data.pwf_tight;
+
+	return r;
+}
diff --git a/polly/lib/External/isl/isl_range.h b/polly/lib/External/isl/isl_range.h
new file mode 100644
index 00000000000..c0970790d9e
--- /dev/null
+++ b/polly/lib/External/isl/isl_range.h
@@ -0,0 +1,6 @@
+#include <isl_bound.h>
+
+int isl_qpolynomial_bound_on_domain_range(__isl_take isl_basic_set *bset,
+	__isl_take isl_qpolynomial *poly, struct isl_bound *bound);
+__isl_give isl_qpolynomial *isl_qpolynomial_terms_of_sign(
+	__isl_keep isl_qpolynomial *poly, int *signs, int sign);
diff --git a/polly/lib/External/isl/isl_reordering.c b/polly/lib/External/isl/isl_reordering.c
new file mode 100644
index 00000000000..ca9c9737f76
--- /dev/null
+++ b/polly/lib/External/isl/isl_reordering.c
@@ -0,0 +1,205 @@
+/*
+ * Copyright 2010      INRIA Saclay
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_space_private.h>
+#include <isl_reordering.h>
+
+__isl_give isl_reordering *isl_reordering_alloc(isl_ctx *ctx, int len)
+{
+	isl_reordering *exp;
+
+	exp = isl_alloc(ctx, struct isl_reordering,
+			sizeof(struct isl_reordering) + (len - 1) * sizeof(int));
+	if (!exp)
+		return NULL;
+
+	exp->ref = 1;
+	exp->len = len;
+	exp->dim = NULL;
+
+	return exp;
+}
+
+__isl_give isl_reordering *isl_reordering_copy(__isl_keep isl_reordering *exp)
+{
+	if (!exp)
+		return NULL;
+
+	exp->ref++;
+	return exp;
+}
+
+__isl_give isl_reordering *isl_reordering_dup(__isl_keep isl_reordering *r)
+{
+	int i;
+	isl_reordering *dup;
+
+	if (!r)
+		return NULL;
+
+	dup = isl_reordering_alloc(r->dim->ctx, r->len);
+	if (!dup)
+		return NULL;
+
+	dup->dim = isl_space_copy(r->dim);
+	if (!dup->dim)
+		return isl_reordering_free(dup);
+	for (i = 0; i < dup->len; ++i)
+		dup->pos[i] = r->pos[i];
+
+	return dup;
+}
+
+__isl_give isl_reordering *isl_reordering_cow(__isl_take isl_reordering *r)
+{
+	if (!r)
+		return NULL;
+
+	if (r->ref == 1)
+		return r;
+	r->ref--;
+	return isl_reordering_dup(r);
+}
+
+void *isl_reordering_free(__isl_take isl_reordering *exp)
+{
+	if (!exp)
+		return NULL;
+
+	if (--exp->ref > 0)
+		return NULL;
+
+	isl_space_free(exp->dim);
+	free(exp);
+	return NULL;
+}
+
+/* Construct a reordering that maps the parameters of "alignee"
+ * to the corresponding parameters in a new dimension specification
+ * that has the parameters of "aligner" first, followed by
+ * any remaining parameters of "alignee" that do not occur in "aligner".
+ */
+__isl_give isl_reordering *isl_parameter_alignment_reordering(
+	__isl_keep isl_space *alignee, __isl_keep isl_space *aligner)
+{
+	int i, j;
+	isl_reordering *exp;
+
+	if (!alignee || !aligner)
+		return NULL;
+
+	exp = isl_reordering_alloc(alignee->ctx, alignee->nparam);
+	if (!exp)
+		return NULL;
+
+	exp->dim = isl_space_copy(aligner);
+
+	for (i = 0; i < alignee->nparam; ++i) {
+		isl_id *id_i;
+		id_i = isl_space_get_dim_id(alignee, isl_dim_param, i);
+		if (!id_i)
+			isl_die(alignee->ctx, isl_error_invalid,
+				"cannot align unnamed parameters", goto error);
+		for (j = 0; j < aligner->nparam; ++j) {
+			isl_id *id_j;
+			id_j = isl_space_get_dim_id(aligner, isl_dim_param, j);
+			isl_id_free(id_j);
+			if (id_i == id_j)
+				break;
+		}
+		if (j < aligner->nparam) {
+			exp->pos[i] = j;
+			isl_id_free(id_i);
+		} else {
+			int pos;
+			pos = isl_space_dim(exp->dim, isl_dim_param);
+			exp->dim = isl_space_add_dims(exp->dim, isl_dim_param, 1);
+			exp->dim = isl_space_set_dim_id(exp->dim,
+						isl_dim_param, pos, id_i);
+			exp->pos[i] = pos;
+		}
+	}
+
+	if (!exp->dim)
+		return isl_reordering_free(exp);
+	return exp;
+error:
+	isl_reordering_free(exp);
+	return NULL;
+}
+
+__isl_give isl_reordering *isl_reordering_extend(__isl_take isl_reordering *exp,
+	unsigned extra)
+{
+	int i;
+	isl_reordering *res;
+	int offset;
+
+	if (!exp)
+		return NULL;
+	if (extra == 0)
+		return exp;
+
+	offset = isl_space_dim(exp->dim, isl_dim_all) - exp->len;
+	res = isl_reordering_alloc(exp->dim->ctx, exp->len + extra);
+	if (!res)
+		goto error;
+	res->dim = isl_space_copy(exp->dim);
+	for (i = 0; i < exp->len; ++i)
+		res->pos[i] = exp->pos[i];
+	for (i = exp->len; i < res->len; ++i)
+		res->pos[i] = offset + i;
+
+	isl_reordering_free(exp);
+
+	return res;
+error:
+	isl_reordering_free(exp);
+	return NULL;
+}
+
+__isl_give isl_reordering *isl_reordering_extend_space(
+	__isl_take isl_reordering *exp, __isl_take isl_space *dim)
+{
+	isl_reordering *res;
+
+	if (!exp || !dim)
+		goto error;
+
+	res = isl_reordering_extend(isl_reordering_copy(exp),
+				    isl_space_dim(dim, isl_dim_all) - exp->len);
+	res = isl_reordering_cow(res);
+	if (!res)
+		goto error;
+	isl_space_free(res->dim);
+	res->dim = isl_space_replace(dim, isl_dim_param, exp->dim);
+
+	isl_reordering_free(exp);
+
+	if (!res->dim)
+		return isl_reordering_free(res);
+
+	return res;
+error:
+	isl_reordering_free(exp);
+	isl_space_free(dim);
+	return NULL;
+}
+
+void isl_reordering_dump(__isl_keep isl_reordering *exp)
+{
+	int i;
+
+	isl_space_dump(exp->dim);
+	for (i = 0; i < exp->len; ++i)
+		fprintf(stderr, "%d -> %d; ", i, exp->pos[i]);
+	fprintf(stderr, "\n");
+}
diff --git a/polly/lib/External/isl/isl_reordering.h b/polly/lib/External/isl/isl_reordering.h
new file mode 100644
index 00000000000..ac5b6facaba
--- /dev/null
+++ b/polly/lib/External/isl/isl_reordering.h
@@ -0,0 +1,31 @@
+#ifndef ISL_REORDERING_H
+#define ISL_REORDERING_H
+
+#include <isl/space.h>
+
+/* pos maps original dimensions to new dimensions.
+ * The final dimension is given by dim.
+ * The number of dimensions (i.e., the range of values) in the result
+ * may be larger than the number of dimensions in the input.
+ * In particular, the possible values of the entries in pos ranges from 0 to
+ * the total dimension of dim - 1, unless isl_reordering_extend
+ * has been called.
+ */
+struct isl_reordering {
+	int ref;
+	isl_space *dim;
+	unsigned len;
+	int pos[1];
+};
+typedef struct isl_reordering isl_reordering;
+
+__isl_give isl_reordering *isl_parameter_alignment_reordering(
+	__isl_keep isl_space *alignee, __isl_keep isl_space *aligner);
+__isl_give isl_reordering *isl_reordering_copy(__isl_keep isl_reordering *exp);
+void *isl_reordering_free(__isl_take isl_reordering *exp);
+__isl_give isl_reordering *isl_reordering_extend_space(
+	__isl_take isl_reordering *exp, __isl_take isl_space *dim);
+__isl_give isl_reordering *isl_reordering_extend(__isl_take isl_reordering *exp,
+	unsigned extra);
+
+#endif
diff --git a/polly/lib/External/isl/isl_sample.c b/polly/lib/External/isl/isl_sample.c
new file mode 100644
index 00000000000..eaa07543140
--- /dev/null
+++ b/polly/lib/External/isl/isl_sample.c
@@ -0,0 +1,1301 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include "isl_sample.h"
+#include <isl/vec.h>
+#include <isl/mat.h>
+#include <isl_seq.h>
+#include "isl_equalities.h"
+#include "isl_tab.h"
+#include "isl_basis_reduction.h"
+#include <isl_factorization.h>
+#include <isl_point_private.h>
+#include <isl_options_private.h>
+#include <isl_vec_private.h>
+
+static struct isl_vec *empty_sample(struct isl_basic_set *bset)
+{
+	struct isl_vec *vec;
+
+	vec = isl_vec_alloc(bset->ctx, 0);
+	isl_basic_set_free(bset);
+	return vec;
+}
+
+/* Construct a zero sample of the same dimension as bset.
+ * As a special case, if bset is zero-dimensional, this
+ * function creates a zero-dimensional sample point.
+ */
+static struct isl_vec *zero_sample(struct isl_basic_set *bset)
+{
+	unsigned dim;
+	struct isl_vec *sample;
+
+	dim = isl_basic_set_total_dim(bset);
+	sample = isl_vec_alloc(bset->ctx, 1 + dim);
+	if (sample) {
+		isl_int_set_si(sample->el[0], 1);
+		isl_seq_clr(sample->el + 1, dim);
+	}
+	isl_basic_set_free(bset);
+	return sample;
+}
+
+static struct isl_vec *interval_sample(struct isl_basic_set *bset)
+{
+	int i;
+	isl_int t;
+	struct isl_vec *sample;
+
+	bset = isl_basic_set_simplify(bset);
+	if (!bset)
+		return NULL;
+	if (isl_basic_set_plain_is_empty(bset))
+		return empty_sample(bset);
+	if (bset->n_eq == 0 && bset->n_ineq == 0)
+		return zero_sample(bset);
+
+	sample = isl_vec_alloc(bset->ctx, 2);
+	if (!sample)
+		goto error;
+	if (!bset)
+		return NULL;
+	isl_int_set_si(sample->block.data[0], 1);
+
+	if (bset->n_eq > 0) {
+		isl_assert(bset->ctx, bset->n_eq == 1, goto error);
+		isl_assert(bset->ctx, bset->n_ineq == 0, goto error);
+		if (isl_int_is_one(bset->eq[0][1]))
+			isl_int_neg(sample->el[1], bset->eq[0][0]);
+		else {
+			isl_assert(bset->ctx, isl_int_is_negone(bset->eq[0][1]),
+				   goto error);
+			isl_int_set(sample->el[1], bset->eq[0][0]);
+		}
+		isl_basic_set_free(bset);
+		return sample;
+	}
+
+	isl_int_init(t);
+	if (isl_int_is_one(bset->ineq[0][1]))
+		isl_int_neg(sample->block.data[1], bset->ineq[0][0]);
+	else
+		isl_int_set(sample->block.data[1], bset->ineq[0][0]);
+	for (i = 1; i < bset->n_ineq; ++i) {
+		isl_seq_inner_product(sample->block.data,
+					bset->ineq[i], 2, &t);
+		if (isl_int_is_neg(t))
+			break;
+	}
+	isl_int_clear(t);
+	if (i < bset->n_ineq) {
+		isl_vec_free(sample);
+		return empty_sample(bset);
+	}
+
+	isl_basic_set_free(bset);
+	return sample;
+error:
+	isl_basic_set_free(bset);
+	isl_vec_free(sample);
+	return NULL;
+}
+
+/* Find a sample integer point, if any, in bset, which is known
+ * to have equalities.  If bset contains no integer points, then
+ * return a zero-length vector.
+ * We simply remove the known equalities, compute a sample
+ * in the resulting bset, using the specified recurse function,
+ * and then transform the sample back to the original space.
+ */
+static struct isl_vec *sample_eq(struct isl_basic_set *bset,
+	struct isl_vec *(*recurse)(struct isl_basic_set *))
+{
+	struct isl_mat *T;
+	struct isl_vec *sample;
+
+	if (!bset)
+		return NULL;
+
+	bset = isl_basic_set_remove_equalities(bset, &T, NULL);
+	sample = recurse(bset);
+	if (!sample || sample->size == 0)
+		isl_mat_free(T);
+	else
+		sample = isl_mat_vec_product(T, sample);
+	return sample;
+}
+
+/* Return a matrix containing the equalities of the tableau
+ * in constraint form.  The tableau is assumed to have
+ * an associated bset that has been kept up-to-date.
+ */
+static struct isl_mat *tab_equalities(struct isl_tab *tab)
+{
+	int i, j;
+	int n_eq;
+	struct isl_mat *eq;
+	struct isl_basic_set *bset;
+
+	if (!tab)
+		return NULL;
+
+	bset = isl_tab_peek_bset(tab);
+	isl_assert(tab->mat->ctx, bset, return NULL);
+
+	n_eq = tab->n_var - tab->n_col + tab->n_dead;
+	if (tab->empty || n_eq == 0)
+		return isl_mat_alloc(tab->mat->ctx, 0, tab->n_var);
+	if (n_eq == tab->n_var)
+		return isl_mat_identity(tab->mat->ctx, tab->n_var);
+
+	eq = isl_mat_alloc(tab->mat->ctx, n_eq, tab->n_var);
+	if (!eq)
+		return NULL;
+	for (i = 0, j = 0; i < tab->n_con; ++i) {
+		if (tab->con[i].is_row)
+			continue;
+		if (tab->con[i].index >= 0 && tab->con[i].index >= tab->n_dead)
+			continue;
+		if (i < bset->n_eq)
+			isl_seq_cpy(eq->row[j], bset->eq[i] + 1, tab->n_var);
+		else
+			isl_seq_cpy(eq->row[j],
+				    bset->ineq[i - bset->n_eq] + 1, tab->n_var);
+		++j;
+	}
+	isl_assert(bset->ctx, j == n_eq, goto error);
+	return eq;
+error:
+	isl_mat_free(eq);
+	return NULL;
+}
+
+/* Compute and return an initial basis for the bounded tableau "tab".
+ *
+ * If the tableau is either full-dimensional or zero-dimensional,
+ * the we simply return an identity matrix.
+ * Otherwise, we construct a basis whose first directions correspond
+ * to equalities.
+ */
+static struct isl_mat *initial_basis(struct isl_tab *tab)
+{
+	int n_eq;
+	struct isl_mat *eq;
+	struct isl_mat *Q;
+
+	tab->n_unbounded = 0;
+	tab->n_zero = n_eq = tab->n_var - tab->n_col + tab->n_dead;
+	if (tab->empty || n_eq == 0 || n_eq == tab->n_var)
+		return isl_mat_identity(tab->mat->ctx, 1 + tab->n_var);
+
+	eq = tab_equalities(tab);
+	eq = isl_mat_left_hermite(eq, 0, NULL, &Q);
+	if (!eq)
+		return NULL;
+	isl_mat_free(eq);
+
+	Q = isl_mat_lin_to_aff(Q);
+	return Q;
+}
+
+/* Compute the minimum of the current ("level") basis row over "tab"
+ * and store the result in position "level" of "min".
+ */
+static enum isl_lp_result compute_min(isl_ctx *ctx, struct isl_tab *tab,
+	__isl_keep isl_vec *min, int level)
+{
+	return isl_tab_min(tab, tab->basis->row[1 + level],
+			    ctx->one, &min->el[level], NULL, 0);
+}
+
+/* Compute the maximum of the current ("level") basis row over "tab"
+ * and store the result in position "level" of "max".
+ */
+static enum isl_lp_result compute_max(isl_ctx *ctx, struct isl_tab *tab,
+	__isl_keep isl_vec *max, int level)
+{
+	enum isl_lp_result res;
+	unsigned dim = tab->n_var;
+
+	isl_seq_neg(tab->basis->row[1 + level] + 1,
+		    tab->basis->row[1 + level] + 1, dim);
+	res = isl_tab_min(tab, tab->basis->row[1 + level],
+		    ctx->one, &max->el[level], NULL, 0);
+	isl_seq_neg(tab->basis->row[1 + level] + 1,
+		    tab->basis->row[1 + level] + 1, dim);
+	isl_int_neg(max->el[level], max->el[level]);
+
+	return res;
+}
+
+/* Perform a greedy search for an integer point in the set represented
+ * by "tab", given that the minimal rational value (rounded up to the
+ * nearest integer) at "level" is smaller than the maximal rational
+ * value (rounded down to the nearest integer).
+ *
+ * Return 1 if we have found an integer point (if tab->n_unbounded > 0
+ * then we may have only found integer values for the bounded dimensions
+ * and it is the responsibility of the caller to extend this solution
+ * to the unbounded dimensions).
+ * Return 0 if greedy search did not result in a solution.
+ * Return -1 if some error occurred.
+ *
+ * We assign a value half-way between the minimum and the maximum
+ * to the current dimension and check if the minimal value of the
+ * next dimension is still smaller than (or equal) to the maximal value.
+ * We continue this process until either
+ * - the minimal value (rounded up) is greater than the maximal value
+ *	(rounded down).  In this case, greedy search has failed.
+ * - we have exhausted all bounded dimensions, meaning that we have
+ *	found a solution.
+ * - the sample value of the tableau is integral.
+ * - some error has occurred.
+ */
+static int greedy_search(isl_ctx *ctx, struct isl_tab *tab,
+	__isl_keep isl_vec *min, __isl_keep isl_vec *max, int level)
+{
+	struct isl_tab_undo *snap;
+	enum isl_lp_result res;
+
+	snap = isl_tab_snap(tab);
+
+	do {
+		isl_int_add(tab->basis->row[1 + level][0],
+			    min->el[level], max->el[level]);
+		isl_int_fdiv_q_ui(tab->basis->row[1 + level][0],
+			    tab->basis->row[1 + level][0], 2);
+		isl_int_neg(tab->basis->row[1 + level][0],
+			    tab->basis->row[1 + level][0]);
+		if (isl_tab_add_valid_eq(tab, tab->basis->row[1 + level]) < 0)
+			return -1;
+		isl_int_set_si(tab->basis->row[1 + level][0], 0);
+
+		if (++level >= tab->n_var - tab->n_unbounded)
+			return 1;
+		if (isl_tab_sample_is_integer(tab))
+			return 1;
+
+		res = compute_min(ctx, tab, min, level);
+		if (res == isl_lp_error)
+			return -1;
+		if (res != isl_lp_ok)
+			isl_die(ctx, isl_error_internal,
+				"expecting bounded rational solution",
+				return -1);
+		res = compute_max(ctx, tab, max, level);
+		if (res == isl_lp_error)
+			return -1;
+		if (res != isl_lp_ok)
+			isl_die(ctx, isl_error_internal,
+				"expecting bounded rational solution",
+				return -1);
+	} while (isl_int_le(min->el[level], max->el[level]));
+
+	if (isl_tab_rollback(tab, snap) < 0)
+		return -1;
+
+	return 0;
+}
+
+/* Given a tableau representing a set, find and return
+ * an integer point in the set, if there is any.
+ *
+ * We perform a depth first search
+ * for an integer point, by scanning all possible values in the range
+ * attained by a basis vector, where an initial basis may have been set
+ * by the calling function.  Otherwise an initial basis that exploits
+ * the equalities in the tableau is created.
+ * tab->n_zero is currently ignored and is clobbered by this function.
+ *
+ * The tableau is allowed to have unbounded direction, but then
+ * the calling function needs to set an initial basis, with the
+ * unbounded directions last and with tab->n_unbounded set
+ * to the number of unbounded directions.
+ * Furthermore, the calling functions needs to add shifted copies
+ * of all constraints involving unbounded directions to ensure
+ * that any feasible rational value in these directions can be rounded
+ * up to yield a feasible integer value.
+ * In particular, let B define the given basis x' = B x
+ * and let T be the inverse of B, i.e., X = T x'.
+ * Let a x + c >= 0 be a constraint of the set represented by the tableau,
+ * or a T x' + c >= 0 in terms of the given basis.  Assume that
+ * the bounded directions have an integer value, then we can safely
+ * round up the values for the unbounded directions if we make sure
+ * that x' not only satisfies the original constraint, but also
+ * the constraint "a T x' + c + s >= 0" with s the sum of all
+ * negative values in the last n_unbounded entries of "a T".
+ * The calling function therefore needs to add the constraint
+ * a x + c + s >= 0.  The current function then scans the first
+ * directions for an integer value and once those have been found,
+ * it can compute "T ceil(B x)" to yield an integer point in the set.
+ * Note that during the search, the first rows of B may be changed
+ * by a basis reduction, but the last n_unbounded rows of B remain
+ * unaltered and are also not mixed into the first rows.
+ *
+ * The search is implemented iteratively.  "level" identifies the current
+ * basis vector.  "init" is true if we want the first value at the current
+ * level and false if we want the next value.
+ *
+ * At the start of each level, we first check if we can find a solution
+ * using greedy search.  If not, we continue with the exhaustive search.
+ *
+ * The initial basis is the identity matrix.  If the range in some direction
+ * contains more than one integer value, we perform basis reduction based
+ * on the value of ctx->opt->gbr
+ *	- ISL_GBR_NEVER:	never perform basis reduction
+ *	- ISL_GBR_ONCE:		only perform basis reduction the first
+ *				time such a range is encountered
+ *	- ISL_GBR_ALWAYS:	always perform basis reduction when
+ *				such a range is encountered
+ *
+ * When ctx->opt->gbr is set to ISL_GBR_ALWAYS, then we allow the basis
+ * reduction computation to return early.  That is, as soon as it
+ * finds a reasonable first direction.
+ */ 
+struct isl_vec *isl_tab_sample(struct isl_tab *tab)
+{
+	unsigned dim;
+	unsigned gbr;
+	struct isl_ctx *ctx;
+	struct isl_vec *sample;
+	struct isl_vec *min;
+	struct isl_vec *max;
+	enum isl_lp_result res;
+	int level;
+	int init;
+	int reduced;
+	struct isl_tab_undo **snap;
+
+	if (!tab)
+		return NULL;
+	if (tab->empty)
+		return isl_vec_alloc(tab->mat->ctx, 0);
+
+	if (!tab->basis)
+		tab->basis = initial_basis(tab);
+	if (!tab->basis)
+		return NULL;
+	isl_assert(tab->mat->ctx, tab->basis->n_row == tab->n_var + 1,
+		    return NULL);
+	isl_assert(tab->mat->ctx, tab->basis->n_col == tab->n_var + 1,
+		    return NULL);
+
+	ctx = tab->mat->ctx;
+	dim = tab->n_var;
+	gbr = ctx->opt->gbr;
+
+	if (tab->n_unbounded == tab->n_var) {
+		sample = isl_tab_get_sample_value(tab);
+		sample = isl_mat_vec_product(isl_mat_copy(tab->basis), sample);
+		sample = isl_vec_ceil(sample);
+		sample = isl_mat_vec_inverse_product(isl_mat_copy(tab->basis),
+							sample);
+		return sample;
+	}
+
+	if (isl_tab_extend_cons(tab, dim + 1) < 0)
+		return NULL;
+
+	min = isl_vec_alloc(ctx, dim);
+	max = isl_vec_alloc(ctx, dim);
+	snap = isl_alloc_array(ctx, struct isl_tab_undo *, dim);
+
+	if (!min || !max || !snap)
+		goto error;
+
+	level = 0;
+	init = 1;
+	reduced = 0;
+
+	while (level >= 0) {
+		if (init) {
+			int choice;
+
+			res = compute_min(ctx, tab, min, level);
+			if (res == isl_lp_error)
+				goto error;
+			if (res != isl_lp_ok)
+				isl_die(ctx, isl_error_internal,
+					"expecting bounded rational solution",
+					goto error);
+			if (isl_tab_sample_is_integer(tab))
+				break;
+			res = compute_max(ctx, tab, max, level);
+			if (res == isl_lp_error)
+				goto error;
+			if (res != isl_lp_ok)
+				isl_die(ctx, isl_error_internal,
+					"expecting bounded rational solution",
+					goto error);
+			if (isl_tab_sample_is_integer(tab))
+				break;
+			choice = isl_int_lt(min->el[level], max->el[level]);
+			if (choice) {
+				int g;
+				g = greedy_search(ctx, tab, min, max, level);
+				if (g < 0)
+					goto error;
+				if (g)
+					break;
+			}
+			if (!reduced && choice &&
+			    ctx->opt->gbr != ISL_GBR_NEVER) {
+				unsigned gbr_only_first;
+				if (ctx->opt->gbr == ISL_GBR_ONCE)
+					ctx->opt->gbr = ISL_GBR_NEVER;
+				tab->n_zero = level;
+				gbr_only_first = ctx->opt->gbr_only_first;
+				ctx->opt->gbr_only_first =
+					ctx->opt->gbr == ISL_GBR_ALWAYS;
+				tab = isl_tab_compute_reduced_basis(tab);
+				ctx->opt->gbr_only_first = gbr_only_first;
+				if (!tab || !tab->basis)
+					goto error;
+				reduced = 1;
+				continue;
+			}
+			reduced = 0;
+			snap[level] = isl_tab_snap(tab);
+		} else
+			isl_int_add_ui(min->el[level], min->el[level], 1);
+
+		if (isl_int_gt(min->el[level], max->el[level])) {
+			level--;
+			init = 0;
+			if (level >= 0)
+				if (isl_tab_rollback(tab, snap[level]) < 0)
+					goto error;
+			continue;
+		}
+		isl_int_neg(tab->basis->row[1 + level][0], min->el[level]);
+		if (isl_tab_add_valid_eq(tab, tab->basis->row[1 + level]) < 0)
+			goto error;
+		isl_int_set_si(tab->basis->row[1 + level][0], 0);
+		if (level + tab->n_unbounded < dim - 1) {
+			++level;
+			init = 1;
+			continue;
+		}
+		break;
+	}
+
+	if (level >= 0) {
+		sample = isl_tab_get_sample_value(tab);
+		if (!sample)
+			goto error;
+		if (tab->n_unbounded && !isl_int_is_one(sample->el[0])) {
+			sample = isl_mat_vec_product(isl_mat_copy(tab->basis),
+						     sample);
+			sample = isl_vec_ceil(sample);
+			sample = isl_mat_vec_inverse_product(
+					isl_mat_copy(tab->basis), sample);
+		}
+	} else
+		sample = isl_vec_alloc(ctx, 0);
+
+	ctx->opt->gbr = gbr;
+	isl_vec_free(min);
+	isl_vec_free(max);
+	free(snap);
+	return sample;
+error:
+	ctx->opt->gbr = gbr;
+	isl_vec_free(min);
+	isl_vec_free(max);
+	free(snap);
+	return NULL;
+}
+
+static struct isl_vec *sample_bounded(struct isl_basic_set *bset);
+
+/* Compute a sample point of the given basic set, based on the given,
+ * non-trivial factorization.
+ */
+static __isl_give isl_vec *factored_sample(__isl_take isl_basic_set *bset,
+	__isl_take isl_factorizer *f)
+{
+	int i, n;
+	isl_vec *sample = NULL;
+	isl_ctx *ctx;
+	unsigned nparam;
+	unsigned nvar;
+
+	ctx = isl_basic_set_get_ctx(bset);
+	if (!ctx)
+		goto error;
+
+	nparam = isl_basic_set_dim(bset, isl_dim_param);
+	nvar = isl_basic_set_dim(bset, isl_dim_set);
+
+	sample = isl_vec_alloc(ctx, 1 + isl_basic_set_total_dim(bset));
+	if (!sample)
+		goto error;
+	isl_int_set_si(sample->el[0], 1);
+
+	bset = isl_morph_basic_set(isl_morph_copy(f->morph), bset);
+
+	for (i = 0, n = 0; i < f->n_group; ++i) {
+		isl_basic_set *bset_i;
+		isl_vec *sample_i;
+
+		bset_i = isl_basic_set_copy(bset);
+		bset_i = isl_basic_set_drop_constraints_involving(bset_i,
+			    nparam + n + f->len[i], nvar - n - f->len[i]);
+		bset_i = isl_basic_set_drop_constraints_involving(bset_i,
+			    nparam, n);
+		bset_i = isl_basic_set_drop(bset_i, isl_dim_set,
+			    n + f->len[i], nvar - n - f->len[i]);
+		bset_i = isl_basic_set_drop(bset_i, isl_dim_set, 0, n);
+
+		sample_i = sample_bounded(bset_i);
+		if (!sample_i)
+			goto error;
+		if (sample_i->size == 0) {
+			isl_basic_set_free(bset);
+			isl_factorizer_free(f);
+			isl_vec_free(sample);
+			return sample_i;
+		}
+		isl_seq_cpy(sample->el + 1 + nparam + n,
+			    sample_i->el + 1, f->len[i]);
+		isl_vec_free(sample_i);
+
+		n += f->len[i];
+	}
+
+	f->morph = isl_morph_inverse(f->morph);
+	sample = isl_morph_vec(isl_morph_copy(f->morph), sample);
+
+	isl_basic_set_free(bset);
+	isl_factorizer_free(f);
+	return sample;
+error:
+	isl_basic_set_free(bset);
+	isl_factorizer_free(f);
+	isl_vec_free(sample);
+	return NULL;
+}
+
+/* Given a basic set that is known to be bounded, find and return
+ * an integer point in the basic set, if there is any.
+ *
+ * After handling some trivial cases, we construct a tableau
+ * and then use isl_tab_sample to find a sample, passing it
+ * the identity matrix as initial basis.
+ */ 
+static struct isl_vec *sample_bounded(struct isl_basic_set *bset)
+{
+	unsigned dim;
+	struct isl_ctx *ctx;
+	struct isl_vec *sample;
+	struct isl_tab *tab = NULL;
+	isl_factorizer *f;
+
+	if (!bset)
+		return NULL;
+
+	if (isl_basic_set_plain_is_empty(bset))
+		return empty_sample(bset);
+
+	dim = isl_basic_set_total_dim(bset);
+	if (dim == 0)
+		return zero_sample(bset);
+	if (dim == 1)
+		return interval_sample(bset);
+	if (bset->n_eq > 0)
+		return sample_eq(bset, sample_bounded);
+
+	f = isl_basic_set_factorizer(bset);
+	if (!f)
+		goto error;
+	if (f->n_group != 0)
+		return factored_sample(bset, f);
+	isl_factorizer_free(f);
+		
+	ctx = bset->ctx;
+
+	tab = isl_tab_from_basic_set(bset, 1);
+	if (tab && tab->empty) {
+		isl_tab_free(tab);
+		ISL_F_SET(bset, ISL_BASIC_SET_EMPTY);
+		sample = isl_vec_alloc(bset->ctx, 0);
+		isl_basic_set_free(bset);
+		return sample;
+	}
+
+	if (!ISL_F_ISSET(bset, ISL_BASIC_SET_NO_IMPLICIT))
+		if (isl_tab_detect_implicit_equalities(tab) < 0)
+			goto error;
+
+	sample = isl_tab_sample(tab);
+	if (!sample)
+		goto error;
+
+	if (sample->size > 0) {
+		isl_vec_free(bset->sample);
+		bset->sample = isl_vec_copy(sample);
+	}
+
+	isl_basic_set_free(bset);
+	isl_tab_free(tab);
+	return sample;
+error:
+	isl_basic_set_free(bset);
+	isl_tab_free(tab);
+	return NULL;
+}
+
+/* Given a basic set "bset" and a value "sample" for the first coordinates
+ * of bset, plug in these values and drop the corresponding coordinates.
+ *
+ * We do this by computing the preimage of the transformation
+ *
+ *	     [ 1 0 ]
+ *	x =  [ s 0 ] x'
+ *	     [ 0 I ]
+ *
+ * where [1 s] is the sample value and I is the identity matrix of the
+ * appropriate dimension.
+ */
+static struct isl_basic_set *plug_in(struct isl_basic_set *bset,
+	struct isl_vec *sample)
+{
+	int i;
+	unsigned total;
+	struct isl_mat *T;
+
+	if (!bset || !sample)
+		goto error;
+
+	total = isl_basic_set_total_dim(bset);
+	T = isl_mat_alloc(bset->ctx, 1 + total, 1 + total - (sample->size - 1));
+	if (!T)
+		goto error;
+
+	for (i = 0; i < sample->size; ++i) {
+		isl_int_set(T->row[i][0], sample->el[i]);
+		isl_seq_clr(T->row[i] + 1, T->n_col - 1);
+	}
+	for (i = 0; i < T->n_col - 1; ++i) {
+		isl_seq_clr(T->row[sample->size + i], T->n_col);
+		isl_int_set_si(T->row[sample->size + i][1 + i], 1);
+	}
+	isl_vec_free(sample);
+
+	bset = isl_basic_set_preimage(bset, T);
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	isl_vec_free(sample);
+	return NULL;
+}
+
+/* Given a basic set "bset", return any (possibly non-integer) point
+ * in the basic set.
+ */
+static struct isl_vec *rational_sample(struct isl_basic_set *bset)
+{
+	struct isl_tab *tab;
+	struct isl_vec *sample;
+
+	if (!bset)
+		return NULL;
+
+	tab = isl_tab_from_basic_set(bset, 0);
+	sample = isl_tab_get_sample_value(tab);
+	isl_tab_free(tab);
+
+	isl_basic_set_free(bset);
+
+	return sample;
+}
+
+/* Given a linear cone "cone" and a rational point "vec",
+ * construct a polyhedron with shifted copies of the constraints in "cone",
+ * i.e., a polyhedron with "cone" as its recession cone, such that each
+ * point x in this polyhedron is such that the unit box positioned at x
+ * lies entirely inside the affine cone 'vec + cone'.
+ * Any rational point in this polyhedron may therefore be rounded up
+ * to yield an integer point that lies inside said affine cone.
+ *
+ * Denote the constraints of cone by "<a_i, x> >= 0" and the rational
+ * point "vec" by v/d.
+ * Let b_i = <a_i, v>.  Then the affine cone 'vec + cone' is given
+ * by <a_i, x> - b/d >= 0.
+ * The polyhedron <a_i, x> - ceil{b/d} >= 0 is a subset of this affine cone.
+ * We prefer this polyhedron over the actual affine cone because it doesn't
+ * require a scaling of the constraints.
+ * If each of the vertices of the unit cube positioned at x lies inside
+ * this polyhedron, then the whole unit cube at x lies inside the affine cone.
+ * We therefore impose that x' = x + \sum e_i, for any selection of unit
+ * vectors lies inside the polyhedron, i.e.,
+ *
+ *	<a_i, x'> - ceil{b/d} = <a_i, x> + sum a_i - ceil{b/d} >= 0
+ *
+ * The most stringent of these constraints is the one that selects
+ * all negative a_i, so the polyhedron we are looking for has constraints
+ *
+ *	<a_i, x> + sum_{a_i < 0} a_i - ceil{b/d} >= 0
+ *
+ * Note that if cone were known to have only non-negative rays
+ * (which can be accomplished by a unimodular transformation),
+ * then we would only have to check the points x' = x + e_i
+ * and we only have to add the smallest negative a_i (if any)
+ * instead of the sum of all negative a_i.
+ */
+static struct isl_basic_set *shift_cone(struct isl_basic_set *cone,
+	struct isl_vec *vec)
+{
+	int i, j, k;
+	unsigned total;
+
+	struct isl_basic_set *shift = NULL;
+
+	if (!cone || !vec)
+		goto error;
+
+	isl_assert(cone->ctx, cone->n_eq == 0, goto error);
+
+	total = isl_basic_set_total_dim(cone);
+
+	shift = isl_basic_set_alloc_space(isl_basic_set_get_space(cone),
+					0, 0, cone->n_ineq);
+
+	for (i = 0; i < cone->n_ineq; ++i) {
+		k = isl_basic_set_alloc_inequality(shift);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(shift->ineq[k] + 1, cone->ineq[i] + 1, total);
+		isl_seq_inner_product(shift->ineq[k] + 1, vec->el + 1, total,
+				      &shift->ineq[k][0]);
+		isl_int_cdiv_q(shift->ineq[k][0],
+			       shift->ineq[k][0], vec->el[0]);
+		isl_int_neg(shift->ineq[k][0], shift->ineq[k][0]);
+		for (j = 0; j < total; ++j) {
+			if (isl_int_is_nonneg(shift->ineq[k][1 + j]))
+				continue;
+			isl_int_add(shift->ineq[k][0],
+				    shift->ineq[k][0], shift->ineq[k][1 + j]);
+		}
+	}
+
+	isl_basic_set_free(cone);
+	isl_vec_free(vec);
+
+	return isl_basic_set_finalize(shift);
+error:
+	isl_basic_set_free(shift);
+	isl_basic_set_free(cone);
+	isl_vec_free(vec);
+	return NULL;
+}
+
+/* Given a rational point vec in a (transformed) basic set,
+ * such that cone is the recession cone of the original basic set,
+ * "round up" the rational point to an integer point.
+ *
+ * We first check if the rational point just happens to be integer.
+ * If not, we transform the cone in the same way as the basic set,
+ * pick a point x in this cone shifted to the rational point such that
+ * the whole unit cube at x is also inside this affine cone.
+ * Then we simply round up the coordinates of x and return the
+ * resulting integer point.
+ */
+static struct isl_vec *round_up_in_cone(struct isl_vec *vec,
+	struct isl_basic_set *cone, struct isl_mat *U)
+{
+	unsigned total;
+
+	if (!vec || !cone || !U)
+		goto error;
+
+	isl_assert(vec->ctx, vec->size != 0, goto error);
+	if (isl_int_is_one(vec->el[0])) {
+		isl_mat_free(U);
+		isl_basic_set_free(cone);
+		return vec;
+	}
+
+	total = isl_basic_set_total_dim(cone);
+	cone = isl_basic_set_preimage(cone, U);
+	cone = isl_basic_set_remove_dims(cone, isl_dim_set,
+					 0, total - (vec->size - 1));
+
+	cone = shift_cone(cone, vec);
+
+	vec = rational_sample(cone);
+	vec = isl_vec_ceil(vec);
+	return vec;
+error:
+	isl_mat_free(U);
+	isl_vec_free(vec);
+	isl_basic_set_free(cone);
+	return NULL;
+}
+
+/* Concatenate two integer vectors, i.e., two vectors with denominator
+ * (stored in element 0) equal to 1.
+ */
+static struct isl_vec *vec_concat(struct isl_vec *vec1, struct isl_vec *vec2)
+{
+	struct isl_vec *vec;
+
+	if (!vec1 || !vec2)
+		goto error;
+	isl_assert(vec1->ctx, vec1->size > 0, goto error);
+	isl_assert(vec2->ctx, vec2->size > 0, goto error);
+	isl_assert(vec1->ctx, isl_int_is_one(vec1->el[0]), goto error);
+	isl_assert(vec2->ctx, isl_int_is_one(vec2->el[0]), goto error);
+
+	vec = isl_vec_alloc(vec1->ctx, vec1->size + vec2->size - 1);
+	if (!vec)
+		goto error;
+
+	isl_seq_cpy(vec->el, vec1->el, vec1->size);
+	isl_seq_cpy(vec->el + vec1->size, vec2->el + 1, vec2->size - 1);
+
+	isl_vec_free(vec1);
+	isl_vec_free(vec2);
+
+	return vec;
+error:
+	isl_vec_free(vec1);
+	isl_vec_free(vec2);
+	return NULL;
+}
+
+/* Give a basic set "bset" with recession cone "cone", compute and
+ * return an integer point in bset, if any.
+ *
+ * If the recession cone is full-dimensional, then we know that
+ * bset contains an infinite number of integer points and it is
+ * fairly easy to pick one of them.
+ * If the recession cone is not full-dimensional, then we first
+ * transform bset such that the bounded directions appear as
+ * the first dimensions of the transformed basic set.
+ * We do this by using a unimodular transformation that transforms
+ * the equalities in the recession cone to equalities on the first
+ * dimensions.
+ *
+ * The transformed set is then projected onto its bounded dimensions.
+ * Note that to compute this projection, we can simply drop all constraints
+ * involving any of the unbounded dimensions since these constraints
+ * cannot be combined to produce a constraint on the bounded dimensions.
+ * To see this, assume that there is such a combination of constraints
+ * that produces a constraint on the bounded dimensions.  This means
+ * that some combination of the unbounded dimensions has both an upper
+ * bound and a lower bound in terms of the bounded dimensions, but then
+ * this combination would be a bounded direction too and would have been
+ * transformed into a bounded dimensions.
+ *
+ * We then compute a sample value in the bounded dimensions.
+ * If no such value can be found, then the original set did not contain
+ * any integer points and we are done.
+ * Otherwise, we plug in the value we found in the bounded dimensions,
+ * project out these bounded dimensions and end up with a set with
+ * a full-dimensional recession cone.
+ * A sample point in this set is computed by "rounding up" any
+ * rational point in the set.
+ *
+ * The sample points in the bounded and unbounded dimensions are
+ * then combined into a single sample point and transformed back
+ * to the original space.
+ */
+__isl_give isl_vec *isl_basic_set_sample_with_cone(
+	__isl_take isl_basic_set *bset, __isl_take isl_basic_set *cone)
+{
+	unsigned total;
+	unsigned cone_dim;
+	struct isl_mat *M, *U;
+	struct isl_vec *sample;
+	struct isl_vec *cone_sample;
+	struct isl_ctx *ctx;
+	struct isl_basic_set *bounded;
+
+	if (!bset || !cone)
+		goto error;
+
+	ctx = bset->ctx;
+	total = isl_basic_set_total_dim(cone);
+	cone_dim = total - cone->n_eq;
+
+	M = isl_mat_sub_alloc6(bset->ctx, cone->eq, 0, cone->n_eq, 1, total);
+	M = isl_mat_left_hermite(M, 0, &U, NULL);
+	if (!M)
+		goto error;
+	isl_mat_free(M);
+
+	U = isl_mat_lin_to_aff(U);
+	bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
+
+	bounded = isl_basic_set_copy(bset);
+	bounded = isl_basic_set_drop_constraints_involving(bounded,
+						   total - cone_dim, cone_dim);
+	bounded = isl_basic_set_drop_dims(bounded, total - cone_dim, cone_dim);
+	sample = sample_bounded(bounded);
+	if (!sample || sample->size == 0) {
+		isl_basic_set_free(bset);
+		isl_basic_set_free(cone);
+		isl_mat_free(U);
+		return sample;
+	}
+	bset = plug_in(bset, isl_vec_copy(sample));
+	cone_sample = rational_sample(bset);
+	cone_sample = round_up_in_cone(cone_sample, cone, isl_mat_copy(U));
+	sample = vec_concat(sample, cone_sample);
+	sample = isl_mat_vec_product(U, sample);
+	return sample;
+error:
+	isl_basic_set_free(cone);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+static void vec_sum_of_neg(struct isl_vec *v, isl_int *s)
+{
+	int i;
+
+	isl_int_set_si(*s, 0);
+
+	for (i = 0; i < v->size; ++i)
+		if (isl_int_is_neg(v->el[i]))
+			isl_int_add(*s, *s, v->el[i]);
+}
+
+/* Given a tableau "tab", a tableau "tab_cone" that corresponds
+ * to the recession cone and the inverse of a new basis U = inv(B),
+ * with the unbounded directions in B last,
+ * add constraints to "tab" that ensure any rational value
+ * in the unbounded directions can be rounded up to an integer value.
+ *
+ * The new basis is given by x' = B x, i.e., x = U x'.
+ * For any rational value of the last tab->n_unbounded coordinates
+ * in the update tableau, the value that is obtained by rounding
+ * up this value should be contained in the original tableau.
+ * For any constraint "a x + c >= 0", we therefore need to add
+ * a constraint "a x + c + s >= 0", with s the sum of all negative
+ * entries in the last elements of "a U".
+ *
+ * Since we are not interested in the first entries of any of the "a U",
+ * we first drop the columns of U that correpond to bounded directions.
+ */
+static int tab_shift_cone(struct isl_tab *tab,
+	struct isl_tab *tab_cone, struct isl_mat *U)
+{
+	int i;
+	isl_int v;
+	struct isl_basic_set *bset = NULL;
+
+	if (tab && tab->n_unbounded == 0) {
+		isl_mat_free(U);
+		return 0;
+	}
+	isl_int_init(v);
+	if (!tab || !tab_cone || !U)
+		goto error;
+	bset = isl_tab_peek_bset(tab_cone);
+	U = isl_mat_drop_cols(U, 0, tab->n_var - tab->n_unbounded);
+	for (i = 0; i < bset->n_ineq; ++i) {
+		int ok;
+		struct isl_vec *row = NULL;
+		if (isl_tab_is_equality(tab_cone, tab_cone->n_eq + i))
+			continue;
+		row = isl_vec_alloc(bset->ctx, tab_cone->n_var);
+		if (!row)
+			goto error;
+		isl_seq_cpy(row->el, bset->ineq[i] + 1, tab_cone->n_var);
+		row = isl_vec_mat_product(row, isl_mat_copy(U));
+		if (!row)
+			goto error;
+		vec_sum_of_neg(row, &v);
+		isl_vec_free(row);
+		if (isl_int_is_zero(v))
+			continue;
+		if (isl_tab_extend_cons(tab, 1) < 0)
+			goto error;
+		isl_int_add(bset->ineq[i][0], bset->ineq[i][0], v);
+		ok = isl_tab_add_ineq(tab, bset->ineq[i]) >= 0;
+		isl_int_sub(bset->ineq[i][0], bset->ineq[i][0], v);
+		if (!ok)
+			goto error;
+	}
+
+	isl_mat_free(U);
+	isl_int_clear(v);
+	return 0;
+error:
+	isl_mat_free(U);
+	isl_int_clear(v);
+	return -1;
+}
+
+/* Compute and return an initial basis for the possibly
+ * unbounded tableau "tab".  "tab_cone" is a tableau
+ * for the corresponding recession cone.
+ * Additionally, add constraints to "tab" that ensure
+ * that any rational value for the unbounded directions
+ * can be rounded up to an integer value.
+ *
+ * If the tableau is bounded, i.e., if the recession cone
+ * is zero-dimensional, then we just use inital_basis.
+ * Otherwise, we construct a basis whose first directions
+ * correspond to equalities, followed by bounded directions,
+ * i.e., equalities in the recession cone.
+ * The remaining directions are then unbounded.
+ */
+int isl_tab_set_initial_basis_with_cone(struct isl_tab *tab,
+	struct isl_tab *tab_cone)
+{
+	struct isl_mat *eq;
+	struct isl_mat *cone_eq;
+	struct isl_mat *U, *Q;
+
+	if (!tab || !tab_cone)
+		return -1;
+
+	if (tab_cone->n_col == tab_cone->n_dead) {
+		tab->basis = initial_basis(tab);
+		return tab->basis ? 0 : -1;
+	}
+
+	eq = tab_equalities(tab);
+	if (!eq)
+		return -1;
+	tab->n_zero = eq->n_row;
+	cone_eq = tab_equalities(tab_cone);
+	eq = isl_mat_concat(eq, cone_eq);
+	if (!eq)
+		return -1;
+	tab->n_unbounded = tab->n_var - (eq->n_row - tab->n_zero);
+	eq = isl_mat_left_hermite(eq, 0, &U, &Q);
+	if (!eq)
+		return -1;
+	isl_mat_free(eq);
+	tab->basis = isl_mat_lin_to_aff(Q);
+	if (tab_shift_cone(tab, tab_cone, U) < 0)
+		return -1;
+	if (!tab->basis)
+		return -1;
+	return 0;
+}
+
+/* Compute and return a sample point in bset using generalized basis
+ * reduction.  We first check if the input set has a non-trivial
+ * recession cone.  If so, we perform some extra preprocessing in
+ * sample_with_cone.  Otherwise, we directly perform generalized basis
+ * reduction.
+ */
+static struct isl_vec *gbr_sample(struct isl_basic_set *bset)
+{
+	unsigned dim;
+	struct isl_basic_set *cone;
+
+	dim = isl_basic_set_total_dim(bset);
+
+	cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));
+	if (!cone)
+		goto error;
+
+	if (cone->n_eq < dim)
+		return isl_basic_set_sample_with_cone(bset, cone);
+
+	isl_basic_set_free(cone);
+	return sample_bounded(bset);
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+static struct isl_vec *basic_set_sample(struct isl_basic_set *bset, int bounded)
+{
+	struct isl_ctx *ctx;
+	unsigned dim;
+	if (!bset)
+		return NULL;
+
+	ctx = bset->ctx;
+	if (isl_basic_set_plain_is_empty(bset))
+		return empty_sample(bset);
+
+	dim = isl_basic_set_n_dim(bset);
+	isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
+	isl_assert(ctx, bset->n_div == 0, goto error);
+
+	if (bset->sample && bset->sample->size == 1 + dim) {
+		int contains = isl_basic_set_contains(bset, bset->sample);
+		if (contains < 0)
+			goto error;
+		if (contains) {
+			struct isl_vec *sample = isl_vec_copy(bset->sample);
+			isl_basic_set_free(bset);
+			return sample;
+		}
+	}
+	isl_vec_free(bset->sample);
+	bset->sample = NULL;
+
+	if (bset->n_eq > 0)
+		return sample_eq(bset, bounded ? isl_basic_set_sample_bounded
+					       : isl_basic_set_sample_vec);
+	if (dim == 0)
+		return zero_sample(bset);
+	if (dim == 1)
+		return interval_sample(bset);
+
+	return bounded ? sample_bounded(bset) : gbr_sample(bset);
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+__isl_give isl_vec *isl_basic_set_sample_vec(__isl_take isl_basic_set *bset)
+{
+	return basic_set_sample(bset, 0);
+}
+
+/* Compute an integer sample in "bset", where the caller guarantees
+ * that "bset" is bounded.
+ */
+struct isl_vec *isl_basic_set_sample_bounded(struct isl_basic_set *bset)
+{
+	return basic_set_sample(bset, 1);
+}
+
+__isl_give isl_basic_set *isl_basic_set_from_vec(__isl_take isl_vec *vec)
+{
+	int i;
+	int k;
+	struct isl_basic_set *bset = NULL;
+	struct isl_ctx *ctx;
+	unsigned dim;
+
+	if (!vec)
+		return NULL;
+	ctx = vec->ctx;
+	isl_assert(ctx, vec->size != 0, goto error);
+
+	bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
+	if (!bset)
+		goto error;
+	dim = isl_basic_set_n_dim(bset);
+	for (i = dim - 1; i >= 0; --i) {
+		k = isl_basic_set_alloc_equality(bset);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(bset->eq[k], 1 + dim);
+		isl_int_neg(bset->eq[k][0], vec->el[1 + i]);
+		isl_int_set(bset->eq[k][1 + i], vec->el[0]);
+	}
+	bset->sample = vec;
+
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	isl_vec_free(vec);
+	return NULL;
+}
+
+__isl_give isl_basic_map *isl_basic_map_sample(__isl_take isl_basic_map *bmap)
+{
+	struct isl_basic_set *bset;
+	struct isl_vec *sample_vec;
+
+	bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
+	sample_vec = isl_basic_set_sample_vec(bset);
+	if (!sample_vec)
+		goto error;
+	if (sample_vec->size == 0) {
+		struct isl_basic_map *sample;
+		sample = isl_basic_map_empty_like(bmap);
+		isl_vec_free(sample_vec);
+		isl_basic_map_free(bmap);
+		return sample;
+	}
+	bset = isl_basic_set_from_vec(sample_vec);
+	return isl_basic_map_overlying_set(bset, bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_basic_set_sample(__isl_take isl_basic_set *bset)
+{
+	return isl_basic_map_sample(bset);
+}
+
+__isl_give isl_basic_map *isl_map_sample(__isl_take isl_map *map)
+{
+	int i;
+	isl_basic_map *sample = NULL;
+
+	if (!map)
+		goto error;
+
+	for (i = 0; i < map->n; ++i) {
+		sample = isl_basic_map_sample(isl_basic_map_copy(map->p[i]));
+		if (!sample)
+			goto error;
+		if (!ISL_F_ISSET(sample, ISL_BASIC_MAP_EMPTY))
+			break;
+		isl_basic_map_free(sample);
+	}
+	if (i == map->n)
+		sample = isl_basic_map_empty_like_map(map);
+	isl_map_free(map);
+	return sample;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_set_sample(__isl_take isl_set *set)
+{
+	return (isl_basic_set *) isl_map_sample((isl_map *)set);
+}
+
+__isl_give isl_point *isl_basic_set_sample_point(__isl_take isl_basic_set *bset)
+{
+	isl_vec *vec;
+	isl_space *dim;
+
+	dim = isl_basic_set_get_space(bset);
+	bset = isl_basic_set_underlying_set(bset);
+	vec = isl_basic_set_sample_vec(bset);
+
+	return isl_point_alloc(dim, vec);
+}
+
+__isl_give isl_point *isl_set_sample_point(__isl_take isl_set *set)
+{
+	int i;
+	isl_point *pnt;
+
+	if (!set)
+		return NULL;
+
+	for (i = 0; i < set->n; ++i) {
+		pnt = isl_basic_set_sample_point(isl_basic_set_copy(set->p[i]));
+		if (!pnt)
+			goto error;
+		if (!isl_point_is_void(pnt))
+			break;
+		isl_point_free(pnt);
+	}
+	if (i == set->n)
+		pnt = isl_point_void(isl_set_get_space(set));
+
+	isl_set_free(set);
+	return pnt;
+error:
+	isl_set_free(set);
+	return NULL;
+}
diff --git a/polly/lib/External/isl/isl_sample.h b/polly/lib/External/isl/isl_sample.h
new file mode 100644
index 00000000000..57898b25ce0
--- /dev/null
+++ b/polly/lib/External/isl/isl_sample.h
@@ -0,0 +1,35 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_SAMPLE_H
+#define ISL_SAMPLE_H
+
+#include <isl/set.h>
+#include <isl_tab.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+__isl_give isl_vec *isl_basic_set_sample_vec(__isl_take isl_basic_set *bset);
+struct isl_vec *isl_basic_set_sample_bounded(struct isl_basic_set *bset);
+__isl_give isl_vec *isl_basic_set_sample_with_cone(
+	__isl_take isl_basic_set *bset, __isl_take isl_basic_set *cone);
+
+__isl_give isl_basic_set *isl_basic_set_from_vec(__isl_take isl_vec *vec);
+
+int isl_tab_set_initial_basis_with_cone(struct isl_tab *tab,
+	struct isl_tab *tab_cone);
+struct isl_vec *isl_tab_sample(struct isl_tab *tab);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/isl_scan.c b/polly/lib/External/isl/isl_scan.c
new file mode 100644
index 00000000000..eb8423c6e35
--- /dev/null
+++ b/polly/lib/External/isl/isl_scan.c
@@ -0,0 +1,324 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include "isl_basis_reduction.h"
+#include "isl_scan.h"
+#include <isl_seq.h>
+#include "isl_tab.h"
+#include <isl_val_private.h>
+#include <isl_vec_private.h>
+
+struct isl_counter {
+	struct isl_scan_callback callback;
+	isl_int count;
+	isl_int max;
+};
+
+static int increment_counter(struct isl_scan_callback *cb,
+	__isl_take isl_vec *sample)
+{
+	struct isl_counter *cnt = (struct isl_counter *)cb;
+
+	isl_int_add_ui(cnt->count, cnt->count, 1);
+
+	isl_vec_free(sample);
+
+	if (isl_int_is_zero(cnt->max) || isl_int_lt(cnt->count, cnt->max))
+		return 0;
+	return -1;
+}
+
+static int increment_range(struct isl_scan_callback *cb, isl_int min, isl_int max)
+{
+	struct isl_counter *cnt = (struct isl_counter *)cb;
+
+	isl_int_add(cnt->count, cnt->count, max);
+	isl_int_sub(cnt->count, cnt->count, min);
+	isl_int_add_ui(cnt->count, cnt->count, 1);
+
+	if (isl_int_is_zero(cnt->max) || isl_int_lt(cnt->count, cnt->max))
+		return 0;
+	isl_int_set(cnt->count, cnt->max);
+	return -1;
+}
+
+/* Call callback->add with the current sample value of the tableau "tab".
+ */
+static int add_solution(struct isl_tab *tab, struct isl_scan_callback *callback)
+{
+	struct isl_vec *sample;
+
+	if (!tab)
+		return -1;
+	sample = isl_tab_get_sample_value(tab);
+	if (!sample)
+		return -1;
+
+	return callback->add(callback, sample);
+}
+
+static int scan_0D(struct isl_basic_set *bset,
+	struct isl_scan_callback *callback)
+{
+	struct isl_vec *sample;
+
+	sample = isl_vec_alloc(bset->ctx, 1);
+	isl_basic_set_free(bset);
+
+	if (!sample)
+		return -1;
+
+	isl_int_set_si(sample->el[0], 1);
+
+	return callback->add(callback, sample);
+}
+
+/* Look for all integer points in "bset", which is assumed to be bounded,
+ * and call callback->add on each of them.
+ *
+ * We first compute a reduced basis for the set and then scan
+ * the set in the directions of this basis.
+ * We basically perform a depth first search, where in each level i
+ * we compute the range in the i-th basis vector direction, given
+ * fixed values in the directions of the previous basis vector.
+ * We then add an equality to the tableau fixing the value in the
+ * direction of the current basis vector to each value in the range
+ * in turn and then continue to the next level.
+ *
+ * The search is implemented iteratively.  "level" identifies the current
+ * basis vector.  "init" is true if we want the first value at the current
+ * level and false if we want the next value.
+ * Solutions are added in the leaves of the search tree, i.e., after
+ * we have fixed a value in each direction of the basis.
+ */
+int isl_basic_set_scan(struct isl_basic_set *bset,
+	struct isl_scan_callback *callback)
+{
+	unsigned dim;
+	struct isl_mat *B = NULL;
+	struct isl_tab *tab = NULL;
+	struct isl_vec *min;
+	struct isl_vec *max;
+	struct isl_tab_undo **snap;
+	int level;
+	int init;
+	enum isl_lp_result res;
+
+	if (!bset)
+		return -1;
+
+	dim = isl_basic_set_total_dim(bset);
+	if (dim == 0)
+		return scan_0D(bset, callback);
+
+	min = isl_vec_alloc(bset->ctx, dim);
+	max = isl_vec_alloc(bset->ctx, dim);
+	snap = isl_alloc_array(bset->ctx, struct isl_tab_undo *, dim);
+
+	if (!min || !max || !snap)
+		goto error;
+
+	tab = isl_tab_from_basic_set(bset, 0);
+	if (!tab)
+		goto error;
+	if (isl_tab_extend_cons(tab, dim + 1) < 0)
+		goto error;
+
+	tab->basis = isl_mat_identity(bset->ctx, 1 + dim);
+	if (1)
+		tab = isl_tab_compute_reduced_basis(tab);
+	if (!tab)
+		goto error;
+	B = isl_mat_copy(tab->basis);
+	if (!B)
+		goto error;
+
+	level = 0;
+	init = 1;
+
+	while (level >= 0) {
+		int empty = 0;
+		if (init) {
+			res = isl_tab_min(tab, B->row[1 + level],
+				    bset->ctx->one, &min->el[level], NULL, 0);
+			if (res == isl_lp_empty)
+				empty = 1;
+			if (res == isl_lp_error || res == isl_lp_unbounded)
+				goto error;
+			isl_seq_neg(B->row[1 + level] + 1,
+				    B->row[1 + level] + 1, dim);
+			res = isl_tab_min(tab, B->row[1 + level],
+				    bset->ctx->one, &max->el[level], NULL, 0);
+			isl_seq_neg(B->row[1 + level] + 1,
+				    B->row[1 + level] + 1, dim);
+			isl_int_neg(max->el[level], max->el[level]);
+			if (res == isl_lp_empty)
+				empty = 1;
+			if (res == isl_lp_error || res == isl_lp_unbounded)
+				goto error;
+			snap[level] = isl_tab_snap(tab);
+		} else
+			isl_int_add_ui(min->el[level], min->el[level], 1);
+
+		if (empty || isl_int_gt(min->el[level], max->el[level])) {
+			level--;
+			init = 0;
+			if (level >= 0)
+				if (isl_tab_rollback(tab, snap[level]) < 0)
+					goto error;
+			continue;
+		}
+		if (level == dim - 1 && callback->add == increment_counter) {
+			if (increment_range(callback,
+					    min->el[level], max->el[level]))
+				goto error;
+			level--;
+			init = 0;
+			if (level >= 0)
+				if (isl_tab_rollback(tab, snap[level]) < 0)
+					goto error;
+			continue;
+		}
+		isl_int_neg(B->row[1 + level][0], min->el[level]);
+		if (isl_tab_add_valid_eq(tab, B->row[1 + level]) < 0)
+			goto error;
+		isl_int_set_si(B->row[1 + level][0], 0);
+		if (level < dim - 1) {
+			++level;
+			init = 1;
+			continue;
+		}
+		if (add_solution(tab, callback) < 0)
+			goto error;
+		init = 0;
+		if (isl_tab_rollback(tab, snap[level]) < 0)
+			goto error;
+	}
+
+	isl_tab_free(tab);
+	free(snap);
+	isl_vec_free(min);
+	isl_vec_free(max);
+	isl_basic_set_free(bset);
+	isl_mat_free(B);
+	return 0;
+error:
+	isl_tab_free(tab);
+	free(snap);
+	isl_vec_free(min);
+	isl_vec_free(max);
+	isl_basic_set_free(bset);
+	isl_mat_free(B);
+	return -1;
+}
+
+int isl_set_scan(__isl_take isl_set *set, struct isl_scan_callback *callback)
+{
+	int i;
+
+	if (!set || !callback)
+		goto error;
+
+	set = isl_set_cow(set);
+	set = isl_set_make_disjoint(set);
+	set = isl_set_compute_divs(set);
+	if (!set)
+		goto error;
+
+	for (i = 0; i < set->n; ++i)
+		if (isl_basic_set_scan(isl_basic_set_copy(set->p[i]),
+					callback) < 0)
+			goto error;
+
+	isl_set_free(set);
+	return 0;
+error:
+	isl_set_free(set);
+	return -1;
+}
+
+int isl_basic_set_count_upto(__isl_keep isl_basic_set *bset,
+	isl_int max, isl_int *count)
+{
+	struct isl_counter cnt = { { &increment_counter } };
+
+	if (!bset)
+		return -1;
+
+	isl_int_init(cnt.count);
+	isl_int_init(cnt.max);
+
+	isl_int_set_si(cnt.count, 0);
+	isl_int_set(cnt.max, max);
+	if (isl_basic_set_scan(isl_basic_set_copy(bset), &cnt.callback) < 0 &&
+	    isl_int_lt(cnt.count, cnt.max))
+		goto error;
+
+	isl_int_set(*count, cnt.count);
+	isl_int_clear(cnt.max);
+	isl_int_clear(cnt.count);
+
+	return 0;
+error:
+	isl_int_clear(cnt.count);
+	return -1;
+}
+
+int isl_set_count_upto(__isl_keep isl_set *set, isl_int max, isl_int *count)
+{
+	struct isl_counter cnt = { { &increment_counter } };
+
+	if (!set)
+		return -1;
+
+	isl_int_init(cnt.count);
+	isl_int_init(cnt.max);
+
+	isl_int_set_si(cnt.count, 0);
+	isl_int_set(cnt.max, max);
+	if (isl_set_scan(isl_set_copy(set), &cnt.callback) < 0 &&
+	    isl_int_lt(cnt.count, cnt.max))
+		goto error;
+
+	isl_int_set(*count, cnt.count);
+	isl_int_clear(cnt.max);
+	isl_int_clear(cnt.count);
+
+	return 0;
+error:
+	isl_int_clear(cnt.count);
+	return -1;
+}
+
+int isl_set_count(__isl_keep isl_set *set, isl_int *count)
+{
+	if (!set)
+		return -1;
+	return isl_set_count_upto(set, set->ctx->zero, count);
+}
+
+/* Count the total number of elements in "set" (in an inefficient way) and
+ * return the result.
+ */
+__isl_give isl_val *isl_set_count_val(__isl_keep isl_set *set)
+{
+	isl_val *v;
+
+	if (!set)
+		return NULL;
+	v = isl_val_zero(isl_set_get_ctx(set));
+	v = isl_val_cow(v);
+	if (!v)
+		return NULL;
+	if (isl_set_count(set, &v->n) < 0)
+		v = isl_val_free(v);
+	return v;
+}
diff --git a/polly/lib/External/isl/isl_scan.h b/polly/lib/External/isl/isl_scan.h
new file mode 100644
index 00000000000..d186df4e53f
--- /dev/null
+++ b/polly/lib/External/isl/isl_scan.h
@@ -0,0 +1,24 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_SCAN_H
+#define ISL_SCAN_H
+
+#include <isl/set.h>
+#include <isl/vec.h>
+
+struct isl_scan_callback {
+	int (*add)(struct isl_scan_callback *cb, __isl_take isl_vec *sample);
+};
+
+int isl_basic_set_scan(struct isl_basic_set *bset,
+	struct isl_scan_callback *callback);
+int isl_set_scan(__isl_take isl_set *set, struct isl_scan_callback *callback);
+
+#endif
diff --git a/polly/lib/External/isl/isl_schedule.c b/polly/lib/External/isl/isl_schedule.c
new file mode 100644
index 00000000000..1ad98d5df4d
--- /dev/null
+++ b/polly/lib/External/isl/isl_schedule.c
@@ -0,0 +1,672 @@
+/*
+ * Copyright 2011      INRIA Saclay
+ * Copyright 2012-2014 Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France
+ * and Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France
+ */
+
+#include <isl/ctx.h>
+#include <isl_aff_private.h>
+#include <isl/map.h>
+#include <isl/set.h>
+#include <isl_sort.h>
+#include <isl_schedule_private.h>
+#include <isl_band_private.h>
+
+__isl_null isl_schedule *isl_schedule_free(__isl_take isl_schedule *sched)
+{
+	int i;
+	if (!sched)
+		return NULL;
+
+	if (--sched->ref > 0)
+		return NULL;
+
+	for (i = 0; i < sched->n; ++i) {
+		isl_multi_aff_free(sched->node[i].sched);
+		free(sched->node[i].band_end);
+		free(sched->node[i].band_id);
+		free(sched->node[i].coincident);
+	}
+	isl_space_free(sched->dim);
+	isl_band_list_free(sched->band_forest);
+	free(sched);
+	return NULL;
+}
+
+isl_ctx *isl_schedule_get_ctx(__isl_keep isl_schedule *schedule)
+{
+	return schedule ? isl_space_get_ctx(schedule->dim) : NULL;
+}
+
+/* Set max_out to the maximal number of output dimensions over
+ * all maps.
+ */
+static int update_max_out(__isl_take isl_map *map, void *user)
+{
+	int *max_out = user;
+	int n_out = isl_map_dim(map, isl_dim_out);
+
+	if (n_out > *max_out)
+		*max_out = n_out;
+
+	isl_map_free(map);
+	return 0;
+}
+
+/* Internal data structure for map_pad_range.
+ *
+ * "max_out" is the maximal schedule dimension.
+ * "res" collects the results.
+ */
+struct isl_pad_schedule_map_data {
+	int max_out;
+	isl_union_map *res;
+};
+
+/* Pad the range of the given map with zeros to data->max_out and
+ * then add the result to data->res.
+ */
+static int map_pad_range(__isl_take isl_map *map, void *user)
+{
+	struct isl_pad_schedule_map_data *data = user;
+	int i;
+	int n_out = isl_map_dim(map, isl_dim_out);
+
+	map = isl_map_add_dims(map, isl_dim_out, data->max_out - n_out);
+	for (i = n_out; i < data->max_out; ++i)
+		map = isl_map_fix_si(map, isl_dim_out, i, 0);
+
+	data->res = isl_union_map_add_map(data->res, map);
+	if (!data->res)
+		return -1;
+
+	return 0;
+}
+
+/* Pad the ranges of the maps in the union map with zeros such they all have
+ * the same dimension.
+ */
+static __isl_give isl_union_map *pad_schedule_map(
+	__isl_take isl_union_map *umap)
+{
+	struct isl_pad_schedule_map_data data;
+
+	if (!umap)
+		return NULL;
+	if (isl_union_map_n_map(umap) <= 1)
+		return umap;
+
+	data.max_out = 0;
+	if (isl_union_map_foreach_map(umap, &update_max_out, &data.max_out) < 0)
+		return isl_union_map_free(umap);
+
+	data.res = isl_union_map_empty(isl_union_map_get_space(umap));
+	if (isl_union_map_foreach_map(umap, &map_pad_range, &data) < 0)
+		data.res = isl_union_map_free(data.res);
+
+	isl_union_map_free(umap);
+	return data.res;
+}
+
+/* Return an isl_union_map of the schedule.  If we have already constructed
+ * a band forest, then this band forest may have been modified so we need
+ * to extract the isl_union_map from the forest rather than from
+ * the originally computed schedule.  This reconstructed schedule map
+ * then needs to be padded with zeros to unify the schedule space
+ * since the result of isl_band_list_get_suffix_schedule may not have
+ * a unified schedule space.
+ */
+__isl_give isl_union_map *isl_schedule_get_map(__isl_keep isl_schedule *sched)
+{
+	int i;
+	isl_union_map *umap;
+
+	if (!sched)
+		return NULL;
+
+	if (sched->band_forest) {
+		umap = isl_band_list_get_suffix_schedule(sched->band_forest);
+		return pad_schedule_map(umap);
+	}
+
+	umap = isl_union_map_empty(isl_space_copy(sched->dim));
+	for (i = 0; i < sched->n; ++i) {
+		isl_multi_aff *ma;
+
+		ma = isl_multi_aff_copy(sched->node[i].sched);
+		umap = isl_union_map_add_map(umap, isl_map_from_multi_aff(ma));
+	}
+
+	return umap;
+}
+
+static __isl_give isl_band_list *construct_band_list(
+	__isl_keep isl_schedule *schedule, __isl_keep isl_band *parent,
+	int band_nr, int *parent_active, int n_active);
+
+/* Construct an isl_band structure for the band in the given schedule
+ * with sequence number band_nr for the n_active nodes marked by active.
+ * If the nodes don't have a band with the given sequence number,
+ * then a band without members is created.
+ *
+ * Because of the way the schedule is constructed, we know that
+ * the position of the band inside the schedule of a node is the same
+ * for all active nodes.
+ *
+ * The partial schedule for the band is created before the children
+ * are created to that construct_band_list can refer to the partial
+ * schedule of the parent.
+ */
+static __isl_give isl_band *construct_band(__isl_keep isl_schedule *schedule,
+	__isl_keep isl_band *parent,
+	int band_nr, int *active, int n_active)
+{
+	int i, j;
+	isl_ctx *ctx = isl_schedule_get_ctx(schedule);
+	isl_band *band;
+	unsigned start, end;
+
+	band = isl_band_alloc(ctx);
+	if (!band)
+		return NULL;
+
+	band->schedule = schedule;
+	band->parent = parent;
+
+	for (i = 0; i < schedule->n; ++i)
+		if (active[i])
+			break;
+
+	if (i >= schedule->n)
+		isl_die(ctx, isl_error_internal,
+			"band without active statements", goto error);
+
+	start = band_nr ? schedule->node[i].band_end[band_nr - 1] : 0;
+	end = band_nr < schedule->node[i].n_band ?
+		schedule->node[i].band_end[band_nr] : start;
+	band->n = end - start;
+
+	band->coincident = isl_alloc_array(ctx, int, band->n);
+	if (band->n && !band->coincident)
+		goto error;
+
+	for (j = 0; j < band->n; ++j)
+		band->coincident[j] = schedule->node[i].coincident[start + j];
+
+	band->pma = isl_union_pw_multi_aff_empty(isl_space_copy(schedule->dim));
+	for (i = 0; i < schedule->n; ++i) {
+		isl_multi_aff *ma;
+		isl_pw_multi_aff *pma;
+		unsigned n_out;
+
+		if (!active[i])
+			continue;
+
+		ma = isl_multi_aff_copy(schedule->node[i].sched);
+		n_out = isl_multi_aff_dim(ma, isl_dim_out);
+		ma = isl_multi_aff_drop_dims(ma, isl_dim_out, end, n_out - end);
+		ma = isl_multi_aff_drop_dims(ma, isl_dim_out, 0, start);
+		pma = isl_pw_multi_aff_from_multi_aff(ma);
+		band->pma = isl_union_pw_multi_aff_add_pw_multi_aff(band->pma,
+								    pma);
+	}
+	if (!band->pma)
+		goto error;
+
+	for (i = 0; i < schedule->n; ++i)
+		if (active[i] && schedule->node[i].n_band > band_nr + 1)
+			break;
+
+	if (i < schedule->n) {
+		band->children = construct_band_list(schedule, band,
+						band_nr + 1, active, n_active);
+		if (!band->children)
+			goto error;
+	}
+
+	return band;
+error:
+	isl_band_free(band);
+	return NULL;
+}
+
+/* Internal data structure used inside cmp_band and pw_multi_aff_extract_int.
+ *
+ * r is set to a negative value if anything goes wrong.
+ *
+ * c1 stores the result of extract_int.
+ * c2 is a temporary value used inside cmp_band_in_ancestor.
+ * t is a temporary value used inside extract_int.
+ *
+ * first and equal are used inside extract_int.
+ * first is set if we are looking at the first isl_multi_aff inside
+ * the isl_union_pw_multi_aff.
+ * equal is set if all the isl_multi_affs have been equal so far.
+ */
+struct isl_cmp_band_data {
+	int r;
+
+	int first;
+	int equal;
+
+	isl_int t;
+	isl_int c1;
+	isl_int c2;
+};
+
+/* Check if "ma" assigns a constant value.
+ * Note that this function is only called on isl_multi_affs
+ * with a single output dimension.
+ *
+ * If "ma" assigns a constant value then we compare it to data->c1
+ * or assign it to data->c1 if this is the first isl_multi_aff we consider.
+ * If "ma" does not assign a constant value or if it assigns a value
+ * that is different from data->c1, then we set data->equal to zero
+ * and terminate the check.
+ */
+static int multi_aff_extract_int(__isl_take isl_set *set,
+	__isl_take isl_multi_aff *ma, void *user)
+{
+	isl_aff *aff;
+	struct isl_cmp_band_data *data = user;
+
+	aff = isl_multi_aff_get_aff(ma, 0);
+	data->r = isl_aff_is_cst(aff);
+	if (data->r >= 0 && data->r) {
+		isl_aff_get_constant(aff, &data->t);
+		if (data->first) {
+			isl_int_set(data->c1, data->t);
+			data->first = 0;
+		} else if (!isl_int_eq(data->c1, data->t))
+			data->equal = 0;
+	} else if (data->r >= 0 && !data->r)
+		data->equal = 0;
+
+	isl_aff_free(aff);
+	isl_set_free(set);
+	isl_multi_aff_free(ma);
+
+	if (data->r < 0)
+		return -1;
+	if (!data->equal)
+		return -1;
+	return 0;
+}
+
+/* This function is called for each isl_pw_multi_aff in
+ * the isl_union_pw_multi_aff checked by extract_int.
+ * Check all the isl_multi_affs inside "pma".
+ */
+static int pw_multi_aff_extract_int(__isl_take isl_pw_multi_aff *pma,
+	void *user)
+{
+	int r;
+
+	r = isl_pw_multi_aff_foreach_piece(pma, &multi_aff_extract_int, user);
+	isl_pw_multi_aff_free(pma);
+
+	return r;
+}
+
+/* Check if "upma" assigns a single constant value to its domain.
+ * If so, return 1 and store the result in data->c1.
+ * If not, return 0.
+ *
+ * A negative return value from isl_union_pw_multi_aff_foreach_pw_multi_aff
+ * means that either an error occurred or that we have broken off the check
+ * because we already know the result is going to be negative.
+ * In the latter case, data->equal is set to zero.
+ */
+static int extract_int(__isl_keep isl_union_pw_multi_aff *upma,
+	struct isl_cmp_band_data *data)
+{
+	data->first = 1;
+	data->equal = 1;
+
+	if (isl_union_pw_multi_aff_foreach_pw_multi_aff(upma,
+					&pw_multi_aff_extract_int, data) < 0) {
+		if (!data->equal)
+			return 0;
+		return -1;
+	}
+
+	return !data->first && data->equal;
+}
+
+/* Compare "b1" and "b2" based on the parent schedule of their ancestor
+ * "ancestor".
+ *
+ * If the parent of "ancestor" also has a single member, then we
+ * first try to compare the two band based on the partial schedule
+ * of this parent.
+ *
+ * Otherwise, or if the result is inconclusive, we look at the partial schedule
+ * of "ancestor" itself.
+ * In particular, we specialize the parent schedule based
+ * on the domains of the child schedules, check if both assign
+ * a single constant value and, if so, compare the two constant values.
+ * If the specialized parent schedules do not assign a constant value,
+ * then they cannot be used to order the two bands and so in this case
+ * we return 0.
+ */
+static int cmp_band_in_ancestor(__isl_keep isl_band *b1,
+	__isl_keep isl_band *b2, struct isl_cmp_band_data *data,
+	__isl_keep isl_band *ancestor)
+{
+	isl_union_pw_multi_aff *upma;
+	isl_union_set *domain;
+	int r;
+
+	if (data->r < 0)
+		return 0;
+
+	if (ancestor->parent && ancestor->parent->n == 1) {
+		r = cmp_band_in_ancestor(b1, b2, data, ancestor->parent);
+		if (data->r < 0)
+			return 0;
+		if (r)
+			return r;
+	}
+
+	upma = isl_union_pw_multi_aff_copy(b1->pma);
+	domain = isl_union_pw_multi_aff_domain(upma);
+	upma = isl_union_pw_multi_aff_copy(ancestor->pma);
+	upma = isl_union_pw_multi_aff_intersect_domain(upma, domain);
+	r = extract_int(upma, data);
+	isl_union_pw_multi_aff_free(upma);
+
+	if (r < 0)
+		data->r = -1;
+	if (r < 0 || !r)
+		return 0;
+
+	isl_int_set(data->c2, data->c1);
+
+	upma = isl_union_pw_multi_aff_copy(b2->pma);
+	domain = isl_union_pw_multi_aff_domain(upma);
+	upma = isl_union_pw_multi_aff_copy(ancestor->pma);
+	upma = isl_union_pw_multi_aff_intersect_domain(upma, domain);
+	r = extract_int(upma, data);
+	isl_union_pw_multi_aff_free(upma);
+
+	if (r < 0)
+		data->r = -1;
+	if (r < 0 || !r)
+		return 0;
+
+	return isl_int_cmp(data->c2, data->c1);
+}
+
+/* Compare "a" and "b" based on the parent schedule of their parent.
+ */
+static int cmp_band(const void *a, const void *b, void *user)
+{
+	isl_band *b1 = *(isl_band * const *) a;
+	isl_band *b2 = *(isl_band * const *) b;
+	struct isl_cmp_band_data *data = user;
+
+	return cmp_band_in_ancestor(b1, b2, data, b1->parent);
+}
+
+/* Sort the elements in "list" based on the partial schedules of its parent
+ * (and ancestors).  In particular if the parent assigns constant values
+ * to the domains of the bands in "list", then the elements are sorted
+ * according to that order.
+ * This order should be a more "natural" order for the user, but otherwise
+ * shouldn't have any effect.
+ * If we would be constructing an isl_band forest directly in
+ * isl_schedule_constraints_compute_schedule then there wouldn't be any need
+ * for a reordering, since the children would be added to the list
+ * in their natural order automatically.
+ *
+ * If there is only one element in the list, then there is no need to sort
+ * anything.
+ * If the partial schedule of the parent has more than one member
+ * (or if there is no parent), then it's
+ * defnitely not assigning constant values to the different children in
+ * the list and so we wouldn't be able to use it to sort the list.
+ */
+static __isl_give isl_band_list *sort_band_list(__isl_take isl_band_list *list,
+	__isl_keep isl_band *parent)
+{
+	struct isl_cmp_band_data data;
+
+	if (!list)
+		return NULL;
+	if (list->n <= 1)
+		return list;
+	if (!parent || parent->n != 1)
+		return list;
+
+	data.r = 0;
+	isl_int_init(data.c1);
+	isl_int_init(data.c2);
+	isl_int_init(data.t);
+	isl_sort(list->p, list->n, sizeof(list->p[0]), &cmp_band, &data);
+	if (data.r < 0)
+		list = isl_band_list_free(list);
+	isl_int_clear(data.c1);
+	isl_int_clear(data.c2);
+	isl_int_clear(data.t);
+
+	return list;
+}
+
+/* Construct a list of bands that start at the same position (with
+ * sequence number band_nr) in the schedules of the nodes that
+ * were active in the parent band.
+ *
+ * A separate isl_band structure is created for each band_id
+ * and for each node that does not have a band with sequence
+ * number band_nr.  In the latter case, a band without members
+ * is created.
+ * This ensures that if a band has any children, then each node
+ * that was active in the band is active in exactly one of the children.
+ */
+static __isl_give isl_band_list *construct_band_list(
+	__isl_keep isl_schedule *schedule, __isl_keep isl_band *parent,
+	int band_nr, int *parent_active, int n_active)
+{
+	int i, j;
+	isl_ctx *ctx = isl_schedule_get_ctx(schedule);
+	int *active;
+	int n_band;
+	isl_band_list *list;
+
+	n_band = 0;
+	for (i = 0; i < n_active; ++i) {
+		for (j = 0; j < schedule->n; ++j) {
+			if (!parent_active[j])
+				continue;
+			if (schedule->node[j].n_band <= band_nr)
+				continue;
+			if (schedule->node[j].band_id[band_nr] == i) {
+				n_band++;
+				break;
+			}
+		}
+	}
+	for (j = 0; j < schedule->n; ++j)
+		if (schedule->node[j].n_band <= band_nr)
+			n_band++;
+
+	if (n_band == 1) {
+		isl_band *band;
+		list = isl_band_list_alloc(ctx, n_band);
+		band = construct_band(schedule, parent, band_nr,
+					parent_active, n_active);
+		return isl_band_list_add(list, band);
+	}
+
+	active = isl_alloc_array(ctx, int, schedule->n);
+	if (schedule->n && !active)
+		return NULL;
+
+	list = isl_band_list_alloc(ctx, n_band);
+
+	for (i = 0; i < n_active; ++i) {
+		int n = 0;
+		isl_band *band;
+
+		for (j = 0; j < schedule->n; ++j) {
+			active[j] = parent_active[j] &&
+					schedule->node[j].n_band > band_nr &&
+					schedule->node[j].band_id[band_nr] == i;
+			if (active[j])
+				n++;
+		}
+		if (n == 0)
+			continue;
+
+		band = construct_band(schedule, parent, band_nr, active, n);
+
+		list = isl_band_list_add(list, band);
+	}
+	for (i = 0; i < schedule->n; ++i) {
+		isl_band *band;
+		if (!parent_active[i])
+			continue;
+		if (schedule->node[i].n_band > band_nr)
+			continue;
+		for (j = 0; j < schedule->n; ++j)
+			active[j] = j == i;
+		band = construct_band(schedule, parent, band_nr, active, 1);
+		list = isl_band_list_add(list, band);
+	}
+
+	free(active);
+
+	list = sort_band_list(list, parent);
+
+	return list;
+}
+
+/* Construct a band forest representation of the schedule and
+ * return the list of roots.
+ */
+static __isl_give isl_band_list *construct_forest(
+	__isl_keep isl_schedule *schedule)
+{
+	int i;
+	isl_ctx *ctx = isl_schedule_get_ctx(schedule);
+	isl_band_list *forest;
+	int *active;
+
+	active = isl_alloc_array(ctx, int, schedule->n);
+	if (schedule->n && !active)
+		return NULL;
+
+	for (i = 0; i < schedule->n; ++i)
+		active[i] = 1;
+
+	forest = construct_band_list(schedule, NULL, 0, active, schedule->n);
+
+	free(active);
+
+	return forest;
+}
+
+/* Return the roots of a band forest representation of the schedule.
+ */
+__isl_give isl_band_list *isl_schedule_get_band_forest(
+	__isl_keep isl_schedule *schedule)
+{
+	if (!schedule)
+		return NULL;
+	if (!schedule->band_forest)
+		schedule->band_forest = construct_forest(schedule);
+	return isl_band_list_dup(schedule->band_forest);
+}
+
+/* Call "fn" on each band in the schedule in depth-first post-order.
+ */
+int isl_schedule_foreach_band(__isl_keep isl_schedule *sched,
+	int (*fn)(__isl_keep isl_band *band, void *user), void *user)
+{
+	int r;
+	isl_band_list *forest;
+
+	if (!sched)
+		return -1;
+
+	forest = isl_schedule_get_band_forest(sched);
+	r = isl_band_list_foreach_band(forest, fn, user);
+	isl_band_list_free(forest);
+
+	return r;
+}
+
+static __isl_give isl_printer *print_band_list(__isl_take isl_printer *p,
+	__isl_keep isl_band_list *list);
+
+static __isl_give isl_printer *print_band(__isl_take isl_printer *p,
+	__isl_keep isl_band *band)
+{
+	isl_band_list *children;
+
+	p = isl_printer_start_line(p);
+	p = isl_printer_print_union_pw_multi_aff(p, band->pma);
+	p = isl_printer_end_line(p);
+
+	if (!isl_band_has_children(band))
+		return p;
+
+	children = isl_band_get_children(band);
+
+	p = isl_printer_indent(p, 4);
+	p = print_band_list(p, children);
+	p = isl_printer_indent(p, -4);
+
+	isl_band_list_free(children);
+
+	return p;
+}
+
+static __isl_give isl_printer *print_band_list(__isl_take isl_printer *p,
+	__isl_keep isl_band_list *list)
+{
+	int i, n;
+
+	n = isl_band_list_n_band(list);
+	for (i = 0; i < n; ++i) {
+		isl_band *band;
+		band = isl_band_list_get_band(list, i);
+		p = print_band(p, band);
+		isl_band_free(band);
+	}
+
+	return p;
+}
+
+__isl_give isl_printer *isl_printer_print_schedule(__isl_take isl_printer *p,
+	__isl_keep isl_schedule *schedule)
+{
+	isl_band_list *forest;
+
+	forest = isl_schedule_get_band_forest(schedule);
+
+	p = print_band_list(p, forest);
+
+	isl_band_list_free(forest);
+
+	return p;
+}
+
+void isl_schedule_dump(__isl_keep isl_schedule *schedule)
+{
+	isl_printer *printer;
+
+	if (!schedule)
+		return;
+
+	printer = isl_printer_to_file(isl_schedule_get_ctx(schedule), stderr);
+	printer = isl_printer_print_schedule(printer, schedule);
+
+	isl_printer_free(printer);
+}
diff --git a/polly/lib/External/isl/isl_schedule_private.h b/polly/lib/External/isl/isl_schedule_private.h
new file mode 100644
index 00000000000..f40a72a0146
--- /dev/null
+++ b/polly/lib/External/isl/isl_schedule_private.h
@@ -0,0 +1,44 @@
+#ifndef ISL_SCHEDLUE_PRIVATE_H
+#define ISL_SCHEDLUE_PRIVATE_H
+
+#include <isl/aff.h>
+#include <isl/schedule.h>
+
+/* The schedule for an individual domain, plus information about the bands
+ * and scheduling dimensions.
+ * In particular, we keep track of the number of bands and for each
+ * band, the starting position of the next band.  The first band starts at
+ * position 0.
+ * For each scheduling dimension, we keep track of whether it satisfies
+ * the coincidence constraints (within its band).
+ */
+struct isl_schedule_node {
+	isl_multi_aff *sched;
+	int	 n_band;
+	int	*band_end;
+	int	*band_id;
+	int	*coincident;
+};
+
+/* Information about the computed schedule.
+ * n is the number of nodes/domains/statements.
+ * n_band is the maximal number of bands.
+ * n_total_row is the number of coordinates of the schedule.
+ * dim contains a description of the parameters.
+ * band_forest points to a band forest representation of the schedule
+ * and may be NULL if the forest hasn't been created yet.
+ */
+struct isl_schedule {
+	int ref;
+
+	int n;
+	int n_band;
+	int n_total_row;
+	isl_space *dim;
+
+	isl_band_list *band_forest;
+
+	struct isl_schedule_node node[1];
+};
+
+#endif
diff --git a/polly/lib/External/isl/isl_scheduler.c b/polly/lib/External/isl/isl_scheduler.c
new file mode 100644
index 00000000000..4a7428045e0
--- /dev/null
+++ b/polly/lib/External/isl/isl_scheduler.c
@@ -0,0 +1,4148 @@
+/*
+ * Copyright 2011      INRIA Saclay
+ * Copyright 2012-2014 Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France
+ * and Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl_space_private.h>
+#include <isl_aff_private.h>
+#include <isl/hash.h>
+#include <isl/constraint.h>
+#include <isl/schedule.h>
+#include <isl_mat_private.h>
+#include <isl_vec_private.h>
+#include <isl/set.h>
+#include <isl_seq.h>
+#include <isl_tab.h>
+#include <isl_dim_map.h>
+#include <isl/map_to_basic_set.h>
+#include <isl_sort.h>
+#include <isl_schedule_private.h>
+#include <isl_options_private.h>
+#include <isl_tarjan.h>
+#include <isl_morph.h>
+
+/*
+ * The scheduling algorithm implemented in this file was inspired by
+ * Bondhugula et al., "Automatic Transformations for Communication-Minimized
+ * Parallelization and Locality Optimization in the Polyhedral Model".
+ */
+
+enum isl_edge_type {
+	isl_edge_validity = 0,
+	isl_edge_first = isl_edge_validity,
+	isl_edge_coincidence,
+	isl_edge_condition,
+	isl_edge_conditional_validity,
+	isl_edge_proximity,
+	isl_edge_last = isl_edge_proximity
+};
+
+/* The constraints that need to be satisfied by a schedule on "domain".
+ *
+ * "validity" constraints map domain elements i to domain elements
+ * that should be scheduled after i.  (Hard constraint)
+ * "proximity" constraints map domain elements i to domains elements
+ * that should be scheduled as early as possible after i (or before i).
+ * (Soft constraint)
+ *
+ * "condition" and "conditional_validity" constraints map possibly "tagged"
+ * domain elements i -> s to "tagged" domain elements j -> t.
+ * The elements of the "conditional_validity" constraints, but without the
+ * tags (i.e., the elements i -> j) are treated as validity constraints,
+ * except that during the construction of a tilable band,
+ * the elements of the "conditional_validity" constraints may be violated
+ * provided that all adjacent elements of the "condition" constraints
+ * are local within the band.
+ * A dependence is local within a band if domain and range are mapped
+ * to the same schedule point by the band.
+ */
+struct isl_schedule_constraints {
+	isl_union_set *domain;
+
+	isl_union_map *constraint[isl_edge_last + 1];
+};
+
+ __isl_give isl_schedule_constraints *isl_schedule_constraints_copy(
+	__isl_keep isl_schedule_constraints *sc)
+{
+	isl_ctx *ctx;
+	isl_schedule_constraints *sc_copy;
+	enum isl_edge_type i;
+
+	ctx = isl_union_set_get_ctx(sc->domain);
+	sc_copy = isl_calloc_type(ctx, struct isl_schedule_constraints);
+	if (!sc_copy)
+		return NULL;
+
+	sc_copy->domain = isl_union_set_copy(sc->domain);
+	if (!sc_copy->domain)
+		return isl_schedule_constraints_free(sc_copy);
+
+	for (i = isl_edge_first; i <= isl_edge_last; ++i) {
+		sc_copy->constraint[i] = isl_union_map_copy(sc->constraint[i]);
+		if (!sc_copy->constraint[i])
+			return isl_schedule_constraints_free(sc_copy);
+	}
+
+	return sc_copy;
+}
+
+
+/* Construct an isl_schedule_constraints object for computing a schedule
+ * on "domain".  The initial object does not impose any constraints.
+ */
+__isl_give isl_schedule_constraints *isl_schedule_constraints_on_domain(
+	__isl_take isl_union_set *domain)
+{
+	isl_ctx *ctx;
+	isl_space *space;
+	isl_schedule_constraints *sc;
+	isl_union_map *empty;
+	enum isl_edge_type i;
+
+	if (!domain)
+		return NULL;
+
+	ctx = isl_union_set_get_ctx(domain);
+	sc = isl_calloc_type(ctx, struct isl_schedule_constraints);
+	if (!sc)
+		goto error;
+
+	space = isl_union_set_get_space(domain);
+	sc->domain = domain;
+	empty = isl_union_map_empty(space);
+	for (i = isl_edge_first; i <= isl_edge_last; ++i) {
+		sc->constraint[i] = isl_union_map_copy(empty);
+		if (!sc->constraint[i])
+			sc->domain = isl_union_set_free(sc->domain);
+	}
+	isl_union_map_free(empty);
+
+	if (!sc->domain)
+		return isl_schedule_constraints_free(sc);
+
+	return sc;
+error:
+	isl_union_set_free(domain);
+	return NULL;
+}
+
+/* Replace the validity constraints of "sc" by "validity".
+ */
+__isl_give isl_schedule_constraints *isl_schedule_constraints_set_validity(
+	__isl_take isl_schedule_constraints *sc,
+	__isl_take isl_union_map *validity)
+{
+	if (!sc || !validity)
+		goto error;
+
+	isl_union_map_free(sc->constraint[isl_edge_validity]);
+	sc->constraint[isl_edge_validity] = validity;
+
+	return sc;
+error:
+	isl_schedule_constraints_free(sc);
+	isl_union_map_free(validity);
+	return NULL;
+}
+
+/* Replace the coincidence constraints of "sc" by "coincidence".
+ */
+__isl_give isl_schedule_constraints *isl_schedule_constraints_set_coincidence(
+	__isl_take isl_schedule_constraints *sc,
+	__isl_take isl_union_map *coincidence)
+{
+	if (!sc || !coincidence)
+		goto error;
+
+	isl_union_map_free(sc->constraint[isl_edge_coincidence]);
+	sc->constraint[isl_edge_coincidence] = coincidence;
+
+	return sc;
+error:
+	isl_schedule_constraints_free(sc);
+	isl_union_map_free(coincidence);
+	return NULL;
+}
+
+/* Replace the proximity constraints of "sc" by "proximity".
+ */
+__isl_give isl_schedule_constraints *isl_schedule_constraints_set_proximity(
+	__isl_take isl_schedule_constraints *sc,
+	__isl_take isl_union_map *proximity)
+{
+	if (!sc || !proximity)
+		goto error;
+
+	isl_union_map_free(sc->constraint[isl_edge_proximity]);
+	sc->constraint[isl_edge_proximity] = proximity;
+
+	return sc;
+error:
+	isl_schedule_constraints_free(sc);
+	isl_union_map_free(proximity);
+	return NULL;
+}
+
+/* Replace the conditional validity constraints of "sc" by "condition"
+ * and "validity".
+ */
+__isl_give isl_schedule_constraints *
+isl_schedule_constraints_set_conditional_validity(
+	__isl_take isl_schedule_constraints *sc,
+	__isl_take isl_union_map *condition,
+	__isl_take isl_union_map *validity)
+{
+	if (!sc || !condition || !validity)
+		goto error;
+
+	isl_union_map_free(sc->constraint[isl_edge_condition]);
+	sc->constraint[isl_edge_condition] = condition;
+	isl_union_map_free(sc->constraint[isl_edge_conditional_validity]);
+	sc->constraint[isl_edge_conditional_validity] = validity;
+
+	return sc;
+error:
+	isl_schedule_constraints_free(sc);
+	isl_union_map_free(condition);
+	isl_union_map_free(validity);
+	return NULL;
+}
+
+__isl_null isl_schedule_constraints *isl_schedule_constraints_free(
+	__isl_take isl_schedule_constraints *sc)
+{
+	enum isl_edge_type i;
+
+	if (!sc)
+		return NULL;
+
+	isl_union_set_free(sc->domain);
+	for (i = isl_edge_first; i <= isl_edge_last; ++i)
+		isl_union_map_free(sc->constraint[i]);
+
+	free(sc);
+
+	return NULL;
+}
+
+isl_ctx *isl_schedule_constraints_get_ctx(
+	__isl_keep isl_schedule_constraints *sc)
+{
+	return sc ? isl_union_set_get_ctx(sc->domain) : NULL;
+}
+
+void isl_schedule_constraints_dump(__isl_keep isl_schedule_constraints *sc)
+{
+	if (!sc)
+		return;
+
+	fprintf(stderr, "domain: ");
+	isl_union_set_dump(sc->domain);
+	fprintf(stderr, "validity: ");
+	isl_union_map_dump(sc->constraint[isl_edge_validity]);
+	fprintf(stderr, "proximity: ");
+	isl_union_map_dump(sc->constraint[isl_edge_proximity]);
+	fprintf(stderr, "coincidence: ");
+	isl_union_map_dump(sc->constraint[isl_edge_coincidence]);
+	fprintf(stderr, "condition: ");
+	isl_union_map_dump(sc->constraint[isl_edge_condition]);
+	fprintf(stderr, "conditional_validity: ");
+	isl_union_map_dump(sc->constraint[isl_edge_conditional_validity]);
+}
+
+/* Align the parameters of the fields of "sc".
+ */
+static __isl_give isl_schedule_constraints *
+isl_schedule_constraints_align_params(__isl_take isl_schedule_constraints *sc)
+{
+	isl_space *space;
+	enum isl_edge_type i;
+
+	if (!sc)
+		return NULL;
+
+	space = isl_union_set_get_space(sc->domain);
+	for (i = isl_edge_first; i <= isl_edge_last; ++i)
+		space = isl_space_align_params(space,
+				    isl_union_map_get_space(sc->constraint[i]));
+
+	for (i = isl_edge_first; i <= isl_edge_last; ++i) {
+		sc->constraint[i] = isl_union_map_align_params(
+				    sc->constraint[i], isl_space_copy(space));
+		if (!sc->constraint[i])
+			space = isl_space_free(space);
+	}
+	sc->domain = isl_union_set_align_params(sc->domain, space);
+	if (!sc->domain)
+		return isl_schedule_constraints_free(sc);
+
+	return sc;
+}
+
+/* Return the total number of isl_maps in the constraints of "sc".
+ */
+static __isl_give int isl_schedule_constraints_n_map(
+	__isl_keep isl_schedule_constraints *sc)
+{
+	enum isl_edge_type i;
+	int n = 0;
+
+	for (i = isl_edge_first; i <= isl_edge_last; ++i)
+		n += isl_union_map_n_map(sc->constraint[i]);
+
+	return n;
+}
+
+/* Internal information about a node that is used during the construction
+ * of a schedule.
+ * space represents the space in which the domain lives
+ * sched is a matrix representation of the schedule being constructed
+ *	for this node; if compressed is set, then this schedule is
+ *	defined over the compressed domain space
+ * sched_map is an isl_map representation of the same (partial) schedule
+ *	sched_map may be NULL; if compressed is set, then this map
+ *	is defined over the uncompressed domain space
+ * rank is the number of linearly independent rows in the linear part
+ *	of sched
+ * the columns of cmap represent a change of basis for the schedule
+ *	coefficients; the first rank columns span the linear part of
+ *	the schedule rows
+ * cinv is the inverse of cmap.
+ * start is the first variable in the LP problem in the sequences that
+ *	represents the schedule coefficients of this node
+ * nvar is the dimension of the domain
+ * nparam is the number of parameters or 0 if we are not constructing
+ *	a parametric schedule
+ *
+ * If compressed is set, then hull represents the constraints
+ * that were used to derive the compression, while compress and
+ * decompress map the original space to the compressed space and
+ * vice versa.
+ *
+ * scc is the index of SCC (or WCC) this node belongs to
+ *
+ * band contains the band index for each of the rows of the schedule.
+ * band_id is used to differentiate between separate bands at the same
+ * level within the same parent band, i.e., bands that are separated
+ * by the parent band or bands that are independent of each other.
+ * coincident contains a boolean for each of the rows of the schedule,
+ * indicating whether the corresponding scheduling dimension satisfies
+ * the coincidence constraints in the sense that the corresponding
+ * dependence distances are zero.
+ */
+struct isl_sched_node {
+	isl_space *space;
+	int	compressed;
+	isl_set	*hull;
+	isl_multi_aff *compress;
+	isl_multi_aff *decompress;
+	isl_mat *sched;
+	isl_map *sched_map;
+	int	 rank;
+	isl_mat *cmap;
+	isl_mat *cinv;
+	int	 start;
+	int	 nvar;
+	int	 nparam;
+
+	int	 scc;
+
+	int	*band;
+	int	*band_id;
+	int	*coincident;
+};
+
+static int node_has_space(const void *entry, const void *val)
+{
+	struct isl_sched_node *node = (struct isl_sched_node *)entry;
+	isl_space *dim = (isl_space *)val;
+
+	return isl_space_is_equal(node->space, dim);
+}
+
+/* An edge in the dependence graph.  An edge may be used to
+ * ensure validity of the generated schedule, to minimize the dependence
+ * distance or both
+ *
+ * map is the dependence relation, with i -> j in the map if j depends on i
+ * tagged_condition and tagged_validity contain the union of all tagged
+ *	condition or conditional validity dependence relations that
+ *	specialize the dependence relation "map"; that is,
+ *	if (i -> a) -> (j -> b) is an element of "tagged_condition"
+ *	or "tagged_validity", then i -> j is an element of "map".
+ *	If these fields are NULL, then they represent the empty relation.
+ * src is the source node
+ * dst is the sink node
+ * validity is set if the edge is used to ensure correctness
+ * coincidence is used to enforce zero dependence distances
+ * proximity is set if the edge is used to minimize dependence distances
+ * condition is set if the edge represents a condition
+ *	for a conditional validity schedule constraint
+ * local can only be set for condition edges and indicates that
+ *	the dependence distance over the edge should be zero
+ * conditional_validity is set if the edge is used to conditionally
+ *	ensure correctness
+ *
+ * For validity edges, start and end mark the sequence of inequality
+ * constraints in the LP problem that encode the validity constraint
+ * corresponding to this edge.
+ */
+struct isl_sched_edge {
+	isl_map *map;
+	isl_union_map *tagged_condition;
+	isl_union_map *tagged_validity;
+
+	struct isl_sched_node *src;
+	struct isl_sched_node *dst;
+
+	unsigned validity : 1;
+	unsigned coincidence : 1;
+	unsigned proximity : 1;
+	unsigned local : 1;
+	unsigned condition : 1;
+	unsigned conditional_validity : 1;
+
+	int start;
+	int end;
+};
+
+/* Internal information about the dependence graph used during
+ * the construction of the schedule.
+ *
+ * intra_hmap is a cache, mapping dependence relations to their dual,
+ *	for dependences from a node to itself
+ * inter_hmap is a cache, mapping dependence relations to their dual,
+ *	for dependences between distinct nodes
+ * if compression is involved then the key for these maps
+ * it the original, uncompressed dependence relation, while
+ * the value is the dual of the compressed dependence relation.
+ *
+ * n is the number of nodes
+ * node is the list of nodes
+ * maxvar is the maximal number of variables over all nodes
+ * max_row is the allocated number of rows in the schedule
+ * n_row is the current (maximal) number of linearly independent
+ *	rows in the node schedules
+ * n_total_row is the current number of rows in the node schedules
+ * n_band is the current number of completed bands
+ * band_start is the starting row in the node schedules of the current band
+ * root is set if this graph is the original dependence graph,
+ *	without any splitting
+ *
+ * sorted contains a list of node indices sorted according to the
+ *	SCC to which a node belongs
+ *
+ * n_edge is the number of edges
+ * edge is the list of edges
+ * max_edge contains the maximal number of edges of each type;
+ *	in particular, it contains the number of edges in the inital graph.
+ * edge_table contains pointers into the edge array, hashed on the source
+ *	and sink spaces; there is one such table for each type;
+ *	a given edge may be referenced from more than one table
+ *	if the corresponding relation appears in more than of the
+ *	sets of dependences
+ *
+ * node_table contains pointers into the node array, hashed on the space
+ *
+ * region contains a list of variable sequences that should be non-trivial
+ *
+ * lp contains the (I)LP problem used to obtain new schedule rows
+ *
+ * src_scc and dst_scc are the source and sink SCCs of an edge with
+ *	conflicting constraints
+ *
+ * scc represents the number of components
+ */
+struct isl_sched_graph {
+	isl_map_to_basic_set *intra_hmap;
+	isl_map_to_basic_set *inter_hmap;
+
+	struct isl_sched_node *node;
+	int n;
+	int maxvar;
+	int max_row;
+	int n_row;
+
+	int *sorted;
+
+	int n_band;
+	int n_total_row;
+	int band_start;
+
+	int root;
+
+	struct isl_sched_edge *edge;
+	int n_edge;
+	int max_edge[isl_edge_last + 1];
+	struct isl_hash_table *edge_table[isl_edge_last + 1];
+
+	struct isl_hash_table *node_table;
+	struct isl_region *region;
+
+	isl_basic_set *lp;
+
+	int src_scc;
+	int dst_scc;
+
+	int scc;
+};
+
+/* Initialize node_table based on the list of nodes.
+ */
+static int graph_init_table(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	int i;
+
+	graph->node_table = isl_hash_table_alloc(ctx, graph->n);
+	if (!graph->node_table)
+		return -1;
+
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_hash_table_entry *entry;
+		uint32_t hash;
+
+		hash = isl_space_get_hash(graph->node[i].space);
+		entry = isl_hash_table_find(ctx, graph->node_table, hash,
+					    &node_has_space,
+					    graph->node[i].space, 1);
+		if (!entry)
+			return -1;
+		entry->data = &graph->node[i];
+	}
+
+	return 0;
+}
+
+/* Return a pointer to the node that lives within the given space,
+ * or NULL if there is no such node.
+ */
+static struct isl_sched_node *graph_find_node(isl_ctx *ctx,
+	struct isl_sched_graph *graph, __isl_keep isl_space *dim)
+{
+	struct isl_hash_table_entry *entry;
+	uint32_t hash;
+
+	hash = isl_space_get_hash(dim);
+	entry = isl_hash_table_find(ctx, graph->node_table, hash,
+				    &node_has_space, dim, 0);
+
+	return entry ? entry->data : NULL;
+}
+
+static int edge_has_src_and_dst(const void *entry, const void *val)
+{
+	const struct isl_sched_edge *edge = entry;
+	const struct isl_sched_edge *temp = val;
+
+	return edge->src == temp->src && edge->dst == temp->dst;
+}
+
+/* Add the given edge to graph->edge_table[type].
+ */
+static int graph_edge_table_add(isl_ctx *ctx, struct isl_sched_graph *graph,
+	enum isl_edge_type type, struct isl_sched_edge *edge)
+{
+	struct isl_hash_table_entry *entry;
+	uint32_t hash;
+
+	hash = isl_hash_init();
+	hash = isl_hash_builtin(hash, edge->src);
+	hash = isl_hash_builtin(hash, edge->dst);
+	entry = isl_hash_table_find(ctx, graph->edge_table[type], hash,
+				    &edge_has_src_and_dst, edge, 1);
+	if (!entry)
+		return -1;
+	entry->data = edge;
+
+	return 0;
+}
+
+/* Allocate the edge_tables based on the maximal number of edges of
+ * each type.
+ */
+static int graph_init_edge_tables(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	int i;
+
+	for (i = 0; i <= isl_edge_last; ++i) {
+		graph->edge_table[i] = isl_hash_table_alloc(ctx,
+							    graph->max_edge[i]);
+		if (!graph->edge_table[i])
+			return -1;
+	}
+
+	return 0;
+}
+
+/* If graph->edge_table[type] contains an edge from the given source
+ * to the given destination, then return the hash table entry of this edge.
+ * Otherwise, return NULL.
+ */
+static struct isl_hash_table_entry *graph_find_edge_entry(
+	struct isl_sched_graph *graph,
+	enum isl_edge_type type,
+	struct isl_sched_node *src, struct isl_sched_node *dst)
+{
+	isl_ctx *ctx = isl_space_get_ctx(src->space);
+	uint32_t hash;
+	struct isl_sched_edge temp = { .src = src, .dst = dst };
+
+	hash = isl_hash_init();
+	hash = isl_hash_builtin(hash, temp.src);
+	hash = isl_hash_builtin(hash, temp.dst);
+	return isl_hash_table_find(ctx, graph->edge_table[type], hash,
+				    &edge_has_src_and_dst, &temp, 0);
+}
+
+
+/* If graph->edge_table[type] contains an edge from the given source
+ * to the given destination, then return this edge.
+ * Otherwise, return NULL.
+ */
+static struct isl_sched_edge *graph_find_edge(struct isl_sched_graph *graph,
+	enum isl_edge_type type,
+	struct isl_sched_node *src, struct isl_sched_node *dst)
+{
+	struct isl_hash_table_entry *entry;
+
+	entry = graph_find_edge_entry(graph, type, src, dst);
+	if (!entry)
+		return NULL;
+
+	return entry->data;
+}
+
+/* Check whether the dependence graph has an edge of the given type
+ * between the given two nodes.
+ */
+static int graph_has_edge(struct isl_sched_graph *graph,
+	enum isl_edge_type type,
+	struct isl_sched_node *src, struct isl_sched_node *dst)
+{
+	struct isl_sched_edge *edge;
+	int empty;
+
+	edge = graph_find_edge(graph, type, src, dst);
+	if (!edge)
+		return 0;
+
+	empty = isl_map_plain_is_empty(edge->map);
+	if (empty < 0)
+		return -1;
+
+	return !empty;
+}
+
+/* Look for any edge with the same src, dst and map fields as "model".
+ *
+ * Return the matching edge if one can be found.
+ * Return "model" if no matching edge is found.
+ * Return NULL on error.
+ */
+static struct isl_sched_edge *graph_find_matching_edge(
+	struct isl_sched_graph *graph, struct isl_sched_edge *model)
+{
+	enum isl_edge_type i;
+	struct isl_sched_edge *edge;
+
+	for (i = isl_edge_first; i <= isl_edge_last; ++i) {
+		int is_equal;
+
+		edge = graph_find_edge(graph, i, model->src, model->dst);
+		if (!edge)
+			continue;
+		is_equal = isl_map_plain_is_equal(model->map, edge->map);
+		if (is_equal < 0)
+			return NULL;
+		if (is_equal)
+			return edge;
+	}
+
+	return model;
+}
+
+/* Remove the given edge from all the edge_tables that refer to it.
+ */
+static void graph_remove_edge(struct isl_sched_graph *graph,
+	struct isl_sched_edge *edge)
+{
+	isl_ctx *ctx = isl_map_get_ctx(edge->map);
+	enum isl_edge_type i;
+
+	for (i = isl_edge_first; i <= isl_edge_last; ++i) {
+		struct isl_hash_table_entry *entry;
+
+		entry = graph_find_edge_entry(graph, i, edge->src, edge->dst);
+		if (!entry)
+			continue;
+		if (entry->data != edge)
+			continue;
+		isl_hash_table_remove(ctx, graph->edge_table[i], entry);
+	}
+}
+
+/* Check whether the dependence graph has any edge
+ * between the given two nodes.
+ */
+static int graph_has_any_edge(struct isl_sched_graph *graph,
+	struct isl_sched_node *src, struct isl_sched_node *dst)
+{
+	enum isl_edge_type i;
+	int r;
+
+	for (i = isl_edge_first; i <= isl_edge_last; ++i) {
+		r = graph_has_edge(graph, i, src, dst);
+		if (r < 0 || r)
+			return r;
+	}
+
+	return r;
+}
+
+/* Check whether the dependence graph has a validity edge
+ * between the given two nodes.
+ *
+ * Conditional validity edges are essentially validity edges that
+ * can be ignored if the corresponding condition edges are iteration private.
+ * Here, we are only checking for the presence of validity
+ * edges, so we need to consider the conditional validity edges too.
+ * In particular, this function is used during the detection
+ * of strongly connected components and we cannot ignore
+ * conditional validity edges during this detection.
+ */
+static int graph_has_validity_edge(struct isl_sched_graph *graph,
+	struct isl_sched_node *src, struct isl_sched_node *dst)
+{
+	int r;
+
+	r = graph_has_edge(graph, isl_edge_validity, src, dst);
+	if (r < 0 || r)
+		return r;
+
+	return graph_has_edge(graph, isl_edge_conditional_validity, src, dst);
+}
+
+static int graph_alloc(isl_ctx *ctx, struct isl_sched_graph *graph,
+	int n_node, int n_edge)
+{
+	int i;
+
+	graph->n = n_node;
+	graph->n_edge = n_edge;
+	graph->node = isl_calloc_array(ctx, struct isl_sched_node, graph->n);
+	graph->sorted = isl_calloc_array(ctx, int, graph->n);
+	graph->region = isl_alloc_array(ctx, struct isl_region, graph->n);
+	graph->edge = isl_calloc_array(ctx,
+					struct isl_sched_edge, graph->n_edge);
+
+	graph->intra_hmap = isl_map_to_basic_set_alloc(ctx, 2 * n_edge);
+	graph->inter_hmap = isl_map_to_basic_set_alloc(ctx, 2 * n_edge);
+
+	if (!graph->node || !graph->region || (graph->n_edge && !graph->edge) ||
+	    !graph->sorted)
+		return -1;
+
+	for(i = 0; i < graph->n; ++i)
+		graph->sorted[i] = i;
+
+	return 0;
+}
+
+static void graph_free(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	int i;
+
+	isl_map_to_basic_set_free(graph->intra_hmap);
+	isl_map_to_basic_set_free(graph->inter_hmap);
+
+	if (graph->node)
+		for (i = 0; i < graph->n; ++i) {
+			isl_space_free(graph->node[i].space);
+			isl_set_free(graph->node[i].hull);
+			isl_multi_aff_free(graph->node[i].compress);
+			isl_multi_aff_free(graph->node[i].decompress);
+			isl_mat_free(graph->node[i].sched);
+			isl_map_free(graph->node[i].sched_map);
+			isl_mat_free(graph->node[i].cmap);
+			isl_mat_free(graph->node[i].cinv);
+			if (graph->root) {
+				free(graph->node[i].band);
+				free(graph->node[i].band_id);
+				free(graph->node[i].coincident);
+			}
+		}
+	free(graph->node);
+	free(graph->sorted);
+	if (graph->edge)
+		for (i = 0; i < graph->n_edge; ++i) {
+			isl_map_free(graph->edge[i].map);
+			isl_union_map_free(graph->edge[i].tagged_condition);
+			isl_union_map_free(graph->edge[i].tagged_validity);
+		}
+	free(graph->edge);
+	free(graph->region);
+	for (i = 0; i <= isl_edge_last; ++i)
+		isl_hash_table_free(ctx, graph->edge_table[i]);
+	isl_hash_table_free(ctx, graph->node_table);
+	isl_basic_set_free(graph->lp);
+}
+
+/* For each "set" on which this function is called, increment
+ * graph->n by one and update graph->maxvar.
+ */
+static int init_n_maxvar(__isl_take isl_set *set, void *user)
+{
+	struct isl_sched_graph *graph = user;
+	int nvar = isl_set_dim(set, isl_dim_set);
+
+	graph->n++;
+	if (nvar > graph->maxvar)
+		graph->maxvar = nvar;
+
+	isl_set_free(set);
+
+	return 0;
+}
+
+/* Add the number of basic maps in "map" to *n.
+ */
+static int add_n_basic_map(__isl_take isl_map *map, void *user)
+{
+	int *n = user;
+
+	*n += isl_map_n_basic_map(map);
+	isl_map_free(map);
+
+	return 0;
+}
+
+/* Compute the number of rows that should be allocated for the schedule.
+ * The graph can be split at most "n - 1" times, there can be at most
+ * one row for each dimension in the iteration domains plus two rows
+ * for each basic map in the dependences (in particular,
+ * we usually have one row, but it may be split by split_scaled),
+ * and there can be one extra row for ordering the statements.
+ * Note that if we have actually split "n - 1" times, then no ordering
+ * is needed, so in principle we could use "graph->n + 2 * graph->maxvar - 1".
+ * It is also practically impossible to exhaust both the number of dependences
+ * and the number of variables.
+ */
+static int compute_max_row(struct isl_sched_graph *graph,
+	__isl_keep isl_schedule_constraints *sc)
+{
+	enum isl_edge_type i;
+	int n_edge;
+
+	graph->n = 0;
+	graph->maxvar = 0;
+	if (isl_union_set_foreach_set(sc->domain, &init_n_maxvar, graph) < 0)
+		return -1;
+	n_edge = 0;
+	for (i = isl_edge_first; i <= isl_edge_last; ++i)
+		if (isl_union_map_foreach_map(sc->constraint[i],
+						&add_n_basic_map, &n_edge) < 0)
+			return -1;
+	graph->max_row = graph->n + 2 * n_edge + graph->maxvar;
+
+	return 0;
+}
+
+/* Does "bset" have any defining equalities for its set variables?
+ */
+static int has_any_defining_equality(__isl_keep isl_basic_set *bset)
+{
+	int i, n;
+
+	if (!bset)
+		return -1;
+
+	n = isl_basic_set_dim(bset, isl_dim_set);
+	for (i = 0; i < n; ++i) {
+		int has;
+
+		has = isl_basic_set_has_defining_equality(bset, isl_dim_set, i,
+							NULL);
+		if (has < 0 || has)
+			return has;
+	}
+
+	return 0;
+}
+
+/* Add a new node to the graph representing the given space.
+ * "nvar" is the (possibly compressed) number of variables and
+ * may be smaller than then number of set variables in "space"
+ * if "compressed" is set.
+ * If "compressed" is set, then "hull" represents the constraints
+ * that were used to derive the compression, while "compress" and
+ * "decompress" map the original space to the compressed space and
+ * vice versa.
+ * If "compressed" is not set, then "hull", "compress" and "decompress"
+ * should be NULL.
+ */
+static int add_node(struct isl_sched_graph *graph, __isl_take isl_space *space,
+	int nvar, int compressed, __isl_take isl_set *hull,
+	__isl_take isl_multi_aff *compress,
+	__isl_take isl_multi_aff *decompress)
+{
+	int nparam;
+	isl_ctx *ctx;
+	isl_mat *sched;
+	int *band, *band_id, *coincident;
+
+	if (!space)
+		return -1;
+
+	ctx = isl_space_get_ctx(space);
+	nparam = isl_space_dim(space, isl_dim_param);
+	if (!ctx->opt->schedule_parametric)
+		nparam = 0;
+	sched = isl_mat_alloc(ctx, 0, 1 + nparam + nvar);
+	graph->node[graph->n].space = space;
+	graph->node[graph->n].nvar = nvar;
+	graph->node[graph->n].nparam = nparam;
+	graph->node[graph->n].sched = sched;
+	graph->node[graph->n].sched_map = NULL;
+	band = isl_alloc_array(ctx, int, graph->max_row);
+	graph->node[graph->n].band = band;
+	band_id = isl_calloc_array(ctx, int, graph->max_row);
+	graph->node[graph->n].band_id = band_id;
+	coincident = isl_calloc_array(ctx, int, graph->max_row);
+	graph->node[graph->n].coincident = coincident;
+	graph->node[graph->n].compressed = compressed;
+	graph->node[graph->n].hull = hull;
+	graph->node[graph->n].compress = compress;
+	graph->node[graph->n].decompress = decompress;
+	graph->n++;
+
+	if (!space || !sched ||
+	    (graph->max_row && (!band || !band_id || !coincident)))
+		return -1;
+	if (compressed && (!hull || !compress || !decompress))
+		return -1;
+
+	return 0;
+}
+
+/* Add a new node to the graph representing the given set.
+ *
+ * If any of the set variables is defined by an equality, then
+ * we perform variable compression such that we can perform
+ * the scheduling on the compressed domain.
+ */
+static int extract_node(__isl_take isl_set *set, void *user)
+{
+	int nvar;
+	int has_equality;
+	isl_space *space;
+	isl_basic_set *hull;
+	isl_set *hull_set;
+	isl_morph *morph;
+	isl_multi_aff *compress, *decompress;
+	struct isl_sched_graph *graph = user;
+
+	space = isl_set_get_space(set);
+	hull = isl_set_affine_hull(set);
+	hull = isl_basic_set_remove_divs(hull);
+	nvar = isl_space_dim(space, isl_dim_set);
+	has_equality = has_any_defining_equality(hull);
+
+	if (has_equality < 0)
+		goto error;
+	if (!has_equality) {
+		isl_basic_set_free(hull);
+		return add_node(graph, space, nvar, 0, NULL, NULL, NULL);
+	}
+
+	morph = isl_basic_set_variable_compression(hull, isl_dim_set);
+	nvar = isl_morph_ran_dim(morph, isl_dim_set);
+	compress = isl_morph_get_var_multi_aff(morph);
+	morph = isl_morph_inverse(morph);
+	decompress = isl_morph_get_var_multi_aff(morph);
+	isl_morph_free(morph);
+
+	hull_set = isl_set_from_basic_set(hull);
+	return add_node(graph, space, nvar, 1, hull_set, compress, decompress);
+error:
+	isl_basic_set_free(hull);
+	isl_space_free(space);
+	return -1;
+}
+
+struct isl_extract_edge_data {
+	enum isl_edge_type type;
+	struct isl_sched_graph *graph;
+};
+
+/* Merge edge2 into edge1, freeing the contents of edge2.
+ * "type" is the type of the schedule constraint from which edge2 was
+ * extracted.
+ * Return 0 on success and -1 on failure.
+ *
+ * edge1 and edge2 are assumed to have the same value for the map field.
+ */
+static int merge_edge(enum isl_edge_type type, struct isl_sched_edge *edge1,
+	struct isl_sched_edge *edge2)
+{
+	edge1->validity |= edge2->validity;
+	edge1->coincidence |= edge2->coincidence;
+	edge1->proximity |= edge2->proximity;
+	edge1->condition |= edge2->condition;
+	edge1->conditional_validity |= edge2->conditional_validity;
+	isl_map_free(edge2->map);
+
+	if (type == isl_edge_condition) {
+		if (!edge1->tagged_condition)
+			edge1->tagged_condition = edge2->tagged_condition;
+		else
+			edge1->tagged_condition =
+				isl_union_map_union(edge1->tagged_condition,
+						    edge2->tagged_condition);
+	}
+
+	if (type == isl_edge_conditional_validity) {
+		if (!edge1->tagged_validity)
+			edge1->tagged_validity = edge2->tagged_validity;
+		else
+			edge1->tagged_validity =
+				isl_union_map_union(edge1->tagged_validity,
+						    edge2->tagged_validity);
+	}
+
+	if (type == isl_edge_condition && !edge1->tagged_condition)
+		return -1;
+	if (type == isl_edge_conditional_validity && !edge1->tagged_validity)
+		return -1;
+
+	return 0;
+}
+
+/* Insert dummy tags in domain and range of "map".
+ *
+ * In particular, if "map" is of the form
+ *
+ *	A -> B
+ *
+ * then return
+ *
+ *	[A -> dummy_tag] -> [B -> dummy_tag]
+ *
+ * where the dummy_tags are identical and equal to any dummy tags
+ * introduced by any other call to this function.
+ */
+static __isl_give isl_map *insert_dummy_tags(__isl_take isl_map *map)
+{
+	static char dummy;
+	isl_ctx *ctx;
+	isl_id *id;
+	isl_space *space;
+	isl_set *domain, *range;
+
+	ctx = isl_map_get_ctx(map);
+
+	id = isl_id_alloc(ctx, NULL, &dummy);
+	space = isl_space_params(isl_map_get_space(map));
+	space = isl_space_set_from_params(space);
+	space = isl_space_set_tuple_id(space, isl_dim_set, id);
+	space = isl_space_map_from_set(space);
+
+	domain = isl_map_wrap(map);
+	range = isl_map_wrap(isl_map_universe(space));
+	map = isl_map_from_domain_and_range(domain, range);
+	map = isl_map_zip(map);
+
+	return map;
+}
+
+/* Given that at least one of "src" or "dst" is compressed, return
+ * a map between the spaces of these nodes restricted to the affine
+ * hull that was used in the compression.
+ */
+static __isl_give isl_map *extract_hull(struct isl_sched_node *src,
+	struct isl_sched_node *dst)
+{
+	isl_set *dom, *ran;
+
+	if (src->compressed)
+		dom = isl_set_copy(src->hull);
+	else
+		dom = isl_set_universe(isl_space_copy(src->space));
+	if (dst->compressed)
+		ran = isl_set_copy(dst->hull);
+	else
+		ran = isl_set_universe(isl_space_copy(dst->space));
+
+	return isl_map_from_domain_and_range(dom, ran);
+}
+
+/* Intersect the domains of the nested relations in domain and range
+ * of "tagged" with "map".
+ */
+static __isl_give isl_map *map_intersect_domains(__isl_take isl_map *tagged,
+	__isl_keep isl_map *map)
+{
+	isl_set *set;
+
+	tagged = isl_map_zip(tagged);
+	set = isl_map_wrap(isl_map_copy(map));
+	tagged = isl_map_intersect_domain(tagged, set);
+	tagged = isl_map_zip(tagged);
+	return tagged;
+}
+
+/* Add a new edge to the graph based on the given map
+ * and add it to data->graph->edge_table[data->type].
+ * If a dependence relation of a given type happens to be identical
+ * to one of the dependence relations of a type that was added before,
+ * then we don't create a new edge, but instead mark the original edge
+ * as also representing a dependence of the current type.
+ *
+ * Edges of type isl_edge_condition or isl_edge_conditional_validity
+ * may be specified as "tagged" dependence relations.  That is, "map"
+ * may contain elements (i -> a) -> (j -> b), where i -> j denotes
+ * the dependence on iterations and a and b are tags.
+ * edge->map is set to the relation containing the elements i -> j,
+ * while edge->tagged_condition and edge->tagged_validity contain
+ * the union of all the "map" relations
+ * for which extract_edge is called that result in the same edge->map.
+ *
+ * If the source or the destination node is compressed, then
+ * intersect both "map" and "tagged" with the constraints that
+ * were used to construct the compression.
+ * This ensures that there are no schedule constraints defined
+ * outside of these domains, while the scheduler no longer has
+ * any control over those outside parts.
+ */
+static int extract_edge(__isl_take isl_map *map, void *user)
+{
+	isl_ctx *ctx = isl_map_get_ctx(map);
+	struct isl_extract_edge_data *data = user;
+	struct isl_sched_graph *graph = data->graph;
+	struct isl_sched_node *src, *dst;
+	isl_space *dim;
+	struct isl_sched_edge *edge;
+	isl_map *tagged = NULL;
+
+	if (data->type == isl_edge_condition ||
+	    data->type == isl_edge_conditional_validity) {
+		if (isl_map_can_zip(map)) {
+			tagged = isl_map_copy(map);
+			map = isl_set_unwrap(isl_map_domain(isl_map_zip(map)));
+		} else {
+			tagged = insert_dummy_tags(isl_map_copy(map));
+		}
+	}
+
+	dim = isl_space_domain(isl_map_get_space(map));
+	src = graph_find_node(ctx, graph, dim);
+	isl_space_free(dim);
+	dim = isl_space_range(isl_map_get_space(map));
+	dst = graph_find_node(ctx, graph, dim);
+	isl_space_free(dim);
+
+	if (!src || !dst) {
+		isl_map_free(map);
+		isl_map_free(tagged);
+		return 0;
+	}
+
+	if (src->compressed || dst->compressed) {
+		isl_map *hull;
+		hull = extract_hull(src, dst);
+		if (tagged)
+			tagged = map_intersect_domains(tagged, hull);
+		map = isl_map_intersect(map, hull);
+	}
+
+	graph->edge[graph->n_edge].src = src;
+	graph->edge[graph->n_edge].dst = dst;
+	graph->edge[graph->n_edge].map = map;
+	graph->edge[graph->n_edge].validity = 0;
+	graph->edge[graph->n_edge].coincidence = 0;
+	graph->edge[graph->n_edge].proximity = 0;
+	graph->edge[graph->n_edge].condition = 0;
+	graph->edge[graph->n_edge].local = 0;
+	graph->edge[graph->n_edge].conditional_validity = 0;
+	graph->edge[graph->n_edge].tagged_condition = NULL;
+	graph->edge[graph->n_edge].tagged_validity = NULL;
+	if (data->type == isl_edge_validity)
+		graph->edge[graph->n_edge].validity = 1;
+	if (data->type == isl_edge_coincidence)
+		graph->edge[graph->n_edge].coincidence = 1;
+	if (data->type == isl_edge_proximity)
+		graph->edge[graph->n_edge].proximity = 1;
+	if (data->type == isl_edge_condition) {
+		graph->edge[graph->n_edge].condition = 1;
+		graph->edge[graph->n_edge].tagged_condition =
+					isl_union_map_from_map(tagged);
+	}
+	if (data->type == isl_edge_conditional_validity) {
+		graph->edge[graph->n_edge].conditional_validity = 1;
+		graph->edge[graph->n_edge].tagged_validity =
+					isl_union_map_from_map(tagged);
+	}
+
+	edge = graph_find_matching_edge(graph, &graph->edge[graph->n_edge]);
+	if (!edge) {
+		graph->n_edge++;
+		return -1;
+	}
+	if (edge == &graph->edge[graph->n_edge])
+		return graph_edge_table_add(ctx, graph, data->type,
+				    &graph->edge[graph->n_edge++]);
+
+	if (merge_edge(data->type, edge, &graph->edge[graph->n_edge]) < 0)
+		return -1;
+
+	return graph_edge_table_add(ctx, graph, data->type, edge);
+}
+
+/* Check whether there is any dependence from node[j] to node[i]
+ * or from node[i] to node[j].
+ */
+static int node_follows_weak(int i, int j, void *user)
+{
+	int f;
+	struct isl_sched_graph *graph = user;
+
+	f = graph_has_any_edge(graph, &graph->node[j], &graph->node[i]);
+	if (f < 0 || f)
+		return f;
+	return graph_has_any_edge(graph, &graph->node[i], &graph->node[j]);
+}
+
+/* Check whether there is a (conditional) validity dependence from node[j]
+ * to node[i], forcing node[i] to follow node[j].
+ */
+static int node_follows_strong(int i, int j, void *user)
+{
+	struct isl_sched_graph *graph = user;
+
+	return graph_has_validity_edge(graph, &graph->node[j], &graph->node[i]);
+}
+
+/* Use Tarjan's algorithm for computing the strongly connected components
+ * in the dependence graph (only validity edges).
+ * If weak is set, we consider the graph to be undirected and
+ * we effectively compute the (weakly) connected components.
+ * Additionally, we also consider other edges when weak is set.
+ */
+static int detect_ccs(isl_ctx *ctx, struct isl_sched_graph *graph, int weak)
+{
+	int i, n;
+	struct isl_tarjan_graph *g = NULL;
+
+	g = isl_tarjan_graph_init(ctx, graph->n,
+		weak ? &node_follows_weak : &node_follows_strong, graph);
+	if (!g)
+		return -1;
+
+	graph->scc = 0;
+	i = 0;
+	n = graph->n;
+	while (n) {
+		while (g->order[i] != -1) {
+			graph->node[g->order[i]].scc = graph->scc;
+			--n;
+			++i;
+		}
+		++i;
+		graph->scc++;
+	}
+
+	isl_tarjan_graph_free(g);
+
+	return 0;
+}
+
+/* Apply Tarjan's algorithm to detect the strongly connected components
+ * in the dependence graph.
+ */
+static int detect_sccs(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	return detect_ccs(ctx, graph, 0);
+}
+
+/* Apply Tarjan's algorithm to detect the (weakly) connected components
+ * in the dependence graph.
+ */
+static int detect_wccs(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	return detect_ccs(ctx, graph, 1);
+}
+
+static int cmp_scc(const void *a, const void *b, void *data)
+{
+	struct isl_sched_graph *graph = data;
+	const int *i1 = a;
+	const int *i2 = b;
+
+	return graph->node[*i1].scc - graph->node[*i2].scc;
+}
+
+/* Sort the elements of graph->sorted according to the corresponding SCCs.
+ */
+static int sort_sccs(struct isl_sched_graph *graph)
+{
+	return isl_sort(graph->sorted, graph->n, sizeof(int), &cmp_scc, graph);
+}
+
+/* Given a dependence relation R from "node" to itself,
+ * construct the set of coefficients of valid constraints for elements
+ * in that dependence relation.
+ * In particular, the result contains tuples of coefficients
+ * c_0, c_n, c_x such that
+ *
+ *	c_0 + c_n n + c_x y - c_x x >= 0 for each (x,y) in R
+ *
+ * or, equivalently,
+ *
+ *	c_0 + c_n n + c_x d >= 0 for each d in delta R = { y - x | (x,y) in R }
+ *
+ * We choose here to compute the dual of delta R.
+ * Alternatively, we could have computed the dual of R, resulting
+ * in a set of tuples c_0, c_n, c_x, c_y, and then
+ * plugged in (c_0, c_n, c_x, -c_x).
+ *
+ * If "node" has been compressed, then the dependence relation
+ * is also compressed before the set of coefficients is computed.
+ */
+static __isl_give isl_basic_set *intra_coefficients(
+	struct isl_sched_graph *graph, struct isl_sched_node *node,
+	__isl_take isl_map *map)
+{
+	isl_set *delta;
+	isl_map *key;
+	isl_basic_set *coef;
+
+	if (isl_map_to_basic_set_has(graph->intra_hmap, map))
+		return isl_map_to_basic_set_get(graph->intra_hmap, map);
+
+	key = isl_map_copy(map);
+	if (node->compressed) {
+		map = isl_map_preimage_domain_multi_aff(map,
+				    isl_multi_aff_copy(node->decompress));
+		map = isl_map_preimage_range_multi_aff(map,
+				    isl_multi_aff_copy(node->decompress));
+	}
+	delta = isl_set_remove_divs(isl_map_deltas(map));
+	coef = isl_set_coefficients(delta);
+	graph->intra_hmap = isl_map_to_basic_set_set(graph->intra_hmap, key,
+					isl_basic_set_copy(coef));
+
+	return coef;
+}
+
+/* Given a dependence relation R, construct the set of coefficients
+ * of valid constraints for elements in that dependence relation.
+ * In particular, the result contains tuples of coefficients
+ * c_0, c_n, c_x, c_y such that
+ *
+ *	c_0 + c_n n + c_x x + c_y y >= 0 for each (x,y) in R
+ *
+ * If the source or destination nodes of "edge" have been compressed,
+ * then the dependence relation is also compressed before
+ * the set of coefficients is computed.
+ */
+static __isl_give isl_basic_set *inter_coefficients(
+	struct isl_sched_graph *graph, struct isl_sched_edge *edge,
+	__isl_take isl_map *map)
+{
+	isl_set *set;
+	isl_map *key;
+	isl_basic_set *coef;
+
+	if (isl_map_to_basic_set_has(graph->inter_hmap, map))
+		return isl_map_to_basic_set_get(graph->inter_hmap, map);
+
+	key = isl_map_copy(map);
+	if (edge->src->compressed)
+		map = isl_map_preimage_domain_multi_aff(map,
+				    isl_multi_aff_copy(edge->src->decompress));
+	if (edge->dst->compressed)
+		map = isl_map_preimage_range_multi_aff(map,
+				    isl_multi_aff_copy(edge->dst->decompress));
+	set = isl_map_wrap(isl_map_remove_divs(map));
+	coef = isl_set_coefficients(set);
+	graph->inter_hmap = isl_map_to_basic_set_set(graph->inter_hmap, key,
+					isl_basic_set_copy(coef));
+
+	return coef;
+}
+
+/* Add constraints to graph->lp that force validity for the given
+ * dependence from a node i to itself.
+ * That is, add constraints that enforce
+ *
+ *	(c_i_0 + c_i_n n + c_i_x y) - (c_i_0 + c_i_n n + c_i_x x)
+ *	= c_i_x (y - x) >= 0
+ *
+ * for each (x,y) in R.
+ * We obtain general constraints on coefficients (c_0, c_n, c_x)
+ * of valid constraints for (y - x) and then plug in (0, 0, c_i_x^+ - c_i_x^-),
+ * where c_i_x = c_i_x^+ - c_i_x^-, with c_i_x^+ and c_i_x^- non-negative.
+ * In graph->lp, the c_i_x^- appear before their c_i_x^+ counterpart.
+ *
+ * Actually, we do not construct constraints for the c_i_x themselves,
+ * but for the coefficients of c_i_x written as a linear combination
+ * of the columns in node->cmap.
+ */
+static int add_intra_validity_constraints(struct isl_sched_graph *graph,
+	struct isl_sched_edge *edge)
+{
+	unsigned total;
+	isl_map *map = isl_map_copy(edge->map);
+	isl_ctx *ctx = isl_map_get_ctx(map);
+	isl_space *dim;
+	isl_dim_map *dim_map;
+	isl_basic_set *coef;
+	struct isl_sched_node *node = edge->src;
+
+	coef = intra_coefficients(graph, node, map);
+
+	dim = isl_space_domain(isl_space_unwrap(isl_basic_set_get_space(coef)));
+
+	coef = isl_basic_set_transform_dims(coef, isl_dim_set,
+		    isl_space_dim(dim, isl_dim_set), isl_mat_copy(node->cmap));
+	if (!coef)
+		goto error;
+
+	total = isl_basic_set_total_dim(graph->lp);
+	dim_map = isl_dim_map_alloc(ctx, total);
+	isl_dim_map_range(dim_map, node->start + 2 * node->nparam + 1, 2,
+			  isl_space_dim(dim, isl_dim_set), 1,
+			  node->nvar, -1);
+	isl_dim_map_range(dim_map, node->start + 2 * node->nparam + 2, 2,
+			  isl_space_dim(dim, isl_dim_set), 1,
+			  node->nvar, 1);
+	graph->lp = isl_basic_set_extend_constraints(graph->lp,
+			coef->n_eq, coef->n_ineq);
+	graph->lp = isl_basic_set_add_constraints_dim_map(graph->lp,
+							   coef, dim_map);
+	isl_space_free(dim);
+
+	return 0;
+error:
+	isl_space_free(dim);
+	return -1;
+}
+
+/* Add constraints to graph->lp that force validity for the given
+ * dependence from node i to node j.
+ * That is, add constraints that enforce
+ *
+ *	(c_j_0 + c_j_n n + c_j_x y) - (c_i_0 + c_i_n n + c_i_x x) >= 0
+ *
+ * for each (x,y) in R.
+ * We obtain general constraints on coefficients (c_0, c_n, c_x, c_y)
+ * of valid constraints for R and then plug in
+ * (c_j_0 - c_i_0, c_j_n^+ - c_j_n^- - (c_i_n^+ - c_i_n^-),
+ *  c_j_x^+ - c_j_x^- - (c_i_x^+ - c_i_x^-)),
+ * where c_* = c_*^+ - c_*^-, with c_*^+ and c_*^- non-negative.
+ * In graph->lp, the c_*^- appear before their c_*^+ counterpart.
+ *
+ * Actually, we do not construct constraints for the c_*_x themselves,
+ * but for the coefficients of c_*_x written as a linear combination
+ * of the columns in node->cmap.
+ */
+static int add_inter_validity_constraints(struct isl_sched_graph *graph,
+	struct isl_sched_edge *edge)
+{
+	unsigned total;
+	isl_map *map = isl_map_copy(edge->map);
+	isl_ctx *ctx = isl_map_get_ctx(map);
+	isl_space *dim;
+	isl_dim_map *dim_map;
+	isl_basic_set *coef;
+	struct isl_sched_node *src = edge->src;
+	struct isl_sched_node *dst = edge->dst;
+
+	coef = inter_coefficients(graph, edge, map);
+
+	dim = isl_space_domain(isl_space_unwrap(isl_basic_set_get_space(coef)));
+
+	coef = isl_basic_set_transform_dims(coef, isl_dim_set,
+		    isl_space_dim(dim, isl_dim_set), isl_mat_copy(src->cmap));
+	coef = isl_basic_set_transform_dims(coef, isl_dim_set,
+		    isl_space_dim(dim, isl_dim_set) + src->nvar,
+		    isl_mat_copy(dst->cmap));
+	if (!coef)
+		goto error;
+
+	total = isl_basic_set_total_dim(graph->lp);
+	dim_map = isl_dim_map_alloc(ctx, total);
+
+	isl_dim_map_range(dim_map, dst->start, 0, 0, 0, 1, 1);
+	isl_dim_map_range(dim_map, dst->start + 1, 2, 1, 1, dst->nparam, -1);
+	isl_dim_map_range(dim_map, dst->start + 2, 2, 1, 1, dst->nparam, 1);
+	isl_dim_map_range(dim_map, dst->start + 2 * dst->nparam + 1, 2,
+			  isl_space_dim(dim, isl_dim_set) + src->nvar, 1,
+			  dst->nvar, -1);
+	isl_dim_map_range(dim_map, dst->start + 2 * dst->nparam + 2, 2,
+			  isl_space_dim(dim, isl_dim_set) + src->nvar, 1,
+			  dst->nvar, 1);
+
+	isl_dim_map_range(dim_map, src->start, 0, 0, 0, 1, -1);
+	isl_dim_map_range(dim_map, src->start + 1, 2, 1, 1, src->nparam, 1);
+	isl_dim_map_range(dim_map, src->start + 2, 2, 1, 1, src->nparam, -1);
+	isl_dim_map_range(dim_map, src->start + 2 * src->nparam + 1, 2,
+			  isl_space_dim(dim, isl_dim_set), 1,
+			  src->nvar, 1);
+	isl_dim_map_range(dim_map, src->start + 2 * src->nparam + 2, 2,
+			  isl_space_dim(dim, isl_dim_set), 1,
+			  src->nvar, -1);
+
+	edge->start = graph->lp->n_ineq;
+	graph->lp = isl_basic_set_extend_constraints(graph->lp,
+			coef->n_eq, coef->n_ineq);
+	graph->lp = isl_basic_set_add_constraints_dim_map(graph->lp,
+							   coef, dim_map);
+	if (!graph->lp)
+		goto error;
+	isl_space_free(dim);
+	edge->end = graph->lp->n_ineq;
+
+	return 0;
+error:
+	isl_space_free(dim);
+	return -1;
+}
+
+/* Add constraints to graph->lp that bound the dependence distance for the given
+ * dependence from a node i to itself.
+ * If s = 1, we add the constraint
+ *
+ *	c_i_x (y - x) <= m_0 + m_n n
+ *
+ * or
+ *
+ *	-c_i_x (y - x) + m_0 + m_n n >= 0
+ *
+ * for each (x,y) in R.
+ * If s = -1, we add the constraint
+ *
+ *	-c_i_x (y - x) <= m_0 + m_n n
+ *
+ * or
+ *
+ *	c_i_x (y - x) + m_0 + m_n n >= 0
+ *
+ * for each (x,y) in R.
+ * We obtain general constraints on coefficients (c_0, c_n, c_x)
+ * of valid constraints for (y - x) and then plug in (m_0, m_n, -s * c_i_x),
+ * with each coefficient (except m_0) represented as a pair of non-negative
+ * coefficients.
+ *
+ * Actually, we do not construct constraints for the c_i_x themselves,
+ * but for the coefficients of c_i_x written as a linear combination
+ * of the columns in node->cmap.
+ *
+ *
+ * If "local" is set, then we add constraints
+ *
+ *	c_i_x (y - x) <= 0
+ *
+ * or
+ *
+ *	-c_i_x (y - x) <= 0
+ *
+ * instead, forcing the dependence distance to be (less than or) equal to 0.
+ * That is, we plug in (0, 0, -s * c_i_x),
+ * Note that dependences marked local are treated as validity constraints
+ * by add_all_validity_constraints and therefore also have
+ * their distances bounded by 0 from below.
+ */
+static int add_intra_proximity_constraints(struct isl_sched_graph *graph,
+	struct isl_sched_edge *edge, int s, int local)
+{
+	unsigned total;
+	unsigned nparam;
+	isl_map *map = isl_map_copy(edge->map);
+	isl_ctx *ctx = isl_map_get_ctx(map);
+	isl_space *dim;
+	isl_dim_map *dim_map;
+	isl_basic_set *coef;
+	struct isl_sched_node *node = edge->src;
+
+	coef = intra_coefficients(graph, node, map);
+
+	dim = isl_space_domain(isl_space_unwrap(isl_basic_set_get_space(coef)));
+
+	coef = isl_basic_set_transform_dims(coef, isl_dim_set,
+		    isl_space_dim(dim, isl_dim_set), isl_mat_copy(node->cmap));
+	if (!coef)
+		goto error;
+
+	nparam = isl_space_dim(node->space, isl_dim_param);
+	total = isl_basic_set_total_dim(graph->lp);
+	dim_map = isl_dim_map_alloc(ctx, total);
+
+	if (!local) {
+		isl_dim_map_range(dim_map, 1, 0, 0, 0, 1, 1);
+		isl_dim_map_range(dim_map, 4, 2, 1, 1, nparam, -1);
+		isl_dim_map_range(dim_map, 5, 2, 1, 1, nparam, 1);
+	}
+	isl_dim_map_range(dim_map, node->start + 2 * node->nparam + 1, 2,
+			  isl_space_dim(dim, isl_dim_set), 1,
+			  node->nvar, s);
+	isl_dim_map_range(dim_map, node->start + 2 * node->nparam + 2, 2,
+			  isl_space_dim(dim, isl_dim_set), 1,
+			  node->nvar, -s);
+	graph->lp = isl_basic_set_extend_constraints(graph->lp,
+			coef->n_eq, coef->n_ineq);
+	graph->lp = isl_basic_set_add_constraints_dim_map(graph->lp,
+							   coef, dim_map);
+	isl_space_free(dim);
+
+	return 0;
+error:
+	isl_space_free(dim);
+	return -1;
+}
+
+/* Add constraints to graph->lp that bound the dependence distance for the given
+ * dependence from node i to node j.
+ * If s = 1, we add the constraint
+ *
+ *	(c_j_0 + c_j_n n + c_j_x y) - (c_i_0 + c_i_n n + c_i_x x)
+ *		<= m_0 + m_n n
+ *
+ * or
+ *
+ *	-(c_j_0 + c_j_n n + c_j_x y) + (c_i_0 + c_i_n n + c_i_x x) +
+ *		m_0 + m_n n >= 0
+ *
+ * for each (x,y) in R.
+ * If s = -1, we add the constraint
+ *
+ *	-((c_j_0 + c_j_n n + c_j_x y) - (c_i_0 + c_i_n n + c_i_x x))
+ *		<= m_0 + m_n n
+ *
+ * or
+ *
+ *	(c_j_0 + c_j_n n + c_j_x y) - (c_i_0 + c_i_n n + c_i_x x) +
+ *		m_0 + m_n n >= 0
+ *
+ * for each (x,y) in R.
+ * We obtain general constraints on coefficients (c_0, c_n, c_x, c_y)
+ * of valid constraints for R and then plug in
+ * (m_0 - s*c_j_0 + s*c_i_0, m_n - s*c_j_n + s*c_i_n,
+ *  -s*c_j_x+s*c_i_x)
+ * with each coefficient (except m_0, c_j_0 and c_i_0)
+ * represented as a pair of non-negative coefficients.
+ *
+ * Actually, we do not construct constraints for the c_*_x themselves,
+ * but for the coefficients of c_*_x written as a linear combination
+ * of the columns in node->cmap.
+ *
+ *
+ * If "local" is set, then we add constraints
+ *
+ *	(c_j_0 + c_j_n n + c_j_x y) - (c_i_0 + c_i_n n + c_i_x x) <= 0
+ *
+ * or
+ *
+ *	-((c_j_0 + c_j_n n + c_j_x y) - (c_i_0 + c_i_n n + c_i_x x)) <= 0
+ *
+ * instead, forcing the dependence distance to be (less than or) equal to 0.
+ * That is, we plug in
+ * (-s*c_j_0 + s*c_i_0, -s*c_j_n + s*c_i_n, -s*c_j_x+s*c_i_x).
+ * Note that dependences marked local are treated as validity constraints
+ * by add_all_validity_constraints and therefore also have
+ * their distances bounded by 0 from below.
+ */
+static int add_inter_proximity_constraints(struct isl_sched_graph *graph,
+	struct isl_sched_edge *edge, int s, int local)
+{
+	unsigned total;
+	unsigned nparam;
+	isl_map *map = isl_map_copy(edge->map);
+	isl_ctx *ctx = isl_map_get_ctx(map);
+	isl_space *dim;
+	isl_dim_map *dim_map;
+	isl_basic_set *coef;
+	struct isl_sched_node *src = edge->src;
+	struct isl_sched_node *dst = edge->dst;
+
+	coef = inter_coefficients(graph, edge, map);
+
+	dim = isl_space_domain(isl_space_unwrap(isl_basic_set_get_space(coef)));
+
+	coef = isl_basic_set_transform_dims(coef, isl_dim_set,
+		    isl_space_dim(dim, isl_dim_set), isl_mat_copy(src->cmap));
+	coef = isl_basic_set_transform_dims(coef, isl_dim_set,
+		    isl_space_dim(dim, isl_dim_set) + src->nvar,
+		    isl_mat_copy(dst->cmap));
+	if (!coef)
+		goto error;
+
+	nparam = isl_space_dim(src->space, isl_dim_param);
+	total = isl_basic_set_total_dim(graph->lp);
+	dim_map = isl_dim_map_alloc(ctx, total);
+
+	if (!local) {
+		isl_dim_map_range(dim_map, 1, 0, 0, 0, 1, 1);
+		isl_dim_map_range(dim_map, 4, 2, 1, 1, nparam, -1);
+		isl_dim_map_range(dim_map, 5, 2, 1, 1, nparam, 1);
+	}
+
+	isl_dim_map_range(dim_map, dst->start, 0, 0, 0, 1, -s);
+	isl_dim_map_range(dim_map, dst->start + 1, 2, 1, 1, dst->nparam, s);
+	isl_dim_map_range(dim_map, dst->start + 2, 2, 1, 1, dst->nparam, -s);
+	isl_dim_map_range(dim_map, dst->start + 2 * dst->nparam + 1, 2,
+			  isl_space_dim(dim, isl_dim_set) + src->nvar, 1,
+			  dst->nvar, s);
+	isl_dim_map_range(dim_map, dst->start + 2 * dst->nparam + 2, 2,
+			  isl_space_dim(dim, isl_dim_set) + src->nvar, 1,
+			  dst->nvar, -s);
+
+	isl_dim_map_range(dim_map, src->start, 0, 0, 0, 1, s);
+	isl_dim_map_range(dim_map, src->start + 1, 2, 1, 1, src->nparam, -s);
+	isl_dim_map_range(dim_map, src->start + 2, 2, 1, 1, src->nparam, s);
+	isl_dim_map_range(dim_map, src->start + 2 * src->nparam + 1, 2,
+			  isl_space_dim(dim, isl_dim_set), 1,
+			  src->nvar, -s);
+	isl_dim_map_range(dim_map, src->start + 2 * src->nparam + 2, 2,
+			  isl_space_dim(dim, isl_dim_set), 1,
+			  src->nvar, s);
+
+	graph->lp = isl_basic_set_extend_constraints(graph->lp,
+			coef->n_eq, coef->n_ineq);
+	graph->lp = isl_basic_set_add_constraints_dim_map(graph->lp,
+							   coef, dim_map);
+	isl_space_free(dim);
+
+	return 0;
+error:
+	isl_space_free(dim);
+	return -1;
+}
+
+/* Add all validity constraints to graph->lp.
+ *
+ * An edge that is forced to be local needs to have its dependence
+ * distances equal to zero.  We take care of bounding them by 0 from below
+ * here.  add_all_proximity_constraints takes care of bounding them by 0
+ * from above.
+ *
+ * If "use_coincidence" is set, then we treat coincidence edges as local edges.
+ * Otherwise, we ignore them.
+ */
+static int add_all_validity_constraints(struct isl_sched_graph *graph,
+	int use_coincidence)
+{
+	int i;
+
+	for (i = 0; i < graph->n_edge; ++i) {
+		struct isl_sched_edge *edge= &graph->edge[i];
+		int local;
+
+		local = edge->local || (edge->coincidence && use_coincidence);
+		if (!edge->validity && !local)
+			continue;
+		if (edge->src != edge->dst)
+			continue;
+		if (add_intra_validity_constraints(graph, edge) < 0)
+			return -1;
+	}
+
+	for (i = 0; i < graph->n_edge; ++i) {
+		struct isl_sched_edge *edge = &graph->edge[i];
+		int local;
+
+		local = edge->local || (edge->coincidence && use_coincidence);
+		if (!edge->validity && !local)
+			continue;
+		if (edge->src == edge->dst)
+			continue;
+		if (add_inter_validity_constraints(graph, edge) < 0)
+			return -1;
+	}
+
+	return 0;
+}
+
+/* Add constraints to graph->lp that bound the dependence distance
+ * for all dependence relations.
+ * If a given proximity dependence is identical to a validity
+ * dependence, then the dependence distance is already bounded
+ * from below (by zero), so we only need to bound the distance
+ * from above.  (This includes the case of "local" dependences
+ * which are treated as validity dependence by add_all_validity_constraints.)
+ * Otherwise, we need to bound the distance both from above and from below.
+ *
+ * If "use_coincidence" is set, then we treat coincidence edges as local edges.
+ * Otherwise, we ignore them.
+ */
+static int add_all_proximity_constraints(struct isl_sched_graph *graph,
+	int use_coincidence)
+{
+	int i;
+
+	for (i = 0; i < graph->n_edge; ++i) {
+		struct isl_sched_edge *edge= &graph->edge[i];
+		int local;
+
+		local = edge->local || (edge->coincidence && use_coincidence);
+		if (!edge->proximity && !local)
+			continue;
+		if (edge->src == edge->dst &&
+		    add_intra_proximity_constraints(graph, edge, 1, local) < 0)
+			return -1;
+		if (edge->src != edge->dst &&
+		    add_inter_proximity_constraints(graph, edge, 1, local) < 0)
+			return -1;
+		if (edge->validity || local)
+			continue;
+		if (edge->src == edge->dst &&
+		    add_intra_proximity_constraints(graph, edge, -1, 0) < 0)
+			return -1;
+		if (edge->src != edge->dst &&
+		    add_inter_proximity_constraints(graph, edge, -1, 0) < 0)
+			return -1;
+	}
+
+	return 0;
+}
+
+/* Compute a basis for the rows in the linear part of the schedule
+ * and extend this basis to a full basis.  The remaining rows
+ * can then be used to force linear independence from the rows
+ * in the schedule.
+ *
+ * In particular, given the schedule rows S, we compute
+ *
+ *	S   = H Q
+ *	S U = H
+ *
+ * with H the Hermite normal form of S.  That is, all but the
+ * first rank columns of H are zero and so each row in S is
+ * a linear combination of the first rank rows of Q.
+ * The matrix Q is then transposed because we will write the
+ * coefficients of the next schedule row as a column vector s
+ * and express this s as a linear combination s = Q c of the
+ * computed basis.
+ * Similarly, the matrix U is transposed such that we can
+ * compute the coefficients c = U s from a schedule row s.
+ */
+static int node_update_cmap(struct isl_sched_node *node)
+{
+	isl_mat *H, *U, *Q;
+	int n_row = isl_mat_rows(node->sched);
+
+	H = isl_mat_sub_alloc(node->sched, 0, n_row,
+			      1 + node->nparam, node->nvar);
+
+	H = isl_mat_left_hermite(H, 0, &U, &Q);
+	isl_mat_free(node->cmap);
+	isl_mat_free(node->cinv);
+	node->cmap = isl_mat_transpose(Q);
+	node->cinv = isl_mat_transpose(U);
+	node->rank = isl_mat_initial_non_zero_cols(H);
+	isl_mat_free(H);
+
+	if (!node->cmap || !node->cinv || node->rank < 0)
+		return -1;
+	return 0;
+}
+
+/* How many times should we count the constraints in "edge"?
+ *
+ * If carry is set, then we are counting the number of
+ * (validity or conditional validity) constraints that will be added
+ * in setup_carry_lp and we count each edge exactly once.
+ *
+ * Otherwise, we count as follows
+ * validity		-> 1 (>= 0)
+ * validity+proximity	-> 2 (>= 0 and upper bound)
+ * proximity		-> 2 (lower and upper bound)
+ * local(+any)		-> 2 (>= 0 and <= 0)
+ *
+ * If an edge is only marked conditional_validity then it counts
+ * as zero since it is only checked afterwards.
+ *
+ * If "use_coincidence" is set, then we treat coincidence edges as local edges.
+ * Otherwise, we ignore them.
+ */
+static int edge_multiplicity(struct isl_sched_edge *edge, int carry,
+	int use_coincidence)
+{
+	if (carry && !edge->validity && !edge->conditional_validity)
+		return 0;
+	if (carry)
+		return 1;
+	if (edge->proximity || edge->local)
+		return 2;
+	if (use_coincidence && edge->coincidence)
+		return 2;
+	if (edge->validity)
+		return 1;
+	return 0;
+}
+
+/* Count the number of equality and inequality constraints
+ * that will be added for the given map.
+ *
+ * "use_coincidence" is set if we should take into account coincidence edges.
+ */
+static int count_map_constraints(struct isl_sched_graph *graph,
+	struct isl_sched_edge *edge, __isl_take isl_map *map,
+	int *n_eq, int *n_ineq, int carry, int use_coincidence)
+{
+	isl_basic_set *coef;
+	int f = edge_multiplicity(edge, carry, use_coincidence);
+
+	if (f == 0) {
+		isl_map_free(map);
+		return 0;
+	}
+
+	if (edge->src == edge->dst)
+		coef = intra_coefficients(graph, edge->src, map);
+	else
+		coef = inter_coefficients(graph, edge, map);
+	if (!coef)
+		return -1;
+	*n_eq += f * coef->n_eq;
+	*n_ineq += f * coef->n_ineq;
+	isl_basic_set_free(coef);
+
+	return 0;
+}
+
+/* Count the number of equality and inequality constraints
+ * that will be added to the main lp problem.
+ * We count as follows
+ * validity		-> 1 (>= 0)
+ * validity+proximity	-> 2 (>= 0 and upper bound)
+ * proximity		-> 2 (lower and upper bound)
+ * local(+any)		-> 2 (>= 0 and <= 0)
+ *
+ * If "use_coincidence" is set, then we treat coincidence edges as local edges.
+ * Otherwise, we ignore them.
+ */
+static int count_constraints(struct isl_sched_graph *graph,
+	int *n_eq, int *n_ineq, int use_coincidence)
+{
+	int i;
+
+	*n_eq = *n_ineq = 0;
+	for (i = 0; i < graph->n_edge; ++i) {
+		struct isl_sched_edge *edge= &graph->edge[i];
+		isl_map *map = isl_map_copy(edge->map);
+
+		if (count_map_constraints(graph, edge, map, n_eq, n_ineq,
+					    0, use_coincidence) < 0)
+			return -1;
+	}
+
+	return 0;
+}
+
+/* Count the number of constraints that will be added by
+ * add_bound_coefficient_constraints and increment *n_eq and *n_ineq
+ * accordingly.
+ *
+ * In practice, add_bound_coefficient_constraints only adds inequalities.
+ */
+static int count_bound_coefficient_constraints(isl_ctx *ctx,
+	struct isl_sched_graph *graph, int *n_eq, int *n_ineq)
+{
+	int i;
+
+	if (ctx->opt->schedule_max_coefficient == -1)
+		return 0;
+
+	for (i = 0; i < graph->n; ++i)
+		*n_ineq += 2 * graph->node[i].nparam + 2 * graph->node[i].nvar;
+
+	return 0;
+}
+
+/* Add constraints that bound the values of the variable and parameter
+ * coefficients of the schedule.
+ *
+ * The maximal value of the coefficients is defined by the option
+ * 'schedule_max_coefficient'.
+ */
+static int add_bound_coefficient_constraints(isl_ctx *ctx,
+	struct isl_sched_graph *graph)
+{
+	int i, j, k;
+	int max_coefficient;
+	int total;
+
+	max_coefficient = ctx->opt->schedule_max_coefficient;
+
+	if (max_coefficient == -1)
+		return 0;
+
+	total = isl_basic_set_total_dim(graph->lp);
+
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[i];
+		for (j = 0; j < 2 * node->nparam + 2 * node->nvar; ++j) {
+			int dim;
+			k = isl_basic_set_alloc_inequality(graph->lp);
+			if (k < 0)
+				return -1;
+			dim = 1 + node->start + 1 + j;
+			isl_seq_clr(graph->lp->ineq[k], 1 +  total);
+			isl_int_set_si(graph->lp->ineq[k][dim], -1);
+			isl_int_set_si(graph->lp->ineq[k][0], max_coefficient);
+		}
+	}
+
+	return 0;
+}
+
+/* Construct an ILP problem for finding schedule coefficients
+ * that result in non-negative, but small dependence distances
+ * over all dependences.
+ * In particular, the dependence distances over proximity edges
+ * are bounded by m_0 + m_n n and we compute schedule coefficients
+ * with small values (preferably zero) of m_n and m_0.
+ *
+ * All variables of the ILP are non-negative.  The actual coefficients
+ * may be negative, so each coefficient is represented as the difference
+ * of two non-negative variables.  The negative part always appears
+ * immediately before the positive part.
+ * Other than that, the variables have the following order
+ *
+ *	- sum of positive and negative parts of m_n coefficients
+ *	- m_0
+ *	- sum of positive and negative parts of all c_n coefficients
+ *		(unconstrained when computing non-parametric schedules)
+ *	- sum of positive and negative parts of all c_x coefficients
+ *	- positive and negative parts of m_n coefficients
+ *	- for each node
+ *		- c_i_0
+ *		- positive and negative parts of c_i_n (if parametric)
+ *		- positive and negative parts of c_i_x
+ *
+ * The c_i_x are not represented directly, but through the columns of
+ * node->cmap.  That is, the computed values are for variable t_i_x
+ * such that c_i_x = Q t_i_x with Q equal to node->cmap.
+ *
+ * The constraints are those from the edges plus two or three equalities
+ * to express the sums.
+ *
+ * If "use_coincidence" is set, then we treat coincidence edges as local edges.
+ * Otherwise, we ignore them.
+ */
+static int setup_lp(isl_ctx *ctx, struct isl_sched_graph *graph,
+	int use_coincidence)
+{
+	int i, j;
+	int k;
+	unsigned nparam;
+	unsigned total;
+	isl_space *dim;
+	int parametric;
+	int param_pos;
+	int n_eq, n_ineq;
+	int max_constant_term;
+
+	max_constant_term = ctx->opt->schedule_max_constant_term;
+
+	parametric = ctx->opt->schedule_parametric;
+	nparam = isl_space_dim(graph->node[0].space, isl_dim_param);
+	param_pos = 4;
+	total = param_pos + 2 * nparam;
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[graph->sorted[i]];
+		if (node_update_cmap(node) < 0)
+			return -1;
+		node->start = total;
+		total += 1 + 2 * (node->nparam + node->nvar);
+	}
+
+	if (count_constraints(graph, &n_eq, &n_ineq, use_coincidence) < 0)
+		return -1;
+	if (count_bound_coefficient_constraints(ctx, graph, &n_eq, &n_ineq) < 0)
+		return -1;
+
+	dim = isl_space_set_alloc(ctx, 0, total);
+	isl_basic_set_free(graph->lp);
+	n_eq += 2 + parametric;
+	if (max_constant_term != -1)
+		n_ineq += graph->n;
+
+	graph->lp = isl_basic_set_alloc_space(dim, 0, n_eq, n_ineq);
+
+	k = isl_basic_set_alloc_equality(graph->lp);
+	if (k < 0)
+		return -1;
+	isl_seq_clr(graph->lp->eq[k], 1 +  total);
+	isl_int_set_si(graph->lp->eq[k][1], -1);
+	for (i = 0; i < 2 * nparam; ++i)
+		isl_int_set_si(graph->lp->eq[k][1 + param_pos + i], 1);
+
+	if (parametric) {
+		k = isl_basic_set_alloc_equality(graph->lp);
+		if (k < 0)
+			return -1;
+		isl_seq_clr(graph->lp->eq[k], 1 +  total);
+		isl_int_set_si(graph->lp->eq[k][3], -1);
+		for (i = 0; i < graph->n; ++i) {
+			int pos = 1 + graph->node[i].start + 1;
+
+			for (j = 0; j < 2 * graph->node[i].nparam; ++j)
+				isl_int_set_si(graph->lp->eq[k][pos + j], 1);
+		}
+	}
+
+	k = isl_basic_set_alloc_equality(graph->lp);
+	if (k < 0)
+		return -1;
+	isl_seq_clr(graph->lp->eq[k], 1 +  total);
+	isl_int_set_si(graph->lp->eq[k][4], -1);
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[i];
+		int pos = 1 + node->start + 1 + 2 * node->nparam;
+
+		for (j = 0; j < 2 * node->nvar; ++j)
+			isl_int_set_si(graph->lp->eq[k][pos + j], 1);
+	}
+
+	if (max_constant_term != -1)
+		for (i = 0; i < graph->n; ++i) {
+			struct isl_sched_node *node = &graph->node[i];
+			k = isl_basic_set_alloc_inequality(graph->lp);
+			if (k < 0)
+				return -1;
+			isl_seq_clr(graph->lp->ineq[k], 1 +  total);
+			isl_int_set_si(graph->lp->ineq[k][1 + node->start], -1);
+			isl_int_set_si(graph->lp->ineq[k][0], max_constant_term);
+		}
+
+	if (add_bound_coefficient_constraints(ctx, graph) < 0)
+		return -1;
+	if (add_all_validity_constraints(graph, use_coincidence) < 0)
+		return -1;
+	if (add_all_proximity_constraints(graph, use_coincidence) < 0)
+		return -1;
+
+	return 0;
+}
+
+/* Analyze the conflicting constraint found by
+ * isl_tab_basic_set_non_trivial_lexmin.  If it corresponds to the validity
+ * constraint of one of the edges between distinct nodes, living, moreover
+ * in distinct SCCs, then record the source and sink SCC as this may
+ * be a good place to cut between SCCs.
+ */
+static int check_conflict(int con, void *user)
+{
+	int i;
+	struct isl_sched_graph *graph = user;
+
+	if (graph->src_scc >= 0)
+		return 0;
+
+	con -= graph->lp->n_eq;
+
+	if (con >= graph->lp->n_ineq)
+		return 0;
+
+	for (i = 0; i < graph->n_edge; ++i) {
+		if (!graph->edge[i].validity)
+			continue;
+		if (graph->edge[i].src == graph->edge[i].dst)
+			continue;
+		if (graph->edge[i].src->scc == graph->edge[i].dst->scc)
+			continue;
+		if (graph->edge[i].start > con)
+			continue;
+		if (graph->edge[i].end <= con)
+			continue;
+		graph->src_scc = graph->edge[i].src->scc;
+		graph->dst_scc = graph->edge[i].dst->scc;
+	}
+
+	return 0;
+}
+
+/* Check whether the next schedule row of the given node needs to be
+ * non-trivial.  Lower-dimensional domains may have some trivial rows,
+ * but as soon as the number of remaining required non-trivial rows
+ * is as large as the number or remaining rows to be computed,
+ * all remaining rows need to be non-trivial.
+ */
+static int needs_row(struct isl_sched_graph *graph, struct isl_sched_node *node)
+{
+	return node->nvar - node->rank >= graph->maxvar - graph->n_row;
+}
+
+/* Solve the ILP problem constructed in setup_lp.
+ * For each node such that all the remaining rows of its schedule
+ * need to be non-trivial, we construct a non-triviality region.
+ * This region imposes that the next row is independent of previous rows.
+ * In particular the coefficients c_i_x are represented by t_i_x
+ * variables with c_i_x = Q t_i_x and Q a unimodular matrix such that
+ * its first columns span the rows of the previously computed part
+ * of the schedule.  The non-triviality region enforces that at least
+ * one of the remaining components of t_i_x is non-zero, i.e.,
+ * that the new schedule row depends on at least one of the remaining
+ * columns of Q.
+ */
+static __isl_give isl_vec *solve_lp(struct isl_sched_graph *graph)
+{
+	int i;
+	isl_vec *sol;
+	isl_basic_set *lp;
+
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[i];
+		int skip = node->rank;
+		graph->region[i].pos = node->start + 1 + 2*(node->nparam+skip);
+		if (needs_row(graph, node))
+			graph->region[i].len = 2 * (node->nvar - skip);
+		else
+			graph->region[i].len = 0;
+	}
+	lp = isl_basic_set_copy(graph->lp);
+	sol = isl_tab_basic_set_non_trivial_lexmin(lp, 2, graph->n,
+				       graph->region, &check_conflict, graph);
+	return sol;
+}
+
+/* Update the schedules of all nodes based on the given solution
+ * of the LP problem.
+ * The new row is added to the current band.
+ * All possibly negative coefficients are encoded as a difference
+ * of two non-negative variables, so we need to perform the subtraction
+ * here.  Moreover, if use_cmap is set, then the solution does
+ * not refer to the actual coefficients c_i_x, but instead to variables
+ * t_i_x such that c_i_x = Q t_i_x and Q is equal to node->cmap.
+ * In this case, we then also need to perform this multiplication
+ * to obtain the values of c_i_x.
+ *
+ * If coincident is set, then the caller guarantees that the new
+ * row satisfies the coincidence constraints.
+ */
+static int update_schedule(struct isl_sched_graph *graph,
+	__isl_take isl_vec *sol, int use_cmap, int coincident)
+{
+	int i, j;
+	isl_vec *csol = NULL;
+
+	if (!sol)
+		goto error;
+	if (sol->size == 0)
+		isl_die(sol->ctx, isl_error_internal,
+			"no solution found", goto error);
+	if (graph->n_total_row >= graph->max_row)
+		isl_die(sol->ctx, isl_error_internal,
+			"too many schedule rows", goto error);
+
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[i];
+		int pos = node->start;
+		int row = isl_mat_rows(node->sched);
+
+		isl_vec_free(csol);
+		csol = isl_vec_alloc(sol->ctx, node->nvar);
+		if (!csol)
+			goto error;
+
+		isl_map_free(node->sched_map);
+		node->sched_map = NULL;
+		node->sched = isl_mat_add_rows(node->sched, 1);
+		if (!node->sched)
+			goto error;
+		node->sched = isl_mat_set_element(node->sched, row, 0,
+						  sol->el[1 + pos]);
+		for (j = 0; j < node->nparam + node->nvar; ++j)
+			isl_int_sub(sol->el[1 + pos + 1 + 2 * j + 1],
+				    sol->el[1 + pos + 1 + 2 * j + 1],
+				    sol->el[1 + pos + 1 + 2 * j]);
+		for (j = 0; j < node->nparam; ++j)
+			node->sched = isl_mat_set_element(node->sched,
+					row, 1 + j, sol->el[1+pos+1+2*j+1]);
+		for (j = 0; j < node->nvar; ++j)
+			isl_int_set(csol->el[j],
+				    sol->el[1+pos+1+2*(node->nparam+j)+1]);
+		if (use_cmap)
+			csol = isl_mat_vec_product(isl_mat_copy(node->cmap),
+						   csol);
+		if (!csol)
+			goto error;
+		for (j = 0; j < node->nvar; ++j)
+			node->sched = isl_mat_set_element(node->sched,
+					row, 1 + node->nparam + j, csol->el[j]);
+		node->band[graph->n_total_row] = graph->n_band;
+		node->coincident[graph->n_total_row] = coincident;
+	}
+	isl_vec_free(sol);
+	isl_vec_free(csol);
+
+	graph->n_row++;
+	graph->n_total_row++;
+
+	return 0;
+error:
+	isl_vec_free(sol);
+	isl_vec_free(csol);
+	return -1;
+}
+
+/* Convert row "row" of node->sched into an isl_aff living in "ls"
+ * and return this isl_aff.
+ */
+static __isl_give isl_aff *extract_schedule_row(__isl_take isl_local_space *ls,
+	struct isl_sched_node *node, int row)
+{
+	int j;
+	isl_int v;
+	isl_aff *aff;
+
+	isl_int_init(v);
+
+	aff = isl_aff_zero_on_domain(ls);
+	isl_mat_get_element(node->sched, row, 0, &v);
+	aff = isl_aff_set_constant(aff, v);
+	for (j = 0; j < node->nparam; ++j) {
+		isl_mat_get_element(node->sched, row, 1 + j, &v);
+		aff = isl_aff_set_coefficient(aff, isl_dim_param, j, v);
+	}
+	for (j = 0; j < node->nvar; ++j) {
+		isl_mat_get_element(node->sched, row, 1 + node->nparam + j, &v);
+		aff = isl_aff_set_coefficient(aff, isl_dim_in, j, v);
+	}
+
+	isl_int_clear(v);
+
+	return aff;
+}
+
+/* Convert node->sched into a multi_aff and return this multi_aff.
+ *
+ * The result is defined over the uncompressed node domain.
+ */
+static __isl_give isl_multi_aff *node_extract_schedule_multi_aff(
+	struct isl_sched_node *node)
+{
+	int i;
+	isl_space *space;
+	isl_local_space *ls;
+	isl_aff *aff;
+	isl_multi_aff *ma;
+	int nrow, ncol;
+
+	nrow = isl_mat_rows(node->sched);
+	ncol = isl_mat_cols(node->sched) - 1;
+	if (node->compressed)
+		space = isl_multi_aff_get_domain_space(node->decompress);
+	else
+		space = isl_space_copy(node->space);
+	ls = isl_local_space_from_space(isl_space_copy(space));
+	space = isl_space_from_domain(space);
+	space = isl_space_add_dims(space, isl_dim_out, nrow);
+	ma = isl_multi_aff_zero(space);
+
+	for (i = 0; i < nrow; ++i) {
+		aff = extract_schedule_row(isl_local_space_copy(ls), node, i);
+		ma = isl_multi_aff_set_aff(ma, i, aff);
+	}
+
+	isl_local_space_free(ls);
+
+	if (node->compressed)
+		ma = isl_multi_aff_pullback_multi_aff(ma,
+					isl_multi_aff_copy(node->compress));
+
+	return ma;
+}
+
+/* Convert node->sched into a map and return this map.
+ *
+ * The result is cached in node->sched_map, which needs to be released
+ * whenever node->sched is updated.
+ * It is defined over the uncompressed node domain.
+ */
+static __isl_give isl_map *node_extract_schedule(struct isl_sched_node *node)
+{
+	if (!node->sched_map) {
+		isl_multi_aff *ma;
+
+		ma = node_extract_schedule_multi_aff(node);
+		node->sched_map = isl_map_from_multi_aff(ma);
+	}
+
+	return isl_map_copy(node->sched_map);
+}
+
+/* Construct a map that can be used to update a dependence relation
+ * based on the current schedule.
+ * That is, construct a map expressing that source and sink
+ * are executed within the same iteration of the current schedule.
+ * This map can then be intersected with the dependence relation.
+ * This is not the most efficient way, but this shouldn't be a critical
+ * operation.
+ */
+static __isl_give isl_map *specializer(struct isl_sched_node *src,
+	struct isl_sched_node *dst)
+{
+	isl_map *src_sched, *dst_sched;
+
+	src_sched = node_extract_schedule(src);
+	dst_sched = node_extract_schedule(dst);
+	return isl_map_apply_range(src_sched, isl_map_reverse(dst_sched));
+}
+
+/* Intersect the domains of the nested relations in domain and range
+ * of "umap" with "map".
+ */
+static __isl_give isl_union_map *intersect_domains(
+	__isl_take isl_union_map *umap, __isl_keep isl_map *map)
+{
+	isl_union_set *uset;
+
+	umap = isl_union_map_zip(umap);
+	uset = isl_union_set_from_set(isl_map_wrap(isl_map_copy(map)));
+	umap = isl_union_map_intersect_domain(umap, uset);
+	umap = isl_union_map_zip(umap);
+	return umap;
+}
+
+/* Update the dependence relation of the given edge based
+ * on the current schedule.
+ * If the dependence is carried completely by the current schedule, then
+ * it is removed from the edge_tables.  It is kept in the list of edges
+ * as otherwise all edge_tables would have to be recomputed.
+ */
+static int update_edge(struct isl_sched_graph *graph,
+	struct isl_sched_edge *edge)
+{
+	isl_map *id;
+
+	id = specializer(edge->src, edge->dst);
+	edge->map = isl_map_intersect(edge->map, isl_map_copy(id));
+	if (!edge->map)
+		goto error;
+
+	if (edge->tagged_condition) {
+		edge->tagged_condition =
+			intersect_domains(edge->tagged_condition, id);
+		if (!edge->tagged_condition)
+			goto error;
+	}
+	if (edge->tagged_validity) {
+		edge->tagged_validity =
+			intersect_domains(edge->tagged_validity, id);
+		if (!edge->tagged_validity)
+			goto error;
+	}
+
+	isl_map_free(id);
+	if (isl_map_plain_is_empty(edge->map))
+		graph_remove_edge(graph, edge);
+
+	return 0;
+error:
+	isl_map_free(id);
+	return -1;
+}
+
+/* Update the dependence relations of all edges based on the current schedule.
+ */
+static int update_edges(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	int i;
+
+	for (i = graph->n_edge - 1; i >= 0; --i) {
+		if (update_edge(graph, &graph->edge[i]) < 0)
+			return -1;
+	}
+
+	return 0;
+}
+
+static void next_band(struct isl_sched_graph *graph)
+{
+	graph->band_start = graph->n_total_row;
+	graph->n_band++;
+}
+
+/* Topologically sort statements mapped to the same schedule iteration
+ * and add a row to the schedule corresponding to this order.
+ */
+static int sort_statements(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	int i, j;
+
+	if (graph->n <= 1)
+		return 0;
+
+	if (update_edges(ctx, graph) < 0)
+		return -1;
+
+	if (graph->n_edge == 0)
+		return 0;
+
+	if (detect_sccs(ctx, graph) < 0)
+		return -1;
+
+	if (graph->n_total_row >= graph->max_row)
+		isl_die(ctx, isl_error_internal,
+			"too many schedule rows", return -1);
+
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[i];
+		int row = isl_mat_rows(node->sched);
+		int cols = isl_mat_cols(node->sched);
+
+		isl_map_free(node->sched_map);
+		node->sched_map = NULL;
+		node->sched = isl_mat_add_rows(node->sched, 1);
+		if (!node->sched)
+			return -1;
+		node->sched = isl_mat_set_element_si(node->sched, row, 0,
+						     node->scc);
+		for (j = 1; j < cols; ++j)
+			node->sched = isl_mat_set_element_si(node->sched,
+							     row, j, 0);
+		node->band[graph->n_total_row] = graph->n_band;
+	}
+
+	graph->n_total_row++;
+	next_band(graph);
+
+	return 0;
+}
+
+/* Construct an isl_schedule based on the computed schedule stored
+ * in graph and with parameters specified by dim.
+ */
+static __isl_give isl_schedule *extract_schedule(struct isl_sched_graph *graph,
+	__isl_take isl_space *dim)
+{
+	int i;
+	isl_ctx *ctx;
+	isl_schedule *sched = NULL;
+
+	if (!dim)
+		return NULL;
+
+	ctx = isl_space_get_ctx(dim);
+	sched = isl_calloc(ctx, struct isl_schedule,
+			   sizeof(struct isl_schedule) +
+			   (graph->n - 1) * sizeof(struct isl_schedule_node));
+	if (!sched)
+		goto error;
+
+	sched->ref = 1;
+	sched->n = graph->n;
+	sched->n_band = graph->n_band;
+	sched->n_total_row = graph->n_total_row;
+
+	for (i = 0; i < sched->n; ++i) {
+		int r, b;
+		int *band_end, *band_id, *coincident;
+
+		sched->node[i].sched =
+			node_extract_schedule_multi_aff(&graph->node[i]);
+		if (!sched->node[i].sched)
+			goto error;
+
+		sched->node[i].n_band = graph->n_band;
+		if (graph->n_band == 0)
+			continue;
+
+		band_end = isl_alloc_array(ctx, int, graph->n_band);
+		band_id = isl_alloc_array(ctx, int, graph->n_band);
+		coincident = isl_alloc_array(ctx, int, graph->n_total_row);
+		sched->node[i].band_end = band_end;
+		sched->node[i].band_id = band_id;
+		sched->node[i].coincident = coincident;
+		if (!band_end || !band_id || !coincident)
+			goto error;
+
+		for (r = 0; r < graph->n_total_row; ++r)
+			coincident[r] = graph->node[i].coincident[r];
+		for (r = b = 0; r < graph->n_total_row; ++r) {
+			if (graph->node[i].band[r] == b)
+				continue;
+			band_end[b++] = r;
+			if (graph->node[i].band[r] == -1)
+				break;
+		}
+		if (r == graph->n_total_row)
+			band_end[b++] = r;
+		sched->node[i].n_band = b;
+		for (--b; b >= 0; --b)
+			band_id[b] = graph->node[i].band_id[b];
+	}
+
+	sched->dim = dim;
+
+	return sched;
+error:
+	isl_space_free(dim);
+	isl_schedule_free(sched);
+	return NULL;
+}
+
+/* Copy nodes that satisfy node_pred from the src dependence graph
+ * to the dst dependence graph.
+ */
+static int copy_nodes(struct isl_sched_graph *dst, struct isl_sched_graph *src,
+	int (*node_pred)(struct isl_sched_node *node, int data), int data)
+{
+	int i;
+
+	dst->n = 0;
+	for (i = 0; i < src->n; ++i) {
+		int j;
+
+		if (!node_pred(&src->node[i], data))
+			continue;
+
+		j = dst->n;
+		dst->node[j].space = isl_space_copy(src->node[i].space);
+		dst->node[j].compressed = src->node[i].compressed;
+		dst->node[j].hull = isl_set_copy(src->node[i].hull);
+		dst->node[j].compress =
+			isl_multi_aff_copy(src->node[i].compress);
+		dst->node[j].decompress =
+			isl_multi_aff_copy(src->node[i].decompress);
+		dst->node[j].nvar = src->node[i].nvar;
+		dst->node[j].nparam = src->node[i].nparam;
+		dst->node[j].sched = isl_mat_copy(src->node[i].sched);
+		dst->node[j].sched_map = isl_map_copy(src->node[i].sched_map);
+		dst->node[j].band = src->node[i].band;
+		dst->node[j].band_id = src->node[i].band_id;
+		dst->node[j].coincident = src->node[i].coincident;
+		dst->n++;
+
+		if (!dst->node[j].space || !dst->node[j].sched)
+			return -1;
+		if (dst->node[j].compressed &&
+		    (!dst->node[j].hull || !dst->node[j].compress ||
+		     !dst->node[j].decompress))
+			return -1;
+	}
+
+	return 0;
+}
+
+/* Copy non-empty edges that satisfy edge_pred from the src dependence graph
+ * to the dst dependence graph.
+ * If the source or destination node of the edge is not in the destination
+ * graph, then it must be a backward proximity edge and it should simply
+ * be ignored.
+ */
+static int copy_edges(isl_ctx *ctx, struct isl_sched_graph *dst,
+	struct isl_sched_graph *src,
+	int (*edge_pred)(struct isl_sched_edge *edge, int data), int data)
+{
+	int i;
+	enum isl_edge_type t;
+
+	dst->n_edge = 0;
+	for (i = 0; i < src->n_edge; ++i) {
+		struct isl_sched_edge *edge = &src->edge[i];
+		isl_map *map;
+		isl_union_map *tagged_condition;
+		isl_union_map *tagged_validity;
+		struct isl_sched_node *dst_src, *dst_dst;
+
+		if (!edge_pred(edge, data))
+			continue;
+
+		if (isl_map_plain_is_empty(edge->map))
+			continue;
+
+		dst_src = graph_find_node(ctx, dst, edge->src->space);
+		dst_dst = graph_find_node(ctx, dst, edge->dst->space);
+		if (!dst_src || !dst_dst) {
+			if (edge->validity || edge->conditional_validity)
+				isl_die(ctx, isl_error_internal,
+					"backward (conditional) validity edge",
+					return -1);
+			continue;
+		}
+
+		map = isl_map_copy(edge->map);
+		tagged_condition = isl_union_map_copy(edge->tagged_condition);
+		tagged_validity = isl_union_map_copy(edge->tagged_validity);
+
+		dst->edge[dst->n_edge].src = dst_src;
+		dst->edge[dst->n_edge].dst = dst_dst;
+		dst->edge[dst->n_edge].map = map;
+		dst->edge[dst->n_edge].tagged_condition = tagged_condition;
+		dst->edge[dst->n_edge].tagged_validity = tagged_validity;
+		dst->edge[dst->n_edge].validity = edge->validity;
+		dst->edge[dst->n_edge].proximity = edge->proximity;
+		dst->edge[dst->n_edge].coincidence = edge->coincidence;
+		dst->edge[dst->n_edge].condition = edge->condition;
+		dst->edge[dst->n_edge].conditional_validity =
+						edge->conditional_validity;
+		dst->n_edge++;
+
+		if (edge->tagged_condition && !tagged_condition)
+			return -1;
+		if (edge->tagged_validity && !tagged_validity)
+			return -1;
+
+		for (t = isl_edge_first; t <= isl_edge_last; ++t) {
+			if (edge !=
+			    graph_find_edge(src, t, edge->src, edge->dst))
+				continue;
+			if (graph_edge_table_add(ctx, dst, t,
+					    &dst->edge[dst->n_edge - 1]) < 0)
+				return -1;
+		}
+	}
+
+	return 0;
+}
+
+/* Given a "src" dependence graph that contains the nodes from "dst"
+ * that satisfy node_pred, copy the schedule computed in "src"
+ * for those nodes back to "dst".
+ */
+static int copy_schedule(struct isl_sched_graph *dst,
+	struct isl_sched_graph *src,
+	int (*node_pred)(struct isl_sched_node *node, int data), int data)
+{
+	int i;
+
+	src->n = 0;
+	for (i = 0; i < dst->n; ++i) {
+		if (!node_pred(&dst->node[i], data))
+			continue;
+		isl_mat_free(dst->node[i].sched);
+		isl_map_free(dst->node[i].sched_map);
+		dst->node[i].sched = isl_mat_copy(src->node[src->n].sched);
+		dst->node[i].sched_map =
+			isl_map_copy(src->node[src->n].sched_map);
+		src->n++;
+	}
+
+	dst->max_row = src->max_row;
+	dst->n_total_row = src->n_total_row;
+	dst->n_band = src->n_band;
+
+	return 0;
+}
+
+/* Compute the maximal number of variables over all nodes.
+ * This is the maximal number of linearly independent schedule
+ * rows that we need to compute.
+ * Just in case we end up in a part of the dependence graph
+ * with only lower-dimensional domains, we make sure we will
+ * compute the required amount of extra linearly independent rows.
+ */
+static int compute_maxvar(struct isl_sched_graph *graph)
+{
+	int i;
+
+	graph->maxvar = 0;
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[i];
+		int nvar;
+
+		if (node_update_cmap(node) < 0)
+			return -1;
+		nvar = node->nvar + graph->n_row - node->rank;
+		if (nvar > graph->maxvar)
+			graph->maxvar = nvar;
+	}
+
+	return 0;
+}
+
+static int compute_schedule(isl_ctx *ctx, struct isl_sched_graph *graph);
+static int compute_schedule_wcc(isl_ctx *ctx, struct isl_sched_graph *graph);
+
+/* Compute a schedule for a subgraph of "graph".  In particular, for
+ * the graph composed of nodes that satisfy node_pred and edges that
+ * that satisfy edge_pred.  The caller should precompute the number
+ * of nodes and edges that satisfy these predicates and pass them along
+ * as "n" and "n_edge".
+ * If the subgraph is known to consist of a single component, then wcc should
+ * be set and then we call compute_schedule_wcc on the constructed subgraph.
+ * Otherwise, we call compute_schedule, which will check whether the subgraph
+ * is connected.
+ */
+static int compute_sub_schedule(isl_ctx *ctx,
+	struct isl_sched_graph *graph, int n, int n_edge,
+	int (*node_pred)(struct isl_sched_node *node, int data),
+	int (*edge_pred)(struct isl_sched_edge *edge, int data),
+	int data, int wcc)
+{
+	struct isl_sched_graph split = { 0 };
+	int t;
+
+	if (graph_alloc(ctx, &split, n, n_edge) < 0)
+		goto error;
+	if (copy_nodes(&split, graph, node_pred, data) < 0)
+		goto error;
+	if (graph_init_table(ctx, &split) < 0)
+		goto error;
+	for (t = 0; t <= isl_edge_last; ++t)
+		split.max_edge[t] = graph->max_edge[t];
+	if (graph_init_edge_tables(ctx, &split) < 0)
+		goto error;
+	if (copy_edges(ctx, &split, graph, edge_pred, data) < 0)
+		goto error;
+	split.n_row = graph->n_row;
+	split.max_row = graph->max_row;
+	split.n_total_row = graph->n_total_row;
+	split.n_band = graph->n_band;
+	split.band_start = graph->band_start;
+
+	if (wcc && compute_schedule_wcc(ctx, &split) < 0)
+		goto error;
+	if (!wcc && compute_schedule(ctx, &split) < 0)
+		goto error;
+
+	copy_schedule(graph, &split, node_pred, data);
+
+	graph_free(ctx, &split);
+	return 0;
+error:
+	graph_free(ctx, &split);
+	return -1;
+}
+
+static int node_scc_exactly(struct isl_sched_node *node, int scc)
+{
+	return node->scc == scc;
+}
+
+static int node_scc_at_most(struct isl_sched_node *node, int scc)
+{
+	return node->scc <= scc;
+}
+
+static int node_scc_at_least(struct isl_sched_node *node, int scc)
+{
+	return node->scc >= scc;
+}
+
+static int edge_scc_exactly(struct isl_sched_edge *edge, int scc)
+{
+	return edge->src->scc == scc && edge->dst->scc == scc;
+}
+
+static int edge_dst_scc_at_most(struct isl_sched_edge *edge, int scc)
+{
+	return edge->dst->scc <= scc;
+}
+
+static int edge_src_scc_at_least(struct isl_sched_edge *edge, int scc)
+{
+	return edge->src->scc >= scc;
+}
+
+/* Pad the schedules of all nodes with zero rows such that in the end
+ * they all have graph->n_total_row rows.
+ * The extra rows don't belong to any band, so they get assigned band number -1.
+ */
+static int pad_schedule(struct isl_sched_graph *graph)
+{
+	int i, j;
+
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[i];
+		int row = isl_mat_rows(node->sched);
+		if (graph->n_total_row > row) {
+			isl_map_free(node->sched_map);
+			node->sched_map = NULL;
+		}
+		node->sched = isl_mat_add_zero_rows(node->sched,
+						    graph->n_total_row - row);
+		if (!node->sched)
+			return -1;
+		for (j = row; j < graph->n_total_row; ++j)
+			node->band[j] = -1;
+	}
+
+	return 0;
+}
+
+/* Reset the current band by dropping all its schedule rows.
+ */
+static int reset_band(struct isl_sched_graph *graph)
+{
+	int i;
+	int drop;
+
+	drop = graph->n_total_row - graph->band_start;
+	graph->n_total_row -= drop;
+	graph->n_row -= drop;
+
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[i];
+
+		isl_map_free(node->sched_map);
+		node->sched_map = NULL;
+
+		node->sched = isl_mat_drop_rows(node->sched,
+						graph->band_start, drop);
+
+		if (!node->sched)
+			return -1;
+	}
+
+	return 0;
+}
+
+/* Split the current graph into two parts and compute a schedule for each
+ * part individually.  In particular, one part consists of all SCCs up
+ * to and including graph->src_scc, while the other part contains the other
+ * SCCS.
+ *
+ * The split is enforced in the schedule by constant rows with two different
+ * values (0 and 1).  These constant rows replace the previously computed rows
+ * in the current band.
+ * It would be possible to reuse them as the first rows in the next
+ * band, but recomputing them may result in better rows as we are looking
+ * at a smaller part of the dependence graph.
+ *
+ * Since we do not enforce coincidence, we conservatively mark the
+ * splitting row as not coincident.
+ *
+ * The band_id of the second group is set to n, where n is the number
+ * of nodes in the first group.  This ensures that the band_ids over
+ * the two groups remain disjoint, even if either or both of the two
+ * groups contain independent components.
+ */
+static int compute_split_schedule(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	int i, j, n, e1, e2;
+	int n_total_row, orig_total_row;
+	int n_band, orig_band;
+
+	if (graph->n_total_row >= graph->max_row)
+		isl_die(ctx, isl_error_internal,
+			"too many schedule rows", return -1);
+
+	if (reset_band(graph) < 0)
+		return -1;
+
+	n = 0;
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[i];
+		int row = isl_mat_rows(node->sched);
+		int cols = isl_mat_cols(node->sched);
+		int before = node->scc <= graph->src_scc;
+
+		if (before)
+			n++;
+
+		isl_map_free(node->sched_map);
+		node->sched_map = NULL;
+		node->sched = isl_mat_add_rows(node->sched, 1);
+		if (!node->sched)
+			return -1;
+		node->sched = isl_mat_set_element_si(node->sched, row, 0,
+						     !before);
+		for (j = 1; j < cols; ++j)
+			node->sched = isl_mat_set_element_si(node->sched,
+							     row, j, 0);
+		node->band[graph->n_total_row] = graph->n_band;
+		node->coincident[graph->n_total_row] = 0;
+	}
+
+	e1 = e2 = 0;
+	for (i = 0; i < graph->n_edge; ++i) {
+		if (graph->edge[i].dst->scc <= graph->src_scc)
+			e1++;
+		if (graph->edge[i].src->scc > graph->src_scc)
+			e2++;
+	}
+
+	graph->n_total_row++;
+	next_band(graph);
+
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[i];
+		if (node->scc > graph->src_scc)
+			node->band_id[graph->n_band] = n;
+	}
+
+	orig_total_row = graph->n_total_row;
+	orig_band = graph->n_band;
+	if (compute_sub_schedule(ctx, graph, n, e1,
+				&node_scc_at_most, &edge_dst_scc_at_most,
+				graph->src_scc, 0) < 0)
+		return -1;
+	n_total_row = graph->n_total_row;
+	graph->n_total_row = orig_total_row;
+	n_band = graph->n_band;
+	graph->n_band = orig_band;
+	if (compute_sub_schedule(ctx, graph, graph->n - n, e2,
+				&node_scc_at_least, &edge_src_scc_at_least,
+				graph->src_scc + 1, 0) < 0)
+		return -1;
+	if (n_total_row > graph->n_total_row)
+		graph->n_total_row = n_total_row;
+	if (n_band > graph->n_band)
+		graph->n_band = n_band;
+
+	return pad_schedule(graph);
+}
+
+/* Compute the next band of the schedule after updating the dependence
+ * relations based on the the current schedule.
+ */
+static int compute_next_band(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	if (update_edges(ctx, graph) < 0)
+		return -1;
+	next_band(graph);
+
+	return compute_schedule(ctx, graph);
+}
+
+/* Add constraints to graph->lp that force the dependence "map" (which
+ * is part of the dependence relation of "edge")
+ * to be respected and attempt to carry it, where the edge is one from
+ * a node j to itself.  "pos" is the sequence number of the given map.
+ * That is, add constraints that enforce
+ *
+ *	(c_j_0 + c_j_n n + c_j_x y) - (c_j_0 + c_j_n n + c_j_x x)
+ *	= c_j_x (y - x) >= e_i
+ *
+ * for each (x,y) in R.
+ * We obtain general constraints on coefficients (c_0, c_n, c_x)
+ * of valid constraints for (y - x) and then plug in (-e_i, 0, c_j_x),
+ * with each coefficient in c_j_x represented as a pair of non-negative
+ * coefficients.
+ */
+static int add_intra_constraints(struct isl_sched_graph *graph,
+	struct isl_sched_edge *edge, __isl_take isl_map *map, int pos)
+{
+	unsigned total;
+	isl_ctx *ctx = isl_map_get_ctx(map);
+	isl_space *dim;
+	isl_dim_map *dim_map;
+	isl_basic_set *coef;
+	struct isl_sched_node *node = edge->src;
+
+	coef = intra_coefficients(graph, node, map);
+	if (!coef)
+		return -1;
+
+	dim = isl_space_domain(isl_space_unwrap(isl_basic_set_get_space(coef)));
+
+	total = isl_basic_set_total_dim(graph->lp);
+	dim_map = isl_dim_map_alloc(ctx, total);
+	isl_dim_map_range(dim_map, 3 + pos, 0, 0, 0, 1, -1);
+	isl_dim_map_range(dim_map, node->start + 2 * node->nparam + 1, 2,
+			  isl_space_dim(dim, isl_dim_set), 1,
+			  node->nvar, -1);
+	isl_dim_map_range(dim_map, node->start + 2 * node->nparam + 2, 2,
+			  isl_space_dim(dim, isl_dim_set), 1,
+			  node->nvar, 1);
+	graph->lp = isl_basic_set_extend_constraints(graph->lp,
+			coef->n_eq, coef->n_ineq);
+	graph->lp = isl_basic_set_add_constraints_dim_map(graph->lp,
+							   coef, dim_map);
+	isl_space_free(dim);
+
+	return 0;
+}
+
+/* Add constraints to graph->lp that force the dependence "map" (which
+ * is part of the dependence relation of "edge")
+ * to be respected and attempt to carry it, where the edge is one from
+ * node j to node k.  "pos" is the sequence number of the given map.
+ * That is, add constraints that enforce
+ *
+ *	(c_k_0 + c_k_n n + c_k_x y) - (c_j_0 + c_j_n n + c_j_x x) >= e_i
+ *
+ * for each (x,y) in R.
+ * We obtain general constraints on coefficients (c_0, c_n, c_x)
+ * of valid constraints for R and then plug in
+ * (-e_i + c_k_0 - c_j_0, c_k_n - c_j_n, c_k_x - c_j_x)
+ * with each coefficient (except e_i, c_k_0 and c_j_0)
+ * represented as a pair of non-negative coefficients.
+ */
+static int add_inter_constraints(struct isl_sched_graph *graph,
+	struct isl_sched_edge *edge, __isl_take isl_map *map, int pos)
+{
+	unsigned total;
+	isl_ctx *ctx = isl_map_get_ctx(map);
+	isl_space *dim;
+	isl_dim_map *dim_map;
+	isl_basic_set *coef;
+	struct isl_sched_node *src = edge->src;
+	struct isl_sched_node *dst = edge->dst;
+
+	coef = inter_coefficients(graph, edge, map);
+	if (!coef)
+		return -1;
+
+	dim = isl_space_domain(isl_space_unwrap(isl_basic_set_get_space(coef)));
+
+	total = isl_basic_set_total_dim(graph->lp);
+	dim_map = isl_dim_map_alloc(ctx, total);
+
+	isl_dim_map_range(dim_map, 3 + pos, 0, 0, 0, 1, -1);
+
+	isl_dim_map_range(dim_map, dst->start, 0, 0, 0, 1, 1);
+	isl_dim_map_range(dim_map, dst->start + 1, 2, 1, 1, dst->nparam, -1);
+	isl_dim_map_range(dim_map, dst->start + 2, 2, 1, 1, dst->nparam, 1);
+	isl_dim_map_range(dim_map, dst->start + 2 * dst->nparam + 1, 2,
+			  isl_space_dim(dim, isl_dim_set) + src->nvar, 1,
+			  dst->nvar, -1);
+	isl_dim_map_range(dim_map, dst->start + 2 * dst->nparam + 2, 2,
+			  isl_space_dim(dim, isl_dim_set) + src->nvar, 1,
+			  dst->nvar, 1);
+
+	isl_dim_map_range(dim_map, src->start, 0, 0, 0, 1, -1);
+	isl_dim_map_range(dim_map, src->start + 1, 2, 1, 1, src->nparam, 1);
+	isl_dim_map_range(dim_map, src->start + 2, 2, 1, 1, src->nparam, -1);
+	isl_dim_map_range(dim_map, src->start + 2 * src->nparam + 1, 2,
+			  isl_space_dim(dim, isl_dim_set), 1,
+			  src->nvar, 1);
+	isl_dim_map_range(dim_map, src->start + 2 * src->nparam + 2, 2,
+			  isl_space_dim(dim, isl_dim_set), 1,
+			  src->nvar, -1);
+
+	graph->lp = isl_basic_set_extend_constraints(graph->lp,
+			coef->n_eq, coef->n_ineq);
+	graph->lp = isl_basic_set_add_constraints_dim_map(graph->lp,
+							   coef, dim_map);
+	isl_space_free(dim);
+
+	return 0;
+}
+
+/* Add constraints to graph->lp that force all (conditional) validity
+ * dependences to be respected and attempt to carry them.
+ */
+static int add_all_constraints(struct isl_sched_graph *graph)
+{
+	int i, j;
+	int pos;
+
+	pos = 0;
+	for (i = 0; i < graph->n_edge; ++i) {
+		struct isl_sched_edge *edge= &graph->edge[i];
+
+		if (!edge->validity && !edge->conditional_validity)
+			continue;
+
+		for (j = 0; j < edge->map->n; ++j) {
+			isl_basic_map *bmap;
+			isl_map *map;
+
+			bmap = isl_basic_map_copy(edge->map->p[j]);
+			map = isl_map_from_basic_map(bmap);
+
+			if (edge->src == edge->dst &&
+			    add_intra_constraints(graph, edge, map, pos) < 0)
+				return -1;
+			if (edge->src != edge->dst &&
+			    add_inter_constraints(graph, edge, map, pos) < 0)
+				return -1;
+			++pos;
+		}
+	}
+
+	return 0;
+}
+
+/* Count the number of equality and inequality constraints
+ * that will be added to the carry_lp problem.
+ * We count each edge exactly once.
+ */
+static int count_all_constraints(struct isl_sched_graph *graph,
+	int *n_eq, int *n_ineq)
+{
+	int i, j;
+
+	*n_eq = *n_ineq = 0;
+	for (i = 0; i < graph->n_edge; ++i) {
+		struct isl_sched_edge *edge= &graph->edge[i];
+		for (j = 0; j < edge->map->n; ++j) {
+			isl_basic_map *bmap;
+			isl_map *map;
+
+			bmap = isl_basic_map_copy(edge->map->p[j]);
+			map = isl_map_from_basic_map(bmap);
+
+			if (count_map_constraints(graph, edge, map,
+						  n_eq, n_ineq, 1, 0) < 0)
+				    return -1;
+		}
+	}
+
+	return 0;
+}
+
+/* Construct an LP problem for finding schedule coefficients
+ * such that the schedule carries as many dependences as possible.
+ * In particular, for each dependence i, we bound the dependence distance
+ * from below by e_i, with 0 <= e_i <= 1 and then maximize the sum
+ * of all e_i's.  Dependence with e_i = 0 in the solution are simply
+ * respected, while those with e_i > 0 (in practice e_i = 1) are carried.
+ * Note that if the dependence relation is a union of basic maps,
+ * then we have to consider each basic map individually as it may only
+ * be possible to carry the dependences expressed by some of those
+ * basic maps and not all off them.
+ * Below, we consider each of those basic maps as a separate "edge".
+ *
+ * All variables of the LP are non-negative.  The actual coefficients
+ * may be negative, so each coefficient is represented as the difference
+ * of two non-negative variables.  The negative part always appears
+ * immediately before the positive part.
+ * Other than that, the variables have the following order
+ *
+ *	- sum of (1 - e_i) over all edges
+ *	- sum of positive and negative parts of all c_n coefficients
+ *		(unconstrained when computing non-parametric schedules)
+ *	- sum of positive and negative parts of all c_x coefficients
+ *	- for each edge
+ *		- e_i
+ *	- for each node
+ *		- c_i_0
+ *		- positive and negative parts of c_i_n (if parametric)
+ *		- positive and negative parts of c_i_x
+ *
+ * The constraints are those from the (validity) edges plus three equalities
+ * to express the sums and n_edge inequalities to express e_i <= 1.
+ */
+static int setup_carry_lp(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	int i, j;
+	int k;
+	isl_space *dim;
+	unsigned total;
+	int n_eq, n_ineq;
+	int n_edge;
+
+	n_edge = 0;
+	for (i = 0; i < graph->n_edge; ++i)
+		n_edge += graph->edge[i].map->n;
+
+	total = 3 + n_edge;
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[graph->sorted[i]];
+		node->start = total;
+		total += 1 + 2 * (node->nparam + node->nvar);
+	}
+
+	if (count_all_constraints(graph, &n_eq, &n_ineq) < 0)
+		return -1;
+	if (count_bound_coefficient_constraints(ctx, graph, &n_eq, &n_ineq) < 0)
+		return -1;
+
+	dim = isl_space_set_alloc(ctx, 0, total);
+	isl_basic_set_free(graph->lp);
+	n_eq += 3;
+	n_ineq += n_edge;
+	graph->lp = isl_basic_set_alloc_space(dim, 0, n_eq, n_ineq);
+	graph->lp = isl_basic_set_set_rational(graph->lp);
+
+	k = isl_basic_set_alloc_equality(graph->lp);
+	if (k < 0)
+		return -1;
+	isl_seq_clr(graph->lp->eq[k], 1 +  total);
+	isl_int_set_si(graph->lp->eq[k][0], -n_edge);
+	isl_int_set_si(graph->lp->eq[k][1], 1);
+	for (i = 0; i < n_edge; ++i)
+		isl_int_set_si(graph->lp->eq[k][4 + i], 1);
+
+	k = isl_basic_set_alloc_equality(graph->lp);
+	if (k < 0)
+		return -1;
+	isl_seq_clr(graph->lp->eq[k], 1 +  total);
+	isl_int_set_si(graph->lp->eq[k][2], -1);
+	for (i = 0; i < graph->n; ++i) {
+		int pos = 1 + graph->node[i].start + 1;
+
+		for (j = 0; j < 2 * graph->node[i].nparam; ++j)
+			isl_int_set_si(graph->lp->eq[k][pos + j], 1);
+	}
+
+	k = isl_basic_set_alloc_equality(graph->lp);
+	if (k < 0)
+		return -1;
+	isl_seq_clr(graph->lp->eq[k], 1 +  total);
+	isl_int_set_si(graph->lp->eq[k][3], -1);
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[i];
+		int pos = 1 + node->start + 1 + 2 * node->nparam;
+
+		for (j = 0; j < 2 * node->nvar; ++j)
+			isl_int_set_si(graph->lp->eq[k][pos + j], 1);
+	}
+
+	for (i = 0; i < n_edge; ++i) {
+		k = isl_basic_set_alloc_inequality(graph->lp);
+		if (k < 0)
+			return -1;
+		isl_seq_clr(graph->lp->ineq[k], 1 +  total);
+		isl_int_set_si(graph->lp->ineq[k][4 + i], -1);
+		isl_int_set_si(graph->lp->ineq[k][0], 1);
+	}
+
+	if (add_bound_coefficient_constraints(ctx, graph) < 0)
+		return -1;
+	if (add_all_constraints(graph) < 0)
+		return -1;
+
+	return 0;
+}
+
+/* If the schedule_split_scaled option is set and if the linear
+ * parts of the scheduling rows for all nodes in the graphs have
+ * non-trivial common divisor, then split off the constant term
+ * from the linear part.
+ * The constant term is then placed in a separate band and
+ * the linear part is reduced.
+ */
+static int split_scaled(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	int i;
+	int row;
+	isl_int gcd, gcd_i;
+
+	if (!ctx->opt->schedule_split_scaled)
+		return 0;
+	if (graph->n <= 1)
+		return 0;
+
+	if (graph->n_total_row >= graph->max_row)
+		isl_die(ctx, isl_error_internal,
+			"too many schedule rows", return -1);
+
+	isl_int_init(gcd);
+	isl_int_init(gcd_i);
+
+	isl_int_set_si(gcd, 0);
+
+	row = isl_mat_rows(graph->node[0].sched) - 1;
+
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[i];
+		int cols = isl_mat_cols(node->sched);
+
+		isl_seq_gcd(node->sched->row[row] + 1, cols - 1, &gcd_i);
+		isl_int_gcd(gcd, gcd, gcd_i);
+	}
+
+	isl_int_clear(gcd_i);
+
+	if (isl_int_cmp_si(gcd, 1) <= 0) {
+		isl_int_clear(gcd);
+		return 0;
+	}
+
+	next_band(graph);
+
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[i];
+
+		isl_map_free(node->sched_map);
+		node->sched_map = NULL;
+		node->sched = isl_mat_add_zero_rows(node->sched, 1);
+		if (!node->sched)
+			goto error;
+		isl_int_fdiv_r(node->sched->row[row + 1][0],
+			       node->sched->row[row][0], gcd);
+		isl_int_fdiv_q(node->sched->row[row][0],
+			       node->sched->row[row][0], gcd);
+		isl_int_mul(node->sched->row[row][0],
+			    node->sched->row[row][0], gcd);
+		node->sched = isl_mat_scale_down_row(node->sched, row, gcd);
+		if (!node->sched)
+			goto error;
+		node->band[graph->n_total_row] = graph->n_band;
+	}
+
+	graph->n_total_row++;
+
+	isl_int_clear(gcd);
+	return 0;
+error:
+	isl_int_clear(gcd);
+	return -1;
+}
+
+static int compute_component_schedule(isl_ctx *ctx,
+	struct isl_sched_graph *graph);
+
+/* Is the schedule row "sol" trivial on node "node"?
+ * That is, is the solution zero on the dimensions orthogonal to
+ * the previously found solutions?
+ * Return 1 if the solution is trivial, 0 if it is not and -1 on error.
+ *
+ * Each coefficient is represented as the difference between
+ * two non-negative values in "sol".  "sol" has been computed
+ * in terms of the original iterators (i.e., without use of cmap).
+ * We construct the schedule row s and write it as a linear
+ * combination of (linear combinations of) previously computed schedule rows.
+ * s = Q c or c = U s.
+ * If the final entries of c are all zero, then the solution is trivial.
+ */
+static int is_trivial(struct isl_sched_node *node, __isl_keep isl_vec *sol)
+{
+	int i;
+	int pos;
+	int trivial;
+	isl_ctx *ctx;
+	isl_vec *node_sol;
+
+	if (!sol)
+		return -1;
+	if (node->nvar == node->rank)
+		return 0;
+
+	ctx = isl_vec_get_ctx(sol);
+	node_sol = isl_vec_alloc(ctx, node->nvar);
+	if (!node_sol)
+		return -1;
+
+	pos = 1 + node->start + 1 + 2 * node->nparam;
+
+	for (i = 0; i < node->nvar; ++i)
+		isl_int_sub(node_sol->el[i],
+			    sol->el[pos + 2 * i + 1], sol->el[pos + 2 * i]);
+
+	node_sol = isl_mat_vec_product(isl_mat_copy(node->cinv), node_sol);
+
+	if (!node_sol)
+		return -1;
+
+	trivial = isl_seq_first_non_zero(node_sol->el + node->rank,
+					node->nvar - node->rank) == -1;
+
+	isl_vec_free(node_sol);
+
+	return trivial;
+}
+
+/* Is the schedule row "sol" trivial on any node where it should
+ * not be trivial?
+ * "sol" has been computed in terms of the original iterators
+ * (i.e., without use of cmap).
+ * Return 1 if any solution is trivial, 0 if they are not and -1 on error.
+ */
+static int is_any_trivial(struct isl_sched_graph *graph,
+	__isl_keep isl_vec *sol)
+{
+	int i;
+
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[i];
+		int trivial;
+
+		if (!needs_row(graph, node))
+			continue;
+		trivial = is_trivial(node, sol);
+		if (trivial < 0 || trivial)
+			return trivial;
+	}
+
+	return 0;
+}
+
+/* Construct a schedule row for each node such that as many dependences
+ * as possible are carried and then continue with the next band.
+ *
+ * If the computed schedule row turns out to be trivial on one or
+ * more nodes where it should not be trivial, then we throw it away
+ * and try again on each component separately.
+ *
+ * If there is only one component, then we accept the schedule row anyway,
+ * but we do not consider it as a complete row and therefore do not
+ * increment graph->n_row.  Note that the ranks of the nodes that
+ * do get a non-trivial schedule part will get updated regardless and
+ * graph->maxvar is computed based on these ranks.  The test for
+ * whether more schedule rows are required in compute_schedule_wcc
+ * is therefore not affected.
+ */
+static int carry_dependences(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	int i;
+	int n_edge;
+	int trivial;
+	isl_vec *sol;
+	isl_basic_set *lp;
+
+	n_edge = 0;
+	for (i = 0; i < graph->n_edge; ++i)
+		n_edge += graph->edge[i].map->n;
+
+	if (setup_carry_lp(ctx, graph) < 0)
+		return -1;
+
+	lp = isl_basic_set_copy(graph->lp);
+	sol = isl_tab_basic_set_non_neg_lexmin(lp);
+	if (!sol)
+		return -1;
+
+	if (sol->size == 0) {
+		isl_vec_free(sol);
+		isl_die(ctx, isl_error_internal,
+			"error in schedule construction", return -1);
+	}
+
+	isl_int_divexact(sol->el[1], sol->el[1], sol->el[0]);
+	if (isl_int_cmp_si(sol->el[1], n_edge) >= 0) {
+		isl_vec_free(sol);
+		isl_die(ctx, isl_error_unknown,
+			"unable to carry dependences", return -1);
+	}
+
+	trivial = is_any_trivial(graph, sol);
+	if (trivial < 0) {
+		sol = isl_vec_free(sol);
+	} else if (trivial && graph->scc > 1) {
+		isl_vec_free(sol);
+		return compute_component_schedule(ctx, graph);
+	}
+
+	if (update_schedule(graph, sol, 0, 0) < 0)
+		return -1;
+	if (trivial)
+		graph->n_row--;
+
+	if (split_scaled(ctx, graph) < 0)
+		return -1;
+
+	return compute_next_band(ctx, graph);
+}
+
+/* Are there any (non-empty) (conditional) validity edges in the graph?
+ */
+static int has_validity_edges(struct isl_sched_graph *graph)
+{
+	int i;
+
+	for (i = 0; i < graph->n_edge; ++i) {
+		int empty;
+
+		empty = isl_map_plain_is_empty(graph->edge[i].map);
+		if (empty < 0)
+			return -1;
+		if (empty)
+			continue;
+		if (graph->edge[i].validity ||
+		    graph->edge[i].conditional_validity)
+			return 1;
+	}
+
+	return 0;
+}
+
+/* Should we apply a Feautrier step?
+ * That is, did the user request the Feautrier algorithm and are
+ * there any validity dependences (left)?
+ */
+static int need_feautrier_step(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	if (ctx->opt->schedule_algorithm != ISL_SCHEDULE_ALGORITHM_FEAUTRIER)
+		return 0;
+
+	return has_validity_edges(graph);
+}
+
+/* Compute a schedule for a connected dependence graph using Feautrier's
+ * multi-dimensional scheduling algorithm.
+ * The original algorithm is described in [1].
+ * The main idea is to minimize the number of scheduling dimensions, by
+ * trying to satisfy as many dependences as possible per scheduling dimension.
+ *
+ * [1] P. Feautrier, Some Efficient Solutions to the Affine Scheduling
+ *     Problem, Part II: Multi-Dimensional Time.
+ *     In Intl. Journal of Parallel Programming, 1992.
+ */
+static int compute_schedule_wcc_feautrier(isl_ctx *ctx,
+	struct isl_sched_graph *graph)
+{
+	return carry_dependences(ctx, graph);
+}
+
+/* Turn off the "local" bit on all (condition) edges.
+ */
+static void clear_local_edges(struct isl_sched_graph *graph)
+{
+	int i;
+
+	for (i = 0; i < graph->n_edge; ++i)
+		if (graph->edge[i].condition)
+			graph->edge[i].local = 0;
+}
+
+/* Does "graph" have both condition and conditional validity edges?
+ */
+static int need_condition_check(struct isl_sched_graph *graph)
+{
+	int i;
+	int any_condition = 0;
+	int any_conditional_validity = 0;
+
+	for (i = 0; i < graph->n_edge; ++i) {
+		if (graph->edge[i].condition)
+			any_condition = 1;
+		if (graph->edge[i].conditional_validity)
+			any_conditional_validity = 1;
+	}
+
+	return any_condition && any_conditional_validity;
+}
+
+/* Does "graph" contain any coincidence edge?
+ */
+static int has_any_coincidence(struct isl_sched_graph *graph)
+{
+	int i;
+
+	for (i = 0; i < graph->n_edge; ++i)
+		if (graph->edge[i].coincidence)
+			return 1;
+
+	return 0;
+}
+
+/* Extract the final schedule row as a map with the iteration domain
+ * of "node" as domain.
+ */
+static __isl_give isl_map *final_row(struct isl_sched_node *node)
+{
+	isl_local_space *ls;
+	isl_aff *aff;
+	int row;
+
+	row = isl_mat_rows(node->sched) - 1;
+	ls = isl_local_space_from_space(isl_space_copy(node->space));
+	aff = extract_schedule_row(ls, node, row);
+	return isl_map_from_aff(aff);
+}
+
+/* Is the conditional validity dependence in the edge with index "edge_index"
+ * violated by the latest (i.e., final) row of the schedule?
+ * That is, is i scheduled after j
+ * for any conditional validity dependence i -> j?
+ */
+static int is_violated(struct isl_sched_graph *graph, int edge_index)
+{
+	isl_map *src_sched, *dst_sched, *map;
+	struct isl_sched_edge *edge = &graph->edge[edge_index];
+	int empty;
+
+	src_sched = final_row(edge->src);
+	dst_sched = final_row(edge->dst);
+	map = isl_map_copy(edge->map);
+	map = isl_map_apply_domain(map, src_sched);
+	map = isl_map_apply_range(map, dst_sched);
+	map = isl_map_order_gt(map, isl_dim_in, 0, isl_dim_out, 0);
+	empty = isl_map_is_empty(map);
+	isl_map_free(map);
+
+	if (empty < 0)
+		return -1;
+
+	return !empty;
+}
+
+/* Does the domain of "umap" intersect "uset"?
+ */
+static int domain_intersects(__isl_keep isl_union_map *umap,
+	__isl_keep isl_union_set *uset)
+{
+	int empty;
+
+	umap = isl_union_map_copy(umap);
+	umap = isl_union_map_intersect_domain(umap, isl_union_set_copy(uset));
+	empty = isl_union_map_is_empty(umap);
+	isl_union_map_free(umap);
+
+	return empty < 0 ? -1 : !empty;
+}
+
+/* Does the range of "umap" intersect "uset"?
+ */
+static int range_intersects(__isl_keep isl_union_map *umap,
+	__isl_keep isl_union_set *uset)
+{
+	int empty;
+
+	umap = isl_union_map_copy(umap);
+	umap = isl_union_map_intersect_range(umap, isl_union_set_copy(uset));
+	empty = isl_union_map_is_empty(umap);
+	isl_union_map_free(umap);
+
+	return empty < 0 ? -1 : !empty;
+}
+
+/* Are the condition dependences of "edge" local with respect to
+ * the current schedule?
+ *
+ * That is, are domain and range of the condition dependences mapped
+ * to the same point?
+ *
+ * In other words, is the condition false?
+ */
+static int is_condition_false(struct isl_sched_edge *edge)
+{
+	isl_union_map *umap;
+	isl_map *map, *sched, *test;
+	int local;
+
+	umap = isl_union_map_copy(edge->tagged_condition);
+	umap = isl_union_map_zip(umap);
+	umap = isl_union_set_unwrap(isl_union_map_domain(umap));
+	map = isl_map_from_union_map(umap);
+
+	sched = node_extract_schedule(edge->src);
+	map = isl_map_apply_domain(map, sched);
+	sched = node_extract_schedule(edge->dst);
+	map = isl_map_apply_range(map, sched);
+
+	test = isl_map_identity(isl_map_get_space(map));
+	local = isl_map_is_subset(map, test);
+	isl_map_free(map);
+	isl_map_free(test);
+
+	return local;
+}
+
+/* Does "graph" have any satisfied condition edges that
+ * are adjacent to the conditional validity constraint with
+ * domain "conditional_source" and range "conditional_sink"?
+ *
+ * A satisfied condition is one that is not local.
+ * If a condition was forced to be local already (i.e., marked as local)
+ * then there is no need to check if it is in fact local.
+ *
+ * Additionally, mark all adjacent condition edges found as local.
+ */
+static int has_adjacent_true_conditions(struct isl_sched_graph *graph,
+	__isl_keep isl_union_set *conditional_source,
+	__isl_keep isl_union_set *conditional_sink)
+{
+	int i;
+	int any = 0;
+
+	for (i = 0; i < graph->n_edge; ++i) {
+		int adjacent, local;
+		isl_union_map *condition;
+
+		if (!graph->edge[i].condition)
+			continue;
+		if (graph->edge[i].local)
+			continue;
+
+		condition = graph->edge[i].tagged_condition;
+		adjacent = domain_intersects(condition, conditional_sink);
+		if (adjacent >= 0 && !adjacent)
+			adjacent = range_intersects(condition,
+							conditional_source);
+		if (adjacent < 0)
+			return -1;
+		if (!adjacent)
+			continue;
+
+		graph->edge[i].local = 1;
+
+		local = is_condition_false(&graph->edge[i]);
+		if (local < 0)
+			return -1;
+		if (!local)
+			any = 1;
+	}
+
+	return any;
+}
+
+/* Are there any violated conditional validity dependences with
+ * adjacent condition dependences that are not local with respect
+ * to the current schedule?
+ * That is, is the conditional validity constraint violated?
+ *
+ * Additionally, mark all those adjacent condition dependences as local.
+ * We also mark those adjacent condition dependences that were not marked
+ * as local before, but just happened to be local already.  This ensures
+ * that they remain local if the schedule is recomputed.
+ *
+ * We first collect domain and range of all violated conditional validity
+ * dependences and then check if there are any adjacent non-local
+ * condition dependences.
+ */
+static int has_violated_conditional_constraint(isl_ctx *ctx,
+	struct isl_sched_graph *graph)
+{
+	int i;
+	int any = 0;
+	isl_union_set *source, *sink;
+
+	source = isl_union_set_empty(isl_space_params_alloc(ctx, 0));
+	sink = isl_union_set_empty(isl_space_params_alloc(ctx, 0));
+	for (i = 0; i < graph->n_edge; ++i) {
+		isl_union_set *uset;
+		isl_union_map *umap;
+		int violated;
+
+		if (!graph->edge[i].conditional_validity)
+			continue;
+
+		violated = is_violated(graph, i);
+		if (violated < 0)
+			goto error;
+		if (!violated)
+			continue;
+
+		any = 1;
+
+		umap = isl_union_map_copy(graph->edge[i].tagged_validity);
+		uset = isl_union_map_domain(umap);
+		source = isl_union_set_union(source, uset);
+		source = isl_union_set_coalesce(source);
+
+		umap = isl_union_map_copy(graph->edge[i].tagged_validity);
+		uset = isl_union_map_range(umap);
+		sink = isl_union_set_union(sink, uset);
+		sink = isl_union_set_coalesce(sink);
+	}
+
+	if (any)
+		any = has_adjacent_true_conditions(graph, source, sink);
+
+	isl_union_set_free(source);
+	isl_union_set_free(sink);
+	return any;
+error:
+	isl_union_set_free(source);
+	isl_union_set_free(sink);
+	return -1;
+}
+
+/* Compute a schedule for a connected dependence graph.
+ * We try to find a sequence of as many schedule rows as possible that result
+ * in non-negative dependence distances (independent of the previous rows
+ * in the sequence, i.e., such that the sequence is tilable), with as
+ * many of the initial rows as possible satisfying the coincidence constraints.
+ * If we can't find any more rows we either
+ * - split between SCCs and start over (assuming we found an interesting
+ *	pair of SCCs between which to split)
+ * - continue with the next band (assuming the current band has at least
+ *	one row)
+ * - try to carry as many dependences as possible and continue with the next
+ *	band
+ *
+ * If Feautrier's algorithm is selected, we first recursively try to satisfy
+ * as many validity dependences as possible. When all validity dependences
+ * are satisfied we extend the schedule to a full-dimensional schedule.
+ *
+ * If we manage to complete the schedule, we finish off by topologically
+ * sorting the statements based on the remaining dependences.
+ *
+ * If ctx->opt->schedule_outer_coincidence is set, then we force the
+ * outermost dimension to satisfy the coincidence constraints.  If this
+ * turns out to be impossible, we fall back on the general scheme above
+ * and try to carry as many dependences as possible.
+ *
+ * If "graph" contains both condition and conditional validity dependences,
+ * then we need to check that that the conditional schedule constraint
+ * is satisfied, i.e., there are no violated conditional validity dependences
+ * that are adjacent to any non-local condition dependences.
+ * If there are, then we mark all those adjacent condition dependences
+ * as local and recompute the current band.  Those dependences that
+ * are marked local will then be forced to be local.
+ * The initial computation is performed with no dependences marked as local.
+ * If we are lucky, then there will be no violated conditional validity
+ * dependences adjacent to any non-local condition dependences.
+ * Otherwise, we mark some additional condition dependences as local and
+ * recompute.  We continue this process until there are no violations left or
+ * until we are no longer able to compute a schedule.
+ * Since there are only a finite number of dependences,
+ * there will only be a finite number of iterations.
+ */
+static int compute_schedule_wcc(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	int has_coincidence;
+	int use_coincidence;
+	int force_coincidence = 0;
+	int check_conditional;
+
+	if (detect_sccs(ctx, graph) < 0)
+		return -1;
+	if (sort_sccs(graph) < 0)
+		return -1;
+
+	if (compute_maxvar(graph) < 0)
+		return -1;
+
+	if (need_feautrier_step(ctx, graph))
+		return compute_schedule_wcc_feautrier(ctx, graph);
+
+	clear_local_edges(graph);
+	check_conditional = need_condition_check(graph);
+	has_coincidence = has_any_coincidence(graph);
+
+	if (ctx->opt->schedule_outer_coincidence)
+		force_coincidence = 1;
+
+	use_coincidence = has_coincidence;
+	while (graph->n_row < graph->maxvar) {
+		isl_vec *sol;
+		int violated;
+		int coincident;
+
+		graph->src_scc = -1;
+		graph->dst_scc = -1;
+
+		if (setup_lp(ctx, graph, use_coincidence) < 0)
+			return -1;
+		sol = solve_lp(graph);
+		if (!sol)
+			return -1;
+		if (sol->size == 0) {
+			int empty = graph->n_total_row == graph->band_start;
+
+			isl_vec_free(sol);
+			if (use_coincidence && (!force_coincidence || !empty)) {
+				use_coincidence = 0;
+				continue;
+			}
+			if (!ctx->opt->schedule_maximize_band_depth && !empty)
+				return compute_next_band(ctx, graph);
+			if (graph->src_scc >= 0)
+				return compute_split_schedule(ctx, graph);
+			if (!empty)
+				return compute_next_band(ctx, graph);
+			return carry_dependences(ctx, graph);
+		}
+		coincident = !has_coincidence || use_coincidence;
+		if (update_schedule(graph, sol, 1, coincident) < 0)
+			return -1;
+
+		if (!check_conditional)
+			continue;
+		violated = has_violated_conditional_constraint(ctx, graph);
+		if (violated < 0)
+			return -1;
+		if (!violated)
+			continue;
+		if (reset_band(graph) < 0)
+			return -1;
+		use_coincidence = has_coincidence;
+	}
+
+	if (graph->n_total_row > graph->band_start)
+		next_band(graph);
+	return sort_statements(ctx, graph);
+}
+
+/* Add a row to the schedules that separates the SCCs and move
+ * to the next band.
+ */
+static int split_on_scc(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	int i;
+
+	if (graph->n_total_row >= graph->max_row)
+		isl_die(ctx, isl_error_internal,
+			"too many schedule rows", return -1);
+
+	for (i = 0; i < graph->n; ++i) {
+		struct isl_sched_node *node = &graph->node[i];
+		int row = isl_mat_rows(node->sched);
+
+		isl_map_free(node->sched_map);
+		node->sched_map = NULL;
+		node->sched = isl_mat_add_zero_rows(node->sched, 1);
+		node->sched = isl_mat_set_element_si(node->sched, row, 0,
+						     node->scc);
+		if (!node->sched)
+			return -1;
+		node->band[graph->n_total_row] = graph->n_band;
+	}
+
+	graph->n_total_row++;
+	next_band(graph);
+
+	return 0;
+}
+
+/* Compute a schedule for each component (identified by node->scc)
+ * of the dependence graph separately and then combine the results.
+ * Depending on the setting of schedule_fuse, a component may be
+ * either weakly or strongly connected.
+ *
+ * The band_id is adjusted such that each component has a separate id.
+ * Note that the band_id may have already been set to a value different
+ * from zero by compute_split_schedule.
+ */
+static int compute_component_schedule(isl_ctx *ctx,
+	struct isl_sched_graph *graph)
+{
+	int wcc, i;
+	int n, n_edge;
+	int n_total_row, orig_total_row;
+	int n_band, orig_band;
+
+	if (ctx->opt->schedule_fuse == ISL_SCHEDULE_FUSE_MIN ||
+	    ctx->opt->schedule_separate_components)
+		if (split_on_scc(ctx, graph) < 0)
+			return -1;
+
+	n_total_row = 0;
+	orig_total_row = graph->n_total_row;
+	n_band = 0;
+	orig_band = graph->n_band;
+	for (i = 0; i < graph->n; ++i)
+		graph->node[i].band_id[graph->n_band] += graph->node[i].scc;
+	for (wcc = 0; wcc < graph->scc; ++wcc) {
+		n = 0;
+		for (i = 0; i < graph->n; ++i)
+			if (graph->node[i].scc == wcc)
+				n++;
+		n_edge = 0;
+		for (i = 0; i < graph->n_edge; ++i)
+			if (graph->edge[i].src->scc == wcc &&
+			    graph->edge[i].dst->scc == wcc)
+				n_edge++;
+
+		if (compute_sub_schedule(ctx, graph, n, n_edge,
+				    &node_scc_exactly,
+				    &edge_scc_exactly, wcc, 1) < 0)
+			return -1;
+		if (graph->n_total_row > n_total_row)
+			n_total_row = graph->n_total_row;
+		graph->n_total_row = orig_total_row;
+		if (graph->n_band > n_band)
+			n_band = graph->n_band;
+		graph->n_band = orig_band;
+	}
+
+	graph->n_total_row = n_total_row;
+	graph->n_band = n_band;
+
+	return pad_schedule(graph);
+}
+
+/* Compute a schedule for the given dependence graph.
+ * We first check if the graph is connected (through validity and conditional
+ * validity dependences) and, if not, compute a schedule
+ * for each component separately.
+ * If schedule_fuse is set to minimal fusion, then we check for strongly
+ * connected components instead and compute a separate schedule for
+ * each such strongly connected component.
+ */
+static int compute_schedule(isl_ctx *ctx, struct isl_sched_graph *graph)
+{
+	if (ctx->opt->schedule_fuse == ISL_SCHEDULE_FUSE_MIN) {
+		if (detect_sccs(ctx, graph) < 0)
+			return -1;
+	} else {
+		if (detect_wccs(ctx, graph) < 0)
+			return -1;
+	}
+
+	if (graph->scc > 1)
+		return compute_component_schedule(ctx, graph);
+
+	return compute_schedule_wcc(ctx, graph);
+}
+
+/* Compute a schedule on sc->domain that respects the given schedule
+ * constraints.
+ *
+ * In particular, the schedule respects all the validity dependences.
+ * If the default isl scheduling algorithm is used, it tries to minimize
+ * the dependence distances over the proximity dependences.
+ * If Feautrier's scheduling algorithm is used, the proximity dependence
+ * distances are only minimized during the extension to a full-dimensional
+ * schedule.
+ *
+ * If there are any condition and conditional validity dependences,
+ * then the conditional validity dependences may be violated inside
+ * a tilable band, provided they have no adjacent non-local
+ * condition dependences.
+ */
+__isl_give isl_schedule *isl_schedule_constraints_compute_schedule(
+	__isl_take isl_schedule_constraints *sc)
+{
+	isl_ctx *ctx = isl_schedule_constraints_get_ctx(sc);
+	struct isl_sched_graph graph = { 0 };
+	isl_schedule *sched;
+	struct isl_extract_edge_data data;
+	enum isl_edge_type i;
+
+	sc = isl_schedule_constraints_align_params(sc);
+	if (!sc)
+		return NULL;
+
+	graph.n = isl_union_set_n_set(sc->domain);
+	if (graph.n == 0)
+		goto empty;
+	if (graph_alloc(ctx, &graph, graph.n,
+	    isl_schedule_constraints_n_map(sc)) < 0)
+		goto error;
+	if (compute_max_row(&graph, sc) < 0)
+		goto error;
+	graph.root = 1;
+	graph.n = 0;
+	if (isl_union_set_foreach_set(sc->domain, &extract_node, &graph) < 0)
+		goto error;
+	if (graph_init_table(ctx, &graph) < 0)
+		goto error;
+	for (i = isl_edge_first; i <= isl_edge_last; ++i)
+		graph.max_edge[i] = isl_union_map_n_map(sc->constraint[i]);
+	if (graph_init_edge_tables(ctx, &graph) < 0)
+		goto error;
+	graph.n_edge = 0;
+	data.graph = &graph;
+	for (i = isl_edge_first; i <= isl_edge_last; ++i) {
+		data.type = i;
+		if (isl_union_map_foreach_map(sc->constraint[i],
+						&extract_edge, &data) < 0)
+			goto error;
+	}
+
+	if (compute_schedule(ctx, &graph) < 0)
+		goto error;
+
+empty:
+	sched = extract_schedule(&graph, isl_union_set_get_space(sc->domain));
+
+	graph_free(ctx, &graph);
+	isl_schedule_constraints_free(sc);
+
+	return sched;
+error:
+	graph_free(ctx, &graph);
+	isl_schedule_constraints_free(sc);
+	return NULL;
+}
+
+/* Compute a schedule for the given union of domains that respects
+ * all the validity dependences and minimizes
+ * the dependence distances over the proximity dependences.
+ *
+ * This function is kept for backward compatibility.
+ */
+__isl_give isl_schedule *isl_union_set_compute_schedule(
+	__isl_take isl_union_set *domain,
+	__isl_take isl_union_map *validity,
+	__isl_take isl_union_map *proximity)
+{
+	isl_schedule_constraints *sc;
+
+	sc = isl_schedule_constraints_on_domain(domain);
+	sc = isl_schedule_constraints_set_validity(sc, validity);
+	sc = isl_schedule_constraints_set_proximity(sc, proximity);
+
+	return isl_schedule_constraints_compute_schedule(sc);
+}
diff --git a/polly/lib/External/isl/isl_seq.c b/polly/lib/External/isl/isl_seq.c
new file mode 100644
index 00000000000..fbbc26f148c
--- /dev/null
+++ b/polly/lib/External/isl/isl_seq.c
@@ -0,0 +1,321 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_seq.h>
+
+void isl_seq_clr(isl_int *p, unsigned len)
+{
+	int i;
+	for (i = 0; i < len; ++i)
+		isl_int_set_si(p[i], 0);
+}
+
+void isl_seq_set_si(isl_int *p, int v, unsigned len)
+{
+	int i;
+	for (i = 0; i < len; ++i)
+		isl_int_set_si(p[i], v);
+}
+
+void isl_seq_set(isl_int *p, isl_int v, unsigned len)
+{
+	int i;
+	for (i = 0; i < len; ++i)
+		isl_int_set(p[i], v);
+}
+
+void isl_seq_neg(isl_int *dst, isl_int *src, unsigned len)
+{
+	int i;
+	for (i = 0; i < len; ++i)
+		isl_int_neg(dst[i], src[i]);
+}
+
+void isl_seq_cpy(isl_int *dst, isl_int *src, unsigned len)
+{
+	int i;
+	for (i = 0; i < len; ++i)
+		isl_int_set(dst[i], src[i]);
+}
+
+void isl_seq_submul(isl_int *dst, isl_int f, isl_int *src, unsigned len)
+{
+	int i;
+	for (i = 0; i < len; ++i)
+		isl_int_submul(dst[i], f, src[i]);
+}
+
+void isl_seq_addmul(isl_int *dst, isl_int f, isl_int *src, unsigned len)
+{
+	int i;
+	for (i = 0; i < len; ++i)
+		isl_int_addmul(dst[i], f, src[i]);
+}
+
+void isl_seq_swp_or_cpy(isl_int *dst, isl_int *src, unsigned len)
+{
+	int i;
+	for (i = 0; i < len; ++i)
+		isl_int_swap_or_set(dst[i], src[i]);
+}
+
+void isl_seq_scale(isl_int *dst, isl_int *src, isl_int m, unsigned len)
+{
+	int i;
+	for (i = 0; i < len; ++i)
+		isl_int_mul(dst[i], src[i], m);
+}
+
+void isl_seq_scale_down(isl_int *dst, isl_int *src, isl_int m, unsigned len)
+{
+	int i;
+	for (i = 0; i < len; ++i)
+		isl_int_divexact(dst[i], src[i], m);
+}
+
+void isl_seq_cdiv_q(isl_int *dst, isl_int *src, isl_int m, unsigned len)
+{
+	int i;
+	for (i = 0; i < len; ++i)
+		isl_int_cdiv_q(dst[i], src[i], m);
+}
+
+void isl_seq_fdiv_q(isl_int *dst, isl_int *src, isl_int m, unsigned len)
+{
+	int i;
+	for (i = 0; i < len; ++i)
+		isl_int_fdiv_q(dst[i], src[i], m);
+}
+
+void isl_seq_fdiv_r(isl_int *dst, isl_int *src, isl_int m, unsigned len)
+{
+	int i;
+	for (i = 0; i < len; ++i)
+		isl_int_fdiv_r(dst[i], src[i], m);
+}
+
+void isl_seq_combine(isl_int *dst, isl_int m1, isl_int *src1,
+			isl_int m2, isl_int *src2, unsigned len)
+{
+	int i;
+	isl_int tmp;
+
+	isl_int_init(tmp);
+	for (i = 0; i < len; ++i) {
+		isl_int_mul(tmp, m1, src1[i]);
+		isl_int_addmul(tmp, m2, src2[i]);
+		isl_int_set(dst[i], tmp);
+	}
+	isl_int_clear(tmp);
+}
+
+/*
+ * Let d = dst[pos] and s = src[pos]
+ * dst is replaced by |s| dst - sgn(s)d src
+ */
+void isl_seq_elim(isl_int *dst, isl_int *src, unsigned pos, unsigned len,
+		  isl_int *m)
+{
+	isl_int a;
+	isl_int b;
+
+	if (isl_int_is_zero(dst[pos]))
+		return;
+
+	isl_int_init(a);
+	isl_int_init(b);
+
+	isl_int_gcd(a, src[pos], dst[pos]);
+	isl_int_divexact(b, dst[pos], a);
+	if (isl_int_is_pos(src[pos]))
+		isl_int_neg(b, b);
+	isl_int_divexact(a, src[pos], a);
+	isl_int_abs(a, a);
+	isl_seq_combine(dst, a, dst, b, src, len);
+
+	if (m)
+		isl_int_mul(*m, *m, a);
+
+	isl_int_clear(a);
+	isl_int_clear(b);
+}
+
+int isl_seq_eq(isl_int *p1, isl_int *p2, unsigned len)
+{
+	int i;
+	for (i = 0; i < len; ++i)
+		if (isl_int_ne(p1[i], p2[i]))
+			return 0;
+	return 1;
+}
+
+int isl_seq_cmp(isl_int *p1, isl_int *p2, unsigned len)
+{
+	int i;
+	int cmp;
+	for (i = 0; i < len; ++i)
+		if ((cmp = isl_int_cmp(p1[i], p2[i])) != 0)
+			return cmp;
+	return 0;
+}
+
+int isl_seq_is_neg(isl_int *p1, isl_int *p2, unsigned len)
+{
+	int i;
+
+	for (i = 0; i < len; ++i) {
+		if (isl_int_abs_ne(p1[i], p2[i]))
+			return 0;
+		if (isl_int_is_zero(p1[i]))
+			continue;
+		if (isl_int_eq(p1[i], p2[i]))
+			return 0;
+	}
+	return 1;
+}
+
+int isl_seq_first_non_zero(isl_int *p, unsigned len)
+{
+	int i;
+
+	for (i = 0; i < len; ++i)
+		if (!isl_int_is_zero(p[i]))
+			return i;
+	return -1;
+}
+
+int isl_seq_last_non_zero(isl_int *p, unsigned len)
+{
+	int i;
+
+	for (i = len - 1; i >= 0; --i)
+		if (!isl_int_is_zero(p[i]))
+			return i;
+	return -1;
+}
+
+void isl_seq_abs_max(isl_int *p, unsigned len, isl_int *max)
+{
+	int i;
+
+	isl_int_set_si(*max, 0);
+
+	for (i = 0; i < len; ++i)
+		if (isl_int_abs_gt(p[i], *max))
+			isl_int_abs(*max, p[i]);
+}
+
+int isl_seq_abs_min_non_zero(isl_int *p, unsigned len)
+{
+	int i, min = isl_seq_first_non_zero(p, len);
+	if (min < 0)
+		return -1;
+	for (i = min + 1; i < len; ++i) {
+		if (isl_int_is_zero(p[i]))
+			continue;
+		if (isl_int_abs_lt(p[i], p[min]))
+			min = i;
+	}
+	return min;
+}
+
+void isl_seq_gcd(isl_int *p, unsigned len, isl_int *gcd)
+{
+	int i, min = isl_seq_abs_min_non_zero(p, len);
+
+	if (min < 0) {
+		isl_int_set_si(*gcd, 0);
+		return;
+	}
+	isl_int_abs(*gcd, p[min]);
+	for (i = 0; isl_int_cmp_si(*gcd, 1) > 0 && i < len; ++i) {
+		if (i == min)
+			continue;
+		if (isl_int_is_zero(p[i]))
+			continue;
+		isl_int_gcd(*gcd, *gcd, p[i]);
+	}
+}
+
+void isl_seq_normalize(struct isl_ctx *ctx, isl_int *p, unsigned len)
+{
+	if (len == 0)
+		return;
+	isl_seq_gcd(p, len, &ctx->normalize_gcd);
+	if (!isl_int_is_zero(ctx->normalize_gcd) &&
+	    !isl_int_is_one(ctx->normalize_gcd))
+		isl_seq_scale_down(p, p, ctx->normalize_gcd, len);
+}
+
+void isl_seq_lcm(isl_int *p, unsigned len, isl_int *lcm)
+{
+	int i;
+
+	if (len == 0) {
+		isl_int_set_si(*lcm, 1);
+		return;
+	}
+	isl_int_set(*lcm, p[0]);
+	for (i = 1; i < len; ++i)
+		isl_int_lcm(*lcm, *lcm, p[i]);
+}
+
+void isl_seq_inner_product(isl_int *p1, isl_int *p2, unsigned len,
+			   isl_int *prod)
+{
+	int i;
+	if (len == 0) {
+		isl_int_set_si(*prod, 0);
+		return;
+	}
+	isl_int_mul(*prod, p1[0], p2[0]);
+	for (i = 1; i < len; ++i)
+		isl_int_addmul(*prod, p1[i], p2[i]);
+}
+
+uint32_t isl_seq_hash(isl_int *p, unsigned len, uint32_t hash)
+{
+	int i;
+	for (i = 0; i < len; ++i) {
+		if (isl_int_is_zero(p[i]))
+			continue;
+		hash *= 16777619;
+		hash ^= (i & 0xFF);
+		hash = isl_int_hash(p[i], hash);
+	}
+	return hash;
+}
+
+uint32_t isl_seq_get_hash(isl_int *p, unsigned len)
+{
+	uint32_t hash = isl_hash_init();
+
+	return isl_seq_hash(p, len, hash);
+}
+
+uint32_t isl_seq_get_hash_bits(isl_int *p, unsigned len, unsigned bits)
+{
+	uint32_t hash;
+
+	hash = isl_seq_get_hash(p, len);
+	return isl_hash_bits(hash, bits);
+}
+
+void isl_seq_dump(isl_int *p, unsigned len)
+{
+	int i;
+
+	for (i = 0; i < len; ++i) {
+		if (i)
+			fprintf(stderr, " ");
+		isl_int_print(stderr, p[i], 0);
+	}
+	fprintf(stderr, "\n");
+}
diff --git a/polly/lib/External/isl/isl_seq.h b/polly/lib/External/isl/isl_seq.h
new file mode 100644
index 00000000000..05825f28720
--- /dev/null
+++ b/polly/lib/External/isl/isl_seq.h
@@ -0,0 +1,60 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_SEQ_H
+#define ISL_SEQ_H
+
+#include <sys/types.h>
+#include <isl_int.h>
+#include <isl/ctx.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+/* Some common operations on sequences of isl_int's */
+
+void isl_seq_clr(isl_int *p, unsigned len);
+void isl_seq_set(isl_int *p, isl_int v, unsigned len);
+void isl_seq_set_si(isl_int *p, int v, unsigned len);
+void isl_seq_neg(isl_int *dst, isl_int *src, unsigned len);
+void isl_seq_cpy(isl_int *dst, isl_int *src, unsigned len);
+void isl_seq_addmul(isl_int *dst, isl_int f, isl_int *src, unsigned len);
+void isl_seq_submul(isl_int *dst, isl_int f, isl_int *src, unsigned len);
+void isl_seq_swp_or_cpy(isl_int *dst, isl_int *src, unsigned len);
+void isl_seq_scale(isl_int *dst, isl_int *src, isl_int f, unsigned len);
+void isl_seq_scale_down(isl_int *dst, isl_int *src, isl_int f, unsigned len);
+void isl_seq_cdiv_q(isl_int *dst, isl_int *src, isl_int m, unsigned len);
+void isl_seq_fdiv_q(isl_int *dst, isl_int *src, isl_int m, unsigned len);
+void isl_seq_fdiv_r(isl_int *dst, isl_int *src, isl_int m, unsigned len);
+void isl_seq_combine(isl_int *dst, isl_int m1, isl_int *src1,
+			isl_int m2, isl_int *src2, unsigned len);
+void isl_seq_elim(isl_int *dst, isl_int *src, unsigned pos, unsigned len,
+		  isl_int *m);
+void isl_seq_abs_max(isl_int *p, unsigned len, isl_int *max);
+void isl_seq_gcd(isl_int *p, unsigned len, isl_int *gcd);
+void isl_seq_lcm(isl_int *p, unsigned len, isl_int *lcm);
+void isl_seq_normalize(struct isl_ctx *ctx, isl_int *p, unsigned len);
+void isl_seq_inner_product(isl_int *p1, isl_int *p2, unsigned len,
+			   isl_int *prod);
+int isl_seq_first_non_zero(isl_int *p, unsigned len);
+int isl_seq_last_non_zero(isl_int *p, unsigned len);
+int isl_seq_abs_min_non_zero(isl_int *p, unsigned len);
+int isl_seq_eq(isl_int *p1, isl_int *p2, unsigned len);
+int isl_seq_cmp(isl_int *p1, isl_int *p2, unsigned len);
+int isl_seq_is_neg(isl_int *p1, isl_int *p2, unsigned len);
+
+uint32_t isl_seq_get_hash(isl_int *p, unsigned len);
+uint32_t isl_seq_get_hash_bits(isl_int *p, unsigned len, unsigned bits);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/polly/lib/External/isl/isl_set_list.c b/polly/lib/External/isl/isl_set_list.c
new file mode 100644
index 00000000000..b27982748a7
--- /dev/null
+++ b/polly/lib/External/isl/isl_set_list.c
@@ -0,0 +1,21 @@
+#include <isl/set.h>
+
+#undef EL
+#define EL isl_basic_set
+
+#include <isl_list_templ.h>
+
+#undef EL
+#define EL isl_set
+
+#include <isl_list_templ.h>
+
+#undef BASE
+#define BASE basic_set
+
+#include <isl_list_templ.c>
+
+#undef BASE
+#define BASE set
+
+#include <isl_list_templ.c>
diff --git a/polly/lib/External/isl/isl_sort.c b/polly/lib/External/isl/isl_sort.c
new file mode 100644
index 00000000000..9ed273caa18
--- /dev/null
+++ b/polly/lib/External/isl/isl_sort.c
@@ -0,0 +1,157 @@
+/*
+ * The code of this file was taken from http://jeffreystedfast.blogspot.be,
+ * where it was posted in 2011 by Jeffrey Stedfast under the MIT license.
+ * The MIT license text is as follows:
+ *
+ * Permission is hereby granted, free of charge, to any person obtaining a copy
+ * of this software and associated documentation files (the "Software"), to
+ * deal in the Software without restriction, including without limitation the
+ * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or
+ * sell copies of the Software, and to permit persons to whom the Software is
+ * furnished to do so, subject to the following conditions:
+ *
+ * The above copyright notice and this permission notice shall be included in
+ * all copies or substantial portions of the Software.
+ *
+ * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+ * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+ * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+ * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+ * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
+ * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS
+ * IN THE SOFTWARE.
+ */
+
+#include <errno.h>
+#include <string.h>
+#include <stdlib.h>
+#include <isl_sort.h>
+
+#define MID(lo, hi) (lo + ((hi - lo) >> 1))
+
+/* The code here is an optimized merge sort. Starting from a generic merge sort
+ * the following optimizations were applied:
+ *
+ * o Batching of memcpy() calls: Instead of calling memcpy() to copy each and
+ *   every element into a temporary buffer, blocks of elements are copied
+ *   at a time.
+ *
+ * o To reduce the number of memcpy() calls further, copying leading
+ *   and trailing elements into our temporary buffer is avoided, in case it is
+ *   not necessary to merge them.
+ *
+ * A further optimization could be to specialize memcpy calls based on the
+ * size of the types we compare. For now, this code does not include the
+ * relevant optimization, as clang e.g. inlines a very efficient memcpy()
+ * implementation. It is not clear, that the specialized version as provided in
+ * the blog post, is really superior to the one that will be inlined by
+ * default. So we decided to keep the code simple until this optimization was
+ * proven to be beneficial.
+ */
+
+static void
+msort (void *array, void *buf, size_t low, size_t high, size_t size,
+       int (* compare) (const void *, const void *, void *), void *arg)
+{
+    char *a1, *al, *am, *ah, *ls, *hs, *lo, *hi, *b;
+    size_t copied = 0;
+    size_t mid;
+
+    mid = MID (low, high);
+
+    if (mid + 1 < high)
+        msort (array, buf, mid + 1, high, size, compare, arg);
+
+    if (mid > low)
+        msort (array, buf, low, mid, size, compare, arg);
+
+    ah = ((char *) array) + ((high + 1) * size);
+    am = ((char *) array) + ((mid + 1) * size);
+    a1 = al = ((char *) array) + (low * size);
+
+    b = (char *) buf;
+    lo = al;
+    hi = am;
+
+    do {
+        ls = lo;
+        hs = hi;
+
+        if (lo > al || hi > am) {
+            /* our last loop already compared lo & hi and found lo <= hi */
+            lo += size;
+        }
+
+        while (lo < am && compare (lo, hi, arg) <= 0)
+            lo += size;
+
+        if (lo < am) {
+            if (copied == 0) {
+                /* avoid copying the leading items */
+                a1 = lo;
+                ls = lo;
+            }
+
+            /* our last compare tells us hi < lo */
+            hi += size;
+
+            while (hi < ah && compare (hi, lo, arg) < 0)
+                hi += size;
+
+            if (lo > ls) {
+                memcpy (b, ls, lo - ls);
+                copied += (lo - ls);
+                b += (lo - ls);
+            }
+
+            memcpy (b, hs, hi - hs);
+            copied += (hi - hs);
+            b += (hi - hs);
+        } else if (copied) {
+            memcpy (b, ls, lo - ls);
+            copied += (lo - ls);
+            b += (lo - ls);
+
+            /* copy everything we needed to re-order back into array */
+            memcpy (a1, buf, copied);
+            return;
+        } else {
+            /* everything already in order */
+            return;
+        }
+    } while (hi < ah);
+
+    if (lo < am) {
+        memcpy (b, lo, am - lo);
+        copied += (am - lo);
+    }
+
+    memcpy (a1, buf, copied);
+}
+
+static int
+MergeSort (void *base, size_t nmemb, size_t size,
+           int (* compare) (const void *, const void *, void *), void *arg)
+{
+    void *tmp;
+
+    if (nmemb < 2)
+        return 0;
+
+    if (!(tmp = malloc (nmemb * size))) {
+        errno = ENOMEM;
+        return -1;
+    }
+
+    msort (base, tmp, 0, nmemb - 1, size, compare, arg);
+
+    free (tmp);
+
+    return 0;
+}
+
+int isl_sort(void *const pbase, size_t total_elems, size_t size,
+	int (*cmp)(const void *, const void *, void *arg), void *arg)
+{
+    return MergeSort (pbase, total_elems, size, cmp, arg);
+}
diff --git a/polly/lib/External/isl/isl_sort.h b/polly/lib/External/isl/isl_sort.h
new file mode 100644
index 00000000000..b69fe01d991
--- /dev/null
+++ b/polly/lib/External/isl/isl_sort.h
@@ -0,0 +1,9 @@
+#ifndef ISL_SORT_H
+#define ISL_SORT_H
+
+#include <stddef.h>
+
+int isl_sort(void *const pbase, size_t total_elems, size_t size,
+	int (*cmp)(const void *, const void *, void *arg), void *arg);
+
+#endif
diff --git a/polly/lib/External/isl/isl_space.c b/polly/lib/External/isl/isl_space.c
new file mode 100644
index 00000000000..1a5d6d3b2d8
--- /dev/null
+++ b/polly/lib/External/isl/isl_space.c
@@ -0,0 +1,2379 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2010      INRIA Saclay
+ * Copyright 2013-2014 Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France 
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+#include <stdlib.h>
+#include <string.h>
+#include <isl_space_private.h>
+#include <isl_id_private.h>
+#include <isl_reordering.h>
+
+isl_ctx *isl_space_get_ctx(__isl_keep isl_space *dim)
+{
+	return dim ? dim->ctx : NULL;
+}
+
+__isl_give isl_space *isl_space_alloc(isl_ctx *ctx,
+			unsigned nparam, unsigned n_in, unsigned n_out)
+{
+	isl_space *dim;
+
+	dim = isl_alloc_type(ctx, struct isl_space);
+	if (!dim)
+		return NULL;
+
+	dim->ctx = ctx;
+	isl_ctx_ref(ctx);
+	dim->ref = 1;
+	dim->nparam = nparam;
+	dim->n_in = n_in;
+	dim->n_out = n_out;
+
+	dim->tuple_id[0] = NULL;
+	dim->tuple_id[1] = NULL;
+
+	dim->nested[0] = NULL;
+	dim->nested[1] = NULL;
+
+	dim->n_id = 0;
+	dim->ids = NULL;
+
+	return dim;
+}
+
+/* Mark the space as being that of a set, by setting the domain tuple
+ * to isl_id_none.
+ */
+static __isl_give isl_space *mark_as_set(__isl_take isl_space *space)
+{
+	space = isl_space_cow(space);
+	if (!space)
+		return NULL;
+	space = isl_space_set_tuple_id(space, isl_dim_in, &isl_id_none);
+	return space;
+}
+
+/* Is the space that of a set?
+ */
+int isl_space_is_set(__isl_keep isl_space *space)
+{
+	if (!space)
+		return -1;
+	if (space->n_in != 0 || space->nested[0])
+		return 0;
+	if (space->tuple_id[0] != &isl_id_none)
+		return 0;
+	return 1;
+}
+
+/* Is the given space that of a map?
+ */
+int isl_space_is_map(__isl_keep isl_space *space)
+{
+	if (!space)
+		return -1;
+	return space->tuple_id[0] != &isl_id_none &&
+		space->tuple_id[1] != &isl_id_none;
+}
+
+__isl_give isl_space *isl_space_set_alloc(isl_ctx *ctx,
+			unsigned nparam, unsigned dim)
+{
+	isl_space *space;
+	space = isl_space_alloc(ctx, nparam, 0, dim);
+	space = mark_as_set(space);
+	return space;
+}
+
+/* Mark the space as being that of a parameter domain, by setting
+ * both tuples to isl_id_none.
+ */
+static __isl_give isl_space *mark_as_params(isl_space *space)
+{
+	if (!space)
+		return NULL;
+	space = isl_space_set_tuple_id(space, isl_dim_in, &isl_id_none);
+	space = isl_space_set_tuple_id(space, isl_dim_out, &isl_id_none);
+	return space;
+}
+
+/* Is the space that of a parameter domain?
+ */
+int isl_space_is_params(__isl_keep isl_space *space)
+{
+	if (!space)
+		return -1;
+	if (space->n_in != 0 || space->nested[0] ||
+	    space->n_out != 0 || space->nested[1])
+		return 0;
+	if (space->tuple_id[0] != &isl_id_none)
+		return 0;
+	if (space->tuple_id[1] != &isl_id_none)
+		return 0;
+	return 1;
+}
+
+/* Create a space for a parameter domain.
+ */
+__isl_give isl_space *isl_space_params_alloc(isl_ctx *ctx, unsigned nparam)
+{
+	isl_space *space;
+	space = isl_space_alloc(ctx, nparam, 0, 0);
+	space = mark_as_params(space);
+	return space;
+}
+
+static unsigned global_pos(__isl_keep isl_space *dim,
+				 enum isl_dim_type type, unsigned pos)
+{
+	struct isl_ctx *ctx = dim->ctx;
+
+	switch (type) {
+	case isl_dim_param:
+		isl_assert(ctx, pos < dim->nparam,
+			    return isl_space_dim(dim, isl_dim_all));
+		return pos;
+	case isl_dim_in:
+		isl_assert(ctx, pos < dim->n_in,
+			    return isl_space_dim(dim, isl_dim_all));
+		return pos + dim->nparam;
+	case isl_dim_out:
+		isl_assert(ctx, pos < dim->n_out,
+			    return isl_space_dim(dim, isl_dim_all));
+		return pos + dim->nparam + dim->n_in;
+	default:
+		isl_assert(ctx, 0, return isl_space_dim(dim, isl_dim_all));
+	}
+	return isl_space_dim(dim, isl_dim_all);
+}
+
+/* Extend length of ids array to the total number of dimensions.
+ */
+static __isl_give isl_space *extend_ids(__isl_take isl_space *dim)
+{
+	isl_id **ids;
+	int i;
+
+	if (isl_space_dim(dim, isl_dim_all) <= dim->n_id)
+		return dim;
+
+	if (!dim->ids) {
+		dim->ids = isl_calloc_array(dim->ctx,
+				isl_id *, isl_space_dim(dim, isl_dim_all));
+		if (!dim->ids)
+			goto error;
+	} else {
+		ids = isl_realloc_array(dim->ctx, dim->ids,
+				isl_id *, isl_space_dim(dim, isl_dim_all));
+		if (!ids)
+			goto error;
+		dim->ids = ids;
+		for (i = dim->n_id; i < isl_space_dim(dim, isl_dim_all); ++i)
+			dim->ids[i] = NULL;
+	}
+
+	dim->n_id = isl_space_dim(dim, isl_dim_all);
+
+	return dim;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+static __isl_give isl_space *set_id(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id)
+{
+	dim = isl_space_cow(dim);
+
+	if (!dim)
+		goto error;
+
+	pos = global_pos(dim, type, pos);
+	if (pos == isl_space_dim(dim, isl_dim_all))
+		goto error;
+
+	if (pos >= dim->n_id) {
+		if (!id)
+			return dim;
+		dim = extend_ids(dim);
+		if (!dim)
+			goto error;
+	}
+
+	dim->ids[pos] = id;
+
+	return dim;
+error:
+	isl_id_free(id);
+	isl_space_free(dim);
+	return NULL;
+}
+
+static __isl_keep isl_id *get_id(__isl_keep isl_space *dim,
+				 enum isl_dim_type type, unsigned pos)
+{
+	if (!dim)
+		return NULL;
+
+	pos = global_pos(dim, type, pos);
+	if (pos == isl_space_dim(dim, isl_dim_all))
+		return NULL;
+	if (pos >= dim->n_id)
+		return NULL;
+	return dim->ids[pos];
+}
+
+static unsigned offset(__isl_keep isl_space *dim, enum isl_dim_type type)
+{
+	switch (type) {
+	case isl_dim_param:	return 0;
+	case isl_dim_in:	return dim->nparam;
+	case isl_dim_out:	return dim->nparam + dim->n_in;
+	default:		return 0;
+	}
+}
+
+static unsigned n(__isl_keep isl_space *dim, enum isl_dim_type type)
+{
+	switch (type) {
+	case isl_dim_param:	return dim->nparam;
+	case isl_dim_in:	return dim->n_in;
+	case isl_dim_out:	return dim->n_out;
+	case isl_dim_all:	return dim->nparam + dim->n_in + dim->n_out;
+	default:		return 0;
+	}
+}
+
+unsigned isl_space_dim(__isl_keep isl_space *dim, enum isl_dim_type type)
+{
+	if (!dim)
+		return 0;
+	return n(dim, type);
+}
+
+unsigned isl_space_offset(__isl_keep isl_space *dim, enum isl_dim_type type)
+{
+	if (!dim)
+		return 0;
+	return offset(dim, type);
+}
+
+static __isl_give isl_space *copy_ids(__isl_take isl_space *dst,
+	enum isl_dim_type dst_type, unsigned offset, __isl_keep isl_space *src,
+	enum isl_dim_type src_type)
+{
+	int i;
+	isl_id *id;
+
+	if (!dst)
+		return NULL;
+
+	for (i = 0; i < n(src, src_type); ++i) {
+		id = get_id(src, src_type, i);
+		if (!id)
+			continue;
+		dst = set_id(dst, dst_type, offset + i, isl_id_copy(id));
+		if (!dst)
+			return NULL;
+	}
+	return dst;
+}
+
+__isl_take isl_space *isl_space_dup(__isl_keep isl_space *dim)
+{
+	isl_space *dup;
+	if (!dim)
+		return NULL;
+	dup = isl_space_alloc(dim->ctx, dim->nparam, dim->n_in, dim->n_out);
+	if (!dup)
+		return NULL;
+	if (dim->tuple_id[0] &&
+	    !(dup->tuple_id[0] = isl_id_copy(dim->tuple_id[0])))
+		goto error;
+	if (dim->tuple_id[1] &&
+	    !(dup->tuple_id[1] = isl_id_copy(dim->tuple_id[1])))
+		goto error;
+	if (dim->nested[0] && !(dup->nested[0] = isl_space_copy(dim->nested[0])))
+		goto error;
+	if (dim->nested[1] && !(dup->nested[1] = isl_space_copy(dim->nested[1])))
+		goto error;
+	if (!dim->ids)
+		return dup;
+	dup = copy_ids(dup, isl_dim_param, 0, dim, isl_dim_param);
+	dup = copy_ids(dup, isl_dim_in, 0, dim, isl_dim_in);
+	dup = copy_ids(dup, isl_dim_out, 0, dim, isl_dim_out);
+	return dup;
+error:
+	isl_space_free(dup);
+	return NULL;
+}
+
+__isl_give isl_space *isl_space_cow(__isl_take isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+
+	if (dim->ref == 1)
+		return dim;
+	dim->ref--;
+	return isl_space_dup(dim);
+}
+
+__isl_give isl_space *isl_space_copy(__isl_keep isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+
+	dim->ref++;
+	return dim;
+}
+
+__isl_null isl_space *isl_space_free(__isl_take isl_space *space)
+{
+	int i;
+
+	if (!space)
+		return NULL;
+
+	if (--space->ref > 0)
+		return NULL;
+
+	isl_id_free(space->tuple_id[0]);
+	isl_id_free(space->tuple_id[1]);
+
+	isl_space_free(space->nested[0]);
+	isl_space_free(space->nested[1]);
+
+	for (i = 0; i < space->n_id; ++i)
+		isl_id_free(space->ids[i]);
+	free(space->ids);
+	isl_ctx_deref(space->ctx);
+	
+	free(space);
+
+	return NULL;
+}
+
+/* Check if "s" is a valid dimension or tuple name.
+ * We currently only forbid names that look like a number.
+ *
+ * s is assumed to be non-NULL.
+ */
+static int name_ok(isl_ctx *ctx, const char *s)
+{
+	char *p;
+	long dummy;
+
+	dummy = strtol(s, &p, 0);
+	if (p != s)
+		isl_die(ctx, isl_error_invalid, "name looks like a number",
+			return 0);
+
+	return 1;
+}
+
+/* Is it possible for the given dimension type to have a tuple id?
+ */
+static int space_can_have_id(__isl_keep isl_space *space,
+	enum isl_dim_type type)
+{
+	if (!space)
+		return 0;
+	if (isl_space_is_params(space))
+		isl_die(space->ctx, isl_error_invalid,
+			"parameter spaces don't have tuple ids", return 0);
+	if (isl_space_is_set(space) && type != isl_dim_set)
+		isl_die(space->ctx, isl_error_invalid,
+			"set spaces can only have a set id", return 0);
+	if (type != isl_dim_in && type != isl_dim_out)
+		isl_die(space->ctx, isl_error_invalid,
+			"only input, output and set tuples can have ids",
+			return 0);
+
+	return 1;
+}
+
+/* Does the tuple have an id?
+ */
+int isl_space_has_tuple_id(__isl_keep isl_space *dim, enum isl_dim_type type)
+{
+	if (!space_can_have_id(dim, type))
+		return -1;
+	return dim->tuple_id[type - isl_dim_in] != NULL;
+}
+
+__isl_give isl_id *isl_space_get_tuple_id(__isl_keep isl_space *dim,
+	enum isl_dim_type type)
+{
+	int has_id;
+
+	if (!dim)
+		return NULL;
+	has_id = isl_space_has_tuple_id(dim, type);
+	if (has_id < 0)
+		return NULL;
+	if (!has_id)
+		isl_die(dim->ctx, isl_error_invalid,
+			"tuple has no id", return NULL);
+	return isl_id_copy(dim->tuple_id[type - isl_dim_in]);
+}
+
+__isl_give isl_space *isl_space_set_tuple_id(__isl_take isl_space *dim,
+	enum isl_dim_type type, __isl_take isl_id *id)
+{
+	dim = isl_space_cow(dim);
+	if (!dim || !id)
+		goto error;
+	if (type != isl_dim_in && type != isl_dim_out)
+		isl_die(dim->ctx, isl_error_invalid,
+			"only input, output and set tuples can have names",
+			goto error);
+
+	isl_id_free(dim->tuple_id[type - isl_dim_in]);
+	dim->tuple_id[type - isl_dim_in] = id;
+
+	return dim;
+error:
+	isl_id_free(id);
+	isl_space_free(dim);
+	return NULL;
+}
+
+__isl_give isl_space *isl_space_reset_tuple_id(__isl_take isl_space *dim,
+	enum isl_dim_type type)
+{
+	dim = isl_space_cow(dim);
+	if (!dim)
+		return NULL;
+	if (type != isl_dim_in && type != isl_dim_out)
+		isl_die(dim->ctx, isl_error_invalid,
+			"only input, output and set tuples can have names",
+			goto error);
+
+	isl_id_free(dim->tuple_id[type - isl_dim_in]);
+	dim->tuple_id[type - isl_dim_in] = NULL;
+
+	return dim;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+/* Set the id of the given dimension of "space" to "id".
+ * If the dimension already has an id, then it is replaced.
+ * If the dimension is a parameter, then we need to change it
+ * in the nested spaces (if any) as well.
+ */
+__isl_give isl_space *isl_space_set_dim_id(__isl_take isl_space *space,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id)
+{
+	space = isl_space_cow(space);
+	if (!space || !id)
+		goto error;
+
+	if (type == isl_dim_param) {
+		int i;
+
+		for (i = 0; i < 2; ++i) {
+			if (!space->nested[i])
+				continue;
+			space->nested[i] =
+				isl_space_set_dim_id(space->nested[i],
+						type, pos, isl_id_copy(id));
+			if (!space->nested[i])
+				goto error;
+		}
+	}
+
+	isl_id_free(get_id(space, type, pos));
+	return set_id(space, type, pos, id);
+error:
+	isl_id_free(id);
+	isl_space_free(space);
+	return NULL;
+}
+
+/* Reset the id of the given dimension of "space".
+ * If the dimension already has an id, then it is removed.
+ * If the dimension is a parameter, then we need to reset it
+ * in the nested spaces (if any) as well.
+ */
+__isl_give isl_space *isl_space_reset_dim_id(__isl_take isl_space *space,
+	enum isl_dim_type type, unsigned pos)
+{
+	space = isl_space_cow(space);
+	if (!space)
+		goto error;
+
+	if (type == isl_dim_param) {
+		int i;
+
+		for (i = 0; i < 2; ++i) {
+			if (!space->nested[i])
+				continue;
+			space->nested[i] =
+				isl_space_reset_dim_id(space->nested[i],
+							type, pos);
+			if (!space->nested[i])
+				goto error;
+		}
+	}
+
+	isl_id_free(get_id(space, type, pos));
+	return set_id(space, type, pos, NULL);
+error:
+	isl_space_free(space);
+	return NULL;
+}
+
+int isl_space_has_dim_id(__isl_keep isl_space *dim,
+	enum isl_dim_type type, unsigned pos)
+{
+	if (!dim)
+		return -1;
+	return get_id(dim, type, pos) != NULL;
+}
+
+__isl_give isl_id *isl_space_get_dim_id(__isl_keep isl_space *dim,
+	enum isl_dim_type type, unsigned pos)
+{
+	if (!dim)
+		return NULL;
+	if (!get_id(dim, type, pos))
+		isl_die(dim->ctx, isl_error_invalid,
+			"dim has no id", return NULL);
+	return isl_id_copy(get_id(dim, type, pos));
+}
+
+__isl_give isl_space *isl_space_set_tuple_name(__isl_take isl_space *dim,
+	enum isl_dim_type type, const char *s)
+{
+	isl_id *id;
+
+	if (!dim)
+		return NULL;
+
+	if (!s)
+		return isl_space_reset_tuple_id(dim, type);
+
+	if (!name_ok(dim->ctx, s))
+		goto error;
+
+	id = isl_id_alloc(dim->ctx, s, NULL);
+	return isl_space_set_tuple_id(dim, type, id);
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+/* Does the tuple have a name?
+ */
+int isl_space_has_tuple_name(__isl_keep isl_space *space,
+	enum isl_dim_type type)
+{
+	isl_id *id;
+
+	if (!space_can_have_id(space, type))
+		return -1;
+	id = space->tuple_id[type - isl_dim_in];
+	return id && id->name;
+}
+
+const char *isl_space_get_tuple_name(__isl_keep isl_space *dim,
+	 enum isl_dim_type type)
+{
+	isl_id *id;
+	if (!dim)
+		return NULL;
+	if (type != isl_dim_in && type != isl_dim_out)
+		return NULL;
+	id = dim->tuple_id[type - isl_dim_in];
+	return id ? id->name : NULL;
+}
+
+__isl_give isl_space *isl_space_set_dim_name(__isl_take isl_space *dim,
+				 enum isl_dim_type type, unsigned pos,
+				 const char *s)
+{
+	isl_id *id;
+
+	if (!dim)
+		return NULL;
+	if (!s)
+		return isl_space_reset_dim_id(dim, type, pos);
+	if (!name_ok(dim->ctx, s))
+		goto error;
+	id = isl_id_alloc(dim->ctx, s, NULL);
+	return isl_space_set_dim_id(dim, type, pos, id);
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+/* Does the given dimension have a name?
+ */
+int isl_space_has_dim_name(__isl_keep isl_space *space,
+	enum isl_dim_type type, unsigned pos)
+{
+	isl_id *id;
+
+	if (!space)
+		return -1;
+	id = get_id(space, type, pos);
+	return id && id->name;
+}
+
+__isl_keep const char *isl_space_get_dim_name(__isl_keep isl_space *dim,
+				 enum isl_dim_type type, unsigned pos)
+{
+	isl_id *id = get_id(dim, type, pos);
+	return id ? id->name : NULL;
+}
+
+int isl_space_find_dim_by_id(__isl_keep isl_space *dim, enum isl_dim_type type,
+	__isl_keep isl_id *id)
+{
+	int i;
+	int offset;
+	int n;
+
+	if (!dim || !id)
+		return -1;
+
+	offset = isl_space_offset(dim, type);
+	n = isl_space_dim(dim, type);
+	for (i = 0; i < n && offset + i < dim->n_id; ++i)
+		if (dim->ids[offset + i] == id)
+			return i;
+
+	return -1;
+}
+
+int isl_space_find_dim_by_name(__isl_keep isl_space *space,
+	enum isl_dim_type type, const char *name)
+{
+	int i;
+	int offset;
+	int n;
+
+	if (!space || !name)
+		return -1;
+
+	offset = isl_space_offset(space, type);
+	n = isl_space_dim(space, type);
+	for (i = 0; i < n && offset + i < space->n_id; ++i)
+		if (space->ids[offset + i]->name &&
+		    !strcmp(space->ids[offset + i]->name, name))
+			return i;
+
+	return -1;
+}
+
+/* Reset the user pointer on all identifiers of parameters and tuples
+ * of "space".
+ */
+__isl_give isl_space *isl_space_reset_user(__isl_take isl_space *space)
+{
+	int i;
+	isl_ctx *ctx;
+	isl_id *id;
+	const char *name;
+
+	if (!space)
+		return NULL;
+
+	ctx = isl_space_get_ctx(space);
+
+	for (i = 0; i < space->nparam && i < space->n_id; ++i) {
+		if (!isl_id_get_user(space->ids[i]))
+			continue;
+		space = isl_space_cow(space);
+		if (!space)
+			return NULL;
+		name = isl_id_get_name(space->ids[i]);
+		id = isl_id_alloc(ctx, name, NULL);
+		isl_id_free(space->ids[i]);
+		space->ids[i] = id;
+		if (!id)
+			return isl_space_free(space);
+	}
+
+	for (i = 0; i < 2; ++i) {
+		if (!space->tuple_id[i])
+			continue;
+		if (!isl_id_get_user(space->tuple_id[i]))
+			continue;
+		space = isl_space_cow(space);
+		if (!space)
+			return NULL;
+		name = isl_id_get_name(space->tuple_id[i]);
+		id = isl_id_alloc(ctx, name, NULL);
+		isl_id_free(space->tuple_id[i]);
+		space->tuple_id[i] = id;
+		if (!id)
+			return isl_space_free(space);
+	}
+
+	for (i = 0; i < 2; ++i) {
+		if (!space->nested[i])
+			continue;
+		space = isl_space_cow(space);
+		if (!space)
+			return NULL;
+		space->nested[i] = isl_space_reset_user(space->nested[i]);
+		if (!space->nested[i])
+			return isl_space_free(space);
+	}
+
+	return space;
+}
+
+static __isl_keep isl_id *tuple_id(__isl_keep isl_space *dim,
+	enum isl_dim_type type)
+{
+	if (!dim)
+		return NULL;
+	if (type == isl_dim_in)
+		return dim->tuple_id[0];
+	if (type == isl_dim_out)
+		return dim->tuple_id[1];
+	return NULL;
+}
+
+static __isl_keep isl_space *nested(__isl_keep isl_space *dim,
+	enum isl_dim_type type)
+{
+	if (!dim)
+		return NULL;
+	if (type == isl_dim_in)
+		return dim->nested[0];
+	if (type == isl_dim_out)
+		return dim->nested[1];
+	return NULL;
+}
+
+/* Are the two spaces the same, apart from positions and names of parameters?
+ */
+static int isl_space_has_equal_tuples(__isl_keep isl_space *space1,
+	__isl_keep isl_space *space2)
+{
+	if (!space1 || !space2)
+		return -1;
+	if (space1 == space2)
+		return 1;
+	return isl_space_tuple_is_equal(space1, isl_dim_in,
+					space2, isl_dim_in) &&
+	       isl_space_tuple_is_equal(space1, isl_dim_out,
+					space2, isl_dim_out);
+}
+
+/* Check if the tuple of type "type1" of "space1" is the same as
+ * the tuple of type "type2" of "space2".
+ *
+ * That is, check if the tuples have the same identifier, the same dimension
+ * and the same internal structure.
+ * The identifiers of the dimensions inside the tuples do not affect the result.
+ *
+ * Note that this function only checks the tuples themselves.
+ * If nested tuples are involved, then we need to be careful not
+ * to have result affected by possibly differing parameters
+ * in those nested tuples.
+ */
+int isl_space_tuple_is_equal(__isl_keep isl_space *space1,
+	enum isl_dim_type type1, __isl_keep isl_space *space2,
+	enum isl_dim_type type2)
+{
+	isl_id *id1, *id2;
+	isl_space *nested1, *nested2;
+
+	if (!space1 || !space2)
+		return -1;
+
+	if (space1 == space2 && type1 == type2)
+		return 1;
+
+	if (n(space1, type1) != n(space2, type2))
+		return 0;
+	id1 = tuple_id(space1, type1);
+	id2 = tuple_id(space2, type2);
+	if (!id1 ^ !id2)
+		return 0;
+	if (id1 && id1 != id2)
+		return 0;
+	nested1 = nested(space1, type1);
+	nested2 = nested(space2, type2);
+	if (!nested1 ^ !nested2)
+		return 0;
+	if (nested1 && !isl_space_has_equal_tuples(nested1, nested2))
+		return 0;
+	return 1;
+}
+
+/* This is the old, undocumented, name for isl_space_tuple_is_equal.
+ * It will be removed at some point.
+ */
+int isl_space_tuple_match(__isl_keep isl_space *space1, enum isl_dim_type type1,
+	__isl_keep isl_space *space2, enum isl_dim_type type2)
+{
+	return isl_space_tuple_is_equal(space1, type1, space2, type2);
+}
+
+static int match(__isl_keep isl_space *dim1, enum isl_dim_type dim1_type,
+	__isl_keep isl_space *dim2, enum isl_dim_type dim2_type)
+{
+	int i;
+
+	if (dim1 == dim2 && dim1_type == dim2_type)
+		return 1;
+
+	if (!isl_space_tuple_is_equal(dim1, dim1_type, dim2, dim2_type))
+		return 0;
+
+	if (!dim1->ids && !dim2->ids)
+		return 1;
+
+	for (i = 0; i < n(dim1, dim1_type); ++i) {
+		if (get_id(dim1, dim1_type, i) != get_id(dim2, dim2_type, i))
+			return 0;
+	}
+	return 1;
+}
+
+int isl_space_match(__isl_keep isl_space *dim1, enum isl_dim_type dim1_type,
+	__isl_keep isl_space *dim2, enum isl_dim_type dim2_type)
+{
+	if (!dim1 || !dim2)
+		return -1;
+
+	return match(dim1, dim1_type, dim2, dim2_type);
+}
+
+static void get_ids(__isl_keep isl_space *dim, enum isl_dim_type type,
+	unsigned first, unsigned n, __isl_keep isl_id **ids)
+{
+	int i;
+
+	for (i = 0; i < n ; ++i)
+		ids[i] = get_id(dim, type, first + i);
+}
+
+__isl_give isl_space *isl_space_extend(__isl_take isl_space *dim,
+			unsigned nparam, unsigned n_in, unsigned n_out)
+{
+	isl_id **ids = NULL;
+
+	if (!dim)
+		return NULL;
+	if (dim->nparam == nparam && dim->n_in == n_in && dim->n_out == n_out)
+		return dim;
+
+	isl_assert(dim->ctx, dim->nparam <= nparam, goto error);
+	isl_assert(dim->ctx, dim->n_in <= n_in, goto error);
+	isl_assert(dim->ctx, dim->n_out <= n_out, goto error);
+
+	dim = isl_space_cow(dim);
+	if (!dim)
+		goto error;
+
+	if (dim->ids) {
+		ids = isl_calloc_array(dim->ctx, isl_id *,
+					 nparam + n_in + n_out);
+		if (!ids)
+			goto error;
+		get_ids(dim, isl_dim_param, 0, dim->nparam, ids);
+		get_ids(dim, isl_dim_in, 0, dim->n_in, ids + nparam);
+		get_ids(dim, isl_dim_out, 0, dim->n_out, ids + nparam + n_in);
+		free(dim->ids);
+		dim->ids = ids;
+		dim->n_id = nparam + n_in + n_out;
+	}
+	dim->nparam = nparam;
+	dim->n_in = n_in;
+	dim->n_out = n_out;
+
+	return dim;
+error:
+	free(ids);
+	isl_space_free(dim);
+	return NULL;
+}
+
+__isl_give isl_space *isl_space_add_dims(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned n)
+{
+	if (!dim)
+		return NULL;
+	dim = isl_space_reset(dim, type);
+	switch (type) {
+	case isl_dim_param:
+		dim = isl_space_extend(dim,
+					dim->nparam + n, dim->n_in, dim->n_out);
+		if (dim && dim->nested[0] &&
+		    !(dim->nested[0] = isl_space_add_dims(dim->nested[0],
+						    isl_dim_param, n)))
+			goto error;
+		if (dim && dim->nested[1] &&
+		    !(dim->nested[1] = isl_space_add_dims(dim->nested[1],
+						    isl_dim_param, n)))
+			goto error;
+		return dim;
+	case isl_dim_in:
+		return isl_space_extend(dim,
+					dim->nparam, dim->n_in + n, dim->n_out);
+	case isl_dim_out:
+		return isl_space_extend(dim,
+					dim->nparam, dim->n_in, dim->n_out + n);
+	default:
+		isl_die(dim->ctx, isl_error_invalid,
+			"cannot add dimensions of specified type", goto error);
+	}
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+static int valid_dim_type(enum isl_dim_type type)
+{
+	switch (type) {
+	case isl_dim_param:
+	case isl_dim_in:
+	case isl_dim_out:
+		return 1;
+	default:
+		return 0;
+	}
+}
+
+/* Insert "n" dimensions of type "type" at position "pos".
+ * If we are inserting parameters, then they are also inserted in
+ * any nested spaces.
+ */
+__isl_give isl_space *isl_space_insert_dims(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned pos, unsigned n)
+{
+	isl_id **ids = NULL;
+
+	if (!dim)
+		return NULL;
+	if (n == 0)
+		return isl_space_reset(dim, type);
+
+	if (!valid_dim_type(type))
+		isl_die(dim->ctx, isl_error_invalid,
+			"cannot insert dimensions of specified type",
+			goto error);
+
+	isl_assert(dim->ctx, pos <= isl_space_dim(dim, type), goto error);
+
+	dim = isl_space_cow(dim);
+	if (!dim)
+		return NULL;
+
+	if (dim->ids) {
+		enum isl_dim_type t, o = isl_dim_param;
+		int off;
+		int s[3];
+		ids = isl_calloc_array(dim->ctx, isl_id *,
+				     dim->nparam + dim->n_in + dim->n_out + n);
+		if (!ids)
+			goto error;
+		off = 0;
+		s[isl_dim_param - o] = dim->nparam;
+		s[isl_dim_in - o] = dim->n_in;
+		s[isl_dim_out - o] = dim->n_out;
+		for (t = isl_dim_param; t <= isl_dim_out; ++t) {
+			if (t != type) {
+				get_ids(dim, t, 0, s[t - o], ids + off);
+				off += s[t - o];
+			} else {
+				get_ids(dim, t, 0, pos, ids + off);
+				off += pos + n;
+				get_ids(dim, t, pos, s[t - o] - pos, ids + off);
+				off += s[t - o] - pos;
+			}
+		}
+		free(dim->ids);
+		dim->ids = ids;
+		dim->n_id = dim->nparam + dim->n_in + dim->n_out + n;
+	}
+	switch (type) {
+	case isl_dim_param:	dim->nparam += n; break;
+	case isl_dim_in:	dim->n_in += n; break;
+	case isl_dim_out:	dim->n_out += n; break;
+	default:		;
+	}
+	dim = isl_space_reset(dim, type);
+
+	if (type == isl_dim_param) {
+		if (dim && dim->nested[0] &&
+		    !(dim->nested[0] = isl_space_insert_dims(dim->nested[0],
+						    isl_dim_param, pos, n)))
+			goto error;
+		if (dim && dim->nested[1] &&
+		    !(dim->nested[1] = isl_space_insert_dims(dim->nested[1],
+						    isl_dim_param, pos, n)))
+			goto error;
+	}
+
+	return dim;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+__isl_give isl_space *isl_space_move_dims(__isl_take isl_space *dim,
+	enum isl_dim_type dst_type, unsigned dst_pos,
+	enum isl_dim_type src_type, unsigned src_pos, unsigned n)
+{
+	int i;
+
+	if (!dim)
+		return NULL;
+	if (n == 0) {
+		dim = isl_space_reset(dim, src_type);
+		return isl_space_reset(dim, dst_type);
+	}
+
+	isl_assert(dim->ctx, src_pos + n <= isl_space_dim(dim, src_type),
+		goto error);
+
+	if (dst_type == src_type && dst_pos == src_pos)
+		return dim;
+
+	isl_assert(dim->ctx, dst_type != src_type, goto error);
+
+	dim = isl_space_reset(dim, src_type);
+	dim = isl_space_reset(dim, dst_type);
+
+	dim = isl_space_cow(dim);
+	if (!dim)
+		return NULL;
+
+	if (dim->ids) {
+		isl_id **ids;
+		enum isl_dim_type t, o = isl_dim_param;
+		int off;
+		int s[3];
+		ids = isl_calloc_array(dim->ctx, isl_id *,
+					 dim->nparam + dim->n_in + dim->n_out);
+		if (!ids)
+			goto error;
+		off = 0;
+		s[isl_dim_param - o] = dim->nparam;
+		s[isl_dim_in - o] = dim->n_in;
+		s[isl_dim_out - o] = dim->n_out;
+		for (t = isl_dim_param; t <= isl_dim_out; ++t) {
+			if (t == dst_type) {
+				get_ids(dim, t, 0, dst_pos, ids + off);
+				off += dst_pos;
+				get_ids(dim, src_type, src_pos, n, ids + off);
+				off += n;
+				get_ids(dim, t, dst_pos, s[t - o] - dst_pos,
+						ids + off);
+				off += s[t - o] - dst_pos;
+			} else if (t == src_type) {
+				get_ids(dim, t, 0, src_pos, ids + off);
+				off += src_pos;
+				get_ids(dim, t, src_pos + n,
+					    s[t - o] - src_pos - n, ids + off);
+				off += s[t - o] - src_pos - n;
+			} else {
+				get_ids(dim, t, 0, s[t - o], ids + off);
+				off += s[t - o];
+			}
+		}
+		free(dim->ids);
+		dim->ids = ids;
+		dim->n_id = dim->nparam + dim->n_in + dim->n_out;
+	}
+
+	switch (dst_type) {
+	case isl_dim_param:	dim->nparam += n; break;
+	case isl_dim_in:	dim->n_in += n; break;
+	case isl_dim_out:	dim->n_out += n; break;
+	default:		;
+	}
+
+	switch (src_type) {
+	case isl_dim_param:	dim->nparam -= n; break;
+	case isl_dim_in:	dim->n_in -= n; break;
+	case isl_dim_out:	dim->n_out -= n; break;
+	default:		;
+	}
+
+	if (dst_type != isl_dim_param && src_type != isl_dim_param)
+		return dim;
+
+	for (i = 0; i < 2; ++i) {
+		if (!dim->nested[i])
+			continue;
+		dim->nested[i] = isl_space_replace(dim->nested[i],
+						 isl_dim_param, dim);
+		if (!dim->nested[i])
+			goto error;
+	}
+
+	return dim;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+__isl_give isl_space *isl_space_join(__isl_take isl_space *left,
+	__isl_take isl_space *right)
+{
+	isl_space *dim;
+
+	if (!left || !right)
+		goto error;
+
+	isl_assert(left->ctx, match(left, isl_dim_param, right, isl_dim_param),
+			goto error);
+	isl_assert(left->ctx,
+		isl_space_tuple_is_equal(left, isl_dim_out, right, isl_dim_in),
+		goto error);
+
+	dim = isl_space_alloc(left->ctx, left->nparam, left->n_in, right->n_out);
+	if (!dim)
+		goto error;
+
+	dim = copy_ids(dim, isl_dim_param, 0, left, isl_dim_param);
+	dim = copy_ids(dim, isl_dim_in, 0, left, isl_dim_in);
+	dim = copy_ids(dim, isl_dim_out, 0, right, isl_dim_out);
+
+	if (dim && left->tuple_id[0] &&
+	    !(dim->tuple_id[0] = isl_id_copy(left->tuple_id[0])))
+		goto error;
+	if (dim && right->tuple_id[1] &&
+	    !(dim->tuple_id[1] = isl_id_copy(right->tuple_id[1])))
+		goto error;
+	if (dim && left->nested[0] &&
+	    !(dim->nested[0] = isl_space_copy(left->nested[0])))
+		goto error;
+	if (dim && right->nested[1] &&
+	    !(dim->nested[1] = isl_space_copy(right->nested[1])))
+		goto error;
+
+	isl_space_free(left);
+	isl_space_free(right);
+
+	return dim;
+error:
+	isl_space_free(left);
+	isl_space_free(right);
+	return NULL;
+}
+
+/* Given two map spaces { A -> C } and { B -> D }, construct the space
+ * { [A -> B] -> [C -> D] }.
+ * Given two set spaces { A } and { B }, construct the space { [A -> B] }.
+ */
+__isl_give isl_space *isl_space_product(__isl_take isl_space *left,
+	__isl_take isl_space *right)
+{
+	isl_space *dom1, *dom2, *nest1, *nest2;
+	int is_set;
+
+	if (!left || !right)
+		goto error;
+
+	is_set = isl_space_is_set(left);
+	if (is_set != isl_space_is_set(right))
+		isl_die(isl_space_get_ctx(left), isl_error_invalid,
+			"expecting either two set spaces or two map spaces",
+			goto error);
+	if (is_set)
+		return isl_space_range_product(left, right);
+
+	isl_assert(left->ctx, match(left, isl_dim_param, right, isl_dim_param),
+			goto error);
+
+	dom1 = isl_space_domain(isl_space_copy(left));
+	dom2 = isl_space_domain(isl_space_copy(right));
+	nest1 = isl_space_wrap(isl_space_join(isl_space_reverse(dom1), dom2));
+
+	dom1 = isl_space_range(left);
+	dom2 = isl_space_range(right);
+	nest2 = isl_space_wrap(isl_space_join(isl_space_reverse(dom1), dom2));
+
+	return isl_space_join(isl_space_reverse(nest1), nest2);
+error:
+	isl_space_free(left);
+	isl_space_free(right);
+	return NULL;
+}
+
+/* Given two spaces { A -> C } and { B -> C }, construct the space
+ * { [A -> B] -> C }
+ */
+__isl_give isl_space *isl_space_domain_product(__isl_take isl_space *left,
+	__isl_take isl_space *right)
+{
+	isl_space *ran, *dom1, *dom2, *nest;
+
+	if (!left || !right)
+		goto error;
+
+	if (!match(left, isl_dim_param, right, isl_dim_param))
+		isl_die(left->ctx, isl_error_invalid,
+			"parameters need to match", goto error);
+	if (!isl_space_tuple_is_equal(left, isl_dim_out, right, isl_dim_out))
+		isl_die(left->ctx, isl_error_invalid,
+			"ranges need to match", goto error);
+
+	ran = isl_space_range(isl_space_copy(left));
+
+	dom1 = isl_space_domain(left);
+	dom2 = isl_space_domain(right);
+	nest = isl_space_wrap(isl_space_join(isl_space_reverse(dom1), dom2));
+
+	return isl_space_join(isl_space_reverse(nest), ran);
+error:
+	isl_space_free(left);
+	isl_space_free(right);
+	return NULL;
+}
+
+__isl_give isl_space *isl_space_range_product(__isl_take isl_space *left,
+	__isl_take isl_space *right)
+{
+	isl_space *dom, *ran1, *ran2, *nest;
+
+	if (!left || !right)
+		goto error;
+
+	isl_assert(left->ctx, match(left, isl_dim_param, right, isl_dim_param),
+			goto error);
+	if (!isl_space_tuple_is_equal(left, isl_dim_in, right, isl_dim_in))
+		isl_die(left->ctx, isl_error_invalid,
+			"domains need to match", goto error);
+
+	dom = isl_space_domain(isl_space_copy(left));
+
+	ran1 = isl_space_range(left);
+	ran2 = isl_space_range(right);
+	nest = isl_space_wrap(isl_space_join(isl_space_reverse(ran1), ran2));
+
+	return isl_space_join(isl_space_reverse(dom), nest);
+error:
+	isl_space_free(left);
+	isl_space_free(right);
+	return NULL;
+}
+
+/* Given a space of the form [A -> B] -> [C -> D], return the space A -> C.
+ */
+__isl_give isl_space *isl_space_factor_domain(__isl_take isl_space *space)
+{
+	space = isl_space_domain_factor_domain(space);
+	space = isl_space_range_factor_domain(space);
+	return space;
+}
+
+/* Given a space of the form [A -> B] -> [C -> D], return the space B -> D.
+ */
+__isl_give isl_space *isl_space_factor_range(__isl_take isl_space *space)
+{
+	space = isl_space_domain_factor_range(space);
+	space = isl_space_range_factor_range(space);
+	return space;
+}
+
+/* Given a space of the form [A -> B] -> C, return the space A -> C.
+ */
+__isl_give isl_space *isl_space_domain_factor_domain(
+	__isl_take isl_space *space)
+{
+	isl_space *nested;
+	isl_space *domain;
+
+	if (!space)
+		return NULL;
+	if (!isl_space_domain_is_wrapping(space))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"domain not a product", return isl_space_free(space));
+
+	nested = space->nested[0];
+	domain = isl_space_copy(space);
+	domain = isl_space_drop_dims(space, isl_dim_in,
+					nested->n_in, nested->n_out);
+	if (!domain)
+		return isl_space_free(space);
+	if (nested->tuple_id[0]) {
+		domain->tuple_id[0] = isl_id_copy(nested->tuple_id[0]);
+		if (!domain->tuple_id[0])
+			goto error;
+	}
+	if (nested->nested[0]) {
+		domain->nested[0] = isl_space_copy(nested->nested[0]);
+		if (!domain->nested[0])
+			goto error;
+	}
+
+	isl_space_free(space);
+	return domain;
+error:
+	isl_space_free(space);
+	isl_space_free(domain);
+	return NULL;
+}
+
+/* Given a space of the form [A -> B] -> C, return the space B -> C.
+ */
+__isl_give isl_space *isl_space_domain_factor_range(
+	__isl_take isl_space *space)
+{
+	isl_space *nested;
+	isl_space *range;
+
+	if (!space)
+		return NULL;
+	if (!isl_space_domain_is_wrapping(space))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"domain not a product", return isl_space_free(space));
+
+	nested = space->nested[0];
+	range = isl_space_copy(space);
+	range = isl_space_drop_dims(space, isl_dim_in, 0, nested->n_in);
+	if (!range)
+		return isl_space_free(space);
+	if (nested->tuple_id[1]) {
+		range->tuple_id[0] = isl_id_copy(nested->tuple_id[1]);
+		if (!range->tuple_id[0])
+			goto error;
+	}
+	if (nested->nested[1]) {
+		range->nested[0] = isl_space_copy(nested->nested[1]);
+		if (!range->nested[0])
+			goto error;
+	}
+
+	isl_space_free(space);
+	return range;
+error:
+	isl_space_free(space);
+	isl_space_free(range);
+	return NULL;
+}
+
+/* Given a space of the form A -> [B -> C], return the space A -> B.
+ */
+__isl_give isl_space *isl_space_range_factor_domain(
+	__isl_take isl_space *space)
+{
+	isl_space *nested;
+	isl_space *domain;
+
+	if (!space)
+		return NULL;
+	if (!isl_space_range_is_wrapping(space))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"range not a product", return isl_space_free(space));
+
+	nested = space->nested[1];
+	domain = isl_space_copy(space);
+	domain = isl_space_drop_dims(space, isl_dim_out,
+					nested->n_in, nested->n_out);
+	if (!domain)
+		return isl_space_free(space);
+	if (nested->tuple_id[0]) {
+		domain->tuple_id[1] = isl_id_copy(nested->tuple_id[0]);
+		if (!domain->tuple_id[1])
+			goto error;
+	}
+	if (nested->nested[0]) {
+		domain->nested[1] = isl_space_copy(nested->nested[0]);
+		if (!domain->nested[1])
+			goto error;
+	}
+
+	isl_space_free(space);
+	return domain;
+error:
+	isl_space_free(space);
+	isl_space_free(domain);
+	return NULL;
+}
+
+/* Given a space of the form A -> [B -> C], return the space A -> C.
+ */
+__isl_give isl_space *isl_space_range_factor_range(
+	__isl_take isl_space *space)
+{
+	isl_space *nested;
+	isl_space *range;
+
+	if (!space)
+		return NULL;
+	if (!isl_space_range_is_wrapping(space))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"range not a product", return isl_space_free(space));
+
+	nested = space->nested[1];
+	range = isl_space_copy(space);
+	range = isl_space_drop_dims(space, isl_dim_out, 0, nested->n_in);
+	if (!range)
+		return isl_space_free(space);
+	if (nested->tuple_id[1]) {
+		range->tuple_id[1] = isl_id_copy(nested->tuple_id[1]);
+		if (!range->tuple_id[1])
+			goto error;
+	}
+	if (nested->nested[1]) {
+		range->nested[1] = isl_space_copy(nested->nested[1]);
+		if (!range->nested[1])
+			goto error;
+	}
+
+	isl_space_free(space);
+	return range;
+error:
+	isl_space_free(space);
+	isl_space_free(range);
+	return NULL;
+}
+
+__isl_give isl_space *isl_space_map_from_set(__isl_take isl_space *dim)
+{
+	isl_ctx *ctx;
+	isl_id **ids = NULL;
+
+	if (!dim)
+		return NULL;
+	ctx = isl_space_get_ctx(dim);
+	if (!isl_space_is_set(dim))
+		isl_die(ctx, isl_error_invalid, "not a set space", goto error);
+	dim = isl_space_cow(dim);
+	if (!dim)
+		return NULL;
+	if (dim->ids) {
+		ids = isl_calloc_array(dim->ctx, isl_id *,
+					dim->nparam + dim->n_out + dim->n_out);
+		if (!ids)
+			goto error;
+		get_ids(dim, isl_dim_param, 0, dim->nparam, ids);
+		get_ids(dim, isl_dim_out, 0, dim->n_out, ids + dim->nparam);
+	}
+	dim->n_in = dim->n_out;
+	if (ids) {
+		free(dim->ids);
+		dim->ids = ids;
+		dim->n_id = dim->nparam + dim->n_out + dim->n_out;
+		dim = copy_ids(dim, isl_dim_out, 0, dim, isl_dim_in);
+	}
+	isl_id_free(dim->tuple_id[0]);
+	dim->tuple_id[0] = isl_id_copy(dim->tuple_id[1]);
+	isl_space_free(dim->nested[0]);
+	dim->nested[0] = isl_space_copy(dim->nested[1]);
+	return dim;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+__isl_give isl_space *isl_space_map_from_domain_and_range(
+	__isl_take isl_space *domain, __isl_take isl_space *range)
+{
+	if (!domain || !range)
+		goto error;
+	if (!isl_space_is_set(domain))
+		isl_die(isl_space_get_ctx(domain), isl_error_invalid,
+			"domain is not a set space", goto error);
+	if (!isl_space_is_set(range))
+		isl_die(isl_space_get_ctx(range), isl_error_invalid,
+			"range is not a set space", goto error);
+	return isl_space_join(isl_space_reverse(domain), range);
+error:
+	isl_space_free(domain);
+	isl_space_free(range);
+	return NULL;
+}
+
+static __isl_give isl_space *set_ids(__isl_take isl_space *dim,
+	enum isl_dim_type type,
+	unsigned first, unsigned n, __isl_take isl_id **ids)
+{
+	int i;
+
+	for (i = 0; i < n ; ++i)
+		dim = set_id(dim, type, first + i, ids[i]);
+
+	return dim;
+}
+
+__isl_give isl_space *isl_space_reverse(__isl_take isl_space *dim)
+{
+	unsigned t;
+	isl_space *nested;
+	isl_id **ids = NULL;
+	isl_id *id;
+
+	if (!dim)
+		return NULL;
+	if (match(dim, isl_dim_in, dim, isl_dim_out))
+		return dim;
+
+	dim = isl_space_cow(dim);
+	if (!dim)
+		return NULL;
+
+	id = dim->tuple_id[0];
+	dim->tuple_id[0] = dim->tuple_id[1];
+	dim->tuple_id[1] = id;
+
+	nested = dim->nested[0];
+	dim->nested[0] = dim->nested[1];
+	dim->nested[1] = nested;
+
+	if (dim->ids) {
+		int n_id = dim->n_in + dim->n_out;
+		ids = isl_alloc_array(dim->ctx, isl_id *, n_id);
+		if (n_id && !ids)
+			goto error;
+		get_ids(dim, isl_dim_in, 0, dim->n_in, ids);
+		get_ids(dim, isl_dim_out, 0, dim->n_out, ids + dim->n_in);
+	}
+
+	t = dim->n_in;
+	dim->n_in = dim->n_out;
+	dim->n_out = t;
+
+	if (dim->ids) {
+		dim = set_ids(dim, isl_dim_out, 0, dim->n_out, ids);
+		dim = set_ids(dim, isl_dim_in, 0, dim->n_in, ids + dim->n_out);
+		free(ids);
+	}
+
+	return dim;
+error:
+	free(ids);
+	isl_space_free(dim);
+	return NULL;
+}
+
+__isl_give isl_space *isl_space_drop_dims(__isl_take isl_space *dim,
+	enum isl_dim_type type, unsigned first, unsigned num)
+{
+	int i;
+
+	if (!dim)
+		return NULL;
+
+	if (num == 0)
+		return isl_space_reset(dim, type);
+
+	if (!valid_dim_type(type))
+		isl_die(dim->ctx, isl_error_invalid,
+			"cannot drop dimensions of specified type", goto error);
+
+	if (first + num > n(dim, type) || first + num < first)
+		isl_die(isl_space_get_ctx(dim), isl_error_invalid,
+			"index out of bounds", return isl_space_free(dim));
+	dim = isl_space_cow(dim);
+	if (!dim)
+		goto error;
+	if (dim->ids) {
+		dim = extend_ids(dim);
+		if (!dim)
+			goto error;
+		for (i = 0; i < num; ++i)
+			isl_id_free(get_id(dim, type, first + i));
+		for (i = first+num; i < n(dim, type); ++i)
+			set_id(dim, type, i - num, get_id(dim, type, i));
+		switch (type) {
+		case isl_dim_param:
+			get_ids(dim, isl_dim_in, 0, dim->n_in,
+				dim->ids + offset(dim, isl_dim_in) - num);
+		case isl_dim_in:
+			get_ids(dim, isl_dim_out, 0, dim->n_out,
+				dim->ids + offset(dim, isl_dim_out) - num);
+		default:
+			;
+		}
+		dim->n_id -= num;
+	}
+	switch (type) {
+	case isl_dim_param:	dim->nparam -= num; break;
+	case isl_dim_in:	dim->n_in -= num; break;
+	case isl_dim_out:	dim->n_out -= num; break;
+	default:		;
+	}
+	dim = isl_space_reset(dim, type);
+	if (type == isl_dim_param) {
+		if (dim && dim->nested[0] &&
+		    !(dim->nested[0] = isl_space_drop_dims(dim->nested[0],
+						    isl_dim_param, first, num)))
+			goto error;
+		if (dim && dim->nested[1] &&
+		    !(dim->nested[1] = isl_space_drop_dims(dim->nested[1],
+						    isl_dim_param, first, num)))
+			goto error;
+	}
+	return dim;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+__isl_give isl_space *isl_space_drop_inputs(__isl_take isl_space *dim,
+		unsigned first, unsigned n)
+{
+	if (!dim)
+		return NULL;
+	return isl_space_drop_dims(dim, isl_dim_in, first, n);
+}
+
+__isl_give isl_space *isl_space_drop_outputs(__isl_take isl_space *dim,
+		unsigned first, unsigned n)
+{
+	if (!dim)
+		return NULL;
+	return isl_space_drop_dims(dim, isl_dim_out, first, n);
+}
+
+__isl_give isl_space *isl_space_domain(__isl_take isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+	dim = isl_space_drop_outputs(dim, 0, dim->n_out);
+	dim = isl_space_reverse(dim);
+	dim = mark_as_set(dim);
+	return dim;
+}
+
+__isl_give isl_space *isl_space_from_domain(__isl_take isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+	if (!isl_space_is_set(dim))
+		isl_die(isl_space_get_ctx(dim), isl_error_invalid,
+			"not a set space", goto error);
+	dim = isl_space_reverse(dim);
+	dim = isl_space_reset(dim, isl_dim_out);
+	return dim;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+__isl_give isl_space *isl_space_range(__isl_take isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+	dim = isl_space_drop_inputs(dim, 0, dim->n_in);
+	dim = mark_as_set(dim);
+	return dim;
+}
+
+__isl_give isl_space *isl_space_from_range(__isl_take isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+	if (!isl_space_is_set(dim))
+		isl_die(isl_space_get_ctx(dim), isl_error_invalid,
+			"not a set space", goto error);
+	return isl_space_reset(dim, isl_dim_in);
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+/* Given a map space A -> B, return the map space [A -> B] -> A.
+ */
+__isl_give isl_space *isl_space_domain_map(__isl_take isl_space *space)
+{
+	isl_space *domain;
+
+	domain = isl_space_from_range(isl_space_domain(isl_space_copy(space)));
+	space = isl_space_from_domain(isl_space_wrap(space));
+	space = isl_space_join(space, domain);
+
+	return space;
+}
+
+/* Given a map space A -> B, return the map space [A -> B] -> B.
+ */
+__isl_give isl_space *isl_space_range_map(__isl_take isl_space *space)
+{
+	isl_space *range;
+
+	range = isl_space_from_range(isl_space_range(isl_space_copy(space)));
+	space = isl_space_from_domain(isl_space_wrap(space));
+	space = isl_space_join(space, range);
+
+	return space;
+}
+
+__isl_give isl_space *isl_space_params(__isl_take isl_space *space)
+{
+	if (isl_space_is_params(space))
+		return space;
+	space = isl_space_drop_dims(space,
+			    isl_dim_in, 0, isl_space_dim(space, isl_dim_in));
+	space = isl_space_drop_dims(space,
+			    isl_dim_out, 0, isl_space_dim(space, isl_dim_out));
+	space = mark_as_params(space);
+	return space;
+}
+
+__isl_give isl_space *isl_space_set_from_params(__isl_take isl_space *space)
+{
+	if (!space)
+		return NULL;
+	if (!isl_space_is_params(space))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"not a parameter space", goto error);
+	return isl_space_reset(space, isl_dim_set);
+error:
+	isl_space_free(space);
+	return NULL;
+}
+
+__isl_give isl_space *isl_space_as_set_space(__isl_take isl_space *dim)
+{
+	dim = isl_space_cow(dim);
+	if (!dim)
+		return NULL;
+
+	dim->n_out += dim->n_in;
+	dim->n_in = 0;
+	dim = isl_space_reset(dim, isl_dim_in);
+	dim = isl_space_reset(dim, isl_dim_out);
+
+	return dim;
+}
+
+__isl_give isl_space *isl_space_underlying(__isl_take isl_space *dim,
+	unsigned n_div)
+{
+	int i;
+
+	if (!dim)
+		return NULL;
+	if (n_div == 0 &&
+	    dim->nparam == 0 && dim->n_in == 0 && dim->n_id == 0)
+		return isl_space_reset(isl_space_reset(dim, isl_dim_in), isl_dim_out);
+	dim = isl_space_cow(dim);
+	if (!dim)
+		return NULL;
+	dim->n_out += dim->nparam + dim->n_in + n_div;
+	dim->nparam = 0;
+	dim->n_in = 0;
+
+	for (i = 0; i < dim->n_id; ++i)
+		isl_id_free(get_id(dim, isl_dim_out, i));
+	dim->n_id = 0;
+	dim = isl_space_reset(dim, isl_dim_in);
+	dim = isl_space_reset(dim, isl_dim_out);
+
+	return dim;
+}
+
+/* Are the two spaces the same, including positions and names of parameters?
+ */
+int isl_space_is_equal(__isl_keep isl_space *dim1, __isl_keep isl_space *dim2)
+{
+	if (!dim1 || !dim2)
+		return -1;
+	if (dim1 == dim2)
+		return 1;
+	return match(dim1, isl_dim_param, dim2, isl_dim_param) &&
+	       isl_space_tuple_is_equal(dim1, isl_dim_in, dim2, isl_dim_in) &&
+	       isl_space_tuple_is_equal(dim1, isl_dim_out, dim2, isl_dim_out);
+}
+
+/* Is space1 equal to the domain of space2?
+ *
+ * In the internal version we also allow space2 to be the space of a set,
+ * provided space1 is a parameter space.
+ */
+int isl_space_is_domain_internal(__isl_keep isl_space *space1,
+	__isl_keep isl_space *space2)
+{
+	if (!space1 || !space2)
+		return -1;
+	if (!isl_space_is_set(space1))
+		return 0;
+	return match(space1, isl_dim_param, space2, isl_dim_param) &&
+	       isl_space_tuple_is_equal(space1, isl_dim_set,
+					space2, isl_dim_in);
+}
+
+/* Is space1 equal to the domain of space2?
+ */
+int isl_space_is_domain(__isl_keep isl_space *space1,
+	__isl_keep isl_space *space2)
+{
+	if (!space2)
+		return -1;
+	if (!isl_space_is_map(space2))
+		return 0;
+	return isl_space_is_domain_internal(space1, space2);
+}
+
+/* Is space1 equal to the range of space2?
+ *
+ * In the internal version, space2 is allowed to be the space of a set,
+ * in which case it should be equal to space1.
+ */
+int isl_space_is_range_internal(__isl_keep isl_space *space1,
+	__isl_keep isl_space *space2)
+{
+	if (!space1 || !space2)
+		return -1;
+	if (!isl_space_is_set(space1))
+		return 0;
+	return match(space1, isl_dim_param, space2, isl_dim_param) &&
+	       isl_space_tuple_is_equal(space1, isl_dim_set,
+					space2, isl_dim_out);
+}
+
+/* Is space1 equal to the range of space2?
+ */
+int isl_space_is_range(__isl_keep isl_space *space1,
+	__isl_keep isl_space *space2)
+{
+	if (!space2)
+		return -1;
+	if (!isl_space_is_map(space2))
+		return 0;
+	return isl_space_is_range_internal(space1, space2);
+}
+
+int isl_space_compatible(__isl_keep isl_space *dim1,
+	__isl_keep isl_space *dim2)
+{
+	return dim1->nparam == dim2->nparam &&
+	       dim1->n_in + dim1->n_out == dim2->n_in + dim2->n_out;
+}
+
+static uint32_t isl_hash_dim(uint32_t hash, __isl_keep isl_space *dim)
+{
+	int i;
+	isl_id *id;
+
+	if (!dim)
+		return hash;
+
+	isl_hash_byte(hash, dim->nparam % 256);
+	isl_hash_byte(hash, dim->n_in % 256);
+	isl_hash_byte(hash, dim->n_out % 256);
+
+	for (i = 0; i < dim->nparam; ++i) {
+		id = get_id(dim, isl_dim_param, i);
+		hash = isl_hash_id(hash, id);
+	}
+
+	id = tuple_id(dim, isl_dim_in);
+	hash = isl_hash_id(hash, id);
+	id = tuple_id(dim, isl_dim_out);
+	hash = isl_hash_id(hash, id);
+
+	hash = isl_hash_dim(hash, dim->nested[0]);
+	hash = isl_hash_dim(hash, dim->nested[1]);
+
+	return hash;
+}
+
+uint32_t isl_space_get_hash(__isl_keep isl_space *dim)
+{
+	uint32_t hash;
+
+	if (!dim)
+		return 0;
+
+	hash = isl_hash_init();
+	hash = isl_hash_dim(hash, dim);
+
+	return hash;
+}
+
+int isl_space_is_wrapping(__isl_keep isl_space *dim)
+{
+	if (!dim)
+		return -1;
+
+	if (!isl_space_is_set(dim))
+		return 0;
+
+	return dim->nested[1] != NULL;
+}
+
+/* Is "space" the space of a map where the domain is a wrapped map space?
+ */
+int isl_space_domain_is_wrapping(__isl_keep isl_space *space)
+{
+	if (!space)
+		return -1;
+
+	if (isl_space_is_set(space))
+		return 0;
+
+	return space->nested[0] != NULL;
+}
+
+/* Is "space" the space of a map where the range is a wrapped map space?
+ */
+int isl_space_range_is_wrapping(__isl_keep isl_space *space)
+{
+	if (!space)
+		return -1;
+
+	if (isl_space_is_set(space))
+		return 0;
+
+	return space->nested[1] != NULL;
+}
+
+__isl_give isl_space *isl_space_wrap(__isl_take isl_space *dim)
+{
+	isl_space *wrap;
+
+	if (!dim)
+		return NULL;
+
+	wrap = isl_space_set_alloc(dim->ctx,
+				    dim->nparam, dim->n_in + dim->n_out);
+
+	wrap = copy_ids(wrap, isl_dim_param, 0, dim, isl_dim_param);
+	wrap = copy_ids(wrap, isl_dim_set, 0, dim, isl_dim_in);
+	wrap = copy_ids(wrap, isl_dim_set, dim->n_in, dim, isl_dim_out);
+
+	if (!wrap)
+		goto error;
+
+	wrap->nested[1] = dim;
+
+	return wrap;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+__isl_give isl_space *isl_space_unwrap(__isl_take isl_space *dim)
+{
+	isl_space *unwrap;
+
+	if (!dim)
+		return NULL;
+
+	if (!isl_space_is_wrapping(dim))
+		isl_die(dim->ctx, isl_error_invalid, "not a wrapping space",
+			goto error);
+
+	unwrap = isl_space_copy(dim->nested[1]);
+	isl_space_free(dim);
+
+	return unwrap;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+int isl_space_is_named_or_nested(__isl_keep isl_space *dim, enum isl_dim_type type)
+{
+	if (type != isl_dim_in && type != isl_dim_out)
+		return 0;
+	if (!dim)
+		return -1;
+	if (dim->tuple_id[type - isl_dim_in])
+		return 1;
+	if (dim->nested[type - isl_dim_in])
+		return 1;
+	return 0;
+}
+
+int isl_space_may_be_set(__isl_keep isl_space *dim)
+{
+	if (!dim)
+		return -1;
+	if (isl_space_is_set(dim))
+		return 1;
+	if (isl_space_dim(dim, isl_dim_in) != 0)
+		return 0;
+	if (isl_space_is_named_or_nested(dim, isl_dim_in))
+		return 0;
+	return 1;
+}
+
+__isl_give isl_space *isl_space_reset(__isl_take isl_space *dim,
+	enum isl_dim_type type)
+{
+	if (!isl_space_is_named_or_nested(dim, type))
+		return dim;
+
+	dim = isl_space_cow(dim);
+	if (!dim)
+		return NULL;
+
+	isl_id_free(dim->tuple_id[type - isl_dim_in]);
+	dim->tuple_id[type - isl_dim_in] = NULL;
+	isl_space_free(dim->nested[type - isl_dim_in]);
+	dim->nested[type - isl_dim_in] = NULL;
+
+	return dim;
+}
+
+__isl_give isl_space *isl_space_flatten(__isl_take isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+	if (!dim->nested[0] && !dim->nested[1])
+		return dim;
+
+	if (dim->nested[0])
+		dim = isl_space_reset(dim, isl_dim_in);
+	if (dim && dim->nested[1])
+		dim = isl_space_reset(dim, isl_dim_out);
+
+	return dim;
+}
+
+__isl_give isl_space *isl_space_flatten_domain(__isl_take isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+	if (!dim->nested[0])
+		return dim;
+
+	return isl_space_reset(dim, isl_dim_in);
+}
+
+__isl_give isl_space *isl_space_flatten_range(__isl_take isl_space *dim)
+{
+	if (!dim)
+		return NULL;
+	if (!dim->nested[1])
+		return dim;
+
+	return isl_space_reset(dim, isl_dim_out);
+}
+
+/* Replace the dimensions of the given type of dst by those of src.
+ */
+__isl_give isl_space *isl_space_replace(__isl_take isl_space *dst,
+	enum isl_dim_type type, __isl_keep isl_space *src)
+{
+	dst = isl_space_cow(dst);
+
+	if (!dst || !src)
+		goto error;
+
+	dst = isl_space_drop_dims(dst, type, 0, isl_space_dim(dst, type));
+	dst = isl_space_add_dims(dst, type, isl_space_dim(src, type));
+	dst = copy_ids(dst, type, 0, src, type);
+
+	if (dst && type == isl_dim_param) {
+		int i;
+		for (i = 0; i <= 1; ++i) {
+			if (!dst->nested[i])
+				continue;
+			dst->nested[i] = isl_space_replace(dst->nested[i],
+							 type, src);
+			if (!dst->nested[i])
+				goto error;
+		}
+	}
+
+	return dst;
+error:
+	isl_space_free(dst);
+	return NULL;
+}
+
+/* Given a dimension specification "dim" of a set, create a dimension
+ * specification for the lift of the set.  In particular, the result
+ * is of the form [dim -> local[..]], with n_local variables in the
+ * range of the wrapped map.
+ */
+__isl_give isl_space *isl_space_lift(__isl_take isl_space *dim, unsigned n_local)
+{
+	isl_space *local_dim;
+
+	if (!dim)
+		return NULL;
+
+	local_dim = isl_space_dup(dim);
+	local_dim = isl_space_drop_dims(local_dim, isl_dim_set, 0, dim->n_out);
+	local_dim = isl_space_add_dims(local_dim, isl_dim_set, n_local);
+	local_dim = isl_space_set_tuple_name(local_dim, isl_dim_set, "local");
+	dim = isl_space_join(isl_space_from_domain(dim),
+			    isl_space_from_range(local_dim));
+	dim = isl_space_wrap(dim);
+	dim = isl_space_set_tuple_name(dim, isl_dim_set, "lifted");
+
+	return dim;
+}
+
+int isl_space_can_zip(__isl_keep isl_space *dim)
+{
+	if (!dim)
+		return -1;
+
+	return dim->nested[0] && dim->nested[1];
+}
+
+__isl_give isl_space *isl_space_zip(__isl_take isl_space *dim)
+{
+	isl_space *dom, *ran;
+	isl_space *dom_dom, *dom_ran, *ran_dom, *ran_ran;
+
+	if (!isl_space_can_zip(dim))
+		isl_die(dim->ctx, isl_error_invalid, "dim cannot be zipped",
+			goto error);
+
+	if (!dim)
+		return NULL;
+	dom = isl_space_unwrap(isl_space_domain(isl_space_copy(dim)));
+	ran = isl_space_unwrap(isl_space_range(dim));
+	dom_dom = isl_space_domain(isl_space_copy(dom));
+	dom_ran = isl_space_range(dom);
+	ran_dom = isl_space_domain(isl_space_copy(ran));
+	ran_ran = isl_space_range(ran);
+	dom = isl_space_join(isl_space_from_domain(dom_dom),
+			   isl_space_from_range(ran_dom));
+	ran = isl_space_join(isl_space_from_domain(dom_ran),
+			   isl_space_from_range(ran_ran));
+	return isl_space_join(isl_space_from_domain(isl_space_wrap(dom)),
+			    isl_space_from_range(isl_space_wrap(ran)));
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+/* Can we apply isl_space_curry to "space"?
+ * That is, does it have a nested relation in its domain?
+ */
+int isl_space_can_curry(__isl_keep isl_space *space)
+{
+	if (!space)
+		return -1;
+
+	return !!space->nested[0];
+}
+
+/* Given a space (A -> B) -> C, return the corresponding space
+ * A -> (B -> C).
+ */
+__isl_give isl_space *isl_space_curry(__isl_take isl_space *space)
+{
+	isl_space *dom, *ran;
+	isl_space *dom_dom, *dom_ran;
+
+	if (!space)
+		return NULL;
+
+	if (!isl_space_can_curry(space))
+		isl_die(space->ctx, isl_error_invalid,
+			"space cannot be curried", goto error);
+
+	dom = isl_space_unwrap(isl_space_domain(isl_space_copy(space)));
+	ran = isl_space_range(space);
+	dom_dom = isl_space_domain(isl_space_copy(dom));
+	dom_ran = isl_space_range(dom);
+	ran = isl_space_join(isl_space_from_domain(dom_ran),
+			   isl_space_from_range(ran));
+	return isl_space_join(isl_space_from_domain(dom_dom),
+			    isl_space_from_range(isl_space_wrap(ran)));
+error:
+	isl_space_free(space);
+	return NULL;
+}
+
+/* Can we apply isl_space_uncurry to "space"?
+ * That is, does it have a nested relation in its range?
+ */
+int isl_space_can_uncurry(__isl_keep isl_space *space)
+{
+	if (!space)
+		return -1;
+
+	return !!space->nested[1];
+}
+
+/* Given a space A -> (B -> C), return the corresponding space
+ * (A -> B) -> C.
+ */
+__isl_give isl_space *isl_space_uncurry(__isl_take isl_space *space)
+{
+	isl_space *dom, *ran;
+	isl_space *ran_dom, *ran_ran;
+
+	if (!space)
+		return NULL;
+
+	if (!isl_space_can_uncurry(space))
+		isl_die(space->ctx, isl_error_invalid,
+			"space cannot be uncurried",
+			return isl_space_free(space));
+
+	dom = isl_space_domain(isl_space_copy(space));
+	ran = isl_space_unwrap(isl_space_range(space));
+	ran_dom = isl_space_domain(isl_space_copy(ran));
+	ran_ran = isl_space_range(ran);
+	dom = isl_space_join(isl_space_from_domain(dom),
+			   isl_space_from_range(ran_dom));
+	return isl_space_join(isl_space_from_domain(isl_space_wrap(dom)),
+			    isl_space_from_range(ran_ran));
+}
+
+int isl_space_has_named_params(__isl_keep isl_space *dim)
+{
+	int i;
+	unsigned off;
+
+	if (!dim)
+		return -1;
+	if (dim->nparam == 0)
+		return 1;
+	off = isl_space_offset(dim, isl_dim_param);
+	if (off + dim->nparam > dim->n_id)
+		return 0;
+	for (i = 0; i < dim->nparam; ++i)
+		if (!dim->ids[off + i])
+			return 0;
+	return 1;
+}
+
+/* Align the initial parameters of dim1 to match the order in dim2.
+ */
+__isl_give isl_space *isl_space_align_params(__isl_take isl_space *dim1,
+	__isl_take isl_space *dim2)
+{
+	isl_reordering *exp;
+
+	if (!isl_space_has_named_params(dim1) || !isl_space_has_named_params(dim2))
+		isl_die(isl_space_get_ctx(dim1), isl_error_invalid,
+			"parameter alignment requires named parameters",
+			goto error);
+
+	dim2 = isl_space_params(dim2);
+	exp = isl_parameter_alignment_reordering(dim1, dim2);
+	exp = isl_reordering_extend_space(exp, dim1);
+	isl_space_free(dim2);
+	if (!exp)
+		return NULL;
+	dim1 = isl_space_copy(exp->dim);
+	isl_reordering_free(exp);
+	return dim1;
+error:
+	isl_space_free(dim1);
+	isl_space_free(dim2);
+	return NULL;
+}
+
+/* Given the space of set (domain), construct a space for a map
+ * with as domain the given space and as range the range of "model".
+ */
+__isl_give isl_space *isl_space_extend_domain_with_range(
+	__isl_take isl_space *space, __isl_take isl_space *model)
+{
+	if (!model)
+		goto error;
+
+	space = isl_space_from_domain(space);
+	space = isl_space_add_dims(space, isl_dim_out,
+				    isl_space_dim(model, isl_dim_out));
+	if (isl_space_has_tuple_id(model, isl_dim_out))
+		space = isl_space_set_tuple_id(space, isl_dim_out,
+				isl_space_get_tuple_id(model, isl_dim_out));
+	if (!space)
+		goto error;
+	if (model->nested[1]) {
+		isl_space *nested = isl_space_copy(model->nested[1]);
+		int n_nested, n_space;
+		nested = isl_space_align_params(nested, isl_space_copy(space));
+		n_nested = isl_space_dim(nested, isl_dim_param);
+		n_space = isl_space_dim(space, isl_dim_param);
+		if (n_nested > n_space)
+			nested = isl_space_drop_dims(nested, isl_dim_param,
+						n_space, n_nested - n_space);
+		if (!nested)
+			goto error;
+		space->nested[1] = nested;
+	}
+	isl_space_free(model);
+	return space;
+error:
+	isl_space_free(model);
+	isl_space_free(space);
+	return NULL;
+}
+
+/* Compare the "type" dimensions of two isl_spaces.
+ *
+ * The order is fairly arbitrary.
+ */
+static int isl_space_cmp_type(__isl_keep isl_space *space1,
+	__isl_keep isl_space *space2, enum isl_dim_type type)
+{
+	int cmp;
+	isl_space *nested1, *nested2;
+
+	if (isl_space_dim(space1, type) != isl_space_dim(space2, type))
+		return isl_space_dim(space1, type) -
+			isl_space_dim(space2, type);
+
+	cmp = isl_id_cmp(tuple_id(space1, type), tuple_id(space2, type));
+	if (cmp != 0)
+		return cmp;
+
+	nested1 = nested(space1, type);
+	nested2 = nested(space2, type);
+	if (!nested1 != !nested2)
+		return !nested1 - !nested2;
+
+	if (nested1)
+		return isl_space_cmp(nested1, nested2);
+
+	return 0;
+}
+
+/* Compare two isl_spaces.
+ *
+ * The order is fairly arbitrary.
+ */
+int isl_space_cmp(__isl_keep isl_space *space1, __isl_keep isl_space *space2)
+{
+	int i;
+	int cmp;
+
+	if (space1 == space2)
+		return 0;
+	if (!space1)
+		return -1;
+	if (!space2)
+		return 1;
+
+	cmp = isl_space_cmp_type(space1, space2, isl_dim_param);
+	if (cmp != 0)
+		return cmp;
+	cmp = isl_space_cmp_type(space1, space2, isl_dim_in);
+	if (cmp != 0)
+		return cmp;
+	cmp = isl_space_cmp_type(space1, space2, isl_dim_out);
+	if (cmp != 0)
+		return cmp;
+
+	if (!space1->ids && !space2->ids)
+		return 0;
+
+	for (i = 0; i < n(space1, isl_dim_param); ++i) {
+		cmp = isl_id_cmp(get_id(space1, isl_dim_param, i),
+				 get_id(space2, isl_dim_param, i));
+		if (cmp != 0)
+			return cmp;
+	}
+
+	return 0;
+}
diff --git a/polly/lib/External/isl/isl_space_private.h b/polly/lib/External/isl/isl_space_private.h
new file mode 100644
index 00000000000..3c892a4986a
--- /dev/null
+++ b/polly/lib/External/isl/isl_space_private.h
@@ -0,0 +1,60 @@
+#ifndef ISL_SPACE_PRIVATE
+#define ISL_SPACE_PRIVATE
+
+#include <isl/space.h>
+#include <isl/hash.h>
+#include <isl/id.h>
+
+struct isl_name;
+struct isl_space {
+	int ref;
+
+	struct isl_ctx *ctx;
+
+	unsigned nparam;
+	unsigned n_in;		/* zero for sets */
+	unsigned n_out;		/* dim for sets */
+
+	isl_id *tuple_id[2];
+	isl_space *nested[2];
+
+	unsigned n_id;
+	isl_id **ids;
+};
+
+__isl_give isl_space *isl_space_cow(__isl_take isl_space *dim);
+
+__isl_give isl_space *isl_space_underlying(__isl_take isl_space *dim,
+	unsigned n_div);
+
+uint32_t isl_space_get_hash(__isl_keep isl_space *dim);
+
+int isl_space_is_domain_internal(__isl_keep isl_space *space1,
+	__isl_keep isl_space *space2);
+int isl_space_is_range_internal(__isl_keep isl_space *space1,
+	__isl_keep isl_space *space2);
+
+__isl_give isl_space *isl_space_as_set_space(__isl_take isl_space *dim);
+
+unsigned isl_space_offset(__isl_keep isl_space *dim, enum isl_dim_type type);
+
+int isl_space_may_be_set(__isl_keep isl_space *dim);
+int isl_space_is_named_or_nested(__isl_keep isl_space *dim, enum isl_dim_type type);
+int isl_space_has_named_params(__isl_keep isl_space *dim);
+__isl_give isl_space *isl_space_reset(__isl_take isl_space *dim,
+	enum isl_dim_type type);
+__isl_give isl_space *isl_space_flatten(__isl_take isl_space *dim);
+__isl_give isl_space *isl_space_flatten_domain(__isl_take isl_space *dim);
+__isl_give isl_space *isl_space_flatten_range(__isl_take isl_space *dim);
+
+__isl_give isl_space *isl_space_replace(__isl_take isl_space *dst,
+	enum isl_dim_type type, __isl_keep isl_space *src);
+
+__isl_give isl_space *isl_space_lift(__isl_take isl_space *dim, unsigned n_local);
+
+__isl_give isl_space *isl_space_extend_domain_with_range(
+	__isl_take isl_space *domain, __isl_take isl_space *model);
+
+int isl_space_cmp(__isl_keep isl_space *space1, __isl_keep isl_space *space2);
+
+#endif
diff --git a/polly/lib/External/isl/isl_stream.c b/polly/lib/External/isl/isl_stream.c
new file mode 100644
index 00000000000..c5b15115668
--- /dev/null
+++ b/polly/lib/External/isl/isl_stream.c
@@ -0,0 +1,778 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <ctype.h>
+#include <string.h>
+#include <strings.h>
+#include <isl/ctx.h>
+#include <isl_stream_private.h>
+#include <isl/map.h>
+#include <isl/aff.h>
+#include <isl_val_private.h>
+
+struct isl_keyword {
+	char			*name;
+	enum isl_token_type	type;
+};
+
+static int same_name(const void *entry, const void *val)
+{
+	const struct isl_keyword *keyword = (const struct isl_keyword *)entry;
+
+	return !strcmp(keyword->name, val);
+}
+
+enum isl_token_type isl_stream_register_keyword(struct isl_stream *s,
+	const char *name)
+{
+	struct isl_hash_table_entry *entry;
+	struct isl_keyword *keyword;
+	uint32_t name_hash;
+
+	if (!s->keywords) {
+		s->keywords = isl_hash_table_alloc(s->ctx, 10);
+		if (!s->keywords)
+			return ISL_TOKEN_ERROR;
+		s->next_type = ISL_TOKEN_LAST;
+	}
+
+	name_hash = isl_hash_string(isl_hash_init(), name);
+
+	entry = isl_hash_table_find(s->ctx, s->keywords, name_hash,
+					same_name, name, 1);
+	if (!entry)
+		return ISL_TOKEN_ERROR;
+	if (entry->data) {
+		keyword = entry->data;
+		return keyword->type;
+	}
+
+	keyword = isl_calloc_type(s->ctx, struct isl_keyword);
+	if (!keyword)
+		return ISL_TOKEN_ERROR;
+	keyword->type = s->next_type++;
+	keyword->name = strdup(name);
+	if (!keyword->name) {
+		free(keyword);
+		return ISL_TOKEN_ERROR;
+	}
+	entry->data = keyword;
+
+	return keyword->type;
+}
+
+struct isl_token *isl_token_new(isl_ctx *ctx,
+	int line, int col, unsigned on_new_line)
+{
+	struct isl_token *tok = isl_alloc_type(ctx, struct isl_token);
+	if (!tok)
+		return NULL;
+	tok->line = line;
+	tok->col = col;
+	tok->on_new_line = on_new_line;
+	tok->is_keyword = 0;
+	tok->u.s = NULL;
+	return tok;
+}
+
+/* Return the type of "tok".
+ */
+int isl_token_get_type(struct isl_token *tok)
+{
+	return tok ? tok->type : ISL_TOKEN_ERROR;
+}
+
+/* Given a token of type ISL_TOKEN_VALUE, return the value it represents.
+ */
+__isl_give isl_val *isl_token_get_val(isl_ctx *ctx, struct isl_token *tok)
+{
+	if (!tok)
+		return NULL;
+	if (tok->type != ISL_TOKEN_VALUE)
+		isl_die(ctx, isl_error_invalid, "not a value token",
+			return NULL);
+
+	return isl_val_int_from_isl_int(ctx, tok->u.v);
+}
+
+/* Given a token of type ISL_TOKEN_STRING, return the string it represents.
+ */
+__isl_give char *isl_token_get_str(isl_ctx *ctx, struct isl_token *tok)
+{
+	if (!tok)
+		return NULL;
+	if (tok->type != ISL_TOKEN_STRING)
+		isl_die(ctx, isl_error_invalid, "not a string token",
+			return NULL);
+
+	return strdup(tok->u.s);
+}
+
+void isl_token_free(struct isl_token *tok)
+{
+	if (!tok)
+		return;
+	if (tok->type == ISL_TOKEN_VALUE)
+		isl_int_clear(tok->u.v);
+	else if (tok->type == ISL_TOKEN_MAP)
+		isl_map_free(tok->u.map);
+	else if (tok->type == ISL_TOKEN_AFF)
+		isl_pw_aff_free(tok->u.pwaff);
+	else
+		free(tok->u.s);
+	free(tok);
+}
+
+void isl_stream_error(struct isl_stream *s, struct isl_token *tok, char *msg)
+{
+	int line = tok ? tok->line : s->line;
+	int col = tok ? tok->col : s->col;
+	fprintf(stderr, "syntax error (%d, %d): %s\n", line, col, msg);
+	if (tok) {
+		if (tok->type < 256)
+			fprintf(stderr, "got '%c'\n", tok->type);
+		else if (tok->type == ISL_TOKEN_IDENT)
+			fprintf(stderr, "got ident '%s'\n", tok->u.s);
+		else if (tok->is_keyword)
+			fprintf(stderr, "got keyword '%s'\n", tok->u.s);
+		else if (tok->type == ISL_TOKEN_VALUE) {
+			fprintf(stderr, "got value '");
+			isl_int_print(stderr, tok->u.v, 0);
+			fprintf(stderr, "'\n");
+		} else if (tok->type == ISL_TOKEN_MAP) {
+			isl_printer *p;
+			fprintf(stderr, "got map '");
+			p = isl_printer_to_file(s->ctx, stderr);
+			p = isl_printer_print_map(p, tok->u.map);
+			isl_printer_free(p);
+			fprintf(stderr, "'\n");
+		} else if (tok->type == ISL_TOKEN_AFF) {
+			isl_printer *p;
+			fprintf(stderr, "got affine expression '");
+			p = isl_printer_to_file(s->ctx, stderr);
+			p = isl_printer_print_pw_aff(p, tok->u.pwaff);
+			isl_printer_free(p);
+			fprintf(stderr, "'\n");
+		} else if (tok->u.s)
+			fprintf(stderr, "got token '%s'\n", tok->u.s);
+		else
+			fprintf(stderr, "got token type %d\n", tok->type);
+	}
+}
+
+static struct isl_stream* isl_stream_new(struct isl_ctx *ctx)
+{
+	int i;
+	struct isl_stream *s = isl_alloc_type(ctx, struct isl_stream);
+	if (!s)
+		return NULL;
+	s->ctx = ctx;
+	isl_ctx_ref(s->ctx);
+	s->file = NULL;
+	s->str = NULL;
+	s->len = 0;
+	s->line = 1;
+	s->col = 0;
+	s->eof = 0;
+	s->c = -1;
+	s->n_un = 0;
+	for (i = 0; i < 5; ++i)
+		s->tokens[i] = NULL;
+	s->n_token = 0;
+	s->keywords = NULL;
+	s->size = 256;
+	s->buffer = isl_alloc_array(ctx, char, s->size);
+	if (!s->buffer)
+		goto error;
+	return s;
+error:
+	isl_stream_free(s);
+	return NULL;
+}
+
+struct isl_stream* isl_stream_new_file(struct isl_ctx *ctx, FILE *file)
+{
+	struct isl_stream *s = isl_stream_new(ctx);
+	if (!s)
+		return NULL;
+	s->file = file;
+	return s;
+}
+
+struct isl_stream* isl_stream_new_str(struct isl_ctx *ctx, const char *str)
+{
+	struct isl_stream *s;
+	if (!str)
+		return NULL;
+	s = isl_stream_new(ctx);
+	if (!s)
+		return NULL;
+	s->str = str;
+	return s;
+}
+
+static int stream_getc(struct isl_stream *s)
+{
+	int c;
+	if (s->eof)
+		return -1;
+	if (s->n_un)
+		return s->c = s->un[--s->n_un];
+	if (s->file)
+		c = fgetc(s->file);
+	else {
+		c = *s->str++;
+		if (c == '\0')
+			c = -1;
+	}
+	if (c == -1)
+		s->eof = 1;
+	if (!s->eof) {
+		if (s->c == '\n') {
+			s->line++;
+			s->col = 0;
+		} else
+			s->col++;
+	}
+	s->c = c;
+	return c;
+}
+
+static void isl_stream_ungetc(struct isl_stream *s, int c)
+{
+	isl_assert(s->ctx, s->n_un < 5, return);
+	s->un[s->n_un++] = c;
+	s->c = -1;
+}
+
+static int isl_stream_getc(struct isl_stream *s)
+{
+	int c;
+
+	do {
+		c = stream_getc(s);
+		if (c != '\\')
+			return c;
+		c = stream_getc(s);
+	} while (c == '\n');
+
+	isl_stream_ungetc(s, c);
+
+	return '\\';
+}
+
+static int isl_stream_push_char(struct isl_stream *s, int c)
+{
+	if (s->len >= s->size) {
+		char *buffer;
+		s->size = (3*s->size)/2;
+		buffer = isl_realloc_array(s->ctx, s->buffer, char, s->size);
+		if (!buffer)
+			return -1;
+		s->buffer = buffer;
+	}
+	s->buffer[s->len++] = c;
+	return 0;
+}
+
+void isl_stream_push_token(struct isl_stream *s, struct isl_token *tok)
+{
+	isl_assert(s->ctx, s->n_token < 5, return);
+	s->tokens[s->n_token++] = tok;
+}
+
+static enum isl_token_type check_keywords(struct isl_stream *s)
+{
+	struct isl_hash_table_entry *entry;
+	struct isl_keyword *keyword;
+	uint32_t name_hash;
+
+	if (!strcasecmp(s->buffer, "exists"))
+		return ISL_TOKEN_EXISTS;
+	if (!strcasecmp(s->buffer, "and"))
+		return ISL_TOKEN_AND;
+	if (!strcasecmp(s->buffer, "or"))
+		return ISL_TOKEN_OR;
+	if (!strcasecmp(s->buffer, "implies"))
+		return ISL_TOKEN_IMPLIES;
+	if (!strcasecmp(s->buffer, "not"))
+		return ISL_TOKEN_NOT;
+	if (!strcasecmp(s->buffer, "infty"))
+		return ISL_TOKEN_INFTY;
+	if (!strcasecmp(s->buffer, "infinity"))
+		return ISL_TOKEN_INFTY;
+	if (!strcasecmp(s->buffer, "NaN"))
+		return ISL_TOKEN_NAN;
+	if (!strcasecmp(s->buffer, "min"))
+		return ISL_TOKEN_MIN;
+	if (!strcasecmp(s->buffer, "max"))
+		return ISL_TOKEN_MAX;
+	if (!strcasecmp(s->buffer, "rat"))
+		return ISL_TOKEN_RAT;
+	if (!strcasecmp(s->buffer, "true"))
+		return ISL_TOKEN_TRUE;
+	if (!strcasecmp(s->buffer, "false"))
+		return ISL_TOKEN_FALSE;
+	if (!strcasecmp(s->buffer, "ceild"))
+		return ISL_TOKEN_CEILD;
+	if (!strcasecmp(s->buffer, "floord"))
+		return ISL_TOKEN_FLOORD;
+	if (!strcasecmp(s->buffer, "mod"))
+		return ISL_TOKEN_MOD;
+	if (!strcasecmp(s->buffer, "ceil"))
+		return ISL_TOKEN_CEIL;
+	if (!strcasecmp(s->buffer, "floor"))
+		return ISL_TOKEN_FLOOR;
+
+	if (!s->keywords)
+		return ISL_TOKEN_IDENT;
+
+	name_hash = isl_hash_string(isl_hash_init(), s->buffer);
+	entry = isl_hash_table_find(s->ctx, s->keywords, name_hash, same_name,
+					s->buffer, 0);
+	if (entry) {
+		keyword = entry->data;
+		return keyword->type;
+	}
+
+	return ISL_TOKEN_IDENT;
+}
+
+int isl_stream_skip_line(struct isl_stream *s)
+{
+	int c;
+
+	while ((c = isl_stream_getc(s)) != -1 && c != '\n')
+		/* nothing */
+		;
+
+	return c == -1 ? -1 : 0;
+}
+
+static struct isl_token *next_token(struct isl_stream *s, int same_line)
+{
+	int c;
+	struct isl_token *tok = NULL;
+	int line, col;
+	int old_line = s->line;
+
+	if (s->n_token) {
+		if (same_line && s->tokens[s->n_token - 1]->on_new_line)
+			return NULL;
+		return s->tokens[--s->n_token];
+	}
+
+	if (same_line && s->c == '\n')
+		return NULL;
+
+	s->len = 0;
+
+	/* skip spaces and comment lines */
+	while ((c = isl_stream_getc(s)) != -1) {
+		if (c == '#') {
+			if (isl_stream_skip_line(s) < 0)
+				break;
+			c = '\n';
+			if (same_line)
+				break;
+		} else if (!isspace(c) || (same_line && c == '\n'))
+			break;
+	}
+
+	line = s->line;
+	col = s->col;
+
+	if (c == -1 || (same_line && c == '\n'))
+		return NULL;
+	if (c == '(' ||
+	    c == ')' ||
+	    c == '+' ||
+	    c == '*' ||
+	    c == '%' ||
+	    c == '?' ||
+	    c == '^' ||
+	    c == '@' ||
+	    c == '$' ||
+	    c == ',' ||
+	    c == '.' ||
+	    c == ';' ||
+	    c == '[' ||
+	    c == ']' ||
+	    c == '{' ||
+	    c == '}') {
+		tok = isl_token_new(s->ctx, line, col, old_line != line);
+		if (!tok)
+			return NULL;
+		tok->type = (enum isl_token_type)c;
+		return tok;
+	}
+	if (c == '-') {
+		int c;
+		if ((c = isl_stream_getc(s)) == '>') {
+			tok = isl_token_new(s->ctx, line, col, old_line != line);
+			if (!tok)
+				return NULL;
+			tok->u.s = strdup("->");
+			tok->type = ISL_TOKEN_TO;
+			return tok;
+		}
+		if (c != -1)
+			isl_stream_ungetc(s, c);
+		if (!isdigit(c)) {
+			tok = isl_token_new(s->ctx, line, col, old_line != line);
+			if (!tok)
+				return NULL;
+			tok->type = (enum isl_token_type) '-';
+			return tok;
+		}
+	}
+	if (c == '-' || isdigit(c)) {
+		int minus = c == '-';
+		tok = isl_token_new(s->ctx, line, col, old_line != line);
+		if (!tok)
+			return NULL;
+		tok->type = ISL_TOKEN_VALUE;
+		isl_int_init(tok->u.v);
+		if (isl_stream_push_char(s, c))
+			goto error;
+		while ((c = isl_stream_getc(s)) != -1 && isdigit(c))
+			if (isl_stream_push_char(s, c))
+				goto error;
+		if (c != -1)
+			isl_stream_ungetc(s, c);
+		isl_stream_push_char(s, '\0');
+		isl_int_read(tok->u.v, s->buffer);
+		if (minus && isl_int_is_zero(tok->u.v)) {
+			tok->col++;
+			tok->on_new_line = 0;
+			isl_stream_push_token(s, tok);
+			tok = isl_token_new(s->ctx, line, col, old_line != line);
+			if (!tok)
+				return NULL;
+			tok->type = (enum isl_token_type) '-';
+		}
+		return tok;
+	}
+	if (isalpha(c) || c == '_') {
+		tok = isl_token_new(s->ctx, line, col, old_line != line);
+		if (!tok)
+			return NULL;
+		isl_stream_push_char(s, c);
+		while ((c = isl_stream_getc(s)) != -1 &&
+				(isalnum(c) || c == '_'))
+			isl_stream_push_char(s, c);
+		if (c != -1)
+			isl_stream_ungetc(s, c);
+		while ((c = isl_stream_getc(s)) != -1 && c == '\'')
+			isl_stream_push_char(s, c);
+		if (c != -1)
+			isl_stream_ungetc(s, c);
+		isl_stream_push_char(s, '\0');
+		tok->type = check_keywords(s);
+		if (tok->type != ISL_TOKEN_IDENT)
+			tok->is_keyword = 1;
+		tok->u.s = strdup(s->buffer);
+		if (!tok->u.s)
+			goto error;
+		return tok;
+	}
+	if (c == '"') {
+		tok = isl_token_new(s->ctx, line, col, old_line != line);
+		if (!tok)
+			return NULL;
+		tok->type = ISL_TOKEN_STRING;
+		tok->u.s = NULL;
+		while ((c = isl_stream_getc(s)) != -1 && c != '"' && c != '\n')
+			isl_stream_push_char(s, c);
+		if (c != '"') {
+			isl_stream_error(s, NULL, "unterminated string");
+			goto error;
+		}
+		isl_stream_push_char(s, '\0');
+		tok->u.s = strdup(s->buffer);
+		return tok;
+	}
+	if (c == '=') {
+		int c;
+		tok = isl_token_new(s->ctx, line, col, old_line != line);
+		if (!tok)
+			return NULL;
+		if ((c = isl_stream_getc(s)) == '=') {
+			tok->u.s = strdup("==");
+			tok->type = ISL_TOKEN_EQ_EQ;
+			return tok;
+		}
+		if (c != -1)
+			isl_stream_ungetc(s, c);
+		tok->type = (enum isl_token_type) '=';
+		return tok;
+	}
+	if (c == ':') {
+		int c;
+		tok = isl_token_new(s->ctx, line, col, old_line != line);
+		if (!tok)
+			return NULL;
+		if ((c = isl_stream_getc(s)) == '=') {
+			tok->u.s = strdup(":=");
+			tok->type = ISL_TOKEN_DEF;
+			return tok;
+		}
+		if (c != -1)
+			isl_stream_ungetc(s, c);
+		tok->type = (enum isl_token_type) ':';
+		return tok;
+	}
+	if (c == '>') {
+		int c;
+		tok = isl_token_new(s->ctx, line, col, old_line != line);
+		if (!tok)
+			return NULL;
+		if ((c = isl_stream_getc(s)) == '=') {
+			tok->u.s = strdup(">=");
+			tok->type = ISL_TOKEN_GE;
+			return tok;
+		} else if (c == '>') {
+			if ((c = isl_stream_getc(s)) == '=') {
+				tok->u.s = strdup(">>=");
+				tok->type = ISL_TOKEN_LEX_GE;
+				return tok;
+			}
+			tok->u.s = strdup(">>");
+			tok->type = ISL_TOKEN_LEX_GT;
+		} else {
+			tok->u.s = strdup(">");
+			tok->type = ISL_TOKEN_GT;
+		}
+		if (c != -1)
+			isl_stream_ungetc(s, c);
+		return tok;
+	}
+	if (c == '<') {
+		int c;
+		tok = isl_token_new(s->ctx, line, col, old_line != line);
+		if (!tok)
+			return NULL;
+		if ((c = isl_stream_getc(s)) == '=') {
+			tok->u.s = strdup("<=");
+			tok->type = ISL_TOKEN_LE;
+			return tok;
+		} else if (c == '<') {
+			if ((c = isl_stream_getc(s)) == '=') {
+				tok->u.s = strdup("<<=");
+				tok->type = ISL_TOKEN_LEX_LE;
+				return tok;
+			}
+			tok->u.s = strdup("<<");
+			tok->type = ISL_TOKEN_LEX_LT;
+		} else {
+			tok->u.s = strdup("<");
+			tok->type = ISL_TOKEN_LT;
+		}
+		if (c != -1)
+			isl_stream_ungetc(s, c);
+		return tok;
+	}
+	if (c == '&') {
+		tok = isl_token_new(s->ctx, line, col, old_line != line);
+		if (!tok)
+			return NULL;
+		tok->type = ISL_TOKEN_AND;
+		if ((c = isl_stream_getc(s)) != '&' && c != -1) {
+			tok->u.s = strdup("&");
+			isl_stream_ungetc(s, c);
+		} else
+			tok->u.s = strdup("&&");
+		return tok;
+	}
+	if (c == '|') {
+		tok = isl_token_new(s->ctx, line, col, old_line != line);
+		if (!tok)
+			return NULL;
+		tok->type = ISL_TOKEN_OR;
+		if ((c = isl_stream_getc(s)) != '|' && c != -1) {
+			tok->u.s = strdup("|");
+			isl_stream_ungetc(s, c);
+		} else
+			tok->u.s = strdup("||");
+		return tok;
+	}
+	if (c == '/') {
+		tok = isl_token_new(s->ctx, line, col, old_line != line);
+		if (!tok)
+			return NULL;
+		if ((c = isl_stream_getc(s)) != '\\' && c != -1) {
+			tok->type = (enum isl_token_type) '/';
+			isl_stream_ungetc(s, c);
+		} else {
+			tok->u.s = strdup("/\\");
+			tok->type = ISL_TOKEN_AND;
+		}
+		return tok;
+	}
+	if (c == '\\') {
+		tok = isl_token_new(s->ctx, line, col, old_line != line);
+		if (!tok)
+			return NULL;
+		if ((c = isl_stream_getc(s)) != '/' && c != -1) {
+			tok->type = (enum isl_token_type) '\\';
+			isl_stream_ungetc(s, c);
+		} else {
+			tok->u.s = strdup("\\/");
+			tok->type = ISL_TOKEN_OR;
+		}
+		return tok;
+	}
+	if (c == '!') {
+		tok = isl_token_new(s->ctx, line, col, old_line != line);
+		if (!tok)
+			return NULL;
+		if ((c = isl_stream_getc(s)) == '=') {
+			tok->u.s = strdup("!=");
+			tok->type = ISL_TOKEN_NE;
+			return tok;
+		} else {
+			tok->type = ISL_TOKEN_NOT;
+			tok->u.s = strdup("!");
+		}
+		if (c != -1)
+			isl_stream_ungetc(s, c);
+		return tok;
+	}
+
+	tok = isl_token_new(s->ctx, line, col, old_line != line);
+	if (!tok)
+		return NULL;
+	tok->type = ISL_TOKEN_UNKNOWN;
+	return tok;
+error:
+	isl_token_free(tok);
+	return NULL;
+}
+
+struct isl_token *isl_stream_next_token(struct isl_stream *s)
+{
+	return next_token(s, 0);
+}
+
+struct isl_token *isl_stream_next_token_on_same_line(struct isl_stream *s)
+{
+	return next_token(s, 1);
+}
+
+int isl_stream_eat_if_available(struct isl_stream *s, int type)
+{
+	struct isl_token *tok;
+
+	tok = isl_stream_next_token(s);
+	if (!tok)
+		return 0;
+	if (tok->type == type) {
+		isl_token_free(tok);
+		return 1;
+	}
+	isl_stream_push_token(s, tok);
+	return 0;
+}
+
+int isl_stream_next_token_is(struct isl_stream *s, int type)
+{
+	struct isl_token *tok;
+	int r;
+
+	tok = isl_stream_next_token(s);
+	if (!tok)
+		return 0;
+	r = tok->type == type;
+	isl_stream_push_token(s, tok);
+	return r;
+}
+
+char *isl_stream_read_ident_if_available(struct isl_stream *s)
+{
+	struct isl_token *tok;
+
+	tok = isl_stream_next_token(s);
+	if (!tok)
+		return NULL;
+	if (tok->type == ISL_TOKEN_IDENT) {
+		char *ident = strdup(tok->u.s);
+		isl_token_free(tok);
+		return ident;
+	}
+	isl_stream_push_token(s, tok);
+	return NULL;
+}
+
+int isl_stream_eat(struct isl_stream *s, int type)
+{
+	struct isl_token *tok;
+
+	tok = isl_stream_next_token(s);
+	if (!tok)
+		return -1;
+	if (tok->type == type) {
+		isl_token_free(tok);
+		return 0;
+	}
+	isl_stream_error(s, tok, "expecting other token");
+	isl_stream_push_token(s, tok);
+	return -1;
+}
+
+int isl_stream_is_empty(struct isl_stream *s)
+{
+	struct isl_token *tok;
+
+	tok = isl_stream_next_token(s);
+
+	if (!tok)
+		return 1;
+
+	isl_stream_push_token(s, tok);
+	return 0;
+}
+
+static int free_keyword(void **p, void *user)
+{
+	struct isl_keyword *keyword = *p;
+
+	free(keyword->name);
+	free(keyword);
+
+	return 0;
+}
+
+void isl_stream_flush_tokens(struct isl_stream *s)
+{
+	int i;
+
+	if (!s)
+		return;
+	for (i = 0; i < s->n_token; ++i)
+		isl_token_free(s->tokens[i]);
+	s->n_token = 0;
+}
+
+void isl_stream_free(struct isl_stream *s)
+{
+	if (!s)
+		return;
+	free(s->buffer);
+	if (s->n_token != 0) {
+		struct isl_token *tok = isl_stream_next_token(s);
+		isl_stream_error(s, tok, "unexpected token");
+		isl_token_free(tok);
+	}
+	if (s->keywords) {
+		isl_hash_table_foreach(s->ctx, s->keywords, &free_keyword, NULL);
+		isl_hash_table_free(s->ctx, s->keywords);
+	}
+	isl_ctx_deref(s->ctx);
+	free(s);
+}
diff --git a/polly/lib/External/isl/isl_stream_private.h b/polly/lib/External/isl/isl_stream_private.h
new file mode 100644
index 00000000000..d5d4422127f
--- /dev/null
+++ b/polly/lib/External/isl/isl_stream_private.h
@@ -0,0 +1,21 @@
+#include <isl_int.h>
+#include <isl/stream.h>
+
+struct isl_token {
+	int type;
+
+	unsigned int on_new_line : 1;
+	unsigned is_keyword : 1;
+	int line;
+	int col;
+
+	union {
+		isl_int	v;
+		char	*s;
+		isl_map *map;
+		isl_pw_aff *pwaff;
+	} u;
+};
+
+struct isl_token *isl_token_new(isl_ctx *ctx,
+	int line, int col, unsigned on_new_line);
diff --git a/polly/lib/External/isl/isl_tab.c b/polly/lib/External/isl/isl_tab.c
new file mode 100644
index 00000000000..915521036a6
--- /dev/null
+++ b/polly/lib/External/isl/isl_tab.c
@@ -0,0 +1,3509 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2013      Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_mat_private.h>
+#include <isl_vec_private.h>
+#include "isl_map_private.h"
+#include "isl_tab.h"
+#include <isl_seq.h>
+#include <isl_config.h>
+
+/*
+ * The implementation of tableaus in this file was inspired by Section 8
+ * of David Detlefs, Greg Nelson and James B. Saxe, "Simplify: a theorem
+ * prover for program checking".
+ */
+
+struct isl_tab *isl_tab_alloc(struct isl_ctx *ctx,
+	unsigned n_row, unsigned n_var, unsigned M)
+{
+	int i;
+	struct isl_tab *tab;
+	unsigned off = 2 + M;
+
+	tab = isl_calloc_type(ctx, struct isl_tab);
+	if (!tab)
+		return NULL;
+	tab->mat = isl_mat_alloc(ctx, n_row, off + n_var);
+	if (!tab->mat)
+		goto error;
+	tab->var = isl_alloc_array(ctx, struct isl_tab_var, n_var);
+	if (n_var && !tab->var)
+		goto error;
+	tab->con = isl_alloc_array(ctx, struct isl_tab_var, n_row);
+	if (n_row && !tab->con)
+		goto error;
+	tab->col_var = isl_alloc_array(ctx, int, n_var);
+	if (n_var && !tab->col_var)
+		goto error;
+	tab->row_var = isl_alloc_array(ctx, int, n_row);
+	if (n_row && !tab->row_var)
+		goto error;
+	for (i = 0; i < n_var; ++i) {
+		tab->var[i].index = i;
+		tab->var[i].is_row = 0;
+		tab->var[i].is_nonneg = 0;
+		tab->var[i].is_zero = 0;
+		tab->var[i].is_redundant = 0;
+		tab->var[i].frozen = 0;
+		tab->var[i].negated = 0;
+		tab->col_var[i] = i;
+	}
+	tab->n_row = 0;
+	tab->n_con = 0;
+	tab->n_eq = 0;
+	tab->max_con = n_row;
+	tab->n_col = n_var;
+	tab->n_var = n_var;
+	tab->max_var = n_var;
+	tab->n_param = 0;
+	tab->n_div = 0;
+	tab->n_dead = 0;
+	tab->n_redundant = 0;
+	tab->strict_redundant = 0;
+	tab->need_undo = 0;
+	tab->rational = 0;
+	tab->empty = 0;
+	tab->in_undo = 0;
+	tab->M = M;
+	tab->cone = 0;
+	tab->bottom.type = isl_tab_undo_bottom;
+	tab->bottom.next = NULL;
+	tab->top = &tab->bottom;
+
+	tab->n_zero = 0;
+	tab->n_unbounded = 0;
+	tab->basis = NULL;
+
+	return tab;
+error:
+	isl_tab_free(tab);
+	return NULL;
+}
+
+isl_ctx *isl_tab_get_ctx(struct isl_tab *tab)
+{
+	return tab ? isl_mat_get_ctx(tab->mat) : NULL;
+}
+
+int isl_tab_extend_cons(struct isl_tab *tab, unsigned n_new)
+{
+	unsigned off;
+
+	if (!tab)
+		return -1;
+
+	off = 2 + tab->M;
+
+	if (tab->max_con < tab->n_con + n_new) {
+		struct isl_tab_var *con;
+
+		con = isl_realloc_array(tab->mat->ctx, tab->con,
+				    struct isl_tab_var, tab->max_con + n_new);
+		if (!con)
+			return -1;
+		tab->con = con;
+		tab->max_con += n_new;
+	}
+	if (tab->mat->n_row < tab->n_row + n_new) {
+		int *row_var;
+
+		tab->mat = isl_mat_extend(tab->mat,
+					tab->n_row + n_new, off + tab->n_col);
+		if (!tab->mat)
+			return -1;
+		row_var = isl_realloc_array(tab->mat->ctx, tab->row_var,
+					    int, tab->mat->n_row);
+		if (!row_var)
+			return -1;
+		tab->row_var = row_var;
+		if (tab->row_sign) {
+			enum isl_tab_row_sign *s;
+			s = isl_realloc_array(tab->mat->ctx, tab->row_sign,
+					enum isl_tab_row_sign, tab->mat->n_row);
+			if (!s)
+				return -1;
+			tab->row_sign = s;
+		}
+	}
+	return 0;
+}
+
+/* Make room for at least n_new extra variables.
+ * Return -1 if anything went wrong.
+ */
+int isl_tab_extend_vars(struct isl_tab *tab, unsigned n_new)
+{
+	struct isl_tab_var *var;
+	unsigned off = 2 + tab->M;
+
+	if (tab->max_var < tab->n_var + n_new) {
+		var = isl_realloc_array(tab->mat->ctx, tab->var,
+				    struct isl_tab_var, tab->n_var + n_new);
+		if (!var)
+			return -1;
+		tab->var = var;
+		tab->max_var += n_new;
+	}
+
+	if (tab->mat->n_col < off + tab->n_col + n_new) {
+		int *p;
+
+		tab->mat = isl_mat_extend(tab->mat,
+				    tab->mat->n_row, off + tab->n_col + n_new);
+		if (!tab->mat)
+			return -1;
+		p = isl_realloc_array(tab->mat->ctx, tab->col_var,
+					    int, tab->n_col + n_new);
+		if (!p)
+			return -1;
+		tab->col_var = p;
+	}
+
+	return 0;
+}
+
+static void free_undo_record(struct isl_tab_undo *undo)
+{
+	switch (undo->type) {
+	case isl_tab_undo_saved_basis:
+		free(undo->u.col_var);
+		break;
+	default:;
+	}
+	free(undo);
+}
+
+static void free_undo(struct isl_tab *tab)
+{
+	struct isl_tab_undo *undo, *next;
+
+	for (undo = tab->top; undo && undo != &tab->bottom; undo = next) {
+		next = undo->next;
+		free_undo_record(undo);
+	}
+	tab->top = undo;
+}
+
+void isl_tab_free(struct isl_tab *tab)
+{
+	if (!tab)
+		return;
+	free_undo(tab);
+	isl_mat_free(tab->mat);
+	isl_vec_free(tab->dual);
+	isl_basic_map_free(tab->bmap);
+	free(tab->var);
+	free(tab->con);
+	free(tab->row_var);
+	free(tab->col_var);
+	free(tab->row_sign);
+	isl_mat_free(tab->samples);
+	free(tab->sample_index);
+	isl_mat_free(tab->basis);
+	free(tab);
+}
+
+struct isl_tab *isl_tab_dup(struct isl_tab *tab)
+{
+	int i;
+	struct isl_tab *dup;
+	unsigned off;
+
+	if (!tab)
+		return NULL;
+
+	off = 2 + tab->M;
+	dup = isl_calloc_type(tab->mat->ctx, struct isl_tab);
+	if (!dup)
+		return NULL;
+	dup->mat = isl_mat_dup(tab->mat);
+	if (!dup->mat)
+		goto error;
+	dup->var = isl_alloc_array(tab->mat->ctx, struct isl_tab_var, tab->max_var);
+	if (tab->max_var && !dup->var)
+		goto error;
+	for (i = 0; i < tab->n_var; ++i)
+		dup->var[i] = tab->var[i];
+	dup->con = isl_alloc_array(tab->mat->ctx, struct isl_tab_var, tab->max_con);
+	if (tab->max_con && !dup->con)
+		goto error;
+	for (i = 0; i < tab->n_con; ++i)
+		dup->con[i] = tab->con[i];
+	dup->col_var = isl_alloc_array(tab->mat->ctx, int, tab->mat->n_col - off);
+	if ((tab->mat->n_col - off) && !dup->col_var)
+		goto error;
+	for (i = 0; i < tab->n_col; ++i)
+		dup->col_var[i] = tab->col_var[i];
+	dup->row_var = isl_alloc_array(tab->mat->ctx, int, tab->mat->n_row);
+	if (tab->mat->n_row && !dup->row_var)
+		goto error;
+	for (i = 0; i < tab->n_row; ++i)
+		dup->row_var[i] = tab->row_var[i];
+	if (tab->row_sign) {
+		dup->row_sign = isl_alloc_array(tab->mat->ctx, enum isl_tab_row_sign,
+						tab->mat->n_row);
+		if (tab->mat->n_row && !dup->row_sign)
+			goto error;
+		for (i = 0; i < tab->n_row; ++i)
+			dup->row_sign[i] = tab->row_sign[i];
+	}
+	if (tab->samples) {
+		dup->samples = isl_mat_dup(tab->samples);
+		if (!dup->samples)
+			goto error;
+		dup->sample_index = isl_alloc_array(tab->mat->ctx, int,
+							tab->samples->n_row);
+		if (tab->samples->n_row && !dup->sample_index)
+			goto error;
+		dup->n_sample = tab->n_sample;
+		dup->n_outside = tab->n_outside;
+	}
+	dup->n_row = tab->n_row;
+	dup->n_con = tab->n_con;
+	dup->n_eq = tab->n_eq;
+	dup->max_con = tab->max_con;
+	dup->n_col = tab->n_col;
+	dup->n_var = tab->n_var;
+	dup->max_var = tab->max_var;
+	dup->n_param = tab->n_param;
+	dup->n_div = tab->n_div;
+	dup->n_dead = tab->n_dead;
+	dup->n_redundant = tab->n_redundant;
+	dup->rational = tab->rational;
+	dup->empty = tab->empty;
+	dup->strict_redundant = 0;
+	dup->need_undo = 0;
+	dup->in_undo = 0;
+	dup->M = tab->M;
+	tab->cone = tab->cone;
+	dup->bottom.type = isl_tab_undo_bottom;
+	dup->bottom.next = NULL;
+	dup->top = &dup->bottom;
+
+	dup->n_zero = tab->n_zero;
+	dup->n_unbounded = tab->n_unbounded;
+	dup->basis = isl_mat_dup(tab->basis);
+
+	return dup;
+error:
+	isl_tab_free(dup);
+	return NULL;
+}
+
+/* Construct the coefficient matrix of the product tableau
+ * of two tableaus.
+ * mat{1,2} is the coefficient matrix of tableau {1,2}
+ * row{1,2} is the number of rows in tableau {1,2}
+ * col{1,2} is the number of columns in tableau {1,2}
+ * off is the offset to the coefficient column (skipping the
+ *	denominator, the constant term and the big parameter if any)
+ * r{1,2} is the number of redundant rows in tableau {1,2}
+ * d{1,2} is the number of dead columns in tableau {1,2}
+ *
+ * The order of the rows and columns in the result is as explained
+ * in isl_tab_product.
+ */
+static struct isl_mat *tab_mat_product(struct isl_mat *mat1,
+	struct isl_mat *mat2, unsigned row1, unsigned row2,
+	unsigned col1, unsigned col2,
+	unsigned off, unsigned r1, unsigned r2, unsigned d1, unsigned d2)
+{
+	int i;
+	struct isl_mat *prod;
+	unsigned n;
+
+	prod = isl_mat_alloc(mat1->ctx, mat1->n_row + mat2->n_row,
+					off + col1 + col2);
+	if (!prod)
+		return NULL;
+
+	n = 0;
+	for (i = 0; i < r1; ++i) {
+		isl_seq_cpy(prod->row[n + i], mat1->row[i], off + d1);
+		isl_seq_clr(prod->row[n + i] + off + d1, d2);
+		isl_seq_cpy(prod->row[n + i] + off + d1 + d2,
+				mat1->row[i] + off + d1, col1 - d1);
+		isl_seq_clr(prod->row[n + i] + off + col1 + d1, col2 - d2);
+	}
+
+	n += r1;
+	for (i = 0; i < r2; ++i) {
+		isl_seq_cpy(prod->row[n + i], mat2->row[i], off);
+		isl_seq_clr(prod->row[n + i] + off, d1);
+		isl_seq_cpy(prod->row[n + i] + off + d1,
+			    mat2->row[i] + off, d2);
+		isl_seq_clr(prod->row[n + i] + off + d1 + d2, col1 - d1);
+		isl_seq_cpy(prod->row[n + i] + off + col1 + d1,
+			    mat2->row[i] + off + d2, col2 - d2);
+	}
+
+	n += r2;
+	for (i = 0; i < row1 - r1; ++i) {
+		isl_seq_cpy(prod->row[n + i], mat1->row[r1 + i], off + d1);
+		isl_seq_clr(prod->row[n + i] + off + d1, d2);
+		isl_seq_cpy(prod->row[n + i] + off + d1 + d2,
+				mat1->row[r1 + i] + off + d1, col1 - d1);
+		isl_seq_clr(prod->row[n + i] + off + col1 + d1, col2 - d2);
+	}
+
+	n += row1 - r1;
+	for (i = 0; i < row2 - r2; ++i) {
+		isl_seq_cpy(prod->row[n + i], mat2->row[r2 + i], off);
+		isl_seq_clr(prod->row[n + i] + off, d1);
+		isl_seq_cpy(prod->row[n + i] + off + d1,
+			    mat2->row[r2 + i] + off, d2);
+		isl_seq_clr(prod->row[n + i] + off + d1 + d2, col1 - d1);
+		isl_seq_cpy(prod->row[n + i] + off + col1 + d1,
+			    mat2->row[r2 + i] + off + d2, col2 - d2);
+	}
+
+	return prod;
+}
+
+/* Update the row or column index of a variable that corresponds
+ * to a variable in the first input tableau.
+ */
+static void update_index1(struct isl_tab_var *var,
+	unsigned r1, unsigned r2, unsigned d1, unsigned d2)
+{
+	if (var->index == -1)
+		return;
+	if (var->is_row && var->index >= r1)
+		var->index += r2;
+	if (!var->is_row && var->index >= d1)
+		var->index += d2;
+}
+
+/* Update the row or column index of a variable that corresponds
+ * to a variable in the second input tableau.
+ */
+static void update_index2(struct isl_tab_var *var,
+	unsigned row1, unsigned col1,
+	unsigned r1, unsigned r2, unsigned d1, unsigned d2)
+{
+	if (var->index == -1)
+		return;
+	if (var->is_row) {
+		if (var->index < r2)
+			var->index += r1;
+		else
+			var->index += row1;
+	} else {
+		if (var->index < d2)
+			var->index += d1;
+		else
+			var->index += col1;
+	}
+}
+
+/* Create a tableau that represents the Cartesian product of the sets
+ * represented by tableaus tab1 and tab2.
+ * The order of the rows in the product is
+ *	- redundant rows of tab1
+ *	- redundant rows of tab2
+ *	- non-redundant rows of tab1
+ *	- non-redundant rows of tab2
+ * The order of the columns is
+ *	- denominator
+ *	- constant term
+ *	- coefficient of big parameter, if any
+ *	- dead columns of tab1
+ *	- dead columns of tab2
+ *	- live columns of tab1
+ *	- live columns of tab2
+ * The order of the variables and the constraints is a concatenation
+ * of order in the two input tableaus.
+ */
+struct isl_tab *isl_tab_product(struct isl_tab *tab1, struct isl_tab *tab2)
+{
+	int i;
+	struct isl_tab *prod;
+	unsigned off;
+	unsigned r1, r2, d1, d2;
+
+	if (!tab1 || !tab2)
+		return NULL;
+
+	isl_assert(tab1->mat->ctx, tab1->M == tab2->M, return NULL);
+	isl_assert(tab1->mat->ctx, tab1->rational == tab2->rational, return NULL);
+	isl_assert(tab1->mat->ctx, tab1->cone == tab2->cone, return NULL);
+	isl_assert(tab1->mat->ctx, !tab1->row_sign, return NULL);
+	isl_assert(tab1->mat->ctx, !tab2->row_sign, return NULL);
+	isl_assert(tab1->mat->ctx, tab1->n_param == 0, return NULL);
+	isl_assert(tab1->mat->ctx, tab2->n_param == 0, return NULL);
+	isl_assert(tab1->mat->ctx, tab1->n_div == 0, return NULL);
+	isl_assert(tab1->mat->ctx, tab2->n_div == 0, return NULL);
+
+	off = 2 + tab1->M;
+	r1 = tab1->n_redundant;
+	r2 = tab2->n_redundant;
+	d1 = tab1->n_dead;
+	d2 = tab2->n_dead;
+	prod = isl_calloc_type(tab1->mat->ctx, struct isl_tab);
+	if (!prod)
+		return NULL;
+	prod->mat = tab_mat_product(tab1->mat, tab2->mat,
+				tab1->n_row, tab2->n_row,
+				tab1->n_col, tab2->n_col, off, r1, r2, d1, d2);
+	if (!prod->mat)
+		goto error;
+	prod->var = isl_alloc_array(tab1->mat->ctx, struct isl_tab_var,
+					tab1->max_var + tab2->max_var);
+	if ((tab1->max_var + tab2->max_var) && !prod->var)
+		goto error;
+	for (i = 0; i < tab1->n_var; ++i) {
+		prod->var[i] = tab1->var[i];
+		update_index1(&prod->var[i], r1, r2, d1, d2);
+	}
+	for (i = 0; i < tab2->n_var; ++i) {
+		prod->var[tab1->n_var + i] = tab2->var[i];
+		update_index2(&prod->var[tab1->n_var + i],
+				tab1->n_row, tab1->n_col,
+				r1, r2, d1, d2);
+	}
+	prod->con = isl_alloc_array(tab1->mat->ctx, struct isl_tab_var,
+					tab1->max_con +  tab2->max_con);
+	if ((tab1->max_con + tab2->max_con) && !prod->con)
+		goto error;
+	for (i = 0; i < tab1->n_con; ++i) {
+		prod->con[i] = tab1->con[i];
+		update_index1(&prod->con[i], r1, r2, d1, d2);
+	}
+	for (i = 0; i < tab2->n_con; ++i) {
+		prod->con[tab1->n_con + i] = tab2->con[i];
+		update_index2(&prod->con[tab1->n_con + i],
+				tab1->n_row, tab1->n_col,
+				r1, r2, d1, d2);
+	}
+	prod->col_var = isl_alloc_array(tab1->mat->ctx, int,
+					tab1->n_col + tab2->n_col);
+	if ((tab1->n_col + tab2->n_col) && !prod->col_var)
+		goto error;
+	for (i = 0; i < tab1->n_col; ++i) {
+		int pos = i < d1 ? i : i + d2;
+		prod->col_var[pos] = tab1->col_var[i];
+	}
+	for (i = 0; i < tab2->n_col; ++i) {
+		int pos = i < d2 ? d1 + i : tab1->n_col + i;
+		int t = tab2->col_var[i];
+		if (t >= 0)
+			t += tab1->n_var;
+		else
+			t -= tab1->n_con;
+		prod->col_var[pos] = t;
+	}
+	prod->row_var = isl_alloc_array(tab1->mat->ctx, int,
+					tab1->mat->n_row + tab2->mat->n_row);
+	if ((tab1->mat->n_row + tab2->mat->n_row) && !prod->row_var)
+		goto error;
+	for (i = 0; i < tab1->n_row; ++i) {
+		int pos = i < r1 ? i : i + r2;
+		prod->row_var[pos] = tab1->row_var[i];
+	}
+	for (i = 0; i < tab2->n_row; ++i) {
+		int pos = i < r2 ? r1 + i : tab1->n_row + i;
+		int t = tab2->row_var[i];
+		if (t >= 0)
+			t += tab1->n_var;
+		else
+			t -= tab1->n_con;
+		prod->row_var[pos] = t;
+	}
+	prod->samples = NULL;
+	prod->sample_index = NULL;
+	prod->n_row = tab1->n_row + tab2->n_row;
+	prod->n_con = tab1->n_con + tab2->n_con;
+	prod->n_eq = 0;
+	prod->max_con = tab1->max_con + tab2->max_con;
+	prod->n_col = tab1->n_col + tab2->n_col;
+	prod->n_var = tab1->n_var + tab2->n_var;
+	prod->max_var = tab1->max_var + tab2->max_var;
+	prod->n_param = 0;
+	prod->n_div = 0;
+	prod->n_dead = tab1->n_dead + tab2->n_dead;
+	prod->n_redundant = tab1->n_redundant + tab2->n_redundant;
+	prod->rational = tab1->rational;
+	prod->empty = tab1->empty || tab2->empty;
+	prod->strict_redundant = tab1->strict_redundant || tab2->strict_redundant;
+	prod->need_undo = 0;
+	prod->in_undo = 0;
+	prod->M = tab1->M;
+	prod->cone = tab1->cone;
+	prod->bottom.type = isl_tab_undo_bottom;
+	prod->bottom.next = NULL;
+	prod->top = &prod->bottom;
+
+	prod->n_zero = 0;
+	prod->n_unbounded = 0;
+	prod->basis = NULL;
+
+	return prod;
+error:
+	isl_tab_free(prod);
+	return NULL;
+}
+
+static struct isl_tab_var *var_from_index(struct isl_tab *tab, int i)
+{
+	if (i >= 0)
+		return &tab->var[i];
+	else
+		return &tab->con[~i];
+}
+
+struct isl_tab_var *isl_tab_var_from_row(struct isl_tab *tab, int i)
+{
+	return var_from_index(tab, tab->row_var[i]);
+}
+
+static struct isl_tab_var *var_from_col(struct isl_tab *tab, int i)
+{
+	return var_from_index(tab, tab->col_var[i]);
+}
+
+/* Check if there are any upper bounds on column variable "var",
+ * i.e., non-negative rows where var appears with a negative coefficient.
+ * Return 1 if there are no such bounds.
+ */
+static int max_is_manifestly_unbounded(struct isl_tab *tab,
+	struct isl_tab_var *var)
+{
+	int i;
+	unsigned off = 2 + tab->M;
+
+	if (var->is_row)
+		return 0;
+	for (i = tab->n_redundant; i < tab->n_row; ++i) {
+		if (!isl_int_is_neg(tab->mat->row[i][off + var->index]))
+			continue;
+		if (isl_tab_var_from_row(tab, i)->is_nonneg)
+			return 0;
+	}
+	return 1;
+}
+
+/* Check if there are any lower bounds on column variable "var",
+ * i.e., non-negative rows where var appears with a positive coefficient.
+ * Return 1 if there are no such bounds.
+ */
+static int min_is_manifestly_unbounded(struct isl_tab *tab,
+	struct isl_tab_var *var)
+{
+	int i;
+	unsigned off = 2 + tab->M;
+
+	if (var->is_row)
+		return 0;
+	for (i = tab->n_redundant; i < tab->n_row; ++i) {
+		if (!isl_int_is_pos(tab->mat->row[i][off + var->index]))
+			continue;
+		if (isl_tab_var_from_row(tab, i)->is_nonneg)
+			return 0;
+	}
+	return 1;
+}
+
+static int row_cmp(struct isl_tab *tab, int r1, int r2, int c, isl_int t)
+{
+	unsigned off = 2 + tab->M;
+
+	if (tab->M) {
+		int s;
+		isl_int_mul(t, tab->mat->row[r1][2], tab->mat->row[r2][off+c]);
+		isl_int_submul(t, tab->mat->row[r2][2], tab->mat->row[r1][off+c]);
+		s = isl_int_sgn(t);
+		if (s)
+			return s;
+	}
+	isl_int_mul(t, tab->mat->row[r1][1], tab->mat->row[r2][off + c]);
+	isl_int_submul(t, tab->mat->row[r2][1], tab->mat->row[r1][off + c]);
+	return isl_int_sgn(t);
+}
+
+/* Given the index of a column "c", return the index of a row
+ * that can be used to pivot the column in, with either an increase
+ * (sgn > 0) or a decrease (sgn < 0) of the corresponding variable.
+ * If "var" is not NULL, then the row returned will be different from
+ * the one associated with "var".
+ *
+ * Each row in the tableau is of the form
+ *
+ *	x_r = a_r0 + \sum_i a_ri x_i
+ *
+ * Only rows with x_r >= 0 and with the sign of a_ri opposite to "sgn"
+ * impose any limit on the increase or decrease in the value of x_c
+ * and this bound is equal to a_r0 / |a_rc|.  We are therefore looking
+ * for the row with the smallest (most stringent) such bound.
+ * Note that the common denominator of each row drops out of the fraction.
+ * To check if row j has a smaller bound than row r, i.e.,
+ * a_j0 / |a_jc| < a_r0 / |a_rc| or a_j0 |a_rc| < a_r0 |a_jc|,
+ * we check if -sign(a_jc) (a_j0 a_rc - a_r0 a_jc) < 0,
+ * where -sign(a_jc) is equal to "sgn".
+ */
+static int pivot_row(struct isl_tab *tab,
+	struct isl_tab_var *var, int sgn, int c)
+{
+	int j, r, tsgn;
+	isl_int t;
+	unsigned off = 2 + tab->M;
+
+	isl_int_init(t);
+	r = -1;
+	for (j = tab->n_redundant; j < tab->n_row; ++j) {
+		if (var && j == var->index)
+			continue;
+		if (!isl_tab_var_from_row(tab, j)->is_nonneg)
+			continue;
+		if (sgn * isl_int_sgn(tab->mat->row[j][off + c]) >= 0)
+			continue;
+		if (r < 0) {
+			r = j;
+			continue;
+		}
+		tsgn = sgn * row_cmp(tab, r, j, c, t);
+		if (tsgn < 0 || (tsgn == 0 &&
+					    tab->row_var[j] < tab->row_var[r]))
+			r = j;
+	}
+	isl_int_clear(t);
+	return r;
+}
+
+/* Find a pivot (row and col) that will increase (sgn > 0) or decrease
+ * (sgn < 0) the value of row variable var.
+ * If not NULL, then skip_var is a row variable that should be ignored
+ * while looking for a pivot row.  It is usually equal to var.
+ *
+ * As the given row in the tableau is of the form
+ *
+ *	x_r = a_r0 + \sum_i a_ri x_i
+ *
+ * we need to find a column such that the sign of a_ri is equal to "sgn"
+ * (such that an increase in x_i will have the desired effect) or a
+ * column with a variable that may attain negative values.
+ * If a_ri is positive, then we need to move x_i in the same direction
+ * to obtain the desired effect.  Otherwise, x_i has to move in the
+ * opposite direction.
+ */
+static void find_pivot(struct isl_tab *tab,
+	struct isl_tab_var *var, struct isl_tab_var *skip_var,
+	int sgn, int *row, int *col)
+{
+	int j, r, c;
+	isl_int *tr;
+
+	*row = *col = -1;
+
+	isl_assert(tab->mat->ctx, var->is_row, return);
+	tr = tab->mat->row[var->index] + 2 + tab->M;
+
+	c = -1;
+	for (j = tab->n_dead; j < tab->n_col; ++j) {
+		if (isl_int_is_zero(tr[j]))
+			continue;
+		if (isl_int_sgn(tr[j]) != sgn &&
+		    var_from_col(tab, j)->is_nonneg)
+			continue;
+		if (c < 0 || tab->col_var[j] < tab->col_var[c])
+			c = j;
+	}
+	if (c < 0)
+		return;
+
+	sgn *= isl_int_sgn(tr[c]);
+	r = pivot_row(tab, skip_var, sgn, c);
+	*row = r < 0 ? var->index : r;
+	*col = c;
+}
+
+/* Return 1 if row "row" represents an obviously redundant inequality.
+ * This means
+ *	- it represents an inequality or a variable
+ *	- that is the sum of a non-negative sample value and a positive
+ *	  combination of zero or more non-negative constraints.
+ */
+int isl_tab_row_is_redundant(struct isl_tab *tab, int row)
+{
+	int i;
+	unsigned off = 2 + tab->M;
+
+	if (tab->row_var[row] < 0 && !isl_tab_var_from_row(tab, row)->is_nonneg)
+		return 0;
+
+	if (isl_int_is_neg(tab->mat->row[row][1]))
+		return 0;
+	if (tab->strict_redundant && isl_int_is_zero(tab->mat->row[row][1]))
+		return 0;
+	if (tab->M && isl_int_is_neg(tab->mat->row[row][2]))
+		return 0;
+
+	for (i = tab->n_dead; i < tab->n_col; ++i) {
+		if (isl_int_is_zero(tab->mat->row[row][off + i]))
+			continue;
+		if (tab->col_var[i] >= 0)
+			return 0;
+		if (isl_int_is_neg(tab->mat->row[row][off + i]))
+			return 0;
+		if (!var_from_col(tab, i)->is_nonneg)
+			return 0;
+	}
+	return 1;
+}
+
+static void swap_rows(struct isl_tab *tab, int row1, int row2)
+{
+	int t;
+	enum isl_tab_row_sign s;
+
+	t = tab->row_var[row1];
+	tab->row_var[row1] = tab->row_var[row2];
+	tab->row_var[row2] = t;
+	isl_tab_var_from_row(tab, row1)->index = row1;
+	isl_tab_var_from_row(tab, row2)->index = row2;
+	tab->mat = isl_mat_swap_rows(tab->mat, row1, row2);
+
+	if (!tab->row_sign)
+		return;
+	s = tab->row_sign[row1];
+	tab->row_sign[row1] = tab->row_sign[row2];
+	tab->row_sign[row2] = s;
+}
+
+static int push_union(struct isl_tab *tab,
+	enum isl_tab_undo_type type, union isl_tab_undo_val u) WARN_UNUSED;
+static int push_union(struct isl_tab *tab,
+	enum isl_tab_undo_type type, union isl_tab_undo_val u)
+{
+	struct isl_tab_undo *undo;
+
+	if (!tab)
+		return -1;
+	if (!tab->need_undo)
+		return 0;
+
+	undo = isl_alloc_type(tab->mat->ctx, struct isl_tab_undo);
+	if (!undo)
+		return -1;
+	undo->type = type;
+	undo->u = u;
+	undo->next = tab->top;
+	tab->top = undo;
+
+	return 0;
+}
+
+int isl_tab_push_var(struct isl_tab *tab,
+	enum isl_tab_undo_type type, struct isl_tab_var *var)
+{
+	union isl_tab_undo_val u;
+	if (var->is_row)
+		u.var_index = tab->row_var[var->index];
+	else
+		u.var_index = tab->col_var[var->index];
+	return push_union(tab, type, u);
+}
+
+int isl_tab_push(struct isl_tab *tab, enum isl_tab_undo_type type)
+{
+	union isl_tab_undo_val u = { 0 };
+	return push_union(tab, type, u);
+}
+
+/* Push a record on the undo stack describing the current basic
+ * variables, so that the this state can be restored during rollback.
+ */
+int isl_tab_push_basis(struct isl_tab *tab)
+{
+	int i;
+	union isl_tab_undo_val u;
+
+	u.col_var = isl_alloc_array(tab->mat->ctx, int, tab->n_col);
+	if (tab->n_col && !u.col_var)
+		return -1;
+	for (i = 0; i < tab->n_col; ++i)
+		u.col_var[i] = tab->col_var[i];
+	return push_union(tab, isl_tab_undo_saved_basis, u);
+}
+
+int isl_tab_push_callback(struct isl_tab *tab, struct isl_tab_callback *callback)
+{
+	union isl_tab_undo_val u;
+	u.callback = callback;
+	return push_union(tab, isl_tab_undo_callback, u);
+}
+
+struct isl_tab *isl_tab_init_samples(struct isl_tab *tab)
+{
+	if (!tab)
+		return NULL;
+
+	tab->n_sample = 0;
+	tab->n_outside = 0;
+	tab->samples = isl_mat_alloc(tab->mat->ctx, 1, 1 + tab->n_var);
+	if (!tab->samples)
+		goto error;
+	tab->sample_index = isl_alloc_array(tab->mat->ctx, int, 1);
+	if (!tab->sample_index)
+		goto error;
+	return tab;
+error:
+	isl_tab_free(tab);
+	return NULL;
+}
+
+int isl_tab_add_sample(struct isl_tab *tab, __isl_take isl_vec *sample)
+{
+	if (!tab || !sample)
+		goto error;
+
+	if (tab->n_sample + 1 > tab->samples->n_row) {
+		int *t = isl_realloc_array(tab->mat->ctx,
+			    tab->sample_index, int, tab->n_sample + 1);
+		if (!t)
+			goto error;
+		tab->sample_index = t;
+	}
+
+	tab->samples = isl_mat_extend(tab->samples,
+				tab->n_sample + 1, tab->samples->n_col);
+	if (!tab->samples)
+		goto error;
+
+	isl_seq_cpy(tab->samples->row[tab->n_sample], sample->el, sample->size);
+	isl_vec_free(sample);
+	tab->sample_index[tab->n_sample] = tab->n_sample;
+	tab->n_sample++;
+
+	return 0;
+error:
+	isl_vec_free(sample);
+	return -1;
+}
+
+struct isl_tab *isl_tab_drop_sample(struct isl_tab *tab, int s)
+{
+	if (s != tab->n_outside) {
+		int t = tab->sample_index[tab->n_outside];
+		tab->sample_index[tab->n_outside] = tab->sample_index[s];
+		tab->sample_index[s] = t;
+		isl_mat_swap_rows(tab->samples, tab->n_outside, s);
+	}
+	tab->n_outside++;
+	if (isl_tab_push(tab, isl_tab_undo_drop_sample) < 0) {
+		isl_tab_free(tab);
+		return NULL;
+	}
+
+	return tab;
+}
+
+/* Record the current number of samples so that we can remove newer
+ * samples during a rollback.
+ */
+int isl_tab_save_samples(struct isl_tab *tab)
+{
+	union isl_tab_undo_val u;
+
+	if (!tab)
+		return -1;
+
+	u.n = tab->n_sample;
+	return push_union(tab, isl_tab_undo_saved_samples, u);
+}
+
+/* Mark row with index "row" as being redundant.
+ * If we may need to undo the operation or if the row represents
+ * a variable of the original problem, the row is kept,
+ * but no longer considered when looking for a pivot row.
+ * Otherwise, the row is simply removed.
+ *
+ * The row may be interchanged with some other row.  If it
+ * is interchanged with a later row, return 1.  Otherwise return 0.
+ * If the rows are checked in order in the calling function,
+ * then a return value of 1 means that the row with the given
+ * row number may now contain a different row that hasn't been checked yet.
+ */
+int isl_tab_mark_redundant(struct isl_tab *tab, int row)
+{
+	struct isl_tab_var *var = isl_tab_var_from_row(tab, row);
+	var->is_redundant = 1;
+	isl_assert(tab->mat->ctx, row >= tab->n_redundant, return -1);
+	if (tab->preserve || tab->need_undo || tab->row_var[row] >= 0) {
+		if (tab->row_var[row] >= 0 && !var->is_nonneg) {
+			var->is_nonneg = 1;
+			if (isl_tab_push_var(tab, isl_tab_undo_nonneg, var) < 0)
+				return -1;
+		}
+		if (row != tab->n_redundant)
+			swap_rows(tab, row, tab->n_redundant);
+		tab->n_redundant++;
+		return isl_tab_push_var(tab, isl_tab_undo_redundant, var);
+	} else {
+		if (row != tab->n_row - 1)
+			swap_rows(tab, row, tab->n_row - 1);
+		isl_tab_var_from_row(tab, tab->n_row - 1)->index = -1;
+		tab->n_row--;
+		return 1;
+	}
+}
+
+int isl_tab_mark_empty(struct isl_tab *tab)
+{
+	if (!tab)
+		return -1;
+	if (!tab->empty && tab->need_undo)
+		if (isl_tab_push(tab, isl_tab_undo_empty) < 0)
+			return -1;
+	tab->empty = 1;
+	return 0;
+}
+
+int isl_tab_freeze_constraint(struct isl_tab *tab, int con)
+{
+	struct isl_tab_var *var;
+
+	if (!tab)
+		return -1;
+
+	var = &tab->con[con];
+	if (var->frozen)
+		return 0;
+	if (var->index < 0)
+		return 0;
+	var->frozen = 1;
+
+	if (tab->need_undo)
+		return isl_tab_push_var(tab, isl_tab_undo_freeze, var);
+
+	return 0;
+}
+
+/* Update the rows signs after a pivot of "row" and "col", with "row_sgn"
+ * the original sign of the pivot element.
+ * We only keep track of row signs during PILP solving and in this case
+ * we only pivot a row with negative sign (meaning the value is always
+ * non-positive) using a positive pivot element.
+ *
+ * For each row j, the new value of the parametric constant is equal to
+ *
+ *	a_j0 - a_jc a_r0/a_rc
+ *
+ * where a_j0 is the original parametric constant, a_rc is the pivot element,
+ * a_r0 is the parametric constant of the pivot row and a_jc is the
+ * pivot column entry of the row j.
+ * Since a_r0 is non-positive and a_rc is positive, the sign of row j
+ * remains the same if a_jc has the same sign as the row j or if
+ * a_jc is zero.  In all other cases, we reset the sign to "unknown".
+ */
+static void update_row_sign(struct isl_tab *tab, int row, int col, int row_sgn)
+{
+	int i;
+	struct isl_mat *mat = tab->mat;
+	unsigned off = 2 + tab->M;
+
+	if (!tab->row_sign)
+		return;
+
+	if (tab->row_sign[row] == 0)
+		return;
+	isl_assert(mat->ctx, row_sgn > 0, return);
+	isl_assert(mat->ctx, tab->row_sign[row] == isl_tab_row_neg, return);
+	tab->row_sign[row] = isl_tab_row_pos;
+	for (i = 0; i < tab->n_row; ++i) {
+		int s;
+		if (i == row)
+			continue;
+		s = isl_int_sgn(mat->row[i][off + col]);
+		if (!s)
+			continue;
+		if (!tab->row_sign[i])
+			continue;
+		if (s < 0 && tab->row_sign[i] == isl_tab_row_neg)
+			continue;
+		if (s > 0 && tab->row_sign[i] == isl_tab_row_pos)
+			continue;
+		tab->row_sign[i] = isl_tab_row_unknown;
+	}
+}
+
+/* Given a row number "row" and a column number "col", pivot the tableau
+ * such that the associated variables are interchanged.
+ * The given row in the tableau expresses
+ *
+ *	x_r = a_r0 + \sum_i a_ri x_i
+ *
+ * or
+ *
+ *	x_c = 1/a_rc x_r - a_r0/a_rc + sum_{i \ne r} -a_ri/a_rc
+ *
+ * Substituting this equality into the other rows
+ *
+ *	x_j = a_j0 + \sum_i a_ji x_i
+ *
+ * with a_jc \ne 0, we obtain
+ *
+ *	x_j = a_jc/a_rc x_r + a_j0 - a_jc a_r0/a_rc + sum a_ji - a_jc a_ri/a_rc 
+ *
+ * The tableau
+ *
+ *	n_rc/d_r		n_ri/d_r
+ *	n_jc/d_j		n_ji/d_j
+ *
+ * where i is any other column and j is any other row,
+ * is therefore transformed into
+ *
+ * s(n_rc)d_r/|n_rc|		-s(n_rc)n_ri/|n_rc|
+ * s(n_rc)d_r n_jc/(|n_rc| d_j)	(n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j)
+ *
+ * The transformation is performed along the following steps
+ *
+ *	d_r/n_rc		n_ri/n_rc
+ *	n_jc/d_j		n_ji/d_j
+ *
+ *	s(n_rc)d_r/|n_rc|	-s(n_rc)n_ri/|n_rc|
+ *	n_jc/d_j		n_ji/d_j
+ *
+ *	s(n_rc)d_r/|n_rc|	-s(n_rc)n_ri/|n_rc|
+ *	n_jc/(|n_rc| d_j)	n_ji/(|n_rc| d_j)
+ *
+ *	s(n_rc)d_r/|n_rc|	-s(n_rc)n_ri/|n_rc|
+ *	n_jc/(|n_rc| d_j)	(n_ji |n_rc|)/(|n_rc| d_j)
+ *
+ *	s(n_rc)d_r/|n_rc|	-s(n_rc)n_ri/|n_rc|
+ *	n_jc/(|n_rc| d_j)	(n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j)
+ *
+ * s(n_rc)d_r/|n_rc|		-s(n_rc)n_ri/|n_rc|
+ * s(n_rc)d_r n_jc/(|n_rc| d_j)	(n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j)
+ *
+ */
+int isl_tab_pivot(struct isl_tab *tab, int row, int col)
+{
+	int i, j;
+	int sgn;
+	int t;
+	isl_ctx *ctx;
+	struct isl_mat *mat = tab->mat;
+	struct isl_tab_var *var;
+	unsigned off = 2 + tab->M;
+
+	ctx = isl_tab_get_ctx(tab);
+	if (isl_ctx_next_operation(ctx) < 0)
+		return -1;
+
+	isl_int_swap(mat->row[row][0], mat->row[row][off + col]);
+	sgn = isl_int_sgn(mat->row[row][0]);
+	if (sgn < 0) {
+		isl_int_neg(mat->row[row][0], mat->row[row][0]);
+		isl_int_neg(mat->row[row][off + col], mat->row[row][off + col]);
+	} else
+		for (j = 0; j < off - 1 + tab->n_col; ++j) {
+			if (j == off - 1 + col)
+				continue;
+			isl_int_neg(mat->row[row][1 + j], mat->row[row][1 + j]);
+		}
+	if (!isl_int_is_one(mat->row[row][0]))
+		isl_seq_normalize(mat->ctx, mat->row[row], off + tab->n_col);
+	for (i = 0; i < tab->n_row; ++i) {
+		if (i == row)
+			continue;
+		if (isl_int_is_zero(mat->row[i][off + col]))
+			continue;
+		isl_int_mul(mat->row[i][0], mat->row[i][0], mat->row[row][0]);
+		for (j = 0; j < off - 1 + tab->n_col; ++j) {
+			if (j == off - 1 + col)
+				continue;
+			isl_int_mul(mat->row[i][1 + j],
+				    mat->row[i][1 + j], mat->row[row][0]);
+			isl_int_addmul(mat->row[i][1 + j],
+				    mat->row[i][off + col], mat->row[row][1 + j]);
+		}
+		isl_int_mul(mat->row[i][off + col],
+			    mat->row[i][off + col], mat->row[row][off + col]);
+		if (!isl_int_is_one(mat->row[i][0]))
+			isl_seq_normalize(mat->ctx, mat->row[i], off + tab->n_col);
+	}
+	t = tab->row_var[row];
+	tab->row_var[row] = tab->col_var[col];
+	tab->col_var[col] = t;
+	var = isl_tab_var_from_row(tab, row);
+	var->is_row = 1;
+	var->index = row;
+	var = var_from_col(tab, col);
+	var->is_row = 0;
+	var->index = col;
+	update_row_sign(tab, row, col, sgn);
+	if (tab->in_undo)
+		return 0;
+	for (i = tab->n_redundant; i < tab->n_row; ++i) {
+		if (isl_int_is_zero(mat->row[i][off + col]))
+			continue;
+		if (!isl_tab_var_from_row(tab, i)->frozen &&
+		    isl_tab_row_is_redundant(tab, i)) {
+			int redo = isl_tab_mark_redundant(tab, i);
+			if (redo < 0)
+				return -1;
+			if (redo)
+				--i;
+		}
+	}
+	return 0;
+}
+
+/* If "var" represents a column variable, then pivot is up (sgn > 0)
+ * or down (sgn < 0) to a row.  The variable is assumed not to be
+ * unbounded in the specified direction.
+ * If sgn = 0, then the variable is unbounded in both directions,
+ * and we pivot with any row we can find.
+ */
+static int to_row(struct isl_tab *tab, struct isl_tab_var *var, int sign) WARN_UNUSED;
+static int to_row(struct isl_tab *tab, struct isl_tab_var *var, int sign)
+{
+	int r;
+	unsigned off = 2 + tab->M;
+
+	if (var->is_row)
+		return 0;
+
+	if (sign == 0) {
+		for (r = tab->n_redundant; r < tab->n_row; ++r)
+			if (!isl_int_is_zero(tab->mat->row[r][off+var->index]))
+				break;
+		isl_assert(tab->mat->ctx, r < tab->n_row, return -1);
+	} else {
+		r = pivot_row(tab, NULL, sign, var->index);
+		isl_assert(tab->mat->ctx, r >= 0, return -1);
+	}
+
+	return isl_tab_pivot(tab, r, var->index);
+}
+
+/* Check whether all variables that are marked as non-negative
+ * also have a non-negative sample value.  This function is not
+ * called from the current code but is useful during debugging.
+ */
+static void check_table(struct isl_tab *tab) __attribute__ ((unused));
+static void check_table(struct isl_tab *tab)
+{
+	int i;
+
+	if (tab->empty)
+		return;
+	for (i = tab->n_redundant; i < tab->n_row; ++i) {
+		struct isl_tab_var *var;
+		var = isl_tab_var_from_row(tab, i);
+		if (!var->is_nonneg)
+			continue;
+		if (tab->M) {
+			isl_assert(tab->mat->ctx,
+				!isl_int_is_neg(tab->mat->row[i][2]), abort());
+			if (isl_int_is_pos(tab->mat->row[i][2]))
+				continue;
+		}
+		isl_assert(tab->mat->ctx, !isl_int_is_neg(tab->mat->row[i][1]),
+				abort());
+	}
+}
+
+/* Return the sign of the maximal value of "var".
+ * If the sign is not negative, then on return from this function,
+ * the sample value will also be non-negative.
+ *
+ * If "var" is manifestly unbounded wrt positive values, we are done.
+ * Otherwise, we pivot the variable up to a row if needed
+ * Then we continue pivoting down until either
+ *	- no more down pivots can be performed
+ *	- the sample value is positive
+ *	- the variable is pivoted into a manifestly unbounded column
+ */
+static int sign_of_max(struct isl_tab *tab, struct isl_tab_var *var)
+{
+	int row, col;
+
+	if (max_is_manifestly_unbounded(tab, var))
+		return 1;
+	if (to_row(tab, var, 1) < 0)
+		return -2;
+	while (!isl_int_is_pos(tab->mat->row[var->index][1])) {
+		find_pivot(tab, var, var, 1, &row, &col);
+		if (row == -1)
+			return isl_int_sgn(tab->mat->row[var->index][1]);
+		if (isl_tab_pivot(tab, row, col) < 0)
+			return -2;
+		if (!var->is_row) /* manifestly unbounded */
+			return 1;
+	}
+	return 1;
+}
+
+int isl_tab_sign_of_max(struct isl_tab *tab, int con)
+{
+	struct isl_tab_var *var;
+
+	if (!tab)
+		return -2;
+
+	var = &tab->con[con];
+	isl_assert(tab->mat->ctx, !var->is_redundant, return -2);
+	isl_assert(tab->mat->ctx, !var->is_zero, return -2);
+
+	return sign_of_max(tab, var);
+}
+
+static int row_is_neg(struct isl_tab *tab, int row)
+{
+	if (!tab->M)
+		return isl_int_is_neg(tab->mat->row[row][1]);
+	if (isl_int_is_pos(tab->mat->row[row][2]))
+		return 0;
+	if (isl_int_is_neg(tab->mat->row[row][2]))
+		return 1;
+	return isl_int_is_neg(tab->mat->row[row][1]);
+}
+
+static int row_sgn(struct isl_tab *tab, int row)
+{
+	if (!tab->M)
+		return isl_int_sgn(tab->mat->row[row][1]);
+	if (!isl_int_is_zero(tab->mat->row[row][2]))
+		return isl_int_sgn(tab->mat->row[row][2]);
+	else
+		return isl_int_sgn(tab->mat->row[row][1]);
+}
+
+/* Perform pivots until the row variable "var" has a non-negative
+ * sample value or until no more upward pivots can be performed.
+ * Return the sign of the sample value after the pivots have been
+ * performed.
+ */
+static int restore_row(struct isl_tab *tab, struct isl_tab_var *var)
+{
+	int row, col;
+
+	while (row_is_neg(tab, var->index)) {
+		find_pivot(tab, var, var, 1, &row, &col);
+		if (row == -1)
+			break;
+		if (isl_tab_pivot(tab, row, col) < 0)
+			return -2;
+		if (!var->is_row) /* manifestly unbounded */
+			return 1;
+	}
+	return row_sgn(tab, var->index);
+}
+
+/* Perform pivots until we are sure that the row variable "var"
+ * can attain non-negative values.  After return from this
+ * function, "var" is still a row variable, but its sample
+ * value may not be non-negative, even if the function returns 1.
+ */
+static int at_least_zero(struct isl_tab *tab, struct isl_tab_var *var)
+{
+	int row, col;
+
+	while (isl_int_is_neg(tab->mat->row[var->index][1])) {
+		find_pivot(tab, var, var, 1, &row, &col);
+		if (row == -1)
+			break;
+		if (row == var->index) /* manifestly unbounded */
+			return 1;
+		if (isl_tab_pivot(tab, row, col) < 0)
+			return -1;
+	}
+	return !isl_int_is_neg(tab->mat->row[var->index][1]);
+}
+
+/* Return a negative value if "var" can attain negative values.
+ * Return a non-negative value otherwise.
+ *
+ * If "var" is manifestly unbounded wrt negative values, we are done.
+ * Otherwise, if var is in a column, we can pivot it down to a row.
+ * Then we continue pivoting down until either
+ *	- the pivot would result in a manifestly unbounded column
+ *	  => we don't perform the pivot, but simply return -1
+ *	- no more down pivots can be performed
+ *	- the sample value is negative
+ * If the sample value becomes negative and the variable is supposed
+ * to be nonnegative, then we undo the last pivot.
+ * However, if the last pivot has made the pivoting variable
+ * obviously redundant, then it may have moved to another row.
+ * In that case we look for upward pivots until we reach a non-negative
+ * value again.
+ */
+static int sign_of_min(struct isl_tab *tab, struct isl_tab_var *var)
+{
+	int row, col;
+	struct isl_tab_var *pivot_var = NULL;
+
+	if (min_is_manifestly_unbounded(tab, var))
+		return -1;
+	if (!var->is_row) {
+		col = var->index;
+		row = pivot_row(tab, NULL, -1, col);
+		pivot_var = var_from_col(tab, col);
+		if (isl_tab_pivot(tab, row, col) < 0)
+			return -2;
+		if (var->is_redundant)
+			return 0;
+		if (isl_int_is_neg(tab->mat->row[var->index][1])) {
+			if (var->is_nonneg) {
+				if (!pivot_var->is_redundant &&
+				    pivot_var->index == row) {
+					if (isl_tab_pivot(tab, row, col) < 0)
+						return -2;
+				} else
+					if (restore_row(tab, var) < -1)
+						return -2;
+			}
+			return -1;
+		}
+	}
+	if (var->is_redundant)
+		return 0;
+	while (!isl_int_is_neg(tab->mat->row[var->index][1])) {
+		find_pivot(tab, var, var, -1, &row, &col);
+		if (row == var->index)
+			return -1;
+		if (row == -1)
+			return isl_int_sgn(tab->mat->row[var->index][1]);
+		pivot_var = var_from_col(tab, col);
+		if (isl_tab_pivot(tab, row, col) < 0)
+			return -2;
+		if (var->is_redundant)
+			return 0;
+	}
+	if (pivot_var && var->is_nonneg) {
+		/* pivot back to non-negative value */
+		if (!pivot_var->is_redundant && pivot_var->index == row) {
+			if (isl_tab_pivot(tab, row, col) < 0)
+				return -2;
+		} else
+			if (restore_row(tab, var) < -1)
+				return -2;
+	}
+	return -1;
+}
+
+static int row_at_most_neg_one(struct isl_tab *tab, int row)
+{
+	if (tab->M) {
+		if (isl_int_is_pos(tab->mat->row[row][2]))
+			return 0;
+		if (isl_int_is_neg(tab->mat->row[row][2]))
+			return 1;
+	}
+	return isl_int_is_neg(tab->mat->row[row][1]) &&
+	       isl_int_abs_ge(tab->mat->row[row][1],
+			      tab->mat->row[row][0]);
+}
+
+/* Return 1 if "var" can attain values <= -1.
+ * Return 0 otherwise.
+ *
+ * The sample value of "var" is assumed to be non-negative when the
+ * the function is called.  If 1 is returned then the constraint
+ * is not redundant and the sample value is made non-negative again before
+ * the function returns.
+ */
+int isl_tab_min_at_most_neg_one(struct isl_tab *tab, struct isl_tab_var *var)
+{
+	int row, col;
+	struct isl_tab_var *pivot_var;
+
+	if (min_is_manifestly_unbounded(tab, var))
+		return 1;
+	if (!var->is_row) {
+		col = var->index;
+		row = pivot_row(tab, NULL, -1, col);
+		pivot_var = var_from_col(tab, col);
+		if (isl_tab_pivot(tab, row, col) < 0)
+			return -1;
+		if (var->is_redundant)
+			return 0;
+		if (row_at_most_neg_one(tab, var->index)) {
+			if (var->is_nonneg) {
+				if (!pivot_var->is_redundant &&
+				    pivot_var->index == row) {
+					if (isl_tab_pivot(tab, row, col) < 0)
+						return -1;
+				} else
+					if (restore_row(tab, var) < -1)
+						return -1;
+			}
+			return 1;
+		}
+	}
+	if (var->is_redundant)
+		return 0;
+	do {
+		find_pivot(tab, var, var, -1, &row, &col);
+		if (row == var->index) {
+			if (restore_row(tab, var) < -1)
+				return -1;
+			return 1;
+		}
+		if (row == -1)
+			return 0;
+		pivot_var = var_from_col(tab, col);
+		if (isl_tab_pivot(tab, row, col) < 0)
+			return -1;
+		if (var->is_redundant)
+			return 0;
+	} while (!row_at_most_neg_one(tab, var->index));
+	if (var->is_nonneg) {
+		/* pivot back to non-negative value */
+		if (!pivot_var->is_redundant && pivot_var->index == row)
+			if (isl_tab_pivot(tab, row, col) < 0)
+				return -1;
+		if (restore_row(tab, var) < -1)
+			return -1;
+	}
+	return 1;
+}
+
+/* Return 1 if "var" can attain values >= 1.
+ * Return 0 otherwise.
+ */
+static int at_least_one(struct isl_tab *tab, struct isl_tab_var *var)
+{
+	int row, col;
+	isl_int *r;
+
+	if (max_is_manifestly_unbounded(tab, var))
+		return 1;
+	if (to_row(tab, var, 1) < 0)
+		return -1;
+	r = tab->mat->row[var->index];
+	while (isl_int_lt(r[1], r[0])) {
+		find_pivot(tab, var, var, 1, &row, &col);
+		if (row == -1)
+			return isl_int_ge(r[1], r[0]);
+		if (row == var->index) /* manifestly unbounded */
+			return 1;
+		if (isl_tab_pivot(tab, row, col) < 0)
+			return -1;
+	}
+	return 1;
+}
+
+static void swap_cols(struct isl_tab *tab, int col1, int col2)
+{
+	int t;
+	unsigned off = 2 + tab->M;
+	t = tab->col_var[col1];
+	tab->col_var[col1] = tab->col_var[col2];
+	tab->col_var[col2] = t;
+	var_from_col(tab, col1)->index = col1;
+	var_from_col(tab, col2)->index = col2;
+	tab->mat = isl_mat_swap_cols(tab->mat, off + col1, off + col2);
+}
+
+/* Mark column with index "col" as representing a zero variable.
+ * If we may need to undo the operation the column is kept,
+ * but no longer considered.
+ * Otherwise, the column is simply removed.
+ *
+ * The column may be interchanged with some other column.  If it
+ * is interchanged with a later column, return 1.  Otherwise return 0.
+ * If the columns are checked in order in the calling function,
+ * then a return value of 1 means that the column with the given
+ * column number may now contain a different column that
+ * hasn't been checked yet.
+ */
+int isl_tab_kill_col(struct isl_tab *tab, int col)
+{
+	var_from_col(tab, col)->is_zero = 1;
+	if (tab->need_undo) {
+		if (isl_tab_push_var(tab, isl_tab_undo_zero,
+					    var_from_col(tab, col)) < 0)
+			return -1;
+		if (col != tab->n_dead)
+			swap_cols(tab, col, tab->n_dead);
+		tab->n_dead++;
+		return 0;
+	} else {
+		if (col != tab->n_col - 1)
+			swap_cols(tab, col, tab->n_col - 1);
+		var_from_col(tab, tab->n_col - 1)->index = -1;
+		tab->n_col--;
+		return 1;
+	}
+}
+
+static int row_is_manifestly_non_integral(struct isl_tab *tab, int row)
+{
+	unsigned off = 2 + tab->M;
+
+	if (tab->M && !isl_int_eq(tab->mat->row[row][2],
+				  tab->mat->row[row][0]))
+		return 0;
+	if (isl_seq_first_non_zero(tab->mat->row[row] + off + tab->n_dead,
+				    tab->n_col - tab->n_dead) != -1)
+		return 0;
+
+	return !isl_int_is_divisible_by(tab->mat->row[row][1],
+					tab->mat->row[row][0]);
+}
+
+/* For integer tableaus, check if any of the coordinates are stuck
+ * at a non-integral value.
+ */
+static int tab_is_manifestly_empty(struct isl_tab *tab)
+{
+	int i;
+
+	if (tab->empty)
+		return 1;
+	if (tab->rational)
+		return 0;
+
+	for (i = 0; i < tab->n_var; ++i) {
+		if (!tab->var[i].is_row)
+			continue;
+		if (row_is_manifestly_non_integral(tab, tab->var[i].index))
+			return 1;
+	}
+
+	return 0;
+}
+
+/* Row variable "var" is non-negative and cannot attain any values
+ * larger than zero.  This means that the coefficients of the unrestricted
+ * column variables are zero and that the coefficients of the non-negative
+ * column variables are zero or negative.
+ * Each of the non-negative variables with a negative coefficient can
+ * then also be written as the negative sum of non-negative variables
+ * and must therefore also be zero.
+ */
+static int close_row(struct isl_tab *tab, struct isl_tab_var *var) WARN_UNUSED;
+static int close_row(struct isl_tab *tab, struct isl_tab_var *var)
+{
+	int j;
+	struct isl_mat *mat = tab->mat;
+	unsigned off = 2 + tab->M;
+
+	isl_assert(tab->mat->ctx, var->is_nonneg, return -1);
+	var->is_zero = 1;
+	if (tab->need_undo)
+		if (isl_tab_push_var(tab, isl_tab_undo_zero, var) < 0)
+			return -1;
+	for (j = tab->n_dead; j < tab->n_col; ++j) {
+		int recheck;
+		if (isl_int_is_zero(mat->row[var->index][off + j]))
+			continue;
+		isl_assert(tab->mat->ctx,
+		    isl_int_is_neg(mat->row[var->index][off + j]), return -1);
+		recheck = isl_tab_kill_col(tab, j);
+		if (recheck < 0)
+			return -1;
+		if (recheck)
+			--j;
+	}
+	if (isl_tab_mark_redundant(tab, var->index) < 0)
+		return -1;
+	if (tab_is_manifestly_empty(tab) && isl_tab_mark_empty(tab) < 0)
+		return -1;
+	return 0;
+}
+
+/* Add a constraint to the tableau and allocate a row for it.
+ * Return the index into the constraint array "con".
+ */
+int isl_tab_allocate_con(struct isl_tab *tab)
+{
+	int r;
+
+	isl_assert(tab->mat->ctx, tab->n_row < tab->mat->n_row, return -1);
+	isl_assert(tab->mat->ctx, tab->n_con < tab->max_con, return -1);
+
+	r = tab->n_con;
+	tab->con[r].index = tab->n_row;
+	tab->con[r].is_row = 1;
+	tab->con[r].is_nonneg = 0;
+	tab->con[r].is_zero = 0;
+	tab->con[r].is_redundant = 0;
+	tab->con[r].frozen = 0;
+	tab->con[r].negated = 0;
+	tab->row_var[tab->n_row] = ~r;
+
+	tab->n_row++;
+	tab->n_con++;
+	if (isl_tab_push_var(tab, isl_tab_undo_allocate, &tab->con[r]) < 0)
+		return -1;
+
+	return r;
+}
+
+/* Add a variable to the tableau and allocate a column for it.
+ * Return the index into the variable array "var".
+ */
+int isl_tab_allocate_var(struct isl_tab *tab)
+{
+	int r;
+	int i;
+	unsigned off = 2 + tab->M;
+
+	isl_assert(tab->mat->ctx, tab->n_col < tab->mat->n_col, return -1);
+	isl_assert(tab->mat->ctx, tab->n_var < tab->max_var, return -1);
+
+	r = tab->n_var;
+	tab->var[r].index = tab->n_col;
+	tab->var[r].is_row = 0;
+	tab->var[r].is_nonneg = 0;
+	tab->var[r].is_zero = 0;
+	tab->var[r].is_redundant = 0;
+	tab->var[r].frozen = 0;
+	tab->var[r].negated = 0;
+	tab->col_var[tab->n_col] = r;
+
+	for (i = 0; i < tab->n_row; ++i)
+		isl_int_set_si(tab->mat->row[i][off + tab->n_col], 0);
+
+	tab->n_var++;
+	tab->n_col++;
+	if (isl_tab_push_var(tab, isl_tab_undo_allocate, &tab->var[r]) < 0)
+		return -1;
+
+	return r;
+}
+
+/* Add a row to the tableau.  The row is given as an affine combination
+ * of the original variables and needs to be expressed in terms of the
+ * column variables.
+ *
+ * We add each term in turn.
+ * If r = n/d_r is the current sum and we need to add k x, then
+ * 	if x is a column variable, we increase the numerator of
+ *		this column by k d_r
+ *	if x = f/d_x is a row variable, then the new representation of r is
+ *
+ *		 n    k f   d_x/g n + d_r/g k f   m/d_r n + m/d_g k f
+ *		--- + --- = ------------------- = -------------------
+ *		d_r   d_r        d_r d_x/g                m
+ *
+ *	with g the gcd of d_r and d_x and m the lcm of d_r and d_x.
+ *
+ * If tab->M is set, then, internally, each variable x is represented
+ * as x' - M.  We then also need no subtract k d_r from the coefficient of M.
+ */
+int isl_tab_add_row(struct isl_tab *tab, isl_int *line)
+{
+	int i;
+	int r;
+	isl_int *row;
+	isl_int a, b;
+	unsigned off = 2 + tab->M;
+
+	r = isl_tab_allocate_con(tab);
+	if (r < 0)
+		return -1;
+
+	isl_int_init(a);
+	isl_int_init(b);
+	row = tab->mat->row[tab->con[r].index];
+	isl_int_set_si(row[0], 1);
+	isl_int_set(row[1], line[0]);
+	isl_seq_clr(row + 2, tab->M + tab->n_col);
+	for (i = 0; i < tab->n_var; ++i) {
+		if (tab->var[i].is_zero)
+			continue;
+		if (tab->var[i].is_row) {
+			isl_int_lcm(a,
+				row[0], tab->mat->row[tab->var[i].index][0]);
+			isl_int_swap(a, row[0]);
+			isl_int_divexact(a, row[0], a);
+			isl_int_divexact(b,
+				row[0], tab->mat->row[tab->var[i].index][0]);
+			isl_int_mul(b, b, line[1 + i]);
+			isl_seq_combine(row + 1, a, row + 1,
+			    b, tab->mat->row[tab->var[i].index] + 1,
+			    1 + tab->M + tab->n_col);
+		} else
+			isl_int_addmul(row[off + tab->var[i].index],
+							line[1 + i], row[0]);
+		if (tab->M && i >= tab->n_param && i < tab->n_var - tab->n_div)
+			isl_int_submul(row[2], line[1 + i], row[0]);
+	}
+	isl_seq_normalize(tab->mat->ctx, row, off + tab->n_col);
+	isl_int_clear(a);
+	isl_int_clear(b);
+
+	if (tab->row_sign)
+		tab->row_sign[tab->con[r].index] = isl_tab_row_unknown;
+
+	return r;
+}
+
+static int drop_row(struct isl_tab *tab, int row)
+{
+	isl_assert(tab->mat->ctx, ~tab->row_var[row] == tab->n_con - 1, return -1);
+	if (row != tab->n_row - 1)
+		swap_rows(tab, row, tab->n_row - 1);
+	tab->n_row--;
+	tab->n_con--;
+	return 0;
+}
+
+static int drop_col(struct isl_tab *tab, int col)
+{
+	isl_assert(tab->mat->ctx, tab->col_var[col] == tab->n_var - 1, return -1);
+	if (col != tab->n_col - 1)
+		swap_cols(tab, col, tab->n_col - 1);
+	tab->n_col--;
+	tab->n_var--;
+	return 0;
+}
+
+/* Add inequality "ineq" and check if it conflicts with the
+ * previously added constraints or if it is obviously redundant.
+ */
+int isl_tab_add_ineq(struct isl_tab *tab, isl_int *ineq)
+{
+	int r;
+	int sgn;
+	isl_int cst;
+
+	if (!tab)
+		return -1;
+	if (tab->bmap) {
+		struct isl_basic_map *bmap = tab->bmap;
+
+		isl_assert(tab->mat->ctx, tab->n_eq == bmap->n_eq, return -1);
+		isl_assert(tab->mat->ctx,
+			    tab->n_con == bmap->n_eq + bmap->n_ineq, return -1);
+		tab->bmap = isl_basic_map_add_ineq(tab->bmap, ineq);
+		if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0)
+			return -1;
+		if (!tab->bmap)
+			return -1;
+	}
+	if (tab->cone) {
+		isl_int_init(cst);
+		isl_int_swap(ineq[0], cst);
+	}
+	r = isl_tab_add_row(tab, ineq);
+	if (tab->cone) {
+		isl_int_swap(ineq[0], cst);
+		isl_int_clear(cst);
+	}
+	if (r < 0)
+		return -1;
+	tab->con[r].is_nonneg = 1;
+	if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
+		return -1;
+	if (isl_tab_row_is_redundant(tab, tab->con[r].index)) {
+		if (isl_tab_mark_redundant(tab, tab->con[r].index) < 0)
+			return -1;
+		return 0;
+	}
+
+	sgn = restore_row(tab, &tab->con[r]);
+	if (sgn < -1)
+		return -1;
+	if (sgn < 0)
+		return isl_tab_mark_empty(tab);
+	if (tab->con[r].is_row && isl_tab_row_is_redundant(tab, tab->con[r].index))
+		if (isl_tab_mark_redundant(tab, tab->con[r].index) < 0)
+			return -1;
+	return 0;
+}
+
+/* Pivot a non-negative variable down until it reaches the value zero
+ * and then pivot the variable into a column position.
+ */
+static int to_col(struct isl_tab *tab, struct isl_tab_var *var) WARN_UNUSED;
+static int to_col(struct isl_tab *tab, struct isl_tab_var *var)
+{
+	int i;
+	int row, col;
+	unsigned off = 2 + tab->M;
+
+	if (!var->is_row)
+		return 0;
+
+	while (isl_int_is_pos(tab->mat->row[var->index][1])) {
+		find_pivot(tab, var, NULL, -1, &row, &col);
+		isl_assert(tab->mat->ctx, row != -1, return -1);
+		if (isl_tab_pivot(tab, row, col) < 0)
+			return -1;
+		if (!var->is_row)
+			return 0;
+	}
+
+	for (i = tab->n_dead; i < tab->n_col; ++i)
+		if (!isl_int_is_zero(tab->mat->row[var->index][off + i]))
+			break;
+
+	isl_assert(tab->mat->ctx, i < tab->n_col, return -1);
+	if (isl_tab_pivot(tab, var->index, i) < 0)
+		return -1;
+
+	return 0;
+}
+
+/* We assume Gaussian elimination has been performed on the equalities.
+ * The equalities can therefore never conflict.
+ * Adding the equalities is currently only really useful for a later call
+ * to isl_tab_ineq_type.
+ */
+static struct isl_tab *add_eq(struct isl_tab *tab, isl_int *eq)
+{
+	int i;
+	int r;
+
+	if (!tab)
+		return NULL;
+	r = isl_tab_add_row(tab, eq);
+	if (r < 0)
+		goto error;
+
+	r = tab->con[r].index;
+	i = isl_seq_first_non_zero(tab->mat->row[r] + 2 + tab->M + tab->n_dead,
+					tab->n_col - tab->n_dead);
+	isl_assert(tab->mat->ctx, i >= 0, goto error);
+	i += tab->n_dead;
+	if (isl_tab_pivot(tab, r, i) < 0)
+		goto error;
+	if (isl_tab_kill_col(tab, i) < 0)
+		goto error;
+	tab->n_eq++;
+
+	return tab;
+error:
+	isl_tab_free(tab);
+	return NULL;
+}
+
+static int row_is_manifestly_zero(struct isl_tab *tab, int row)
+{
+	unsigned off = 2 + tab->M;
+
+	if (!isl_int_is_zero(tab->mat->row[row][1]))
+		return 0;
+	if (tab->M && !isl_int_is_zero(tab->mat->row[row][2]))
+		return 0;
+	return isl_seq_first_non_zero(tab->mat->row[row] + off + tab->n_dead,
+					tab->n_col - tab->n_dead) == -1;
+}
+
+/* Add an equality that is known to be valid for the given tableau.
+ */
+int isl_tab_add_valid_eq(struct isl_tab *tab, isl_int *eq)
+{
+	struct isl_tab_var *var;
+	int r;
+
+	if (!tab)
+		return -1;
+	r = isl_tab_add_row(tab, eq);
+	if (r < 0)
+		return -1;
+
+	var = &tab->con[r];
+	r = var->index;
+	if (row_is_manifestly_zero(tab, r)) {
+		var->is_zero = 1;
+		if (isl_tab_mark_redundant(tab, r) < 0)
+			return -1;
+		return 0;
+	}
+
+	if (isl_int_is_neg(tab->mat->row[r][1])) {
+		isl_seq_neg(tab->mat->row[r] + 1, tab->mat->row[r] + 1,
+			    1 + tab->n_col);
+		var->negated = 1;
+	}
+	var->is_nonneg = 1;
+	if (to_col(tab, var) < 0)
+		return -1;
+	var->is_nonneg = 0;
+	if (isl_tab_kill_col(tab, var->index) < 0)
+		return -1;
+
+	return 0;
+}
+
+static int add_zero_row(struct isl_tab *tab)
+{
+	int r;
+	isl_int *row;
+
+	r = isl_tab_allocate_con(tab);
+	if (r < 0)
+		return -1;
+
+	row = tab->mat->row[tab->con[r].index];
+	isl_seq_clr(row + 1, 1 + tab->M + tab->n_col);
+	isl_int_set_si(row[0], 1);
+
+	return r;
+}
+
+/* Add equality "eq" and check if it conflicts with the
+ * previously added constraints or if it is obviously redundant.
+ */
+int isl_tab_add_eq(struct isl_tab *tab, isl_int *eq)
+{
+	struct isl_tab_undo *snap = NULL;
+	struct isl_tab_var *var;
+	int r;
+	int row;
+	int sgn;
+	isl_int cst;
+
+	if (!tab)
+		return -1;
+	isl_assert(tab->mat->ctx, !tab->M, return -1);
+
+	if (tab->need_undo)
+		snap = isl_tab_snap(tab);
+
+	if (tab->cone) {
+		isl_int_init(cst);
+		isl_int_swap(eq[0], cst);
+	}
+	r = isl_tab_add_row(tab, eq);
+	if (tab->cone) {
+		isl_int_swap(eq[0], cst);
+		isl_int_clear(cst);
+	}
+	if (r < 0)
+		return -1;
+
+	var = &tab->con[r];
+	row = var->index;
+	if (row_is_manifestly_zero(tab, row)) {
+		if (snap) {
+			if (isl_tab_rollback(tab, snap) < 0)
+				return -1;
+		} else
+			drop_row(tab, row);
+		return 0;
+	}
+
+	if (tab->bmap) {
+		tab->bmap = isl_basic_map_add_ineq(tab->bmap, eq);
+		if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0)
+			return -1;
+		isl_seq_neg(eq, eq, 1 + tab->n_var);
+		tab->bmap = isl_basic_map_add_ineq(tab->bmap, eq);
+		isl_seq_neg(eq, eq, 1 + tab->n_var);
+		if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0)
+			return -1;
+		if (!tab->bmap)
+			return -1;
+		if (add_zero_row(tab) < 0)
+			return -1;
+	}
+
+	sgn = isl_int_sgn(tab->mat->row[row][1]);
+
+	if (sgn > 0) {
+		isl_seq_neg(tab->mat->row[row] + 1, tab->mat->row[row] + 1,
+			    1 + tab->n_col);
+		var->negated = 1;
+		sgn = -1;
+	}
+
+	if (sgn < 0) {
+		sgn = sign_of_max(tab, var);
+		if (sgn < -1)
+			return -1;
+		if (sgn < 0) {
+			if (isl_tab_mark_empty(tab) < 0)
+				return -1;
+			return 0;
+		}
+	}
+
+	var->is_nonneg = 1;
+	if (to_col(tab, var) < 0)
+		return -1;
+	var->is_nonneg = 0;
+	if (isl_tab_kill_col(tab, var->index) < 0)
+		return -1;
+
+	return 0;
+}
+
+/* Construct and return an inequality that expresses an upper bound
+ * on the given div.
+ * In particular, if the div is given by
+ *
+ *	d = floor(e/m)
+ *
+ * then the inequality expresses
+ *
+ *	m d <= e
+ */
+static struct isl_vec *ineq_for_div(struct isl_basic_map *bmap, unsigned div)
+{
+	unsigned total;
+	unsigned div_pos;
+	struct isl_vec *ineq;
+
+	if (!bmap)
+		return NULL;
+
+	total = isl_basic_map_total_dim(bmap);
+	div_pos = 1 + total - bmap->n_div + div;
+
+	ineq = isl_vec_alloc(bmap->ctx, 1 + total);
+	if (!ineq)
+		return NULL;
+
+	isl_seq_cpy(ineq->el, bmap->div[div] + 1, 1 + total);
+	isl_int_neg(ineq->el[div_pos], bmap->div[div][0]);
+	return ineq;
+}
+
+/* For a div d = floor(f/m), add the constraints
+ *
+ *		f - m d >= 0
+ *		-(f-(m-1)) + m d >= 0
+ *
+ * Note that the second constraint is the negation of
+ *
+ *		f - m d >= m
+ *
+ * If add_ineq is not NULL, then this function is used
+ * instead of isl_tab_add_ineq to effectively add the inequalities.
+ */
+static int add_div_constraints(struct isl_tab *tab, unsigned div,
+	int (*add_ineq)(void *user, isl_int *), void *user)
+{
+	unsigned total;
+	unsigned div_pos;
+	struct isl_vec *ineq;
+
+	total = isl_basic_map_total_dim(tab->bmap);
+	div_pos = 1 + total - tab->bmap->n_div + div;
+
+	ineq = ineq_for_div(tab->bmap, div);
+	if (!ineq)
+		goto error;
+
+	if (add_ineq) {
+		if (add_ineq(user, ineq->el) < 0)
+			goto error;
+	} else {
+		if (isl_tab_add_ineq(tab, ineq->el) < 0)
+			goto error;
+	}
+
+	isl_seq_neg(ineq->el, tab->bmap->div[div] + 1, 1 + total);
+	isl_int_set(ineq->el[div_pos], tab->bmap->div[div][0]);
+	isl_int_add(ineq->el[0], ineq->el[0], ineq->el[div_pos]);
+	isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
+
+	if (add_ineq) {
+		if (add_ineq(user, ineq->el) < 0)
+			goto error;
+	} else {
+		if (isl_tab_add_ineq(tab, ineq->el) < 0)
+			goto error;
+	}
+
+	isl_vec_free(ineq);
+
+	return 0;
+error:
+	isl_vec_free(ineq);
+	return -1;
+}
+
+/* Check whether the div described by "div" is obviously non-negative.
+ * If we are using a big parameter, then we will encode the div
+ * as div' = M + div, which is always non-negative.
+ * Otherwise, we check whether div is a non-negative affine combination
+ * of non-negative variables.
+ */
+static int div_is_nonneg(struct isl_tab *tab, __isl_keep isl_vec *div)
+{
+	int i;
+
+	if (tab->M)
+		return 1;
+
+	if (isl_int_is_neg(div->el[1]))
+		return 0;
+
+	for (i = 0; i < tab->n_var; ++i) {
+		if (isl_int_is_neg(div->el[2 + i]))
+			return 0;
+		if (isl_int_is_zero(div->el[2 + i]))
+			continue;
+		if (!tab->var[i].is_nonneg)
+			return 0;
+	}
+
+	return 1;
+}
+
+/* Add an extra div, prescribed by "div" to the tableau and
+ * the associated bmap (which is assumed to be non-NULL).
+ *
+ * If add_ineq is not NULL, then this function is used instead
+ * of isl_tab_add_ineq to add the div constraints.
+ * This complication is needed because the code in isl_tab_pip
+ * wants to perform some extra processing when an inequality
+ * is added to the tableau.
+ */
+int isl_tab_add_div(struct isl_tab *tab, __isl_keep isl_vec *div,
+	int (*add_ineq)(void *user, isl_int *), void *user)
+{
+	int r;
+	int k;
+	int nonneg;
+
+	if (!tab || !div)
+		return -1;
+
+	isl_assert(tab->mat->ctx, tab->bmap, return -1);
+
+	nonneg = div_is_nonneg(tab, div);
+
+	if (isl_tab_extend_cons(tab, 3) < 0)
+		return -1;
+	if (isl_tab_extend_vars(tab, 1) < 0)
+		return -1;
+	r = isl_tab_allocate_var(tab);
+	if (r < 0)
+		return -1;
+
+	if (nonneg)
+		tab->var[r].is_nonneg = 1;
+
+	tab->bmap = isl_basic_map_extend_space(tab->bmap,
+		isl_basic_map_get_space(tab->bmap), 1, 0, 2);
+	k = isl_basic_map_alloc_div(tab->bmap);
+	if (k < 0)
+		return -1;
+	isl_seq_cpy(tab->bmap->div[k], div->el, div->size);
+	if (isl_tab_push(tab, isl_tab_undo_bmap_div) < 0)
+		return -1;
+
+	if (add_div_constraints(tab, k, add_ineq, user) < 0)
+		return -1;
+
+	return r;
+}
+
+/* If "track" is set, then we want to keep track of all constraints in tab
+ * in its bmap field.  This field is initialized from a copy of "bmap",
+ * so we need to make sure that all constraints in "bmap" also appear
+ * in the constructed tab.
+ */
+__isl_give struct isl_tab *isl_tab_from_basic_map(
+	__isl_keep isl_basic_map *bmap, int track)
+{
+	int i;
+	struct isl_tab *tab;
+
+	if (!bmap)
+		return NULL;
+	tab = isl_tab_alloc(bmap->ctx,
+			    isl_basic_map_total_dim(bmap) + bmap->n_ineq + 1,
+			    isl_basic_map_total_dim(bmap), 0);
+	if (!tab)
+		return NULL;
+	tab->preserve = track;
+	tab->rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) {
+		if (isl_tab_mark_empty(tab) < 0)
+			goto error;
+		goto done;
+	}
+	for (i = 0; i < bmap->n_eq; ++i) {
+		tab = add_eq(tab, bmap->eq[i]);
+		if (!tab)
+			return tab;
+	}
+	for (i = 0; i < bmap->n_ineq; ++i) {
+		if (isl_tab_add_ineq(tab, bmap->ineq[i]) < 0)
+			goto error;
+		if (tab->empty)
+			goto done;
+	}
+done:
+	if (track && isl_tab_track_bmap(tab, isl_basic_map_copy(bmap)) < 0)
+		goto error;
+	return tab;
+error:
+	isl_tab_free(tab);
+	return NULL;
+}
+
+__isl_give struct isl_tab *isl_tab_from_basic_set(
+	__isl_keep isl_basic_set *bset, int track)
+{
+	return isl_tab_from_basic_map(bset, track);
+}
+
+/* Construct a tableau corresponding to the recession cone of "bset".
+ */
+struct isl_tab *isl_tab_from_recession_cone(__isl_keep isl_basic_set *bset,
+	int parametric)
+{
+	isl_int cst;
+	int i;
+	struct isl_tab *tab;
+	unsigned offset = 0;
+
+	if (!bset)
+		return NULL;
+	if (parametric)
+		offset = isl_basic_set_dim(bset, isl_dim_param);
+	tab = isl_tab_alloc(bset->ctx, bset->n_eq + bset->n_ineq,
+				isl_basic_set_total_dim(bset) - offset, 0);
+	if (!tab)
+		return NULL;
+	tab->rational = ISL_F_ISSET(bset, ISL_BASIC_SET_RATIONAL);
+	tab->cone = 1;
+
+	isl_int_init(cst);
+	for (i = 0; i < bset->n_eq; ++i) {
+		isl_int_swap(bset->eq[i][offset], cst);
+		if (offset > 0) {
+			if (isl_tab_add_eq(tab, bset->eq[i] + offset) < 0)
+				goto error;
+		} else
+			tab = add_eq(tab, bset->eq[i]);
+		isl_int_swap(bset->eq[i][offset], cst);
+		if (!tab)
+			goto done;
+	}
+	for (i = 0; i < bset->n_ineq; ++i) {
+		int r;
+		isl_int_swap(bset->ineq[i][offset], cst);
+		r = isl_tab_add_row(tab, bset->ineq[i] + offset);
+		isl_int_swap(bset->ineq[i][offset], cst);
+		if (r < 0)
+			goto error;
+		tab->con[r].is_nonneg = 1;
+		if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
+			goto error;
+	}
+done:
+	isl_int_clear(cst);
+	return tab;
+error:
+	isl_int_clear(cst);
+	isl_tab_free(tab);
+	return NULL;
+}
+
+/* Assuming "tab" is the tableau of a cone, check if the cone is
+ * bounded, i.e., if it is empty or only contains the origin.
+ */
+int isl_tab_cone_is_bounded(struct isl_tab *tab)
+{
+	int i;
+
+	if (!tab)
+		return -1;
+	if (tab->empty)
+		return 1;
+	if (tab->n_dead == tab->n_col)
+		return 1;
+
+	for (;;) {
+		for (i = tab->n_redundant; i < tab->n_row; ++i) {
+			struct isl_tab_var *var;
+			int sgn;
+			var = isl_tab_var_from_row(tab, i);
+			if (!var->is_nonneg)
+				continue;
+			sgn = sign_of_max(tab, var);
+			if (sgn < -1)
+				return -1;
+			if (sgn != 0)
+				return 0;
+			if (close_row(tab, var) < 0)
+				return -1;
+			break;
+		}
+		if (tab->n_dead == tab->n_col)
+			return 1;
+		if (i == tab->n_row)
+			return 0;
+	}
+}
+
+int isl_tab_sample_is_integer(struct isl_tab *tab)
+{
+	int i;
+
+	if (!tab)
+		return -1;
+
+	for (i = 0; i < tab->n_var; ++i) {
+		int row;
+		if (!tab->var[i].is_row)
+			continue;
+		row = tab->var[i].index;
+		if (!isl_int_is_divisible_by(tab->mat->row[row][1],
+						tab->mat->row[row][0]))
+			return 0;
+	}
+	return 1;
+}
+
+static struct isl_vec *extract_integer_sample(struct isl_tab *tab)
+{
+	int i;
+	struct isl_vec *vec;
+
+	vec = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_var);
+	if (!vec)
+		return NULL;
+
+	isl_int_set_si(vec->block.data[0], 1);
+	for (i = 0; i < tab->n_var; ++i) {
+		if (!tab->var[i].is_row)
+			isl_int_set_si(vec->block.data[1 + i], 0);
+		else {
+			int row = tab->var[i].index;
+			isl_int_divexact(vec->block.data[1 + i],
+				tab->mat->row[row][1], tab->mat->row[row][0]);
+		}
+	}
+
+	return vec;
+}
+
+struct isl_vec *isl_tab_get_sample_value(struct isl_tab *tab)
+{
+	int i;
+	struct isl_vec *vec;
+	isl_int m;
+
+	if (!tab)
+		return NULL;
+
+	vec = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_var);
+	if (!vec)
+		return NULL;
+
+	isl_int_init(m);
+
+	isl_int_set_si(vec->block.data[0], 1);
+	for (i = 0; i < tab->n_var; ++i) {
+		int row;
+		if (!tab->var[i].is_row) {
+			isl_int_set_si(vec->block.data[1 + i], 0);
+			continue;
+		}
+		row = tab->var[i].index;
+		isl_int_gcd(m, vec->block.data[0], tab->mat->row[row][0]);
+		isl_int_divexact(m, tab->mat->row[row][0], m);
+		isl_seq_scale(vec->block.data, vec->block.data, m, 1 + i);
+		isl_int_divexact(m, vec->block.data[0], tab->mat->row[row][0]);
+		isl_int_mul(vec->block.data[1 + i], m, tab->mat->row[row][1]);
+	}
+	vec = isl_vec_normalize(vec);
+
+	isl_int_clear(m);
+	return vec;
+}
+
+/* Update "bmap" based on the results of the tableau "tab".
+ * In particular, implicit equalities are made explicit, redundant constraints
+ * are removed and if the sample value happens to be integer, it is stored
+ * in "bmap" (unless "bmap" already had an integer sample).
+ *
+ * The tableau is assumed to have been created from "bmap" using
+ * isl_tab_from_basic_map.
+ */
+struct isl_basic_map *isl_basic_map_update_from_tab(struct isl_basic_map *bmap,
+	struct isl_tab *tab)
+{
+	int i;
+	unsigned n_eq;
+
+	if (!bmap)
+		return NULL;
+	if (!tab)
+		return bmap;
+
+	n_eq = tab->n_eq;
+	if (tab->empty)
+		bmap = isl_basic_map_set_to_empty(bmap);
+	else
+		for (i = bmap->n_ineq - 1; i >= 0; --i) {
+			if (isl_tab_is_equality(tab, n_eq + i))
+				isl_basic_map_inequality_to_equality(bmap, i);
+			else if (isl_tab_is_redundant(tab, n_eq + i))
+				isl_basic_map_drop_inequality(bmap, i);
+		}
+	if (bmap->n_eq != n_eq)
+		isl_basic_map_gauss(bmap, NULL);
+	if (!tab->rational &&
+	    !bmap->sample && isl_tab_sample_is_integer(tab))
+		bmap->sample = extract_integer_sample(tab);
+	return bmap;
+}
+
+struct isl_basic_set *isl_basic_set_update_from_tab(struct isl_basic_set *bset,
+	struct isl_tab *tab)
+{
+	return (struct isl_basic_set *)isl_basic_map_update_from_tab(
+		(struct isl_basic_map *)bset, tab);
+}
+
+/* Given a non-negative variable "var", add a new non-negative variable
+ * that is the opposite of "var", ensuring that var can only attain the
+ * value zero.
+ * If var = n/d is a row variable, then the new variable = -n/d.
+ * If var is a column variables, then the new variable = -var.
+ * If the new variable cannot attain non-negative values, then
+ * the resulting tableau is empty.
+ * Otherwise, we know the value will be zero and we close the row.
+ */
+static int cut_to_hyperplane(struct isl_tab *tab, struct isl_tab_var *var)
+{
+	unsigned r;
+	isl_int *row;
+	int sgn;
+	unsigned off = 2 + tab->M;
+
+	if (var->is_zero)
+		return 0;
+	isl_assert(tab->mat->ctx, !var->is_redundant, return -1);
+	isl_assert(tab->mat->ctx, var->is_nonneg, return -1);
+
+	if (isl_tab_extend_cons(tab, 1) < 0)
+		return -1;
+
+	r = tab->n_con;
+	tab->con[r].index = tab->n_row;
+	tab->con[r].is_row = 1;
+	tab->con[r].is_nonneg = 0;
+	tab->con[r].is_zero = 0;
+	tab->con[r].is_redundant = 0;
+	tab->con[r].frozen = 0;
+	tab->con[r].negated = 0;
+	tab->row_var[tab->n_row] = ~r;
+	row = tab->mat->row[tab->n_row];
+
+	if (var->is_row) {
+		isl_int_set(row[0], tab->mat->row[var->index][0]);
+		isl_seq_neg(row + 1,
+			    tab->mat->row[var->index] + 1, 1 + tab->n_col);
+	} else {
+		isl_int_set_si(row[0], 1);
+		isl_seq_clr(row + 1, 1 + tab->n_col);
+		isl_int_set_si(row[off + var->index], -1);
+	}
+
+	tab->n_row++;
+	tab->n_con++;
+	if (isl_tab_push_var(tab, isl_tab_undo_allocate, &tab->con[r]) < 0)
+		return -1;
+
+	sgn = sign_of_max(tab, &tab->con[r]);
+	if (sgn < -1)
+		return -1;
+	if (sgn < 0) {
+		if (isl_tab_mark_empty(tab) < 0)
+			return -1;
+		return 0;
+	}
+	tab->con[r].is_nonneg = 1;
+	if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
+		return -1;
+	/* sgn == 0 */
+	if (close_row(tab, &tab->con[r]) < 0)
+		return -1;
+
+	return 0;
+}
+
+/* Given a tableau "tab" and an inequality constraint "con" of the tableau,
+ * relax the inequality by one.  That is, the inequality r >= 0 is replaced
+ * by r' = r + 1 >= 0.
+ * If r is a row variable, we simply increase the constant term by one
+ * (taking into account the denominator).
+ * If r is a column variable, then we need to modify each row that
+ * refers to r = r' - 1 by substituting this equality, effectively
+ * subtracting the coefficient of the column from the constant.
+ * We should only do this if the minimum is manifestly unbounded,
+ * however.  Otherwise, we may end up with negative sample values
+ * for non-negative variables.
+ * So, if r is a column variable with a minimum that is not
+ * manifestly unbounded, then we need to move it to a row.
+ * However, the sample value of this row may be negative,
+ * even after the relaxation, so we need to restore it.
+ * We therefore prefer to pivot a column up to a row, if possible.
+ */
+struct isl_tab *isl_tab_relax(struct isl_tab *tab, int con)
+{
+	struct isl_tab_var *var;
+	unsigned off = 2 + tab->M;
+
+	if (!tab)
+		return NULL;
+
+	var = &tab->con[con];
+
+	if (var->is_row && (var->index < 0 || var->index < tab->n_redundant))
+		isl_die(tab->mat->ctx, isl_error_invalid,
+			"cannot relax redundant constraint", goto error);
+	if (!var->is_row && (var->index < 0 || var->index < tab->n_dead))
+		isl_die(tab->mat->ctx, isl_error_invalid,
+			"cannot relax dead constraint", goto error);
+
+	if (!var->is_row && !max_is_manifestly_unbounded(tab, var))
+		if (to_row(tab, var, 1) < 0)
+			goto error;
+	if (!var->is_row && !min_is_manifestly_unbounded(tab, var))
+		if (to_row(tab, var, -1) < 0)
+			goto error;
+
+	if (var->is_row) {
+		isl_int_add(tab->mat->row[var->index][1],
+		    tab->mat->row[var->index][1], tab->mat->row[var->index][0]);
+		if (restore_row(tab, var) < 0)
+			goto error;
+	} else {
+		int i;
+
+		for (i = 0; i < tab->n_row; ++i) {
+			if (isl_int_is_zero(tab->mat->row[i][off + var->index]))
+				continue;
+			isl_int_sub(tab->mat->row[i][1], tab->mat->row[i][1],
+			    tab->mat->row[i][off + var->index]);
+		}
+
+	}
+
+	if (isl_tab_push_var(tab, isl_tab_undo_relax, var) < 0)
+		goto error;
+
+	return tab;
+error:
+	isl_tab_free(tab);
+	return NULL;
+}
+
+/* Remove the sign constraint from constraint "con".
+ *
+ * If the constraint variable was originally marked non-negative,
+ * then we make sure we mark it non-negative again during rollback.
+ */
+int isl_tab_unrestrict(struct isl_tab *tab, int con)
+{
+	struct isl_tab_var *var;
+
+	if (!tab)
+		return -1;
+
+	var = &tab->con[con];
+	if (!var->is_nonneg)
+		return 0;
+
+	var->is_nonneg = 0;
+	if (isl_tab_push_var(tab, isl_tab_undo_unrestrict, var) < 0)
+		return -1;
+
+	return 0;
+}
+
+int isl_tab_select_facet(struct isl_tab *tab, int con)
+{
+	if (!tab)
+		return -1;
+
+	return cut_to_hyperplane(tab, &tab->con[con]);
+}
+
+static int may_be_equality(struct isl_tab *tab, int row)
+{
+	return tab->rational ? isl_int_is_zero(tab->mat->row[row][1])
+			     : isl_int_lt(tab->mat->row[row][1],
+					    tab->mat->row[row][0]);
+}
+
+/* Check for (near) equalities among the constraints.
+ * A constraint is an equality if it is non-negative and if
+ * its maximal value is either
+ *	- zero (in case of rational tableaus), or
+ *	- strictly less than 1 (in case of integer tableaus)
+ *
+ * We first mark all non-redundant and non-dead variables that
+ * are not frozen and not obviously not an equality.
+ * Then we iterate over all marked variables if they can attain
+ * any values larger than zero or at least one.
+ * If the maximal value is zero, we mark any column variables
+ * that appear in the row as being zero and mark the row as being redundant.
+ * Otherwise, if the maximal value is strictly less than one (and the
+ * tableau is integer), then we restrict the value to being zero
+ * by adding an opposite non-negative variable.
+ */
+int isl_tab_detect_implicit_equalities(struct isl_tab *tab)
+{
+	int i;
+	unsigned n_marked;
+
+	if (!tab)
+		return -1;
+	if (tab->empty)
+		return 0;
+	if (tab->n_dead == tab->n_col)
+		return 0;
+
+	n_marked = 0;
+	for (i = tab->n_redundant; i < tab->n_row; ++i) {
+		struct isl_tab_var *var = isl_tab_var_from_row(tab, i);
+		var->marked = !var->frozen && var->is_nonneg &&
+			may_be_equality(tab, i);
+		if (var->marked)
+			n_marked++;
+	}
+	for (i = tab->n_dead; i < tab->n_col; ++i) {
+		struct isl_tab_var *var = var_from_col(tab, i);
+		var->marked = !var->frozen && var->is_nonneg;
+		if (var->marked)
+			n_marked++;
+	}
+	while (n_marked) {
+		struct isl_tab_var *var;
+		int sgn;
+		for (i = tab->n_redundant; i < tab->n_row; ++i) {
+			var = isl_tab_var_from_row(tab, i);
+			if (var->marked)
+				break;
+		}
+		if (i == tab->n_row) {
+			for (i = tab->n_dead; i < tab->n_col; ++i) {
+				var = var_from_col(tab, i);
+				if (var->marked)
+					break;
+			}
+			if (i == tab->n_col)
+				break;
+		}
+		var->marked = 0;
+		n_marked--;
+		sgn = sign_of_max(tab, var);
+		if (sgn < 0)
+			return -1;
+		if (sgn == 0) {
+			if (close_row(tab, var) < 0)
+				return -1;
+		} else if (!tab->rational && !at_least_one(tab, var)) {
+			if (cut_to_hyperplane(tab, var) < 0)
+				return -1;
+			return isl_tab_detect_implicit_equalities(tab);
+		}
+		for (i = tab->n_redundant; i < tab->n_row; ++i) {
+			var = isl_tab_var_from_row(tab, i);
+			if (!var->marked)
+				continue;
+			if (may_be_equality(tab, i))
+				continue;
+			var->marked = 0;
+			n_marked--;
+		}
+	}
+
+	return 0;
+}
+
+/* Update the element of row_var or col_var that corresponds to
+ * constraint tab->con[i] to a move from position "old" to position "i".
+ */
+static int update_con_after_move(struct isl_tab *tab, int i, int old)
+{
+	int *p;
+	int index;
+
+	index = tab->con[i].index;
+	if (index == -1)
+		return 0;
+	p = tab->con[i].is_row ? tab->row_var : tab->col_var;
+	if (p[index] != ~old)
+		isl_die(tab->mat->ctx, isl_error_internal,
+			"broken internal state", return -1);
+	p[index] = ~i;
+
+	return 0;
+}
+
+/* Rotate the "n" constraints starting at "first" to the right,
+ * putting the last constraint in the position of the first constraint.
+ */
+static int rotate_constraints(struct isl_tab *tab, int first, int n)
+{
+	int i, last;
+	struct isl_tab_var var;
+
+	if (n <= 1)
+		return 0;
+
+	last = first + n - 1;
+	var = tab->con[last];
+	for (i = last; i > first; --i) {
+		tab->con[i] = tab->con[i - 1];
+		if (update_con_after_move(tab, i, i - 1) < 0)
+			return -1;
+	}
+	tab->con[first] = var;
+	if (update_con_after_move(tab, first, last) < 0)
+		return -1;
+
+	return 0;
+}
+
+/* Make the equalities that are implicit in "bmap" but that have been
+ * detected in the corresponding "tab" explicit in "bmap" and update
+ * "tab" to reflect the new order of the constraints.
+ *
+ * In particular, if inequality i is an implicit equality then
+ * isl_basic_map_inequality_to_equality will move the inequality
+ * in front of the other equality and it will move the last inequality
+ * in the position of inequality i.
+ * In the tableau, the inequalities of "bmap" are stored after the equalities
+ * and so the original order
+ *
+ *		E E E E E A A A I B B B B L
+ *
+ * is changed into
+ *
+ *		I E E E E E A A A L B B B B
+ *
+ * where I is the implicit equality, the E are equalities,
+ * the A inequalities before I, the B inequalities after I and
+ * L the last inequality.
+ * We therefore need to rotate to the right two sets of constraints,
+ * those up to and including I and those after I.
+ *
+ * If "tab" contains any constraints that are not in "bmap" then they
+ * appear after those in "bmap" and they should be left untouched.
+ *
+ * Note that this function leaves "bmap" in a temporary state
+ * as it does not call isl_basic_map_gauss.  Calling this function
+ * is the responsibility of the caller.
+ */
+__isl_give isl_basic_map *isl_tab_make_equalities_explicit(struct isl_tab *tab,
+	__isl_take isl_basic_map *bmap)
+{
+	int i;
+
+	if (!tab || !bmap)
+		return isl_basic_map_free(bmap);
+	if (tab->empty)
+		return bmap;
+
+	for (i = bmap->n_ineq - 1; i >= 0; --i) {
+		if (!isl_tab_is_equality(tab, bmap->n_eq + i))
+			continue;
+		isl_basic_map_inequality_to_equality(bmap, i);
+		if (rotate_constraints(tab, 0, tab->n_eq + i + 1) < 0)
+			return isl_basic_map_free(bmap);
+		if (rotate_constraints(tab, tab->n_eq + i + 1,
+					bmap->n_ineq - i) < 0)
+			return isl_basic_map_free(bmap);
+		tab->n_eq++;
+	}
+
+	return bmap;
+}
+
+static int con_is_redundant(struct isl_tab *tab, struct isl_tab_var *var)
+{
+	if (!tab)
+		return -1;
+	if (tab->rational) {
+		int sgn = sign_of_min(tab, var);
+		if (sgn < -1)
+			return -1;
+		return sgn >= 0;
+	} else {
+		int irred = isl_tab_min_at_most_neg_one(tab, var);
+		if (irred < 0)
+			return -1;
+		return !irred;
+	}
+}
+
+/* Check for (near) redundant constraints.
+ * A constraint is redundant if it is non-negative and if
+ * its minimal value (temporarily ignoring the non-negativity) is either
+ *	- zero (in case of rational tableaus), or
+ *	- strictly larger than -1 (in case of integer tableaus)
+ *
+ * We first mark all non-redundant and non-dead variables that
+ * are not frozen and not obviously negatively unbounded.
+ * Then we iterate over all marked variables if they can attain
+ * any values smaller than zero or at most negative one.
+ * If not, we mark the row as being redundant (assuming it hasn't
+ * been detected as being obviously redundant in the mean time).
+ */
+int isl_tab_detect_redundant(struct isl_tab *tab)
+{
+	int i;
+	unsigned n_marked;
+
+	if (!tab)
+		return -1;
+	if (tab->empty)
+		return 0;
+	if (tab->n_redundant == tab->n_row)
+		return 0;
+
+	n_marked = 0;
+	for (i = tab->n_redundant; i < tab->n_row; ++i) {
+		struct isl_tab_var *var = isl_tab_var_from_row(tab, i);
+		var->marked = !var->frozen && var->is_nonneg;
+		if (var->marked)
+			n_marked++;
+	}
+	for (i = tab->n_dead; i < tab->n_col; ++i) {
+		struct isl_tab_var *var = var_from_col(tab, i);
+		var->marked = !var->frozen && var->is_nonneg &&
+			!min_is_manifestly_unbounded(tab, var);
+		if (var->marked)
+			n_marked++;
+	}
+	while (n_marked) {
+		struct isl_tab_var *var;
+		int red;
+		for (i = tab->n_redundant; i < tab->n_row; ++i) {
+			var = isl_tab_var_from_row(tab, i);
+			if (var->marked)
+				break;
+		}
+		if (i == tab->n_row) {
+			for (i = tab->n_dead; i < tab->n_col; ++i) {
+				var = var_from_col(tab, i);
+				if (var->marked)
+					break;
+			}
+			if (i == tab->n_col)
+				break;
+		}
+		var->marked = 0;
+		n_marked--;
+		red = con_is_redundant(tab, var);
+		if (red < 0)
+			return -1;
+		if (red && !var->is_redundant)
+			if (isl_tab_mark_redundant(tab, var->index) < 0)
+				return -1;
+		for (i = tab->n_dead; i < tab->n_col; ++i) {
+			var = var_from_col(tab, i);
+			if (!var->marked)
+				continue;
+			if (!min_is_manifestly_unbounded(tab, var))
+				continue;
+			var->marked = 0;
+			n_marked--;
+		}
+	}
+
+	return 0;
+}
+
+int isl_tab_is_equality(struct isl_tab *tab, int con)
+{
+	int row;
+	unsigned off;
+
+	if (!tab)
+		return -1;
+	if (tab->con[con].is_zero)
+		return 1;
+	if (tab->con[con].is_redundant)
+		return 0;
+	if (!tab->con[con].is_row)
+		return tab->con[con].index < tab->n_dead;
+
+	row = tab->con[con].index;
+
+	off = 2 + tab->M;
+	return isl_int_is_zero(tab->mat->row[row][1]) &&
+		(!tab->M || isl_int_is_zero(tab->mat->row[row][2])) &&
+		isl_seq_first_non_zero(tab->mat->row[row] + off + tab->n_dead,
+					tab->n_col - tab->n_dead) == -1;
+}
+
+/* Return the minimal value of the affine expression "f" with denominator
+ * "denom" in *opt, *opt_denom, assuming the tableau is not empty and
+ * the expression cannot attain arbitrarily small values.
+ * If opt_denom is NULL, then *opt is rounded up to the nearest integer.
+ * The return value reflects the nature of the result (empty, unbounded,
+ * minimal value returned in *opt).
+ */
+enum isl_lp_result isl_tab_min(struct isl_tab *tab,
+	isl_int *f, isl_int denom, isl_int *opt, isl_int *opt_denom,
+	unsigned flags)
+{
+	int r;
+	enum isl_lp_result res = isl_lp_ok;
+	struct isl_tab_var *var;
+	struct isl_tab_undo *snap;
+
+	if (!tab)
+		return isl_lp_error;
+
+	if (tab->empty)
+		return isl_lp_empty;
+
+	snap = isl_tab_snap(tab);
+	r = isl_tab_add_row(tab, f);
+	if (r < 0)
+		return isl_lp_error;
+	var = &tab->con[r];
+	for (;;) {
+		int row, col;
+		find_pivot(tab, var, var, -1, &row, &col);
+		if (row == var->index) {
+			res = isl_lp_unbounded;
+			break;
+		}
+		if (row == -1)
+			break;
+		if (isl_tab_pivot(tab, row, col) < 0)
+			return isl_lp_error;
+	}
+	isl_int_mul(tab->mat->row[var->index][0],
+		    tab->mat->row[var->index][0], denom);
+	if (ISL_FL_ISSET(flags, ISL_TAB_SAVE_DUAL)) {
+		int i;
+
+		isl_vec_free(tab->dual);
+		tab->dual = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_con);
+		if (!tab->dual)
+			return isl_lp_error;
+		isl_int_set(tab->dual->el[0], tab->mat->row[var->index][0]);
+		for (i = 0; i < tab->n_con; ++i) {
+			int pos;
+			if (tab->con[i].is_row) {
+				isl_int_set_si(tab->dual->el[1 + i], 0);
+				continue;
+			}
+			pos = 2 + tab->M + tab->con[i].index;
+			if (tab->con[i].negated)
+				isl_int_neg(tab->dual->el[1 + i],
+					    tab->mat->row[var->index][pos]);
+			else
+				isl_int_set(tab->dual->el[1 + i],
+					    tab->mat->row[var->index][pos]);
+		}
+	}
+	if (opt && res == isl_lp_ok) {
+		if (opt_denom) {
+			isl_int_set(*opt, tab->mat->row[var->index][1]);
+			isl_int_set(*opt_denom, tab->mat->row[var->index][0]);
+		} else
+			isl_int_cdiv_q(*opt, tab->mat->row[var->index][1],
+					     tab->mat->row[var->index][0]);
+	}
+	if (isl_tab_rollback(tab, snap) < 0)
+		return isl_lp_error;
+	return res;
+}
+
+int isl_tab_is_redundant(struct isl_tab *tab, int con)
+{
+	if (!tab)
+		return -1;
+	if (tab->con[con].is_zero)
+		return 0;
+	if (tab->con[con].is_redundant)
+		return 1;
+	return tab->con[con].is_row && tab->con[con].index < tab->n_redundant;
+}
+
+/* Take a snapshot of the tableau that can be restored by s call to
+ * isl_tab_rollback.
+ */
+struct isl_tab_undo *isl_tab_snap(struct isl_tab *tab)
+{
+	if (!tab)
+		return NULL;
+	tab->need_undo = 1;
+	return tab->top;
+}
+
+/* Undo the operation performed by isl_tab_relax.
+ */
+static int unrelax(struct isl_tab *tab, struct isl_tab_var *var) WARN_UNUSED;
+static int unrelax(struct isl_tab *tab, struct isl_tab_var *var)
+{
+	unsigned off = 2 + tab->M;
+
+	if (!var->is_row && !max_is_manifestly_unbounded(tab, var))
+		if (to_row(tab, var, 1) < 0)
+			return -1;
+
+	if (var->is_row) {
+		isl_int_sub(tab->mat->row[var->index][1],
+		    tab->mat->row[var->index][1], tab->mat->row[var->index][0]);
+		if (var->is_nonneg) {
+			int sgn = restore_row(tab, var);
+			isl_assert(tab->mat->ctx, sgn >= 0, return -1);
+		}
+	} else {
+		int i;
+
+		for (i = 0; i < tab->n_row; ++i) {
+			if (isl_int_is_zero(tab->mat->row[i][off + var->index]))
+				continue;
+			isl_int_add(tab->mat->row[i][1], tab->mat->row[i][1],
+			    tab->mat->row[i][off + var->index]);
+		}
+
+	}
+
+	return 0;
+}
+
+/* Undo the operation performed by isl_tab_unrestrict.
+ *
+ * In particular, mark the variable as being non-negative and make
+ * sure the sample value respects this constraint.
+ */
+static int ununrestrict(struct isl_tab *tab, struct isl_tab_var *var)
+{
+	var->is_nonneg = 1;
+
+	if (var->is_row && restore_row(tab, var) < -1)
+		return -1;
+
+	return 0;
+}
+
+static int perform_undo_var(struct isl_tab *tab, struct isl_tab_undo *undo) WARN_UNUSED;
+static int perform_undo_var(struct isl_tab *tab, struct isl_tab_undo *undo)
+{
+	struct isl_tab_var *var = var_from_index(tab, undo->u.var_index);
+	switch (undo->type) {
+	case isl_tab_undo_nonneg:
+		var->is_nonneg = 0;
+		break;
+	case isl_tab_undo_redundant:
+		var->is_redundant = 0;
+		tab->n_redundant--;
+		restore_row(tab, isl_tab_var_from_row(tab, tab->n_redundant));
+		break;
+	case isl_tab_undo_freeze:
+		var->frozen = 0;
+		break;
+	case isl_tab_undo_zero:
+		var->is_zero = 0;
+		if (!var->is_row)
+			tab->n_dead--;
+		break;
+	case isl_tab_undo_allocate:
+		if (undo->u.var_index >= 0) {
+			isl_assert(tab->mat->ctx, !var->is_row, return -1);
+			drop_col(tab, var->index);
+			break;
+		}
+		if (!var->is_row) {
+			if (!max_is_manifestly_unbounded(tab, var)) {
+				if (to_row(tab, var, 1) < 0)
+					return -1;
+			} else if (!min_is_manifestly_unbounded(tab, var)) {
+				if (to_row(tab, var, -1) < 0)
+					return -1;
+			} else
+				if (to_row(tab, var, 0) < 0)
+					return -1;
+		}
+		drop_row(tab, var->index);
+		break;
+	case isl_tab_undo_relax:
+		return unrelax(tab, var);
+	case isl_tab_undo_unrestrict:
+		return ununrestrict(tab, var);
+	default:
+		isl_die(tab->mat->ctx, isl_error_internal,
+			"perform_undo_var called on invalid undo record",
+			return -1);
+	}
+
+	return 0;
+}
+
+/* Restore the tableau to the state where the basic variables
+ * are those in "col_var".
+ * We first construct a list of variables that are currently in
+ * the basis, but shouldn't.  Then we iterate over all variables
+ * that should be in the basis and for each one that is currently
+ * not in the basis, we exchange it with one of the elements of the
+ * list constructed before.
+ * We can always find an appropriate variable to pivot with because
+ * the current basis is mapped to the old basis by a non-singular
+ * matrix and so we can never end up with a zero row.
+ */
+static int restore_basis(struct isl_tab *tab, int *col_var)
+{
+	int i, j;
+	int n_extra = 0;
+	int *extra = NULL;	/* current columns that contain bad stuff */
+	unsigned off = 2 + tab->M;
+
+	extra = isl_alloc_array(tab->mat->ctx, int, tab->n_col);
+	if (tab->n_col && !extra)
+		goto error;
+	for (i = 0; i < tab->n_col; ++i) {
+		for (j = 0; j < tab->n_col; ++j)
+			if (tab->col_var[i] == col_var[j])
+				break;
+		if (j < tab->n_col)
+			continue;
+		extra[n_extra++] = i;
+	}
+	for (i = 0; i < tab->n_col && n_extra > 0; ++i) {
+		struct isl_tab_var *var;
+		int row;
+
+		for (j = 0; j < tab->n_col; ++j)
+			if (col_var[i] == tab->col_var[j])
+				break;
+		if (j < tab->n_col)
+			continue;
+		var = var_from_index(tab, col_var[i]);
+		row = var->index;
+		for (j = 0; j < n_extra; ++j)
+			if (!isl_int_is_zero(tab->mat->row[row][off+extra[j]]))
+				break;
+		isl_assert(tab->mat->ctx, j < n_extra, goto error);
+		if (isl_tab_pivot(tab, row, extra[j]) < 0)
+			goto error;
+		extra[j] = extra[--n_extra];
+	}
+
+	free(extra);
+	return 0;
+error:
+	free(extra);
+	return -1;
+}
+
+/* Remove all samples with index n or greater, i.e., those samples
+ * that were added since we saved this number of samples in
+ * isl_tab_save_samples.
+ */
+static void drop_samples_since(struct isl_tab *tab, int n)
+{
+	int i;
+
+	for (i = tab->n_sample - 1; i >= 0 && tab->n_sample > n; --i) {
+		if (tab->sample_index[i] < n)
+			continue;
+
+		if (i != tab->n_sample - 1) {
+			int t = tab->sample_index[tab->n_sample-1];
+			tab->sample_index[tab->n_sample-1] = tab->sample_index[i];
+			tab->sample_index[i] = t;
+			isl_mat_swap_rows(tab->samples, tab->n_sample-1, i);
+		}
+		tab->n_sample--;
+	}
+}
+
+static int perform_undo(struct isl_tab *tab, struct isl_tab_undo *undo) WARN_UNUSED;
+static int perform_undo(struct isl_tab *tab, struct isl_tab_undo *undo)
+{
+	switch (undo->type) {
+	case isl_tab_undo_empty:
+		tab->empty = 0;
+		break;
+	case isl_tab_undo_nonneg:
+	case isl_tab_undo_redundant:
+	case isl_tab_undo_freeze:
+	case isl_tab_undo_zero:
+	case isl_tab_undo_allocate:
+	case isl_tab_undo_relax:
+	case isl_tab_undo_unrestrict:
+		return perform_undo_var(tab, undo);
+	case isl_tab_undo_bmap_eq:
+		return isl_basic_map_free_equality(tab->bmap, 1);
+	case isl_tab_undo_bmap_ineq:
+		return isl_basic_map_free_inequality(tab->bmap, 1);
+	case isl_tab_undo_bmap_div:
+		if (isl_basic_map_free_div(tab->bmap, 1) < 0)
+			return -1;
+		if (tab->samples)
+			tab->samples->n_col--;
+		break;
+	case isl_tab_undo_saved_basis:
+		if (restore_basis(tab, undo->u.col_var) < 0)
+			return -1;
+		break;
+	case isl_tab_undo_drop_sample:
+		tab->n_outside--;
+		break;
+	case isl_tab_undo_saved_samples:
+		drop_samples_since(tab, undo->u.n);
+		break;
+	case isl_tab_undo_callback:
+		return undo->u.callback->run(undo->u.callback);
+	default:
+		isl_assert(tab->mat->ctx, 0, return -1);
+	}
+	return 0;
+}
+
+/* Return the tableau to the state it was in when the snapshot "snap"
+ * was taken.
+ */
+int isl_tab_rollback(struct isl_tab *tab, struct isl_tab_undo *snap)
+{
+	struct isl_tab_undo *undo, *next;
+
+	if (!tab)
+		return -1;
+
+	tab->in_undo = 1;
+	for (undo = tab->top; undo && undo != &tab->bottom; undo = next) {
+		next = undo->next;
+		if (undo == snap)
+			break;
+		if (perform_undo(tab, undo) < 0) {
+			tab->top = undo;
+			free_undo(tab);
+			tab->in_undo = 0;
+			return -1;
+		}
+		free_undo_record(undo);
+	}
+	tab->in_undo = 0;
+	tab->top = undo;
+	if (!undo)
+		return -1;
+	return 0;
+}
+
+/* The given row "row" represents an inequality violated by all
+ * points in the tableau.  Check for some special cases of such
+ * separating constraints.
+ * In particular, if the row has been reduced to the constant -1,
+ * then we know the inequality is adjacent (but opposite) to
+ * an equality in the tableau.
+ * If the row has been reduced to r = c*(-1 -r'), with r' an inequality
+ * of the tableau and c a positive constant, then the inequality
+ * is adjacent (but opposite) to the inequality r'.
+ */
+static enum isl_ineq_type separation_type(struct isl_tab *tab, unsigned row)
+{
+	int pos;
+	unsigned off = 2 + tab->M;
+
+	if (tab->rational)
+		return isl_ineq_separate;
+
+	if (!isl_int_is_one(tab->mat->row[row][0]))
+		return isl_ineq_separate;
+
+	pos = isl_seq_first_non_zero(tab->mat->row[row] + off + tab->n_dead,
+					tab->n_col - tab->n_dead);
+	if (pos == -1) {
+		if (isl_int_is_negone(tab->mat->row[row][1]))
+			return isl_ineq_adj_eq;
+		else
+			return isl_ineq_separate;
+	}
+
+	if (!isl_int_eq(tab->mat->row[row][1],
+			tab->mat->row[row][off + tab->n_dead + pos]))
+		return isl_ineq_separate;
+
+	pos = isl_seq_first_non_zero(
+			tab->mat->row[row] + off + tab->n_dead + pos + 1,
+			tab->n_col - tab->n_dead - pos - 1);
+
+	return pos == -1 ? isl_ineq_adj_ineq : isl_ineq_separate;
+}
+
+/* Check the effect of inequality "ineq" on the tableau "tab".
+ * The result may be
+ *	isl_ineq_redundant:	satisfied by all points in the tableau
+ *	isl_ineq_separate:	satisfied by no point in the tableau
+ *	isl_ineq_cut:		satisfied by some by not all points
+ *	isl_ineq_adj_eq:	adjacent to an equality
+ *	isl_ineq_adj_ineq:	adjacent to an inequality.
+ */
+enum isl_ineq_type isl_tab_ineq_type(struct isl_tab *tab, isl_int *ineq)
+{
+	enum isl_ineq_type type = isl_ineq_error;
+	struct isl_tab_undo *snap = NULL;
+	int con;
+	int row;
+
+	if (!tab)
+		return isl_ineq_error;
+
+	if (isl_tab_extend_cons(tab, 1) < 0)
+		return isl_ineq_error;
+
+	snap = isl_tab_snap(tab);
+
+	con = isl_tab_add_row(tab, ineq);
+	if (con < 0)
+		goto error;
+
+	row = tab->con[con].index;
+	if (isl_tab_row_is_redundant(tab, row))
+		type = isl_ineq_redundant;
+	else if (isl_int_is_neg(tab->mat->row[row][1]) &&
+		 (tab->rational ||
+		    isl_int_abs_ge(tab->mat->row[row][1],
+				   tab->mat->row[row][0]))) {
+		int nonneg = at_least_zero(tab, &tab->con[con]);
+		if (nonneg < 0)
+			goto error;
+		if (nonneg)
+			type = isl_ineq_cut;
+		else
+			type = separation_type(tab, row);
+	} else {
+		int red = con_is_redundant(tab, &tab->con[con]);
+		if (red < 0)
+			goto error;
+		if (!red)
+			type = isl_ineq_cut;
+		else
+			type = isl_ineq_redundant;
+	}
+
+	if (isl_tab_rollback(tab, snap))
+		return isl_ineq_error;
+	return type;
+error:
+	return isl_ineq_error;
+}
+
+int isl_tab_track_bmap(struct isl_tab *tab, __isl_take isl_basic_map *bmap)
+{
+	bmap = isl_basic_map_cow(bmap);
+	if (!tab || !bmap)
+		goto error;
+
+	if (tab->empty) {
+		bmap = isl_basic_map_set_to_empty(bmap);
+		if (!bmap)
+			goto error;
+		tab->bmap = bmap;
+		return 0;
+	}
+
+	isl_assert(tab->mat->ctx, tab->n_eq == bmap->n_eq, goto error);
+	isl_assert(tab->mat->ctx,
+		    tab->n_con == bmap->n_eq + bmap->n_ineq, goto error);
+
+	tab->bmap = bmap;
+
+	return 0;
+error:
+	isl_basic_map_free(bmap);
+	return -1;
+}
+
+int isl_tab_track_bset(struct isl_tab *tab, __isl_take isl_basic_set *bset)
+{
+	return isl_tab_track_bmap(tab, (isl_basic_map *)bset);
+}
+
+__isl_keep isl_basic_set *isl_tab_peek_bset(struct isl_tab *tab)
+{
+	if (!tab)
+		return NULL;
+
+	return (isl_basic_set *)tab->bmap;
+}
+
+static void isl_tab_print_internal(__isl_keep struct isl_tab *tab,
+	FILE *out, int indent)
+{
+	unsigned r, c;
+	int i;
+
+	if (!tab) {
+		fprintf(out, "%*snull tab\n", indent, "");
+		return;
+	}
+	fprintf(out, "%*sn_redundant: %d, n_dead: %d", indent, "",
+		tab->n_redundant, tab->n_dead);
+	if (tab->rational)
+		fprintf(out, ", rational");
+	if (tab->empty)
+		fprintf(out, ", empty");
+	fprintf(out, "\n");
+	fprintf(out, "%*s[", indent, "");
+	for (i = 0; i < tab->n_var; ++i) {
+		if (i)
+			fprintf(out, (i == tab->n_param ||
+				      i == tab->n_var - tab->n_div) ? "; "
+								    : ", ");
+		fprintf(out, "%c%d%s", tab->var[i].is_row ? 'r' : 'c',
+					tab->var[i].index,
+					tab->var[i].is_zero ? " [=0]" :
+					tab->var[i].is_redundant ? " [R]" : "");
+	}
+	fprintf(out, "]\n");
+	fprintf(out, "%*s[", indent, "");
+	for (i = 0; i < tab->n_con; ++i) {
+		if (i)
+			fprintf(out, ", ");
+		fprintf(out, "%c%d%s", tab->con[i].is_row ? 'r' : 'c',
+					tab->con[i].index,
+					tab->con[i].is_zero ? " [=0]" :
+					tab->con[i].is_redundant ? " [R]" : "");
+	}
+	fprintf(out, "]\n");
+	fprintf(out, "%*s[", indent, "");
+	for (i = 0; i < tab->n_row; ++i) {
+		const char *sign = "";
+		if (i)
+			fprintf(out, ", ");
+		if (tab->row_sign) {
+			if (tab->row_sign[i] == isl_tab_row_unknown)
+				sign = "?";
+			else if (tab->row_sign[i] == isl_tab_row_neg)
+				sign = "-";
+			else if (tab->row_sign[i] == isl_tab_row_pos)
+				sign = "+";
+			else
+				sign = "+-";
+		}
+		fprintf(out, "r%d: %d%s%s", i, tab->row_var[i],
+		    isl_tab_var_from_row(tab, i)->is_nonneg ? " [>=0]" : "", sign);
+	}
+	fprintf(out, "]\n");
+	fprintf(out, "%*s[", indent, "");
+	for (i = 0; i < tab->n_col; ++i) {
+		if (i)
+			fprintf(out, ", ");
+		fprintf(out, "c%d: %d%s", i, tab->col_var[i],
+		    var_from_col(tab, i)->is_nonneg ? " [>=0]" : "");
+	}
+	fprintf(out, "]\n");
+	r = tab->mat->n_row;
+	tab->mat->n_row = tab->n_row;
+	c = tab->mat->n_col;
+	tab->mat->n_col = 2 + tab->M + tab->n_col;
+	isl_mat_print_internal(tab->mat, out, indent);
+	tab->mat->n_row = r;
+	tab->mat->n_col = c;
+	if (tab->bmap)
+		isl_basic_map_print_internal(tab->bmap, out, indent);
+}
+
+void isl_tab_dump(__isl_keep struct isl_tab *tab)
+{
+	isl_tab_print_internal(tab, stderr, 0);
+}
diff --git a/polly/lib/External/isl/isl_tab.h b/polly/lib/External/isl/isl_tab.h
new file mode 100644
index 00000000000..cfae0c919f8
--- /dev/null
+++ b/polly/lib/External/isl/isl_tab.h
@@ -0,0 +1,304 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#ifndef ISL_TAB_H
+#define ISL_TAB_H
+
+#include <isl/lp.h>
+#include <isl/map.h>
+#include <isl/mat.h>
+#include <isl/set.h>
+#include <isl_config.h>
+
+struct isl_tab_var {
+	int index;
+	unsigned is_row : 1;
+	unsigned is_nonneg : 1;
+	unsigned is_zero : 1;
+	unsigned is_redundant : 1;
+	unsigned marked : 1;
+	unsigned frozen : 1;
+	unsigned negated : 1;
+};
+
+enum isl_tab_undo_type {
+	isl_tab_undo_bottom,
+	isl_tab_undo_empty,
+	isl_tab_undo_nonneg,
+	isl_tab_undo_redundant,
+	isl_tab_undo_freeze,
+	isl_tab_undo_zero,
+	isl_tab_undo_allocate,
+	isl_tab_undo_relax,
+	isl_tab_undo_unrestrict,
+	isl_tab_undo_bmap_ineq,
+	isl_tab_undo_bmap_eq,
+	isl_tab_undo_bmap_div,
+	isl_tab_undo_saved_basis,
+	isl_tab_undo_drop_sample,
+	isl_tab_undo_saved_samples,
+	isl_tab_undo_callback,
+};
+
+struct isl_tab_callback {
+	int (*run)(struct isl_tab_callback *cb);
+};
+
+union isl_tab_undo_val {
+	int		var_index;
+	int		*col_var;
+	int		n;
+	struct isl_tab_callback	*callback;
+};
+
+struct isl_tab_undo {
+	enum isl_tab_undo_type	type;
+	union isl_tab_undo_val	u;
+	struct isl_tab_undo	*next;
+};
+
+/* The tableau maintains equality relations.
+ * Each column and each row is associated to a variable or a constraint.
+ * The "value" of an inequality constraint is the value of the corresponding
+ * slack variable.
+ * The "row_var" and "col_var" arrays map column and row indices
+ * to indices in the "var" and "con" arrays.  The elements of these
+ * arrays maintain extra information about the variables and the constraints.
+ * Each row expresses the corresponding row variable as an affine expression
+ * of the column variables.
+ * The first two columns in the matrix contain the common denominator of
+ * the row and the numerator of the constant term.
+ * If "M" is set, then the third column represents the "big parameter".
+ * The third (M = 0) or fourth (M = 1) column
+ * in the matrix is called column 0 with respect to the col_var array.
+ * The sample value of the tableau is the value that assigns zero
+ * to all the column variables and the constant term of each affine
+ * expression to the corresponding row variable.
+ * The operations on the tableau maintain the property that the sample
+ * value satisfies the non-negativity constraints (usually on the slack
+ * variables).
+ *
+ * The big parameter represents an arbitrarily big (and divisible)
+ * positive number.  If present, then the sign of a row is determined
+ * lexicographically, with the sign of the big parameter coefficient
+ * considered first.  The big parameter is only used while
+ * solving PILP problems.
+ *
+ * The first n_dead column variables have their values fixed to zero.
+ * The corresponding tab_vars are flagged "is_zero".
+ * Some of the rows that have have zero coefficients in all but
+ * the dead columns are also flagged "is_zero".
+ *
+ * The first n_redundant rows correspond to inequality constraints
+ * that are always satisfied for any value satisfying the non-redundant
+ * rows.  The corresponding tab_vars are flagged "is_redundant".
+ * A row variable that is flagged "is_zero" is also flagged "is_redundant"
+ * since the constraint has been reduced to 0 = 0 and is therefore always
+ * satisfied.
+ *
+ * There are "n_var" variables in total.  The first "n_param" of these
+ * are called parameters and the last "n_div" of these are called divs.
+ * The basic tableau operations makes no distinction between different
+ * kinds of variables.  These special variables are only used while
+ * solving PILP problems.
+ *
+ * Dead columns and redundant rows are detected on the fly.
+ * However, the basic operations do not ensure that all dead columns
+ * or all redundant rows are detected.
+ * isl_tab_detect_implicit_equalities and isl_tab_detect_redundant can be used
+ * to perform and exhaustive search for dead columns and redundant rows.
+ *
+ * The samples matrix contains "n_sample" integer points that have at some
+ * point been elements satisfying the tableau.  The first "n_outside"
+ * of them no longer satisfy the tableau.  They are kept because they
+ * can be reinstated during rollback when the constraint that cut them
+ * out is removed.  These samples are only maintained for the context
+ * tableau while solving PILP problems.
+ *
+ * If "preserve" is set, then we want to keep all constraints in the
+ * tableau, even if they turn out to be redundant.
+ */
+enum isl_tab_row_sign {
+	isl_tab_row_unknown = 0,
+	isl_tab_row_pos,
+	isl_tab_row_neg,
+	isl_tab_row_any,
+};
+struct isl_tab {
+	struct isl_mat *mat;
+
+	unsigned n_row;
+	unsigned n_col;
+	unsigned n_dead;
+	unsigned n_redundant;
+
+	unsigned n_var;
+	unsigned n_param;
+	unsigned n_div;
+	unsigned max_var;
+	unsigned n_con;
+	unsigned n_eq;
+	unsigned max_con;
+	struct isl_tab_var *var;
+	struct isl_tab_var *con;
+	int *row_var;	/* v >= 0 -> var v;	v < 0 -> con ~v */
+	int *col_var;	/* v >= 0 -> var v;	v < 0 -> con ~v */
+	enum isl_tab_row_sign *row_sign;
+
+	struct isl_tab_undo bottom;
+	struct isl_tab_undo *top;
+
+	struct isl_vec *dual;
+	struct isl_basic_map *bmap;
+
+	unsigned n_sample;
+	unsigned n_outside;
+	int *sample_index;
+	struct isl_mat *samples;
+
+	int n_zero;
+	int n_unbounded;
+	struct isl_mat *basis;
+
+	int (*conflict)(int con, void *user);
+	void *conflict_user;
+
+	unsigned strict_redundant : 1;
+	unsigned need_undo : 1;
+	unsigned preserve : 1;
+	unsigned rational : 1;
+	unsigned empty : 1;
+	unsigned in_undo : 1;
+	unsigned M : 1;
+	unsigned cone : 1;
+};
+
+struct isl_tab *isl_tab_alloc(struct isl_ctx *ctx,
+	unsigned n_row, unsigned n_var, unsigned M);
+void isl_tab_free(struct isl_tab *tab);
+
+isl_ctx *isl_tab_get_ctx(struct isl_tab *tab);
+
+__isl_give struct isl_tab *isl_tab_from_basic_map(
+	__isl_keep isl_basic_map *bmap, int track);
+__isl_give struct isl_tab *isl_tab_from_basic_set(
+	__isl_keep isl_basic_set *bset, int track);
+struct isl_tab *isl_tab_from_recession_cone(struct isl_basic_set *bset,
+	int parametric);
+int isl_tab_cone_is_bounded(struct isl_tab *tab);
+struct isl_basic_map *isl_basic_map_update_from_tab(struct isl_basic_map *bmap,
+	struct isl_tab *tab);
+struct isl_basic_set *isl_basic_set_update_from_tab(struct isl_basic_set *bset,
+	struct isl_tab *tab);
+int isl_tab_detect_implicit_equalities(struct isl_tab *tab) WARN_UNUSED;
+__isl_give isl_basic_map *isl_tab_make_equalities_explicit(struct isl_tab *tab,
+	__isl_take isl_basic_map *bmap);
+int isl_tab_detect_redundant(struct isl_tab *tab) WARN_UNUSED;
+#define ISL_TAB_SAVE_DUAL	(1 << 0)
+enum isl_lp_result isl_tab_min(struct isl_tab *tab,
+	isl_int *f, isl_int denom, isl_int *opt, isl_int *opt_denom,
+	unsigned flags) WARN_UNUSED;
+
+int isl_tab_add_ineq(struct isl_tab *tab, isl_int *ineq) WARN_UNUSED;
+int isl_tab_add_eq(struct isl_tab *tab, isl_int *eq) WARN_UNUSED;
+int isl_tab_add_valid_eq(struct isl_tab *tab, isl_int *eq) WARN_UNUSED;
+
+int isl_tab_freeze_constraint(struct isl_tab *tab, int con) WARN_UNUSED;
+
+int isl_tab_track_bmap(struct isl_tab *tab, __isl_take isl_basic_map *bmap) WARN_UNUSED;
+int isl_tab_track_bset(struct isl_tab *tab, __isl_take isl_basic_set *bset) WARN_UNUSED;
+__isl_keep isl_basic_set *isl_tab_peek_bset(struct isl_tab *tab);
+
+int isl_tab_is_equality(struct isl_tab *tab, int con);
+int isl_tab_is_redundant(struct isl_tab *tab, int con);
+
+int isl_tab_sample_is_integer(struct isl_tab *tab);
+struct isl_vec *isl_tab_get_sample_value(struct isl_tab *tab);
+
+enum isl_ineq_type {
+	isl_ineq_error = -1,
+	isl_ineq_redundant,
+	isl_ineq_separate,
+	isl_ineq_cut,
+	isl_ineq_adj_eq,
+	isl_ineq_adj_ineq,
+};
+
+enum isl_ineq_type isl_tab_ineq_type(struct isl_tab *tab, isl_int *ineq);
+
+struct isl_tab_undo *isl_tab_snap(struct isl_tab *tab);
+int isl_tab_rollback(struct isl_tab *tab, struct isl_tab_undo *snap) WARN_UNUSED;
+
+struct isl_tab *isl_tab_relax(struct isl_tab *tab, int con) WARN_UNUSED;
+int isl_tab_select_facet(struct isl_tab *tab, int con) WARN_UNUSED;
+int isl_tab_unrestrict(struct isl_tab *tab, int con) WARN_UNUSED;
+
+void isl_tab_dump(__isl_keep struct isl_tab *tab);
+
+struct isl_map *isl_tab_basic_map_partial_lexopt(
+		struct isl_basic_map *bmap, struct isl_basic_set *dom,
+		struct isl_set **empty, int max);
+__isl_give isl_pw_multi_aff *isl_basic_map_partial_lexopt_pw_multi_aff(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty, int max);
+
+/* An isl_region represents a sequence of consecutive variables.
+ * pos is the location (starting at 0) of the first variable in the sequence.
+ */
+struct isl_region {
+	int pos;
+	int len;
+};
+
+__isl_give isl_vec *isl_tab_basic_set_non_trivial_lexmin(
+	__isl_take isl_basic_set *bset, int n_op, int n_region,
+	struct isl_region *region,
+	int (*conflict)(int con, void *user), void *user);
+__isl_give isl_vec *isl_tab_basic_set_non_neg_lexmin(
+	__isl_take isl_basic_set *bset);
+
+/* private */
+
+struct isl_tab_var *isl_tab_var_from_row(struct isl_tab *tab, int i);
+int isl_tab_mark_redundant(struct isl_tab *tab, int row) WARN_UNUSED;
+int isl_tab_mark_empty(struct isl_tab *tab) WARN_UNUSED;
+struct isl_tab *isl_tab_dup(struct isl_tab *tab);
+struct isl_tab *isl_tab_product(struct isl_tab *tab1, struct isl_tab *tab2);
+int isl_tab_extend_cons(struct isl_tab *tab, unsigned n_new) WARN_UNUSED;
+int isl_tab_allocate_con(struct isl_tab *tab) WARN_UNUSED;
+int isl_tab_extend_vars(struct isl_tab *tab, unsigned n_new) WARN_UNUSED;
+int isl_tab_allocate_var(struct isl_tab *tab) WARN_UNUSED;
+int isl_tab_pivot(struct isl_tab *tab, int row, int col) WARN_UNUSED;
+int isl_tab_add_row(struct isl_tab *tab, isl_int *line) WARN_UNUSED;
+int isl_tab_row_is_redundant(struct isl_tab *tab, int row);
+int isl_tab_min_at_most_neg_one(struct isl_tab *tab, struct isl_tab_var *var);
+int isl_tab_sign_of_max(struct isl_tab *tab, int con);
+int isl_tab_kill_col(struct isl_tab *tab, int col) WARN_UNUSED;
+
+int isl_tab_push(struct isl_tab *tab, enum isl_tab_undo_type type) WARN_UNUSED;
+int isl_tab_push_var(struct isl_tab *tab,
+	enum isl_tab_undo_type type, struct isl_tab_var *var) WARN_UNUSED;
+int isl_tab_push_basis(struct isl_tab *tab) WARN_UNUSED;
+
+struct isl_tab *isl_tab_init_samples(struct isl_tab *tab) WARN_UNUSED;
+int isl_tab_add_sample(struct isl_tab *tab,
+	__isl_take isl_vec *sample) WARN_UNUSED;
+struct isl_tab *isl_tab_drop_sample(struct isl_tab *tab, int s);
+int isl_tab_save_samples(struct isl_tab *tab) WARN_UNUSED;
+
+struct isl_tab *isl_tab_detect_equalities(struct isl_tab *tab,
+	struct isl_tab *tab_cone) WARN_UNUSED;
+
+int isl_tab_push_callback(struct isl_tab *tab,
+	struct isl_tab_callback *callback) WARN_UNUSED;
+
+int isl_tab_add_div(struct isl_tab *tab, __isl_keep isl_vec *div,
+	int (*add_ineq)(void *user, isl_int *), void *user);
+
+#endif
diff --git a/polly/lib/External/isl/isl_tab_pip.c b/polly/lib/External/isl/isl_tab_pip.c
new file mode 100644
index 00000000000..2a4a4f731fd
--- /dev/null
+++ b/polly/lib/External/isl/isl_tab_pip.c
@@ -0,0 +1,5681 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2010      INRIA Saclay
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France 
+ */
+
+#include <isl_ctx_private.h>
+#include "isl_map_private.h"
+#include <isl_seq.h>
+#include "isl_tab.h"
+#include "isl_sample.h"
+#include <isl_mat_private.h>
+#include <isl_vec_private.h>
+#include <isl_aff_private.h>
+#include <isl_options_private.h>
+#include <isl_config.h>
+
+/*
+ * The implementation of parametric integer linear programming in this file
+ * was inspired by the paper "Parametric Integer Programming" and the
+ * report "Solving systems of affine (in)equalities" by Paul Feautrier
+ * (and others).
+ *
+ * The strategy used for obtaining a feasible solution is different
+ * from the one used in isl_tab.c.  In particular, in isl_tab.c,
+ * upon finding a constraint that is not yet satisfied, we pivot
+ * in a row that increases the constant term of the row holding the
+ * constraint, making sure the sample solution remains feasible
+ * for all the constraints it already satisfied.
+ * Here, we always pivot in the row holding the constraint,
+ * choosing a column that induces the lexicographically smallest
+ * increment to the sample solution.
+ *
+ * By starting out from a sample value that is lexicographically
+ * smaller than any integer point in the problem space, the first
+ * feasible integer sample point we find will also be the lexicographically
+ * smallest.  If all variables can be assumed to be non-negative,
+ * then the initial sample value may be chosen equal to zero.
+ * However, we will not make this assumption.  Instead, we apply
+ * the "big parameter" trick.  Any variable x is then not directly
+ * used in the tableau, but instead it is represented by another
+ * variable x' = M + x, where M is an arbitrarily large (positive)
+ * value.  x' is therefore always non-negative, whatever the value of x.
+ * Taking as initial sample value x' = 0 corresponds to x = -M,
+ * which is always smaller than any possible value of x.
+ *
+ * The big parameter trick is used in the main tableau and
+ * also in the context tableau if isl_context_lex is used.
+ * In this case, each tableaus has its own big parameter.
+ * Before doing any real work, we check if all the parameters
+ * happen to be non-negative.  If so, we drop the column corresponding
+ * to M from the initial context tableau.
+ * If isl_context_gbr is used, then the big parameter trick is only
+ * used in the main tableau.
+ */
+
+struct isl_context;
+struct isl_context_op {
+	/* detect nonnegative parameters in context and mark them in tab */
+	struct isl_tab *(*detect_nonnegative_parameters)(
+			struct isl_context *context, struct isl_tab *tab);
+	/* return temporary reference to basic set representation of context */
+	struct isl_basic_set *(*peek_basic_set)(struct isl_context *context);
+	/* return temporary reference to tableau representation of context */
+	struct isl_tab *(*peek_tab)(struct isl_context *context);
+	/* add equality; check is 1 if eq may not be valid;
+	 * update is 1 if we may want to call ineq_sign on context later.
+	 */
+	void (*add_eq)(struct isl_context *context, isl_int *eq,
+			int check, int update);
+	/* add inequality; check is 1 if ineq may not be valid;
+	 * update is 1 if we may want to call ineq_sign on context later.
+	 */
+	void (*add_ineq)(struct isl_context *context, isl_int *ineq,
+			int check, int update);
+	/* check sign of ineq based on previous information.
+	 * strict is 1 if saturation should be treated as a positive sign.
+	 */
+	enum isl_tab_row_sign (*ineq_sign)(struct isl_context *context,
+			isl_int *ineq, int strict);
+	/* check if inequality maintains feasibility */
+	int (*test_ineq)(struct isl_context *context, isl_int *ineq);
+	/* return index of a div that corresponds to "div" */
+	int (*get_div)(struct isl_context *context, struct isl_tab *tab,
+			struct isl_vec *div);
+	/* add div "div" to context and return non-negativity */
+	int (*add_div)(struct isl_context *context, struct isl_vec *div);
+	int (*detect_equalities)(struct isl_context *context,
+			struct isl_tab *tab);
+	/* return row index of "best" split */
+	int (*best_split)(struct isl_context *context, struct isl_tab *tab);
+	/* check if context has already been determined to be empty */
+	int (*is_empty)(struct isl_context *context);
+	/* check if context is still usable */
+	int (*is_ok)(struct isl_context *context);
+	/* save a copy/snapshot of context */
+	void *(*save)(struct isl_context *context);
+	/* restore saved context */
+	void (*restore)(struct isl_context *context, void *);
+	/* discard saved context */
+	void (*discard)(void *);
+	/* invalidate context */
+	void (*invalidate)(struct isl_context *context);
+	/* free context */
+	void (*free)(struct isl_context *context);
+};
+
+struct isl_context {
+	struct isl_context_op *op;
+};
+
+struct isl_context_lex {
+	struct isl_context context;
+	struct isl_tab *tab;
+};
+
+/* A stack (linked list) of solutions of subtrees of the search space.
+ *
+ * "M" describes the solution in terms of the dimensions of "dom".
+ * The number of columns of "M" is one more than the total number
+ * of dimensions of "dom".
+ *
+ * If "M" is NULL, then there is no solution on "dom".
+ */
+struct isl_partial_sol {
+	int level;
+	struct isl_basic_set *dom;
+	struct isl_mat *M;
+
+	struct isl_partial_sol *next;
+};
+
+struct isl_sol;
+struct isl_sol_callback {
+	struct isl_tab_callback callback;
+	struct isl_sol *sol;
+};
+
+/* isl_sol is an interface for constructing a solution to
+ * a parametric integer linear programming problem.
+ * Every time the algorithm reaches a state where a solution
+ * can be read off from the tableau (including cases where the tableau
+ * is empty), the function "add" is called on the isl_sol passed
+ * to find_solutions_main.
+ *
+ * The context tableau is owned by isl_sol and is updated incrementally.
+ *
+ * There are currently two implementations of this interface,
+ * isl_sol_map, which simply collects the solutions in an isl_map
+ * and (optionally) the parts of the context where there is no solution
+ * in an isl_set, and
+ * isl_sol_for, which calls a user-defined function for each part of
+ * the solution.
+ */
+struct isl_sol {
+	int error;
+	int rational;
+	int level;
+	int max;
+	int n_out;
+	struct isl_context *context;
+	struct isl_partial_sol *partial;
+	void (*add)(struct isl_sol *sol,
+			    struct isl_basic_set *dom, struct isl_mat *M);
+	void (*add_empty)(struct isl_sol *sol, struct isl_basic_set *bset);
+	void (*free)(struct isl_sol *sol);
+	struct isl_sol_callback	dec_level;
+};
+
+static void sol_free(struct isl_sol *sol)
+{
+	struct isl_partial_sol *partial, *next;
+	if (!sol)
+		return;
+	for (partial = sol->partial; partial; partial = next) {
+		next = partial->next;
+		isl_basic_set_free(partial->dom);
+		isl_mat_free(partial->M);
+		free(partial);
+	}
+	sol->free(sol);
+}
+
+/* Push a partial solution represented by a domain and mapping M
+ * onto the stack of partial solutions.
+ */
+static void sol_push_sol(struct isl_sol *sol,
+	struct isl_basic_set *dom, struct isl_mat *M)
+{
+	struct isl_partial_sol *partial;
+
+	if (sol->error || !dom)
+		goto error;
+
+	partial = isl_alloc_type(dom->ctx, struct isl_partial_sol);
+	if (!partial)
+		goto error;
+
+	partial->level = sol->level;
+	partial->dom = dom;
+	partial->M = M;
+	partial->next = sol->partial;
+
+	sol->partial = partial;
+
+	return;
+error:
+	isl_basic_set_free(dom);
+	isl_mat_free(M);
+	sol->error = 1;
+}
+
+/* Pop one partial solution from the partial solution stack and
+ * pass it on to sol->add or sol->add_empty.
+ */
+static void sol_pop_one(struct isl_sol *sol)
+{
+	struct isl_partial_sol *partial;
+
+	partial = sol->partial;
+	sol->partial = partial->next;
+
+	if (partial->M)
+		sol->add(sol, partial->dom, partial->M);
+	else
+		sol->add_empty(sol, partial->dom);
+	free(partial);
+}
+
+/* Return a fresh copy of the domain represented by the context tableau.
+ */
+static struct isl_basic_set *sol_domain(struct isl_sol *sol)
+{
+	struct isl_basic_set *bset;
+
+	if (sol->error)
+		return NULL;
+
+	bset = isl_basic_set_dup(sol->context->op->peek_basic_set(sol->context));
+	bset = isl_basic_set_update_from_tab(bset,
+			sol->context->op->peek_tab(sol->context));
+
+	return bset;
+}
+
+/* Check whether two partial solutions have the same mapping, where n_div
+ * is the number of divs that the two partial solutions have in common.
+ */
+static int same_solution(struct isl_partial_sol *s1, struct isl_partial_sol *s2,
+	unsigned n_div)
+{
+	int i;
+	unsigned dim;
+
+	if (!s1->M != !s2->M)
+		return 0;
+	if (!s1->M)
+		return 1;
+
+	dim = isl_basic_set_total_dim(s1->dom) - s1->dom->n_div;
+
+	for (i = 0; i < s1->M->n_row; ++i) {
+		if (isl_seq_first_non_zero(s1->M->row[i]+1+dim+n_div,
+					    s1->M->n_col-1-dim-n_div) != -1)
+			return 0;
+		if (isl_seq_first_non_zero(s2->M->row[i]+1+dim+n_div,
+					    s2->M->n_col-1-dim-n_div) != -1)
+			return 0;
+		if (!isl_seq_eq(s1->M->row[i], s2->M->row[i], 1+dim+n_div))
+			return 0;
+	}
+	return 1;
+}
+
+/* Pop all solutions from the partial solution stack that were pushed onto
+ * the stack at levels that are deeper than the current level.
+ * If the two topmost elements on the stack have the same level
+ * and represent the same solution, then their domains are combined.
+ * This combined domain is the same as the current context domain
+ * as sol_pop is called each time we move back to a higher level.
+ */
+static void sol_pop(struct isl_sol *sol)
+{
+	struct isl_partial_sol *partial;
+	unsigned n_div;
+
+	if (sol->error)
+		return;
+
+	if (sol->level == 0) {
+		for (partial = sol->partial; partial; partial = sol->partial)
+			sol_pop_one(sol);
+		return;
+	}
+
+	partial = sol->partial;
+	if (!partial)
+		return;
+
+	if (partial->level <= sol->level)
+		return;
+
+	if (partial->next && partial->next->level == partial->level) {
+		n_div = isl_basic_set_dim(
+				sol->context->op->peek_basic_set(sol->context),
+				isl_dim_div);
+
+		if (!same_solution(partial, partial->next, n_div)) {
+			sol_pop_one(sol);
+			sol_pop_one(sol);
+		} else {
+			struct isl_basic_set *bset;
+			isl_mat *M;
+			unsigned n;
+
+			n = isl_basic_set_dim(partial->next->dom, isl_dim_div);
+			n -= n_div;
+			bset = sol_domain(sol);
+			isl_basic_set_free(partial->next->dom);
+			partial->next->dom = bset;
+			M = partial->next->M;
+			if (M) {
+				M = isl_mat_drop_cols(M, M->n_col - n, n);
+				partial->next->M = M;
+				if (!M)
+					goto error;
+			}
+			partial->next->level = sol->level;
+
+			if (!bset)
+				goto error;
+
+			sol->partial = partial->next;
+			isl_basic_set_free(partial->dom);
+			isl_mat_free(partial->M);
+			free(partial);
+		}
+	} else
+		sol_pop_one(sol);
+
+	if (0)
+error:		sol->error = 1;
+}
+
+static void sol_dec_level(struct isl_sol *sol)
+{
+	if (sol->error)
+		return;
+
+	sol->level--;
+
+	sol_pop(sol);
+}
+
+static int sol_dec_level_wrap(struct isl_tab_callback *cb)
+{
+	struct isl_sol_callback *callback = (struct isl_sol_callback *)cb;
+
+	sol_dec_level(callback->sol);
+
+	return callback->sol->error ? -1 : 0;
+}
+
+/* Move down to next level and push callback onto context tableau
+ * to decrease the level again when it gets rolled back across
+ * the current state.  That is, dec_level will be called with
+ * the context tableau in the same state as it is when inc_level
+ * is called.
+ */
+static void sol_inc_level(struct isl_sol *sol)
+{
+	struct isl_tab *tab;
+
+	if (sol->error)
+		return;
+
+	sol->level++;
+	tab = sol->context->op->peek_tab(sol->context);
+	if (isl_tab_push_callback(tab, &sol->dec_level.callback) < 0)
+		sol->error = 1;
+}
+
+static void scale_rows(struct isl_mat *mat, isl_int m, int n_row)
+{
+	int i;
+
+	if (isl_int_is_one(m))
+		return;
+
+	for (i = 0; i < n_row; ++i)
+		isl_seq_scale(mat->row[i], mat->row[i], m, mat->n_col);
+}
+
+/* Add the solution identified by the tableau and the context tableau.
+ *
+ * The layout of the variables is as follows.
+ *	tab->n_var is equal to the total number of variables in the input
+ *			map (including divs that were copied from the context)
+ *			+ the number of extra divs constructed
+ *      Of these, the first tab->n_param and the last tab->n_div variables
+ *	correspond to the variables in the context, i.e.,
+ *		tab->n_param + tab->n_div = context_tab->n_var
+ *	tab->n_param is equal to the number of parameters and input
+ *			dimensions in the input map
+ *	tab->n_div is equal to the number of divs in the context
+ *
+ * If there is no solution, then call add_empty with a basic set
+ * that corresponds to the context tableau.  (If add_empty is NULL,
+ * then do nothing).
+ *
+ * If there is a solution, then first construct a matrix that maps
+ * all dimensions of the context to the output variables, i.e.,
+ * the output dimensions in the input map.
+ * The divs in the input map (if any) that do not correspond to any
+ * div in the context do not appear in the solution.
+ * The algorithm will make sure that they have an integer value,
+ * but these values themselves are of no interest.
+ * We have to be careful not to drop or rearrange any divs in the
+ * context because that would change the meaning of the matrix.
+ *
+ * To extract the value of the output variables, it should be noted
+ * that we always use a big parameter M in the main tableau and so
+ * the variable stored in this tableau is not an output variable x itself, but
+ *	x' = M + x (in case of minimization)
+ * or
+ *	x' = M - x (in case of maximization)
+ * If x' appears in a column, then its optimal value is zero,
+ * which means that the optimal value of x is an unbounded number
+ * (-M for minimization and M for maximization).
+ * We currently assume that the output dimensions in the original map
+ * are bounded, so this cannot occur.
+ * Similarly, when x' appears in a row, then the coefficient of M in that
+ * row is necessarily 1.
+ * If the row in the tableau represents
+ *	d x' = c + d M + e(y)
+ * then, in case of minimization, the corresponding row in the matrix
+ * will be
+ *	a c + a e(y)
+ * with a d = m, the (updated) common denominator of the matrix.
+ * In case of maximization, the row will be
+ *	-a c - a e(y)
+ */
+static void sol_add(struct isl_sol *sol, struct isl_tab *tab)
+{
+	struct isl_basic_set *bset = NULL;
+	struct isl_mat *mat = NULL;
+	unsigned off;
+	int row;
+	isl_int m;
+
+	if (sol->error || !tab)
+		goto error;
+
+	if (tab->empty && !sol->add_empty)
+		return;
+	if (sol->context->op->is_empty(sol->context))
+		return;
+
+	bset = sol_domain(sol);
+
+	if (tab->empty) {
+		sol_push_sol(sol, bset, NULL);
+		return;
+	}
+
+	off = 2 + tab->M;
+
+	mat = isl_mat_alloc(tab->mat->ctx, 1 + sol->n_out,
+					    1 + tab->n_param + tab->n_div);
+	if (!mat)
+		goto error;
+
+	isl_int_init(m);
+
+	isl_seq_clr(mat->row[0] + 1, mat->n_col - 1);
+	isl_int_set_si(mat->row[0][0], 1);
+	for (row = 0; row < sol->n_out; ++row) {
+		int i = tab->n_param + row;
+		int r, j;
+
+		isl_seq_clr(mat->row[1 + row], mat->n_col);
+		if (!tab->var[i].is_row) {
+			if (tab->M)
+				isl_die(mat->ctx, isl_error_invalid,
+					"unbounded optimum", goto error2);
+			continue;
+		}
+
+		r = tab->var[i].index;
+		if (tab->M &&
+		    isl_int_ne(tab->mat->row[r][2], tab->mat->row[r][0]))
+			isl_die(mat->ctx, isl_error_invalid,
+				"unbounded optimum", goto error2);
+		isl_int_gcd(m, mat->row[0][0], tab->mat->row[r][0]);
+		isl_int_divexact(m, tab->mat->row[r][0], m);
+		scale_rows(mat, m, 1 + row);
+		isl_int_divexact(m, mat->row[0][0], tab->mat->row[r][0]);
+		isl_int_mul(mat->row[1 + row][0], m, tab->mat->row[r][1]);
+		for (j = 0; j < tab->n_param; ++j) {
+			int col;
+			if (tab->var[j].is_row)
+				continue;
+			col = tab->var[j].index;
+			isl_int_mul(mat->row[1 + row][1 + j], m,
+				    tab->mat->row[r][off + col]);
+		}
+		for (j = 0; j < tab->n_div; ++j) {
+			int col;
+			if (tab->var[tab->n_var - tab->n_div+j].is_row)
+				continue;
+			col = tab->var[tab->n_var - tab->n_div+j].index;
+			isl_int_mul(mat->row[1 + row][1 + tab->n_param + j], m,
+				    tab->mat->row[r][off + col]);
+		}
+		if (sol->max)
+			isl_seq_neg(mat->row[1 + row], mat->row[1 + row],
+				    mat->n_col);
+	}
+
+	isl_int_clear(m);
+
+	sol_push_sol(sol, bset, mat);
+	return;
+error2:
+	isl_int_clear(m);
+error:
+	isl_basic_set_free(bset);
+	isl_mat_free(mat);
+	sol->error = 1;
+}
+
+struct isl_sol_map {
+	struct isl_sol	sol;
+	struct isl_map	*map;
+	struct isl_set	*empty;
+};
+
+static void sol_map_free(struct isl_sol_map *sol_map)
+{
+	if (!sol_map)
+		return;
+	if (sol_map->sol.context)
+		sol_map->sol.context->op->free(sol_map->sol.context);
+	isl_map_free(sol_map->map);
+	isl_set_free(sol_map->empty);
+	free(sol_map);
+}
+
+static void sol_map_free_wrap(struct isl_sol *sol)
+{
+	sol_map_free((struct isl_sol_map *)sol);
+}
+
+/* This function is called for parts of the context where there is
+ * no solution, with "bset" corresponding to the context tableau.
+ * Simply add the basic set to the set "empty".
+ */
+static void sol_map_add_empty(struct isl_sol_map *sol,
+	struct isl_basic_set *bset)
+{
+	if (!bset)
+		goto error;
+	isl_assert(bset->ctx, sol->empty, goto error);
+
+	sol->empty = isl_set_grow(sol->empty, 1);
+	bset = isl_basic_set_simplify(bset);
+	bset = isl_basic_set_finalize(bset);
+	sol->empty = isl_set_add_basic_set(sol->empty, isl_basic_set_copy(bset));
+	if (!sol->empty)
+		goto error;
+	isl_basic_set_free(bset);
+	return;
+error:
+	isl_basic_set_free(bset);
+	sol->sol.error = 1;
+}
+
+static void sol_map_add_empty_wrap(struct isl_sol *sol,
+	struct isl_basic_set *bset)
+{
+	sol_map_add_empty((struct isl_sol_map *)sol, bset);
+}
+
+/* Given a basic map "dom" that represents the context and an affine
+ * matrix "M" that maps the dimensions of the context to the
+ * output variables, construct a basic map with the same parameters
+ * and divs as the context, the dimensions of the context as input
+ * dimensions and a number of output dimensions that is equal to
+ * the number of output dimensions in the input map.
+ *
+ * The constraints and divs of the context are simply copied
+ * from "dom".  For each row
+ *	x = c + e(y)
+ * an equality
+ *	c + e(y) - d x = 0
+ * is added, with d the common denominator of M.
+ */
+static void sol_map_add(struct isl_sol_map *sol,
+	struct isl_basic_set *dom, struct isl_mat *M)
+{
+	int i;
+	struct isl_basic_map *bmap = NULL;
+	unsigned n_eq;
+	unsigned n_ineq;
+	unsigned nparam;
+	unsigned total;
+	unsigned n_div;
+	unsigned n_out;
+
+	if (sol->sol.error || !dom || !M)
+		goto error;
+
+	n_out = sol->sol.n_out;
+	n_eq = dom->n_eq + n_out;
+	n_ineq = dom->n_ineq;
+	n_div = dom->n_div;
+	nparam = isl_basic_set_total_dim(dom) - n_div;
+	total = isl_map_dim(sol->map, isl_dim_all);
+	bmap = isl_basic_map_alloc_space(isl_map_get_space(sol->map),
+					n_div, n_eq, 2 * n_div + n_ineq);
+	if (!bmap)
+		goto error;
+	if (sol->sol.rational)
+		ISL_F_SET(bmap, ISL_BASIC_MAP_RATIONAL);
+	for (i = 0; i < dom->n_div; ++i) {
+		int k = isl_basic_map_alloc_div(bmap);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(bmap->div[k], dom->div[i], 1 + 1 + nparam);
+		isl_seq_clr(bmap->div[k] + 1 + 1 + nparam, total - nparam);
+		isl_seq_cpy(bmap->div[k] + 1 + 1 + total,
+			    dom->div[i] + 1 + 1 + nparam, i);
+	}
+	for (i = 0; i < dom->n_eq; ++i) {
+		int k = isl_basic_map_alloc_equality(bmap);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(bmap->eq[k], dom->eq[i], 1 + nparam);
+		isl_seq_clr(bmap->eq[k] + 1 + nparam, total - nparam);
+		isl_seq_cpy(bmap->eq[k] + 1 + total,
+			    dom->eq[i] + 1 + nparam, n_div);
+	}
+	for (i = 0; i < dom->n_ineq; ++i) {
+		int k = isl_basic_map_alloc_inequality(bmap);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(bmap->ineq[k], dom->ineq[i], 1 + nparam);
+		isl_seq_clr(bmap->ineq[k] + 1 + nparam, total - nparam);
+		isl_seq_cpy(bmap->ineq[k] + 1 + total,
+			dom->ineq[i] + 1 + nparam, n_div);
+	}
+	for (i = 0; i < M->n_row - 1; ++i) {
+		int k = isl_basic_map_alloc_equality(bmap);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(bmap->eq[k], M->row[1 + i], 1 + nparam);
+		isl_seq_clr(bmap->eq[k] + 1 + nparam, n_out);
+		isl_int_neg(bmap->eq[k][1 + nparam + i], M->row[0][0]);
+		isl_seq_cpy(bmap->eq[k] + 1 + nparam + n_out,
+			    M->row[1 + i] + 1 + nparam, n_div);
+	}
+	bmap = isl_basic_map_simplify(bmap);
+	bmap = isl_basic_map_finalize(bmap);
+	sol->map = isl_map_grow(sol->map, 1);
+	sol->map = isl_map_add_basic_map(sol->map, bmap);
+	isl_basic_set_free(dom);
+	isl_mat_free(M);
+	if (!sol->map)
+		sol->sol.error = 1;
+	return;
+error:
+	isl_basic_set_free(dom);
+	isl_mat_free(M);
+	isl_basic_map_free(bmap);
+	sol->sol.error = 1;
+}
+
+static void sol_map_add_wrap(struct isl_sol *sol,
+	struct isl_basic_set *dom, struct isl_mat *M)
+{
+	sol_map_add((struct isl_sol_map *)sol, dom, M);
+}
+
+
+/* Store the "parametric constant" of row "row" of tableau "tab" in "line",
+ * i.e., the constant term and the coefficients of all variables that
+ * appear in the context tableau.
+ * Note that the coefficient of the big parameter M is NOT copied.
+ * The context tableau may not have a big parameter and even when it
+ * does, it is a different big parameter.
+ */
+static void get_row_parameter_line(struct isl_tab *tab, int row, isl_int *line)
+{
+	int i;
+	unsigned off = 2 + tab->M;
+
+	isl_int_set(line[0], tab->mat->row[row][1]);
+	for (i = 0; i < tab->n_param; ++i) {
+		if (tab->var[i].is_row)
+			isl_int_set_si(line[1 + i], 0);
+		else {
+			int col = tab->var[i].index;
+			isl_int_set(line[1 + i], tab->mat->row[row][off + col]);
+		}
+	}
+	for (i = 0; i < tab->n_div; ++i) {
+		if (tab->var[tab->n_var - tab->n_div + i].is_row)
+			isl_int_set_si(line[1 + tab->n_param + i], 0);
+		else {
+			int col = tab->var[tab->n_var - tab->n_div + i].index;
+			isl_int_set(line[1 + tab->n_param + i],
+				    tab->mat->row[row][off + col]);
+		}
+	}
+}
+
+/* Check if rows "row1" and "row2" have identical "parametric constants",
+ * as explained above.
+ * In this case, we also insist that the coefficients of the big parameter
+ * be the same as the values of the constants will only be the same
+ * if these coefficients are also the same.
+ */
+static int identical_parameter_line(struct isl_tab *tab, int row1, int row2)
+{
+	int i;
+	unsigned off = 2 + tab->M;
+
+	if (isl_int_ne(tab->mat->row[row1][1], tab->mat->row[row2][1]))
+		return 0;
+
+	if (tab->M && isl_int_ne(tab->mat->row[row1][2],
+				 tab->mat->row[row2][2]))
+		return 0;
+
+	for (i = 0; i < tab->n_param + tab->n_div; ++i) {
+		int pos = i < tab->n_param ? i :
+			tab->n_var - tab->n_div + i - tab->n_param;
+		int col;
+
+		if (tab->var[pos].is_row)
+			continue;
+		col = tab->var[pos].index;
+		if (isl_int_ne(tab->mat->row[row1][off + col],
+			       tab->mat->row[row2][off + col]))
+			return 0;
+	}
+	return 1;
+}
+
+/* Return an inequality that expresses that the "parametric constant"
+ * should be non-negative.
+ * This function is only called when the coefficient of the big parameter
+ * is equal to zero.
+ */
+static struct isl_vec *get_row_parameter_ineq(struct isl_tab *tab, int row)
+{
+	struct isl_vec *ineq;
+
+	ineq = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_param + tab->n_div);
+	if (!ineq)
+		return NULL;
+
+	get_row_parameter_line(tab, row, ineq->el);
+	if (ineq)
+		ineq = isl_vec_normalize(ineq);
+
+	return ineq;
+}
+
+/* Normalize a div expression of the form
+ *
+ *	[(g*f(x) + c)/(g * m)]
+ *
+ * with c the constant term and f(x) the remaining coefficients, to
+ *
+ *	[(f(x) + [c/g])/m]
+ */
+static void normalize_div(__isl_keep isl_vec *div)
+{
+	isl_ctx *ctx = isl_vec_get_ctx(div);
+	int len = div->size - 2;
+
+	isl_seq_gcd(div->el + 2, len, &ctx->normalize_gcd);
+	isl_int_gcd(ctx->normalize_gcd, ctx->normalize_gcd, div->el[0]);
+
+	if (isl_int_is_one(ctx->normalize_gcd))
+		return;
+
+	isl_int_divexact(div->el[0], div->el[0], ctx->normalize_gcd);
+	isl_int_fdiv_q(div->el[1], div->el[1], ctx->normalize_gcd);
+	isl_seq_scale_down(div->el + 2, div->el + 2, ctx->normalize_gcd, len);
+}
+
+/* Return a integer division for use in a parametric cut based on the given row.
+ * In particular, let the parametric constant of the row be
+ *
+ *		\sum_i a_i y_i
+ *
+ * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
+ * The div returned is equal to
+ *
+ *		floor(\sum_i {-a_i} y_i) = floor((\sum_i (-a_i mod d) y_i)/d)
+ */
+static struct isl_vec *get_row_parameter_div(struct isl_tab *tab, int row)
+{
+	struct isl_vec *div;
+
+	div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div);
+	if (!div)
+		return NULL;
+
+	isl_int_set(div->el[0], tab->mat->row[row][0]);
+	get_row_parameter_line(tab, row, div->el + 1);
+	isl_seq_neg(div->el + 1, div->el + 1, div->size - 1);
+	normalize_div(div);
+	isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1);
+
+	return div;
+}
+
+/* Return a integer division for use in transferring an integrality constraint
+ * to the context.
+ * In particular, let the parametric constant of the row be
+ *
+ *		\sum_i a_i y_i
+ *
+ * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
+ * The the returned div is equal to
+ *
+ *		floor(\sum_i {a_i} y_i) = floor((\sum_i (a_i mod d) y_i)/d)
+ */
+static struct isl_vec *get_row_split_div(struct isl_tab *tab, int row)
+{
+	struct isl_vec *div;
+
+	div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div);
+	if (!div)
+		return NULL;
+
+	isl_int_set(div->el[0], tab->mat->row[row][0]);
+	get_row_parameter_line(tab, row, div->el + 1);
+	normalize_div(div);
+	isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1);
+
+	return div;
+}
+
+/* Construct and return an inequality that expresses an upper bound
+ * on the given div.
+ * In particular, if the div is given by
+ *
+ *	d = floor(e/m)
+ *
+ * then the inequality expresses
+ *
+ *	m d <= e
+ */
+static struct isl_vec *ineq_for_div(struct isl_basic_set *bset, unsigned div)
+{
+	unsigned total;
+	unsigned div_pos;
+	struct isl_vec *ineq;
+
+	if (!bset)
+		return NULL;
+
+	total = isl_basic_set_total_dim(bset);
+	div_pos = 1 + total - bset->n_div + div;
+
+	ineq = isl_vec_alloc(bset->ctx, 1 + total);
+	if (!ineq)
+		return NULL;
+
+	isl_seq_cpy(ineq->el, bset->div[div] + 1, 1 + total);
+	isl_int_neg(ineq->el[div_pos], bset->div[div][0]);
+	return ineq;
+}
+
+/* Given a row in the tableau and a div that was created
+ * using get_row_split_div and that has been constrained to equality, i.e.,
+ *
+ *		d = floor(\sum_i {a_i} y_i) = \sum_i {a_i} y_i
+ *
+ * replace the expression "\sum_i {a_i} y_i" in the row by d,
+ * i.e., we subtract "\sum_i {a_i} y_i" and add 1 d.
+ * The coefficients of the non-parameters in the tableau have been
+ * verified to be integral.  We can therefore simply replace coefficient b
+ * by floor(b).  For the coefficients of the parameters we have
+ * floor(a_i) = a_i - {a_i}, while for the other coefficients, we have
+ * floor(b) = b.
+ */
+static struct isl_tab *set_row_cst_to_div(struct isl_tab *tab, int row, int div)
+{
+	isl_seq_fdiv_q(tab->mat->row[row] + 1, tab->mat->row[row] + 1,
+			tab->mat->row[row][0], 1 + tab->M + tab->n_col);
+
+	isl_int_set_si(tab->mat->row[row][0], 1);
+
+	if (tab->var[tab->n_var - tab->n_div + div].is_row) {
+		int drow = tab->var[tab->n_var - tab->n_div + div].index;
+
+		isl_assert(tab->mat->ctx,
+			isl_int_is_one(tab->mat->row[drow][0]), goto error);
+		isl_seq_combine(tab->mat->row[row] + 1,
+			tab->mat->ctx->one, tab->mat->row[row] + 1,
+			tab->mat->ctx->one, tab->mat->row[drow] + 1,
+			1 + tab->M + tab->n_col);
+	} else {
+		int dcol = tab->var[tab->n_var - tab->n_div + div].index;
+
+		isl_int_add_ui(tab->mat->row[row][2 + tab->M + dcol],
+				tab->mat->row[row][2 + tab->M + dcol], 1);
+	}
+
+	return tab;
+error:
+	isl_tab_free(tab);
+	return NULL;
+}
+
+/* Check if the (parametric) constant of the given row is obviously
+ * negative, meaning that we don't need to consult the context tableau.
+ * If there is a big parameter and its coefficient is non-zero,
+ * then this coefficient determines the outcome.
+ * Otherwise, we check whether the constant is negative and
+ * all non-zero coefficients of parameters are negative and
+ * belong to non-negative parameters.
+ */
+static int is_obviously_neg(struct isl_tab *tab, int row)
+{
+	int i;
+	int col;
+	unsigned off = 2 + tab->M;
+
+	if (tab->M) {
+		if (isl_int_is_pos(tab->mat->row[row][2]))
+			return 0;
+		if (isl_int_is_neg(tab->mat->row[row][2]))
+			return 1;
+	}
+
+	if (isl_int_is_nonneg(tab->mat->row[row][1]))
+		return 0;
+	for (i = 0; i < tab->n_param; ++i) {
+		/* Eliminated parameter */
+		if (tab->var[i].is_row)
+			continue;
+		col = tab->var[i].index;
+		if (isl_int_is_zero(tab->mat->row[row][off + col]))
+			continue;
+		if (!tab->var[i].is_nonneg)
+			return 0;
+		if (isl_int_is_pos(tab->mat->row[row][off + col]))
+			return 0;
+	}
+	for (i = 0; i < tab->n_div; ++i) {
+		if (tab->var[tab->n_var - tab->n_div + i].is_row)
+			continue;
+		col = tab->var[tab->n_var - tab->n_div + i].index;
+		if (isl_int_is_zero(tab->mat->row[row][off + col]))
+			continue;
+		if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg)
+			return 0;
+		if (isl_int_is_pos(tab->mat->row[row][off + col]))
+			return 0;
+	}
+	return 1;
+}
+
+/* Check if the (parametric) constant of the given row is obviously
+ * non-negative, meaning that we don't need to consult the context tableau.
+ * If there is a big parameter and its coefficient is non-zero,
+ * then this coefficient determines the outcome.
+ * Otherwise, we check whether the constant is non-negative and
+ * all non-zero coefficients of parameters are positive and
+ * belong to non-negative parameters.
+ */
+static int is_obviously_nonneg(struct isl_tab *tab, int row)
+{
+	int i;
+	int col;
+	unsigned off = 2 + tab->M;
+
+	if (tab->M) {
+		if (isl_int_is_pos(tab->mat->row[row][2]))
+			return 1;
+		if (isl_int_is_neg(tab->mat->row[row][2]))
+			return 0;
+	}
+
+	if (isl_int_is_neg(tab->mat->row[row][1]))
+		return 0;
+	for (i = 0; i < tab->n_param; ++i) {
+		/* Eliminated parameter */
+		if (tab->var[i].is_row)
+			continue;
+		col = tab->var[i].index;
+		if (isl_int_is_zero(tab->mat->row[row][off + col]))
+			continue;
+		if (!tab->var[i].is_nonneg)
+			return 0;
+		if (isl_int_is_neg(tab->mat->row[row][off + col]))
+			return 0;
+	}
+	for (i = 0; i < tab->n_div; ++i) {
+		if (tab->var[tab->n_var - tab->n_div + i].is_row)
+			continue;
+		col = tab->var[tab->n_var - tab->n_div + i].index;
+		if (isl_int_is_zero(tab->mat->row[row][off + col]))
+			continue;
+		if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg)
+			return 0;
+		if (isl_int_is_neg(tab->mat->row[row][off + col]))
+			return 0;
+	}
+	return 1;
+}
+
+/* Given a row r and two columns, return the column that would
+ * lead to the lexicographically smallest increment in the sample
+ * solution when leaving the basis in favor of the row.
+ * Pivoting with column c will increment the sample value by a non-negative
+ * constant times a_{V,c}/a_{r,c}, with a_{V,c} the elements of column c
+ * corresponding to the non-parametric variables.
+ * If variable v appears in a column c_v, the a_{v,c} = 1 iff c = c_v,
+ * with all other entries in this virtual row equal to zero.
+ * If variable v appears in a row, then a_{v,c} is the element in column c
+ * of that row.
+ *
+ * Let v be the first variable with a_{v,c1}/a_{r,c1} != a_{v,c2}/a_{r,c2}.
+ * Then if a_{v,c1}/a_{r,c1} < a_{v,c2}/a_{r,c2}, i.e.,
+ * a_{v,c2} a_{r,c1} - a_{v,c1} a_{r,c2} > 0, c1 results in the minimal
+ * increment.  Otherwise, it's c2.
+ */
+static int lexmin_col_pair(struct isl_tab *tab,
+	int row, int col1, int col2, isl_int tmp)
+{
+	int i;
+	isl_int *tr;
+
+	tr = tab->mat->row[row] + 2 + tab->M;
+
+	for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
+		int s1, s2;
+		isl_int *r;
+
+		if (!tab->var[i].is_row) {
+			if (tab->var[i].index == col1)
+				return col2;
+			if (tab->var[i].index == col2)
+				return col1;
+			continue;
+		}
+
+		if (tab->var[i].index == row)
+			continue;
+
+		r = tab->mat->row[tab->var[i].index] + 2 + tab->M;
+		s1 = isl_int_sgn(r[col1]);
+		s2 = isl_int_sgn(r[col2]);
+		if (s1 == 0 && s2 == 0)
+			continue;
+		if (s1 < s2)
+			return col1;
+		if (s2 < s1)
+			return col2;
+
+		isl_int_mul(tmp, r[col2], tr[col1]);
+		isl_int_submul(tmp, r[col1], tr[col2]);
+		if (isl_int_is_pos(tmp))
+			return col1;
+		if (isl_int_is_neg(tmp))
+			return col2;
+	}
+	return -1;
+}
+
+/* Given a row in the tableau, find and return the column that would
+ * result in the lexicographically smallest, but positive, increment
+ * in the sample point.
+ * If there is no such column, then return tab->n_col.
+ * If anything goes wrong, return -1.
+ */
+static int lexmin_pivot_col(struct isl_tab *tab, int row)
+{
+	int j;
+	int col = tab->n_col;
+	isl_int *tr;
+	isl_int tmp;
+
+	tr = tab->mat->row[row] + 2 + tab->M;
+
+	isl_int_init(tmp);
+
+	for (j = tab->n_dead; j < tab->n_col; ++j) {
+		if (tab->col_var[j] >= 0 &&
+		    (tab->col_var[j] < tab->n_param  ||
+		    tab->col_var[j] >= tab->n_var - tab->n_div))
+			continue;
+
+		if (!isl_int_is_pos(tr[j]))
+			continue;
+
+		if (col == tab->n_col)
+			col = j;
+		else
+			col = lexmin_col_pair(tab, row, col, j, tmp);
+		isl_assert(tab->mat->ctx, col >= 0, goto error);
+	}
+
+	isl_int_clear(tmp);
+	return col;
+error:
+	isl_int_clear(tmp);
+	return -1;
+}
+
+/* Return the first known violated constraint, i.e., a non-negative
+ * constraint that currently has an either obviously negative value
+ * or a previously determined to be negative value.
+ *
+ * If any constraint has a negative coefficient for the big parameter,
+ * if any, then we return one of these first.
+ */
+static int first_neg(struct isl_tab *tab)
+{
+	int row;
+
+	if (tab->M)
+		for (row = tab->n_redundant; row < tab->n_row; ++row) {
+			if (!isl_tab_var_from_row(tab, row)->is_nonneg)
+				continue;
+			if (!isl_int_is_neg(tab->mat->row[row][2]))
+				continue;
+			if (tab->row_sign)
+				tab->row_sign[row] = isl_tab_row_neg;
+			return row;
+		}
+	for (row = tab->n_redundant; row < tab->n_row; ++row) {
+		if (!isl_tab_var_from_row(tab, row)->is_nonneg)
+			continue;
+		if (tab->row_sign) {
+			if (tab->row_sign[row] == 0 &&
+			    is_obviously_neg(tab, row))
+				tab->row_sign[row] = isl_tab_row_neg;
+			if (tab->row_sign[row] != isl_tab_row_neg)
+				continue;
+		} else if (!is_obviously_neg(tab, row))
+			continue;
+		return row;
+	}
+	return -1;
+}
+
+/* Check whether the invariant that all columns are lexico-positive
+ * is satisfied.  This function is not called from the current code
+ * but is useful during debugging.
+ */
+static void check_lexpos(struct isl_tab *tab) __attribute__ ((unused));
+static void check_lexpos(struct isl_tab *tab)
+{
+	unsigned off = 2 + tab->M;
+	int col;
+	int var;
+	int row;
+
+	for (col = tab->n_dead; col < tab->n_col; ++col) {
+		if (tab->col_var[col] >= 0 &&
+		    (tab->col_var[col] < tab->n_param ||
+		     tab->col_var[col] >= tab->n_var - tab->n_div))
+			continue;
+		for (var = tab->n_param; var < tab->n_var - tab->n_div; ++var) {
+			if (!tab->var[var].is_row) {
+				if (tab->var[var].index == col)
+					break;
+				else
+					continue;
+			}
+			row = tab->var[var].index;
+			if (isl_int_is_zero(tab->mat->row[row][off + col]))
+				continue;
+			if (isl_int_is_pos(tab->mat->row[row][off + col]))
+				break;
+			fprintf(stderr, "lexneg column %d (row %d)\n",
+				col, row);
+		}
+		if (var >= tab->n_var - tab->n_div)
+			fprintf(stderr, "zero column %d\n", col);
+	}
+}
+
+/* Report to the caller that the given constraint is part of an encountered
+ * conflict.
+ */
+static int report_conflicting_constraint(struct isl_tab *tab, int con)
+{
+	return tab->conflict(con, tab->conflict_user);
+}
+
+/* Given a conflicting row in the tableau, report all constraints
+ * involved in the row to the caller.  That is, the row itself
+ * (if it represents a constraint) and all constraint columns with
+ * non-zero (and therefore negative) coefficients.
+ */
+static int report_conflict(struct isl_tab *tab, int row)
+{
+	int j;
+	isl_int *tr;
+
+	if (!tab->conflict)
+		return 0;
+
+	if (tab->row_var[row] < 0 &&
+	    report_conflicting_constraint(tab, ~tab->row_var[row]) < 0)
+		return -1;
+
+	tr = tab->mat->row[row] + 2 + tab->M;
+
+	for (j = tab->n_dead; j < tab->n_col; ++j) {
+		if (tab->col_var[j] >= 0 &&
+		    (tab->col_var[j] < tab->n_param  ||
+		    tab->col_var[j] >= tab->n_var - tab->n_div))
+			continue;
+
+		if (!isl_int_is_neg(tr[j]))
+			continue;
+
+		if (tab->col_var[j] < 0 &&
+		    report_conflicting_constraint(tab, ~tab->col_var[j]) < 0)
+			return -1;
+	}
+
+	return 0;
+}
+
+/* Resolve all known or obviously violated constraints through pivoting.
+ * In particular, as long as we can find any violated constraint, we
+ * look for a pivoting column that would result in the lexicographically
+ * smallest increment in the sample point.  If there is no such column
+ * then the tableau is infeasible.
+ */
+static int restore_lexmin(struct isl_tab *tab) WARN_UNUSED;
+static int restore_lexmin(struct isl_tab *tab)
+{
+	int row, col;
+
+	if (!tab)
+		return -1;
+	if (tab->empty)
+		return 0;
+	while ((row = first_neg(tab)) != -1) {
+		col = lexmin_pivot_col(tab, row);
+		if (col >= tab->n_col) {
+			if (report_conflict(tab, row) < 0)
+				return -1;
+			if (isl_tab_mark_empty(tab) < 0)
+				return -1;
+			return 0;
+		}
+		if (col < 0)
+			return -1;
+		if (isl_tab_pivot(tab, row, col) < 0)
+			return -1;
+	}
+	return 0;
+}
+
+/* Given a row that represents an equality, look for an appropriate
+ * pivoting column.
+ * In particular, if there are any non-zero coefficients among
+ * the non-parameter variables, then we take the last of these
+ * variables.  Eliminating this variable in terms of the other
+ * variables and/or parameters does not influence the property
+ * that all column in the initial tableau are lexicographically
+ * positive.  The row corresponding to the eliminated variable
+ * will only have non-zero entries below the diagonal of the
+ * initial tableau.  That is, we transform
+ *
+ *		I				I
+ *		  1		into		a
+ *		    I				  I
+ *
+ * If there is no such non-parameter variable, then we are dealing with
+ * pure parameter equality and we pick any parameter with coefficient 1 or -1
+ * for elimination.  This will ensure that the eliminated parameter
+ * always has an integer value whenever all the other parameters are integral.
+ * If there is no such parameter then we return -1.
+ */
+static int last_var_col_or_int_par_col(struct isl_tab *tab, int row)
+{
+	unsigned off = 2 + tab->M;
+	int i;
+
+	for (i = tab->n_var - tab->n_div - 1; i >= 0 && i >= tab->n_param; --i) {
+		int col;
+		if (tab->var[i].is_row)
+			continue;
+		col = tab->var[i].index;
+		if (col <= tab->n_dead)
+			continue;
+		if (!isl_int_is_zero(tab->mat->row[row][off + col]))
+			return col;
+	}
+	for (i = tab->n_dead; i < tab->n_col; ++i) {
+		if (isl_int_is_one(tab->mat->row[row][off + i]))
+			return i;
+		if (isl_int_is_negone(tab->mat->row[row][off + i]))
+			return i;
+	}
+	return -1;
+}
+
+/* Add an equality that is known to be valid to the tableau.
+ * We first check if we can eliminate a variable or a parameter.
+ * If not, we add the equality as two inequalities.
+ * In this case, the equality was a pure parameter equality and there
+ * is no need to resolve any constraint violations.
+ */
+static struct isl_tab *add_lexmin_valid_eq(struct isl_tab *tab, isl_int *eq)
+{
+	int i;
+	int r;
+
+	if (!tab)
+		return NULL;
+	r = isl_tab_add_row(tab, eq);
+	if (r < 0)
+		goto error;
+
+	r = tab->con[r].index;
+	i = last_var_col_or_int_par_col(tab, r);
+	if (i < 0) {
+		tab->con[r].is_nonneg = 1;
+		if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
+			goto error;
+		isl_seq_neg(eq, eq, 1 + tab->n_var);
+		r = isl_tab_add_row(tab, eq);
+		if (r < 0)
+			goto error;
+		tab->con[r].is_nonneg = 1;
+		if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
+			goto error;
+	} else {
+		if (isl_tab_pivot(tab, r, i) < 0)
+			goto error;
+		if (isl_tab_kill_col(tab, i) < 0)
+			goto error;
+		tab->n_eq++;
+	}
+
+	return tab;
+error:
+	isl_tab_free(tab);
+	return NULL;
+}
+
+/* Check if the given row is a pure constant.
+ */
+static int is_constant(struct isl_tab *tab, int row)
+{
+	unsigned off = 2 + tab->M;
+
+	return isl_seq_first_non_zero(tab->mat->row[row] + off + tab->n_dead,
+					tab->n_col - tab->n_dead) == -1;
+}
+
+/* Add an equality that may or may not be valid to the tableau.
+ * If the resulting row is a pure constant, then it must be zero.
+ * Otherwise, the resulting tableau is empty.
+ *
+ * If the row is not a pure constant, then we add two inequalities,
+ * each time checking that they can be satisfied.
+ * In the end we try to use one of the two constraints to eliminate
+ * a column.
+ */
+static int add_lexmin_eq(struct isl_tab *tab, isl_int *eq) WARN_UNUSED;
+static int add_lexmin_eq(struct isl_tab *tab, isl_int *eq)
+{
+	int r1, r2;
+	int row;
+	struct isl_tab_undo *snap;
+
+	if (!tab)
+		return -1;
+	snap = isl_tab_snap(tab);
+	r1 = isl_tab_add_row(tab, eq);
+	if (r1 < 0)
+		return -1;
+	tab->con[r1].is_nonneg = 1;
+	if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r1]) < 0)
+		return -1;
+
+	row = tab->con[r1].index;
+	if (is_constant(tab, row)) {
+		if (!isl_int_is_zero(tab->mat->row[row][1]) ||
+		    (tab->M && !isl_int_is_zero(tab->mat->row[row][2]))) {
+			if (isl_tab_mark_empty(tab) < 0)
+				return -1;
+			return 0;
+		}
+		if (isl_tab_rollback(tab, snap) < 0)
+			return -1;
+		return 0;
+	}
+
+	if (restore_lexmin(tab) < 0)
+		return -1;
+	if (tab->empty)
+		return 0;
+
+	isl_seq_neg(eq, eq, 1 + tab->n_var);
+
+	r2 = isl_tab_add_row(tab, eq);
+	if (r2 < 0)
+		return -1;
+	tab->con[r2].is_nonneg = 1;
+	if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r2]) < 0)
+		return -1;
+
+	if (restore_lexmin(tab) < 0)
+		return -1;
+	if (tab->empty)
+		return 0;
+
+	if (!tab->con[r1].is_row) {
+		if (isl_tab_kill_col(tab, tab->con[r1].index) < 0)
+			return -1;
+	} else if (!tab->con[r2].is_row) {
+		if (isl_tab_kill_col(tab, tab->con[r2].index) < 0)
+			return -1;
+	}
+
+	if (tab->bmap) {
+		tab->bmap = isl_basic_map_add_ineq(tab->bmap, eq);
+		if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0)
+			return -1;
+		isl_seq_neg(eq, eq, 1 + tab->n_var);
+		tab->bmap = isl_basic_map_add_ineq(tab->bmap, eq);
+		isl_seq_neg(eq, eq, 1 + tab->n_var);
+		if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0)
+			return -1;
+		if (!tab->bmap)
+			return -1;
+	}
+
+	return 0;
+}
+
+/* Add an inequality to the tableau, resolving violations using
+ * restore_lexmin.
+ */
+static struct isl_tab *add_lexmin_ineq(struct isl_tab *tab, isl_int *ineq)
+{
+	int r;
+
+	if (!tab)
+		return NULL;
+	if (tab->bmap) {
+		tab->bmap = isl_basic_map_add_ineq(tab->bmap, ineq);
+		if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0)
+			goto error;
+		if (!tab->bmap)
+			goto error;
+	}
+	r = isl_tab_add_row(tab, ineq);
+	if (r < 0)
+		goto error;
+	tab->con[r].is_nonneg = 1;
+	if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
+		goto error;
+	if (isl_tab_row_is_redundant(tab, tab->con[r].index)) {
+		if (isl_tab_mark_redundant(tab, tab->con[r].index) < 0)
+			goto error;
+		return tab;
+	}
+
+	if (restore_lexmin(tab) < 0)
+		goto error;
+	if (!tab->empty && tab->con[r].is_row &&
+		 isl_tab_row_is_redundant(tab, tab->con[r].index))
+		if (isl_tab_mark_redundant(tab, tab->con[r].index) < 0)
+			goto error;
+	return tab;
+error:
+	isl_tab_free(tab);
+	return NULL;
+}
+
+/* Check if the coefficients of the parameters are all integral.
+ */
+static int integer_parameter(struct isl_tab *tab, int row)
+{
+	int i;
+	int col;
+	unsigned off = 2 + tab->M;
+
+	for (i = 0; i < tab->n_param; ++i) {
+		/* Eliminated parameter */
+		if (tab->var[i].is_row)
+			continue;
+		col = tab->var[i].index;
+		if (!isl_int_is_divisible_by(tab->mat->row[row][off + col],
+						tab->mat->row[row][0]))
+			return 0;
+	}
+	for (i = 0; i < tab->n_div; ++i) {
+		if (tab->var[tab->n_var - tab->n_div + i].is_row)
+			continue;
+		col = tab->var[tab->n_var - tab->n_div + i].index;
+		if (!isl_int_is_divisible_by(tab->mat->row[row][off + col],
+						tab->mat->row[row][0]))
+			return 0;
+	}
+	return 1;
+}
+
+/* Check if the coefficients of the non-parameter variables are all integral.
+ */
+static int integer_variable(struct isl_tab *tab, int row)
+{
+	int i;
+	unsigned off = 2 + tab->M;
+
+	for (i = tab->n_dead; i < tab->n_col; ++i) {
+		if (tab->col_var[i] >= 0 &&
+		    (tab->col_var[i] < tab->n_param ||
+		     tab->col_var[i] >= tab->n_var - tab->n_div))
+			continue;
+		if (!isl_int_is_divisible_by(tab->mat->row[row][off + i],
+						tab->mat->row[row][0]))
+			return 0;
+	}
+	return 1;
+}
+
+/* Check if the constant term is integral.
+ */
+static int integer_constant(struct isl_tab *tab, int row)
+{
+	return isl_int_is_divisible_by(tab->mat->row[row][1],
+					tab->mat->row[row][0]);
+}
+
+#define I_CST	1 << 0
+#define I_PAR	1 << 1
+#define I_VAR	1 << 2
+
+/* Check for next (non-parameter) variable after "var" (first if var == -1)
+ * that is non-integer and therefore requires a cut and return
+ * the index of the variable.
+ * For parametric tableaus, there are three parts in a row,
+ * the constant, the coefficients of the parameters and the rest.
+ * For each part, we check whether the coefficients in that part
+ * are all integral and if so, set the corresponding flag in *f.
+ * If the constant and the parameter part are integral, then the
+ * current sample value is integral and no cut is required
+ * (irrespective of whether the variable part is integral).
+ */
+static int next_non_integer_var(struct isl_tab *tab, int var, int *f)
+{
+	var = var < 0 ? tab->n_param : var + 1;
+
+	for (; var < tab->n_var - tab->n_div; ++var) {
+		int flags = 0;
+		int row;
+		if (!tab->var[var].is_row)
+			continue;
+		row = tab->var[var].index;
+		if (integer_constant(tab, row))
+			ISL_FL_SET(flags, I_CST);
+		if (integer_parameter(tab, row))
+			ISL_FL_SET(flags, I_PAR);
+		if (ISL_FL_ISSET(flags, I_CST) && ISL_FL_ISSET(flags, I_PAR))
+			continue;
+		if (integer_variable(tab, row))
+			ISL_FL_SET(flags, I_VAR);
+		*f = flags;
+		return var;
+	}
+	return -1;
+}
+
+/* Check for first (non-parameter) variable that is non-integer and
+ * therefore requires a cut and return the corresponding row.
+ * For parametric tableaus, there are three parts in a row,
+ * the constant, the coefficients of the parameters and the rest.
+ * For each part, we check whether the coefficients in that part
+ * are all integral and if so, set the corresponding flag in *f.
+ * If the constant and the parameter part are integral, then the
+ * current sample value is integral and no cut is required
+ * (irrespective of whether the variable part is integral).
+ */
+static int first_non_integer_row(struct isl_tab *tab, int *f)
+{
+	int var = next_non_integer_var(tab, -1, f);
+
+	return var < 0 ? -1 : tab->var[var].index;
+}
+
+/* Add a (non-parametric) cut to cut away the non-integral sample
+ * value of the given row.
+ *
+ * If the row is given by
+ *
+ *	m r = f + \sum_i a_i y_i
+ *
+ * then the cut is
+ *
+ *	c = - {-f/m} + \sum_i {a_i/m} y_i >= 0
+ *
+ * The big parameter, if any, is ignored, since it is assumed to be big
+ * enough to be divisible by any integer.
+ * If the tableau is actually a parametric tableau, then this function
+ * is only called when all coefficients of the parameters are integral.
+ * The cut therefore has zero coefficients for the parameters.
+ *
+ * The current value is known to be negative, so row_sign, if it
+ * exists, is set accordingly.
+ *
+ * Return the row of the cut or -1.
+ */
+static int add_cut(struct isl_tab *tab, int row)
+{
+	int i;
+	int r;
+	isl_int *r_row;
+	unsigned off = 2 + tab->M;
+
+	if (isl_tab_extend_cons(tab, 1) < 0)
+		return -1;
+	r = isl_tab_allocate_con(tab);
+	if (r < 0)
+		return -1;
+
+	r_row = tab->mat->row[tab->con[r].index];
+	isl_int_set(r_row[0], tab->mat->row[row][0]);
+	isl_int_neg(r_row[1], tab->mat->row[row][1]);
+	isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]);
+	isl_int_neg(r_row[1], r_row[1]);
+	if (tab->M)
+		isl_int_set_si(r_row[2], 0);
+	for (i = 0; i < tab->n_col; ++i)
+		isl_int_fdiv_r(r_row[off + i],
+			tab->mat->row[row][off + i], tab->mat->row[row][0]);
+
+	tab->con[r].is_nonneg = 1;
+	if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
+		return -1;
+	if (tab->row_sign)
+		tab->row_sign[tab->con[r].index] = isl_tab_row_neg;
+
+	return tab->con[r].index;
+}
+
+#define CUT_ALL 1
+#define CUT_ONE 0
+
+/* Given a non-parametric tableau, add cuts until an integer
+ * sample point is obtained or until the tableau is determined
+ * to be integer infeasible.
+ * As long as there is any non-integer value in the sample point,
+ * we add appropriate cuts, if possible, for each of these
+ * non-integer values and then resolve the violated
+ * cut constraints using restore_lexmin.
+ * If one of the corresponding rows is equal to an integral
+ * combination of variables/constraints plus a non-integral constant,
+ * then there is no way to obtain an integer point and we return
+ * a tableau that is marked empty.
+ * The parameter cutting_strategy controls the strategy used when adding cuts
+ * to remove non-integer points. CUT_ALL adds all possible cuts
+ * before continuing the search. CUT_ONE adds only one cut at a time.
+ */
+static struct isl_tab *cut_to_integer_lexmin(struct isl_tab *tab,
+	int cutting_strategy)
+{
+	int var;
+	int row;
+	int flags;
+
+	if (!tab)
+		return NULL;
+	if (tab->empty)
+		return tab;
+
+	while ((var = next_non_integer_var(tab, -1, &flags)) != -1) {
+		do {
+			if (ISL_FL_ISSET(flags, I_VAR)) {
+				if (isl_tab_mark_empty(tab) < 0)
+					goto error;
+				return tab;
+			}
+			row = tab->var[var].index;
+			row = add_cut(tab, row);
+			if (row < 0)
+				goto error;
+			if (cutting_strategy == CUT_ONE)
+				break;
+		} while ((var = next_non_integer_var(tab, var, &flags)) != -1);
+		if (restore_lexmin(tab) < 0)
+			goto error;
+		if (tab->empty)
+			break;
+	}
+	return tab;
+error:
+	isl_tab_free(tab);
+	return NULL;
+}
+
+/* Check whether all the currently active samples also satisfy the inequality
+ * "ineq" (treated as an equality if eq is set).
+ * Remove those samples that do not.
+ */
+static struct isl_tab *check_samples(struct isl_tab *tab, isl_int *ineq, int eq)
+{
+	int i;
+	isl_int v;
+
+	if (!tab)
+		return NULL;
+
+	isl_assert(tab->mat->ctx, tab->bmap, goto error);
+	isl_assert(tab->mat->ctx, tab->samples, goto error);
+	isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, goto error);
+
+	isl_int_init(v);
+	for (i = tab->n_outside; i < tab->n_sample; ++i) {
+		int sgn;
+		isl_seq_inner_product(ineq, tab->samples->row[i],
+					1 + tab->n_var, &v);
+		sgn = isl_int_sgn(v);
+		if (eq ? (sgn == 0) : (sgn >= 0))
+			continue;
+		tab = isl_tab_drop_sample(tab, i);
+		if (!tab)
+			break;
+	}
+	isl_int_clear(v);
+
+	return tab;
+error:
+	isl_tab_free(tab);
+	return NULL;
+}
+
+/* Check whether the sample value of the tableau is finite,
+ * i.e., either the tableau does not use a big parameter, or
+ * all values of the variables are equal to the big parameter plus
+ * some constant.  This constant is the actual sample value.
+ */
+static int sample_is_finite(struct isl_tab *tab)
+{
+	int i;
+
+	if (!tab->M)
+		return 1;
+
+	for (i = 0; i < tab->n_var; ++i) {
+		int row;
+		if (!tab->var[i].is_row)
+			return 0;
+		row = tab->var[i].index;
+		if (isl_int_ne(tab->mat->row[row][0], tab->mat->row[row][2]))
+			return 0;
+	}
+	return 1;
+}
+
+/* Check if the context tableau of sol has any integer points.
+ * Leave tab in empty state if no integer point can be found.
+ * If an integer point can be found and if moreover it is finite,
+ * then it is added to the list of sample values.
+ *
+ * This function is only called when none of the currently active sample
+ * values satisfies the most recently added constraint.
+ */
+static struct isl_tab *check_integer_feasible(struct isl_tab *tab)
+{
+	struct isl_tab_undo *snap;
+
+	if (!tab)
+		return NULL;
+
+	snap = isl_tab_snap(tab);
+	if (isl_tab_push_basis(tab) < 0)
+		goto error;
+
+	tab = cut_to_integer_lexmin(tab, CUT_ALL);
+	if (!tab)
+		goto error;
+
+	if (!tab->empty && sample_is_finite(tab)) {
+		struct isl_vec *sample;
+
+		sample = isl_tab_get_sample_value(tab);
+
+		if (isl_tab_add_sample(tab, sample) < 0)
+			goto error;
+	}
+
+	if (!tab->empty && isl_tab_rollback(tab, snap) < 0)
+		goto error;
+
+	return tab;
+error:
+	isl_tab_free(tab);
+	return NULL;
+}
+
+/* Check if any of the currently active sample values satisfies
+ * the inequality "ineq" (an equality if eq is set).
+ */
+static int tab_has_valid_sample(struct isl_tab *tab, isl_int *ineq, int eq)
+{
+	int i;
+	isl_int v;
+
+	if (!tab)
+		return -1;
+
+	isl_assert(tab->mat->ctx, tab->bmap, return -1);
+	isl_assert(tab->mat->ctx, tab->samples, return -1);
+	isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, return -1);
+
+	isl_int_init(v);
+	for (i = tab->n_outside; i < tab->n_sample; ++i) {
+		int sgn;
+		isl_seq_inner_product(ineq, tab->samples->row[i],
+					1 + tab->n_var, &v);
+		sgn = isl_int_sgn(v);
+		if (eq ? (sgn == 0) : (sgn >= 0))
+			break;
+	}
+	isl_int_clear(v);
+
+	return i < tab->n_sample;
+}
+
+/* Add a div specified by "div" to the tableau "tab" and return
+ * 1 if the div is obviously non-negative.
+ */
+static int context_tab_add_div(struct isl_tab *tab, struct isl_vec *div,
+	int (*add_ineq)(void *user, isl_int *), void *user)
+{
+	int i;
+	int r;
+	struct isl_mat *samples;
+	int nonneg;
+
+	r = isl_tab_add_div(tab, div, add_ineq, user);
+	if (r < 0)
+		return -1;
+	nonneg = tab->var[r].is_nonneg;
+	tab->var[r].frozen = 1;
+
+	samples = isl_mat_extend(tab->samples,
+			tab->n_sample, 1 + tab->n_var);
+	tab->samples = samples;
+	if (!samples)
+		return -1;
+	for (i = tab->n_outside; i < samples->n_row; ++i) {
+		isl_seq_inner_product(div->el + 1, samples->row[i],
+			div->size - 1, &samples->row[i][samples->n_col - 1]);
+		isl_int_fdiv_q(samples->row[i][samples->n_col - 1],
+			       samples->row[i][samples->n_col - 1], div->el[0]);
+	}
+
+	return nonneg;
+}
+
+/* Add a div specified by "div" to both the main tableau and
+ * the context tableau.  In case of the main tableau, we only
+ * need to add an extra div.  In the context tableau, we also
+ * need to express the meaning of the div.
+ * Return the index of the div or -1 if anything went wrong.
+ */
+static int add_div(struct isl_tab *tab, struct isl_context *context,
+	struct isl_vec *div)
+{
+	int r;
+	int nonneg;
+
+	if ((nonneg = context->op->add_div(context, div)) < 0)
+		goto error;
+
+	if (!context->op->is_ok(context))
+		goto error;
+
+	if (isl_tab_extend_vars(tab, 1) < 0)
+		goto error;
+	r = isl_tab_allocate_var(tab);
+	if (r < 0)
+		goto error;
+	if (nonneg)
+		tab->var[r].is_nonneg = 1;
+	tab->var[r].frozen = 1;
+	tab->n_div++;
+
+	return tab->n_div - 1;
+error:
+	context->op->invalidate(context);
+	return -1;
+}
+
+static int find_div(struct isl_tab *tab, isl_int *div, isl_int denom)
+{
+	int i;
+	unsigned total = isl_basic_map_total_dim(tab->bmap);
+
+	for (i = 0; i < tab->bmap->n_div; ++i) {
+		if (isl_int_ne(tab->bmap->div[i][0], denom))
+			continue;
+		if (!isl_seq_eq(tab->bmap->div[i] + 1, div, 1 + total))
+			continue;
+		return i;
+	}
+	return -1;
+}
+
+/* Return the index of a div that corresponds to "div".
+ * We first check if we already have such a div and if not, we create one.
+ */
+static int get_div(struct isl_tab *tab, struct isl_context *context,
+	struct isl_vec *div)
+{
+	int d;
+	struct isl_tab *context_tab = context->op->peek_tab(context);
+
+	if (!context_tab)
+		return -1;
+
+	d = find_div(context_tab, div->el + 1, div->el[0]);
+	if (d != -1)
+		return d;
+
+	return add_div(tab, context, div);
+}
+
+/* Add a parametric cut to cut away the non-integral sample value
+ * of the give row.
+ * Let a_i be the coefficients of the constant term and the parameters
+ * and let b_i be the coefficients of the variables or constraints
+ * in basis of the tableau.
+ * Let q be the div q = floor(\sum_i {-a_i} y_i).
+ *
+ * The cut is expressed as
+ *
+ *	c = \sum_i -{-a_i} y_i + \sum_i {b_i} x_i + q >= 0
+ *
+ * If q did not already exist in the context tableau, then it is added first.
+ * If q is in a column of the main tableau then the "+ q" can be accomplished
+ * by setting the corresponding entry to the denominator of the constraint.
+ * If q happens to be in a row of the main tableau, then the corresponding
+ * row needs to be added instead (taking care of the denominators).
+ * Note that this is very unlikely, but perhaps not entirely impossible.
+ *
+ * The current value of the cut is known to be negative (or at least
+ * non-positive), so row_sign is set accordingly.
+ *
+ * Return the row of the cut or -1.
+ */
+static int add_parametric_cut(struct isl_tab *tab, int row,
+	struct isl_context *context)
+{
+	struct isl_vec *div;
+	int d;
+	int i;
+	int r;
+	isl_int *r_row;
+	int col;
+	int n;
+	unsigned off = 2 + tab->M;
+
+	if (!context)
+		return -1;
+
+	div = get_row_parameter_div(tab, row);
+	if (!div)
+		return -1;
+
+	n = tab->n_div;
+	d = context->op->get_div(context, tab, div);
+	isl_vec_free(div);
+	if (d < 0)
+		return -1;
+
+	if (isl_tab_extend_cons(tab, 1) < 0)
+		return -1;
+	r = isl_tab_allocate_con(tab);
+	if (r < 0)
+		return -1;
+
+	r_row = tab->mat->row[tab->con[r].index];
+	isl_int_set(r_row[0], tab->mat->row[row][0]);
+	isl_int_neg(r_row[1], tab->mat->row[row][1]);
+	isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]);
+	isl_int_neg(r_row[1], r_row[1]);
+	if (tab->M)
+		isl_int_set_si(r_row[2], 0);
+	for (i = 0; i < tab->n_param; ++i) {
+		if (tab->var[i].is_row)
+			continue;
+		col = tab->var[i].index;
+		isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]);
+		isl_int_fdiv_r(r_row[off + col], r_row[off + col],
+				tab->mat->row[row][0]);
+		isl_int_neg(r_row[off + col], r_row[off + col]);
+	}
+	for (i = 0; i < tab->n_div; ++i) {
+		if (tab->var[tab->n_var - tab->n_div + i].is_row)
+			continue;
+		col = tab->var[tab->n_var - tab->n_div + i].index;
+		isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]);
+		isl_int_fdiv_r(r_row[off + col], r_row[off + col],
+				tab->mat->row[row][0]);
+		isl_int_neg(r_row[off + col], r_row[off + col]);
+	}
+	for (i = 0; i < tab->n_col; ++i) {
+		if (tab->col_var[i] >= 0 &&
+		    (tab->col_var[i] < tab->n_param ||
+		     tab->col_var[i] >= tab->n_var - tab->n_div))
+			continue;
+		isl_int_fdiv_r(r_row[off + i],
+			tab->mat->row[row][off + i], tab->mat->row[row][0]);
+	}
+	if (tab->var[tab->n_var - tab->n_div + d].is_row) {
+		isl_int gcd;
+		int d_row = tab->var[tab->n_var - tab->n_div + d].index;
+		isl_int_init(gcd);
+		isl_int_gcd(gcd, tab->mat->row[d_row][0], r_row[0]);
+		isl_int_divexact(r_row[0], r_row[0], gcd);
+		isl_int_divexact(gcd, tab->mat->row[d_row][0], gcd);
+		isl_seq_combine(r_row + 1, gcd, r_row + 1,
+				r_row[0], tab->mat->row[d_row] + 1,
+				off - 1 + tab->n_col);
+		isl_int_mul(r_row[0], r_row[0], tab->mat->row[d_row][0]);
+		isl_int_clear(gcd);
+	} else {
+		col = tab->var[tab->n_var - tab->n_div + d].index;
+		isl_int_set(r_row[off + col], tab->mat->row[row][0]);
+	}
+
+	tab->con[r].is_nonneg = 1;
+	if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
+		return -1;
+	if (tab->row_sign)
+		tab->row_sign[tab->con[r].index] = isl_tab_row_neg;
+
+	row = tab->con[r].index;
+
+	if (d >= n && context->op->detect_equalities(context, tab) < 0)
+		return -1;
+
+	return row;
+}
+
+/* Construct a tableau for bmap that can be used for computing
+ * the lexicographic minimum (or maximum) of bmap.
+ * If not NULL, then dom is the domain where the minimum
+ * should be computed.  In this case, we set up a parametric
+ * tableau with row signs (initialized to "unknown").
+ * If M is set, then the tableau will use a big parameter.
+ * If max is set, then a maximum should be computed instead of a minimum.
+ * This means that for each variable x, the tableau will contain the variable
+ * x' = M - x, rather than x' = M + x.  This in turn means that the coefficient
+ * of the variables in all constraints are negated prior to adding them
+ * to the tableau.
+ */
+static struct isl_tab *tab_for_lexmin(struct isl_basic_map *bmap,
+	struct isl_basic_set *dom, unsigned M, int max)
+{
+	int i;
+	struct isl_tab *tab;
+	unsigned n_var;
+	unsigned o_var;
+
+	tab = isl_tab_alloc(bmap->ctx, 2 * bmap->n_eq + bmap->n_ineq + 1,
+			    isl_basic_map_total_dim(bmap), M);
+	if (!tab)
+		return NULL;
+
+	tab->rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
+	if (dom) {
+		tab->n_param = isl_basic_set_total_dim(dom) - dom->n_div;
+		tab->n_div = dom->n_div;
+		tab->row_sign = isl_calloc_array(bmap->ctx,
+					enum isl_tab_row_sign, tab->mat->n_row);
+		if (tab->mat->n_row && !tab->row_sign)
+			goto error;
+	}
+	if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) {
+		if (isl_tab_mark_empty(tab) < 0)
+			goto error;
+		return tab;
+	}
+
+	for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
+		tab->var[i].is_nonneg = 1;
+		tab->var[i].frozen = 1;
+	}
+	o_var = 1 + tab->n_param;
+	n_var = tab->n_var - tab->n_param - tab->n_div;
+	for (i = 0; i < bmap->n_eq; ++i) {
+		if (max)
+			isl_seq_neg(bmap->eq[i] + o_var,
+				    bmap->eq[i] + o_var, n_var);
+		tab = add_lexmin_valid_eq(tab, bmap->eq[i]);
+		if (max)
+			isl_seq_neg(bmap->eq[i] + o_var,
+				    bmap->eq[i] + o_var, n_var);
+		if (!tab || tab->empty)
+			return tab;
+	}
+	if (bmap->n_eq && restore_lexmin(tab) < 0)
+		goto error;
+	for (i = 0; i < bmap->n_ineq; ++i) {
+		if (max)
+			isl_seq_neg(bmap->ineq[i] + o_var,
+				    bmap->ineq[i] + o_var, n_var);
+		tab = add_lexmin_ineq(tab, bmap->ineq[i]);
+		if (max)
+			isl_seq_neg(bmap->ineq[i] + o_var,
+				    bmap->ineq[i] + o_var, n_var);
+		if (!tab || tab->empty)
+			return tab;
+	}
+	return tab;
+error:
+	isl_tab_free(tab);
+	return NULL;
+}
+
+/* Given a main tableau where more than one row requires a split,
+ * determine and return the "best" row to split on.
+ *
+ * Given two rows in the main tableau, if the inequality corresponding
+ * to the first row is redundant with respect to that of the second row
+ * in the current tableau, then it is better to split on the second row,
+ * since in the positive part, both row will be positive.
+ * (In the negative part a pivot will have to be performed and just about
+ * anything can happen to the sign of the other row.)
+ *
+ * As a simple heuristic, we therefore select the row that makes the most
+ * of the other rows redundant.
+ *
+ * Perhaps it would also be useful to look at the number of constraints
+ * that conflict with any given constraint.
+ *
+ * best is the best row so far (-1 when we have not found any row yet).
+ * best_r is the number of other rows made redundant by row best.
+ * When best is still -1, bset_r is meaningless, but it is initialized
+ * to some arbitrary value (0) anyway.  Without this redundant initialization
+ * valgrind may warn about uninitialized memory accesses when isl
+ * is compiled with some versions of gcc.
+ */
+static int best_split(struct isl_tab *tab, struct isl_tab *context_tab)
+{
+	struct isl_tab_undo *snap;
+	int split;
+	int row;
+	int best = -1;
+	int best_r = 0;
+
+	if (isl_tab_extend_cons(context_tab, 2) < 0)
+		return -1;
+
+	snap = isl_tab_snap(context_tab);
+
+	for (split = tab->n_redundant; split < tab->n_row; ++split) {
+		struct isl_tab_undo *snap2;
+		struct isl_vec *ineq = NULL;
+		int r = 0;
+		int ok;
+
+		if (!isl_tab_var_from_row(tab, split)->is_nonneg)
+			continue;
+		if (tab->row_sign[split] != isl_tab_row_any)
+			continue;
+
+		ineq = get_row_parameter_ineq(tab, split);
+		if (!ineq)
+			return -1;
+		ok = isl_tab_add_ineq(context_tab, ineq->el) >= 0;
+		isl_vec_free(ineq);
+		if (!ok)
+			return -1;
+
+		snap2 = isl_tab_snap(context_tab);
+
+		for (row = tab->n_redundant; row < tab->n_row; ++row) {
+			struct isl_tab_var *var;
+
+			if (row == split)
+				continue;
+			if (!isl_tab_var_from_row(tab, row)->is_nonneg)
+				continue;
+			if (tab->row_sign[row] != isl_tab_row_any)
+				continue;
+
+			ineq = get_row_parameter_ineq(tab, row);
+			if (!ineq)
+				return -1;
+			ok = isl_tab_add_ineq(context_tab, ineq->el) >= 0;
+			isl_vec_free(ineq);
+			if (!ok)
+				return -1;
+			var = &context_tab->con[context_tab->n_con - 1];
+			if (!context_tab->empty &&
+			    !isl_tab_min_at_most_neg_one(context_tab, var))
+				r++;
+			if (isl_tab_rollback(context_tab, snap2) < 0)
+				return -1;
+		}
+		if (best == -1 || r > best_r) {
+			best = split;
+			best_r = r;
+		}
+		if (isl_tab_rollback(context_tab, snap) < 0)
+			return -1;
+	}
+
+	return best;
+}
+
+static struct isl_basic_set *context_lex_peek_basic_set(
+	struct isl_context *context)
+{
+	struct isl_context_lex *clex = (struct isl_context_lex *)context;
+	if (!clex->tab)
+		return NULL;
+	return isl_tab_peek_bset(clex->tab);
+}
+
+static struct isl_tab *context_lex_peek_tab(struct isl_context *context)
+{
+	struct isl_context_lex *clex = (struct isl_context_lex *)context;
+	return clex->tab;
+}
+
+static void context_lex_add_eq(struct isl_context *context, isl_int *eq,
+		int check, int update)
+{
+	struct isl_context_lex *clex = (struct isl_context_lex *)context;
+	if (isl_tab_extend_cons(clex->tab, 2) < 0)
+		goto error;
+	if (add_lexmin_eq(clex->tab, eq) < 0)
+		goto error;
+	if (check) {
+		int v = tab_has_valid_sample(clex->tab, eq, 1);
+		if (v < 0)
+			goto error;
+		if (!v)
+			clex->tab = check_integer_feasible(clex->tab);
+	}
+	if (update)
+		clex->tab = check_samples(clex->tab, eq, 1);
+	return;
+error:
+	isl_tab_free(clex->tab);
+	clex->tab = NULL;
+}
+
+static void context_lex_add_ineq(struct isl_context *context, isl_int *ineq,
+		int check, int update)
+{
+	struct isl_context_lex *clex = (struct isl_context_lex *)context;
+	if (isl_tab_extend_cons(clex->tab, 1) < 0)
+		goto error;
+	clex->tab = add_lexmin_ineq(clex->tab, ineq);
+	if (check) {
+		int v = tab_has_valid_sample(clex->tab, ineq, 0);
+		if (v < 0)
+			goto error;
+		if (!v)
+			clex->tab = check_integer_feasible(clex->tab);
+	}
+	if (update)
+		clex->tab = check_samples(clex->tab, ineq, 0);
+	return;
+error:
+	isl_tab_free(clex->tab);
+	clex->tab = NULL;
+}
+
+static int context_lex_add_ineq_wrap(void *user, isl_int *ineq)
+{
+	struct isl_context *context = (struct isl_context *)user;
+	context_lex_add_ineq(context, ineq, 0, 0);
+	return context->op->is_ok(context) ? 0 : -1;
+}
+
+/* Check which signs can be obtained by "ineq" on all the currently
+ * active sample values.  See row_sign for more information.
+ */
+static enum isl_tab_row_sign tab_ineq_sign(struct isl_tab *tab, isl_int *ineq,
+	int strict)
+{
+	int i;
+	int sgn;
+	isl_int tmp;
+	enum isl_tab_row_sign res = isl_tab_row_unknown;
+
+	isl_assert(tab->mat->ctx, tab->samples, return isl_tab_row_unknown);
+	isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var,
+			return isl_tab_row_unknown);
+
+	isl_int_init(tmp);
+	for (i = tab->n_outside; i < tab->n_sample; ++i) {
+		isl_seq_inner_product(tab->samples->row[i], ineq,
+					1 + tab->n_var, &tmp);
+		sgn = isl_int_sgn(tmp);
+		if (sgn > 0 || (sgn == 0 && strict)) {
+			if (res == isl_tab_row_unknown)
+				res = isl_tab_row_pos;
+			if (res == isl_tab_row_neg)
+				res = isl_tab_row_any;
+		}
+		if (sgn < 0) {
+			if (res == isl_tab_row_unknown)
+				res = isl_tab_row_neg;
+			if (res == isl_tab_row_pos)
+				res = isl_tab_row_any;
+		}
+		if (res == isl_tab_row_any)
+			break;
+	}
+	isl_int_clear(tmp);
+
+	return res;
+}
+
+static enum isl_tab_row_sign context_lex_ineq_sign(struct isl_context *context,
+			isl_int *ineq, int strict)
+{
+	struct isl_context_lex *clex = (struct isl_context_lex *)context;
+	return tab_ineq_sign(clex->tab, ineq, strict);
+}
+
+/* Check whether "ineq" can be added to the tableau without rendering
+ * it infeasible.
+ */
+static int context_lex_test_ineq(struct isl_context *context, isl_int *ineq)
+{
+	struct isl_context_lex *clex = (struct isl_context_lex *)context;
+	struct isl_tab_undo *snap;
+	int feasible;
+
+	if (!clex->tab)
+		return -1;
+
+	if (isl_tab_extend_cons(clex->tab, 1) < 0)
+		return -1;
+
+	snap = isl_tab_snap(clex->tab);
+	if (isl_tab_push_basis(clex->tab) < 0)
+		return -1;
+	clex->tab = add_lexmin_ineq(clex->tab, ineq);
+	clex->tab = check_integer_feasible(clex->tab);
+	if (!clex->tab)
+		return -1;
+	feasible = !clex->tab->empty;
+	if (isl_tab_rollback(clex->tab, snap) < 0)
+		return -1;
+
+	return feasible;
+}
+
+static int context_lex_get_div(struct isl_context *context, struct isl_tab *tab,
+		struct isl_vec *div)
+{
+	return get_div(tab, context, div);
+}
+
+/* Add a div specified by "div" to the context tableau and return
+ * 1 if the div is obviously non-negative.
+ * context_tab_add_div will always return 1, because all variables
+ * in a isl_context_lex tableau are non-negative.
+ * However, if we are using a big parameter in the context, then this only
+ * reflects the non-negativity of the variable used to _encode_ the
+ * div, i.e., div' = M + div, so we can't draw any conclusions.
+ */
+static int context_lex_add_div(struct isl_context *context, struct isl_vec *div)
+{
+	struct isl_context_lex *clex = (struct isl_context_lex *)context;
+	int nonneg;
+	nonneg = context_tab_add_div(clex->tab, div,
+					context_lex_add_ineq_wrap, context);
+	if (nonneg < 0)
+		return -1;
+	if (clex->tab->M)
+		return 0;
+	return nonneg;
+}
+
+static int context_lex_detect_equalities(struct isl_context *context,
+		struct isl_tab *tab)
+{
+	return 0;
+}
+
+static int context_lex_best_split(struct isl_context *context,
+		struct isl_tab *tab)
+{
+	struct isl_context_lex *clex = (struct isl_context_lex *)context;
+	struct isl_tab_undo *snap;
+	int r;
+
+	snap = isl_tab_snap(clex->tab);
+	if (isl_tab_push_basis(clex->tab) < 0)
+		return -1;
+	r = best_split(tab, clex->tab);
+
+	if (r >= 0 && isl_tab_rollback(clex->tab, snap) < 0)
+		return -1;
+
+	return r;
+}
+
+static int context_lex_is_empty(struct isl_context *context)
+{
+	struct isl_context_lex *clex = (struct isl_context_lex *)context;
+	if (!clex->tab)
+		return -1;
+	return clex->tab->empty;
+}
+
+static void *context_lex_save(struct isl_context *context)
+{
+	struct isl_context_lex *clex = (struct isl_context_lex *)context;
+	struct isl_tab_undo *snap;
+
+	snap = isl_tab_snap(clex->tab);
+	if (isl_tab_push_basis(clex->tab) < 0)
+		return NULL;
+	if (isl_tab_save_samples(clex->tab) < 0)
+		return NULL;
+
+	return snap;
+}
+
+static void context_lex_restore(struct isl_context *context, void *save)
+{
+	struct isl_context_lex *clex = (struct isl_context_lex *)context;
+	if (isl_tab_rollback(clex->tab, (struct isl_tab_undo *)save) < 0) {
+		isl_tab_free(clex->tab);
+		clex->tab = NULL;
+	}
+}
+
+static void context_lex_discard(void *save)
+{
+}
+
+static int context_lex_is_ok(struct isl_context *context)
+{
+	struct isl_context_lex *clex = (struct isl_context_lex *)context;
+	return !!clex->tab;
+}
+
+/* For each variable in the context tableau, check if the variable can
+ * only attain non-negative values.  If so, mark the parameter as non-negative
+ * in the main tableau.  This allows for a more direct identification of some
+ * cases of violated constraints.
+ */
+static struct isl_tab *tab_detect_nonnegative_parameters(struct isl_tab *tab,
+	struct isl_tab *context_tab)
+{
+	int i;
+	struct isl_tab_undo *snap;
+	struct isl_vec *ineq = NULL;
+	struct isl_tab_var *var;
+	int n;
+
+	if (context_tab->n_var == 0)
+		return tab;
+
+	ineq = isl_vec_alloc(tab->mat->ctx, 1 + context_tab->n_var);
+	if (!ineq)
+		goto error;
+
+	if (isl_tab_extend_cons(context_tab, 1) < 0)
+		goto error;
+
+	snap = isl_tab_snap(context_tab);
+
+	n = 0;
+	isl_seq_clr(ineq->el, ineq->size);
+	for (i = 0; i < context_tab->n_var; ++i) {
+		isl_int_set_si(ineq->el[1 + i], 1);
+		if (isl_tab_add_ineq(context_tab, ineq->el) < 0)
+			goto error;
+		var = &context_tab->con[context_tab->n_con - 1];
+		if (!context_tab->empty &&
+		    !isl_tab_min_at_most_neg_one(context_tab, var)) {
+			int j = i;
+			if (i >= tab->n_param)
+				j = i - tab->n_param + tab->n_var - tab->n_div;
+			tab->var[j].is_nonneg = 1;
+			n++;
+		}
+		isl_int_set_si(ineq->el[1 + i], 0);
+		if (isl_tab_rollback(context_tab, snap) < 0)
+			goto error;
+	}
+
+	if (context_tab->M && n == context_tab->n_var) {
+		context_tab->mat = isl_mat_drop_cols(context_tab->mat, 2, 1);
+		context_tab->M = 0;
+	}
+
+	isl_vec_free(ineq);
+	return tab;
+error:
+	isl_vec_free(ineq);
+	isl_tab_free(tab);
+	return NULL;
+}
+
+static struct isl_tab *context_lex_detect_nonnegative_parameters(
+	struct isl_context *context, struct isl_tab *tab)
+{
+	struct isl_context_lex *clex = (struct isl_context_lex *)context;
+	struct isl_tab_undo *snap;
+
+	if (!tab)
+		return NULL;
+
+	snap = isl_tab_snap(clex->tab);
+	if (isl_tab_push_basis(clex->tab) < 0)
+		goto error;
+
+	tab = tab_detect_nonnegative_parameters(tab, clex->tab);
+
+	if (isl_tab_rollback(clex->tab, snap) < 0)
+		goto error;
+
+	return tab;
+error:
+	isl_tab_free(tab);
+	return NULL;
+}
+
+static void context_lex_invalidate(struct isl_context *context)
+{
+	struct isl_context_lex *clex = (struct isl_context_lex *)context;
+	isl_tab_free(clex->tab);
+	clex->tab = NULL;
+}
+
+static void context_lex_free(struct isl_context *context)
+{
+	struct isl_context_lex *clex = (struct isl_context_lex *)context;
+	isl_tab_free(clex->tab);
+	free(clex);
+}
+
+struct isl_context_op isl_context_lex_op = {
+	context_lex_detect_nonnegative_parameters,
+	context_lex_peek_basic_set,
+	context_lex_peek_tab,
+	context_lex_add_eq,
+	context_lex_add_ineq,
+	context_lex_ineq_sign,
+	context_lex_test_ineq,
+	context_lex_get_div,
+	context_lex_add_div,
+	context_lex_detect_equalities,
+	context_lex_best_split,
+	context_lex_is_empty,
+	context_lex_is_ok,
+	context_lex_save,
+	context_lex_restore,
+	context_lex_discard,
+	context_lex_invalidate,
+	context_lex_free,
+};
+
+static struct isl_tab *context_tab_for_lexmin(struct isl_basic_set *bset)
+{
+	struct isl_tab *tab;
+
+	if (!bset)
+		return NULL;
+	tab = tab_for_lexmin((struct isl_basic_map *)bset, NULL, 1, 0);
+	if (!tab)
+		goto error;
+	if (isl_tab_track_bset(tab, bset) < 0)
+		goto error;
+	tab = isl_tab_init_samples(tab);
+	return tab;
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+static struct isl_context *isl_context_lex_alloc(struct isl_basic_set *dom)
+{
+	struct isl_context_lex *clex;
+
+	if (!dom)
+		return NULL;
+
+	clex = isl_alloc_type(dom->ctx, struct isl_context_lex);
+	if (!clex)
+		return NULL;
+
+	clex->context.op = &isl_context_lex_op;
+
+	clex->tab = context_tab_for_lexmin(isl_basic_set_copy(dom));
+	if (restore_lexmin(clex->tab) < 0)
+		goto error;
+	clex->tab = check_integer_feasible(clex->tab);
+	if (!clex->tab)
+		goto error;
+
+	return &clex->context;
+error:
+	clex->context.op->free(&clex->context);
+	return NULL;
+}
+
+/* Representation of the context when using generalized basis reduction.
+ *
+ * "shifted" contains the offsets of the unit hypercubes that lie inside the
+ * context.  Any rational point in "shifted" can therefore be rounded
+ * up to an integer point in the context.
+ * If the context is constrained by any equality, then "shifted" is not used
+ * as it would be empty.
+ */
+struct isl_context_gbr {
+	struct isl_context context;
+	struct isl_tab *tab;
+	struct isl_tab *shifted;
+	struct isl_tab *cone;
+};
+
+static struct isl_tab *context_gbr_detect_nonnegative_parameters(
+	struct isl_context *context, struct isl_tab *tab)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+	if (!tab)
+		return NULL;
+	return tab_detect_nonnegative_parameters(tab, cgbr->tab);
+}
+
+static struct isl_basic_set *context_gbr_peek_basic_set(
+	struct isl_context *context)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+	if (!cgbr->tab)
+		return NULL;
+	return isl_tab_peek_bset(cgbr->tab);
+}
+
+static struct isl_tab *context_gbr_peek_tab(struct isl_context *context)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+	return cgbr->tab;
+}
+
+/* Initialize the "shifted" tableau of the context, which
+ * contains the constraints of the original tableau shifted
+ * by the sum of all negative coefficients.  This ensures
+ * that any rational point in the shifted tableau can
+ * be rounded up to yield an integer point in the original tableau.
+ */
+static void gbr_init_shifted(struct isl_context_gbr *cgbr)
+{
+	int i, j;
+	struct isl_vec *cst;
+	struct isl_basic_set *bset = isl_tab_peek_bset(cgbr->tab);
+	unsigned dim = isl_basic_set_total_dim(bset);
+
+	cst = isl_vec_alloc(cgbr->tab->mat->ctx, bset->n_ineq);
+	if (!cst)
+		return;
+
+	for (i = 0; i < bset->n_ineq; ++i) {
+		isl_int_set(cst->el[i], bset->ineq[i][0]);
+		for (j = 0; j < dim; ++j) {
+			if (!isl_int_is_neg(bset->ineq[i][1 + j]))
+				continue;
+			isl_int_add(bset->ineq[i][0], bset->ineq[i][0],
+				    bset->ineq[i][1 + j]);
+		}
+	}
+
+	cgbr->shifted = isl_tab_from_basic_set(bset, 0);
+
+	for (i = 0; i < bset->n_ineq; ++i)
+		isl_int_set(bset->ineq[i][0], cst->el[i]);
+
+	isl_vec_free(cst);
+}
+
+/* Check if the shifted tableau is non-empty, and if so
+ * use the sample point to construct an integer point
+ * of the context tableau.
+ */
+static struct isl_vec *gbr_get_shifted_sample(struct isl_context_gbr *cgbr)
+{
+	struct isl_vec *sample;
+
+	if (!cgbr->shifted)
+		gbr_init_shifted(cgbr);
+	if (!cgbr->shifted)
+		return NULL;
+	if (cgbr->shifted->empty)
+		return isl_vec_alloc(cgbr->tab->mat->ctx, 0);
+
+	sample = isl_tab_get_sample_value(cgbr->shifted);
+	sample = isl_vec_ceil(sample);
+
+	return sample;
+}
+
+static struct isl_basic_set *drop_constant_terms(struct isl_basic_set *bset)
+{
+	int i;
+
+	if (!bset)
+		return NULL;
+
+	for (i = 0; i < bset->n_eq; ++i)
+		isl_int_set_si(bset->eq[i][0], 0);
+
+	for (i = 0; i < bset->n_ineq; ++i)
+		isl_int_set_si(bset->ineq[i][0], 0);
+
+	return bset;
+}
+
+static int use_shifted(struct isl_context_gbr *cgbr)
+{
+	if (!cgbr->tab)
+		return 0;
+	return cgbr->tab->bmap->n_eq == 0 && cgbr->tab->bmap->n_div == 0;
+}
+
+static struct isl_vec *gbr_get_sample(struct isl_context_gbr *cgbr)
+{
+	struct isl_basic_set *bset;
+	struct isl_basic_set *cone;
+
+	if (isl_tab_sample_is_integer(cgbr->tab))
+		return isl_tab_get_sample_value(cgbr->tab);
+
+	if (use_shifted(cgbr)) {
+		struct isl_vec *sample;
+
+		sample = gbr_get_shifted_sample(cgbr);
+		if (!sample || sample->size > 0)
+			return sample;
+
+		isl_vec_free(sample);
+	}
+
+	if (!cgbr->cone) {
+		bset = isl_tab_peek_bset(cgbr->tab);
+		cgbr->cone = isl_tab_from_recession_cone(bset, 0);
+		if (!cgbr->cone)
+			return NULL;
+		if (isl_tab_track_bset(cgbr->cone,
+					isl_basic_set_copy(bset)) < 0)
+			return NULL;
+	}
+	if (isl_tab_detect_implicit_equalities(cgbr->cone) < 0)
+		return NULL;
+
+	if (cgbr->cone->n_dead == cgbr->cone->n_col) {
+		struct isl_vec *sample;
+		struct isl_tab_undo *snap;
+
+		if (cgbr->tab->basis) {
+			if (cgbr->tab->basis->n_col != 1 + cgbr->tab->n_var) {
+				isl_mat_free(cgbr->tab->basis);
+				cgbr->tab->basis = NULL;
+			}
+			cgbr->tab->n_zero = 0;
+			cgbr->tab->n_unbounded = 0;
+		}
+
+		snap = isl_tab_snap(cgbr->tab);
+
+		sample = isl_tab_sample(cgbr->tab);
+
+		if (isl_tab_rollback(cgbr->tab, snap) < 0) {
+			isl_vec_free(sample);
+			return NULL;
+		}
+
+		return sample;
+	}
+
+	cone = isl_basic_set_dup(isl_tab_peek_bset(cgbr->cone));
+	cone = drop_constant_terms(cone);
+	cone = isl_basic_set_update_from_tab(cone, cgbr->cone);
+	cone = isl_basic_set_underlying_set(cone);
+	cone = isl_basic_set_gauss(cone, NULL);
+
+	bset = isl_basic_set_dup(isl_tab_peek_bset(cgbr->tab));
+	bset = isl_basic_set_update_from_tab(bset, cgbr->tab);
+	bset = isl_basic_set_underlying_set(bset);
+	bset = isl_basic_set_gauss(bset, NULL);
+
+	return isl_basic_set_sample_with_cone(bset, cone);
+}
+
+static void check_gbr_integer_feasible(struct isl_context_gbr *cgbr)
+{
+	struct isl_vec *sample;
+
+	if (!cgbr->tab)
+		return;
+
+	if (cgbr->tab->empty)
+		return;
+
+	sample = gbr_get_sample(cgbr);
+	if (!sample)
+		goto error;
+
+	if (sample->size == 0) {
+		isl_vec_free(sample);
+		if (isl_tab_mark_empty(cgbr->tab) < 0)
+			goto error;
+		return;
+	}
+
+	if (isl_tab_add_sample(cgbr->tab, sample) < 0)
+		goto error;
+
+	return;
+error:
+	isl_tab_free(cgbr->tab);
+	cgbr->tab = NULL;
+}
+
+static struct isl_tab *add_gbr_eq(struct isl_tab *tab, isl_int *eq)
+{
+	if (!tab)
+		return NULL;
+
+	if (isl_tab_extend_cons(tab, 2) < 0)
+		goto error;
+
+	if (isl_tab_add_eq(tab, eq) < 0)
+		goto error;
+
+	return tab;
+error:
+	isl_tab_free(tab);
+	return NULL;
+}
+
+/* Add the equality described by "eq" to the context.
+ * If "check" is set, then we check if the context is empty after
+ * adding the equality.
+ * If "update" is set, then we check if the samples are still valid.
+ *
+ * We do not explicitly add shifted copies of the equality to
+ * cgbr->shifted since they would conflict with each other.
+ * Instead, we directly mark cgbr->shifted empty.
+ */
+static void context_gbr_add_eq(struct isl_context *context, isl_int *eq,
+		int check, int update)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+
+	cgbr->tab = add_gbr_eq(cgbr->tab, eq);
+
+	if (cgbr->shifted && !cgbr->shifted->empty && use_shifted(cgbr)) {
+		if (isl_tab_mark_empty(cgbr->shifted) < 0)
+			goto error;
+	}
+
+	if (cgbr->cone && cgbr->cone->n_col != cgbr->cone->n_dead) {
+		if (isl_tab_extend_cons(cgbr->cone, 2) < 0)
+			goto error;
+		if (isl_tab_add_eq(cgbr->cone, eq) < 0)
+			goto error;
+	}
+
+	if (check) {
+		int v = tab_has_valid_sample(cgbr->tab, eq, 1);
+		if (v < 0)
+			goto error;
+		if (!v)
+			check_gbr_integer_feasible(cgbr);
+	}
+	if (update)
+		cgbr->tab = check_samples(cgbr->tab, eq, 1);
+	return;
+error:
+	isl_tab_free(cgbr->tab);
+	cgbr->tab = NULL;
+}
+
+static void add_gbr_ineq(struct isl_context_gbr *cgbr, isl_int *ineq)
+{
+	if (!cgbr->tab)
+		return;
+
+	if (isl_tab_extend_cons(cgbr->tab, 1) < 0)
+		goto error;
+
+	if (isl_tab_add_ineq(cgbr->tab, ineq) < 0)
+		goto error;
+
+	if (cgbr->shifted && !cgbr->shifted->empty && use_shifted(cgbr)) {
+		int i;
+		unsigned dim;
+		dim = isl_basic_map_total_dim(cgbr->tab->bmap);
+
+		if (isl_tab_extend_cons(cgbr->shifted, 1) < 0)
+			goto error;
+
+		for (i = 0; i < dim; ++i) {
+			if (!isl_int_is_neg(ineq[1 + i]))
+				continue;
+			isl_int_add(ineq[0], ineq[0], ineq[1 + i]);
+		}
+
+		if (isl_tab_add_ineq(cgbr->shifted, ineq) < 0)
+			goto error;
+
+		for (i = 0; i < dim; ++i) {
+			if (!isl_int_is_neg(ineq[1 + i]))
+				continue;
+			isl_int_sub(ineq[0], ineq[0], ineq[1 + i]);
+		}
+	}
+
+	if (cgbr->cone && cgbr->cone->n_col != cgbr->cone->n_dead) {
+		if (isl_tab_extend_cons(cgbr->cone, 1) < 0)
+			goto error;
+		if (isl_tab_add_ineq(cgbr->cone, ineq) < 0)
+			goto error;
+	}
+
+	return;
+error:
+	isl_tab_free(cgbr->tab);
+	cgbr->tab = NULL;
+}
+
+static void context_gbr_add_ineq(struct isl_context *context, isl_int *ineq,
+		int check, int update)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+
+	add_gbr_ineq(cgbr, ineq);
+	if (!cgbr->tab)
+		return;
+
+	if (check) {
+		int v = tab_has_valid_sample(cgbr->tab, ineq, 0);
+		if (v < 0)
+			goto error;
+		if (!v)
+			check_gbr_integer_feasible(cgbr);
+	}
+	if (update)
+		cgbr->tab = check_samples(cgbr->tab, ineq, 0);
+	return;
+error:
+	isl_tab_free(cgbr->tab);
+	cgbr->tab = NULL;
+}
+
+static int context_gbr_add_ineq_wrap(void *user, isl_int *ineq)
+{
+	struct isl_context *context = (struct isl_context *)user;
+	context_gbr_add_ineq(context, ineq, 0, 0);
+	return context->op->is_ok(context) ? 0 : -1;
+}
+
+static enum isl_tab_row_sign context_gbr_ineq_sign(struct isl_context *context,
+			isl_int *ineq, int strict)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+	return tab_ineq_sign(cgbr->tab, ineq, strict);
+}
+
+/* Check whether "ineq" can be added to the tableau without rendering
+ * it infeasible.
+ */
+static int context_gbr_test_ineq(struct isl_context *context, isl_int *ineq)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+	struct isl_tab_undo *snap;
+	struct isl_tab_undo *shifted_snap = NULL;
+	struct isl_tab_undo *cone_snap = NULL;
+	int feasible;
+
+	if (!cgbr->tab)
+		return -1;
+
+	if (isl_tab_extend_cons(cgbr->tab, 1) < 0)
+		return -1;
+
+	snap = isl_tab_snap(cgbr->tab);
+	if (cgbr->shifted)
+		shifted_snap = isl_tab_snap(cgbr->shifted);
+	if (cgbr->cone)
+		cone_snap = isl_tab_snap(cgbr->cone);
+	add_gbr_ineq(cgbr, ineq);
+	check_gbr_integer_feasible(cgbr);
+	if (!cgbr->tab)
+		return -1;
+	feasible = !cgbr->tab->empty;
+	if (isl_tab_rollback(cgbr->tab, snap) < 0)
+		return -1;
+	if (shifted_snap) {
+		if (isl_tab_rollback(cgbr->shifted, shifted_snap))
+			return -1;
+	} else if (cgbr->shifted) {
+		isl_tab_free(cgbr->shifted);
+		cgbr->shifted = NULL;
+	}
+	if (cone_snap) {
+		if (isl_tab_rollback(cgbr->cone, cone_snap))
+			return -1;
+	} else if (cgbr->cone) {
+		isl_tab_free(cgbr->cone);
+		cgbr->cone = NULL;
+	}
+
+	return feasible;
+}
+
+/* Return the column of the last of the variables associated to
+ * a column that has a non-zero coefficient.
+ * This function is called in a context where only coefficients
+ * of parameters or divs can be non-zero.
+ */
+static int last_non_zero_var_col(struct isl_tab *tab, isl_int *p)
+{
+	int i;
+	int col;
+
+	if (tab->n_var == 0)
+		return -1;
+
+	for (i = tab->n_var - 1; i >= 0; --i) {
+		if (i >= tab->n_param && i < tab->n_var - tab->n_div)
+			continue;
+		if (tab->var[i].is_row)
+			continue;
+		col = tab->var[i].index;
+		if (!isl_int_is_zero(p[col]))
+			return col;
+	}
+
+	return -1;
+}
+
+/* Look through all the recently added equalities in the context
+ * to see if we can propagate any of them to the main tableau.
+ *
+ * The newly added equalities in the context are encoded as pairs
+ * of inequalities starting at inequality "first".
+ *
+ * We tentatively add each of these equalities to the main tableau
+ * and if this happens to result in a row with a final coefficient
+ * that is one or negative one, we use it to kill a column
+ * in the main tableau.  Otherwise, we discard the tentatively
+ * added row.
+ *
+ * Return 0 on success and -1 on failure.
+ */
+static int propagate_equalities(struct isl_context_gbr *cgbr,
+	struct isl_tab *tab, unsigned first)
+{
+	int i;
+	struct isl_vec *eq = NULL;
+
+	eq = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_var);
+	if (!eq)
+		goto error;
+
+	if (isl_tab_extend_cons(tab, (cgbr->tab->bmap->n_ineq - first)/2) < 0)
+		goto error;
+
+	isl_seq_clr(eq->el + 1 + tab->n_param,
+		    tab->n_var - tab->n_param - tab->n_div);
+	for (i = first; i < cgbr->tab->bmap->n_ineq; i += 2) {
+		int j;
+		int r;
+		struct isl_tab_undo *snap;
+		snap = isl_tab_snap(tab);
+
+		isl_seq_cpy(eq->el, cgbr->tab->bmap->ineq[i], 1 + tab->n_param);
+		isl_seq_cpy(eq->el + 1 + tab->n_var - tab->n_div,
+			    cgbr->tab->bmap->ineq[i] + 1 + tab->n_param,
+			    tab->n_div);
+
+		r = isl_tab_add_row(tab, eq->el);
+		if (r < 0)
+			goto error;
+		r = tab->con[r].index;
+		j = last_non_zero_var_col(tab, tab->mat->row[r] + 2 + tab->M);
+		if (j < 0 || j < tab->n_dead ||
+		    !isl_int_is_one(tab->mat->row[r][0]) ||
+		    (!isl_int_is_one(tab->mat->row[r][2 + tab->M + j]) &&
+		     !isl_int_is_negone(tab->mat->row[r][2 + tab->M + j]))) {
+			if (isl_tab_rollback(tab, snap) < 0)
+				goto error;
+			continue;
+		}
+		if (isl_tab_pivot(tab, r, j) < 0)
+			goto error;
+		if (isl_tab_kill_col(tab, j) < 0)
+			goto error;
+
+		if (restore_lexmin(tab) < 0)
+			goto error;
+	}
+
+	isl_vec_free(eq);
+
+	return 0;
+error:
+	isl_vec_free(eq);
+	isl_tab_free(cgbr->tab);
+	cgbr->tab = NULL;
+	return -1;
+}
+
+static int context_gbr_detect_equalities(struct isl_context *context,
+	struct isl_tab *tab)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+	struct isl_ctx *ctx;
+	unsigned n_ineq;
+
+	ctx = cgbr->tab->mat->ctx;
+
+	if (!cgbr->cone) {
+		struct isl_basic_set *bset = isl_tab_peek_bset(cgbr->tab);
+		cgbr->cone = isl_tab_from_recession_cone(bset, 0);
+		if (!cgbr->cone)
+			goto error;
+		if (isl_tab_track_bset(cgbr->cone,
+					isl_basic_set_copy(bset)) < 0)
+			goto error;
+	}
+	if (isl_tab_detect_implicit_equalities(cgbr->cone) < 0)
+		goto error;
+
+	n_ineq = cgbr->tab->bmap->n_ineq;
+	cgbr->tab = isl_tab_detect_equalities(cgbr->tab, cgbr->cone);
+	if (!cgbr->tab)
+		return -1;
+	if (cgbr->tab->bmap->n_ineq > n_ineq &&
+	    propagate_equalities(cgbr, tab, n_ineq) < 0)
+		return -1;
+
+	return 0;
+error:
+	isl_tab_free(cgbr->tab);
+	cgbr->tab = NULL;
+	return -1;
+}
+
+static int context_gbr_get_div(struct isl_context *context, struct isl_tab *tab,
+		struct isl_vec *div)
+{
+	return get_div(tab, context, div);
+}
+
+static int context_gbr_add_div(struct isl_context *context, struct isl_vec *div)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+	if (cgbr->cone) {
+		int k;
+
+		if (isl_tab_extend_cons(cgbr->cone, 3) < 0)
+			return -1;
+		if (isl_tab_extend_vars(cgbr->cone, 1) < 0)
+			return -1;
+		if (isl_tab_allocate_var(cgbr->cone) <0)
+			return -1;
+
+		cgbr->cone->bmap = isl_basic_map_extend_space(cgbr->cone->bmap,
+			isl_basic_map_get_space(cgbr->cone->bmap), 1, 0, 2);
+		k = isl_basic_map_alloc_div(cgbr->cone->bmap);
+		if (k < 0)
+			return -1;
+		isl_seq_cpy(cgbr->cone->bmap->div[k], div->el, div->size);
+		if (isl_tab_push(cgbr->cone, isl_tab_undo_bmap_div) < 0)
+			return -1;
+	}
+	return context_tab_add_div(cgbr->tab, div,
+					context_gbr_add_ineq_wrap, context);
+}
+
+static int context_gbr_best_split(struct isl_context *context,
+		struct isl_tab *tab)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+	struct isl_tab_undo *snap;
+	int r;
+
+	snap = isl_tab_snap(cgbr->tab);
+	r = best_split(tab, cgbr->tab);
+
+	if (r >= 0 && isl_tab_rollback(cgbr->tab, snap) < 0)
+		return -1;
+
+	return r;
+}
+
+static int context_gbr_is_empty(struct isl_context *context)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+	if (!cgbr->tab)
+		return -1;
+	return cgbr->tab->empty;
+}
+
+struct isl_gbr_tab_undo {
+	struct isl_tab_undo *tab_snap;
+	struct isl_tab_undo *shifted_snap;
+	struct isl_tab_undo *cone_snap;
+};
+
+static void *context_gbr_save(struct isl_context *context)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+	struct isl_gbr_tab_undo *snap;
+
+	if (!cgbr->tab)
+		return NULL;
+
+	snap = isl_alloc_type(cgbr->tab->mat->ctx, struct isl_gbr_tab_undo);
+	if (!snap)
+		return NULL;
+
+	snap->tab_snap = isl_tab_snap(cgbr->tab);
+	if (isl_tab_save_samples(cgbr->tab) < 0)
+		goto error;
+
+	if (cgbr->shifted)
+		snap->shifted_snap = isl_tab_snap(cgbr->shifted);
+	else
+		snap->shifted_snap = NULL;
+
+	if (cgbr->cone)
+		snap->cone_snap = isl_tab_snap(cgbr->cone);
+	else
+		snap->cone_snap = NULL;
+
+	return snap;
+error:
+	free(snap);
+	return NULL;
+}
+
+static void context_gbr_restore(struct isl_context *context, void *save)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+	struct isl_gbr_tab_undo *snap = (struct isl_gbr_tab_undo *)save;
+	if (!snap)
+		goto error;
+	if (isl_tab_rollback(cgbr->tab, snap->tab_snap) < 0) {
+		isl_tab_free(cgbr->tab);
+		cgbr->tab = NULL;
+	}
+
+	if (snap->shifted_snap) {
+		if (isl_tab_rollback(cgbr->shifted, snap->shifted_snap) < 0)
+			goto error;
+	} else if (cgbr->shifted) {
+		isl_tab_free(cgbr->shifted);
+		cgbr->shifted = NULL;
+	}
+
+	if (snap->cone_snap) {
+		if (isl_tab_rollback(cgbr->cone, snap->cone_snap) < 0)
+			goto error;
+	} else if (cgbr->cone) {
+		isl_tab_free(cgbr->cone);
+		cgbr->cone = NULL;
+	}
+
+	free(snap);
+
+	return;
+error:
+	free(snap);
+	isl_tab_free(cgbr->tab);
+	cgbr->tab = NULL;
+}
+
+static void context_gbr_discard(void *save)
+{
+	struct isl_gbr_tab_undo *snap = (struct isl_gbr_tab_undo *)save;
+	free(snap);
+}
+
+static int context_gbr_is_ok(struct isl_context *context)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+	return !!cgbr->tab;
+}
+
+static void context_gbr_invalidate(struct isl_context *context)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+	isl_tab_free(cgbr->tab);
+	cgbr->tab = NULL;
+}
+
+static void context_gbr_free(struct isl_context *context)
+{
+	struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
+	isl_tab_free(cgbr->tab);
+	isl_tab_free(cgbr->shifted);
+	isl_tab_free(cgbr->cone);
+	free(cgbr);
+}
+
+struct isl_context_op isl_context_gbr_op = {
+	context_gbr_detect_nonnegative_parameters,
+	context_gbr_peek_basic_set,
+	context_gbr_peek_tab,
+	context_gbr_add_eq,
+	context_gbr_add_ineq,
+	context_gbr_ineq_sign,
+	context_gbr_test_ineq,
+	context_gbr_get_div,
+	context_gbr_add_div,
+	context_gbr_detect_equalities,
+	context_gbr_best_split,
+	context_gbr_is_empty,
+	context_gbr_is_ok,
+	context_gbr_save,
+	context_gbr_restore,
+	context_gbr_discard,
+	context_gbr_invalidate,
+	context_gbr_free,
+};
+
+static struct isl_context *isl_context_gbr_alloc(struct isl_basic_set *dom)
+{
+	struct isl_context_gbr *cgbr;
+
+	if (!dom)
+		return NULL;
+
+	cgbr = isl_calloc_type(dom->ctx, struct isl_context_gbr);
+	if (!cgbr)
+		return NULL;
+
+	cgbr->context.op = &isl_context_gbr_op;
+
+	cgbr->shifted = NULL;
+	cgbr->cone = NULL;
+	cgbr->tab = isl_tab_from_basic_set(dom, 1);
+	cgbr->tab = isl_tab_init_samples(cgbr->tab);
+	if (!cgbr->tab)
+		goto error;
+	check_gbr_integer_feasible(cgbr);
+
+	return &cgbr->context;
+error:
+	cgbr->context.op->free(&cgbr->context);
+	return NULL;
+}
+
+static struct isl_context *isl_context_alloc(struct isl_basic_set *dom)
+{
+	if (!dom)
+		return NULL;
+
+	if (dom->ctx->opt->context == ISL_CONTEXT_LEXMIN)
+		return isl_context_lex_alloc(dom);
+	else
+		return isl_context_gbr_alloc(dom);
+}
+
+/* Construct an isl_sol_map structure for accumulating the solution.
+ * If track_empty is set, then we also keep track of the parts
+ * of the context where there is no solution.
+ * If max is set, then we are solving a maximization, rather than
+ * a minimization problem, which means that the variables in the
+ * tableau have value "M - x" rather than "M + x".
+ */
+static struct isl_sol *sol_map_init(struct isl_basic_map *bmap,
+	struct isl_basic_set *dom, int track_empty, int max)
+{
+	struct isl_sol_map *sol_map = NULL;
+
+	if (!bmap)
+		goto error;
+
+	sol_map = isl_calloc_type(bmap->ctx, struct isl_sol_map);
+	if (!sol_map)
+		goto error;
+
+	sol_map->sol.rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
+	sol_map->sol.dec_level.callback.run = &sol_dec_level_wrap;
+	sol_map->sol.dec_level.sol = &sol_map->sol;
+	sol_map->sol.max = max;
+	sol_map->sol.n_out = isl_basic_map_dim(bmap, isl_dim_out);
+	sol_map->sol.add = &sol_map_add_wrap;
+	sol_map->sol.add_empty = track_empty ? &sol_map_add_empty_wrap : NULL;
+	sol_map->sol.free = &sol_map_free_wrap;
+	sol_map->map = isl_map_alloc_space(isl_basic_map_get_space(bmap), 1,
+					    ISL_MAP_DISJOINT);
+	if (!sol_map->map)
+		goto error;
+
+	sol_map->sol.context = isl_context_alloc(dom);
+	if (!sol_map->sol.context)
+		goto error;
+
+	if (track_empty) {
+		sol_map->empty = isl_set_alloc_space(isl_basic_set_get_space(dom),
+							1, ISL_SET_DISJOINT);
+		if (!sol_map->empty)
+			goto error;
+	}
+
+	isl_basic_set_free(dom);
+	return &sol_map->sol;
+error:
+	isl_basic_set_free(dom);
+	sol_map_free(sol_map);
+	return NULL;
+}
+
+/* Check whether all coefficients of (non-parameter) variables
+ * are non-positive, meaning that no pivots can be performed on the row.
+ */
+static int is_critical(struct isl_tab *tab, int row)
+{
+	int j;
+	unsigned off = 2 + tab->M;
+
+	for (j = tab->n_dead; j < tab->n_col; ++j) {
+		if (tab->col_var[j] >= 0 &&
+		    (tab->col_var[j] < tab->n_param  ||
+		    tab->col_var[j] >= tab->n_var - tab->n_div))
+			continue;
+
+		if (isl_int_is_pos(tab->mat->row[row][off + j]))
+			return 0;
+	}
+
+	return 1;
+}
+
+/* Check whether the inequality represented by vec is strict over the integers,
+ * i.e., there are no integer values satisfying the constraint with
+ * equality.  This happens if the gcd of the coefficients is not a divisor
+ * of the constant term.  If so, scale the constraint down by the gcd
+ * of the coefficients.
+ */
+static int is_strict(struct isl_vec *vec)
+{
+	isl_int gcd;
+	int strict = 0;
+
+	isl_int_init(gcd);
+	isl_seq_gcd(vec->el + 1, vec->size - 1, &gcd);
+	if (!isl_int_is_one(gcd)) {
+		strict = !isl_int_is_divisible_by(vec->el[0], gcd);
+		isl_int_fdiv_q(vec->el[0], vec->el[0], gcd);
+		isl_seq_scale_down(vec->el + 1, vec->el + 1, gcd, vec->size-1);
+	}
+	isl_int_clear(gcd);
+
+	return strict;
+}
+
+/* Determine the sign of the given row of the main tableau.
+ * The result is one of
+ *	isl_tab_row_pos: always non-negative; no pivot needed
+ *	isl_tab_row_neg: always non-positive; pivot
+ *	isl_tab_row_any: can be both positive and negative; split
+ *
+ * We first handle some simple cases
+ *	- the row sign may be known already
+ *	- the row may be obviously non-negative
+ *	- the parametric constant may be equal to that of another row
+ *	  for which we know the sign.  This sign will be either "pos" or
+ *	  "any".  If it had been "neg" then we would have pivoted before.
+ *
+ * If none of these cases hold, we check the value of the row for each
+ * of the currently active samples.  Based on the signs of these values
+ * we make an initial determination of the sign of the row.
+ *
+ *	all zero			->	unk(nown)
+ *	all non-negative		->	pos
+ *	all non-positive		->	neg
+ *	both negative and positive	->	all
+ *
+ * If we end up with "all", we are done.
+ * Otherwise, we perform a check for positive and/or negative
+ * values as follows.
+ *
+ *	samples	       neg	       unk	       pos
+ *	<0 ?			    Y        N	    Y        N
+ *					    pos    any      pos
+ *	>0 ?	     Y      N	 Y     N
+ *		    any    neg  any   neg
+ *
+ * There is no special sign for "zero", because we can usually treat zero
+ * as either non-negative or non-positive, whatever works out best.
+ * However, if the row is "critical", meaning that pivoting is impossible
+ * then we don't want to limp zero with the non-positive case, because
+ * then we we would lose the solution for those values of the parameters
+ * where the value of the row is zero.  Instead, we treat 0 as non-negative
+ * ensuring a split if the row can attain both zero and negative values.
+ * The same happens when the original constraint was one that could not
+ * be satisfied with equality by any integer values of the parameters.
+ * In this case, we normalize the constraint, but then a value of zero
+ * for the normalized constraint is actually a positive value for the
+ * original constraint, so again we need to treat zero as non-negative.
+ * In both these cases, we have the following decision tree instead:
+ *
+ *	all non-negative		->	pos
+ *	all negative			->	neg
+ *	both negative and non-negative	->	all
+ *
+ *	samples	       neg	          	       pos
+ *	<0 ?			             	    Y        N
+ *					           any      pos
+ *	>=0 ?	     Y      N
+ *		    any    neg
+ */
+static enum isl_tab_row_sign row_sign(struct isl_tab *tab,
+	struct isl_sol *sol, int row)
+{
+	struct isl_vec *ineq = NULL;
+	enum isl_tab_row_sign res = isl_tab_row_unknown;
+	int critical;
+	int strict;
+	int row2;
+
+	if (tab->row_sign[row] != isl_tab_row_unknown)
+		return tab->row_sign[row];
+	if (is_obviously_nonneg(tab, row))
+		return isl_tab_row_pos;
+	for (row2 = tab->n_redundant; row2 < tab->n_row; ++row2) {
+		if (tab->row_sign[row2] == isl_tab_row_unknown)
+			continue;
+		if (identical_parameter_line(tab, row, row2))
+			return tab->row_sign[row2];
+	}
+
+	critical = is_critical(tab, row);
+
+	ineq = get_row_parameter_ineq(tab, row);
+	if (!ineq)
+		goto error;
+
+	strict = is_strict(ineq);
+
+	res = sol->context->op->ineq_sign(sol->context, ineq->el,
+					  critical || strict);
+
+	if (res == isl_tab_row_unknown || res == isl_tab_row_pos) {
+		/* test for negative values */
+		int feasible;
+		isl_seq_neg(ineq->el, ineq->el, ineq->size);
+		isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
+
+		feasible = sol->context->op->test_ineq(sol->context, ineq->el);
+		if (feasible < 0)
+			goto error;
+		if (!feasible)
+			res = isl_tab_row_pos;
+		else
+			res = (res == isl_tab_row_unknown) ? isl_tab_row_neg
+							   : isl_tab_row_any;
+		if (res == isl_tab_row_neg) {
+			isl_seq_neg(ineq->el, ineq->el, ineq->size);
+			isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
+		}
+	}
+
+	if (res == isl_tab_row_neg) {
+		/* test for positive values */
+		int feasible;
+		if (!critical && !strict)
+			isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
+
+		feasible = sol->context->op->test_ineq(sol->context, ineq->el);
+		if (feasible < 0)
+			goto error;
+		if (feasible)
+			res = isl_tab_row_any;
+	}
+
+	isl_vec_free(ineq);
+	return res;
+error:
+	isl_vec_free(ineq);
+	return isl_tab_row_unknown;
+}
+
+static void find_solutions(struct isl_sol *sol, struct isl_tab *tab);
+
+/* Find solutions for values of the parameters that satisfy the given
+ * inequality.
+ *
+ * We currently take a snapshot of the context tableau that is reset
+ * when we return from this function, while we make a copy of the main
+ * tableau, leaving the original main tableau untouched.
+ * These are fairly arbitrary choices.  Making a copy also of the context
+ * tableau would obviate the need to undo any changes made to it later,
+ * while taking a snapshot of the main tableau could reduce memory usage.
+ * If we were to switch to taking a snapshot of the main tableau,
+ * we would have to keep in mind that we need to save the row signs
+ * and that we need to do this before saving the current basis
+ * such that the basis has been restore before we restore the row signs.
+ */
+static void find_in_pos(struct isl_sol *sol, struct isl_tab *tab, isl_int *ineq)
+{
+	void *saved;
+
+	if (!sol->context)
+		goto error;
+	saved = sol->context->op->save(sol->context);
+
+	tab = isl_tab_dup(tab);
+	if (!tab)
+		goto error;
+
+	sol->context->op->add_ineq(sol->context, ineq, 0, 1);
+
+	find_solutions(sol, tab);
+
+	if (!sol->error)
+		sol->context->op->restore(sol->context, saved);
+	else
+		sol->context->op->discard(saved);
+	return;
+error:
+	sol->error = 1;
+}
+
+/* Record the absence of solutions for those values of the parameters
+ * that do not satisfy the given inequality with equality.
+ */
+static void no_sol_in_strict(struct isl_sol *sol,
+	struct isl_tab *tab, struct isl_vec *ineq)
+{
+	int empty;
+	void *saved;
+
+	if (!sol->context || sol->error)
+		goto error;
+	saved = sol->context->op->save(sol->context);
+
+	isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
+
+	sol->context->op->add_ineq(sol->context, ineq->el, 1, 0);
+	if (!sol->context)
+		goto error;
+
+	empty = tab->empty;
+	tab->empty = 1;
+	sol_add(sol, tab);
+	tab->empty = empty;
+
+	isl_int_add_ui(ineq->el[0], ineq->el[0], 1);
+
+	sol->context->op->restore(sol->context, saved);
+	return;
+error:
+	sol->error = 1;
+}
+
+/* Compute the lexicographic minimum of the set represented by the main
+ * tableau "tab" within the context "sol->context_tab".
+ * On entry the sample value of the main tableau is lexicographically
+ * less than or equal to this lexicographic minimum.
+ * Pivots are performed until a feasible point is found, which is then
+ * necessarily equal to the minimum, or until the tableau is found to
+ * be infeasible.  Some pivots may need to be performed for only some
+ * feasible values of the context tableau.  If so, the context tableau
+ * is split into a part where the pivot is needed and a part where it is not.
+ *
+ * Whenever we enter the main loop, the main tableau is such that no
+ * "obvious" pivots need to be performed on it, where "obvious" means
+ * that the given row can be seen to be negative without looking at
+ * the context tableau.  In particular, for non-parametric problems,
+ * no pivots need to be performed on the main tableau.
+ * The caller of find_solutions is responsible for making this property
+ * hold prior to the first iteration of the loop, while restore_lexmin
+ * is called before every other iteration.
+ *
+ * Inside the main loop, we first examine the signs of the rows of
+ * the main tableau within the context of the context tableau.
+ * If we find a row that is always non-positive for all values of
+ * the parameters satisfying the context tableau and negative for at
+ * least one value of the parameters, we perform the appropriate pivot
+ * and start over.  An exception is the case where no pivot can be
+ * performed on the row.  In this case, we require that the sign of
+ * the row is negative for all values of the parameters (rather than just
+ * non-positive).  This special case is handled inside row_sign, which
+ * will say that the row can have any sign if it determines that it can
+ * attain both negative and zero values.
+ *
+ * If we can't find a row that always requires a pivot, but we can find
+ * one or more rows that require a pivot for some values of the parameters
+ * (i.e., the row can attain both positive and negative signs), then we split
+ * the context tableau into two parts, one where we force the sign to be
+ * non-negative and one where we force is to be negative.
+ * The non-negative part is handled by a recursive call (through find_in_pos).
+ * Upon returning from this call, we continue with the negative part and
+ * perform the required pivot.
+ *
+ * If no such rows can be found, all rows are non-negative and we have
+ * found a (rational) feasible point.  If we only wanted a rational point
+ * then we are done.
+ * Otherwise, we check if all values of the sample point of the tableau
+ * are integral for the variables.  If so, we have found the minimal
+ * integral point and we are done.
+ * If the sample point is not integral, then we need to make a distinction
+ * based on whether the constant term is non-integral or the coefficients
+ * of the parameters.  Furthermore, in order to decide how to handle
+ * the non-integrality, we also need to know whether the coefficients
+ * of the other columns in the tableau are integral.  This leads
+ * to the following table.  The first two rows do not correspond
+ * to a non-integral sample point and are only mentioned for completeness.
+ *
+ *	constant	parameters	other
+ *
+ *	int		int		int	|
+ *	int		int		rat	| -> no problem
+ *
+ *	rat		int		int	  -> fail
+ *
+ *	rat		int		rat	  -> cut
+ *
+ *	int		rat		rat	|
+ *	rat		rat		rat	| -> parametric cut
+ *
+ *	int		rat		int	|
+ *	rat		rat		int	| -> split context
+ *
+ * If the parametric constant is completely integral, then there is nothing
+ * to be done.  If the constant term is non-integral, but all the other
+ * coefficient are integral, then there is nothing that can be done
+ * and the tableau has no integral solution.
+ * If, on the other hand, one or more of the other columns have rational
+ * coefficients, but the parameter coefficients are all integral, then
+ * we can perform a regular (non-parametric) cut.
+ * Finally, if there is any parameter coefficient that is non-integral,
+ * then we need to involve the context tableau.  There are two cases here.
+ * If at least one other column has a rational coefficient, then we
+ * can perform a parametric cut in the main tableau by adding a new
+ * integer division in the context tableau.
+ * If all other columns have integral coefficients, then we need to
+ * enforce that the rational combination of parameters (c + \sum a_i y_i)/m
+ * is always integral.  We do this by introducing an integer division
+ * q = floor((c + \sum a_i y_i)/m) and stipulating that its argument should
+ * always be integral in the context tableau, i.e., m q = c + \sum a_i y_i.
+ * Since q is expressed in the tableau as
+ *	c + \sum a_i y_i - m q >= 0
+ *	-c - \sum a_i y_i + m q + m - 1 >= 0
+ * it is sufficient to add the inequality
+ *	-c - \sum a_i y_i + m q >= 0
+ * In the part of the context where this inequality does not hold, the
+ * main tableau is marked as being empty.
+ */
+static void find_solutions(struct isl_sol *sol, struct isl_tab *tab)
+{
+	struct isl_context *context;
+	int r;
+
+	if (!tab || sol->error)
+		goto error;
+
+	context = sol->context;
+
+	if (tab->empty)
+		goto done;
+	if (context->op->is_empty(context))
+		goto done;
+
+	for (r = 0; r >= 0 && tab && !tab->empty; r = restore_lexmin(tab)) {
+		int flags;
+		int row;
+		enum isl_tab_row_sign sgn;
+		int split = -1;
+		int n_split = 0;
+
+		for (row = tab->n_redundant; row < tab->n_row; ++row) {
+			if (!isl_tab_var_from_row(tab, row)->is_nonneg)
+				continue;
+			sgn = row_sign(tab, sol, row);
+			if (!sgn)
+				goto error;
+			tab->row_sign[row] = sgn;
+			if (sgn == isl_tab_row_any)
+				n_split++;
+			if (sgn == isl_tab_row_any && split == -1)
+				split = row;
+			if (sgn == isl_tab_row_neg)
+				break;
+		}
+		if (row < tab->n_row)
+			continue;
+		if (split != -1) {
+			struct isl_vec *ineq;
+			if (n_split != 1)
+				split = context->op->best_split(context, tab);
+			if (split < 0)
+				goto error;
+			ineq = get_row_parameter_ineq(tab, split);
+			if (!ineq)
+				goto error;
+			is_strict(ineq);
+			for (row = tab->n_redundant; row < tab->n_row; ++row) {
+				if (!isl_tab_var_from_row(tab, row)->is_nonneg)
+					continue;
+				if (tab->row_sign[row] == isl_tab_row_any)
+					tab->row_sign[row] = isl_tab_row_unknown;
+			}
+			tab->row_sign[split] = isl_tab_row_pos;
+			sol_inc_level(sol);
+			find_in_pos(sol, tab, ineq->el);
+			tab->row_sign[split] = isl_tab_row_neg;
+			row = split;
+			isl_seq_neg(ineq->el, ineq->el, ineq->size);
+			isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
+			if (!sol->error)
+				context->op->add_ineq(context, ineq->el, 0, 1);
+			isl_vec_free(ineq);
+			if (sol->error)
+				goto error;
+			continue;
+		}
+		if (tab->rational)
+			break;
+		row = first_non_integer_row(tab, &flags);
+		if (row < 0)
+			break;
+		if (ISL_FL_ISSET(flags, I_PAR)) {
+			if (ISL_FL_ISSET(flags, I_VAR)) {
+				if (isl_tab_mark_empty(tab) < 0)
+					goto error;
+				break;
+			}
+			row = add_cut(tab, row);
+		} else if (ISL_FL_ISSET(flags, I_VAR)) {
+			struct isl_vec *div;
+			struct isl_vec *ineq;
+			int d;
+			div = get_row_split_div(tab, row);
+			if (!div)
+				goto error;
+			d = context->op->get_div(context, tab, div);
+			isl_vec_free(div);
+			if (d < 0)
+				goto error;
+			ineq = ineq_for_div(context->op->peek_basic_set(context), d);
+			if (!ineq)
+				goto error;
+			sol_inc_level(sol);
+			no_sol_in_strict(sol, tab, ineq);
+			isl_seq_neg(ineq->el, ineq->el, ineq->size);
+			context->op->add_ineq(context, ineq->el, 1, 1);
+			isl_vec_free(ineq);
+			if (sol->error || !context->op->is_ok(context))
+				goto error;
+			tab = set_row_cst_to_div(tab, row, d);
+			if (context->op->is_empty(context))
+				break;
+		} else
+			row = add_parametric_cut(tab, row, context);
+		if (row < 0)
+			goto error;
+	}
+	if (r < 0)
+		goto error;
+done:
+	sol_add(sol, tab);
+	isl_tab_free(tab);
+	return;
+error:
+	isl_tab_free(tab);
+	sol->error = 1;
+}
+
+/* Does "sol" contain a pair of partial solutions that could potentially
+ * be merged?
+ *
+ * We currently only check that "sol" is not in an error state
+ * and that there are at least two partial solutions of which the final two
+ * are defined at the same level.
+ */
+static int sol_has_mergeable_solutions(struct isl_sol *sol)
+{
+	if (sol->error)
+		return 0;
+	if (!sol->partial)
+		return 0;
+	if (!sol->partial->next)
+		return 0;
+	return sol->partial->level == sol->partial->next->level;
+}
+
+/* Compute the lexicographic minimum of the set represented by the main
+ * tableau "tab" within the context "sol->context_tab".
+ *
+ * As a preprocessing step, we first transfer all the purely parametric
+ * equalities from the main tableau to the context tableau, i.e.,
+ * parameters that have been pivoted to a row.
+ * These equalities are ignored by the main algorithm, because the
+ * corresponding rows may not be marked as being non-negative.
+ * In parts of the context where the added equality does not hold,
+ * the main tableau is marked as being empty.
+ *
+ * Before we embark on the actual computation, we save a copy
+ * of the context.  When we return, we check if there are any
+ * partial solutions that can potentially be merged.  If so,
+ * we perform a rollback to the initial state of the context.
+ * The merging of partial solutions happens inside calls to
+ * sol_dec_level that are pushed onto the undo stack of the context.
+ * If there are no partial solutions that can potentially be merged
+ * then the rollback is skipped as it would just be wasted effort.
+ */
+static void find_solutions_main(struct isl_sol *sol, struct isl_tab *tab)
+{
+	int row;
+	void *saved;
+
+	if (!tab)
+		goto error;
+
+	sol->level = 0;
+
+	for (row = tab->n_redundant; row < tab->n_row; ++row) {
+		int p;
+		struct isl_vec *eq;
+
+		if (tab->row_var[row] < 0)
+			continue;
+		if (tab->row_var[row] >= tab->n_param &&
+		    tab->row_var[row] < tab->n_var - tab->n_div)
+			continue;
+		if (tab->row_var[row] < tab->n_param)
+			p = tab->row_var[row];
+		else
+			p = tab->row_var[row]
+				+ tab->n_param - (tab->n_var - tab->n_div);
+
+		eq = isl_vec_alloc(tab->mat->ctx, 1+tab->n_param+tab->n_div);
+		if (!eq)
+			goto error;
+		get_row_parameter_line(tab, row, eq->el);
+		isl_int_neg(eq->el[1 + p], tab->mat->row[row][0]);
+		eq = isl_vec_normalize(eq);
+
+		sol_inc_level(sol);
+		no_sol_in_strict(sol, tab, eq);
+
+		isl_seq_neg(eq->el, eq->el, eq->size);
+		sol_inc_level(sol);
+		no_sol_in_strict(sol, tab, eq);
+		isl_seq_neg(eq->el, eq->el, eq->size);
+
+		sol->context->op->add_eq(sol->context, eq->el, 1, 1);
+
+		isl_vec_free(eq);
+
+		if (isl_tab_mark_redundant(tab, row) < 0)
+			goto error;
+
+		if (sol->context->op->is_empty(sol->context))
+			break;
+
+		row = tab->n_redundant - 1;
+	}
+
+	saved = sol->context->op->save(sol->context);
+
+	find_solutions(sol, tab);
+
+	if (sol_has_mergeable_solutions(sol))
+		sol->context->op->restore(sol->context, saved);
+	else
+		sol->context->op->discard(saved);
+
+	sol->level = 0;
+	sol_pop(sol);
+
+	return;
+error:
+	isl_tab_free(tab);
+	sol->error = 1;
+}
+
+/* Check if integer division "div" of "dom" also occurs in "bmap".
+ * If so, return its position within the divs.
+ * If not, return -1.
+ */
+static int find_context_div(struct isl_basic_map *bmap,
+	struct isl_basic_set *dom, unsigned div)
+{
+	int i;
+	unsigned b_dim = isl_space_dim(bmap->dim, isl_dim_all);
+	unsigned d_dim = isl_space_dim(dom->dim, isl_dim_all);
+
+	if (isl_int_is_zero(dom->div[div][0]))
+		return -1;
+	if (isl_seq_first_non_zero(dom->div[div] + 2 + d_dim, dom->n_div) != -1)
+		return -1;
+
+	for (i = 0; i < bmap->n_div; ++i) {
+		if (isl_int_is_zero(bmap->div[i][0]))
+			continue;
+		if (isl_seq_first_non_zero(bmap->div[i] + 2 + d_dim,
+					   (b_dim - d_dim) + bmap->n_div) != -1)
+			continue;
+		if (isl_seq_eq(bmap->div[i], dom->div[div], 2 + d_dim))
+			return i;
+	}
+	return -1;
+}
+
+/* The correspondence between the variables in the main tableau,
+ * the context tableau, and the input map and domain is as follows.
+ * The first n_param and the last n_div variables of the main tableau
+ * form the variables of the context tableau.
+ * In the basic map, these n_param variables correspond to the
+ * parameters and the input dimensions.  In the domain, they correspond
+ * to the parameters and the set dimensions.
+ * The n_div variables correspond to the integer divisions in the domain.
+ * To ensure that everything lines up, we may need to copy some of the
+ * integer divisions of the domain to the map.  These have to be placed
+ * in the same order as those in the context and they have to be placed
+ * after any other integer divisions that the map may have.
+ * This function performs the required reordering.
+ */
+static struct isl_basic_map *align_context_divs(struct isl_basic_map *bmap,
+	struct isl_basic_set *dom)
+{
+	int i;
+	int common = 0;
+	int other;
+
+	for (i = 0; i < dom->n_div; ++i)
+		if (find_context_div(bmap, dom, i) != -1)
+			common++;
+	other = bmap->n_div - common;
+	if (dom->n_div - common > 0) {
+		bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim),
+				dom->n_div - common, 0, 0);
+		if (!bmap)
+			return NULL;
+	}
+	for (i = 0; i < dom->n_div; ++i) {
+		int pos = find_context_div(bmap, dom, i);
+		if (pos < 0) {
+			pos = isl_basic_map_alloc_div(bmap);
+			if (pos < 0)
+				goto error;
+			isl_int_set_si(bmap->div[pos][0], 0);
+		}
+		if (pos != other + i)
+			isl_basic_map_swap_div(bmap, pos, other + i);
+	}
+	return bmap;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Base case of isl_tab_basic_map_partial_lexopt, after removing
+ * some obvious symmetries.
+ *
+ * We make sure the divs in the domain are properly ordered,
+ * because they will be added one by one in the given order
+ * during the construction of the solution map.
+ */
+static struct isl_sol *basic_map_partial_lexopt_base(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty, int max,
+	struct isl_sol *(*init)(__isl_keep isl_basic_map *bmap,
+		    __isl_take isl_basic_set *dom, int track_empty, int max))
+{
+	struct isl_tab *tab;
+	struct isl_sol *sol = NULL;
+	struct isl_context *context;
+
+	if (dom->n_div) {
+		dom = isl_basic_set_order_divs(dom);
+		bmap = align_context_divs(bmap, dom);
+	}
+	sol = init(bmap, dom, !!empty, max);
+	if (!sol)
+		goto error;
+
+	context = sol->context;
+	if (isl_basic_set_plain_is_empty(context->op->peek_basic_set(context)))
+		/* nothing */;
+	else if (isl_basic_map_plain_is_empty(bmap)) {
+		if (sol->add_empty)
+			sol->add_empty(sol,
+		    isl_basic_set_copy(context->op->peek_basic_set(context)));
+	} else {
+		tab = tab_for_lexmin(bmap,
+				    context->op->peek_basic_set(context), 1, max);
+		tab = context->op->detect_nonnegative_parameters(context, tab);
+		find_solutions_main(sol, tab);
+	}
+	if (sol->error)
+		goto error;
+
+	isl_basic_map_free(bmap);
+	return sol;
+error:
+	sol_free(sol);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Base case of isl_tab_basic_map_partial_lexopt, after removing
+ * some obvious symmetries.
+ *
+ * We call basic_map_partial_lexopt_base and extract the results.
+ */
+static __isl_give isl_map *basic_map_partial_lexopt_base_map(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty, int max)
+{
+	isl_map *result = NULL;
+	struct isl_sol *sol;
+	struct isl_sol_map *sol_map;
+
+	sol = basic_map_partial_lexopt_base(bmap, dom, empty, max,
+					    &sol_map_init);
+	if (!sol)
+		return NULL;
+	sol_map = (struct isl_sol_map *) sol;
+
+	result = isl_map_copy(sol_map->map);
+	if (empty)
+		*empty = isl_set_copy(sol_map->empty);
+	sol_free(&sol_map->sol);
+	return result;
+}
+
+/* Structure used during detection of parallel constraints.
+ * n_in: number of "input" variables: isl_dim_param + isl_dim_in
+ * n_out: number of "output" variables: isl_dim_out + isl_dim_div
+ * val: the coefficients of the output variables
+ */
+struct isl_constraint_equal_info {
+	isl_basic_map *bmap;
+	unsigned n_in;
+	unsigned n_out;
+	isl_int *val;
+};
+
+/* Check whether the coefficients of the output variables
+ * of the constraint in "entry" are equal to info->val.
+ */
+static int constraint_equal(const void *entry, const void *val)
+{
+	isl_int **row = (isl_int **)entry;
+	const struct isl_constraint_equal_info *info = val;
+
+	return isl_seq_eq((*row) + 1 + info->n_in, info->val, info->n_out);
+}
+
+/* Check whether "bmap" has a pair of constraints that have
+ * the same coefficients for the output variables.
+ * Note that the coefficients of the existentially quantified
+ * variables need to be zero since the existentially quantified
+ * of the result are usually not the same as those of the input.
+ * the isl_dim_out and isl_dim_div dimensions.
+ * If so, return 1 and return the row indices of the two constraints
+ * in *first and *second.
+ */
+static int parallel_constraints(__isl_keep isl_basic_map *bmap,
+	int *first, int *second)
+{
+	int i;
+	isl_ctx *ctx = isl_basic_map_get_ctx(bmap);
+	struct isl_hash_table *table = NULL;
+	struct isl_hash_table_entry *entry;
+	struct isl_constraint_equal_info info;
+	unsigned n_out;
+	unsigned n_div;
+
+	ctx = isl_basic_map_get_ctx(bmap);
+	table = isl_hash_table_alloc(ctx, bmap->n_ineq);
+	if (!table)
+		goto error;
+
+	info.n_in = isl_basic_map_dim(bmap, isl_dim_param) +
+		    isl_basic_map_dim(bmap, isl_dim_in);
+	info.bmap = bmap;
+	n_out = isl_basic_map_dim(bmap, isl_dim_out);
+	n_div = isl_basic_map_dim(bmap, isl_dim_div);
+	info.n_out = n_out + n_div;
+	for (i = 0; i < bmap->n_ineq; ++i) {
+		uint32_t hash;
+
+		info.val = bmap->ineq[i] + 1 + info.n_in;
+		if (isl_seq_first_non_zero(info.val, n_out) < 0)
+			continue;
+		if (isl_seq_first_non_zero(info.val + n_out, n_div) >= 0)
+			continue;
+		hash = isl_seq_get_hash(info.val, info.n_out);
+		entry = isl_hash_table_find(ctx, table, hash,
+					    constraint_equal, &info, 1);
+		if (!entry)
+			goto error;
+		if (entry->data)
+			break;
+		entry->data = &bmap->ineq[i];
+	}
+
+	if (i < bmap->n_ineq) {
+		*first = ((isl_int **)entry->data) - bmap->ineq; 
+		*second = i;
+	}
+
+	isl_hash_table_free(ctx, table);
+
+	return i < bmap->n_ineq;
+error:
+	isl_hash_table_free(ctx, table);
+	return -1;
+}
+
+/* Given a set of upper bounds in "var", add constraints to "bset"
+ * that make the i-th bound smallest.
+ *
+ * In particular, if there are n bounds b_i, then add the constraints
+ *
+ *	b_i <= b_j	for j > i
+ *	b_i <  b_j	for j < i
+ */
+static __isl_give isl_basic_set *select_minimum(__isl_take isl_basic_set *bset,
+	__isl_keep isl_mat *var, int i)
+{
+	isl_ctx *ctx;
+	int j, k;
+
+	ctx = isl_mat_get_ctx(var);
+
+	for (j = 0; j < var->n_row; ++j) {
+		if (j == i)
+			continue;
+		k = isl_basic_set_alloc_inequality(bset);
+		if (k < 0)
+			goto error;
+		isl_seq_combine(bset->ineq[k], ctx->one, var->row[j],
+				ctx->negone, var->row[i], var->n_col);
+		isl_int_set_si(bset->ineq[k][var->n_col], 0);
+		if (j < i)
+			isl_int_sub_ui(bset->ineq[k][0], bset->ineq[k][0], 1);
+	}
+
+	bset = isl_basic_set_finalize(bset);
+
+	return bset;
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Given a set of upper bounds on the last "input" variable m,
+ * construct a set that assigns the minimal upper bound to m, i.e.,
+ * construct a set that divides the space into cells where one
+ * of the upper bounds is smaller than all the others and assign
+ * this upper bound to m.
+ *
+ * In particular, if there are n bounds b_i, then the result
+ * consists of n basic sets, each one of the form
+ *
+ *	m = b_i
+ *	b_i <= b_j	for j > i
+ *	b_i <  b_j	for j < i
+ */
+static __isl_give isl_set *set_minimum(__isl_take isl_space *dim,
+	__isl_take isl_mat *var)
+{
+	int i, k;
+	isl_basic_set *bset = NULL;
+	isl_ctx *ctx;
+	isl_set *set = NULL;
+
+	if (!dim || !var)
+		goto error;
+
+	ctx = isl_space_get_ctx(dim);
+	set = isl_set_alloc_space(isl_space_copy(dim),
+				var->n_row, ISL_SET_DISJOINT);
+
+	for (i = 0; i < var->n_row; ++i) {
+		bset = isl_basic_set_alloc_space(isl_space_copy(dim), 0,
+					       1, var->n_row - 1);
+		k = isl_basic_set_alloc_equality(bset);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(bset->eq[k], var->row[i], var->n_col);
+		isl_int_set_si(bset->eq[k][var->n_col], -1);
+		bset = select_minimum(bset, var, i);
+		set = isl_set_add_basic_set(set, bset);
+	}
+
+	isl_space_free(dim);
+	isl_mat_free(var);
+	return set;
+error:
+	isl_basic_set_free(bset);
+	isl_set_free(set);
+	isl_space_free(dim);
+	isl_mat_free(var);
+	return NULL;
+}
+
+/* Given that the last input variable of "bmap" represents the minimum
+ * of the bounds in "cst", check whether we need to split the domain
+ * based on which bound attains the minimum.
+ *
+ * A split is needed when the minimum appears in an integer division
+ * or in an equality.  Otherwise, it is only needed if it appears in
+ * an upper bound that is different from the upper bounds on which it
+ * is defined.
+ */
+static int need_split_basic_map(__isl_keep isl_basic_map *bmap,
+	__isl_keep isl_mat *cst)
+{
+	int i, j;
+	unsigned total;
+	unsigned pos;
+
+	pos = cst->n_col - 1;
+	total = isl_basic_map_dim(bmap, isl_dim_all);
+
+	for (i = 0; i < bmap->n_div; ++i)
+		if (!isl_int_is_zero(bmap->div[i][2 + pos]))
+			return 1;
+
+	for (i = 0; i < bmap->n_eq; ++i)
+		if (!isl_int_is_zero(bmap->eq[i][1 + pos]))
+			return 1;
+
+	for (i = 0; i < bmap->n_ineq; ++i) {
+		if (isl_int_is_nonneg(bmap->ineq[i][1 + pos]))
+			continue;
+		if (!isl_int_is_negone(bmap->ineq[i][1 + pos]))
+			return 1;
+		if (isl_seq_first_non_zero(bmap->ineq[i] + 1 + pos + 1,
+					   total - pos - 1) >= 0)
+			return 1;
+
+		for (j = 0; j < cst->n_row; ++j)
+			if (isl_seq_eq(bmap->ineq[i], cst->row[j], cst->n_col))
+				break;
+		if (j >= cst->n_row)
+			return 1;
+	}
+
+	return 0;
+}
+
+/* Given that the last set variable of "bset" represents the minimum
+ * of the bounds in "cst", check whether we need to split the domain
+ * based on which bound attains the minimum.
+ *
+ * We simply call need_split_basic_map here.  This is safe because
+ * the position of the minimum is computed from "cst" and not
+ * from "bmap".
+ */
+static int need_split_basic_set(__isl_keep isl_basic_set *bset,
+	__isl_keep isl_mat *cst)
+{
+	return need_split_basic_map((isl_basic_map *)bset, cst);
+}
+
+/* Given that the last set variable of "set" represents the minimum
+ * of the bounds in "cst", check whether we need to split the domain
+ * based on which bound attains the minimum.
+ */
+static int need_split_set(__isl_keep isl_set *set, __isl_keep isl_mat *cst)
+{
+	int i;
+
+	for (i = 0; i < set->n; ++i)
+		if (need_split_basic_set(set->p[i], cst))
+			return 1;
+
+	return 0;
+}
+
+/* Given a set of which the last set variable is the minimum
+ * of the bounds in "cst", split each basic set in the set
+ * in pieces where one of the bounds is (strictly) smaller than the others.
+ * This subdivision is given in "min_expr".
+ * The variable is subsequently projected out.
+ *
+ * We only do the split when it is needed.
+ * For example if the last input variable m = min(a,b) and the only
+ * constraints in the given basic set are lower bounds on m,
+ * i.e., l <= m = min(a,b), then we can simply project out m
+ * to obtain l <= a and l <= b, without having to split on whether
+ * m is equal to a or b.
+ */
+static __isl_give isl_set *split(__isl_take isl_set *empty,
+	__isl_take isl_set *min_expr, __isl_take isl_mat *cst)
+{
+	int n_in;
+	int i;
+	isl_space *dim;
+	isl_set *res;
+
+	if (!empty || !min_expr || !cst)
+		goto error;
+
+	n_in = isl_set_dim(empty, isl_dim_set);
+	dim = isl_set_get_space(empty);
+	dim = isl_space_drop_dims(dim, isl_dim_set, n_in - 1, 1);
+	res = isl_set_empty(dim);
+
+	for (i = 0; i < empty->n; ++i) {
+		isl_set *set;
+
+		set = isl_set_from_basic_set(isl_basic_set_copy(empty->p[i]));
+		if (need_split_basic_set(empty->p[i], cst))
+			set = isl_set_intersect(set, isl_set_copy(min_expr));
+		set = isl_set_remove_dims(set, isl_dim_set, n_in - 1, 1);
+
+		res = isl_set_union_disjoint(res, set);
+	}
+
+	isl_set_free(empty);
+	isl_set_free(min_expr);
+	isl_mat_free(cst);
+	return res;
+error:
+	isl_set_free(empty);
+	isl_set_free(min_expr);
+	isl_mat_free(cst);
+	return NULL;
+}
+
+/* Given a map of which the last input variable is the minimum
+ * of the bounds in "cst", split each basic set in the set
+ * in pieces where one of the bounds is (strictly) smaller than the others.
+ * This subdivision is given in "min_expr".
+ * The variable is subsequently projected out.
+ *
+ * The implementation is essentially the same as that of "split".
+ */
+static __isl_give isl_map *split_domain(__isl_take isl_map *opt,
+	__isl_take isl_set *min_expr, __isl_take isl_mat *cst)
+{
+	int n_in;
+	int i;
+	isl_space *dim;
+	isl_map *res;
+
+	if (!opt || !min_expr || !cst)
+		goto error;
+
+	n_in = isl_map_dim(opt, isl_dim_in);
+	dim = isl_map_get_space(opt);
+	dim = isl_space_drop_dims(dim, isl_dim_in, n_in - 1, 1);
+	res = isl_map_empty(dim);
+
+	for (i = 0; i < opt->n; ++i) {
+		isl_map *map;
+
+		map = isl_map_from_basic_map(isl_basic_map_copy(opt->p[i]));
+		if (need_split_basic_map(opt->p[i], cst))
+			map = isl_map_intersect_domain(map,
+						       isl_set_copy(min_expr));
+		map = isl_map_remove_dims(map, isl_dim_in, n_in - 1, 1);
+
+		res = isl_map_union_disjoint(res, map);
+	}
+
+	isl_map_free(opt);
+	isl_set_free(min_expr);
+	isl_mat_free(cst);
+	return res;
+error:
+	isl_map_free(opt);
+	isl_set_free(min_expr);
+	isl_mat_free(cst);
+	return NULL;
+}
+
+static __isl_give isl_map *basic_map_partial_lexopt(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty, int max);
+
+union isl_lex_res {
+	void *p;
+	isl_map *map;
+	isl_pw_multi_aff *pma;
+};
+
+/* This function is called from basic_map_partial_lexopt_symm.
+ * The last variable of "bmap" and "dom" corresponds to the minimum
+ * of the bounds in "cst".  "map_space" is the space of the original
+ * input relation (of basic_map_partial_lexopt_symm) and "set_space"
+ * is the space of the original domain.
+ *
+ * We recursively call basic_map_partial_lexopt and then plug in
+ * the definition of the minimum in the result.
+ */
+static __isl_give union isl_lex_res basic_map_partial_lexopt_symm_map_core(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty, int max, __isl_take isl_mat *cst,
+	__isl_take isl_space *map_space, __isl_take isl_space *set_space)
+{
+	isl_map *opt;
+	isl_set *min_expr;
+	union isl_lex_res res;
+
+	min_expr = set_minimum(isl_basic_set_get_space(dom), isl_mat_copy(cst));
+
+	opt = basic_map_partial_lexopt(bmap, dom, empty, max);
+
+	if (empty) {
+		*empty = split(*empty,
+			       isl_set_copy(min_expr), isl_mat_copy(cst));
+		*empty = isl_set_reset_space(*empty, set_space);
+	}
+
+	opt = split_domain(opt, min_expr, cst);
+	opt = isl_map_reset_space(opt, map_space);
+
+	res.map = opt;
+	return res;
+}
+
+/* Given a basic map with at least two parallel constraints (as found
+ * by the function parallel_constraints), first look for more constraints
+ * parallel to the two constraint and replace the found list of parallel
+ * constraints by a single constraint with as "input" part the minimum
+ * of the input parts of the list of constraints.  Then, recursively call
+ * basic_map_partial_lexopt (possibly finding more parallel constraints)
+ * and plug in the definition of the minimum in the result.
+ *
+ * More specifically, given a set of constraints
+ *
+ *	a x + b_i(p) >= 0
+ *
+ * Replace this set by a single constraint
+ *
+ *	a x + u >= 0
+ *
+ * with u a new parameter with constraints
+ *
+ *	u <= b_i(p)
+ *
+ * Any solution to the new system is also a solution for the original system
+ * since
+ *
+ *	a x >= -u >= -b_i(p)
+ *
+ * Moreover, m = min_i(b_i(p)) satisfies the constraints on u and can
+ * therefore be plugged into the solution.
+ */
+static union isl_lex_res basic_map_partial_lexopt_symm(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty, int max, int first, int second,
+	__isl_give union isl_lex_res (*core)(__isl_take isl_basic_map *bmap,
+					    __isl_take isl_basic_set *dom,
+					    __isl_give isl_set **empty,
+					    int max, __isl_take isl_mat *cst,
+					    __isl_take isl_space *map_space,
+					    __isl_take isl_space *set_space))
+{
+	int i, n, k;
+	int *list = NULL;
+	unsigned n_in, n_out, n_div;
+	isl_ctx *ctx;
+	isl_vec *var = NULL;
+	isl_mat *cst = NULL;
+	isl_space *map_space, *set_space;
+	union isl_lex_res res;
+
+	map_space = isl_basic_map_get_space(bmap);
+	set_space = empty ? isl_basic_set_get_space(dom) : NULL;
+
+	n_in = isl_basic_map_dim(bmap, isl_dim_param) +
+	       isl_basic_map_dim(bmap, isl_dim_in);
+	n_out = isl_basic_map_dim(bmap, isl_dim_all) - n_in;
+
+	ctx = isl_basic_map_get_ctx(bmap);
+	list = isl_alloc_array(ctx, int, bmap->n_ineq);
+	var = isl_vec_alloc(ctx, n_out);
+	if ((bmap->n_ineq && !list) || (n_out && !var))
+		goto error;
+
+	list[0] = first;
+	list[1] = second;
+	isl_seq_cpy(var->el, bmap->ineq[first] + 1 + n_in, n_out);
+	for (i = second + 1, n = 2; i < bmap->n_ineq; ++i) {
+		if (isl_seq_eq(var->el, bmap->ineq[i] + 1 + n_in, n_out))
+			list[n++] = i;
+	}
+
+	cst = isl_mat_alloc(ctx, n, 1 + n_in);
+	if (!cst)
+		goto error;
+
+	for (i = 0; i < n; ++i)
+		isl_seq_cpy(cst->row[i], bmap->ineq[list[i]], 1 + n_in);
+
+	bmap = isl_basic_map_cow(bmap);
+	if (!bmap)
+		goto error;
+	for (i = n - 1; i >= 0; --i)
+		if (isl_basic_map_drop_inequality(bmap, list[i]) < 0)
+			goto error;
+
+	bmap = isl_basic_map_add(bmap, isl_dim_in, 1);
+	bmap = isl_basic_map_extend_constraints(bmap, 0, 1);
+	k = isl_basic_map_alloc_inequality(bmap);
+	if (k < 0)
+		goto error;
+	isl_seq_clr(bmap->ineq[k], 1 + n_in);
+	isl_int_set_si(bmap->ineq[k][1 + n_in], 1);
+	isl_seq_cpy(bmap->ineq[k] + 1 + n_in + 1, var->el, n_out);
+	bmap = isl_basic_map_finalize(bmap);
+
+	n_div = isl_basic_set_dim(dom, isl_dim_div);
+	dom = isl_basic_set_add_dims(dom, isl_dim_set, 1);
+	dom = isl_basic_set_extend_constraints(dom, 0, n);
+	for (i = 0; i < n; ++i) {
+		k = isl_basic_set_alloc_inequality(dom);
+		if (k < 0)
+			goto error;
+		isl_seq_cpy(dom->ineq[k], cst->row[i], 1 + n_in);
+		isl_int_set_si(dom->ineq[k][1 + n_in], -1);
+		isl_seq_clr(dom->ineq[k] + 1 + n_in + 1, n_div);
+	}
+
+	isl_vec_free(var);
+	free(list);
+
+	return core(bmap, dom, empty, max, cst, map_space, set_space);
+error:
+	isl_space_free(map_space);
+	isl_space_free(set_space);
+	isl_mat_free(cst);
+	isl_vec_free(var);
+	free(list);
+	isl_basic_set_free(dom);
+	isl_basic_map_free(bmap);
+	res.p = NULL;
+	return res;
+}
+
+static __isl_give isl_map *basic_map_partial_lexopt_symm_map(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty, int max, int first, int second)
+{
+	return basic_map_partial_lexopt_symm(bmap, dom, empty, max,
+		    first, second, &basic_map_partial_lexopt_symm_map_core).map;
+}
+
+/* Recursive part of isl_tab_basic_map_partial_lexopt, after detecting
+ * equalities and removing redundant constraints.
+ *
+ * We first check if there are any parallel constraints (left).
+ * If not, we are in the base case.
+ * If there are parallel constraints, we replace them by a single
+ * constraint in basic_map_partial_lexopt_symm and then call
+ * this function recursively to look for more parallel constraints.
+ */
+static __isl_give isl_map *basic_map_partial_lexopt(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty, int max)
+{
+	int par = 0;
+	int first, second;
+
+	if (!bmap)
+		goto error;
+
+	if (bmap->ctx->opt->pip_symmetry)
+		par = parallel_constraints(bmap, &first, &second);
+	if (par < 0)
+		goto error;
+	if (!par)
+		return basic_map_partial_lexopt_base_map(bmap, dom, empty, max);
+	
+	return basic_map_partial_lexopt_symm_map(bmap, dom, empty, max,
+						 first, second);
+error:
+	isl_basic_set_free(dom);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Compute the lexicographic minimum (or maximum if "max" is set)
+ * of "bmap" over the domain "dom" and return the result as a map.
+ * If "empty" is not NULL, then *empty is assigned a set that
+ * contains those parts of the domain where there is no solution.
+ * If "bmap" is marked as rational (ISL_BASIC_MAP_RATIONAL),
+ * then we compute the rational optimum.  Otherwise, we compute
+ * the integral optimum.
+ *
+ * We perform some preprocessing.  As the PILP solver does not
+ * handle implicit equalities very well, we first make sure all
+ * the equalities are explicitly available.
+ *
+ * We also add context constraints to the basic map and remove
+ * redundant constraints.  This is only needed because of the
+ * way we handle simple symmetries.  In particular, we currently look
+ * for symmetries on the constraints, before we set up the main tableau.
+ * It is then no good to look for symmetries on possibly redundant constraints.
+ */
+struct isl_map *isl_tab_basic_map_partial_lexopt(
+		struct isl_basic_map *bmap, struct isl_basic_set *dom,
+		struct isl_set **empty, int max)
+{
+	if (empty)
+		*empty = NULL;
+	if (!bmap || !dom)
+		goto error;
+
+	isl_assert(bmap->ctx,
+	    isl_basic_map_compatible_domain(bmap, dom), goto error);
+
+	if (isl_basic_set_dim(dom, isl_dim_all) == 0)
+		return basic_map_partial_lexopt(bmap, dom, empty, max);
+
+	bmap = isl_basic_map_intersect_domain(bmap, isl_basic_set_copy(dom));
+	bmap = isl_basic_map_detect_equalities(bmap);
+	bmap = isl_basic_map_remove_redundancies(bmap);
+
+	return basic_map_partial_lexopt(bmap, dom, empty, max);
+error:
+	isl_basic_set_free(dom);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+struct isl_sol_for {
+	struct isl_sol	sol;
+	int		(*fn)(__isl_take isl_basic_set *dom,
+				__isl_take isl_aff_list *list, void *user);
+	void		*user;
+};
+
+static void sol_for_free(struct isl_sol_for *sol_for)
+{
+	if (!sol_for)
+		return;
+	if (sol_for->sol.context)
+		sol_for->sol.context->op->free(sol_for->sol.context);
+	free(sol_for);
+}
+
+static void sol_for_free_wrap(struct isl_sol *sol)
+{
+	sol_for_free((struct isl_sol_for *)sol);
+}
+
+/* Add the solution identified by the tableau and the context tableau.
+ *
+ * See documentation of sol_add for more details.
+ *
+ * Instead of constructing a basic map, this function calls a user
+ * defined function with the current context as a basic set and
+ * a list of affine expressions representing the relation between
+ * the input and output.  The space over which the affine expressions
+ * are defined is the same as that of the domain.  The number of
+ * affine expressions in the list is equal to the number of output variables.
+ */
+static void sol_for_add(struct isl_sol_for *sol,
+	struct isl_basic_set *dom, struct isl_mat *M)
+{
+	int i;
+	isl_ctx *ctx;
+	isl_local_space *ls;
+	isl_aff *aff;
+	isl_aff_list *list;
+
+	if (sol->sol.error || !dom || !M)
+		goto error;
+
+	ctx = isl_basic_set_get_ctx(dom);
+	ls = isl_basic_set_get_local_space(dom);
+	list = isl_aff_list_alloc(ctx, M->n_row - 1);
+	for (i = 1; i < M->n_row; ++i) {
+		aff = isl_aff_alloc(isl_local_space_copy(ls));
+		if (aff) {
+			isl_int_set(aff->v->el[0], M->row[0][0]);
+			isl_seq_cpy(aff->v->el + 1, M->row[i], M->n_col);
+		}
+		aff = isl_aff_normalize(aff);
+		list = isl_aff_list_add(list, aff);
+	}
+	isl_local_space_free(ls);
+
+	dom = isl_basic_set_finalize(dom);
+
+	if (sol->fn(isl_basic_set_copy(dom), list, sol->user) < 0)
+		goto error;
+
+	isl_basic_set_free(dom);
+	isl_mat_free(M);
+	return;
+error:
+	isl_basic_set_free(dom);
+	isl_mat_free(M);
+	sol->sol.error = 1;
+}
+
+static void sol_for_add_wrap(struct isl_sol *sol,
+	struct isl_basic_set *dom, struct isl_mat *M)
+{
+	sol_for_add((struct isl_sol_for *)sol, dom, M);
+}
+
+static struct isl_sol_for *sol_for_init(struct isl_basic_map *bmap, int max,
+	int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_aff_list *list,
+		  void *user),
+	void *user)
+{
+	struct isl_sol_for *sol_for = NULL;
+	isl_space *dom_dim;
+	struct isl_basic_set *dom = NULL;
+
+	sol_for = isl_calloc_type(bmap->ctx, struct isl_sol_for);
+	if (!sol_for)
+		goto error;
+
+	dom_dim = isl_space_domain(isl_space_copy(bmap->dim));
+	dom = isl_basic_set_universe(dom_dim);
+
+	sol_for->sol.rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
+	sol_for->sol.dec_level.callback.run = &sol_dec_level_wrap;
+	sol_for->sol.dec_level.sol = &sol_for->sol;
+	sol_for->fn = fn;
+	sol_for->user = user;
+	sol_for->sol.max = max;
+	sol_for->sol.n_out = isl_basic_map_dim(bmap, isl_dim_out);
+	sol_for->sol.add = &sol_for_add_wrap;
+	sol_for->sol.add_empty = NULL;
+	sol_for->sol.free = &sol_for_free_wrap;
+
+	sol_for->sol.context = isl_context_alloc(dom);
+	if (!sol_for->sol.context)
+		goto error;
+
+	isl_basic_set_free(dom);
+	return sol_for;
+error:
+	isl_basic_set_free(dom);
+	sol_for_free(sol_for);
+	return NULL;
+}
+
+static void sol_for_find_solutions(struct isl_sol_for *sol_for,
+	struct isl_tab *tab)
+{
+	find_solutions_main(&sol_for->sol, tab);
+}
+
+int isl_basic_map_foreach_lexopt(__isl_keep isl_basic_map *bmap, int max,
+	int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_aff_list *list,
+		  void *user),
+	void *user)
+{
+	struct isl_sol_for *sol_for = NULL;
+
+	bmap = isl_basic_map_copy(bmap);
+	bmap = isl_basic_map_detect_equalities(bmap);
+	if (!bmap)
+		return -1;
+
+	sol_for = sol_for_init(bmap, max, fn, user);
+	if (!sol_for)
+		goto error;
+
+	if (isl_basic_map_plain_is_empty(bmap))
+		/* nothing */;
+	else {
+		struct isl_tab *tab;
+		struct isl_context *context = sol_for->sol.context;
+		tab = tab_for_lexmin(bmap,
+				context->op->peek_basic_set(context), 1, max);
+		tab = context->op->detect_nonnegative_parameters(context, tab);
+		sol_for_find_solutions(sol_for, tab);
+		if (sol_for->sol.error)
+			goto error;
+	}
+
+	sol_free(&sol_for->sol);
+	isl_basic_map_free(bmap);
+	return 0;
+error:
+	sol_free(&sol_for->sol);
+	isl_basic_map_free(bmap);
+	return -1;
+}
+
+int isl_basic_set_foreach_lexopt(__isl_keep isl_basic_set *bset, int max,
+	int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_aff_list *list,
+		  void *user),
+	void *user)
+{
+	return isl_basic_map_foreach_lexopt(bset, max, fn, user);
+}
+
+/* Check if the given sequence of len variables starting at pos
+ * represents a trivial (i.e., zero) solution.
+ * The variables are assumed to be non-negative and to come in pairs,
+ * with each pair representing a variable of unrestricted sign.
+ * The solution is trivial if each such pair in the sequence consists
+ * of two identical values, meaning that the variable being represented
+ * has value zero.
+ */
+static int region_is_trivial(struct isl_tab *tab, int pos, int len)
+{
+	int i;
+
+	if (len == 0)
+		return 0;
+
+	for (i = 0; i < len; i +=  2) {
+		int neg_row;
+		int pos_row;
+
+		neg_row = tab->var[pos + i].is_row ?
+				tab->var[pos + i].index : -1;
+		pos_row = tab->var[pos + i + 1].is_row ?
+				tab->var[pos + i + 1].index : -1;
+
+		if ((neg_row < 0 ||
+		     isl_int_is_zero(tab->mat->row[neg_row][1])) &&
+		    (pos_row < 0 ||
+		     isl_int_is_zero(tab->mat->row[pos_row][1])))
+			continue;
+
+		if (neg_row < 0 || pos_row < 0)
+			return 0;
+		if (isl_int_ne(tab->mat->row[neg_row][1],
+			       tab->mat->row[pos_row][1]))
+			return 0;
+	}
+
+	return 1;
+}
+
+/* Return the index of the first trivial region or -1 if all regions
+ * are non-trivial.
+ */
+static int first_trivial_region(struct isl_tab *tab,
+	int n_region, struct isl_region *region)
+{
+	int i;
+
+	for (i = 0; i < n_region; ++i) {
+		if (region_is_trivial(tab, region[i].pos, region[i].len))
+			return i;
+	}
+
+	return -1;
+}
+
+/* Check if the solution is optimal, i.e., whether the first
+ * n_op entries are zero.
+ */
+static int is_optimal(__isl_keep isl_vec *sol, int n_op)
+{
+	int i;
+
+	for (i = 0; i < n_op; ++i)
+		if (!isl_int_is_zero(sol->el[1 + i]))
+			return 0;
+	return 1;
+}
+
+/* Add constraints to "tab" that ensure that any solution is significantly
+ * better that that represented by "sol".  That is, find the first
+ * relevant (within first n_op) non-zero coefficient and force it (along
+ * with all previous coefficients) to be zero.
+ * If the solution is already optimal (all relevant coefficients are zero),
+ * then just mark the table as empty.
+ */
+static int force_better_solution(struct isl_tab *tab,
+	__isl_keep isl_vec *sol, int n_op)
+{
+	int i;
+	isl_ctx *ctx;
+	isl_vec *v = NULL;
+
+	if (!sol)
+		return -1;
+
+	for (i = 0; i < n_op; ++i)
+		if (!isl_int_is_zero(sol->el[1 + i]))
+			break;
+
+	if (i == n_op) {
+		if (isl_tab_mark_empty(tab) < 0)
+			return -1;
+		return 0;
+	}
+
+	ctx = isl_vec_get_ctx(sol);
+	v = isl_vec_alloc(ctx, 1 + tab->n_var);
+	if (!v)
+		return -1;
+
+	for (; i >= 0; --i) {
+		v = isl_vec_clr(v);
+		isl_int_set_si(v->el[1 + i], -1);
+		if (add_lexmin_eq(tab, v->el) < 0)
+			goto error;
+	}
+
+	isl_vec_free(v);
+	return 0;
+error:
+	isl_vec_free(v);
+	return -1;
+}
+
+struct isl_trivial {
+	int update;
+	int region;
+	int side;
+	struct isl_tab_undo *snap;
+};
+
+/* Return the lexicographically smallest non-trivial solution of the
+ * given ILP problem.
+ *
+ * All variables are assumed to be non-negative.
+ *
+ * n_op is the number of initial coordinates to optimize.
+ * That is, once a solution has been found, we will only continue looking
+ * for solution that result in significantly better values for those
+ * initial coordinates.  That is, we only continue looking for solutions
+ * that increase the number of initial zeros in this sequence.
+ *
+ * A solution is non-trivial, if it is non-trivial on each of the
+ * specified regions.  Each region represents a sequence of pairs
+ * of variables.  A solution is non-trivial on such a region if
+ * at least one of these pairs consists of different values, i.e.,
+ * such that the non-negative variable represented by the pair is non-zero.
+ *
+ * Whenever a conflict is encountered, all constraints involved are
+ * reported to the caller through a call to "conflict".
+ *
+ * We perform a simple branch-and-bound backtracking search.
+ * Each level in the search represents initially trivial region that is forced
+ * to be non-trivial.
+ * At each level we consider n cases, where n is the length of the region.
+ * In terms of the n/2 variables of unrestricted signs being encoded by
+ * the region, we consider the cases
+ *	x_0 >= 1
+ *	x_0 <= -1
+ *	x_0 = 0 and x_1 >= 1
+ *	x_0 = 0 and x_1 <= -1
+ *	x_0 = 0 and x_1 = 0 and x_2 >= 1
+ *	x_0 = 0 and x_1 = 0 and x_2 <= -1
+ *	...
+ * The cases are considered in this order, assuming that each pair
+ * x_i_a x_i_b represents the value x_i_b - x_i_a.
+ * That is, x_0 >= 1 is enforced by adding the constraint
+ *	x_0_b - x_0_a >= 1
+ */
+__isl_give isl_vec *isl_tab_basic_set_non_trivial_lexmin(
+	__isl_take isl_basic_set *bset, int n_op, int n_region,
+	struct isl_region *region,
+	int (*conflict)(int con, void *user), void *user)
+{
+	int i, j;
+	int r;
+	isl_ctx *ctx;
+	isl_vec *v = NULL;
+	isl_vec *sol = NULL;
+	struct isl_tab *tab;
+	struct isl_trivial *triv = NULL;
+	int level, init;
+
+	if (!bset)
+		return NULL;
+
+	ctx = isl_basic_set_get_ctx(bset);
+	sol = isl_vec_alloc(ctx, 0);
+
+	tab = tab_for_lexmin(bset, NULL, 0, 0);
+	if (!tab)
+		goto error;
+	tab->conflict = conflict;
+	tab->conflict_user = user;
+
+	v = isl_vec_alloc(ctx, 1 + tab->n_var);
+	triv = isl_calloc_array(ctx, struct isl_trivial, n_region);
+	if (!v || (n_region && !triv))
+		goto error;
+
+	level = 0;
+	init = 1;
+
+	while (level >= 0) {
+		int side, base;
+
+		if (init) {
+			tab = cut_to_integer_lexmin(tab, CUT_ONE);
+			if (!tab)
+				goto error;
+			if (tab->empty)
+				goto backtrack;
+			r = first_trivial_region(tab, n_region, region);
+			if (r < 0) {
+				for (i = 0; i < level; ++i)
+					triv[i].update = 1;
+				isl_vec_free(sol);
+				sol = isl_tab_get_sample_value(tab);
+				if (!sol)
+					goto error;
+				if (is_optimal(sol, n_op))
+					break;
+				goto backtrack;
+			}
+			if (level >= n_region)
+				isl_die(ctx, isl_error_internal,
+					"nesting level too deep", goto error);
+			if (isl_tab_extend_cons(tab,
+					    2 * region[r].len + 2 * n_op) < 0)
+				goto error;
+			triv[level].region = r;
+			triv[level].side = 0;
+		}
+
+		r = triv[level].region;
+		side = triv[level].side;
+		base = 2 * (side/2);
+
+		if (side >= region[r].len) {
+backtrack:
+			level--;
+			init = 0;
+			if (level >= 0)
+				if (isl_tab_rollback(tab, triv[level].snap) < 0)
+					goto error;
+			continue;
+		}
+
+		if (triv[level].update) {
+			if (force_better_solution(tab, sol, n_op) < 0)
+				goto error;
+			triv[level].update = 0;
+		}
+
+		if (side == base && base >= 2) {
+			for (j = base - 2; j < base; ++j) {
+				v = isl_vec_clr(v);
+				isl_int_set_si(v->el[1 + region[r].pos + j], 1);
+				if (add_lexmin_eq(tab, v->el) < 0)
+					goto error;
+			}
+		}
+
+		triv[level].snap = isl_tab_snap(tab);
+		if (isl_tab_push_basis(tab) < 0)
+			goto error;
+
+		v = isl_vec_clr(v);
+		isl_int_set_si(v->el[0], -1);
+		isl_int_set_si(v->el[1 + region[r].pos + side], -1);
+		isl_int_set_si(v->el[1 + region[r].pos + (side ^ 1)], 1);
+		tab = add_lexmin_ineq(tab, v->el);
+
+		triv[level].side++;
+		level++;
+		init = 1;
+	}
+
+	free(triv);
+	isl_vec_free(v);
+	isl_tab_free(tab);
+	isl_basic_set_free(bset);
+
+	return sol;
+error:
+	free(triv);
+	isl_vec_free(v);
+	isl_tab_free(tab);
+	isl_basic_set_free(bset);
+	isl_vec_free(sol);
+	return NULL;
+}
+
+/* Return the lexicographically smallest rational point in "bset",
+ * assuming that all variables are non-negative.
+ * If "bset" is empty, then return a zero-length vector.
+ */
+__isl_give isl_vec *isl_tab_basic_set_non_neg_lexmin(
+	__isl_take isl_basic_set *bset)
+{
+	struct isl_tab *tab;
+	isl_ctx *ctx = isl_basic_set_get_ctx(bset);
+	isl_vec *sol;
+
+	if (!bset)
+		return NULL;
+
+	tab = tab_for_lexmin(bset, NULL, 0, 0);
+	if (!tab)
+		goto error;
+	if (tab->empty)
+		sol = isl_vec_alloc(ctx, 0);
+	else
+		sol = isl_tab_get_sample_value(tab);
+	isl_tab_free(tab);
+	isl_basic_set_free(bset);
+	return sol;
+error:
+	isl_tab_free(tab);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+struct isl_sol_pma {
+	struct isl_sol	sol;
+	isl_pw_multi_aff *pma;
+	isl_set *empty;
+};
+
+static void sol_pma_free(struct isl_sol_pma *sol_pma)
+{
+	if (!sol_pma)
+		return;
+	if (sol_pma->sol.context)
+		sol_pma->sol.context->op->free(sol_pma->sol.context);
+	isl_pw_multi_aff_free(sol_pma->pma);
+	isl_set_free(sol_pma->empty);
+	free(sol_pma);
+}
+
+/* This function is called for parts of the context where there is
+ * no solution, with "bset" corresponding to the context tableau.
+ * Simply add the basic set to the set "empty".
+ */
+static void sol_pma_add_empty(struct isl_sol_pma *sol,
+	__isl_take isl_basic_set *bset)
+{
+	if (!bset)
+		goto error;
+	isl_assert(bset->ctx, sol->empty, goto error);
+
+	sol->empty = isl_set_grow(sol->empty, 1);
+	bset = isl_basic_set_simplify(bset);
+	bset = isl_basic_set_finalize(bset);
+	sol->empty = isl_set_add_basic_set(sol->empty, bset);
+	if (!sol->empty)
+		sol->sol.error = 1;
+	return;
+error:
+	isl_basic_set_free(bset);
+	sol->sol.error = 1;
+}
+
+/* Given a basic map "dom" that represents the context and an affine
+ * matrix "M" that maps the dimensions of the context to the
+ * output variables, construct an isl_pw_multi_aff with a single
+ * cell corresponding to "dom" and affine expressions copied from "M".
+ */
+static void sol_pma_add(struct isl_sol_pma *sol,
+	__isl_take isl_basic_set *dom, __isl_take isl_mat *M)
+{
+	int i;
+	isl_local_space *ls;
+	isl_aff *aff;
+	isl_multi_aff *maff;
+	isl_pw_multi_aff *pma;
+
+	maff = isl_multi_aff_alloc(isl_pw_multi_aff_get_space(sol->pma));
+	ls = isl_basic_set_get_local_space(dom);
+	for (i = 1; i < M->n_row; ++i) {
+		aff = isl_aff_alloc(isl_local_space_copy(ls));
+		if (aff) {
+			isl_int_set(aff->v->el[0], M->row[0][0]);
+			isl_seq_cpy(aff->v->el + 1, M->row[i], M->n_col);
+		}
+		aff = isl_aff_normalize(aff);
+		maff = isl_multi_aff_set_aff(maff, i - 1, aff);
+	}
+	isl_local_space_free(ls);
+	isl_mat_free(M);
+	dom = isl_basic_set_simplify(dom);
+	dom = isl_basic_set_finalize(dom);
+	pma = isl_pw_multi_aff_alloc(isl_set_from_basic_set(dom), maff);
+	sol->pma = isl_pw_multi_aff_add_disjoint(sol->pma, pma);
+	if (!sol->pma)
+		sol->sol.error = 1;
+}
+
+static void sol_pma_free_wrap(struct isl_sol *sol)
+{
+	sol_pma_free((struct isl_sol_pma *)sol);
+}
+
+static void sol_pma_add_empty_wrap(struct isl_sol *sol,
+	__isl_take isl_basic_set *bset)
+{
+	sol_pma_add_empty((struct isl_sol_pma *)sol, bset);
+}
+
+static void sol_pma_add_wrap(struct isl_sol *sol,
+	__isl_take isl_basic_set *dom, __isl_take isl_mat *M)
+{
+	sol_pma_add((struct isl_sol_pma *)sol, dom, M);
+}
+
+/* Construct an isl_sol_pma structure for accumulating the solution.
+ * If track_empty is set, then we also keep track of the parts
+ * of the context where there is no solution.
+ * If max is set, then we are solving a maximization, rather than
+ * a minimization problem, which means that the variables in the
+ * tableau have value "M - x" rather than "M + x".
+ */
+static struct isl_sol *sol_pma_init(__isl_keep isl_basic_map *bmap,
+	__isl_take isl_basic_set *dom, int track_empty, int max)
+{
+	struct isl_sol_pma *sol_pma = NULL;
+
+	if (!bmap)
+		goto error;
+
+	sol_pma = isl_calloc_type(bmap->ctx, struct isl_sol_pma);
+	if (!sol_pma)
+		goto error;
+
+	sol_pma->sol.rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
+	sol_pma->sol.dec_level.callback.run = &sol_dec_level_wrap;
+	sol_pma->sol.dec_level.sol = &sol_pma->sol;
+	sol_pma->sol.max = max;
+	sol_pma->sol.n_out = isl_basic_map_dim(bmap, isl_dim_out);
+	sol_pma->sol.add = &sol_pma_add_wrap;
+	sol_pma->sol.add_empty = track_empty ? &sol_pma_add_empty_wrap : NULL;
+	sol_pma->sol.free = &sol_pma_free_wrap;
+	sol_pma->pma = isl_pw_multi_aff_empty(isl_basic_map_get_space(bmap));
+	if (!sol_pma->pma)
+		goto error;
+
+	sol_pma->sol.context = isl_context_alloc(dom);
+	if (!sol_pma->sol.context)
+		goto error;
+
+	if (track_empty) {
+		sol_pma->empty = isl_set_alloc_space(isl_basic_set_get_space(dom),
+							1, ISL_SET_DISJOINT);
+		if (!sol_pma->empty)
+			goto error;
+	}
+
+	isl_basic_set_free(dom);
+	return &sol_pma->sol;
+error:
+	isl_basic_set_free(dom);
+	sol_pma_free(sol_pma);
+	return NULL;
+}
+
+/* Base case of isl_tab_basic_map_partial_lexopt, after removing
+ * some obvious symmetries.
+ *
+ * We call basic_map_partial_lexopt_base and extract the results.
+ */
+static __isl_give isl_pw_multi_aff *basic_map_partial_lexopt_base_pma(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty, int max)
+{
+	isl_pw_multi_aff *result = NULL;
+	struct isl_sol *sol;
+	struct isl_sol_pma *sol_pma;
+
+	sol = basic_map_partial_lexopt_base(bmap, dom, empty, max,
+					    &sol_pma_init);
+	if (!sol)
+		return NULL;
+	sol_pma = (struct isl_sol_pma *) sol;
+
+	result = isl_pw_multi_aff_copy(sol_pma->pma);
+	if (empty)
+		*empty = isl_set_copy(sol_pma->empty);
+	sol_free(&sol_pma->sol);
+	return result;
+}
+
+/* Given that the last input variable of "maff" represents the minimum
+ * of some bounds, check whether we need to plug in the expression
+ * of the minimum.
+ *
+ * In particular, check if the last input variable appears in any
+ * of the expressions in "maff".
+ */
+static int need_substitution(__isl_keep isl_multi_aff *maff)
+{
+	int i;
+	unsigned pos;
+
+	pos = isl_multi_aff_dim(maff, isl_dim_in) - 1;
+
+	for (i = 0; i < maff->n; ++i)
+		if (isl_aff_involves_dims(maff->p[i], isl_dim_in, pos, 1))
+			return 1;
+
+	return 0;
+}
+
+/* Given a set of upper bounds on the last "input" variable m,
+ * construct a piecewise affine expression that selects
+ * the minimal upper bound to m, i.e.,
+ * divide the space into cells where one
+ * of the upper bounds is smaller than all the others and select
+ * this upper bound on that cell.
+ *
+ * In particular, if there are n bounds b_i, then the result
+ * consists of n cell, each one of the form
+ *
+ *	b_i <= b_j	for j > i
+ *	b_i <  b_j	for j < i
+ *
+ * The affine expression on this cell is
+ *
+ *	b_i
+ */
+static __isl_give isl_pw_aff *set_minimum_pa(__isl_take isl_space *space,
+	__isl_take isl_mat *var)
+{
+	int i;
+	isl_aff *aff = NULL;
+	isl_basic_set *bset = NULL;
+	isl_ctx *ctx;
+	isl_pw_aff *paff = NULL;
+	isl_space *pw_space;
+	isl_local_space *ls = NULL;
+
+	if (!space || !var)
+		goto error;
+
+	ctx = isl_space_get_ctx(space);
+	ls = isl_local_space_from_space(isl_space_copy(space));
+	pw_space = isl_space_copy(space);
+	pw_space = isl_space_from_domain(pw_space);
+	pw_space = isl_space_add_dims(pw_space, isl_dim_out, 1);
+	paff = isl_pw_aff_alloc_size(pw_space, var->n_row);
+
+	for (i = 0; i < var->n_row; ++i) {
+		isl_pw_aff *paff_i;
+
+		aff = isl_aff_alloc(isl_local_space_copy(ls));
+		bset = isl_basic_set_alloc_space(isl_space_copy(space), 0,
+					       0, var->n_row - 1);
+		if (!aff || !bset)
+			goto error;
+		isl_int_set_si(aff->v->el[0], 1);
+		isl_seq_cpy(aff->v->el + 1, var->row[i], var->n_col);
+		isl_int_set_si(aff->v->el[1 + var->n_col], 0);
+		bset = select_minimum(bset, var, i);
+		paff_i = isl_pw_aff_alloc(isl_set_from_basic_set(bset), aff);
+		paff = isl_pw_aff_add_disjoint(paff, paff_i);
+	}
+
+	isl_local_space_free(ls);
+	isl_space_free(space);
+	isl_mat_free(var);
+	return paff;
+error:
+	isl_aff_free(aff);
+	isl_basic_set_free(bset);
+	isl_pw_aff_free(paff);
+	isl_local_space_free(ls);
+	isl_space_free(space);
+	isl_mat_free(var);
+	return NULL;
+}
+
+/* Given a piecewise multi-affine expression of which the last input variable
+ * is the minimum of the bounds in "cst", plug in the value of the minimum.
+ * This minimum expression is given in "min_expr_pa".
+ * The set "min_expr" contains the same information, but in the form of a set.
+ * The variable is subsequently projected out.
+ *
+ * The implementation is similar to those of "split" and "split_domain".
+ * If the variable appears in a given expression, then minimum expression
+ * is plugged in.  Otherwise, if the variable appears in the constraints
+ * and a split is required, then the domain is split.  Otherwise, no split
+ * is performed.
+ */
+static __isl_give isl_pw_multi_aff *split_domain_pma(
+	__isl_take isl_pw_multi_aff *opt, __isl_take isl_pw_aff *min_expr_pa,
+	__isl_take isl_set *min_expr, __isl_take isl_mat *cst)
+{
+	int n_in;
+	int i;
+	isl_space *space;
+	isl_pw_multi_aff *res;
+
+	if (!opt || !min_expr || !cst)
+		goto error;
+
+	n_in = isl_pw_multi_aff_dim(opt, isl_dim_in);
+	space = isl_pw_multi_aff_get_space(opt);
+	space = isl_space_drop_dims(space, isl_dim_in, n_in - 1, 1);
+	res = isl_pw_multi_aff_empty(space);
+
+	for (i = 0; i < opt->n; ++i) {
+		isl_pw_multi_aff *pma;
+
+		pma = isl_pw_multi_aff_alloc(isl_set_copy(opt->p[i].set),
+					 isl_multi_aff_copy(opt->p[i].maff));
+		if (need_substitution(opt->p[i].maff))
+			pma = isl_pw_multi_aff_substitute(pma,
+					isl_dim_in, n_in - 1, min_expr_pa);
+		else if (need_split_set(opt->p[i].set, cst))
+			pma = isl_pw_multi_aff_intersect_domain(pma,
+						       isl_set_copy(min_expr));
+		pma = isl_pw_multi_aff_project_out(pma,
+						    isl_dim_in, n_in - 1, 1);
+
+		res = isl_pw_multi_aff_add_disjoint(res, pma);
+	}
+
+	isl_pw_multi_aff_free(opt);
+	isl_pw_aff_free(min_expr_pa);
+	isl_set_free(min_expr);
+	isl_mat_free(cst);
+	return res;
+error:
+	isl_pw_multi_aff_free(opt);
+	isl_pw_aff_free(min_expr_pa);
+	isl_set_free(min_expr);
+	isl_mat_free(cst);
+	return NULL;
+}
+
+static __isl_give isl_pw_multi_aff *basic_map_partial_lexopt_pma(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty, int max);
+
+/* This function is called from basic_map_partial_lexopt_symm.
+ * The last variable of "bmap" and "dom" corresponds to the minimum
+ * of the bounds in "cst".  "map_space" is the space of the original
+ * input relation (of basic_map_partial_lexopt_symm) and "set_space"
+ * is the space of the original domain.
+ *
+ * We recursively call basic_map_partial_lexopt and then plug in
+ * the definition of the minimum in the result.
+ */
+static __isl_give union isl_lex_res basic_map_partial_lexopt_symm_pma_core(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty, int max, __isl_take isl_mat *cst,
+	__isl_take isl_space *map_space, __isl_take isl_space *set_space)
+{
+	isl_pw_multi_aff *opt;
+	isl_pw_aff *min_expr_pa;
+	isl_set *min_expr;
+	union isl_lex_res res;
+
+	min_expr = set_minimum(isl_basic_set_get_space(dom), isl_mat_copy(cst));
+	min_expr_pa = set_minimum_pa(isl_basic_set_get_space(dom),
+					isl_mat_copy(cst));
+
+	opt = basic_map_partial_lexopt_pma(bmap, dom, empty, max);
+
+	if (empty) {
+		*empty = split(*empty,
+			       isl_set_copy(min_expr), isl_mat_copy(cst));
+		*empty = isl_set_reset_space(*empty, set_space);
+	}
+
+	opt = split_domain_pma(opt, min_expr_pa, min_expr, cst);
+	opt = isl_pw_multi_aff_reset_space(opt, map_space);
+
+	res.pma = opt;
+	return res;
+}
+
+static __isl_give isl_pw_multi_aff *basic_map_partial_lexopt_symm_pma(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty, int max, int first, int second)
+{
+	return basic_map_partial_lexopt_symm(bmap, dom, empty, max,
+		    first, second, &basic_map_partial_lexopt_symm_pma_core).pma;
+}
+
+/* Recursive part of isl_basic_map_partial_lexopt_pw_multi_aff, after detecting
+ * equalities and removing redundant constraints.
+ *
+ * We first check if there are any parallel constraints (left).
+ * If not, we are in the base case.
+ * If there are parallel constraints, we replace them by a single
+ * constraint in basic_map_partial_lexopt_symm_pma and then call
+ * this function recursively to look for more parallel constraints.
+ */
+static __isl_give isl_pw_multi_aff *basic_map_partial_lexopt_pma(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty, int max)
+{
+	int par = 0;
+	int first, second;
+
+	if (!bmap)
+		goto error;
+
+	if (bmap->ctx->opt->pip_symmetry)
+		par = parallel_constraints(bmap, &first, &second);
+	if (par < 0)
+		goto error;
+	if (!par)
+		return basic_map_partial_lexopt_base_pma(bmap, dom, empty, max);
+	
+	return basic_map_partial_lexopt_symm_pma(bmap, dom, empty, max,
+						 first, second);
+error:
+	isl_basic_set_free(dom);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Compute the lexicographic minimum (or maximum if "max" is set)
+ * of "bmap" over the domain "dom" and return the result as a piecewise
+ * multi-affine expression.
+ * If "empty" is not NULL, then *empty is assigned a set that
+ * contains those parts of the domain where there is no solution.
+ * If "bmap" is marked as rational (ISL_BASIC_MAP_RATIONAL),
+ * then we compute the rational optimum.  Otherwise, we compute
+ * the integral optimum.
+ *
+ * We perform some preprocessing.  As the PILP solver does not
+ * handle implicit equalities very well, we first make sure all
+ * the equalities are explicitly available.
+ *
+ * We also add context constraints to the basic map and remove
+ * redundant constraints.  This is only needed because of the
+ * way we handle simple symmetries.  In particular, we currently look
+ * for symmetries on the constraints, before we set up the main tableau.
+ * It is then no good to look for symmetries on possibly redundant constraints.
+ */
+__isl_give isl_pw_multi_aff *isl_basic_map_partial_lexopt_pw_multi_aff(
+	__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
+	__isl_give isl_set **empty, int max)
+{
+	if (empty)
+		*empty = NULL;
+	if (!bmap || !dom)
+		goto error;
+
+	isl_assert(bmap->ctx,
+	    isl_basic_map_compatible_domain(bmap, dom), goto error);
+
+	if (isl_basic_set_dim(dom, isl_dim_all) == 0)
+		return basic_map_partial_lexopt_pma(bmap, dom, empty, max);
+
+	bmap = isl_basic_map_intersect_domain(bmap, isl_basic_set_copy(dom));
+	bmap = isl_basic_map_detect_equalities(bmap);
+	bmap = isl_basic_map_remove_redundancies(bmap);
+
+	return basic_map_partial_lexopt_pma(bmap, dom, empty, max);
+error:
+	isl_basic_set_free(dom);
+	isl_basic_map_free(bmap);
+	return NULL;
+}
diff --git a/polly/lib/External/isl/isl_tarjan.c b/polly/lib/External/isl/isl_tarjan.c
new file mode 100644
index 00000000000..61e6b6e2f0d
--- /dev/null
+++ b/polly/lib/External/isl/isl_tarjan.c
@@ -0,0 +1,138 @@
+/*
+ * Copyright 2010-2011 INRIA Saclay
+ * Copyright 2012      Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+#include <stdlib.h>
+#include <isl/ctx.h>
+#include <isl_tarjan.h>
+
+void isl_tarjan_graph_free(struct isl_tarjan_graph *g)
+{
+	if (!g)
+		return;
+	free(g->node);
+	free(g->stack);
+	free(g->order);
+	free(g);
+}
+
+static struct isl_tarjan_graph *isl_tarjan_graph_alloc(isl_ctx *ctx, int len)
+{
+	struct isl_tarjan_graph *g;
+	int i;
+
+	g = isl_calloc_type(ctx, struct isl_tarjan_graph);
+	if (!g)
+		return NULL;
+	g->len = len;
+	g->node = isl_alloc_array(ctx, struct isl_tarjan_node, len);
+	if (len && !g->node)
+		goto error;
+	for (i = 0; i < len; ++i)
+		g->node[i].index = -1;
+	g->stack = isl_alloc_array(ctx, int, len);
+	if (len && !g->stack)
+		goto error;
+	g->order = isl_alloc_array(ctx, int, 2 * len);
+	if (len && !g->order)
+		goto error;
+
+	g->sp = 0;
+	g->index = 0;
+	g->op = 0;
+
+	return g;
+error:
+	isl_tarjan_graph_free(g);
+	return NULL;
+}
+
+/* Perform Tarjan's algorithm for computing the strongly connected components
+ * in the graph with g->len nodes and with edges defined by "follows".
+ */
+static int isl_tarjan_components(struct isl_tarjan_graph *g, int i,
+	int (*follows)(int i, int j, void *user), void *user)
+{
+	int j;
+
+	g->node[i].index = g->index;
+	g->node[i].min_index = g->index;
+	g->node[i].on_stack = 1;
+	g->index++;
+	g->stack[g->sp++] = i;
+
+	for (j = g->len - 1; j >= 0; --j) {
+		int f;
+
+		if (j == i)
+			continue;
+		if (g->node[j].index >= 0 &&
+			(!g->node[j].on_stack ||
+			 g->node[j].index > g->node[i].min_index))
+			continue;
+
+		f = follows(i, j, user);
+		if (f < 0)
+			return -1;
+		if (!f)
+			continue;
+
+		if (g->node[j].index < 0) {
+			isl_tarjan_components(g, j, follows, user);
+			if (g->node[j].min_index < g->node[i].min_index)
+				g->node[i].min_index = g->node[j].min_index;
+		} else if (g->node[j].index < g->node[i].min_index)
+			g->node[i].min_index = g->node[j].index;
+	}
+
+	if (g->node[i].index != g->node[i].min_index)
+		return 0;
+
+	do {
+		j = g->stack[--g->sp];
+		g->node[j].on_stack = 0;
+		g->order[g->op++] = j;
+	} while (j != i);
+	g->order[g->op++] = -1;
+
+	return 0;
+}
+
+/* Decompose the graph with "len" nodes and edges defined by "follows"
+ * into strongly connected components (SCCs).
+ * follows(i, j, user) should return 1 if "i" follows "j" and 0 otherwise.
+ * It should return -1 on error.
+ *
+ * If SCC a contains a node i that follows a node j in another SCC b
+ * (i.e., follows(i, j, user) returns 1), then SCC a will appear after SCC b
+ * in the result.
+ */
+struct isl_tarjan_graph *isl_tarjan_graph_init(isl_ctx *ctx, int len,
+	int (*follows)(int i, int j, void *user), void *user)
+{
+	int i;
+	struct isl_tarjan_graph *g = NULL;
+
+	g = isl_tarjan_graph_alloc(ctx, len);
+	if (!g)
+		return NULL;
+	for (i = len - 1; i >= 0; --i) {
+		if (g->node[i].index >= 0)
+			continue;
+		if (isl_tarjan_components(g, i, follows, user) < 0)
+			goto error;
+	}
+
+	return g;
+error:
+	isl_tarjan_graph_free(g);
+	return NULL;
+}
diff --git a/polly/lib/External/isl/isl_tarjan.h b/polly/lib/External/isl/isl_tarjan.h
new file mode 100644
index 00000000000..0767d0916de
--- /dev/null
+++ b/polly/lib/External/isl/isl_tarjan.h
@@ -0,0 +1,40 @@
+#ifndef ISL_TARJAN_H
+#define ISL_TARJAN_H
+
+/* Structure for representing the nodes in the graph being traversed
+ * using Tarjan's algorithm.
+ * index represents the order in which nodes are visited.
+ * min_index is the index of the root of a (sub)component.
+ * on_stack indicates whether the node is currently on the stack.
+ */
+struct isl_tarjan_node {
+	int index;
+	int min_index;
+	int on_stack;
+};
+
+/* Structure for representing the graph being traversed
+ * using Tarjan's algorithm.
+ * len is the number of nodes
+ * node is an array of nodes
+ * stack contains the nodes on the path from the root to the current node
+ * sp is the stack pointer
+ * index is the index of the last node visited
+ * order contains the elements of the components separated by -1
+ * op represents the current position in order
+ */
+struct isl_tarjan_graph {
+	int len;
+	struct isl_tarjan_node *node;
+	int *stack;
+	int sp;
+	int index;
+	int *order;
+	int op;
+};
+
+struct isl_tarjan_graph *isl_tarjan_graph_init(isl_ctx *ctx, int len,
+	int (*follows)(int i, int j, void *user), void *user);
+void isl_tarjan_graph_free(struct isl_tarjan_graph *g);
+
+#endif
diff --git a/polly/lib/External/isl/isl_test.c b/polly/lib/External/isl/isl_test.c
new file mode 100644
index 00000000000..cb9075ad94b
--- /dev/null
+++ b/polly/lib/External/isl/isl_test.c
@@ -0,0 +1,5176 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2010      INRIA Saclay
+ * Copyright 2012-2013 Ecole Normale Superieure
+ * Copyright 2014      INRIA Rocquencourt
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
+ * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
+ * B.P. 105 - 78153 Le Chesnay, France
+ */
+
+#include <assert.h>
+#include <stdio.h>
+#include <limits.h>
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl_aff_private.h>
+#include <isl/set.h>
+#include <isl/flow.h>
+#include <isl_constraint_private.h>
+#include <isl/polynomial.h>
+#include <isl/union_map.h>
+#include <isl_factorization.h>
+#include <isl/schedule.h>
+#include <isl_options_private.h>
+#include <isl/vertices.h>
+#include <isl/ast_build.h>
+#include <isl/val.h>
+#include <isl/ilp.h>
+#include <isl_ast_build_expr.h>
+#include <isl/options.h>
+
+#define ARRAY_SIZE(array) (sizeof(array)/sizeof(*array))
+
+static char *srcdir;
+
+static char *get_filename(isl_ctx *ctx, const char *name, const char *suffix) {
+	char *filename;
+	int length;
+	char *pattern = "%s/test_inputs/%s.%s";
+
+	length = strlen(pattern) - 6 + strlen(srcdir) + strlen(name)
+		+ strlen(suffix) + 1;
+	filename = isl_alloc_array(ctx, char, length);
+
+	if (!filename)
+		return NULL;
+
+	sprintf(filename, pattern, srcdir, name, suffix);
+
+	return filename;
+}
+
+void test_parse_map(isl_ctx *ctx, const char *str)
+{
+	isl_map *map;
+
+	map = isl_map_read_from_str(ctx, str);
+	assert(map);
+	isl_map_free(map);
+}
+
+int test_parse_map_equal(isl_ctx *ctx, const char *str, const char *str2)
+{
+	isl_map *map, *map2;
+	int equal;
+
+	map = isl_map_read_from_str(ctx, str);
+	map2 = isl_map_read_from_str(ctx, str2);
+	equal = isl_map_is_equal(map, map2);
+	isl_map_free(map);
+	isl_map_free(map2);
+
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown, "maps not equal",
+			return -1);
+
+	return 0;
+}
+
+void test_parse_pwqp(isl_ctx *ctx, const char *str)
+{
+	isl_pw_qpolynomial *pwqp;
+
+	pwqp = isl_pw_qpolynomial_read_from_str(ctx, str);
+	assert(pwqp);
+	isl_pw_qpolynomial_free(pwqp);
+}
+
+static void test_parse_pwaff(isl_ctx *ctx, const char *str)
+{
+	isl_pw_aff *pwaff;
+
+	pwaff = isl_pw_aff_read_from_str(ctx, str);
+	assert(pwaff);
+	isl_pw_aff_free(pwaff);
+}
+
+int test_parse(struct isl_ctx *ctx)
+{
+	isl_map *map, *map2;
+	const char *str, *str2;
+
+	str = "{ [i] -> [-i] }";
+	map = isl_map_read_from_str(ctx, str);
+	assert(map);
+	isl_map_free(map);
+
+	str = "{ A[i] -> L[([i/3])] }";
+	map = isl_map_read_from_str(ctx, str);
+	assert(map);
+	isl_map_free(map);
+
+	test_parse_map(ctx, "{[[s] -> A[i]] -> [[s+1] -> A[i]]}");
+	test_parse_map(ctx, "{ [p1, y1, y2] -> [2, y1, y2] : "
+				"p1 = 1 && (y1 <= y2 || y2 = 0) }");
+
+	str = "{ [x,y]  : [([x/2]+y)/3] >= 1 }";
+	str2 = "{ [x, y] : 2y >= 6 - x }";
+	if (test_parse_map_equal(ctx, str, str2) < 0)
+		return -1;
+
+	if (test_parse_map_equal(ctx, "{ [x,y] : x <= min(y, 2*y+3) }",
+				      "{ [x,y] : x <= y, 2*y + 3 }") < 0)
+		return -1;
+	str = "{ [x, y] : (y <= x and y >= -3) or (2y <= -3 + x and y <= -4) }";
+	if (test_parse_map_equal(ctx, "{ [x,y] : x >= min(y, 2*y+3) }",
+					str) < 0)
+		return -1;
+
+	str = "{[new,old] -> [new+1-2*[(new+1)/2],old+1-2*[(old+1)/2]]}";
+	map = isl_map_read_from_str(ctx, str);
+	str = "{ [new, old] -> [o0, o1] : "
+	       "exists (e0 = [(-1 - new + o0)/2], e1 = [(-1 - old + o1)/2]: "
+	       "2e0 = -1 - new + o0 and 2e1 = -1 - old + o1 and o0 >= 0 and "
+	       "o0 <= 1 and o1 >= 0 and o1 <= 1) }";
+	map2 = isl_map_read_from_str(ctx, str);
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map);
+	isl_map_free(map2);
+
+	str = "{[new,old] -> [new+1-2*[(new+1)/2],old+1-2*[(old+1)/2]]}";
+	map = isl_map_read_from_str(ctx, str);
+	str = "{[new,old] -> [(new+1)%2,(old+1)%2]}";
+	map2 = isl_map_read_from_str(ctx, str);
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map);
+	isl_map_free(map2);
+
+	str = "[n] -> { [c1] : c1>=0 and c1<=floord(n-4,3) }";
+	str2 = "[n] -> { [c1] : c1 >= 0 and 3c1 <= -4 + n }";
+	if (test_parse_map_equal(ctx, str, str2) < 0)
+		return -1;
+
+	str = "{ [i,j] -> [i] : i < j; [i,j] -> [j] : j <= i }";
+	str2 = "{ [i,j] -> [min(i,j)] }";
+	if (test_parse_map_equal(ctx, str, str2) < 0)
+		return -1;
+
+	str = "{ [i,j] : i != j }";
+	str2 = "{ [i,j] : i < j or i > j }";
+	if (test_parse_map_equal(ctx, str, str2) < 0)
+		return -1;
+
+	str = "{ [i,j] : (i+1)*2 >= j }";
+	str2 = "{ [i, j] : j <= 2 + 2i }";
+	if (test_parse_map_equal(ctx, str, str2) < 0)
+		return -1;
+
+	str = "{ [i] -> [i > 0 ? 4 : 5] }";
+	str2 = "{ [i] -> [5] : i <= 0; [i] -> [4] : i >= 1 }";
+	if (test_parse_map_equal(ctx, str, str2) < 0)
+		return -1;
+
+	str = "[N=2,M] -> { [i=[(M+N)/4]] }";
+	str2 = "[N, M] -> { [i] : N = 2 and 4i <= 2 + M and 4i >= -1 + M }";
+	if (test_parse_map_equal(ctx, str, str2) < 0)
+		return -1;
+
+	str = "{ [x] : x >= 0 }";
+	str2 = "{ [x] : x-0 >= 0 }";
+	if (test_parse_map_equal(ctx, str, str2) < 0)
+		return -1;
+
+	str = "{ [i] : ((i > 10)) }";
+	str2 = "{ [i] : i >= 11 }";
+	if (test_parse_map_equal(ctx, str, str2) < 0)
+		return -1;
+
+	str = "{ [i] -> [0] }";
+	str2 = "{ [i] -> [0 * i] }";
+	if (test_parse_map_equal(ctx, str, str2) < 0)
+		return -1;
+
+	test_parse_pwqp(ctx, "{ [i] -> i + [ (i + [i/3])/2 ] }");
+	test_parse_map(ctx, "{ S1[i] -> [([i/10]),i%10] : 0 <= i <= 45 }");
+	test_parse_pwaff(ctx, "{ [i] -> [i + 1] : i > 0; [a] -> [a] : a < 0 }");
+	test_parse_pwqp(ctx, "{ [x] -> ([(x)/2] * [(x)/3]) }");
+
+	if (test_parse_map_equal(ctx, "{ [a] -> [b] : (not false) }",
+				      "{ [a] -> [b] : true }") < 0)
+		return -1;
+
+	if (test_parse_map_equal(ctx, "{ [i] : i/2 <= 5 }",
+				      "{ [i] : i <= 10 }") < 0)
+		return -1;
+
+	if (test_parse_map_equal(ctx, "{Sym=[n] [i] : i <= n }",
+				      "[n] -> { [i] : i <= n }") < 0)
+		return -1;
+
+	if (test_parse_map_equal(ctx, "{ [*] }", "{ [a] }") < 0)
+		return -1;
+
+	if (test_parse_map_equal(ctx, "{ [i] : 2*floor(i/2) = i }",
+				      "{ [i] : exists a : i = 2 a }") < 0)
+		return -1;
+
+	if (test_parse_map_equal(ctx, "{ [a] -> [b] : a = 5 implies b = 5 }",
+				      "{ [a] -> [b] : a != 5 or b = 5 }") < 0)
+		return -1;
+
+	if (test_parse_map_equal(ctx, "{ [a] -> [a - 1 : a > 0] }",
+				      "{ [a] -> [a - 1] : a > 0 }") < 0)
+		return -1;
+	if (test_parse_map_equal(ctx,
+	    "{ [a] -> [a - 1 : a > 0; a : a <= 0] }",
+	    "{ [a] -> [a - 1] : a > 0; [a] -> [a] : a <= 0 }") < 0)
+		return -1;
+	if (test_parse_map_equal(ctx,
+	    "{ [a] -> [(a) * 2 : a >= 0; 0 : a < 0] }",
+	    "{ [a] -> [2a] : a >= 0; [a] -> [0] : a < 0 }") < 0)
+		return -1;
+	if (test_parse_map_equal(ctx,
+	    "{ [a] -> [(a * 2) : a >= 0; 0 : a < 0] }",
+	    "{ [a] -> [2a] : a >= 0; [a] -> [0] : a < 0 }") < 0)
+		return -1;
+	if (test_parse_map_equal(ctx,
+	    "{ [a] -> [(a * 2 : a >= 0); 0 : a < 0] }",
+	    "{ [a] -> [2a] : a >= 0; [a] -> [0] : a < 0 }") < 0)
+		return -1;
+	if (test_parse_map_equal(ctx,
+	    "{ [a] -> [(a * 2 : a >= 0; 0 : a < 0)] }",
+	    "{ [a] -> [2a] : a >= 0; [a] -> [0] : a < 0 }") < 0)
+		return -1;
+
+	return 0;
+}
+
+void test_read(struct isl_ctx *ctx)
+{
+	char *filename;
+	FILE *input;
+	struct isl_basic_set *bset1, *bset2;
+	const char *str = "{[y]: Exists ( alpha : 2alpha = y)}";
+
+	filename = get_filename(ctx, "set", "omega");
+	assert(filename);
+	input = fopen(filename, "r");
+	assert(input);
+
+	bset1 = isl_basic_set_read_from_file(ctx, input);
+	bset2 = isl_basic_set_read_from_str(ctx, str);
+
+	assert(isl_basic_set_is_equal(bset1, bset2) == 1);
+
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	free(filename);
+
+	fclose(input);
+}
+
+void test_bounded(struct isl_ctx *ctx)
+{
+	isl_set *set;
+	int bounded;
+
+	set = isl_set_read_from_str(ctx, "[n] -> {[i] : 0 <= i <= n }");
+	bounded = isl_set_is_bounded(set);
+	assert(bounded);
+	isl_set_free(set);
+
+	set = isl_set_read_from_str(ctx, "{[n, i] : 0 <= i <= n }");
+	bounded = isl_set_is_bounded(set);
+	assert(!bounded);
+	isl_set_free(set);
+
+	set = isl_set_read_from_str(ctx, "[n] -> {[i] : i <= n }");
+	bounded = isl_set_is_bounded(set);
+	assert(!bounded);
+	isl_set_free(set);
+}
+
+/* Construct the basic set { [i] : 5 <= i <= N } */
+void test_construction(struct isl_ctx *ctx)
+{
+	isl_int v;
+	isl_space *dim;
+	isl_local_space *ls;
+	struct isl_basic_set *bset;
+	struct isl_constraint *c;
+
+	isl_int_init(v);
+
+	dim = isl_space_set_alloc(ctx, 1, 1);
+	bset = isl_basic_set_universe(isl_space_copy(dim));
+	ls = isl_local_space_from_space(dim);
+
+	c = isl_inequality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, -1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, 1);
+	isl_constraint_set_coefficient(c, isl_dim_param, 0, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	c = isl_inequality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, 1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, -5);
+	isl_constraint_set_constant(c, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	isl_local_space_free(ls);
+	isl_basic_set_free(bset);
+
+	isl_int_clear(v);
+}
+
+void test_dim(struct isl_ctx *ctx)
+{
+	const char *str;
+	isl_map *map1, *map2;
+
+	map1 = isl_map_read_from_str(ctx,
+	    "[n] -> { [i] -> [j] : exists (a = [i/10] : i - 10a <= n ) }");
+	map1 = isl_map_add_dims(map1, isl_dim_in, 1);
+	map2 = isl_map_read_from_str(ctx,
+	    "[n] -> { [i,k] -> [j] : exists (a = [i/10] : i - 10a <= n ) }");
+	assert(isl_map_is_equal(map1, map2));
+	isl_map_free(map2);
+
+	map1 = isl_map_project_out(map1, isl_dim_in, 0, 1);
+	map2 = isl_map_read_from_str(ctx, "[n] -> { [i] -> [j] : n >= 0 }");
+	assert(isl_map_is_equal(map1, map2));
+
+	isl_map_free(map1);
+	isl_map_free(map2);
+
+	str = "[n] -> { [i] -> [] : exists a : 0 <= i <= n and i = 2 a }";
+	map1 = isl_map_read_from_str(ctx, str);
+	str = "{ [i] -> [j] : exists a : 0 <= i <= j and i = 2 a }";
+	map2 = isl_map_read_from_str(ctx, str);
+	map1 = isl_map_move_dims(map1, isl_dim_out, 0, isl_dim_param, 0, 1);
+	assert(isl_map_is_equal(map1, map2));
+
+	isl_map_free(map1);
+	isl_map_free(map2);
+}
+
+struct {
+	__isl_give isl_val *(*op)(__isl_take isl_val *v);
+	const char *arg;
+	const char *res;
+} val_un_tests[] = {
+	{ &isl_val_neg, "0", "0" },
+	{ &isl_val_abs, "0", "0" },
+	{ &isl_val_2exp, "0", "1" },
+	{ &isl_val_floor, "0", "0" },
+	{ &isl_val_ceil, "0", "0" },
+	{ &isl_val_neg, "1", "-1" },
+	{ &isl_val_neg, "-1", "1" },
+	{ &isl_val_neg, "1/2", "-1/2" },
+	{ &isl_val_neg, "-1/2", "1/2" },
+	{ &isl_val_neg, "infty", "-infty" },
+	{ &isl_val_neg, "-infty", "infty" },
+	{ &isl_val_neg, "NaN", "NaN" },
+	{ &isl_val_abs, "1", "1" },
+	{ &isl_val_abs, "-1", "1" },
+	{ &isl_val_abs, "1/2", "1/2" },
+	{ &isl_val_abs, "-1/2", "1/2" },
+	{ &isl_val_abs, "infty", "infty" },
+	{ &isl_val_abs, "-infty", "infty" },
+	{ &isl_val_abs, "NaN", "NaN" },
+	{ &isl_val_floor, "1", "1" },
+	{ &isl_val_floor, "-1", "-1" },
+	{ &isl_val_floor, "1/2", "0" },
+	{ &isl_val_floor, "-1/2", "-1" },
+	{ &isl_val_floor, "infty", "infty" },
+	{ &isl_val_floor, "-infty", "-infty" },
+	{ &isl_val_floor, "NaN", "NaN" },
+	{ &isl_val_ceil, "1", "1" },
+	{ &isl_val_ceil, "-1", "-1" },
+	{ &isl_val_ceil, "1/2", "1" },
+	{ &isl_val_ceil, "-1/2", "0" },
+	{ &isl_val_ceil, "infty", "infty" },
+	{ &isl_val_ceil, "-infty", "-infty" },
+	{ &isl_val_ceil, "NaN", "NaN" },
+	{ &isl_val_2exp, "-3", "1/8" },
+	{ &isl_val_2exp, "-1", "1/2" },
+	{ &isl_val_2exp, "1", "2" },
+	{ &isl_val_2exp, "2", "4" },
+	{ &isl_val_2exp, "3", "8" },
+	{ &isl_val_inv, "1", "1" },
+	{ &isl_val_inv, "2", "1/2" },
+	{ &isl_val_inv, "1/2", "2" },
+	{ &isl_val_inv, "-2", "-1/2" },
+	{ &isl_val_inv, "-1/2", "-2" },
+	{ &isl_val_inv, "0", "NaN" },
+	{ &isl_val_inv, "NaN", "NaN" },
+	{ &isl_val_inv, "infty", "0" },
+	{ &isl_val_inv, "-infty", "0" },
+};
+
+/* Perform some basic tests of unary operations on isl_val objects.
+ */
+static int test_un_val(isl_ctx *ctx)
+{
+	int i;
+	isl_val *v, *res;
+	__isl_give isl_val *(*fn)(__isl_take isl_val *v);
+	int ok;
+
+	for (i = 0; i < ARRAY_SIZE(val_un_tests); ++i) {
+		v = isl_val_read_from_str(ctx, val_un_tests[i].arg);
+		res = isl_val_read_from_str(ctx, val_un_tests[i].res);
+		fn = val_un_tests[i].op;
+		v = fn(v);
+		if (isl_val_is_nan(res))
+			ok = isl_val_is_nan(v);
+		else
+			ok = isl_val_eq(v, res);
+		isl_val_free(v);
+		isl_val_free(res);
+		if (ok < 0)
+			return -1;
+		if (!ok)
+			isl_die(ctx, isl_error_unknown,
+				"unexpected result", return -1);
+	}
+
+	return 0;
+}
+
+struct {
+	__isl_give isl_val *(*fn)(__isl_take isl_val *v1,
+				__isl_take isl_val *v2);
+} val_bin_op[] = {
+	['+'] = { &isl_val_add },
+	['-'] = { &isl_val_sub },
+	['*'] = { &isl_val_mul },
+	['/'] = { &isl_val_div },
+	['g'] = { &isl_val_gcd },
+	['m'] = { &isl_val_min },
+	['M'] = { &isl_val_max },
+};
+
+struct {
+	const char *arg1;
+	unsigned char op;
+	const char *arg2;
+	const char *res;
+} val_bin_tests[] = {
+	{ "0", '+', "0", "0" },
+	{ "1", '+', "0", "1" },
+	{ "1", '+', "1", "2" },
+	{ "1", '-', "1", "0" },
+	{ "1", '*', "1", "1" },
+	{ "1", '/', "1", "1" },
+	{ "2", '*', "3", "6" },
+	{ "2", '*', "1/2", "1" },
+	{ "2", '*', "1/3", "2/3" },
+	{ "2/3", '*', "3/5", "2/5" },
+	{ "2/3", '*', "7/5", "14/15" },
+	{ "2", '/', "1/2", "4" },
+	{ "-2", '/', "-1/2", "4" },
+	{ "-2", '/', "1/2", "-4" },
+	{ "2", '/', "-1/2", "-4" },
+	{ "2", '/', "2", "1" },
+	{ "2", '/', "3", "2/3" },
+	{ "2/3", '/', "5/3", "2/5" },
+	{ "2/3", '/', "5/7", "14/15" },
+	{ "0", '/', "0", "NaN" },
+	{ "42", '/', "0", "NaN" },
+	{ "-42", '/', "0", "NaN" },
+	{ "infty", '/', "0", "NaN" },
+	{ "-infty", '/', "0", "NaN" },
+	{ "NaN", '/', "0", "NaN" },
+	{ "0", '/', "NaN", "NaN" },
+	{ "42", '/', "NaN", "NaN" },
+	{ "-42", '/', "NaN", "NaN" },
+	{ "infty", '/', "NaN", "NaN" },
+	{ "-infty", '/', "NaN", "NaN" },
+	{ "NaN", '/', "NaN", "NaN" },
+	{ "0", '/', "infty", "0" },
+	{ "42", '/', "infty", "0" },
+	{ "-42", '/', "infty", "0" },
+	{ "infty", '/', "infty", "NaN" },
+	{ "-infty", '/', "infty", "NaN" },
+	{ "NaN", '/', "infty", "NaN" },
+	{ "0", '/', "-infty", "0" },
+	{ "42", '/', "-infty", "0" },
+	{ "-42", '/', "-infty", "0" },
+	{ "infty", '/', "-infty", "NaN" },
+	{ "-infty", '/', "-infty", "NaN" },
+	{ "NaN", '/', "-infty", "NaN" },
+	{ "1", '-', "1/3", "2/3" },
+	{ "1/3", '+', "1/2", "5/6" },
+	{ "1/2", '+', "1/2", "1" },
+	{ "3/4", '-', "1/4", "1/2" },
+	{ "1/2", '-', "1/3", "1/6" },
+	{ "infty", '+', "42", "infty" },
+	{ "infty", '+', "infty", "infty" },
+	{ "42", '+', "infty", "infty" },
+	{ "infty", '-', "infty", "NaN" },
+	{ "infty", '*', "infty", "infty" },
+	{ "infty", '*', "-infty", "-infty" },
+	{ "-infty", '*', "infty", "-infty" },
+	{ "-infty", '*', "-infty", "infty" },
+	{ "0", '*', "infty", "NaN" },
+	{ "1", '*', "infty", "infty" },
+	{ "infty", '*', "0", "NaN" },
+	{ "infty", '*', "42", "infty" },
+	{ "42", '-', "infty", "-infty" },
+	{ "infty", '+', "-infty", "NaN" },
+	{ "4", 'g', "6", "2" },
+	{ "5", 'g', "6", "1" },
+	{ "42", 'm', "3", "3" },
+	{ "42", 'M', "3", "42" },
+	{ "3", 'm', "42", "3" },
+	{ "3", 'M', "42", "42" },
+	{ "42", 'm', "infty", "42" },
+	{ "42", 'M', "infty", "infty" },
+	{ "42", 'm', "-infty", "-infty" },
+	{ "42", 'M', "-infty", "42" },
+	{ "42", 'm', "NaN", "NaN" },
+	{ "42", 'M', "NaN", "NaN" },
+	{ "infty", 'm', "-infty", "-infty" },
+	{ "infty", 'M', "-infty", "infty" },
+};
+
+/* Perform some basic tests of binary operations on isl_val objects.
+ */
+static int test_bin_val(isl_ctx *ctx)
+{
+	int i;
+	isl_val *v1, *v2, *res;
+	__isl_give isl_val *(*fn)(__isl_take isl_val *v1,
+				__isl_take isl_val *v2);
+	int ok;
+
+	for (i = 0; i < ARRAY_SIZE(val_bin_tests); ++i) {
+		v1 = isl_val_read_from_str(ctx, val_bin_tests[i].arg1);
+		v2 = isl_val_read_from_str(ctx, val_bin_tests[i].arg2);
+		res = isl_val_read_from_str(ctx, val_bin_tests[i].res);
+		fn = val_bin_op[val_bin_tests[i].op].fn;
+		v1 = fn(v1, v2);
+		if (isl_val_is_nan(res))
+			ok = isl_val_is_nan(v1);
+		else
+			ok = isl_val_eq(v1, res);
+		isl_val_free(v1);
+		isl_val_free(res);
+		if (ok < 0)
+			return -1;
+		if (!ok)
+			isl_die(ctx, isl_error_unknown,
+				"unexpected result", return -1);
+	}
+
+	return 0;
+}
+
+/* Perform some basic tests on isl_val objects.
+ */
+static int test_val(isl_ctx *ctx)
+{
+	if (test_un_val(ctx) < 0)
+		return -1;
+	if (test_bin_val(ctx) < 0)
+		return -1;
+	return 0;
+}
+
+static int test_div(isl_ctx *ctx)
+{
+	unsigned n;
+	const char *str;
+	int empty;
+	isl_int v;
+	isl_space *dim;
+	isl_set *set;
+	isl_local_space *ls;
+	struct isl_basic_set *bset;
+	struct isl_constraint *c;
+
+	isl_int_init(v);
+
+	/* test 1 */
+	dim = isl_space_set_alloc(ctx, 0, 3);
+	bset = isl_basic_set_universe(isl_space_copy(dim));
+	ls = isl_local_space_from_space(dim);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, -1);
+	isl_constraint_set_constant(c, v);
+	isl_int_set_si(v, 1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, 3);
+	isl_constraint_set_coefficient(c, isl_dim_set, 1, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, 1);
+	isl_constraint_set_constant(c, v);
+	isl_int_set_si(v, -1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, 3);
+	isl_constraint_set_coefficient(c, isl_dim_set, 2, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	bset = isl_basic_set_project_out(bset, isl_dim_set, 1, 2);
+
+	assert(bset && bset->n_div == 1);
+	isl_local_space_free(ls);
+	isl_basic_set_free(bset);
+
+	/* test 2 */
+	dim = isl_space_set_alloc(ctx, 0, 3);
+	bset = isl_basic_set_universe(isl_space_copy(dim));
+	ls = isl_local_space_from_space(dim);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, 1);
+	isl_constraint_set_constant(c, v);
+	isl_int_set_si(v, -1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, 3);
+	isl_constraint_set_coefficient(c, isl_dim_set, 1, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, -1);
+	isl_constraint_set_constant(c, v);
+	isl_int_set_si(v, 1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, 3);
+	isl_constraint_set_coefficient(c, isl_dim_set, 2, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	bset = isl_basic_set_project_out(bset, isl_dim_set, 1, 2);
+
+	assert(bset && bset->n_div == 1);
+	isl_local_space_free(ls);
+	isl_basic_set_free(bset);
+
+	/* test 3 */
+	dim = isl_space_set_alloc(ctx, 0, 3);
+	bset = isl_basic_set_universe(isl_space_copy(dim));
+	ls = isl_local_space_from_space(dim);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, 1);
+	isl_constraint_set_constant(c, v);
+	isl_int_set_si(v, -1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, 3);
+	isl_constraint_set_coefficient(c, isl_dim_set, 1, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, -3);
+	isl_constraint_set_constant(c, v);
+	isl_int_set_si(v, 1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, 4);
+	isl_constraint_set_coefficient(c, isl_dim_set, 2, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	bset = isl_basic_set_project_out(bset, isl_dim_set, 1, 2);
+
+	assert(bset && bset->n_div == 1);
+	isl_local_space_free(ls);
+	isl_basic_set_free(bset);
+
+	/* test 4 */
+	dim = isl_space_set_alloc(ctx, 0, 3);
+	bset = isl_basic_set_universe(isl_space_copy(dim));
+	ls = isl_local_space_from_space(dim);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, 2);
+	isl_constraint_set_constant(c, v);
+	isl_int_set_si(v, -1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, 3);
+	isl_constraint_set_coefficient(c, isl_dim_set, 1, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, -1);
+	isl_constraint_set_constant(c, v);
+	isl_int_set_si(v, 1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, 6);
+	isl_constraint_set_coefficient(c, isl_dim_set, 2, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	bset = isl_basic_set_project_out(bset, isl_dim_set, 1, 2);
+
+	assert(isl_basic_set_is_empty(bset));
+	isl_local_space_free(ls);
+	isl_basic_set_free(bset);
+
+	/* test 5 */
+	dim = isl_space_set_alloc(ctx, 0, 3);
+	bset = isl_basic_set_universe(isl_space_copy(dim));
+	ls = isl_local_space_from_space(dim);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, -1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, 3);
+	isl_constraint_set_coefficient(c, isl_dim_set, 2, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, 1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, -3);
+	isl_constraint_set_coefficient(c, isl_dim_set, 1, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	bset = isl_basic_set_project_out(bset, isl_dim_set, 2, 1);
+
+	assert(bset && bset->n_div == 0);
+	isl_basic_set_free(bset);
+	isl_local_space_free(ls);
+
+	/* test 6 */
+	dim = isl_space_set_alloc(ctx, 0, 3);
+	bset = isl_basic_set_universe(isl_space_copy(dim));
+	ls = isl_local_space_from_space(dim);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, -1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, 6);
+	isl_constraint_set_coefficient(c, isl_dim_set, 2, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, 1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, -3);
+	isl_constraint_set_coefficient(c, isl_dim_set, 1, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	bset = isl_basic_set_project_out(bset, isl_dim_set, 2, 1);
+
+	assert(bset && bset->n_div == 1);
+	isl_basic_set_free(bset);
+	isl_local_space_free(ls);
+
+	/* test 7 */
+	/* This test is a bit tricky.  We set up an equality
+	 *		a + 3b + 3c = 6 e0
+	 * Normalization of divs creates _two_ divs
+	 *		a = 3 e0
+	 *		c - b - e0 = 2 e1
+	 * Afterwards e0 is removed again because it has coefficient -1
+	 * and we end up with the original equality and div again.
+	 * Perhaps we can avoid the introduction of this temporary div.
+	 */
+	dim = isl_space_set_alloc(ctx, 0, 4);
+	bset = isl_basic_set_universe(isl_space_copy(dim));
+	ls = isl_local_space_from_space(dim);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, -1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, -3);
+	isl_constraint_set_coefficient(c, isl_dim_set, 1, v);
+	isl_int_set_si(v, -3);
+	isl_constraint_set_coefficient(c, isl_dim_set, 2, v);
+	isl_int_set_si(v, 6);
+	isl_constraint_set_coefficient(c, isl_dim_set, 3, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	bset = isl_basic_set_project_out(bset, isl_dim_set, 3, 1);
+
+	/* Test disabled for now */
+	/*
+	assert(bset && bset->n_div == 1);
+	*/
+	isl_local_space_free(ls);
+	isl_basic_set_free(bset);
+
+	/* test 8 */
+	dim = isl_space_set_alloc(ctx, 0, 5);
+	bset = isl_basic_set_universe(isl_space_copy(dim));
+	ls = isl_local_space_from_space(dim);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, -1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, -3);
+	isl_constraint_set_coefficient(c, isl_dim_set, 1, v);
+	isl_int_set_si(v, -3);
+	isl_constraint_set_coefficient(c, isl_dim_set, 3, v);
+	isl_int_set_si(v, 6);
+	isl_constraint_set_coefficient(c, isl_dim_set, 4, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, -1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, 1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 2, v);
+	isl_int_set_si(v, 1);
+	isl_constraint_set_constant(c, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	bset = isl_basic_set_project_out(bset, isl_dim_set, 4, 1);
+
+	/* Test disabled for now */
+	/*
+	assert(bset && bset->n_div == 1);
+	*/
+	isl_local_space_free(ls);
+	isl_basic_set_free(bset);
+
+	/* test 9 */
+	dim = isl_space_set_alloc(ctx, 0, 4);
+	bset = isl_basic_set_universe(isl_space_copy(dim));
+	ls = isl_local_space_from_space(dim);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, 1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, -1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 1, v);
+	isl_int_set_si(v, -2);
+	isl_constraint_set_coefficient(c, isl_dim_set, 2, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, -1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, 3);
+	isl_constraint_set_coefficient(c, isl_dim_set, 3, v);
+	isl_int_set_si(v, 2);
+	isl_constraint_set_constant(c, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	bset = isl_basic_set_project_out(bset, isl_dim_set, 2, 2);
+
+	bset = isl_basic_set_fix_si(bset, isl_dim_set, 0, 2);
+
+	assert(!isl_basic_set_is_empty(bset));
+
+	isl_local_space_free(ls);
+	isl_basic_set_free(bset);
+
+	/* test 10 */
+	dim = isl_space_set_alloc(ctx, 0, 3);
+	bset = isl_basic_set_universe(isl_space_copy(dim));
+	ls = isl_local_space_from_space(dim);
+
+	c = isl_equality_alloc(isl_local_space_copy(ls));
+	isl_int_set_si(v, 1);
+	isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
+	isl_int_set_si(v, -2);
+	isl_constraint_set_coefficient(c, isl_dim_set, 2, v);
+	bset = isl_basic_set_add_constraint(bset, c);
+
+	bset = isl_basic_set_project_out(bset, isl_dim_set, 2, 1);
+
+	bset = isl_basic_set_fix_si(bset, isl_dim_set, 0, 2);
+
+	isl_local_space_free(ls);
+	isl_basic_set_free(bset);
+
+	isl_int_clear(v);
+
+	str = "{ [i] : exists (e0, e1: 3e1 >= 1 + 2e0 and "
+	    "8e1 <= -1 + 5i - 5e0 and 2e1 >= 1 + 2i - 5e0) }";
+	set = isl_set_read_from_str(ctx, str);
+	set = isl_set_compute_divs(set);
+	isl_set_free(set);
+	if (!set)
+		return -1;
+
+	str = "{ [i,j] : 2*[i/2] + 3 * [j/4] <= 10 and 2 i = j }";
+	bset = isl_basic_set_read_from_str(ctx, str);
+	n = isl_basic_set_dim(bset, isl_dim_div);
+	isl_basic_set_free(bset);
+	if (!bset)
+		return -1;
+	if (n != 0)
+		isl_die(ctx, isl_error_unknown,
+			"expecting no existentials", return -1);
+
+	str = "{ [i,j,k] : 3 + i + 2j >= 0 and 2 * [(i+2j)/4] <= k }";
+	set = isl_set_read_from_str(ctx, str);
+	set = isl_set_remove_divs_involving_dims(set, isl_dim_set, 0, 2);
+	set = isl_set_fix_si(set, isl_dim_set, 2, -3);
+	empty = isl_set_is_empty(set);
+	isl_set_free(set);
+	if (empty < 0)
+		return -1;
+	if (!empty)
+		isl_die(ctx, isl_error_unknown,
+			"result not as accurate as expected", return -1);
+
+	return 0;
+}
+
+void test_application_case(struct isl_ctx *ctx, const char *name)
+{
+	char *filename;
+	FILE *input;
+	struct isl_basic_set *bset1, *bset2;
+	struct isl_basic_map *bmap;
+
+	filename = get_filename(ctx, name, "omega");
+	assert(filename);
+	input = fopen(filename, "r");
+	assert(input);
+
+	bset1 = isl_basic_set_read_from_file(ctx, input);
+	bmap = isl_basic_map_read_from_file(ctx, input);
+
+	bset1 = isl_basic_set_apply(bset1, bmap);
+
+	bset2 = isl_basic_set_read_from_file(ctx, input);
+
+	assert(isl_basic_set_is_equal(bset1, bset2) == 1);
+
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	free(filename);
+
+	fclose(input);
+}
+
+void test_application(struct isl_ctx *ctx)
+{
+	test_application_case(ctx, "application");
+	test_application_case(ctx, "application2");
+}
+
+void test_affine_hull_case(struct isl_ctx *ctx, const char *name)
+{
+	char *filename;
+	FILE *input;
+	struct isl_basic_set *bset1, *bset2;
+
+	filename = get_filename(ctx, name, "polylib");
+	assert(filename);
+	input = fopen(filename, "r");
+	assert(input);
+
+	bset1 = isl_basic_set_read_from_file(ctx, input);
+	bset2 = isl_basic_set_read_from_file(ctx, input);
+
+	bset1 = isl_basic_set_affine_hull(bset1);
+
+	assert(isl_basic_set_is_equal(bset1, bset2) == 1);
+
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	free(filename);
+
+	fclose(input);
+}
+
+int test_affine_hull(struct isl_ctx *ctx)
+{
+	const char *str;
+	isl_set *set;
+	isl_basic_set *bset, *bset2;
+	int n;
+	int subset;
+
+	test_affine_hull_case(ctx, "affine2");
+	test_affine_hull_case(ctx, "affine");
+	test_affine_hull_case(ctx, "affine3");
+
+	str = "[m] -> { [i0] : exists (e0, e1: e1 <= 1 + i0 and "
+			"m >= 3 and 4i0 <= 2 + m and e1 >= i0 and "
+			"e1 >= 0 and e1 <= 2 and e1 >= 1 + 2e0 and "
+			"2e1 <= 1 + m + 4e0 and 2e1 >= 2 - m + 4i0 - 4e0) }";
+	set = isl_set_read_from_str(ctx, str);
+	bset = isl_set_affine_hull(set);
+	n = isl_basic_set_dim(bset, isl_dim_div);
+	isl_basic_set_free(bset);
+	if (n != 0)
+		isl_die(ctx, isl_error_unknown, "not expecting any divs",
+			return -1);
+
+	/* Check that isl_map_affine_hull is not confused by
+	 * the reordering of divs in isl_map_align_divs.
+	 */
+	str = "{ [a, b, c, 0] : exists (e0 = [(b)/32], e1 = [(c)/32]: "
+				"32e0 = b and 32e1 = c); "
+		"[a, 0, c, 0] : exists (e0 = [(c)/32]: 32e0 = c) }";
+	set = isl_set_read_from_str(ctx, str);
+	bset = isl_set_affine_hull(set);
+	isl_basic_set_free(bset);
+	if (!bset)
+		return -1;
+
+	str = "{ [a] : exists e0, e1, e2: 32e1 = 31 + 31a + 31e0 and "
+			"32e2 = 31 + 31e0 }";
+	set = isl_set_read_from_str(ctx, str);
+	bset = isl_set_affine_hull(set);
+	str = "{ [a] : exists e : a = 32 e }";
+	bset2 = isl_basic_set_read_from_str(ctx, str);
+	subset = isl_basic_set_is_subset(bset, bset2);
+	isl_basic_set_free(bset);
+	isl_basic_set_free(bset2);
+	if (subset < 0)
+		return -1;
+	if (!subset)
+		isl_die(ctx, isl_error_unknown, "not as accurate as expected",
+			return -1);
+
+	return 0;
+}
+
+void test_convex_hull_case(struct isl_ctx *ctx, const char *name)
+{
+	char *filename;
+	FILE *input;
+	struct isl_basic_set *bset1, *bset2;
+	struct isl_set *set;
+
+	filename = get_filename(ctx, name, "polylib");
+	assert(filename);
+	input = fopen(filename, "r");
+	assert(input);
+
+	bset1 = isl_basic_set_read_from_file(ctx, input);
+	bset2 = isl_basic_set_read_from_file(ctx, input);
+
+	set = isl_basic_set_union(bset1, bset2);
+	bset1 = isl_set_convex_hull(set);
+
+	bset2 = isl_basic_set_read_from_file(ctx, input);
+
+	assert(isl_basic_set_is_equal(bset1, bset2) == 1);
+
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	free(filename);
+
+	fclose(input);
+}
+
+struct {
+	const char *set;
+	const char *hull;
+} convex_hull_tests[] = {
+	{ "{ [i0, i1, i2] : (i2 = 1 and i0 = 0 and i1 >= 0) or "
+	       "(i0 = 1 and i1 = 0 and i2 = 1) or "
+	       "(i0 = 0 and i1 = 0 and i2 = 0) }",
+	  "{ [i0, i1, i2] : i0 >= 0 and i2 >= i0 and i2 <= 1 and i1 >= 0 }" },
+	{ "[n] -> { [i0, i1, i0] : i0 <= -4 + n; "
+	    "[i0, i0, i2] : n = 6 and i0 >= 0 and i2 <= 7 - i0 and "
+	    "i2 <= 5 and i2 >= 4; "
+	    "[3, i1, 3] : n = 5 and i1 <= 2 and i1 >= 0 }",
+	  "[n] -> { [i0, i1, i2] : i2 <= -1 + n and 2i2 <= -6 + 3n - i0 and "
+	    "i2 <= 5 + i0 and i2 >= i0 }" },
+	{ "{ [x, y] : 3y <= 2x and y >= -2 + 2x and 2y >= 2 - x }",
+	    "{ [x, y] : 1 = 0 }" },
+};
+
+void test_convex_hull_algo(struct isl_ctx *ctx, int convex)
+{
+	int i;
+	int orig_convex = ctx->opt->convex;
+	ctx->opt->convex = convex;
+
+	test_convex_hull_case(ctx, "convex0");
+	test_convex_hull_case(ctx, "convex1");
+	test_convex_hull_case(ctx, "convex2");
+	test_convex_hull_case(ctx, "convex3");
+	test_convex_hull_case(ctx, "convex4");
+	test_convex_hull_case(ctx, "convex5");
+	test_convex_hull_case(ctx, "convex6");
+	test_convex_hull_case(ctx, "convex7");
+	test_convex_hull_case(ctx, "convex8");
+	test_convex_hull_case(ctx, "convex9");
+	test_convex_hull_case(ctx, "convex10");
+	test_convex_hull_case(ctx, "convex11");
+	test_convex_hull_case(ctx, "convex12");
+	test_convex_hull_case(ctx, "convex13");
+	test_convex_hull_case(ctx, "convex14");
+	test_convex_hull_case(ctx, "convex15");
+
+	for (i = 0; i < ARRAY_SIZE(convex_hull_tests); ++i) {
+		isl_set *set1, *set2;
+
+		set1 = isl_set_read_from_str(ctx, convex_hull_tests[i].set);
+		set2 = isl_set_read_from_str(ctx, convex_hull_tests[i].hull);
+		set1 = isl_set_from_basic_set(isl_set_convex_hull(set1));
+		assert(isl_set_is_equal(set1, set2));
+		isl_set_free(set1);
+		isl_set_free(set2);
+	}
+
+	ctx->opt->convex = orig_convex;
+}
+
+void test_convex_hull(struct isl_ctx *ctx)
+{
+	test_convex_hull_algo(ctx, ISL_CONVEX_HULL_FM);
+	test_convex_hull_algo(ctx, ISL_CONVEX_HULL_WRAP);
+}
+
+void test_gist_case(struct isl_ctx *ctx, const char *name)
+{
+	char *filename;
+	FILE *input;
+	struct isl_basic_set *bset1, *bset2;
+
+	filename = get_filename(ctx, name, "polylib");
+	assert(filename);
+	input = fopen(filename, "r");
+	assert(input);
+
+	bset1 = isl_basic_set_read_from_file(ctx, input);
+	bset2 = isl_basic_set_read_from_file(ctx, input);
+
+	bset1 = isl_basic_set_gist(bset1, bset2);
+
+	bset2 = isl_basic_set_read_from_file(ctx, input);
+
+	assert(isl_basic_set_is_equal(bset1, bset2) == 1);
+
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	free(filename);
+
+	fclose(input);
+}
+
+struct {
+	const char *set;
+	const char *context;
+	const char *gist;
+} gist_tests[] = {
+	{ "{ [a, b, c] : a <= 15 and a >= 1 }",
+	  "{ [a, b, c] : exists (e0 = floor((-1 + a)/16): a >= 1 and "
+			"c <= 30 and 32e0 >= -62 + 2a + 2b - c and b >= 0) }",
+	  "{ [a, b, c] : a <= 15 }" },
+	{ "{ : }", "{ : 1 = 0 }", "{ : }" },
+	{ "{ : 1 = 0 }", "{ : 1 = 0 }", "{ : }" },
+	{ "[M] -> { [x] : exists (e0 = floor((-2 + x)/3): 3e0 = -2 + x) }",
+	  "[M] -> { [3M] }" , "[M] -> { [x] : 1 = 0 }" },
+};
+
+static int test_gist(struct isl_ctx *ctx)
+{
+	int i;
+	const char *str;
+	isl_basic_set *bset1, *bset2;
+	isl_map *map1, *map2;
+	int equal;
+
+	for (i = 0; i < ARRAY_SIZE(gist_tests); ++i) {
+		int equal_input;
+		isl_basic_set *copy;
+
+		bset1 = isl_basic_set_read_from_str(ctx, gist_tests[i].set);
+		bset2 = isl_basic_set_read_from_str(ctx, gist_tests[i].context);
+		copy = isl_basic_set_copy(bset1);
+		bset1 = isl_basic_set_gist(bset1, bset2);
+		bset2 = isl_basic_set_read_from_str(ctx, gist_tests[i].gist);
+		equal = isl_basic_set_is_equal(bset1, bset2);
+		isl_basic_set_free(bset1);
+		isl_basic_set_free(bset2);
+		bset1 = isl_basic_set_read_from_str(ctx, gist_tests[i].set);
+		equal_input = isl_basic_set_is_equal(bset1, copy);
+		isl_basic_set_free(bset1);
+		isl_basic_set_free(copy);
+		if (equal < 0 || equal_input < 0)
+			return -1;
+		if (!equal)
+			isl_die(ctx, isl_error_unknown,
+				"incorrect gist result", return -1);
+		if (!equal_input)
+			isl_die(ctx, isl_error_unknown,
+				"gist modified input", return -1);
+	}
+
+	test_gist_case(ctx, "gist1");
+
+	str = "[p0, p2, p3, p5, p6, p10] -> { [] : "
+	    "exists (e0 = [(15 + p0 + 15p6 + 15p10)/16], e1 = [(p5)/8], "
+	    "e2 = [(p6)/128], e3 = [(8p2 - p5)/128], "
+	    "e4 = [(128p3 - p6)/4096]: 8e1 = p5 and 128e2 = p6 and "
+	    "128e3 = 8p2 - p5 and 4096e4 = 128p3 - p6 and p2 >= 0 and "
+	    "16e0 >= 16 + 16p6 + 15p10 and  p2 <= 15 and p3 >= 0 and "
+	    "p3 <= 31 and  p6 >= 128p3 and p5 >= 8p2 and p10 >= 0 and "
+	    "16e0 <= 15 + p0 + 15p6 + 15p10 and 16e0 >= p0 + 15p6 + 15p10 and "
+	    "p10 <= 15 and p10 <= -1 + p0 - p6) }";
+	bset1 = isl_basic_set_read_from_str(ctx, str);
+	str = "[p0, p2, p3, p5, p6, p10] -> { [] : exists (e0 = [(p5)/8], "
+	    "e1 = [(p6)/128], e2 = [(8p2 - p5)/128], "
+	    "e3 = [(128p3 - p6)/4096]: 8e0 = p5 and 128e1 = p6 and "
+	    "128e2 = 8p2 - p5 and 4096e3 = 128p3 - p6 and p5 >= -7 and "
+	    "p2 >= 0 and 8p2 <= -1 + p0 and p2 <= 15 and p3 >= 0 and "
+	    "p3 <= 31 and 128p3 <= -1 + p0 and p6 >= -127 and "
+	    "p5 <= -1 + p0 and p6 <= -1 + p0 and p6 >= 128p3 and "
+	    "p0 >= 1 and p5 >= 8p2 and p10 >= 0 and p10 <= 15 ) }";
+	bset2 = isl_basic_set_read_from_str(ctx, str);
+	bset1 = isl_basic_set_gist(bset1, bset2);
+	assert(bset1 && bset1->n_div == 0);
+	isl_basic_set_free(bset1);
+
+	/* Check that the integer divisions of the second disjunct
+	 * do not spread to the first disjunct.
+	 */
+	str = "[t1] -> { S_0[] -> A[o0] : (exists (e0 = [(-t1 + o0)/16]: "
+		"16e0 = -t1 + o0 and o0 >= 0 and o0 <= 15 and t1 >= 0)) or "
+		"(exists (e0 = [(-1 + t1)/16], "
+			"e1 = [(-16 + t1 - 16e0)/4294967296]: "
+			"4294967296e1 = -16 + t1 - o0 - 16e0 and "
+			"16e0 <= -1 + t1 and 16e0 >= -16 + t1 and o0 >= 0 and "
+			"o0 <= 4294967295 and t1 <= -1)) }";
+	map1 = isl_map_read_from_str(ctx, str);
+	str = "[t1] -> { S_0[] -> A[o0] : t1 >= 0 and t1 <= 4294967295 }";
+	map2 = isl_map_read_from_str(ctx, str);
+	map1 = isl_map_gist(map1, map2);
+	if (!map1)
+		return -1;
+	if (map1->n != 1)
+		isl_die(ctx, isl_error_unknown, "expecting single disjunct",
+			isl_map_free(map1); return -1);
+	if (isl_basic_map_dim(map1->p[0], isl_dim_div) != 1)
+		isl_die(ctx, isl_error_unknown, "expecting single div",
+			isl_map_free(map1); return -1);
+	isl_map_free(map1);
+
+	return 0;
+}
+
+int test_coalesce_set(isl_ctx *ctx, const char *str, int check_one)
+{
+	isl_set *set, *set2;
+	int equal;
+	int one;
+
+	set = isl_set_read_from_str(ctx, str);
+	set = isl_set_coalesce(set);
+	set2 = isl_set_read_from_str(ctx, str);
+	equal = isl_set_is_equal(set, set2);
+	one = set && set->n == 1;
+	isl_set_free(set);
+	isl_set_free(set2);
+
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown,
+			"coalesced set not equal to input", return -1);
+	if (check_one && !one)
+		isl_die(ctx, isl_error_unknown,
+			"coalesced set should not be a union", return -1);
+
+	return 0;
+}
+
+int test_coalesce_unbounded_wrapping(isl_ctx *ctx)
+{
+	int r = 0;
+	int bounded;
+
+	bounded = isl_options_get_coalesce_bounded_wrapping(ctx);
+	isl_options_set_coalesce_bounded_wrapping(ctx, 0);
+
+	if (test_coalesce_set(ctx,
+		"{[x,y,z] : y + 2 >= 0 and x - y + 1 >= 0 and "
+			"-x - y + 1 >= 0 and -3 <= z <= 3;"
+		"[x,y,z] : -x+z + 20 >= 0 and -x-z + 20 >= 0 and "
+			"x-z + 20 >= 0 and x+z + 20 >= 0 and "
+			"-10 <= y <= 0}", 1) < 0)
+		goto error;
+	if (test_coalesce_set(ctx,
+		"{[x,y] : 0 <= x,y <= 10; [5,y]: 4 <=y <= 11}", 1) < 0)
+		goto error;
+	if (test_coalesce_set(ctx,
+		"{[x,0,0] : -5 <= x <= 5; [0,y,1] : -5 <= y <= 5 }", 1) < 0)
+		goto error;
+
+	if (0) {
+error:
+		r = -1;
+	}
+
+	isl_options_set_coalesce_bounded_wrapping(ctx, bounded);
+
+	return r;
+}
+
+/* Inputs for coalescing tests.
+ * "str" is a string representation of the input set.
+ * "single_disjunct" is set if we expect the result to consist of
+ *	a single disjunct.
+ */
+struct {
+	int single_disjunct;
+	const char *str;
+} coalesce_tests[] = {
+	{ 1, "{[x,y]: x >= 0 & x <= 10 & y >= 0 & y <= 10 or "
+		       "y >= x & x >= 2 & 5 >= y }" },
+	{ 1, "{[x,y]: y >= 0 & 2x + y <= 30 & y <= 10 & x >= 0 or "
+		       "x + y >= 10 & y <= x & x + y <= 20 & y >= 0}" },
+	{ 0, "{[x,y]: y >= 0 & 2x + y <= 30 & y <= 10 & x >= 0 or "
+		       "x + y >= 10 & y <= x & x + y <= 19 & y >= 0}" },
+	{ 1, "{[x,y]: y >= 0 & x <= 5 & y <= x or "
+		       "y >= 0 & x >= 6 & x <= 10 & y <= x}" },
+	{ 0, "{[x,y]: y >= 0 & x <= 5 & y <= x or "
+		       "y >= 0 & x >= 7 & x <= 10 & y <= x}" },
+	{ 0, "{[x,y]: y >= 0 & x <= 5 & y <= x or "
+		       "y >= 0 & x >= 6 & x <= 10 & y + 1 <= x}" },
+	{ 1, "{[x,y]: y >= 0 & x <= 5 & y <= x or y >= 0 & x = 6 & y <= 6}" },
+	{ 0, "{[x,y]: y >= 0 & x <= 5 & y <= x or y >= 0 & x = 7 & y <= 6}" },
+	{ 1, "{[x,y]: y >= 0 & x <= 5 & y <= x or y >= 0 & x = 6 & y <= 5}" },
+	{ 0, "{[x,y]: y >= 0 & x <= 5 & y <= x or y >= 0 & x = 6 & y <= 7}" },
+	{ 1, "[n] -> { [i] : i = 1 and n >= 2 or 2 <= i and i <= n }" },
+	{ 0, "{[x,y] : x >= 0 and y >= 0 or 0 <= y and y <= 5 and x = -1}" },
+	{ 1, "[n] -> { [i] : 1 <= i and i <= n - 1 or 2 <= i and i <= n }" },
+	{ 0, "[n] -> { [[i0] -> [o0]] : exists (e0 = [(i0)/4], e1 = [(o0)/4], "
+		"e2 = [(n)/2], e3 = [(-2 + i0)/4], e4 = [(-2 + o0)/4], "
+		"e5 = [(-2n + i0)/4]: 2e2 = n and 4e3 = -2 + i0 and "
+		"4e4 = -2 + o0 and i0 >= 8 + 2n and o0 >= 2 + i0 and "
+		"o0 <= 56 + 2n and o0 <= -12 + 4n and i0 <= 57 + 2n and "
+		"i0 <= -11 + 4n and o0 >= 6 + 2n and 4e0 <= i0 and "
+		"4e0 >= -3 + i0 and 4e1 <= o0 and 4e1 >= -3 + o0 and "
+		"4e5 <= -2n + i0 and 4e5 >= -3 - 2n + i0);"
+		"[[i0] -> [o0]] : exists (e0 = [(i0)/4], e1 = [(o0)/4], "
+		"e2 = [(n)/2], e3 = [(-2 + i0)/4], e4 = [(-2 + o0)/4], "
+		"e5 = [(-2n + i0)/4]: 2e2 = n and 4e3 = -2 + i0 and "
+		"4e4 = -2 + o0 and 2e0 >= 3 + n and e0 <= -4 + n and "
+		"2e0 <= 27 + n and e1 <= -4 + n and 2e1 <= 27 + n and "
+		"2e1 >= 2 + n and e1 >= 1 + e0 and i0 >= 7 + 2n and "
+		"i0 <= -11 + 4n and i0 <= 57 + 2n and 4e0 <= -2 + i0 and "
+		"4e0 >= -3 + i0 and o0 >= 6 + 2n and o0 <= -11 + 4n and "
+		"o0 <= 57 + 2n and 4e1 <= -2 + o0 and 4e1 >= -3 + o0 and "
+		"4e5 <= -2n + i0 and 4e5 >= -3 - 2n + i0 ) }" },
+	{ 0, "[n, m] -> { [o0, o2, o3] : (o3 = 1 and o0 >= 1 + m and "
+	      "o0 <= n + m and o2 <= m and o0 >= 2 + n and o2 >= 3) or "
+	      "(o0 >= 2 + n and o0 >= 1 + m and o0 <= n + m and n >= 1 and "
+	      "o3 <= -1 + o2 and o3 >= 1 - m + o2 and o3 >= 2 and o3 <= n) }" },
+	{ 0, "[M, N] -> { [[i0, i1, i2, i3, i4, i5, i6] -> "
+	  "[o0, o1, o2, o3, o4, o5, o6]] : "
+	  "(o6 <= -4 + 2M - 2N + i0 + i1 - i2 + i6 - o0 - o1 + o2 and "
+	  "o3 <= -2 + i3 and o6 >= 2 + i0 + i3 + i6 - o0 - o3 and "
+	  "o6 >= 2 - M + N + i3 + i4 + i6 - o3 - o4 and o0 <= -1 + i0 and "
+	  "o4 >= 4 - 3M + 3N - i0 - i1 + i2 + 2i3 + i4 + o0 + o1 - o2 - 2o3 "
+	  "and o6 <= -3 + 2M - 2N + i3 + i4 - i5 + i6 - o3 - o4 + o5 and "
+	  "2o6 <= -5 + 5M - 5N + 2i0 + i1 - i2 - i5 + 2i6 - 2o0 - o1 + o2 + o5 "
+	  "and o6 >= 2i0 + i1 + i6 - 2o0 - o1 and "
+	  "3o6 <= -5 + 4M - 4N + 2i0 + i1 - i2 + 2i3 + i4 - i5 + 3i6 "
+	  "- 2o0 - o1 + o2 - 2o3 - o4 + o5) or "
+	  "(N >= 2 and o3 <= -1 + i3 and o0 <= -1 + i0 and "
+	  "o6 >= i3 + i6 - o3 and M >= 0 and "
+	  "2o6 >= 1 + i0 + i3 + 2i6 - o0 - o3 and "
+	  "o6 >= 1 - M + i0 + i6 - o0 and N >= 2M and o6 >= i0 + i6 - o0) }" },
+	{ 0, "[M, N] -> { [o0] : (o0 = 0 and M >= 1 and N >= 2) or "
+		"(o0 = 0 and M >= 1 and N >= 2M and N >= 2 + M) or "
+		"(o0 = 0 and M >= 2 and N >= 3) or "
+		"(M = 0 and o0 = 0 and N >= 3) }" },
+	{ 0, "{ [i0, i1, i2, i3] : (i1 = 10i0 and i0 >= 1 and 10i0 <= 100 and "
+	    "i3 <= 9 + 10 i2 and i3 >= 1 + 10i2 and i3 >= 0) or "
+	    "(i1 <= 9 + 10i0 and i1 >= 1 + 10i0 and i2 >= 0 and "
+	    "i0 >= 0 and i1 <= 100 and i3 <= 9 + 10i2 and i3 >= 1 + 10i2) }" },
+	{ 0, "[M] -> { [i1] : (i1 >= 2 and i1 <= M) or (i1 = M and M >= 1) }" },
+	{ 0, "{[x,y] : x,y >= 0; [x,y] : 10 <= x <= 20 and y >= -1 }" },
+	{ 1, "{ [x, y] : (x >= 1 and y >= 1 and x <= 2 and y <= 2) or "
+		"(y = 3 and x = 1) }" },
+	{ 1, "[M] -> { [i0, i1, i2, i3, i4] : (i1 >= 3 and i4 >= 2 + i2 and "
+		"i2 >= 2 and i0 >= 2 and i3 >= 1 + i2 and i0 <= M and "
+		"i1 <= M and i3 <= M and i4 <= M) or "
+		"(i1 >= 2 and i4 >= 1 + i2 and i2 >= 2 and i0 >= 2 and "
+		"i3 >= 1 + i2 and i0 <= M and i1 <= -1 + M and i3 <= M and "
+		"i4 <= -1 + M) }" },
+	{ 1, "{ [x, y] : (x >= 0 and y >= 0 and x <= 10 and y <= 10) or "
+		"(x >= 1 and y >= 1 and x <= 11 and y <= 11) }" },
+	{ 0, "{[x,0] : x >= 0; [x,1] : x <= 20}" },
+	{ 1, "{ [x, 1 - x] : 0 <= x <= 1; [0,0] }" },
+	{ 1, "{ [0,0]; [i,i] : 1 <= i <= 10 }" },
+	{ 0, "{ [0,0]; [i,j] : 1 <= i,j <= 10 }" },
+	{ 1, "{ [0,0]; [i,2i] : 1 <= i <= 10 }" },
+	{ 0, "{ [0,0]; [i,2i] : 2 <= i <= 10 }" },
+	{ 0, "{ [1,0]; [i,2i] : 1 <= i <= 10 }" },
+	{ 0, "{ [0,1]; [i,2i] : 1 <= i <= 10 }" },
+	{ 0, "{ [a, b] : exists e : 2e = a and "
+		    "a >= 0 and (a <= 3 or (b <= 0 and b >= -4 + a)) }" },
+	{ 0, "{ [i, j, i', j'] : i <= 2 and j <= 2 and "
+			"j' >= -1 + 2i + j - 2i' and i' <= -1 + i and "
+			"j >= 1 and j' <= i + j - i' and i >= 1; "
+		"[1, 1, 1, 1] }" },
+	{ 1, "{ [i,j] : exists a,b : i = 2a and j = 3b; "
+		 "[i,j] : exists a : j = 3a }" },
+	{ 1, "{ [a, b, c] : (c <= 7 - b and b <= 1 and b >= 0 and "
+			"c >= 3 + b and b <= 3 + 8a and b >= -26 + 8a and "
+			"a >= 3) or "
+		    "(b <= 1 and c <= 7 and b >= 0 and c >= 4 + b and "
+			"b <= 3 + 8a and b >= -26 + 8a and a >= 3) }" },
+	{ 1, "{ [a, 0, c] : c >= 1 and c <= 29 and c >= -1 + 8a and "
+				"c <= 6 + 8a and a >= 3; "
+		"[a, -1, c] : c >= 1 and c <= 30 and c >= 8a and "
+				"c <= 7 + 8a and a >= 3 and a <= 4 }" },
+	{ 1, "{ [x,y] : 0 <= x <= 2 and y >= 0 and x + 2y <= 4; "
+		"[x,0] : 3 <= x <= 4 }" },
+	{ 1, "{ [x,y] : 0 <= x <= 3 and y >= 0 and x + 3y <= 6; "
+		"[x,0] : 4 <= x <= 5 }" },
+	{ 0, "{ [x,y] : 0 <= x <= 2 and y >= 0 and x + 2y <= 4; "
+		"[x,0] : 3 <= x <= 5 }" },
+	{ 0, "{ [x,y] : 0 <= x <= 2 and y >= 0 and x + y <= 4; "
+		"[x,0] : 3 <= x <= 4 }" },
+	{ 1 , "{ [i0, i1] : i0 <= 122 and i0 >= 1 and 128i1 >= -249 + i0 and "
+			"i1 <= 0; "
+		"[i0, 0] : i0 >= 123 and i0 <= 124 }" },
+};
+
+/* Test the functionality of isl_set_coalesce.
+ * That is, check that the output is always equal to the input
+ * and in some cases that the result consists of a single disjunct.
+ */
+static int test_coalesce(struct isl_ctx *ctx)
+{
+	int i;
+
+	for (i = 0; i < ARRAY_SIZE(coalesce_tests); ++i) {
+		const char *str = coalesce_tests[i].str;
+		int check_one = coalesce_tests[i].single_disjunct;
+		if (test_coalesce_set(ctx, str, check_one) < 0)
+			return -1;
+	}
+
+	if (test_coalesce_unbounded_wrapping(ctx) < 0)
+		return -1;
+
+	return 0;
+}
+
+void test_closure(struct isl_ctx *ctx)
+{
+	const char *str;
+	isl_set *dom;
+	isl_map *up, *right;
+	isl_map *map, *map2;
+	int exact;
+
+	/* COCOA example 1 */
+	map = isl_map_read_from_str(ctx,
+		"[n] -> { [i,j] -> [i2,j2] : i2 = i + 1 and j2 = j + 1 and "
+			"1 <= i and i < n and 1 <= j and j < n or "
+			"i2 = i + 1 and j2 = j - 1 and "
+			"1 <= i and i < n and 2 <= j and j <= n }");
+	map = isl_map_power(map, &exact);
+	assert(exact);
+	isl_map_free(map);
+
+	/* COCOA example 1 */
+	map = isl_map_read_from_str(ctx,
+		"[n] -> { [i,j] -> [i2,j2] : i2 = i + 1 and j2 = j + 1 and "
+			"1 <= i and i < n and 1 <= j and j < n or "
+			"i2 = i + 1 and j2 = j - 1 and "
+			"1 <= i and i < n and 2 <= j and j <= n }");
+	map = isl_map_transitive_closure(map, &exact);
+	assert(exact);
+	map2 = isl_map_read_from_str(ctx,
+		"[n] -> { [i,j] -> [i2,j2] : exists (k1,k2,k : "
+			"1 <= i and i < n and 1 <= j and j <= n and "
+			"2 <= i2 and i2 <= n and 1 <= j2 and j2 <= n and "
+			"i2 = i + k1 + k2 and j2 = j + k1 - k2 and "
+			"k1 >= 0 and k2 >= 0 and k1 + k2 = k and k >= 1 )}");
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map2);
+	isl_map_free(map);
+
+	map = isl_map_read_from_str(ctx,
+		"[n] -> { [x] -> [y] : y = x + 1 and 0 <= x and x <= n and "
+				     " 0 <= y and y <= n }");
+	map = isl_map_transitive_closure(map, &exact);
+	map2 = isl_map_read_from_str(ctx,
+		"[n] -> { [x] -> [y] : y > x and 0 <= x and x <= n and "
+				     " 0 <= y and y <= n }");
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map2);
+	isl_map_free(map);
+
+	/* COCOA example 2 */
+	map = isl_map_read_from_str(ctx,
+		"[n] -> { [i,j] -> [i2,j2] : i2 = i + 2 and j2 = j + 2 and "
+			"1 <= i and i < n - 1 and 1 <= j and j < n - 1 or "
+			"i2 = i + 2 and j2 = j - 2 and "
+			"1 <= i and i < n - 1 and 3 <= j and j <= n }");
+	map = isl_map_transitive_closure(map, &exact);
+	assert(exact);
+	map2 = isl_map_read_from_str(ctx,
+		"[n] -> { [i,j] -> [i2,j2] : exists (k1,k2,k : "
+			"1 <= i and i < n - 1 and 1 <= j and j <= n and "
+			"3 <= i2 and i2 <= n and 1 <= j2 and j2 <= n and "
+			"i2 = i + 2 k1 + 2 k2 and j2 = j + 2 k1 - 2 k2 and "
+			"k1 >= 0 and k2 >= 0 and k1 + k2 = k and k >= 1) }");
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map);
+	isl_map_free(map2);
+
+	/* COCOA Fig.2 left */
+	map = isl_map_read_from_str(ctx,
+		"[n] -> { [i,j] -> [i2,j2] : i2 = i + 2 and j2 = j and "
+			"i <= 2 j - 3 and i <= n - 2 and j <= 2 i - 1 and "
+			"j <= n or "
+			"i2 = i and j2 = j + 2 and i <= 2 j - 1 and i <= n and "
+			"j <= 2 i - 3 and j <= n - 2 or "
+			"i2 = i + 1 and j2 = j + 1 and i <= 2 j - 1 and "
+			"i <= n - 1 and j <= 2 i - 1 and j <= n - 1 }");
+	map = isl_map_transitive_closure(map, &exact);
+	assert(exact);
+	isl_map_free(map);
+
+	/* COCOA Fig.2 right */
+	map = isl_map_read_from_str(ctx,
+		"[n] -> { [i,j] -> [i2,j2] : i2 = i + 3 and j2 = j and "
+			"i <= 2 j - 4 and i <= n - 3 and j <= 2 i - 1 and "
+			"j <= n or "
+			"i2 = i and j2 = j + 3 and i <= 2 j - 1 and i <= n and "
+			"j <= 2 i - 4 and j <= n - 3 or "
+			"i2 = i + 1 and j2 = j + 1 and i <= 2 j - 1 and "
+			"i <= n - 1 and j <= 2 i - 1 and j <= n - 1 }");
+	map = isl_map_power(map, &exact);
+	assert(exact);
+	isl_map_free(map);
+
+	/* COCOA Fig.2 right */
+	map = isl_map_read_from_str(ctx,
+		"[n] -> { [i,j] -> [i2,j2] : i2 = i + 3 and j2 = j and "
+			"i <= 2 j - 4 and i <= n - 3 and j <= 2 i - 1 and "
+			"j <= n or "
+			"i2 = i and j2 = j + 3 and i <= 2 j - 1 and i <= n and "
+			"j <= 2 i - 4 and j <= n - 3 or "
+			"i2 = i + 1 and j2 = j + 1 and i <= 2 j - 1 and "
+			"i <= n - 1 and j <= 2 i - 1 and j <= n - 1 }");
+	map = isl_map_transitive_closure(map, &exact);
+	assert(exact);
+	map2 = isl_map_read_from_str(ctx,
+		"[n] -> { [i,j] -> [i2,j2] : exists (k1,k2,k3,k : "
+			"i <= 2 j - 1 and i <= n and j <= 2 i - 1 and "
+			"j <= n and 3 + i + 2 j <= 3 n and "
+			"3 + 2 i + j <= 3n and i2 <= 2 j2 -1 and i2 <= n and "
+			"i2 <= 3 j2 - 4 and j2 <= 2 i2 -1 and j2 <= n and "
+			"13 + 4 j2 <= 11 i2 and i2 = i + 3 k1 + k3 and "
+			"j2 = j + 3 k2 + k3 and k1 >= 0 and k2 >= 0 and "
+			"k3 >= 0 and k1 + k2 + k3 = k and k > 0) }");
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map2);
+	isl_map_free(map);
+
+	/* COCOA Fig.1 right */
+	dom = isl_set_read_from_str(ctx,
+		"{ [x,y] : x >= 0 and -2 x + 3 y >= 0 and x <= 3 and "
+			"2 x - 3 y + 3 >= 0 }");
+	right = isl_map_read_from_str(ctx,
+		"{ [x,y] -> [x2,y2] : x2 = x + 1 and y2 = y }");
+	up = isl_map_read_from_str(ctx,
+		"{ [x,y] -> [x2,y2] : x2 = x and y2 = y + 1 }");
+	right = isl_map_intersect_domain(right, isl_set_copy(dom));
+	right = isl_map_intersect_range(right, isl_set_copy(dom));
+	up = isl_map_intersect_domain(up, isl_set_copy(dom));
+	up = isl_map_intersect_range(up, dom);
+	map = isl_map_union(up, right);
+	map = isl_map_transitive_closure(map, &exact);
+	assert(exact);
+	map2 = isl_map_read_from_str(ctx,
+		"{ [0,0] -> [0,1]; [0,0] -> [1,1]; [0,1] -> [1,1]; "
+		"  [2,2] -> [3,2]; [2,2] -> [3,3]; [3,2] -> [3,3] }");
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map2);
+	isl_map_free(map);
+
+	/* COCOA Theorem 1 counter example */
+	map = isl_map_read_from_str(ctx,
+		"{ [i,j] -> [i2,j2] : i = 0 and 0 <= j and j <= 1 and "
+			"i2 = 1 and j2 = j or "
+			"i = 0 and j = 0 and i2 = 0 and j2 = 1 }");
+	map = isl_map_transitive_closure(map, &exact);
+	assert(exact);
+	isl_map_free(map);
+
+	map = isl_map_read_from_str(ctx,
+		"[m,n] -> { [i,j] -> [i2,j2] : i2 = i and j2 = j + 2 and "
+			"1 <= i,i2 <= n and 1 <= j,j2 <= m or "
+			"i2 = i + 1 and 3 <= j2 - j <= 4 and "
+			"1 <= i,i2 <= n and 1 <= j,j2 <= m }");
+	map = isl_map_transitive_closure(map, &exact);
+	assert(exact);
+	isl_map_free(map);
+
+	/* Kelly et al 1996, fig 12 */
+	map = isl_map_read_from_str(ctx,
+		"[n] -> { [i,j] -> [i2,j2] : i2 = i and j2 = j + 1 and "
+			"1 <= i,j,j+1 <= n or "
+			"j = n and j2 = 1 and i2 = i + 1 and "
+			"1 <= i,i+1 <= n }");
+	map = isl_map_transitive_closure(map, &exact);
+	assert(exact);
+	map2 = isl_map_read_from_str(ctx,
+		"[n] -> { [i,j] -> [i2,j2] : 1 <= j < j2 <= n and "
+			"1 <= i <= n and i = i2 or "
+			"1 <= i < i2 <= n and 1 <= j <= n and "
+			"1 <= j2 <= n }");
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map2);
+	isl_map_free(map);
+
+	/* Omega's closure4 */
+	map = isl_map_read_from_str(ctx,
+		"[m,n] -> { [x,y] -> [x2,y2] : x2 = x and y2 = y + 1 and "
+			"1 <= x,y <= 10 or "
+			"x2 = x + 1 and y2 = y and "
+			"1 <= x <= 20 && 5 <= y <= 15 }");
+	map = isl_map_transitive_closure(map, &exact);
+	assert(exact);
+	isl_map_free(map);
+
+	map = isl_map_read_from_str(ctx,
+		"[n] -> { [x] -> [y]: 1 <= n <= y - x <= 10 }");
+	map = isl_map_transitive_closure(map, &exact);
+	assert(!exact);
+	map2 = isl_map_read_from_str(ctx,
+		"[n] -> { [x] -> [y] : 1 <= n <= 10 and y >= n + x }");
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map);
+	isl_map_free(map2);
+
+	str = "[n, m] -> { [i0, i1, i2, i3] -> [o0, o1, o2, o3] : "
+	    "i3 = 1 and o0 = i0 and o1 = -1 + i1 and o2 = -1 + i2 and "
+	    "o3 = -2 + i2 and i1 <= -1 + i0 and i1 >= 1 - m + i0 and "
+	    "i1 >= 2 and i1 <= n and i2 >= 3 and i2 <= 1 + n and i2 <= m }";
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_transitive_closure(map, &exact);
+	assert(exact);
+	map2 = isl_map_read_from_str(ctx, str);
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map);
+	isl_map_free(map2);
+
+	str = "{[0] -> [1]; [2] -> [3]}";
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_transitive_closure(map, &exact);
+	assert(exact);
+	map2 = isl_map_read_from_str(ctx, str);
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map);
+	isl_map_free(map2);
+
+	str = "[n] -> { [[i0, i1, 1, 0, i0] -> [i5, 1]] -> "
+	    "[[i0, -1 + i1, 2, 0, i0] -> [-1 + i5, 2]] : "
+	    "exists (e0 = [(3 - n)/3]: i5 >= 2 and i1 >= 2 and "
+	    "3i0 <= -1 + n and i1 <= -1 + n and i5 <= -1 + n and "
+	    "3e0 >= 1 - n and 3e0 <= 2 - n and 3i0 >= -2 + n); "
+	    "[[i0, i1, 2, 0, i0] -> [i5, 1]] -> "
+	    "[[i0, i1, 1, 0, i0] -> [-1 + i5, 2]] : "
+	    "exists (e0 = [(3 - n)/3]: i5 >= 2 and i1 >= 1 and "
+	    "3i0 <= -1 + n and i1 <= -1 + n and i5 <= -1 + n and "
+	    "3e0 >= 1 - n and 3e0 <= 2 - n and 3i0 >= -2 + n); "
+	    "[[i0, i1, 1, 0, i0] -> [i5, 2]] -> "
+	    "[[i0, -1 + i1, 2, 0, i0] -> [i5, 1]] : "
+	    "exists (e0 = [(3 - n)/3]: i1 >= 2 and i5 >= 1 and "
+	    "3i0 <= -1 + n and i1 <= -1 + n and i5 <= -1 + n and "
+	    "3e0 >= 1 - n and 3e0 <= 2 - n and 3i0 >= -2 + n); "
+	    "[[i0, i1, 2, 0, i0] -> [i5, 2]] -> "
+	    "[[i0, i1, 1, 0, i0] -> [i5, 1]] : "
+	    "exists (e0 = [(3 - n)/3]: i5 >= 1 and i1 >= 1 and "
+	    "3i0 <= -1 + n and i1 <= -1 + n and i5 <= -1 + n and "
+	    "3e0 >= 1 - n and 3e0 <= 2 - n and 3i0 >= -2 + n) }";
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_transitive_closure(map, NULL);
+	assert(map);
+	isl_map_free(map);
+}
+
+void test_lex(struct isl_ctx *ctx)
+{
+	isl_space *dim;
+	isl_map *map;
+
+	dim = isl_space_set_alloc(ctx, 0, 0);
+	map = isl_map_lex_le(dim);
+	assert(!isl_map_is_empty(map));
+	isl_map_free(map);
+}
+
+static int test_lexmin(struct isl_ctx *ctx)
+{
+	int equal;
+	const char *str;
+	isl_basic_map *bmap;
+	isl_map *map, *map2;
+	isl_set *set;
+	isl_set *set2;
+	isl_pw_multi_aff *pma;
+
+	str = "[p0, p1] -> { [] -> [] : "
+	    "exists (e0 = [(2p1)/3], e1, e2, e3 = [(3 - p1 + 3e0)/3], "
+	    "e4 = [(p1)/3], e5 = [(p1 + 3e4)/3]: "
+	    "3e0 >= -2 + 2p1 and 3e0 >= p1 and 3e3 >= 1 - p1 + 3e0 and "
+	    "3e0 <= 2p1 and 3e3 >= -2 + p1 and 3e3 <= -1 + p1 and p1 >= 3 and "
+	    "3e5 >= -2 + 2p1 and 3e5 >= p1 and 3e5 <= -1 + p1 + 3e4 and "
+	    "3e4 <= p1 and 3e4 >= -2 + p1 and e3 <= -1 + e0 and "
+	    "3e4 >= 6 - p1 + 3e1 and 3e1 >= p1 and 3e5 >= -2 + p1 + 3e4 and "
+	    "2e4 >= 3 - p1 + 2e1 and e4 <= e1 and 3e3 <= 2 - p1 + 3e0 and "
+	    "e5 >= 1 + e1 and 3e4 >= 6 - 2p1 + 3e1 and "
+	    "p0 >= 2 and p1 >= p0 and 3e2 >= p1 and 3e4 >= 6 - p1 + 3e2 and "
+	    "e2 <= e1 and e3 >= 1 and e4 <= e2) }";
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_lexmin(map);
+	isl_map_free(map);
+
+	str = "[C] -> { [obj,a,b,c] : obj <= 38 a + 7 b + 10 c and "
+	    "a + b <= 1 and c <= 10 b and c <= C and a,b,c,C >= 0 }";
+	set = isl_set_read_from_str(ctx, str);
+	set = isl_set_lexmax(set);
+	str = "[C] -> { [obj,a,b,c] : C = 8 }";
+	set2 = isl_set_read_from_str(ctx, str);
+	set = isl_set_intersect(set, set2);
+	assert(!isl_set_is_empty(set));
+	isl_set_free(set);
+
+	str = "{ [x] -> [y] : x <= y <= 10; [x] -> [5] : -8 <= x <= 8 }";
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_lexmin(map);
+	str = "{ [x] -> [5] : 6 <= x <= 8; "
+		"[x] -> [x] : x <= 5 or (9 <= x <= 10) }";
+	map2 = isl_map_read_from_str(ctx, str);
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map);
+	isl_map_free(map2);
+
+	str = "{ [x] -> [y] : 4y = x or 4y = -1 + x or 4y = -2 + x }";
+	map = isl_map_read_from_str(ctx, str);
+	map2 = isl_map_copy(map);
+	map = isl_map_lexmin(map);
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map);
+	isl_map_free(map2);
+
+	str = "{ [x] -> [y] : x = 4y; [x] -> [y] : x = 2y }";
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_lexmin(map);
+	str = "{ [x] -> [y] : (4y = x and x >= 0) or "
+		"(exists (e0 = [(x)/4], e1 = [(-2 + x)/4]: 2y = x and "
+		"4e1 = -2 + x and 4e0 <= -1 + x and 4e0 >= -3 + x)) or "
+		"(exists (e0 = [(x)/4]: 2y = x and 4e0 = x and x <= -4)) }";
+	map2 = isl_map_read_from_str(ctx, str);
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map);
+	isl_map_free(map2);
+
+	str = "{ [i] -> [i', j] : j = i - 8i' and i' >= 0 and i' <= 7 and "
+				" 8i' <= i and 8i' >= -7 + i }";
+	bmap = isl_basic_map_read_from_str(ctx, str);
+	pma = isl_basic_map_lexmin_pw_multi_aff(isl_basic_map_copy(bmap));
+	map2 = isl_map_from_pw_multi_aff(pma);
+	map = isl_map_from_basic_map(bmap);
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map);
+	isl_map_free(map2);
+
+	str = "{ T[a] -> S[b, c] : a = 4b-2c and c >= b }";
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_lexmin(map);
+	str = "{ T[a] -> S[b, c] : 2b = a and 2c = a }";
+	map2 = isl_map_read_from_str(ctx, str);
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map);
+	isl_map_free(map2);
+
+	/* Check that empty pieces are properly combined. */
+	str = "[K, N] -> { [x, y] -> [a, b] : K+2<=N<=K+4 and x>=4 and "
+		"2N-6<=x<K+N and N-1<=a<=K+N-1 and N+b-6<=a<=2N-4 and "
+		"b<=2N-3K+a and 3b<=4N-K+1 and b>=N and a>=x+1 }";
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_lexmin(map);
+	str = "[K, N] -> { [x, y] -> [1 + x, N] : x >= -6 + 2N and "
+		"x <= -5 + 2N and x >= -1 + 3K - N and x <= -2 + K + N and "
+		"x >= 4 }";
+	map2 = isl_map_read_from_str(ctx, str);
+	assert(isl_map_is_equal(map, map2));
+	isl_map_free(map);
+	isl_map_free(map2);
+
+	str = "[i] -> { [i', j] : j = i - 8i' and i' >= 0 and i' <= 7 and "
+				" 8i' <= i and 8i' >= -7 + i }";
+	set = isl_set_read_from_str(ctx, str);
+	pma = isl_set_lexmin_pw_multi_aff(isl_set_copy(set));
+	set2 = isl_set_from_pw_multi_aff(pma);
+	equal = isl_set_is_equal(set, set2);
+	isl_set_free(set);
+	isl_set_free(set2);
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown,
+			"unexpected difference between set and "
+			"piecewise affine expression", return -1);
+
+	return 0;
+}
+
+/* Check that isl_set_min_val and isl_set_max_val compute the correct
+ * result on non-convex inputs.
+ */
+static int test_min(struct isl_ctx *ctx)
+{
+	isl_set *set;
+	isl_aff *aff;
+	isl_val *val;
+	int min_ok, max_ok;
+
+	set = isl_set_read_from_str(ctx, "{ [-1]; [1] }");
+	aff = isl_aff_read_from_str(ctx, "{ [x] -> [x] }");
+	val = isl_set_min_val(set, aff);
+	min_ok = isl_val_is_negone(val);
+	isl_val_free(val);
+	val = isl_set_max_val(set, aff);
+	max_ok = isl_val_is_one(val);
+	isl_val_free(val);
+	isl_aff_free(aff);
+	isl_set_free(set);
+
+	if (min_ok < 0 || max_ok < 0)
+		return -1;
+	if (!min_ok)
+		isl_die(ctx, isl_error_unknown,
+			"unexpected minimum", return -1);
+	if (!max_ok)
+		isl_die(ctx, isl_error_unknown,
+			"unexpected maximum", return -1);
+
+	return 0;
+}
+
+struct must_may {
+	isl_map *must;
+	isl_map *may;
+};
+
+static int collect_must_may(__isl_take isl_map *dep, int must,
+	void *dep_user, void *user)
+{
+	struct must_may *mm = (struct must_may *)user;
+
+	if (must)
+		mm->must = isl_map_union(mm->must, dep);
+	else
+		mm->may = isl_map_union(mm->may, dep);
+
+	return 0;
+}
+
+static int common_space(void *first, void *second)
+{
+	int depth = *(int *)first;
+	return 2 * depth;
+}
+
+static int map_is_equal(__isl_keep isl_map *map, const char *str)
+{
+	isl_map *map2;
+	int equal;
+
+	if (!map)
+		return -1;
+
+	map2 = isl_map_read_from_str(map->ctx, str);
+	equal = isl_map_is_equal(map, map2);
+	isl_map_free(map2);
+
+	return equal;
+}
+
+static int map_check_equal(__isl_keep isl_map *map, const char *str)
+{
+	int equal;
+
+	equal = map_is_equal(map, str);
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(isl_map_get_ctx(map), isl_error_unknown,
+			"result not as expected", return -1);
+	return 0;
+}
+
+void test_dep(struct isl_ctx *ctx)
+{
+	const char *str;
+	isl_space *dim;
+	isl_map *map;
+	isl_access_info *ai;
+	isl_flow *flow;
+	int depth;
+	struct must_may mm;
+
+	depth = 3;
+
+	str = "{ [2,i,0] -> [i] : 0 <= i <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_alloc(map, &depth, &common_space, 2);
+
+	str = "{ [0,i,0] -> [i] : 0 <= i <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_add_source(ai, map, 1, &depth);
+
+	str = "{ [1,i,0] -> [5] : 0 <= i <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_add_source(ai, map, 1, &depth);
+
+	flow = isl_access_info_compute_flow(ai);
+	dim = isl_space_alloc(ctx, 0, 3, 3);
+	mm.must = isl_map_empty(isl_space_copy(dim));
+	mm.may = isl_map_empty(dim);
+
+	isl_flow_foreach(flow, collect_must_may, &mm);
+
+	str = "{ [0,i,0] -> [2,i,0] : (0 <= i <= 4) or (6 <= i <= 10); "
+	      "  [1,10,0] -> [2,5,0] }";
+	assert(map_is_equal(mm.must, str));
+	str = "{ [i,j,k] -> [l,m,n] : 1 = 0 }";
+	assert(map_is_equal(mm.may, str));
+
+	isl_map_free(mm.must);
+	isl_map_free(mm.may);
+	isl_flow_free(flow);
+
+
+	str = "{ [2,i,0] -> [i] : 0 <= i <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_alloc(map, &depth, &common_space, 2);
+
+	str = "{ [0,i,0] -> [i] : 0 <= i <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_add_source(ai, map, 1, &depth);
+
+	str = "{ [1,i,0] -> [5] : 0 <= i <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_add_source(ai, map, 0, &depth);
+
+	flow = isl_access_info_compute_flow(ai);
+	dim = isl_space_alloc(ctx, 0, 3, 3);
+	mm.must = isl_map_empty(isl_space_copy(dim));
+	mm.may = isl_map_empty(dim);
+
+	isl_flow_foreach(flow, collect_must_may, &mm);
+
+	str = "{ [0,i,0] -> [2,i,0] : (0 <= i <= 4) or (6 <= i <= 10) }";
+	assert(map_is_equal(mm.must, str));
+	str = "{ [0,5,0] -> [2,5,0]; [1,i,0] -> [2,5,0] : 0 <= i <= 10 }";
+	assert(map_is_equal(mm.may, str));
+
+	isl_map_free(mm.must);
+	isl_map_free(mm.may);
+	isl_flow_free(flow);
+
+
+	str = "{ [2,i,0] -> [i] : 0 <= i <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_alloc(map, &depth, &common_space, 2);
+
+	str = "{ [0,i,0] -> [i] : 0 <= i <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_add_source(ai, map, 0, &depth);
+
+	str = "{ [1,i,0] -> [5] : 0 <= i <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_add_source(ai, map, 0, &depth);
+
+	flow = isl_access_info_compute_flow(ai);
+	dim = isl_space_alloc(ctx, 0, 3, 3);
+	mm.must = isl_map_empty(isl_space_copy(dim));
+	mm.may = isl_map_empty(dim);
+
+	isl_flow_foreach(flow, collect_must_may, &mm);
+
+	str = "{ [0,i,0] -> [2,i,0] : 0 <= i <= 10; "
+	      "  [1,i,0] -> [2,5,0] : 0 <= i <= 10 }";
+	assert(map_is_equal(mm.may, str));
+	str = "{ [i,j,k] -> [l,m,n] : 1 = 0 }";
+	assert(map_is_equal(mm.must, str));
+
+	isl_map_free(mm.must);
+	isl_map_free(mm.may);
+	isl_flow_free(flow);
+
+
+	str = "{ [0,i,2] -> [i] : 0 <= i <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_alloc(map, &depth, &common_space, 2);
+
+	str = "{ [0,i,0] -> [i] : 0 <= i <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_add_source(ai, map, 0, &depth);
+
+	str = "{ [0,i,1] -> [5] : 0 <= i <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_add_source(ai, map, 0, &depth);
+
+	flow = isl_access_info_compute_flow(ai);
+	dim = isl_space_alloc(ctx, 0, 3, 3);
+	mm.must = isl_map_empty(isl_space_copy(dim));
+	mm.may = isl_map_empty(dim);
+
+	isl_flow_foreach(flow, collect_must_may, &mm);
+
+	str = "{ [0,i,0] -> [0,i,2] : 0 <= i <= 10; "
+	      "  [0,i,1] -> [0,5,2] : 0 <= i <= 5 }";
+	assert(map_is_equal(mm.may, str));
+	str = "{ [i,j,k] -> [l,m,n] : 1 = 0 }";
+	assert(map_is_equal(mm.must, str));
+
+	isl_map_free(mm.must);
+	isl_map_free(mm.may);
+	isl_flow_free(flow);
+
+
+	str = "{ [0,i,1] -> [i] : 0 <= i <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_alloc(map, &depth, &common_space, 2);
+
+	str = "{ [0,i,0] -> [i] : 0 <= i <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_add_source(ai, map, 0, &depth);
+
+	str = "{ [0,i,2] -> [5] : 0 <= i <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_add_source(ai, map, 0, &depth);
+
+	flow = isl_access_info_compute_flow(ai);
+	dim = isl_space_alloc(ctx, 0, 3, 3);
+	mm.must = isl_map_empty(isl_space_copy(dim));
+	mm.may = isl_map_empty(dim);
+
+	isl_flow_foreach(flow, collect_must_may, &mm);
+
+	str = "{ [0,i,0] -> [0,i,1] : 0 <= i <= 10; "
+	      "  [0,i,2] -> [0,5,1] : 0 <= i <= 4 }";
+	assert(map_is_equal(mm.may, str));
+	str = "{ [i,j,k] -> [l,m,n] : 1 = 0 }";
+	assert(map_is_equal(mm.must, str));
+
+	isl_map_free(mm.must);
+	isl_map_free(mm.may);
+	isl_flow_free(flow);
+
+
+	depth = 5;
+
+	str = "{ [1,i,0,0,0] -> [i,j] : 0 <= i <= 10 and 0 <= j <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_alloc(map, &depth, &common_space, 1);
+
+	str = "{ [0,i,0,j,0] -> [i,j] : 0 <= i <= 10 and 0 <= j <= 10 }";
+	map = isl_map_read_from_str(ctx, str);
+	ai = isl_access_info_add_source(ai, map, 1, &depth);
+
+	flow = isl_access_info_compute_flow(ai);
+	dim = isl_space_alloc(ctx, 0, 5, 5);
+	mm.must = isl_map_empty(isl_space_copy(dim));
+	mm.may = isl_map_empty(dim);
+
+	isl_flow_foreach(flow, collect_must_may, &mm);
+
+	str = "{ [0,i,0,j,0] -> [1,i,0,0,0] : 0 <= i,j <= 10 }";
+	assert(map_is_equal(mm.must, str));
+	str = "{ [0,0,0,0,0] -> [0,0,0,0,0] : 1 = 0 }";
+	assert(map_is_equal(mm.may, str));
+
+	isl_map_free(mm.must);
+	isl_map_free(mm.may);
+	isl_flow_free(flow);
+}
+
+/* Check that the dependence analysis proceeds without errors.
+ * Earlier versions of isl would break down during the analysis
+ * due to the use of the wrong spaces.
+ */
+static int test_flow(isl_ctx *ctx)
+{
+	const char *str;
+	isl_union_map *access, *schedule;
+	isl_union_map *must_dep, *may_dep;
+	int r;
+
+	str = "{ S0[j] -> i[]; S1[j,i] -> i[]; S2[] -> i[]; S3[] -> i[] }";
+	access = isl_union_map_read_from_str(ctx, str);
+	str = "{ S0[j] -> [0,j,0,0] : 0 <= j < 10; "
+		"S1[j,i] -> [0,j,1,i] : 0 <= j < i < 10; "
+		"S2[] -> [1,0,0,0]; "
+		"S3[] -> [-1,0,0,0] }";
+	schedule = isl_union_map_read_from_str(ctx, str);
+	r = isl_union_map_compute_flow(access, isl_union_map_copy(access),
+					isl_union_map_copy(access), schedule,
+					&must_dep, &may_dep, NULL, NULL);
+	isl_union_map_free(may_dep);
+	isl_union_map_free(must_dep);
+
+	return r;
+}
+
+struct {
+	const char *map;
+	int sv;
+} sv_tests[] = {
+	{ "[N] -> { [i] -> [f] : 0 <= i <= N and 0 <= i - 10 f <= 9 }", 1 },
+	{ "[N] -> { [i] -> [f] : 0 <= i <= N and 0 <= i - 10 f <= 10 }", 0 },
+	{ "{ [i] -> [3*floor(i/2) + 5*floor(i/3)] }", 1 },
+	{ "{ S1[i] -> [i] : 0 <= i <= 9; S2[i] -> [i] : 0 <= i <= 9 }", 1 },
+	{ "{ [i] -> S1[i] : 0 <= i <= 9; [i] -> S2[i] : 0 <= i <= 9 }", 0 },
+	{ "{ A[i] -> [i]; B[i] -> [i]; B[i] -> [i + 1] }", 0 },
+	{ "{ A[i] -> [i]; B[i] -> [i] : i < 0; B[i] -> [i + 1] : i > 0 }", 1 },
+	{ "{ A[i] -> [i]; B[i] -> A[i] : i < 0; B[i] -> [i + 1] : i > 0 }", 1 },
+	{ "{ A[i] -> [i]; B[i] -> [j] : i - 1 <= j <= i }", 0 },
+};
+
+int test_sv(isl_ctx *ctx)
+{
+	isl_union_map *umap;
+	int i;
+	int sv;
+
+	for (i = 0; i < ARRAY_SIZE(sv_tests); ++i) {
+		umap = isl_union_map_read_from_str(ctx, sv_tests[i].map);
+		sv = isl_union_map_is_single_valued(umap);
+		isl_union_map_free(umap);
+		if (sv < 0)
+			return -1;
+		if (sv_tests[i].sv && !sv)
+			isl_die(ctx, isl_error_internal,
+				"map not detected as single valued", return -1);
+		if (!sv_tests[i].sv && sv)
+			isl_die(ctx, isl_error_internal,
+				"map detected as single valued", return -1);
+	}
+
+	return 0;
+}
+
+void test_bijective_case(struct isl_ctx *ctx, const char *str, int bijective)
+{
+	isl_map *map;
+
+	map = isl_map_read_from_str(ctx, str);
+	if (bijective)
+		assert(isl_map_is_bijective(map));
+	else
+		assert(!isl_map_is_bijective(map));
+	isl_map_free(map);
+}
+
+void test_bijective(struct isl_ctx *ctx)
+{
+	test_bijective_case(ctx, "[N,M]->{[i,j] -> [i]}", 0);
+	test_bijective_case(ctx, "[N,M]->{[i,j] -> [i] : j=i}", 1);
+	test_bijective_case(ctx, "[N,M]->{[i,j] -> [i] : j=0}", 1);
+	test_bijective_case(ctx, "[N,M]->{[i,j] -> [i] : j=N}", 1);
+	test_bijective_case(ctx, "[N,M]->{[i,j] -> [j,i]}", 1);
+	test_bijective_case(ctx, "[N,M]->{[i,j] -> [i+j]}", 0);
+	test_bijective_case(ctx, "[N,M]->{[i,j] -> []}", 0);
+	test_bijective_case(ctx, "[N,M]->{[i,j] -> [i,j,N]}", 1);
+	test_bijective_case(ctx, "[N,M]->{[i,j] -> [2i]}", 0);
+	test_bijective_case(ctx, "[N,M]->{[i,j] -> [i,i]}", 0);
+	test_bijective_case(ctx, "[N,M]->{[i,j] -> [2i,i]}", 0);
+	test_bijective_case(ctx, "[N,M]->{[i,j] -> [2i,j]}", 1);
+	test_bijective_case(ctx, "[N,M]->{[i,j] -> [x,y] : 2x=i & y =j}", 1);
+}
+
+/* Inputs for isl_pw_qpolynomial_gist tests.
+ * "pwqp" is the input, "set" is the context and "gist" is the expected result.
+ */
+struct {
+	const char *pwqp;
+	const char *set;
+	const char *gist;
+} pwqp_gist_tests[] = {
+	{ "{ [i] -> i }", "{ [k] : exists a : k = 2a }", "{ [i] -> i }" },
+	{ "{ [i] -> i + [ (i + [i/3])/2 ] }", "{ [10] }", "{ [i] -> 16 }" },
+	{ "{ [i] -> ([(i)/2]) }", "{ [k] : exists a : k = 2a+1 }",
+	  "{ [i] -> -1/2 + 1/2 * i }" },
+	{ "{ [i] -> i^2 : i != 0 }", "{ [i] : i != 0 }", "{ [i] -> i^2 }" },
+};
+
+static int test_pwqp(struct isl_ctx *ctx)
+{
+	int i;
+	const char *str;
+	isl_set *set;
+	isl_pw_qpolynomial *pwqp1, *pwqp2;
+	int equal;
+
+	str = "{ [i,j,k] -> 1 + 9 * [i/5] + 7 * [j/11] + 4 * [k/13] }";
+	pwqp1 = isl_pw_qpolynomial_read_from_str(ctx, str);
+
+	pwqp1 = isl_pw_qpolynomial_move_dims(pwqp1, isl_dim_param, 0,
+						isl_dim_in, 1, 1);
+
+	str = "[j] -> { [i,k] -> 1 + 9 * [i/5] + 7 * [j/11] + 4 * [k/13] }";
+	pwqp2 = isl_pw_qpolynomial_read_from_str(ctx, str);
+
+	pwqp1 = isl_pw_qpolynomial_sub(pwqp1, pwqp2);
+
+	assert(isl_pw_qpolynomial_is_zero(pwqp1));
+
+	isl_pw_qpolynomial_free(pwqp1);
+
+	for (i = 0; i < ARRAY_SIZE(pwqp_gist_tests); ++i) {
+		str = pwqp_gist_tests[i].pwqp;
+		pwqp1 = isl_pw_qpolynomial_read_from_str(ctx, str);
+		str = pwqp_gist_tests[i].set;
+		set = isl_set_read_from_str(ctx, str);
+		pwqp1 = isl_pw_qpolynomial_gist(pwqp1, set);
+		str = pwqp_gist_tests[i].gist;
+		pwqp2 = isl_pw_qpolynomial_read_from_str(ctx, str);
+		pwqp1 = isl_pw_qpolynomial_sub(pwqp1, pwqp2);
+		equal = isl_pw_qpolynomial_is_zero(pwqp1);
+		isl_pw_qpolynomial_free(pwqp1);
+
+		if (equal < 0)
+			return -1;
+		if (!equal)
+			isl_die(ctx, isl_error_unknown,
+				"unexpected result", return -1);
+	}
+
+	str = "{ [i] -> ([([i/2] + [i/2])/5]) }";
+	pwqp1 = isl_pw_qpolynomial_read_from_str(ctx, str);
+	str = "{ [i] -> ([(2 * [i/2])/5]) }";
+	pwqp2 = isl_pw_qpolynomial_read_from_str(ctx, str);
+
+	pwqp1 = isl_pw_qpolynomial_sub(pwqp1, pwqp2);
+
+	assert(isl_pw_qpolynomial_is_zero(pwqp1));
+
+	isl_pw_qpolynomial_free(pwqp1);
+
+	str = "{ [x] -> ([x/2] + [(x+1)/2]) }";
+	pwqp1 = isl_pw_qpolynomial_read_from_str(ctx, str);
+	str = "{ [x] -> x }";
+	pwqp2 = isl_pw_qpolynomial_read_from_str(ctx, str);
+
+	pwqp1 = isl_pw_qpolynomial_sub(pwqp1, pwqp2);
+
+	assert(isl_pw_qpolynomial_is_zero(pwqp1));
+
+	isl_pw_qpolynomial_free(pwqp1);
+
+	str = "{ [i] -> ([i/2]) : i >= 0; [i] -> ([i/3]) : i < 0 }";
+	pwqp1 = isl_pw_qpolynomial_read_from_str(ctx, str);
+	pwqp2 = isl_pw_qpolynomial_read_from_str(ctx, str);
+	pwqp1 = isl_pw_qpolynomial_coalesce(pwqp1);
+	pwqp1 = isl_pw_qpolynomial_sub(pwqp1, pwqp2);
+	assert(isl_pw_qpolynomial_is_zero(pwqp1));
+	isl_pw_qpolynomial_free(pwqp1);
+
+	str = "{ [a,b,a] -> (([(2*[a/3]+b)/5]) * ([(2*[a/3]+b)/5])) }";
+	pwqp2 = isl_pw_qpolynomial_read_from_str(ctx, str);
+	str = "{ [a,b,c] -> (([(2*[a/3]+b)/5]) * ([(2*[c/3]+b)/5])) }";
+	pwqp1 = isl_pw_qpolynomial_read_from_str(ctx, str);
+	set = isl_set_read_from_str(ctx, "{ [a,b,a] }");
+	pwqp1 = isl_pw_qpolynomial_intersect_domain(pwqp1, set);
+	equal = isl_pw_qpolynomial_plain_is_equal(pwqp1, pwqp2);
+	isl_pw_qpolynomial_free(pwqp1);
+	isl_pw_qpolynomial_free(pwqp2);
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown, "unexpected result", return -1);
+
+	str = "{ [a,b,c] -> (([(2*[a/3]+1)/5]) * ([(2*[c/3]+1)/5])) : b = 1 }";
+	pwqp2 = isl_pw_qpolynomial_read_from_str(ctx, str);
+	str = "{ [a,b,c] -> (([(2*[a/3]+b)/5]) * ([(2*[c/3]+b)/5])) }";
+	pwqp1 = isl_pw_qpolynomial_read_from_str(ctx, str);
+	pwqp1 = isl_pw_qpolynomial_fix_val(pwqp1, isl_dim_set, 1,
+						isl_val_one(ctx));
+	equal = isl_pw_qpolynomial_plain_is_equal(pwqp1, pwqp2);
+	isl_pw_qpolynomial_free(pwqp1);
+	isl_pw_qpolynomial_free(pwqp2);
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown, "unexpected result", return -1);
+
+	return 0;
+}
+
+void test_split_periods(isl_ctx *ctx)
+{
+	const char *str;
+	isl_pw_qpolynomial *pwqp;
+
+	str = "{ [U,V] -> 1/3 * U + 2/3 * V - [(U + 2V)/3] + [U/2] : "
+		"U + 2V + 3 >= 0 and - U -2V  >= 0 and - U + 10 >= 0 and "
+		"U  >= 0; [U,V] -> U^2 : U >= 100 }";
+	pwqp = isl_pw_qpolynomial_read_from_str(ctx, str);
+
+	pwqp = isl_pw_qpolynomial_split_periods(pwqp, 2);
+	assert(pwqp);
+
+	isl_pw_qpolynomial_free(pwqp);
+}
+
+void test_union(isl_ctx *ctx)
+{
+	const char *str;
+	isl_union_set *uset1, *uset2;
+	isl_union_map *umap1, *umap2;
+
+	str = "{ [i] : 0 <= i <= 1 }";
+	uset1 = isl_union_set_read_from_str(ctx, str);
+	str = "{ [1] -> [0] }";
+	umap1 = isl_union_map_read_from_str(ctx, str);
+
+	umap2 = isl_union_set_lex_gt_union_set(isl_union_set_copy(uset1), uset1);
+	assert(isl_union_map_is_equal(umap1, umap2));
+
+	isl_union_map_free(umap1);
+	isl_union_map_free(umap2);
+
+	str = "{ A[i] -> B[i]; B[i] -> C[i]; A[0] -> C[1] }";
+	umap1 = isl_union_map_read_from_str(ctx, str);
+	str = "{ A[i]; B[i] }";
+	uset1 = isl_union_set_read_from_str(ctx, str);
+
+	uset2 = isl_union_map_domain(umap1);
+
+	assert(isl_union_set_is_equal(uset1, uset2));
+
+	isl_union_set_free(uset1);
+	isl_union_set_free(uset2);
+}
+
+void test_bound(isl_ctx *ctx)
+{
+	const char *str;
+	isl_pw_qpolynomial *pwqp;
+	isl_pw_qpolynomial_fold *pwf;
+
+	str = "{ [[a, b, c, d] -> [e]] -> 0 }";
+	pwqp = isl_pw_qpolynomial_read_from_str(ctx, str);
+	pwf = isl_pw_qpolynomial_bound(pwqp, isl_fold_max, NULL);
+	assert(isl_pw_qpolynomial_fold_dim(pwf, isl_dim_in) == 4);
+	isl_pw_qpolynomial_fold_free(pwf);
+
+	str = "{ [[x]->[x]] -> 1 : exists a : x = 2 a }";
+	pwqp = isl_pw_qpolynomial_read_from_str(ctx, str);
+	pwf = isl_pw_qpolynomial_bound(pwqp, isl_fold_max, NULL);
+	assert(isl_pw_qpolynomial_fold_dim(pwf, isl_dim_in) == 1);
+	isl_pw_qpolynomial_fold_free(pwf);
+}
+
+void test_lift(isl_ctx *ctx)
+{
+	const char *str;
+	isl_basic_map *bmap;
+	isl_basic_set *bset;
+
+	str = "{ [i0] : exists e0 : i0 = 4e0 }";
+	bset = isl_basic_set_read_from_str(ctx, str);
+	bset = isl_basic_set_lift(bset);
+	bmap = isl_basic_map_from_range(bset);
+	bset = isl_basic_map_domain(bmap);
+	isl_basic_set_free(bset);
+}
+
+struct {
+	const char *set1;
+	const char *set2;
+	int subset;
+} subset_tests[] = {
+	{ "{ [112, 0] }",
+	  "{ [i0, i1] : exists (e0 = [(i0 - i1)/16], e1: "
+		"16e0 <= i0 - i1 and 16e0 >= -15 + i0 - i1 and "
+		"16e1 <= i1 and 16e0 >= -i1 and 16e1 >= -i0 + i1) }", 1 },
+	{ "{ [65] }",
+	  "{ [i] : exists (e0 = [(255i)/256], e1 = [(127i + 65e0)/191], "
+		"e2 = [(3i + 61e1)/65], e3 = [(52i + 12e2)/61], "
+		"e4 = [(2i + e3)/3], e5 = [(4i + e3)/4], e6 = [(8i + e3)/12]: "
+		    "3e4 = 2i + e3 and 4e5 = 4i + e3 and 12e6 = 8i + e3 and "
+		    "i <= 255 and 64e3 >= -45 + 67i and i >= 0 and "
+		    "256e0 <= 255i and 256e0 >= -255 + 255i and "
+		    "191e1 <= 127i + 65e0 and 191e1 >= -190 + 127i + 65e0 and "
+		    "65e2 <= 3i + 61e1 and 65e2 >= -64 + 3i + 61e1 and "
+		    "61e3 <= 52i + 12e2 and 61e3 >= -60 + 52i + 12e2) }", 1 },
+	{ "{ [i] : 0 <= i <= 10 }", "{ rat: [i] : 0 <= i <= 10 }", 1 },
+	{ "{ rat: [i] : 0 <= i <= 10 }", "{ [i] : 0 <= i <= 10 }", 0 },
+	{ "{ rat: [0] }", "{ [i] : 0 <= i <= 10 }", 1 },
+	{ "{ rat: [(1)/2] }", "{ [i] : 0 <= i <= 10 }", 0 },
+	{ "{ [t, i] : (exists (e0 = [(2 + t)/4]: 4e0 <= 2 + t and "
+			"4e0 >= -1 + t and i >= 57 and i <= 62 and "
+			"4e0 <= 62 + t - i and 4e0 >= -61 + t + i and "
+			"t >= 0 and t <= 511 and 4e0 <= -57 + t + i and "
+			"4e0 >= 58 + t - i and i >= 58 + t and i >= 62 - t)) }",
+	  "{ [i0, i1] : (exists (e0 = [(4 + i0)/4]: 4e0 <= 62 + i0 - i1 and "
+			"4e0 >= 1 + i0 and i0 >= 0 and i0 <= 511 and "
+			"4e0 <= -57 + i0 + i1)) or "
+		"(exists (e0 = [(2 + i0)/4]: 4e0 <= i0 and "
+			"4e0 >= 58 + i0 - i1 and i0 >= 2 and i0 <= 511 and "
+			"4e0 >= -61 + i0 + i1)) or "
+		"(i1 <= 66 - i0 and i0 >= 2 and i1 >= 59 + i0) }", 1 },
+};
+
+static int test_subset(isl_ctx *ctx)
+{
+	int i;
+	isl_set *set1, *set2;
+	int subset;
+
+	for (i = 0; i < ARRAY_SIZE(subset_tests); ++i) {
+		set1 = isl_set_read_from_str(ctx, subset_tests[i].set1);
+		set2 = isl_set_read_from_str(ctx, subset_tests[i].set2);
+		subset = isl_set_is_subset(set1, set2);
+		isl_set_free(set1);
+		isl_set_free(set2);
+		if (subset < 0)
+			return -1;
+		if (subset != subset_tests[i].subset)
+			isl_die(ctx, isl_error_unknown,
+				"incorrect subset result", return -1);
+	}
+
+	return 0;
+}
+
+struct {
+	const char *minuend;
+	const char *subtrahend;
+	const char *difference;
+} subtract_domain_tests[] = {
+	{ "{ A[i] -> B[i] }", "{ A[i] }", "{ }" },
+	{ "{ A[i] -> B[i] }", "{ B[i] }", "{ A[i] -> B[i] }" },
+	{ "{ A[i] -> B[i] }", "{ A[i] : i > 0 }", "{ A[i] -> B[i] : i <= 0 }" },
+};
+
+static int test_subtract(isl_ctx *ctx)
+{
+	int i;
+	isl_union_map *umap1, *umap2;
+	isl_union_set *uset;
+	int equal;
+
+	for (i = 0; i < ARRAY_SIZE(subtract_domain_tests); ++i) {
+		umap1 = isl_union_map_read_from_str(ctx,
+				subtract_domain_tests[i].minuend);
+		uset = isl_union_set_read_from_str(ctx,
+				subtract_domain_tests[i].subtrahend);
+		umap2 = isl_union_map_read_from_str(ctx,
+				subtract_domain_tests[i].difference);
+		umap1 = isl_union_map_subtract_domain(umap1, uset);
+		equal = isl_union_map_is_equal(umap1, umap2);
+		isl_union_map_free(umap1);
+		isl_union_map_free(umap2);
+		if (equal < 0)
+			return -1;
+		if (!equal)
+			isl_die(ctx, isl_error_unknown,
+				"incorrect subtract domain result", return -1);
+	}
+
+	return 0;
+}
+
+int test_factorize(isl_ctx *ctx)
+{
+	const char *str;
+	isl_basic_set *bset;
+	isl_factorizer *f;
+
+	str = "{ [i0, i1, i2, i3, i4, i5, i6, i7] : 3i5 <= 2 - 2i0 and "
+	    "i0 >= -2 and i6 >= 1 + i3 and i7 >= 0 and 3i5 >= -2i0 and "
+	    "2i4 <= i2 and i6 >= 1 + 2i0 + 3i1 and i4 <= -1 and "
+	    "i6 >= 1 + 2i0 + 3i5 and i6 <= 2 + 2i0 + 3i5 and "
+	    "3i5 <= 2 - 2i0 - i2 + 3i4 and i6 <= 2 + 2i0 + 3i1 and "
+	    "i0 <= -1 and i7 <= i2 + i3 - 3i4 - i6 and "
+	    "3i5 >= -2i0 - i2 + 3i4 }";
+	bset = isl_basic_set_read_from_str(ctx, str);
+	f = isl_basic_set_factorizer(bset);
+	isl_basic_set_free(bset);
+	isl_factorizer_free(f);
+	if (!f)
+		isl_die(ctx, isl_error_unknown,
+			"failed to construct factorizer", return -1);
+
+	str = "{ [i0, i1, i2, i3, i4, i5, i6, i7, i8, i9, i10, i11, i12] : "
+	    "i12 <= 2 + i0 - i11 and 2i8 >= -i4 and i11 >= i1 and "
+	    "3i5 <= -i2 and 2i11 >= -i4 - 2i7 and i11 <= 3 + i0 + 3i9 and "
+	    "i11 <= -i4 - 2i7 and i12 >= -i10 and i2 >= -2 and "
+	    "i11 >= i1 + 3i10 and i11 >= 1 + i0 + 3i9 and "
+	    "i11 <= 1 - i4 - 2i8 and 6i6 <= 6 - i2 and 3i6 >= 1 - i2 and "
+	    "i11 <= 2 + i1 and i12 <= i4 + i11 and i12 >= i0 - i11 and "
+	    "3i5 >= -2 - i2 and i12 >= -1 + i4 + i11 and 3i3 <= 3 - i2 and "
+	    "9i6 <= 11 - i2 + 6i5 and 3i3 >= 1 - i2 and "
+	    "9i6 <= 5 - i2 + 6i3 and i12 <= -1 and i2 <= 0 }";
+	bset = isl_basic_set_read_from_str(ctx, str);
+	f = isl_basic_set_factorizer(bset);
+	isl_basic_set_free(bset);
+	isl_factorizer_free(f);
+	if (!f)
+		isl_die(ctx, isl_error_unknown,
+			"failed to construct factorizer", return -1);
+
+	return 0;
+}
+
+static int check_injective(__isl_take isl_map *map, void *user)
+{
+	int *injective = user;
+
+	*injective = isl_map_is_injective(map);
+	isl_map_free(map);
+
+	if (*injective < 0 || !*injective)
+		return -1;
+
+	return 0;
+}
+
+int test_one_schedule(isl_ctx *ctx, const char *d, const char *w,
+	const char *r, const char *s, int tilable, int parallel)
+{
+	int i;
+	isl_union_set *D;
+	isl_union_map *W, *R, *S;
+	isl_union_map *empty;
+	isl_union_map *dep_raw, *dep_war, *dep_waw, *dep;
+	isl_union_map *validity, *proximity, *coincidence;
+	isl_union_map *schedule;
+	isl_union_map *test;
+	isl_union_set *delta;
+	isl_union_set *domain;
+	isl_set *delta_set;
+	isl_set *slice;
+	isl_set *origin;
+	isl_schedule_constraints *sc;
+	isl_schedule *sched;
+	int is_nonneg, is_parallel, is_tilable, is_injection, is_complete;
+
+	D = isl_union_set_read_from_str(ctx, d);
+	W = isl_union_map_read_from_str(ctx, w);
+	R = isl_union_map_read_from_str(ctx, r);
+	S = isl_union_map_read_from_str(ctx, s);
+
+	W = isl_union_map_intersect_domain(W, isl_union_set_copy(D));
+	R = isl_union_map_intersect_domain(R, isl_union_set_copy(D));
+
+	empty = isl_union_map_empty(isl_union_map_get_space(S));
+        isl_union_map_compute_flow(isl_union_map_copy(R),
+				   isl_union_map_copy(W), empty,
+				   isl_union_map_copy(S),
+				   &dep_raw, NULL, NULL, NULL);
+        isl_union_map_compute_flow(isl_union_map_copy(W),
+				   isl_union_map_copy(W),
+				   isl_union_map_copy(R),
+				   isl_union_map_copy(S),
+				   &dep_waw, &dep_war, NULL, NULL);
+
+	dep = isl_union_map_union(dep_waw, dep_war);
+	dep = isl_union_map_union(dep, dep_raw);
+	validity = isl_union_map_copy(dep);
+	coincidence = isl_union_map_copy(dep);
+	proximity = isl_union_map_copy(dep);
+
+	sc = isl_schedule_constraints_on_domain(isl_union_set_copy(D));
+	sc = isl_schedule_constraints_set_validity(sc, validity);
+	sc = isl_schedule_constraints_set_coincidence(sc, coincidence);
+	sc = isl_schedule_constraints_set_proximity(sc, proximity);
+	sched = isl_schedule_constraints_compute_schedule(sc);
+	schedule = isl_schedule_get_map(sched);
+	isl_schedule_free(sched);
+	isl_union_map_free(W);
+	isl_union_map_free(R);
+	isl_union_map_free(S);
+
+	is_injection = 1;
+	isl_union_map_foreach_map(schedule, &check_injective, &is_injection);
+
+	domain = isl_union_map_domain(isl_union_map_copy(schedule));
+	is_complete = isl_union_set_is_subset(D, domain);
+	isl_union_set_free(D);
+	isl_union_set_free(domain);
+
+	test = isl_union_map_reverse(isl_union_map_copy(schedule));
+	test = isl_union_map_apply_range(test, dep);
+	test = isl_union_map_apply_range(test, schedule);
+
+	delta = isl_union_map_deltas(test);
+	if (isl_union_set_n_set(delta) == 0) {
+		is_tilable = 1;
+		is_parallel = 1;
+		is_nonneg = 1;
+		isl_union_set_free(delta);
+	} else {
+		delta_set = isl_set_from_union_set(delta);
+
+		slice = isl_set_universe(isl_set_get_space(delta_set));
+		for (i = 0; i < tilable; ++i)
+			slice = isl_set_lower_bound_si(slice, isl_dim_set, i, 0);
+		is_tilable = isl_set_is_subset(delta_set, slice);
+		isl_set_free(slice);
+
+		slice = isl_set_universe(isl_set_get_space(delta_set));
+		for (i = 0; i < parallel; ++i)
+			slice = isl_set_fix_si(slice, isl_dim_set, i, 0);
+		is_parallel = isl_set_is_subset(delta_set, slice);
+		isl_set_free(slice);
+
+		origin = isl_set_universe(isl_set_get_space(delta_set));
+		for (i = 0; i < isl_set_dim(origin, isl_dim_set); ++i)
+			origin = isl_set_fix_si(origin, isl_dim_set, i, 0);
+
+		delta_set = isl_set_union(delta_set, isl_set_copy(origin));
+		delta_set = isl_set_lexmin(delta_set);
+
+		is_nonneg = isl_set_is_equal(delta_set, origin);
+
+		isl_set_free(origin);
+		isl_set_free(delta_set);
+	}
+
+	if (is_nonneg < 0 || is_parallel < 0 || is_tilable < 0 ||
+	    is_injection < 0 || is_complete < 0)
+		return -1;
+	if (!is_complete)
+		isl_die(ctx, isl_error_unknown,
+			"generated schedule incomplete", return -1);
+	if (!is_injection)
+		isl_die(ctx, isl_error_unknown,
+			"generated schedule not injective on each statement",
+			return -1);
+	if (!is_nonneg)
+		isl_die(ctx, isl_error_unknown,
+			"negative dependences in generated schedule",
+			return -1);
+	if (!is_tilable)
+		isl_die(ctx, isl_error_unknown,
+			"generated schedule not as tilable as expected",
+			return -1);
+	if (!is_parallel)
+		isl_die(ctx, isl_error_unknown,
+			"generated schedule not as parallel as expected",
+			return -1);
+
+	return 0;
+}
+
+static __isl_give isl_union_map *compute_schedule(isl_ctx *ctx,
+	const char *domain, const char *validity, const char *proximity)
+{
+	isl_union_set *dom;
+	isl_union_map *dep;
+	isl_union_map *prox;
+	isl_schedule_constraints *sc;
+	isl_schedule *schedule;
+	isl_union_map *sched;
+
+	dom = isl_union_set_read_from_str(ctx, domain);
+	dep = isl_union_map_read_from_str(ctx, validity);
+	prox = isl_union_map_read_from_str(ctx, proximity);
+	sc = isl_schedule_constraints_on_domain(dom);
+	sc = isl_schedule_constraints_set_validity(sc, dep);
+	sc = isl_schedule_constraints_set_proximity(sc, prox);
+	schedule = isl_schedule_constraints_compute_schedule(sc);
+	sched = isl_schedule_get_map(schedule);
+	isl_schedule_free(schedule);
+
+	return sched;
+}
+
+/* Check that a schedule can be constructed on the given domain
+ * with the given validity and proximity constraints.
+ */
+static int test_has_schedule(isl_ctx *ctx, const char *domain,
+	const char *validity, const char *proximity)
+{
+	isl_union_map *sched;
+
+	sched = compute_schedule(ctx, domain, validity, proximity);
+	if (!sched)
+		return -1;
+
+	isl_union_map_free(sched);
+	return 0;
+}
+
+int test_special_schedule(isl_ctx *ctx, const char *domain,
+	const char *validity, const char *proximity, const char *expected_sched)
+{
+	isl_union_map *sched1, *sched2;
+	int equal;
+
+	sched1 = compute_schedule(ctx, domain, validity, proximity);
+	sched2 = isl_union_map_read_from_str(ctx, expected_sched);
+
+	equal = isl_union_map_is_equal(sched1, sched2);
+	isl_union_map_free(sched1);
+	isl_union_map_free(sched2);
+
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown, "unexpected schedule",
+			return -1);
+
+	return 0;
+}
+
+/* Check that the schedule map is properly padded, even after being
+ * reconstructed from the band forest.
+ */
+static int test_padded_schedule(isl_ctx *ctx)
+{
+	const char *str;
+	isl_union_set *D;
+	isl_union_map *validity, *proximity;
+	isl_schedule_constraints *sc;
+	isl_schedule *sched;
+	isl_union_map *map1, *map2;
+	isl_band_list *list;
+	int equal;
+
+	str = "[N] -> { S0[i] : 0 <= i <= N; S1[i, j] : 0 <= i, j <= N }";
+	D = isl_union_set_read_from_str(ctx, str);
+	validity = isl_union_map_empty(isl_union_set_get_space(D));
+	proximity = isl_union_map_copy(validity);
+	sc = isl_schedule_constraints_on_domain(D);
+	sc = isl_schedule_constraints_set_validity(sc, validity);
+	sc = isl_schedule_constraints_set_proximity(sc, proximity);
+	sched = isl_schedule_constraints_compute_schedule(sc);
+	map1 = isl_schedule_get_map(sched);
+	list = isl_schedule_get_band_forest(sched);
+	isl_band_list_free(list);
+	map2 = isl_schedule_get_map(sched);
+	isl_schedule_free(sched);
+	equal = isl_union_map_is_equal(map1, map2);
+	isl_union_map_free(map1);
+	isl_union_map_free(map2);
+
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown,
+			"reconstructed schedule map not the same as original",
+			return -1);
+
+	return 0;
+}
+
+/* Input for testing of schedule construction based on
+ * conditional constraints.
+ *
+ * domain is the iteration domain
+ * flow are the flow dependences, which determine the validity and
+ * 	proximity constraints
+ * condition are the conditions on the conditional validity constraints
+ * conditional_validity are the conditional validity constraints
+ * outer_band_n is the expected number of members in the outer band
+ */
+struct {
+	const char *domain;
+	const char *flow;
+	const char *condition;
+	const char *conditional_validity;
+	int outer_band_n;
+} live_range_tests[] = {
+	/* Contrived example that illustrates that we need to keep
+	 * track of tagged condition dependences and
+	 * tagged conditional validity dependences
+	 * in isl_sched_edge separately.
+	 * In particular, the conditional validity constraints on A
+	 * cannot be satisfied,
+	 * but they can be ignored because there are no corresponding
+	 * condition constraints.  However, we do have an additional
+	 * conditional validity constraint that maps to the same
+	 * dependence relation
+	 * as the condition constraint on B.  If we did not make a distinction
+	 * between tagged condition and tagged conditional validity
+	 * dependences, then we
+	 * could end up treating this shared dependence as an condition
+	 * constraint on A, forcing a localization of the conditions,
+	 * which is impossible.
+	 */
+	{ "{ S[i] : 0 <= 1 < 100; T[i] : 0 <= 1 < 100 }",
+	  "{ S[i] -> S[i+1] : 0 <= i < 99 }",
+	  "{ [S[i] -> B[]] -> [S[i+1] -> B[]] : 0 <= i < 99 }",
+	  "{ [S[i] -> A[]] -> [T[i'] -> A[]] : 0 <= i', i < 100 and i != i';"
+	    "[T[i] -> A[]] -> [S[i'] -> A[]] : 0 <= i', i < 100 and i != i';"
+	    "[S[i] -> A[]] -> [S[i+1] -> A[]] : 0 <= i < 99 }",
+	  1
+	},
+	/* TACO 2013 Fig. 7 */
+	{ "[n] -> { S1[i,j] : 0 <= i,j < n; S2[i,j] : 0 <= i,j < n }",
+	  "[n] -> { S1[i,j] -> S2[i,j] : 0 <= i,j < n;"
+		   "S2[i,j] -> S2[i,j+1] : 0 <= i < n and 0 <= j < n - 1 }",
+	  "[n] -> { [S1[i,j] -> t[]] -> [S2[i,j] -> t[]] : 0 <= i,j < n;"
+		   "[S2[i,j] -> x1[]] -> [S2[i,j+1] -> x1[]] : "
+				"0 <= i < n and 0 <= j < n - 1 }",
+	  "[n] -> { [S2[i,j] -> t[]] -> [S1[i,j'] -> t[]] : "
+				"0 <= i < n and 0 <= j < j' < n;"
+		   "[S2[i,j] -> t[]] -> [S1[i',j'] -> t[]] : "
+				"0 <= i < i' < n and 0 <= j,j' < n;"
+		   "[S2[i,j] -> x1[]] -> [S2[i,j'] -> x1[]] : "
+				"0 <= i,j,j' < n and j < j' }",
+	    2
+	},
+	/* TACO 2013 Fig. 7, without tags */
+	{ "[n] -> { S1[i,j] : 0 <= i,j < n; S2[i,j] : 0 <= i,j < n }",
+	  "[n] -> { S1[i,j] -> S2[i,j] : 0 <= i,j < n;"
+		   "S2[i,j] -> S2[i,j+1] : 0 <= i < n and 0 <= j < n - 1 }",
+	  "[n] -> { S1[i,j] -> S2[i,j] : 0 <= i,j < n;"
+		   "S2[i,j] -> S2[i,j+1] : 0 <= i < n and 0 <= j < n - 1 }",
+	  "[n] -> { S2[i,j] -> S1[i,j'] : 0 <= i < n and 0 <= j < j' < n;"
+		   "S2[i,j] -> S1[i',j'] : 0 <= i < i' < n and 0 <= j,j' < n;"
+		   "S2[i,j] -> S2[i,j'] : 0 <= i,j,j' < n and j < j' }",
+	   1
+	},
+	/* TACO 2013 Fig. 12 */
+	{ "{ S1[i,0] : 0 <= i <= 1; S2[i,j] : 0 <= i <= 1 and 1 <= j <= 2;"
+	    "S3[i,3] : 0 <= i <= 1 }",
+	  "{ S1[i,0] -> S2[i,1] : 0 <= i <= 1;"
+	    "S2[i,1] -> S2[i,2] : 0 <= i <= 1;"
+	    "S2[i,2] -> S3[i,3] : 0 <= i <= 1 }",
+	  "{ [S1[i,0]->t[]] -> [S2[i,1]->t[]] : 0 <= i <= 1;"
+	    "[S2[i,1]->t[]] -> [S2[i,2]->t[]] : 0 <= i <= 1;"
+	    "[S2[i,2]->t[]] -> [S3[i,3]->t[]] : 0 <= i <= 1 }",
+	  "{ [S2[i,1]->t[]] -> [S2[i,2]->t[]] : 0 <= i <= 1;"
+	    "[S2[0,j]->t[]] -> [S2[1,j']->t[]] : 1 <= j,j' <= 2;"
+	    "[S2[0,j]->t[]] -> [S1[1,0]->t[]] : 1 <= j <= 2;"
+	    "[S3[0,3]->t[]] -> [S2[1,j]->t[]] : 1 <= j <= 2;"
+	    "[S3[0,3]->t[]] -> [S1[1,0]->t[]] }",
+	   1
+	}
+};
+
+/* Test schedule construction based on conditional constraints.
+ * In particular, check the number of members in the outer band
+ * as an indication of whether tiling is possible or not.
+ */
+static int test_conditional_schedule_constraints(isl_ctx *ctx)
+{
+	int i;
+	isl_union_set *domain;
+	isl_union_map *condition;
+	isl_union_map *flow;
+	isl_union_map *validity;
+	isl_schedule_constraints *sc;
+	isl_schedule *schedule;
+	isl_band_list *list;
+	isl_band *band;
+	int n_member;
+
+	for (i = 0; i < ARRAY_SIZE(live_range_tests); ++i) {
+		domain = isl_union_set_read_from_str(ctx,
+				live_range_tests[i].domain);
+		flow = isl_union_map_read_from_str(ctx,
+				live_range_tests[i].flow);
+		condition = isl_union_map_read_from_str(ctx,
+				live_range_tests[i].condition);
+		validity = isl_union_map_read_from_str(ctx,
+				live_range_tests[i].conditional_validity);
+		sc = isl_schedule_constraints_on_domain(domain);
+		sc = isl_schedule_constraints_set_validity(sc,
+				isl_union_map_copy(flow));
+		sc = isl_schedule_constraints_set_proximity(sc, flow);
+		sc = isl_schedule_constraints_set_conditional_validity(sc,
+				condition, validity);
+		schedule = isl_schedule_constraints_compute_schedule(sc);
+		list = isl_schedule_get_band_forest(schedule);
+		band = isl_band_list_get_band(list, 0);
+		n_member = isl_band_n_member(band);
+		isl_band_free(band);
+		isl_band_list_free(list);
+		isl_schedule_free(schedule);
+
+		if (!schedule)
+			return -1;
+		if (n_member != live_range_tests[i].outer_band_n)
+			isl_die(ctx, isl_error_unknown,
+				"unexpected number of members in outer band",
+				return -1);
+	}
+	return 0;
+}
+
+int test_schedule(isl_ctx *ctx)
+{
+	const char *D, *W, *R, *V, *P, *S;
+
+	/* Handle resulting schedule with zero bands. */
+	if (test_one_schedule(ctx, "{[]}", "{}", "{}", "{[] -> []}", 0, 0) < 0)
+		return -1;
+
+	/* Jacobi */
+	D = "[T,N] -> { S1[t,i] : 1 <= t <= T and 2 <= i <= N - 1 }";
+	W = "{ S1[t,i] -> a[t,i] }";
+	R = "{ S1[t,i] -> a[t-1,i]; S1[t,i] -> a[t-1,i-1]; "
+	    	"S1[t,i] -> a[t-1,i+1] }";
+	S = "{ S1[t,i] -> [t,i] }";
+	if (test_one_schedule(ctx, D, W, R, S, 2, 0) < 0)
+		return -1;
+
+	/* Fig. 5 of CC2008 */
+	D = "[N] -> { S_0[i, j] : i >= 0 and i <= -1 + N and j >= 2 and "
+				"j <= -1 + N }";
+	W = "[N] -> { S_0[i, j] -> a[i, j] : i >= 0 and i <= -1 + N and "
+				"j >= 2 and j <= -1 + N }";
+	R = "[N] -> { S_0[i, j] -> a[j, i] : i >= 0 and i <= -1 + N and "
+				"j >= 2 and j <= -1 + N; "
+		    "S_0[i, j] -> a[i, -1 + j] : i >= 0 and i <= -1 + N and "
+				"j >= 2 and j <= -1 + N }";
+	S = "[N] -> { S_0[i, j] -> [0, i, 0, j, 0] }";
+	if (test_one_schedule(ctx, D, W, R, S, 2, 0) < 0)
+		return -1;
+
+	D = "{ S1[i] : 0 <= i <= 10; S2[i] : 0 <= i <= 9 }";
+	W = "{ S1[i] -> a[i] }";
+	R = "{ S2[i] -> a[i+1] }";
+	S = "{ S1[i] -> [0,i]; S2[i] -> [1,i] }";
+	if (test_one_schedule(ctx, D, W, R, S, 1, 1) < 0)
+		return -1;
+
+	D = "{ S1[i] : 0 <= i < 10; S2[i] : 0 <= i < 10 }";
+	W = "{ S1[i] -> a[i] }";
+	R = "{ S2[i] -> a[9-i] }";
+	S = "{ S1[i] -> [0,i]; S2[i] -> [1,i] }";
+	if (test_one_schedule(ctx, D, W, R, S, 1, 1) < 0)
+		return -1;
+
+	D = "[N] -> { S1[i] : 0 <= i < N; S2[i] : 0 <= i < N }";
+	W = "{ S1[i] -> a[i] }";
+	R = "[N] -> { S2[i] -> a[N-1-i] }";
+	S = "{ S1[i] -> [0,i]; S2[i] -> [1,i] }";
+	if (test_one_schedule(ctx, D, W, R, S, 1, 1) < 0)
+		return -1;
+	
+	D = "{ S1[i] : 0 < i < 10; S2[i] : 0 <= i < 10 }";
+	W = "{ S1[i] -> a[i]; S2[i] -> b[i] }";
+	R = "{ S2[i] -> a[i]; S1[i] -> b[i-1] }";
+	S = "{ S1[i] -> [i,0]; S2[i] -> [i,1] }";
+	if (test_one_schedule(ctx, D, W, R, S, 0, 0) < 0)
+		return -1;
+
+	D = "[N] -> { S1[i] : 1 <= i <= N; S2[i,j] : 1 <= i,j <= N }";
+	W = "{ S1[i] -> a[0,i]; S2[i,j] -> a[i,j] }";
+	R = "{ S2[i,j] -> a[i-1,j] }";
+	S = "{ S1[i] -> [0,i,0]; S2[i,j] -> [1,i,j] }";
+	if (test_one_schedule(ctx, D, W, R, S, 2, 1) < 0)
+		return -1;
+
+	D = "[N] -> { S1[i] : 1 <= i <= N; S2[i,j] : 1 <= i,j <= N }";
+	W = "{ S1[i] -> a[i,0]; S2[i,j] -> a[i,j] }";
+	R = "{ S2[i,j] -> a[i,j-1] }";
+	S = "{ S1[i] -> [0,i,0]; S2[i,j] -> [1,i,j] }";
+	if (test_one_schedule(ctx, D, W, R, S, 2, 1) < 0)
+		return -1;
+
+	D = "[N] -> { S_0[]; S_1[i] : i >= 0 and i <= -1 + N; S_2[] }";
+	W = "[N] -> { S_0[] -> a[0]; S_2[] -> b[0]; "
+		    "S_1[i] -> a[1 + i] : i >= 0 and i <= -1 + N }";
+	R = "[N] -> { S_2[] -> a[N]; S_1[i] -> a[i] : i >= 0 and i <= -1 + N }";
+	S = "[N] -> { S_1[i] -> [1, i, 0]; S_2[] -> [2, 0, 1]; "
+		    "S_0[] -> [0, 0, 0] }";
+	if (test_one_schedule(ctx, D, W, R, S, 1, 0) < 0)
+		return -1;
+	ctx->opt->schedule_parametric = 0;
+	if (test_one_schedule(ctx, D, W, R, S, 0, 0) < 0)
+		return -1;
+	ctx->opt->schedule_parametric = 1;
+
+	D = "[N] -> { S1[i] : 1 <= i <= N; S2[i] : 1 <= i <= N; "
+		    "S3[i,j] : 1 <= i,j <= N; S4[i] : 1 <= i <= N }";
+	W = "{ S1[i] -> a[i,0]; S2[i] -> a[0,i]; S3[i,j] -> a[i,j] }";
+	R = "[N] -> { S3[i,j] -> a[i-1,j]; S3[i,j] -> a[i,j-1]; "
+		    "S4[i] -> a[i,N] }";
+	S = "{ S1[i] -> [0,i,0]; S2[i] -> [1,i,0]; S3[i,j] -> [2,i,j]; "
+		"S4[i] -> [4,i,0] }";
+	if (test_one_schedule(ctx, D, W, R, S, 2, 0) < 0)
+		return -1;
+
+	D = "[N] -> { S_0[i, j] : i >= 1 and i <= N and j >= 1 and j <= N }";
+	W = "[N] -> { S_0[i, j] -> s[0] : i >= 1 and i <= N and j >= 1 and "
+					"j <= N }";
+	R = "[N] -> { S_0[i, j] -> s[0] : i >= 1 and i <= N and j >= 1 and "
+					"j <= N; "
+		    "S_0[i, j] -> a[i, j] : i >= 1 and i <= N and j >= 1 and "
+					"j <= N }";
+	S = "[N] -> { S_0[i, j] -> [0, i, 0, j, 0] }";
+	if (test_one_schedule(ctx, D, W, R, S, 0, 0) < 0)
+		return -1;
+
+	D = "[N] -> { S_0[t] : t >= 0 and t <= -1 + N; "
+		    " S_2[t] : t >= 0 and t <= -1 + N; "
+		    " S_1[t, i] : t >= 0 and t <= -1 + N and i >= 0 and "
+				"i <= -1 + N }";
+	W = "[N] -> { S_0[t] -> a[t, 0] : t >= 0 and t <= -1 + N; "
+		    " S_2[t] -> b[t] : t >= 0 and t <= -1 + N; "
+		    " S_1[t, i] -> a[t, 1 + i] : t >= 0 and t <= -1 + N and "
+						"i >= 0 and i <= -1 + N }";
+	R = "[N] -> { S_1[t, i] -> a[t, i] : t >= 0 and t <= -1 + N and "
+					    "i >= 0 and i <= -1 + N; "
+		    " S_2[t] -> a[t, N] : t >= 0 and t <= -1 + N }";
+	S = "[N] -> { S_2[t] -> [0, t, 2]; S_1[t, i] -> [0, t, 1, i, 0]; "
+		    " S_0[t] -> [0, t, 0] }";
+
+	if (test_one_schedule(ctx, D, W, R, S, 2, 1) < 0)
+		return -1;
+	ctx->opt->schedule_parametric = 0;
+	if (test_one_schedule(ctx, D, W, R, S, 0, 0) < 0)
+		return -1;
+	ctx->opt->schedule_parametric = 1;
+
+	D = "[N] -> { S1[i,j] : 0 <= i,j < N; S2[i,j] : 0 <= i,j < N }";
+	S = "{ S1[i,j] -> [0,i,j]; S2[i,j] -> [1,i,j] }";
+	if (test_one_schedule(ctx, D, "{}", "{}", S, 2, 2) < 0)
+		return -1;
+
+	D = "[M, N] -> { S_1[i] : i >= 0 and i <= -1 + M; "
+	    "S_0[i, j] : i >= 0 and i <= -1 + M and j >= 0 and j <= -1 + N }";
+	W = "[M, N] -> { S_0[i, j] -> a[j] : i >= 0 and i <= -1 + M and "
+					    "j >= 0 and j <= -1 + N; "
+			"S_1[i] -> b[0] : i >= 0 and i <= -1 + M }";
+	R = "[M, N] -> { S_0[i, j] -> a[0] : i >= 0 and i <= -1 + M and "
+					    "j >= 0 and j <= -1 + N; "
+			"S_1[i] -> b[0] : i >= 0 and i <= -1 + M }";
+	S = "[M, N] -> { S_1[i] -> [1, i, 0]; S_0[i, j] -> [0, i, 0, j, 0] }";
+	if (test_one_schedule(ctx, D, W, R, S, 0, 0) < 0)
+		return -1;
+
+	D = "{ S_0[i] : i >= 0 }";
+	W = "{ S_0[i] -> a[i] : i >= 0 }";
+	R = "{ S_0[i] -> a[0] : i >= 0 }";
+	S = "{ S_0[i] -> [0, i, 0] }";
+	if (test_one_schedule(ctx, D, W, R, S, 0, 0) < 0)
+		return -1;
+
+	D = "{ S_0[i] : i >= 0; S_1[i] : i >= 0 }";
+	W = "{ S_0[i] -> a[i] : i >= 0; S_1[i] -> b[i] : i >= 0 }";
+	R = "{ S_0[i] -> b[0] : i >= 0; S_1[i] -> a[i] : i >= 0 }";
+	S = "{ S_1[i] -> [0, i, 1]; S_0[i] -> [0, i, 0] }";
+	if (test_one_schedule(ctx, D, W, R, S, 0, 0) < 0)
+		return -1;
+
+	D = "[n] -> { S_0[j, k] : j <= -1 + n and j >= 0 and "
+				"k <= -1 + n and k >= 0 }";
+	W = "[n] -> { S_0[j, k] -> B[j] : j <= -1 + n and j >= 0 and "							"k <= -1 + n and k >= 0 }";
+	R = "[n] -> { S_0[j, k] -> B[j] : j <= -1 + n and j >= 0 and "
+					"k <= -1 + n and k >= 0; "
+		    "S_0[j, k] -> B[k] : j <= -1 + n and j >= 0 and "
+					"k <= -1 + n and k >= 0; "
+		    "S_0[j, k] -> A[k] : j <= -1 + n and j >= 0 and "
+					"k <= -1 + n and k >= 0 }";
+	S = "[n] -> { S_0[j, k] -> [2, j, k] }";
+	ctx->opt->schedule_outer_coincidence = 1;
+	if (test_one_schedule(ctx, D, W, R, S, 0, 0) < 0)
+		return -1;
+	ctx->opt->schedule_outer_coincidence = 0;
+
+	D = "{Stmt_for_body24[i0, i1, i2, i3]:"
+		"i0 >= 0 and i0 <= 1 and i1 >= 0 and i1 <= 6 and i2 >= 2 and "
+		"i2 <= 6 - i1 and i3 >= 0 and i3 <= -1 + i2;"
+	     "Stmt_for_body24[i0, i1, 1, 0]:"
+		"i0 >= 0 and i0 <= 1 and i1 >= 0 and i1 <= 5;"
+	     "Stmt_for_body7[i0, i1, i2]:"
+		"i0 >= 0 and i0 <= 1 and i1 >= 0 and i1 <= 7 and i2 >= 0 and "
+		"i2 <= 7 }";
+
+	V = "{Stmt_for_body24[0, i1, i2, i3] -> "
+		"Stmt_for_body24[1, i1, i2, i3]:"
+		"i3 >= 0 and i3 <= -1 + i2 and i1 >= 0 and i2 <= 6 - i1 and "
+		"i2 >= 1;"
+	     "Stmt_for_body24[0, i1, i2, i3] -> "
+		"Stmt_for_body7[1, 1 + i1 + i3, 1 + i1 + i2]:"
+		"i3 <= -1 + i2 and i2 <= 6 - i1 and i2 >= 1 and i1 >= 0 and "
+		"i3 >= 0;"
+	      "Stmt_for_body24[0, i1, i2, i3] ->"
+		"Stmt_for_body7[1, i1, 1 + i1 + i3]:"
+		"i3 >= 0 and i2 <= 6 - i1 and i1 >= 0 and i3 <= -1 + i2;"
+	      "Stmt_for_body7[0, i1, i2] -> Stmt_for_body7[1, i1, i2]:"
+		"(i2 >= 1 + i1 and i2 <= 6 and i1 >= 0 and i1 <= 4) or "
+		"(i2 >= 3 and i2 <= 7 and i1 >= 1 and i2 >= 1 + i1) or "
+		"(i2 >= 0 and i2 <= i1 and i2 >= -7 + i1 and i1 <= 7);"
+	      "Stmt_for_body7[0, i1, 1 + i1] -> Stmt_for_body7[1, i1, 1 + i1]:"
+		"i1 <= 6 and i1 >= 0;"
+	      "Stmt_for_body7[0, 0, 7] -> Stmt_for_body7[1, 0, 7];"
+	      "Stmt_for_body7[i0, i1, i2] -> "
+		"Stmt_for_body24[i0, o1, -1 + i2 - o1, -1 + i1 - o1]:"
+		"i0 >= 0 and i0 <= 1 and o1 >= 0 and i2 >= 1 + i1 and "
+		"o1 <= -2 + i2 and i2 <= 7 and o1 <= -1 + i1;"
+	      "Stmt_for_body7[i0, i1, i2] -> "
+		"Stmt_for_body24[i0, i1, o2, -1 - i1 + i2]:"
+		"i0 >= 0 and i0 <= 1 and i1 >= 0 and o2 >= -i1 + i2 and "
+		"o2 >= 1 and o2 <= 6 - i1 and i2 >= 1 + i1 }";
+	P = V;
+	S = "{ Stmt_for_body24[i0, i1, i2, i3] -> "
+		"[i0, 5i0 + i1, 6i0 + i1 + i2, 1 + 6i0 + i1 + i2 + i3, 1];"
+	    "Stmt_for_body7[i0, i1, i2] -> [0, 5i0, 6i0 + i1, 6i0 + i2, 0] }";
+
+	if (test_special_schedule(ctx, D, V, P, S) < 0)
+		return -1;
+
+	D = "{ S_0[i, j] : i >= 1 and i <= 10 and j >= 1 and j <= 8 }";
+	V = "{ S_0[i, j] -> S_0[i, 1 + j] : i >= 1 and i <= 10 and "
+					   "j >= 1 and j <= 7;"
+		"S_0[i, j] -> S_0[1 + i, j] : i >= 1 and i <= 9 and "
+					     "j >= 1 and j <= 8 }";
+	P = "{ }";
+	S = "{ S_0[i, j] -> [i + j, j] }";
+	ctx->opt->schedule_algorithm = ISL_SCHEDULE_ALGORITHM_FEAUTRIER;
+	if (test_special_schedule(ctx, D, V, P, S) < 0)
+		return -1;
+	ctx->opt->schedule_algorithm = ISL_SCHEDULE_ALGORITHM_ISL;
+
+	/* Fig. 1 from Feautrier's "Some Efficient Solutions..." pt. 2, 1992 */
+	D = "[N] -> { S_0[i, j] : i >= 0 and i <= -1 + N and "
+				 "j >= 0 and j <= -1 + i }";
+	V = "[N] -> { S_0[i, j] -> S_0[i, 1 + j] : j <= -2 + i and "
+					"i <= -1 + N and j >= 0;"
+		     "S_0[i, -1 + i] -> S_0[1 + i, 0] : i >= 1 and "
+					"i <= -2 + N }";
+	P = "{ }";
+	S = "{ S_0[i, j] -> [i, j] }";
+	ctx->opt->schedule_algorithm = ISL_SCHEDULE_ALGORITHM_FEAUTRIER;
+	if (test_special_schedule(ctx, D, V, P, S) < 0)
+		return -1;
+	ctx->opt->schedule_algorithm = ISL_SCHEDULE_ALGORITHM_ISL;
+
+	/* Test both algorithms on a case with only proximity dependences. */
+	D = "{ S[i,j] : 0 <= i <= 10 }";
+	V = "{ }";
+	P = "{ S[i,j] -> S[i+1,j] : 0 <= i,j <= 10 }";
+	S = "{ S[i, j] -> [j, i] }";
+	ctx->opt->schedule_algorithm = ISL_SCHEDULE_ALGORITHM_FEAUTRIER;
+	if (test_special_schedule(ctx, D, V, P, S) < 0)
+		return -1;
+	ctx->opt->schedule_algorithm = ISL_SCHEDULE_ALGORITHM_ISL;
+	if (test_special_schedule(ctx, D, V, P, S) < 0)
+		return -1;
+	
+	D = "{ A[a]; B[] }";
+	V = "{}";
+	P = "{ A[a] -> B[] }";
+	if (test_has_schedule(ctx, D, V, P) < 0)
+		return -1;
+
+	if (test_padded_schedule(ctx) < 0)
+		return -1;
+
+	/* Check that check for progress is not confused by rational
+	 * solution.
+	 */
+	D = "[N] -> { S0[i, j] : i >= 0 and i <= N and j >= 0 and j <= N }";
+	V = "[N] -> { S0[i0, -1 + N] -> S0[2 + i0, 0] : i0 >= 0 and "
+							"i0 <= -2 + N; "
+			"S0[i0, i1] -> S0[i0, 1 + i1] : i0 >= 0 and "
+				"i0 <= N and i1 >= 0 and i1 <= -1 + N }";
+	P = "{}";
+	ctx->opt->schedule_algorithm = ISL_SCHEDULE_ALGORITHM_FEAUTRIER;
+	if (test_has_schedule(ctx, D, V, P) < 0)
+		return -1;
+	ctx->opt->schedule_algorithm = ISL_SCHEDULE_ALGORITHM_ISL;
+
+	/* Check that we allow schedule rows that are only non-trivial
+	 * on some full-dimensional domains.
+	 */
+	D = "{ S1[j] : 0 <= j <= 1; S0[]; S2[k] : 0 <= k <= 1 }";
+	V = "{ S0[] -> S1[j] : 0 <= j <= 1; S2[0] -> S0[];"
+		"S1[j] -> S2[1] : 0 <= j <= 1 }";
+	P = "{}";
+	ctx->opt->schedule_algorithm = ISL_SCHEDULE_ALGORITHM_FEAUTRIER;
+	if (test_has_schedule(ctx, D, V, P) < 0)
+		return -1;
+	ctx->opt->schedule_algorithm = ISL_SCHEDULE_ALGORITHM_ISL;
+
+	if (test_conditional_schedule_constraints(ctx) < 0)
+		return -1;
+
+	return 0;
+}
+
+int test_plain_injective(isl_ctx *ctx, const char *str, int injective)
+{
+	isl_union_map *umap;
+	int test;
+
+	umap = isl_union_map_read_from_str(ctx, str);
+	test = isl_union_map_plain_is_injective(umap);
+	isl_union_map_free(umap);
+	if (test < 0)
+		return -1;
+	if (test == injective)
+		return 0;
+	if (injective)
+		isl_die(ctx, isl_error_unknown,
+			"map not detected as injective", return -1);
+	else
+		isl_die(ctx, isl_error_unknown,
+			"map detected as injective", return -1);
+}
+
+int test_injective(isl_ctx *ctx)
+{
+	const char *str;
+
+	if (test_plain_injective(ctx, "{S[i,j] -> A[0]; T[i,j] -> B[1]}", 0))
+		return -1;
+	if (test_plain_injective(ctx, "{S[] -> A[0]; T[] -> B[0]}", 1))
+		return -1;
+	if (test_plain_injective(ctx, "{S[] -> A[0]; T[] -> A[1]}", 1))
+		return -1;
+	if (test_plain_injective(ctx, "{S[] -> A[0]; T[] -> A[0]}", 0))
+		return -1;
+	if (test_plain_injective(ctx, "{S[i] -> A[i,0]; T[i] -> A[i,1]}", 1))
+		return -1;
+	if (test_plain_injective(ctx, "{S[i] -> A[i]; T[i] -> A[i]}", 0))
+		return -1;
+	if (test_plain_injective(ctx, "{S[] -> A[0,0]; T[] -> A[0,1]}", 1))
+		return -1;
+	if (test_plain_injective(ctx, "{S[] -> A[0,0]; T[] -> A[1,0]}", 1))
+		return -1;
+
+	str = "{S[] -> A[0,0]; T[] -> A[0,1]; U[] -> A[1,0]}";
+	if (test_plain_injective(ctx, str, 1))
+		return -1;
+	str = "{S[] -> A[0,0]; T[] -> A[0,1]; U[] -> A[0,0]}";
+	if (test_plain_injective(ctx, str, 0))
+		return -1;
+
+	return 0;
+}
+
+static int aff_plain_is_equal(__isl_keep isl_aff *aff, const char *str)
+{
+	isl_aff *aff2;
+	int equal;
+
+	if (!aff)
+		return -1;
+
+	aff2 = isl_aff_read_from_str(isl_aff_get_ctx(aff), str);
+	equal = isl_aff_plain_is_equal(aff, aff2);
+	isl_aff_free(aff2);
+
+	return equal;
+}
+
+static int aff_check_plain_equal(__isl_keep isl_aff *aff, const char *str)
+{
+	int equal;
+
+	equal = aff_plain_is_equal(aff, str);
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(isl_aff_get_ctx(aff), isl_error_unknown,
+			"result not as expected", return -1);
+	return 0;
+}
+
+struct {
+	__isl_give isl_aff *(*fn)(__isl_take isl_aff *aff1,
+				__isl_take isl_aff *aff2);
+} aff_bin_op[] = {
+	['+'] = { &isl_aff_add },
+	['-'] = { &isl_aff_sub },
+	['*'] = { &isl_aff_mul },
+	['/'] = { &isl_aff_div },
+};
+
+struct {
+	const char *arg1;
+	unsigned char op;
+	const char *arg2;
+	const char *res;
+} aff_bin_tests[] = {
+	{ "{ [i] -> [i] }", '+', "{ [i] -> [i] }",
+	  "{ [i] -> [2i] }" },
+	{ "{ [i] -> [i] }", '-', "{ [i] -> [i] }",
+	  "{ [i] -> [0] }" },
+	{ "{ [i] -> [i] }", '*', "{ [i] -> [2] }",
+	  "{ [i] -> [2i] }" },
+	{ "{ [i] -> [2] }", '*', "{ [i] -> [i] }",
+	  "{ [i] -> [2i] }" },
+	{ "{ [i] -> [i] }", '/', "{ [i] -> [2] }",
+	  "{ [i] -> [i/2] }" },
+	{ "{ [i] -> [2i] }", '/', "{ [i] -> [2] }",
+	  "{ [i] -> [i] }" },
+	{ "{ [i] -> [i] }", '+', "{ [i] -> [NaN] }",
+	  "{ [i] -> [NaN] }" },
+	{ "{ [i] -> [i] }", '-', "{ [i] -> [NaN] }",
+	  "{ [i] -> [NaN] }" },
+	{ "{ [i] -> [i] }", '*', "{ [i] -> [NaN] }",
+	  "{ [i] -> [NaN] }" },
+	{ "{ [i] -> [2] }", '*', "{ [i] -> [NaN] }",
+	  "{ [i] -> [NaN] }" },
+	{ "{ [i] -> [i] }", '/', "{ [i] -> [NaN] }",
+	  "{ [i] -> [NaN] }" },
+	{ "{ [i] -> [2] }", '/', "{ [i] -> [NaN] }",
+	  "{ [i] -> [NaN] }" },
+	{ "{ [i] -> [NaN] }", '+', "{ [i] -> [i] }",
+	  "{ [i] -> [NaN] }" },
+	{ "{ [i] -> [NaN] }", '-', "{ [i] -> [i] }",
+	  "{ [i] -> [NaN] }" },
+	{ "{ [i] -> [NaN] }", '*', "{ [i] -> [2] }",
+	  "{ [i] -> [NaN] }" },
+	{ "{ [i] -> [NaN] }", '*', "{ [i] -> [i] }",
+	  "{ [i] -> [NaN] }" },
+	{ "{ [i] -> [NaN] }", '/', "{ [i] -> [2] }",
+	  "{ [i] -> [NaN] }" },
+	{ "{ [i] -> [NaN] }", '/', "{ [i] -> [i] }",
+	  "{ [i] -> [NaN] }" },
+};
+
+/* Perform some basic tests of binary operations on isl_aff objects.
+ */
+static int test_bin_aff(isl_ctx *ctx)
+{
+	int i;
+	isl_aff *aff1, *aff2, *res;
+	__isl_give isl_aff *(*fn)(__isl_take isl_aff *aff1,
+				__isl_take isl_aff *aff2);
+	int ok;
+
+	for (i = 0; i < ARRAY_SIZE(aff_bin_tests); ++i) {
+		aff1 = isl_aff_read_from_str(ctx, aff_bin_tests[i].arg1);
+		aff2 = isl_aff_read_from_str(ctx, aff_bin_tests[i].arg2);
+		res = isl_aff_read_from_str(ctx, aff_bin_tests[i].res);
+		fn = aff_bin_op[aff_bin_tests[i].op].fn;
+		aff1 = fn(aff1, aff2);
+		if (isl_aff_is_nan(res))
+			ok = isl_aff_is_nan(aff1);
+		else
+			ok = isl_aff_plain_is_equal(aff1, res);
+		isl_aff_free(aff1);
+		isl_aff_free(res);
+		if (ok < 0)
+			return -1;
+		if (!ok)
+			isl_die(ctx, isl_error_unknown,
+				"unexpected result", return -1);
+	}
+
+	return 0;
+}
+
+struct {
+	__isl_give isl_union_pw_multi_aff *(*fn)(
+		__isl_take isl_union_pw_multi_aff *upma1,
+		__isl_take isl_union_pw_multi_aff *upma2);
+	const char *arg1;
+	const char *arg2;
+	const char *res;
+} upma_bin_tests[] = {
+	{ &isl_union_pw_multi_aff_add, "{ A[] -> [0]; B[0] -> [1] }",
+	  "{ B[x] -> [2] : x >= 0 }", "{ B[0] -> [3] }" },
+	{ &isl_union_pw_multi_aff_union_add, "{ A[] -> [0]; B[0] -> [1] }",
+	  "{ B[x] -> [2] : x >= 0 }",
+	  "{ A[] -> [0]; B[0] -> [3]; B[x] -> [2] : x >= 1 }" },
+};
+
+/* Perform some basic tests of binary operations on
+ * isl_union_pw_multi_aff objects.
+ */
+static int test_bin_upma(isl_ctx *ctx)
+{
+	int i;
+	isl_union_pw_multi_aff *upma1, *upma2, *res;
+	int ok;
+
+	for (i = 0; i < ARRAY_SIZE(upma_bin_tests); ++i) {
+		upma1 = isl_union_pw_multi_aff_read_from_str(ctx,
+							upma_bin_tests[i].arg1);
+		upma2 = isl_union_pw_multi_aff_read_from_str(ctx,
+							upma_bin_tests[i].arg2);
+		res = isl_union_pw_multi_aff_read_from_str(ctx,
+							upma_bin_tests[i].res);
+		upma1 = upma_bin_tests[i].fn(upma1, upma2);
+		ok = isl_union_pw_multi_aff_plain_is_equal(upma1, res);
+		isl_union_pw_multi_aff_free(upma1);
+		isl_union_pw_multi_aff_free(res);
+		if (ok < 0)
+			return -1;
+		if (!ok)
+			isl_die(ctx, isl_error_unknown,
+				"unexpected result", return -1);
+	}
+
+	return 0;
+}
+
+int test_aff(isl_ctx *ctx)
+{
+	const char *str;
+	isl_set *set;
+	isl_space *space;
+	isl_local_space *ls;
+	isl_aff *aff;
+	int zero, equal;
+
+	if (test_bin_aff(ctx) < 0)
+		return -1;
+	if (test_bin_upma(ctx) < 0)
+		return -1;
+
+	space = isl_space_set_alloc(ctx, 0, 1);
+	ls = isl_local_space_from_space(space);
+	aff = isl_aff_zero_on_domain(ls);
+
+	aff = isl_aff_add_coefficient_si(aff, isl_dim_in, 0, 1);
+	aff = isl_aff_scale_down_ui(aff, 3);
+	aff = isl_aff_floor(aff);
+	aff = isl_aff_add_coefficient_si(aff, isl_dim_in, 0, 1);
+	aff = isl_aff_scale_down_ui(aff, 2);
+	aff = isl_aff_floor(aff);
+	aff = isl_aff_add_coefficient_si(aff, isl_dim_in, 0, 1);
+
+	str = "{ [10] }";
+	set = isl_set_read_from_str(ctx, str);
+	aff = isl_aff_gist(aff, set);
+
+	aff = isl_aff_add_constant_si(aff, -16);
+	zero = isl_aff_plain_is_zero(aff);
+	isl_aff_free(aff);
+
+	if (zero < 0)
+		return -1;
+	if (!zero)
+		isl_die(ctx, isl_error_unknown, "unexpected result", return -1);
+
+	aff = isl_aff_read_from_str(ctx, "{ [-1] }");
+	aff = isl_aff_scale_down_ui(aff, 64);
+	aff = isl_aff_floor(aff);
+	equal = aff_check_plain_equal(aff, "{ [-1] }");
+	isl_aff_free(aff);
+	if (equal < 0)
+		return -1;
+
+	return 0;
+}
+
+int test_dim_max(isl_ctx *ctx)
+{
+	int equal;
+	const char *str;
+	isl_set *set1, *set2;
+	isl_set *set;
+	isl_map *map;
+	isl_pw_aff *pwaff;
+
+	str = "[N] -> { [i] : 0 <= i <= min(N,10) }";
+	set = isl_set_read_from_str(ctx, str);
+	pwaff = isl_set_dim_max(set, 0);
+	set1 = isl_set_from_pw_aff(pwaff);
+	str = "[N] -> { [10] : N >= 10; [N] : N <= 9 and N >= 0 }";
+	set2 = isl_set_read_from_str(ctx, str);
+	equal = isl_set_is_equal(set1, set2);
+	isl_set_free(set1);
+	isl_set_free(set2);
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown, "unexpected result", return -1);
+
+	str = "[N] -> { [i] : 0 <= i <= max(2N,N+6) }";
+	set = isl_set_read_from_str(ctx, str);
+	pwaff = isl_set_dim_max(set, 0);
+	set1 = isl_set_from_pw_aff(pwaff);
+	str = "[N] -> { [6 + N] : -6 <= N <= 5; [2N] : N >= 6 }";
+	set2 = isl_set_read_from_str(ctx, str);
+	equal = isl_set_is_equal(set1, set2);
+	isl_set_free(set1);
+	isl_set_free(set2);
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown, "unexpected result", return -1);
+
+	str = "[N] -> { [i] : 0 <= i <= 2N or 0 <= i <= N+6 }";
+	set = isl_set_read_from_str(ctx, str);
+	pwaff = isl_set_dim_max(set, 0);
+	set1 = isl_set_from_pw_aff(pwaff);
+	str = "[N] -> { [6 + N] : -6 <= N <= 5; [2N] : N >= 6 }";
+	set2 = isl_set_read_from_str(ctx, str);
+	equal = isl_set_is_equal(set1, set2);
+	isl_set_free(set1);
+	isl_set_free(set2);
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown, "unexpected result", return -1);
+
+	str = "[N,M] -> { [i,j] -> [([i/16]), i%16, ([j/16]), j%16] : "
+			"0 <= i < N and 0 <= j < M }";
+	map = isl_map_read_from_str(ctx, str);
+	set = isl_map_range(map);
+
+	pwaff = isl_set_dim_max(isl_set_copy(set), 0);
+	set1 = isl_set_from_pw_aff(pwaff);
+	str = "[N,M] -> { [([(N-1)/16])] : M,N > 0 }";
+	set2 = isl_set_read_from_str(ctx, str);
+	equal = isl_set_is_equal(set1, set2);
+	isl_set_free(set1);
+	isl_set_free(set2);
+
+	pwaff = isl_set_dim_max(isl_set_copy(set), 3);
+	set1 = isl_set_from_pw_aff(pwaff);
+	str = "[N,M] -> { [t] : t = min(M-1,15) and M,N > 0 }";
+	set2 = isl_set_read_from_str(ctx, str);
+	if (equal >= 0 && equal)
+		equal = isl_set_is_equal(set1, set2);
+	isl_set_free(set1);
+	isl_set_free(set2);
+
+	isl_set_free(set);
+
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown, "unexpected result", return -1);
+
+	/* Check that solutions are properly merged. */
+	str = "[n] -> { [a, b, c] : c >= -4a - 2b and "
+				"c <= -1 + n - 4a - 2b and c >= -2b and "
+				"4a >= -4 + n and c >= 0 }";
+	set = isl_set_read_from_str(ctx, str);
+	pwaff = isl_set_dim_min(set, 2);
+	set1 = isl_set_from_pw_aff(pwaff);
+	str = "[n] -> { [(0)] : n >= 1 }";
+	set2 = isl_set_read_from_str(ctx, str);
+	equal = isl_set_is_equal(set1, set2);
+	isl_set_free(set1);
+	isl_set_free(set2);
+
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown, "unexpected result", return -1);
+
+	/* Check that empty solution lie in the right space. */
+	str = "[n] -> { [t,a] : 1 = 0 }";
+	set = isl_set_read_from_str(ctx, str);
+	pwaff = isl_set_dim_max(set, 0);
+	set1 = isl_set_from_pw_aff(pwaff);
+	str = "[n] -> { [t] : 1 = 0 }";
+	set2 = isl_set_read_from_str(ctx, str);
+	equal = isl_set_is_equal(set1, set2);
+	isl_set_free(set1);
+	isl_set_free(set2);
+
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown, "unexpected result", return -1);
+
+	return 0;
+}
+
+/* Is "pma" obviously equal to the isl_pw_multi_aff represented by "str"?
+ */
+static int pw_multi_aff_plain_is_equal(__isl_keep isl_pw_multi_aff *pma,
+	const char *str)
+{
+	isl_ctx *ctx;
+	isl_pw_multi_aff *pma2;
+	int equal;
+
+	if (!pma)
+		return -1;
+
+	ctx = isl_pw_multi_aff_get_ctx(pma);
+	pma2 = isl_pw_multi_aff_read_from_str(ctx, str);
+	equal = isl_pw_multi_aff_plain_is_equal(pma, pma2);
+	isl_pw_multi_aff_free(pma2);
+
+	return equal;
+}
+
+/* Check that "pma" is obviously equal to the isl_pw_multi_aff
+ * represented by "str".
+ */
+static int pw_multi_aff_check_plain_equal(__isl_keep isl_pw_multi_aff *pma,
+	const char *str)
+{
+	int equal;
+
+	equal = pw_multi_aff_plain_is_equal(pma, str);
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(isl_pw_multi_aff_get_ctx(pma), isl_error_unknown,
+			"result not as expected", return -1);
+	return 0;
+}
+
+/* Basic test for isl_pw_multi_aff_product.
+ *
+ * Check that multiple pieces are properly handled.
+ */
+static int test_product_pma(isl_ctx *ctx)
+{
+	int equal;
+	const char *str;
+	isl_pw_multi_aff *pma1, *pma2;
+
+	str = "{ A[i] -> B[1] : i < 0; A[i] -> B[2] : i >= 0 }";
+	pma1 = isl_pw_multi_aff_read_from_str(ctx, str);
+	str = "{ C[] -> D[] }";
+	pma2 = isl_pw_multi_aff_read_from_str(ctx, str);
+	pma1 = isl_pw_multi_aff_product(pma1, pma2);
+	str = "{ [A[i] -> C[]] -> [B[(1)] -> D[]] : i < 0;"
+		"[A[i] -> C[]] -> [B[(2)] -> D[]] : i >= 0 }";
+	equal = pw_multi_aff_check_plain_equal(pma1, str);
+	isl_pw_multi_aff_free(pma1);
+	if (equal < 0)
+		return -1;
+
+	return 0;
+}
+
+int test_product(isl_ctx *ctx)
+{
+	const char *str;
+	isl_set *set;
+	isl_union_set *uset1, *uset2;
+	int ok;
+
+	str = "{ A[i] }";
+	set = isl_set_read_from_str(ctx, str);
+	set = isl_set_product(set, isl_set_copy(set));
+	ok = isl_set_is_wrapping(set);
+	isl_set_free(set);
+	if (ok < 0)
+		return -1;
+	if (!ok)
+		isl_die(ctx, isl_error_unknown, "unexpected result", return -1);
+
+	str = "{ [] }";
+	uset1 = isl_union_set_read_from_str(ctx, str);
+	uset1 = isl_union_set_product(uset1, isl_union_set_copy(uset1));
+	str = "{ [[] -> []] }";
+	uset2 = isl_union_set_read_from_str(ctx, str);
+	ok = isl_union_set_is_equal(uset1, uset2);
+	isl_union_set_free(uset1);
+	isl_union_set_free(uset2);
+	if (ok < 0)
+		return -1;
+	if (!ok)
+		isl_die(ctx, isl_error_unknown, "unexpected result", return -1);
+
+	if (test_product_pma(ctx) < 0)
+		return -1;
+
+	return 0;
+}
+
+/* Check that two sets are not considered disjoint just because
+ * they have a different set of (named) parameters.
+ */
+static int test_disjoint(isl_ctx *ctx)
+{
+	const char *str;
+	isl_set *set, *set2;
+	int disjoint;
+
+	str = "[n] -> { [[]->[]] }";
+	set = isl_set_read_from_str(ctx, str);
+	str = "{ [[]->[]] }";
+	set2 = isl_set_read_from_str(ctx, str);
+	disjoint = isl_set_is_disjoint(set, set2);
+	isl_set_free(set);
+	isl_set_free(set2);
+	if (disjoint < 0)
+		return -1;
+	if (disjoint)
+		isl_die(ctx, isl_error_unknown, "unexpected result", return -1);
+
+	return 0;
+}
+
+int test_equal(isl_ctx *ctx)
+{
+	const char *str;
+	isl_set *set, *set2;
+	int equal;
+
+	str = "{ S_6[i] }";
+	set = isl_set_read_from_str(ctx, str);
+	str = "{ S_7[i] }";
+	set2 = isl_set_read_from_str(ctx, str);
+	equal = isl_set_is_equal(set, set2);
+	isl_set_free(set);
+	isl_set_free(set2);
+	if (equal < 0)
+		return -1;
+	if (equal)
+		isl_die(ctx, isl_error_unknown, "unexpected result", return -1);
+
+	return 0;
+}
+
+static int test_plain_fixed(isl_ctx *ctx, __isl_take isl_map *map,
+	enum isl_dim_type type, unsigned pos, int fixed)
+{
+	int test;
+
+	test = isl_map_plain_is_fixed(map, type, pos, NULL);
+	isl_map_free(map);
+	if (test < 0)
+		return -1;
+	if (test == fixed)
+		return 0;
+	if (fixed)
+		isl_die(ctx, isl_error_unknown,
+			"map not detected as fixed", return -1);
+	else
+		isl_die(ctx, isl_error_unknown,
+			"map detected as fixed", return -1);
+}
+
+int test_fixed(isl_ctx *ctx)
+{
+	const char *str;
+	isl_map *map;
+
+	str = "{ [i] -> [i] }";
+	map = isl_map_read_from_str(ctx, str);
+	if (test_plain_fixed(ctx, map, isl_dim_out, 0, 0))
+		return -1;
+	str = "{ [i] -> [1] }";
+	map = isl_map_read_from_str(ctx, str);
+	if (test_plain_fixed(ctx, map, isl_dim_out, 0, 1))
+		return -1;
+	str = "{ S_1[p1] -> [o0] : o0 = -2 and p1 >= 1 and p1 <= 7 }";
+	map = isl_map_read_from_str(ctx, str);
+	if (test_plain_fixed(ctx, map, isl_dim_out, 0, 1))
+		return -1;
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_neg(map);
+	if (test_plain_fixed(ctx, map, isl_dim_out, 0, 1))
+		return -1;
+
+	return 0;
+}
+
+struct isl_vertices_test_data {
+	const char *set;
+	int n;
+	const char *vertex[2];
+} vertices_tests[] = {
+	{ "{ A[t, i] : t = 12 and i >= 4 and i <= 12 }",
+	  2, { "{ A[12, 4] }", "{ A[12, 12] }" } },
+	{ "{ A[t, i] : t = 14 and i = 1 }",
+	  1, { "{ A[14, 1] }" } },
+};
+
+/* Check that "vertex" corresponds to one of the vertices in data->vertex.
+ */
+static int find_vertex(__isl_take isl_vertex *vertex, void *user)
+{
+	struct isl_vertices_test_data *data = user;
+	isl_ctx *ctx;
+	isl_multi_aff *ma;
+	isl_basic_set *bset;
+	isl_pw_multi_aff *pma;
+	int i;
+	int equal;
+
+	ctx = isl_vertex_get_ctx(vertex);
+	bset = isl_vertex_get_domain(vertex);
+	ma = isl_vertex_get_expr(vertex);
+	pma = isl_pw_multi_aff_alloc(isl_set_from_basic_set(bset), ma);
+
+	for (i = 0; i < data->n; ++i) {
+		isl_pw_multi_aff *pma_i;
+
+		pma_i = isl_pw_multi_aff_read_from_str(ctx, data->vertex[i]);
+		equal = isl_pw_multi_aff_plain_is_equal(pma, pma_i);
+		isl_pw_multi_aff_free(pma_i);
+
+		if (equal < 0 || equal)
+			break;
+	}
+
+	isl_pw_multi_aff_free(pma);
+	isl_vertex_free(vertex);
+
+	if (equal < 0)
+		return -1;
+
+	return equal ? 0 : - 1;
+}
+
+int test_vertices(isl_ctx *ctx)
+{
+	int i;
+
+	for (i = 0; i < ARRAY_SIZE(vertices_tests); ++i) {
+		isl_basic_set *bset;
+		isl_vertices *vertices;
+		int ok = 1;
+		int n;
+
+		bset = isl_basic_set_read_from_str(ctx, vertices_tests[i].set);
+		vertices = isl_basic_set_compute_vertices(bset);
+		n = isl_vertices_get_n_vertices(vertices);
+		if (vertices_tests[i].n != n)
+			ok = 0;
+		if (isl_vertices_foreach_vertex(vertices, &find_vertex,
+						&vertices_tests[i]) < 0)
+			ok = 0;
+		isl_vertices_free(vertices);
+		isl_basic_set_free(bset);
+
+		if (!vertices)
+			return -1;
+		if (!ok)
+			isl_die(ctx, isl_error_unknown, "unexpected vertices",
+				return -1);
+	}
+
+	return 0;
+}
+
+int test_union_pw(isl_ctx *ctx)
+{
+	int equal;
+	const char *str;
+	isl_union_set *uset;
+	isl_union_pw_qpolynomial *upwqp1, *upwqp2;
+
+	str = "{ [x] -> x^2 }";
+	upwqp1 = isl_union_pw_qpolynomial_read_from_str(ctx, str);
+	upwqp2 = isl_union_pw_qpolynomial_copy(upwqp1);
+	uset = isl_union_pw_qpolynomial_domain(upwqp1);
+	upwqp1 = isl_union_pw_qpolynomial_copy(upwqp2);
+	upwqp1 = isl_union_pw_qpolynomial_intersect_domain(upwqp1, uset);
+	equal = isl_union_pw_qpolynomial_plain_is_equal(upwqp1, upwqp2);
+	isl_union_pw_qpolynomial_free(upwqp1);
+	isl_union_pw_qpolynomial_free(upwqp2);
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown, "unexpected result", return -1);
+
+	return 0;
+}
+
+int test_output(isl_ctx *ctx)
+{
+	char *s;
+	const char *str;
+	isl_pw_aff *pa;
+	isl_printer *p;
+	int equal;
+
+	str = "[x] -> { [1] : x % 4 <= 2; [2] : x = 3 }";
+	pa = isl_pw_aff_read_from_str(ctx, str);
+
+	p = isl_printer_to_str(ctx);
+	p = isl_printer_set_output_format(p, ISL_FORMAT_C);
+	p = isl_printer_print_pw_aff(p, pa);
+	s = isl_printer_get_str(p);
+	isl_printer_free(p);
+	isl_pw_aff_free(pa);
+	if (!s)
+		equal = -1;
+	else
+		equal = !strcmp(s, "4 * floord(x, 4) + 2 >= x ? 1 : 2");
+	free(s);
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown, "unexpected result", return -1);
+
+	return 0;
+}
+
+int test_sample(isl_ctx *ctx)
+{
+	const char *str;
+	isl_basic_set *bset1, *bset2;
+	int empty, subset;
+
+	str = "{ [a, b, c, d, e, f, g, h, i, j, k] : "
+	    "3i >= 1073741823b - c - 1073741823e + f and c >= 0 and "
+	    "3i >= -1 + 3221225466b + c + d - 3221225466e - f and "
+	    "2e >= a - b and 3e <= 2a and 3k <= -a and f <= -1 + a and "
+	    "3i <= 4 - a + 4b + 2c - e - 2f and 3k <= -a + c - f and "
+	    "3h >= -2 + a and 3g >= -3 - a and 3k >= -2 - a and "
+	    "3i >= -2 - a - 2c + 3e + 2f and 3h <= a + c - f and "
+	    "3h >= a + 2147483646b + 2c - 2147483646e - 2f and "
+	    "3g <= -1 - a and 3i <= 1 + c + d - f and a <= 1073741823 and "
+	    "f >= 1 - a + 1073741822b + c + d - 1073741822e and "
+	    "3i >= 1 + 2b - 2c + e + 2f + 3g and "
+	    "1073741822f <= 1073741822 - a + 1073741821b + 1073741822c +"
+		"d - 1073741821e and "
+	    "3j <= 3 - a + 3b and 3g <= -2 - 2b + c + d - e - f and "
+	    "3j >= 1 - a + b + 2e and "
+	    "3f >= -3 + a + 3221225462b + 3c + d - 3221225465e and "
+	    "3i <= 4 - a + 4b - e and "
+	    "f <= 1073741822 + 1073741822b - 1073741822e and 3h <= a and "
+	    "f >= 0 and 2e <= 4 - a + 5b - d and 2e <= a - b + d and "
+	    "c <= -1 + a and 3i >= -2 - a + 3e and "
+	    "1073741822e <= 1073741823 - a + 1073741822b + c and "
+	    "3g >= -4 + 3221225464b + 3c + d - 3221225467e - 3f and "
+	    "3i >= -1 + 3221225466b + 3c + d - 3221225466e - 3f and "
+	    "1073741823e >= 1 + 1073741823b - d and "
+	    "3i >= 1073741823b + c - 1073741823e - f and "
+	    "3i >= 1 + 2b + e + 3g }";
+	bset1 = isl_basic_set_read_from_str(ctx, str);
+	bset2 = isl_basic_set_sample(isl_basic_set_copy(bset1));
+	empty = isl_basic_set_is_empty(bset2);
+	subset = isl_basic_set_is_subset(bset2, bset1);
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	if (empty < 0 || subset < 0)
+		return -1;
+	if (empty)
+		isl_die(ctx, isl_error_unknown, "point not found", return -1);
+	if (!subset)
+		isl_die(ctx, isl_error_unknown, "bad point found", return -1);
+
+	return 0;
+}
+
+int test_fixed_power(isl_ctx *ctx)
+{
+	const char *str;
+	isl_map *map;
+	isl_int exp;
+	int equal;
+
+	isl_int_init(exp);
+	str = "{ [i] -> [i + 1] }";
+	map = isl_map_read_from_str(ctx, str);
+	isl_int_set_si(exp, 23);
+	map = isl_map_fixed_power(map, exp);
+	equal = map_check_equal(map, "{ [i] -> [i + 23] }");
+	isl_int_clear(exp);
+	isl_map_free(map);
+	if (equal < 0)
+		return -1;
+
+	return 0;
+}
+
+int test_slice(isl_ctx *ctx)
+{
+	const char *str;
+	isl_map *map;
+	int equal;
+
+	str = "{ [i] -> [j] }";
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_equate(map, isl_dim_in, 0, isl_dim_out, 0);
+	equal = map_check_equal(map, "{ [i] -> [i] }");
+	isl_map_free(map);
+	if (equal < 0)
+		return -1;
+
+	str = "{ [i] -> [j] }";
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_equate(map, isl_dim_in, 0, isl_dim_in, 0);
+	equal = map_check_equal(map, "{ [i] -> [j] }");
+	isl_map_free(map);
+	if (equal < 0)
+		return -1;
+
+	str = "{ [i] -> [j] }";
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_oppose(map, isl_dim_in, 0, isl_dim_out, 0);
+	equal = map_check_equal(map, "{ [i] -> [-i] }");
+	isl_map_free(map);
+	if (equal < 0)
+		return -1;
+
+	str = "{ [i] -> [j] }";
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_oppose(map, isl_dim_in, 0, isl_dim_in, 0);
+	equal = map_check_equal(map, "{ [0] -> [j] }");
+	isl_map_free(map);
+	if (equal < 0)
+		return -1;
+
+	str = "{ [i] -> [j] }";
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_order_gt(map, isl_dim_in, 0, isl_dim_out, 0);
+	equal = map_check_equal(map, "{ [i] -> [j] : i > j }");
+	isl_map_free(map);
+	if (equal < 0)
+		return -1;
+
+	str = "{ [i] -> [j] }";
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_order_gt(map, isl_dim_in, 0, isl_dim_in, 0);
+	equal = map_check_equal(map, "{ [i] -> [j] : false }");
+	isl_map_free(map);
+	if (equal < 0)
+		return -1;
+
+	return 0;
+}
+
+int test_eliminate(isl_ctx *ctx)
+{
+	const char *str;
+	isl_map *map;
+	int equal;
+
+	str = "{ [i] -> [j] : i = 2j }";
+	map = isl_map_read_from_str(ctx, str);
+	map = isl_map_eliminate(map, isl_dim_out, 0, 1);
+	equal = map_check_equal(map, "{ [i] -> [j] : exists a : i = 2a }");
+	isl_map_free(map);
+	if (equal < 0)
+		return -1;
+
+	return 0;
+}
+
+/* Check that isl_set_dim_residue_class detects that the values of j
+ * in the set below are all odd and that it does not detect any spurious
+ * strides.
+ */
+static int test_residue_class(isl_ctx *ctx)
+{
+	const char *str;
+	isl_set *set;
+	isl_int m, r;
+	int res;
+
+	str = "{ [i,j] : j = 4 i + 1 and 0 <= i <= 100; "
+		"[i,j] : j = 4 i + 3 and 500 <= i <= 600 }";
+	set = isl_set_read_from_str(ctx, str);
+	isl_int_init(m);
+	isl_int_init(r);
+	res = isl_set_dim_residue_class(set, 1, &m, &r);
+	if (res >= 0 &&
+	    (isl_int_cmp_si(m, 2) != 0 || isl_int_cmp_si(r, 1) != 0))
+		isl_die(ctx, isl_error_unknown, "incorrect residue class",
+			res = -1);
+	isl_int_clear(r);
+	isl_int_clear(m);
+	isl_set_free(set);
+
+	return res;
+}
+
+int test_align_parameters(isl_ctx *ctx)
+{
+	const char *str;
+	isl_space *space;
+	isl_multi_aff *ma1, *ma2;
+	int equal;
+
+	str = "{ A[B[] -> C[]] -> D[E[] -> F[]] }";
+	ma1 = isl_multi_aff_read_from_str(ctx, str);
+
+	space = isl_space_params_alloc(ctx, 1);
+	space = isl_space_set_dim_name(space, isl_dim_param, 0, "N");
+	ma1 = isl_multi_aff_align_params(ma1, space);
+
+	str = "[N] -> { A[B[] -> C[]] -> D[E[] -> F[]] }";
+	ma2 = isl_multi_aff_read_from_str(ctx, str);
+
+	equal = isl_multi_aff_plain_is_equal(ma1, ma2);
+
+	isl_multi_aff_free(ma1);
+	isl_multi_aff_free(ma2);
+
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown,
+			"result not as expected", return -1);
+
+	return 0;
+}
+
+static int test_list(isl_ctx *ctx)
+{
+	isl_id *a, *b, *c, *d, *id;
+	isl_id_list *list;
+	int ok;
+
+	a = isl_id_alloc(ctx, "a", NULL);
+	b = isl_id_alloc(ctx, "b", NULL);
+	c = isl_id_alloc(ctx, "c", NULL);
+	d = isl_id_alloc(ctx, "d", NULL);
+
+	list = isl_id_list_alloc(ctx, 4);
+	list = isl_id_list_add(list, a);
+	list = isl_id_list_add(list, b);
+	list = isl_id_list_add(list, c);
+	list = isl_id_list_add(list, d);
+	list = isl_id_list_drop(list, 1, 1);
+
+	if (isl_id_list_n_id(list) != 3) {
+		isl_id_list_free(list);
+		isl_die(ctx, isl_error_unknown,
+			"unexpected number of elements in list", return -1);
+	}
+
+	id = isl_id_list_get_id(list, 0);
+	ok = id == a;
+	isl_id_free(id);
+	id = isl_id_list_get_id(list, 1);
+	ok = ok && id == c;
+	isl_id_free(id);
+	id = isl_id_list_get_id(list, 2);
+	ok = ok && id == d;
+	isl_id_free(id);
+
+	isl_id_list_free(list);
+
+	if (!ok)
+		isl_die(ctx, isl_error_unknown,
+			"unexpected elements in list", return -1);
+
+	return 0;
+}
+
+const char *set_conversion_tests[] = {
+	"[N] -> { [i] : N - 1 <= 2 i <= N }",
+	"[N] -> { [i] : exists a : i = 4 a and N - 1 <= i <= N }",
+	"[N] -> { [i,j] : exists a : i = 4 a and N - 1 <= i, 2j <= N }",
+	"[N] -> { [[i]->[j]] : exists a : i = 4 a and N - 1 <= i, 2j <= N }",
+	"[N] -> { [3*floor(N/2) + 5*floor(N/3)] }",
+};
+
+/* Check that converting from isl_set to isl_pw_multi_aff and back
+ * to isl_set produces the original isl_set.
+ */
+static int test_set_conversion(isl_ctx *ctx)
+{
+	int i;
+	const char *str;
+	isl_set *set1, *set2;
+	isl_pw_multi_aff *pma;
+	int equal;
+
+	for (i = 0; i < ARRAY_SIZE(set_conversion_tests); ++i) {
+		str = set_conversion_tests[i];
+		set1 = isl_set_read_from_str(ctx, str);
+		pma = isl_pw_multi_aff_from_set(isl_set_copy(set1));
+		set2 = isl_set_from_pw_multi_aff(pma);
+		equal = isl_set_is_equal(set1, set2);
+		isl_set_free(set1);
+		isl_set_free(set2);
+
+		if (equal < 0)
+			return -1;
+		if (!equal)
+			isl_die(ctx, isl_error_unknown, "bad conversion",
+				return -1);
+	}
+
+	return 0;
+}
+
+/* Check that converting from isl_map to isl_pw_multi_aff and back
+ * to isl_map produces the original isl_map.
+ */
+static int test_map_conversion(isl_ctx *ctx)
+{
+	const char *str;
+	isl_map *map1, *map2;
+	isl_pw_multi_aff *pma;
+	int equal;
+
+	str = "{ [a, b, c, d] -> s0[a, b, e, f] : "
+		"exists (e0 = [(a - 2c)/3], e1 = [(-4 + b - 5d)/9], "
+		"e2 = [(-d + f)/9]: 3e0 = a - 2c and 9e1 = -4 + b - 5d and "
+		"9e2 = -d + f and f >= 0 and f <= 8 and 9e >= -5 - 2a and "
+		"9e <= -2 - 2a) }";
+	map1 = isl_map_read_from_str(ctx, str);
+	pma = isl_pw_multi_aff_from_map(isl_map_copy(map1));
+	map2 = isl_map_from_pw_multi_aff(pma);
+	equal = isl_map_is_equal(map1, map2);
+	isl_map_free(map1);
+	isl_map_free(map2);
+
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown, "bad conversion", return -1);
+
+	return 0;
+}
+
+static int test_conversion(isl_ctx *ctx)
+{
+	if (test_set_conversion(ctx) < 0)
+		return -1;
+	if (test_map_conversion(ctx) < 0)
+		return -1;
+	return 0;
+}
+
+/* Check that isl_basic_map_curry does not modify input.
+ */
+static int test_curry(isl_ctx *ctx)
+{
+	const char *str;
+	isl_basic_map *bmap1, *bmap2;
+	int equal;
+
+	str = "{ [A[] -> B[]] -> C[] }";
+	bmap1 = isl_basic_map_read_from_str(ctx, str);
+	bmap2 = isl_basic_map_curry(isl_basic_map_copy(bmap1));
+	equal = isl_basic_map_is_equal(bmap1, bmap2);
+	isl_basic_map_free(bmap1);
+	isl_basic_map_free(bmap2);
+
+	if (equal < 0)
+		return -1;
+	if (equal)
+		isl_die(ctx, isl_error_unknown,
+			"curried map should not be equal to original",
+			return -1);
+
+	return 0;
+}
+
+struct {
+	const char *set;
+	const char *ma;
+	const char *res;
+} preimage_tests[] = {
+	{ "{ B[i,j] : 0 <= i < 10 and 0 <= j < 100 }",
+	  "{ A[j,i] -> B[i,j] }",
+	  "{ A[j,i] : 0 <= i < 10 and 0 <= j < 100 }" },
+	{ "{ rat: B[i,j] : 0 <= i, j and 3 i + 5 j <= 100 }",
+	  "{ A[a,b] -> B[a/2,b/6] }",
+	  "{ rat: A[a,b] : 0 <= a, b and 9 a + 5 b <= 600 }" },
+	{ "{ B[i,j] : 0 <= i, j and 3 i + 5 j <= 100 }",
+	  "{ A[a,b] -> B[a/2,b/6] }",
+	  "{ A[a,b] : 0 <= a, b and 9 a + 5 b <= 600 and "
+		    "exists i,j : a = 2 i and b = 6 j }" },
+	{ "[n] -> { S[i] : 0 <= i <= 100 }", "[n] -> { S[n] }",
+	  "[n] -> { : 0 <= n <= 100 }" },
+	{ "{ B[i] : 0 <= i < 100 and exists a : i = 4 a }",
+	  "{ A[a] -> B[2a] }",
+	  "{ A[a] : 0 <= a < 50 and exists b : a = 2 b }" },
+	{ "{ B[i] : 0 <= i < 100 and exists a : i = 4 a }",
+	  "{ A[a] -> B[([a/2])] }",
+	  "{ A[a] : 0 <= a < 200 and exists b : [a/2] = 4 b }" },
+	{ "{ B[i,j,k] : 0 <= i,j,k <= 100 }",
+	  "{ A[a] -> B[a,a,a/3] }",
+	  "{ A[a] : 0 <= a <= 100 and exists b : a = 3 b }" },
+	{ "{ B[i,j] : j = [(i)/2] } ", "{ A[i,j] -> B[i/3,j] }",
+	  "{ A[i,j] : j = [(i)/6] and exists a : i = 3 a }" },
+};
+
+static int test_preimage_basic_set(isl_ctx *ctx)
+{
+	int i;
+	isl_basic_set *bset1, *bset2;
+	isl_multi_aff *ma;
+	int equal;
+
+	for (i = 0; i < ARRAY_SIZE(preimage_tests); ++i) {
+		bset1 = isl_basic_set_read_from_str(ctx, preimage_tests[i].set);
+		ma = isl_multi_aff_read_from_str(ctx, preimage_tests[i].ma);
+		bset2 = isl_basic_set_read_from_str(ctx, preimage_tests[i].res);
+		bset1 = isl_basic_set_preimage_multi_aff(bset1, ma);
+		equal = isl_basic_set_is_equal(bset1, bset2);
+		isl_basic_set_free(bset1);
+		isl_basic_set_free(bset2);
+		if (equal < 0)
+			return -1;
+		if (!equal)
+			isl_die(ctx, isl_error_unknown, "bad preimage",
+				return -1);
+	}
+
+	return 0;
+}
+
+struct {
+	const char *map;
+	const char *ma;
+	const char *res;
+} preimage_domain_tests[] = {
+	{ "{ B[i,j] -> C[2i + 3j] : 0 <= i < 10 and 0 <= j < 100 }",
+	  "{ A[j,i] -> B[i,j] }",
+	  "{ A[j,i] -> C[2i + 3j] : 0 <= i < 10 and 0 <= j < 100 }" },
+	{ "{ B[i] -> C[i]; D[i] -> E[i] }",
+	  "{ A[i] -> B[i + 1] }",
+	  "{ A[i] -> C[i + 1] }" },
+	{ "{ B[i] -> C[i]; B[i] -> E[i] }",
+	  "{ A[i] -> B[i + 1] }",
+	  "{ A[i] -> C[i + 1]; A[i] -> E[i + 1] }" },
+	{ "{ B[i] -> C[([i/2])] }",
+	  "{ A[i] -> B[2i] }",
+	  "{ A[i] -> C[i] }" },
+	{ "{ B[i,j] -> C[([i/2]), ([(i+j)/3])] }",
+	  "{ A[i] -> B[([i/5]), ([i/7])] }",
+	  "{ A[i] -> C[([([i/5])/2]), ([(([i/5])+([i/7]))/3])] }" },
+	{ "[N] -> { B[i] -> C[([N/2]), i, ([N/3])] }",
+	  "[N] -> { A[] -> B[([N/5])] }",
+	  "[N] -> { A[] -> C[([N/2]), ([N/5]), ([N/3])] }" },
+	{ "{ B[i] -> C[i] : exists a : i = 5 a }",
+	  "{ A[i] -> B[2i] }",
+	  "{ A[i] -> C[2i] : exists a : 2i = 5 a }" },
+	{ "{ B[i] -> C[i] : exists a : i = 2 a; "
+	    "B[i] -> D[i] : exists a : i = 2 a + 1 }",
+	  "{ A[i] -> B[2i] }",
+	  "{ A[i] -> C[2i] }" },
+};
+
+static int test_preimage_union_map(isl_ctx *ctx)
+{
+	int i;
+	isl_union_map *umap1, *umap2;
+	isl_multi_aff *ma;
+	int equal;
+
+	for (i = 0; i < ARRAY_SIZE(preimage_domain_tests); ++i) {
+		umap1 = isl_union_map_read_from_str(ctx,
+						preimage_domain_tests[i].map);
+		ma = isl_multi_aff_read_from_str(ctx,
+						preimage_domain_tests[i].ma);
+		umap2 = isl_union_map_read_from_str(ctx,
+						preimage_domain_tests[i].res);
+		umap1 = isl_union_map_preimage_domain_multi_aff(umap1, ma);
+		equal = isl_union_map_is_equal(umap1, umap2);
+		isl_union_map_free(umap1);
+		isl_union_map_free(umap2);
+		if (equal < 0)
+			return -1;
+		if (!equal)
+			isl_die(ctx, isl_error_unknown, "bad preimage",
+				return -1);
+	}
+
+	return 0;
+}
+
+static int test_preimage(isl_ctx *ctx)
+{
+	if (test_preimage_basic_set(ctx) < 0)
+		return -1;
+	if (test_preimage_union_map(ctx) < 0)
+		return -1;
+
+	return 0;
+}
+
+struct {
+	const char *ma1;
+	const char *ma;
+	const char *res;
+} pullback_tests[] = {
+	{ "{ B[i,j] -> C[i + 2j] }" , "{ A[a,b] -> B[b,a] }",
+	  "{ A[a,b] -> C[b + 2a] }" },
+	{ "{ B[i] -> C[2i] }", "{ A[a] -> B[(a)/2] }", "{ A[a] -> C[a] }" },
+	{ "{ B[i] -> C[(i)/2] }", "{ A[a] -> B[2a] }", "{ A[a] -> C[a] }" },
+	{ "{ B[i] -> C[(i)/2] }", "{ A[a] -> B[(a)/3] }",
+	  "{ A[a] -> C[(a)/6] }" },
+	{ "{ B[i] -> C[2i] }", "{ A[a] -> B[5a] }", "{ A[a] -> C[10a] }" },
+	{ "{ B[i] -> C[2i] }", "{ A[a] -> B[(a)/3] }",
+	  "{ A[a] -> C[(2a)/3] }" },
+	{ "{ B[i,j] -> C[i + j] }", "{ A[a] -> B[a,a] }", "{ A[a] -> C[2a] }"},
+	{ "{ B[a] -> C[a,a] }", "{ A[i,j] -> B[i + j] }",
+	  "{ A[i,j] -> C[i + j, i + j] }"},
+	{ "{ B[i] -> C[([i/2])] }", "{ B[5] }", "{ C[2] }" },
+	{ "[n] -> { B[i,j] -> C[([i/2]) + 2j] }",
+	  "[n] -> { B[n,[n/3]] }", "[n] -> { C[([n/2]) + 2*[n/3]] }", },
+};
+
+static int test_pullback(isl_ctx *ctx)
+{
+	int i;
+	isl_multi_aff *ma1, *ma2;
+	isl_multi_aff *ma;
+	int equal;
+
+	for (i = 0; i < ARRAY_SIZE(pullback_tests); ++i) {
+		ma1 = isl_multi_aff_read_from_str(ctx, pullback_tests[i].ma1);
+		ma = isl_multi_aff_read_from_str(ctx, pullback_tests[i].ma);
+		ma2 = isl_multi_aff_read_from_str(ctx, pullback_tests[i].res);
+		ma1 = isl_multi_aff_pullback_multi_aff(ma1, ma);
+		equal = isl_multi_aff_plain_is_equal(ma1, ma2);
+		isl_multi_aff_free(ma1);
+		isl_multi_aff_free(ma2);
+		if (equal < 0)
+			return -1;
+		if (!equal)
+			isl_die(ctx, isl_error_unknown, "bad pullback",
+				return -1);
+	}
+
+	return 0;
+}
+
+/* Check that negation is printed correctly.
+ */
+static int test_ast(isl_ctx *ctx)
+{
+	isl_ast_expr *expr, *expr1, *expr2, *expr3;
+	char *str;
+	int ok;
+
+	expr1 = isl_ast_expr_from_id(isl_id_alloc(ctx, "A", NULL));
+	expr2 = isl_ast_expr_from_id(isl_id_alloc(ctx, "B", NULL));
+	expr = isl_ast_expr_add(expr1, expr2);
+	expr = isl_ast_expr_neg(expr);
+	str = isl_ast_expr_to_str(expr);
+	ok = str ? !strcmp(str, "-(A + B)") : -1;
+	free(str);
+	isl_ast_expr_free(expr);
+
+	if (ok < 0)
+		return -1;
+	if (!ok)
+		isl_die(ctx, isl_error_unknown,
+			"isl_ast_expr printed incorrectly", return -1);
+
+	expr1 = isl_ast_expr_from_id(isl_id_alloc(ctx, "A", NULL));
+	expr2 = isl_ast_expr_from_id(isl_id_alloc(ctx, "B", NULL));
+	expr = isl_ast_expr_add(expr1, expr2);
+	expr3 = isl_ast_expr_from_id(isl_id_alloc(ctx, "C", NULL));
+	expr = isl_ast_expr_sub(expr3, expr);
+	str = isl_ast_expr_to_str(expr);
+	ok = str ? !strcmp(str, "C - (A + B)") : -1;
+	free(str);
+	isl_ast_expr_free(expr);
+
+	if (ok < 0)
+		return -1;
+	if (!ok)
+		isl_die(ctx, isl_error_unknown,
+			"isl_ast_expr printed incorrectly", return -1);
+
+	return 0;
+}
+
+/* Check that isl_ast_build_expr_from_set returns a valid expression
+ * for an empty set.  Note that isl_ast_build_expr_from_set getting
+ * called on an empty set probably indicates a bug in the caller.
+ */
+static int test_ast_build(isl_ctx *ctx)
+{
+	isl_set *set;
+	isl_ast_build *build;
+	isl_ast_expr *expr;
+
+	set = isl_set_universe(isl_space_params_alloc(ctx, 0));
+	build = isl_ast_build_from_context(set);
+
+	set = isl_set_empty(isl_space_params_alloc(ctx, 0));
+	expr = isl_ast_build_expr_from_set(build, set);
+
+	isl_ast_expr_free(expr);
+	isl_ast_build_free(build);
+
+	if (!expr)
+		return -1;
+
+	return 0;
+}
+
+/* Internal data structure for before_for and after_for callbacks.
+ *
+ * depth is the current depth
+ * before is the number of times before_for has been called
+ * after is the number of times after_for has been called
+ */
+struct isl_test_codegen_data {
+	int depth;
+	int before;
+	int after;
+};
+
+/* This function is called before each for loop in the AST generated
+ * from test_ast_gen1.
+ *
+ * Increment the number of calls and the depth.
+ * Check that the space returned by isl_ast_build_get_schedule_space
+ * matches the target space of the schedule returned by
+ * isl_ast_build_get_schedule.
+ * Return an isl_id that is checked by the corresponding call
+ * to after_for.
+ */
+static __isl_give isl_id *before_for(__isl_keep isl_ast_build *build,
+	void *user)
+{
+	struct isl_test_codegen_data *data = user;
+	isl_ctx *ctx;
+	isl_space *space;
+	isl_union_map *schedule;
+	isl_union_set *uset;
+	isl_set *set;
+	int empty;
+	char name[] = "d0";
+
+	ctx = isl_ast_build_get_ctx(build);
+
+	if (data->before >= 3)
+		isl_die(ctx, isl_error_unknown,
+			"unexpected number of for nodes", return NULL);
+	if (data->depth >= 2)
+		isl_die(ctx, isl_error_unknown,
+			"unexpected depth", return NULL);
+
+	snprintf(name, sizeof(name), "d%d", data->depth);
+	data->before++;
+	data->depth++;
+
+	schedule = isl_ast_build_get_schedule(build);
+	uset = isl_union_map_range(schedule);
+	if (!uset)
+		return NULL;
+	if (isl_union_set_n_set(uset) != 1) {
+		isl_union_set_free(uset);
+		isl_die(ctx, isl_error_unknown,
+			"expecting single range space", return NULL);
+	}
+
+	space = isl_ast_build_get_schedule_space(build);
+	set = isl_union_set_extract_set(uset, space);
+	isl_union_set_free(uset);
+	empty = isl_set_is_empty(set);
+	isl_set_free(set);
+
+	if (empty < 0)
+		return NULL;
+	if (empty)
+		isl_die(ctx, isl_error_unknown,
+			"spaces don't match", return NULL);
+
+	return isl_id_alloc(ctx, name, NULL);
+}
+
+/* This function is called after each for loop in the AST generated
+ * from test_ast_gen1.
+ *
+ * Increment the number of calls and decrement the depth.
+ * Check that the annotation attached to the node matches
+ * the isl_id returned by the corresponding call to before_for.
+ */
+static __isl_give isl_ast_node *after_for(__isl_take isl_ast_node *node,
+	__isl_keep isl_ast_build *build, void *user)
+{
+	struct isl_test_codegen_data *data = user;
+	isl_id *id;
+	const char *name;
+	int valid;
+
+	data->after++;
+	data->depth--;
+
+	if (data->after > data->before)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_unknown,
+			"mismatch in number of for nodes",
+			return isl_ast_node_free(node));
+
+	id = isl_ast_node_get_annotation(node);
+	if (!id)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_unknown,
+			"missing annotation", return isl_ast_node_free(node));
+
+	name = isl_id_get_name(id);
+	valid = name && atoi(name + 1) == data->depth;
+	isl_id_free(id);
+
+	if (!valid)
+		isl_die(isl_ast_node_get_ctx(node), isl_error_unknown,
+			"wrong annotation", return isl_ast_node_free(node));
+
+	return node;
+}
+
+/* Check that the before_each_for and after_each_for callbacks
+ * are called for each for loop in the generated code,
+ * that they are called in the right order and that the isl_id
+ * returned from the before_each_for callback is attached to
+ * the isl_ast_node passed to the corresponding after_each_for call.
+ */
+static int test_ast_gen1(isl_ctx *ctx)
+{
+	const char *str;
+	isl_set *set;
+	isl_union_map *schedule;
+	isl_ast_build *build;
+	isl_ast_node *tree;
+	struct isl_test_codegen_data data;
+
+	str = "[N] -> { : N >= 10 }";
+	set = isl_set_read_from_str(ctx, str);
+	str = "[N] -> { A[i,j] -> S[8,i,3,j] : 0 <= i,j <= N; "
+		    "B[i,j] -> S[8,j,9,i] : 0 <= i,j <= N }";
+	schedule = isl_union_map_read_from_str(ctx, str);
+
+	data.before = 0;
+	data.after = 0;
+	data.depth = 0;
+	build = isl_ast_build_from_context(set);
+	build = isl_ast_build_set_before_each_for(build,
+			&before_for, &data);
+	build = isl_ast_build_set_after_each_for(build,
+			&after_for, &data);
+	tree = isl_ast_build_ast_from_schedule(build, schedule);
+	isl_ast_build_free(build);
+	if (!tree)
+		return -1;
+
+	isl_ast_node_free(tree);
+
+	if (data.before != 3 || data.after != 3)
+		isl_die(ctx, isl_error_unknown,
+			"unexpected number of for nodes", return -1);
+
+	return 0;
+}
+
+/* Check that the AST generator handles domains that are integrally disjoint
+ * but not ratinoally disjoint.
+ */
+static int test_ast_gen2(isl_ctx *ctx)
+{
+	const char *str;
+	isl_set *set;
+	isl_union_map *schedule;
+	isl_union_map *options;
+	isl_ast_build *build;
+	isl_ast_node *tree;
+
+	str = "{ A[i,j] -> [i,j] : 0 <= i,j <= 1 }";
+	schedule = isl_union_map_read_from_str(ctx, str);
+	set = isl_set_universe(isl_space_params_alloc(ctx, 0));
+	build = isl_ast_build_from_context(set);
+
+	str = "{ [i,j] -> atomic[1] : i + j = 1; [i,j] -> unroll[1] : i = j }";
+	options = isl_union_map_read_from_str(ctx, str);
+	build = isl_ast_build_set_options(build, options);
+	tree = isl_ast_build_ast_from_schedule(build, schedule);
+	isl_ast_build_free(build);
+	if (!tree)
+		return -1;
+	isl_ast_node_free(tree);
+
+	return 0;
+}
+
+/* Increment *user on each call.
+ */
+static __isl_give isl_ast_node *count_domains(__isl_take isl_ast_node *node,
+	__isl_keep isl_ast_build *build, void *user)
+{
+	int *n = user;
+
+	(*n)++;
+
+	return node;
+}
+
+/* Test that unrolling tries to minimize the number of instances.
+ * In particular, for the schedule given below, make sure it generates
+ * 3 nodes (rather than 101).
+ */
+static int test_ast_gen3(isl_ctx *ctx)
+{
+	const char *str;
+	isl_set *set;
+	isl_union_map *schedule;
+	isl_union_map *options;
+	isl_ast_build *build;
+	isl_ast_node *tree;
+	int n_domain = 0;
+
+	str = "[n] -> { A[i] -> [i] : 0 <= i <= 100 and n <= i <= n + 2 }";
+	schedule = isl_union_map_read_from_str(ctx, str);
+	set = isl_set_universe(isl_space_params_alloc(ctx, 0));
+
+	str = "{ [i] -> unroll[0] }";
+	options = isl_union_map_read_from_str(ctx, str);
+
+	build = isl_ast_build_from_context(set);
+	build = isl_ast_build_set_options(build, options);
+	build = isl_ast_build_set_at_each_domain(build,
+			&count_domains, &n_domain);
+	tree = isl_ast_build_ast_from_schedule(build, schedule);
+	isl_ast_build_free(build);
+	if (!tree)
+		return -1;
+
+	isl_ast_node_free(tree);
+
+	if (n_domain != 3)
+		isl_die(ctx, isl_error_unknown,
+			"unexpected number of for nodes", return -1);
+
+	return 0;
+}
+
+/* Check that if the ast_build_exploit_nested_bounds options is set,
+ * we do not get an outer if node in the generated AST,
+ * while we do get such an outer if node if the options is not set.
+ */
+static int test_ast_gen4(isl_ctx *ctx)
+{
+	const char *str;
+	isl_set *set;
+	isl_union_map *schedule;
+	isl_ast_build *build;
+	isl_ast_node *tree;
+	enum isl_ast_node_type type;
+	int enb;
+
+	enb = isl_options_get_ast_build_exploit_nested_bounds(ctx);
+	str = "[N,M] -> { A[i,j] -> [i,j] : 0 <= i <= N and 0 <= j <= M }";
+
+	isl_options_set_ast_build_exploit_nested_bounds(ctx, 1);
+
+	schedule = isl_union_map_read_from_str(ctx, str);
+	set = isl_set_universe(isl_space_params_alloc(ctx, 0));
+	build = isl_ast_build_from_context(set);
+	tree = isl_ast_build_ast_from_schedule(build, schedule);
+	isl_ast_build_free(build);
+	if (!tree)
+		return -1;
+
+	type = isl_ast_node_get_type(tree);
+	isl_ast_node_free(tree);
+
+	if (type == isl_ast_node_if)
+		isl_die(ctx, isl_error_unknown,
+			"not expecting if node", return -1);
+
+	isl_options_set_ast_build_exploit_nested_bounds(ctx, 0);
+
+	schedule = isl_union_map_read_from_str(ctx, str);
+	set = isl_set_universe(isl_space_params_alloc(ctx, 0));
+	build = isl_ast_build_from_context(set);
+	tree = isl_ast_build_ast_from_schedule(build, schedule);
+	isl_ast_build_free(build);
+	if (!tree)
+		return -1;
+
+	type = isl_ast_node_get_type(tree);
+	isl_ast_node_free(tree);
+
+	if (type != isl_ast_node_if)
+		isl_die(ctx, isl_error_unknown,
+			"expecting if node", return -1);
+
+	isl_options_set_ast_build_exploit_nested_bounds(ctx, enb);
+
+	return 0;
+}
+
+/* This function is called for each leaf in the AST generated
+ * from test_ast_gen5.
+ *
+ * We finalize the AST generation by extending the outer schedule
+ * with a zero-dimensional schedule.  If this results in any for loops,
+ * then this means that we did not pass along enough information
+ * about the outer schedule to the inner AST generation.
+ */
+static __isl_give isl_ast_node *create_leaf(__isl_take isl_ast_build *build,
+	void *user)
+{
+	isl_union_map *schedule, *extra;
+	isl_ast_node *tree;
+
+	schedule = isl_ast_build_get_schedule(build);
+	extra = isl_union_map_copy(schedule);
+	extra = isl_union_map_from_domain(isl_union_map_domain(extra));
+	schedule = isl_union_map_range_product(schedule, extra);
+	tree = isl_ast_build_ast_from_schedule(build, schedule);
+	isl_ast_build_free(build);
+
+	if (!tree)
+		return NULL;
+
+	if (isl_ast_node_get_type(tree) == isl_ast_node_for)
+		isl_die(isl_ast_node_get_ctx(tree), isl_error_unknown,
+			"code should not contain any for loop",
+			return isl_ast_node_free(tree));
+
+	return tree;
+}
+
+/* Check that we do not lose any information when going back and
+ * forth between internal and external schedule.
+ *
+ * In particular, we create an AST where we unroll the only
+ * non-constant dimension in the schedule.  We therefore do
+ * not expect any for loops in the AST.  However, older versions
+ * of isl would not pass along enough information about the outer
+ * schedule when performing an inner code generation from a create_leaf
+ * callback, resulting in the inner code generation producing a for loop.
+ */
+static int test_ast_gen5(isl_ctx *ctx)
+{
+	const char *str;
+	isl_set *set;
+	isl_union_map *schedule, *options;
+	isl_ast_build *build;
+	isl_ast_node *tree;
+
+	str = "{ A[] -> [1, 1, 2]; B[i] -> [1, i, 0] : i >= 1 and i <= 2 }";
+	schedule = isl_union_map_read_from_str(ctx, str);
+
+	str = "{ [a, b, c] -> unroll[1] : exists (e0 = [(a)/4]: "
+				"4e0 >= -1 + a - b and 4e0 <= -2 + a + b) }";
+	options = isl_union_map_read_from_str(ctx, str);
+
+	set = isl_set_universe(isl_space_params_alloc(ctx, 0));
+	build = isl_ast_build_from_context(set);
+	build = isl_ast_build_set_options(build, options);
+        build = isl_ast_build_set_create_leaf(build, &create_leaf, NULL);
+	tree = isl_ast_build_ast_from_schedule(build, schedule);
+	isl_ast_build_free(build);
+	isl_ast_node_free(tree);
+	if (!tree)
+		return -1;
+
+	return 0;
+}
+
+static int test_ast_gen(isl_ctx *ctx)
+{
+	if (test_ast_gen1(ctx) < 0)
+		return -1;
+	if (test_ast_gen2(ctx) < 0)
+		return -1;
+	if (test_ast_gen3(ctx) < 0)
+		return -1;
+	if (test_ast_gen4(ctx) < 0)
+		return -1;
+	if (test_ast_gen5(ctx) < 0)
+		return -1;
+	return 0;
+}
+
+/* Check if dropping output dimensions from an isl_pw_multi_aff
+ * works properly.
+ */
+static int test_pw_multi_aff(isl_ctx *ctx)
+{
+	const char *str;
+	isl_pw_multi_aff *pma1, *pma2;
+	int equal;
+
+	str = "{ [i,j] -> [i+j, 4i-j] }";
+	pma1 = isl_pw_multi_aff_read_from_str(ctx, str);
+	str = "{ [i,j] -> [4i-j] }";
+	pma2 = isl_pw_multi_aff_read_from_str(ctx, str);
+
+	pma1 = isl_pw_multi_aff_drop_dims(pma1, isl_dim_out, 0, 1);
+
+	equal = isl_pw_multi_aff_plain_is_equal(pma1, pma2);
+
+	isl_pw_multi_aff_free(pma1);
+	isl_pw_multi_aff_free(pma2);
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown,
+			"expressions not equal", return -1);
+
+	return 0;
+}
+
+/* Check that we can properly parse multi piecewise affine expressions
+ * where the piecewise affine expressions have different domains.
+ */
+static int test_multi_pw_aff(isl_ctx *ctx)
+{
+	const char *str;
+	isl_set *dom, *dom2;
+	isl_multi_pw_aff *mpa1, *mpa2;
+	isl_pw_aff *pa;
+	int equal;
+	int equal_domain;
+
+	mpa1 = isl_multi_pw_aff_read_from_str(ctx, "{ [i] -> [i] }");
+	dom = isl_set_read_from_str(ctx, "{ [i] : i > 0 }");
+	mpa1 = isl_multi_pw_aff_intersect_domain(mpa1, dom);
+	mpa2 = isl_multi_pw_aff_read_from_str(ctx, "{ [i] -> [2i] }");
+	mpa2 = isl_multi_pw_aff_flat_range_product(mpa1, mpa2);
+	str = "{ [i] -> [(i : i > 0), 2i] }";
+	mpa1 = isl_multi_pw_aff_read_from_str(ctx, str);
+
+	equal = isl_multi_pw_aff_plain_is_equal(mpa1, mpa2);
+
+	pa = isl_multi_pw_aff_get_pw_aff(mpa1, 0);
+	dom = isl_pw_aff_domain(pa);
+	pa = isl_multi_pw_aff_get_pw_aff(mpa1, 1);
+	dom2 = isl_pw_aff_domain(pa);
+	equal_domain = isl_set_is_equal(dom, dom2);
+
+	isl_set_free(dom);
+	isl_set_free(dom2);
+	isl_multi_pw_aff_free(mpa1);
+	isl_multi_pw_aff_free(mpa2);
+
+	if (equal < 0)
+		return -1;
+	if (!equal)
+		isl_die(ctx, isl_error_unknown,
+			"expressions not equal", return -1);
+
+	if (equal_domain < 0)
+		return -1;
+	if (equal_domain)
+		isl_die(ctx, isl_error_unknown,
+			"domains unexpectedly equal", return -1);
+
+	return 0;
+}
+
+/* This is a regression test for a bug where isl_basic_map_simplify
+ * would end up in an infinite loop.  In particular, we construct
+ * an empty basic set that is not obviously empty.
+ * isl_basic_set_is_empty marks the basic set as empty.
+ * After projecting out i3, the variable can be dropped completely,
+ * but isl_basic_map_simplify refrains from doing so if the basic set
+ * is empty and would end up in an infinite loop if it didn't test
+ * explicitly for empty basic maps in the outer loop.
+ */
+static int test_simplify(isl_ctx *ctx)
+{
+	const char *str;
+	isl_basic_set *bset;
+	int empty;
+
+	str = "{ [i0, i1, i2, i3] : i0 >= -2 and 6i2 <= 4 + i0 + 5i1 and "
+		"i2 <= 22 and 75i2 <= 111 + 13i0 + 60i1 and "
+		"25i2 >= 38 + 6i0 + 20i1 and i0 <= -1 and i2 >= 20 and "
+		"i3 >= i2 }";
+	bset = isl_basic_set_read_from_str(ctx, str);
+	empty = isl_basic_set_is_empty(bset);
+	bset = isl_basic_set_project_out(bset, isl_dim_set, 3, 1);
+	isl_basic_set_free(bset);
+	if (!bset)
+		return -1;
+	if (!empty)
+		isl_die(ctx, isl_error_unknown,
+			"basic set should be empty", return -1);
+
+	return 0;
+}
+
+/* This is a regression test for a bug where isl_tab_basic_map_partial_lexopt
+ * with gbr context would fail to disable the use of the shifted tableau
+ * when transferring equalities for the input to the context, resulting
+ * in invalid sample values.
+ */
+static int test_partial_lexmin(isl_ctx *ctx)
+{
+	const char *str;
+	isl_basic_set *bset;
+	isl_basic_map *bmap;
+	isl_map *map;
+
+	str = "{ [1, b, c, 1 - c] -> [e] : 2e <= -c and 2e >= -3 + c }";
+	bmap = isl_basic_map_read_from_str(ctx, str);
+	str = "{ [a, b, c, d] : c <= 1 and 2d >= 6 - 4b - c }";
+	bset = isl_basic_set_read_from_str(ctx, str);
+	map = isl_basic_map_partial_lexmin(bmap, bset, NULL);
+	isl_map_free(map);
+
+	if (!map)
+		return -1;
+
+	return 0;
+}
+
+/* Check that the variable compression performed on the existentially
+ * quantified variables inside isl_basic_set_compute_divs is not confused
+ * by the implicit equalities among the parameters.
+ */
+static int test_compute_divs(isl_ctx *ctx)
+{
+	const char *str;
+	isl_basic_set *bset;
+	isl_set *set;
+
+	str = "[a, b, c, d, e] -> { [] : exists (e0: 2d = b and a <= 124 and "
+		"b <= 2046 and b >= 0 and b <= 60 + 64a and 2e >= b + 2c and "
+		"2e >= b and 2e <= 1 + b and 2e <= 1 + b + 2c and "
+		"32768e0 >= -124 + a and 2097152e0 <= 60 + 64a - b) }";
+	bset = isl_basic_set_read_from_str(ctx, str);
+	set = isl_basic_set_compute_divs(bset);
+	isl_set_free(set);
+	if (!set)
+		return -1;
+
+	return 0;
+}
+
+struct {
+	const char *set;
+	const char *dual;
+} coef_tests[] = {
+	{ "{ rat: [i] : 0 <= i <= 10 }",
+	  "{ rat: coefficients[[cst] -> [a]] : cst >= 0 and 10a + cst >= 0 }" },
+	{ "{ rat: [i] : FALSE }",
+	  "{ rat: coefficients[[cst] -> [a]] }" },
+	{ "{ rat: [i] : }",
+	  "{ rat: coefficients[[cst] -> [0]] : cst >= 0 }" },
+};
+
+struct {
+	const char *set;
+	const char *dual;
+} sol_tests[] = {
+	{ "{ rat: coefficients[[cst] -> [a]] : cst >= 0 and 10a + cst >= 0 }",
+	  "{ rat: [i] : 0 <= i <= 10 }" },
+	{ "{ rat: coefficients[[cst] -> [a]] : FALSE }",
+	  "{ rat: [i] }" },
+	{ "{ rat: coefficients[[cst] -> [a]] }",
+	  "{ rat: [i] : FALSE }" },
+};
+
+/* Test the basic functionality of isl_basic_set_coefficients and
+ * isl_basic_set_solutions.
+ */
+static int test_dual(isl_ctx *ctx)
+{
+	int i;
+
+	for (i = 0; i < ARRAY_SIZE(coef_tests); ++i) {
+		int equal;
+		isl_basic_set *bset1, *bset2;
+
+		bset1 = isl_basic_set_read_from_str(ctx, coef_tests[i].set);
+		bset2 = isl_basic_set_read_from_str(ctx, coef_tests[i].dual);
+		bset1 = isl_basic_set_coefficients(bset1);
+		equal = isl_basic_set_is_equal(bset1, bset2);
+		isl_basic_set_free(bset1);
+		isl_basic_set_free(bset2);
+		if (equal < 0)
+			return -1;
+		if (!equal)
+			isl_die(ctx, isl_error_unknown,
+				"incorrect dual", return -1);
+	}
+
+	for (i = 0; i < ARRAY_SIZE(sol_tests); ++i) {
+		int equal;
+		isl_basic_set *bset1, *bset2;
+
+		bset1 = isl_basic_set_read_from_str(ctx, sol_tests[i].set);
+		bset2 = isl_basic_set_read_from_str(ctx, sol_tests[i].dual);
+		bset1 = isl_basic_set_solutions(bset1);
+		equal = isl_basic_set_is_equal(bset1, bset2);
+		isl_basic_set_free(bset1);
+		isl_basic_set_free(bset2);
+		if (equal < 0)
+			return -1;
+		if (!equal)
+			isl_die(ctx, isl_error_unknown,
+				"incorrect dual", return -1);
+	}
+
+	return 0;
+}
+
+struct {
+	const char *name;
+	int (*fn)(isl_ctx *ctx);
+} tests [] = {
+	{ "dual", &test_dual },
+	{ "dependence analysis", &test_flow },
+	{ "val", &test_val },
+	{ "compute divs", &test_compute_divs },
+	{ "partial lexmin", &test_partial_lexmin },
+	{ "simplify", &test_simplify },
+	{ "curry", &test_curry },
+	{ "piecewise multi affine expressions", &test_pw_multi_aff },
+	{ "multi piecewise affine expressions", &test_multi_pw_aff },
+	{ "conversion", &test_conversion },
+	{ "list", &test_list },
+	{ "align parameters", &test_align_parameters },
+	{ "preimage", &test_preimage },
+	{ "pullback", &test_pullback },
+	{ "AST", &test_ast },
+	{ "AST build", &test_ast_build },
+	{ "AST generation", &test_ast_gen },
+	{ "eliminate", &test_eliminate },
+	{ "residue class", &test_residue_class },
+	{ "div", &test_div },
+	{ "slice", &test_slice },
+	{ "fixed power", &test_fixed_power },
+	{ "sample", &test_sample },
+	{ "output", &test_output },
+	{ "vertices", &test_vertices },
+	{ "fixed", &test_fixed },
+	{ "equal", &test_equal },
+	{ "disjoint", &test_disjoint },
+	{ "product", &test_product },
+	{ "dim_max", &test_dim_max },
+	{ "affine", &test_aff },
+	{ "injective", &test_injective },
+	{ "schedule", &test_schedule },
+	{ "union_pw", &test_union_pw },
+	{ "parse", &test_parse },
+	{ "single-valued", &test_sv },
+	{ "affine hull", &test_affine_hull },
+	{ "coalesce", &test_coalesce },
+	{ "factorize", &test_factorize },
+	{ "subset", &test_subset },
+	{ "subtract", &test_subtract },
+	{ "lexmin", &test_lexmin },
+	{ "min", &test_min },
+	{ "gist", &test_gist },
+	{ "piecewise quasi-polynomials", &test_pwqp },
+};
+
+int main(int argc, char **argv)
+{
+	int i;
+	struct isl_ctx *ctx;
+	struct isl_options *options;
+
+	srcdir = getenv("srcdir");
+	assert(srcdir);
+
+	options = isl_options_new_with_defaults();
+	assert(options);
+	argc = isl_options_parse(options, argc, argv, ISL_ARG_ALL);
+
+	ctx = isl_ctx_alloc_with_options(&isl_options_args, options);
+	for (i = 0; i < ARRAY_SIZE(tests); ++i) {
+		printf("%s\n", tests[i].name);
+		if (tests[i].fn(ctx) < 0)
+			goto error;
+	}
+	test_lift(ctx);
+	test_bound(ctx);
+	test_union(ctx);
+	test_split_periods(ctx);
+	test_lex(ctx);
+	test_bijective(ctx);
+	test_dep(ctx);
+	test_read(ctx);
+	test_bounded(ctx);
+	test_construction(ctx);
+	test_dim(ctx);
+	test_application(ctx);
+	test_convex_hull(ctx);
+	test_closure(ctx);
+	isl_ctx_free(ctx);
+	return 0;
+error:
+	isl_ctx_free(ctx);
+	return -1;
+}
diff --git a/polly/lib/External/isl/isl_transitive_closure.c b/polly/lib/External/isl/isl_transitive_closure.c
new file mode 100644
index 00000000000..d91a58f3787
--- /dev/null
+++ b/polly/lib/External/isl/isl_transitive_closure.c
@@ -0,0 +1,2981 @@
+/*
+ * Copyright 2010      INRIA Saclay
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France 
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_map_private.h>
+#include <isl/map.h>
+#include <isl_seq.h>
+#include <isl_space_private.h>
+#include <isl_lp_private.h>
+#include <isl/union_map.h>
+#include <isl_mat_private.h>
+#include <isl_vec_private.h>
+#include <isl_options_private.h>
+#include <isl_tarjan.h>
+
+int isl_map_is_transitively_closed(__isl_keep isl_map *map)
+{
+	isl_map *map2;
+	int closed;
+
+	map2 = isl_map_apply_range(isl_map_copy(map), isl_map_copy(map));
+	closed = isl_map_is_subset(map2, map);
+	isl_map_free(map2);
+
+	return closed;
+}
+
+int isl_union_map_is_transitively_closed(__isl_keep isl_union_map *umap)
+{
+	isl_union_map *umap2;
+	int closed;
+
+	umap2 = isl_union_map_apply_range(isl_union_map_copy(umap),
+					  isl_union_map_copy(umap));
+	closed = isl_union_map_is_subset(umap2, umap);
+	isl_union_map_free(umap2);
+
+	return closed;
+}
+ 
+/* Given a map that represents a path with the length of the path
+ * encoded as the difference between the last output coordindate
+ * and the last input coordinate, set this length to either
+ * exactly "length" (if "exactly" is set) or at least "length"
+ * (if "exactly" is not set).
+ */
+static __isl_give isl_map *set_path_length(__isl_take isl_map *map,
+	int exactly, int length)
+{
+	isl_space *dim;
+	struct isl_basic_map *bmap;
+	unsigned d;
+	unsigned nparam;
+	int k;
+	isl_int *c;
+
+	if (!map)
+		return NULL;
+
+	dim = isl_map_get_space(map);
+	d = isl_space_dim(dim, isl_dim_in);
+	nparam = isl_space_dim(dim, isl_dim_param);
+	bmap = isl_basic_map_alloc_space(dim, 0, 1, 1);
+	if (exactly) {
+		k = isl_basic_map_alloc_equality(bmap);
+		c = bmap->eq[k];
+	} else {
+		k = isl_basic_map_alloc_inequality(bmap);
+		c = bmap->ineq[k];
+	}
+	if (k < 0)
+		goto error;
+	isl_seq_clr(c, 1 + isl_basic_map_total_dim(bmap));
+	isl_int_set_si(c[0], -length);
+	isl_int_set_si(c[1 + nparam + d - 1], -1);
+	isl_int_set_si(c[1 + nparam + d + d - 1], 1);
+
+	bmap = isl_basic_map_finalize(bmap);
+	map = isl_map_intersect(map, isl_map_from_basic_map(bmap));
+
+	return map;
+error:
+	isl_basic_map_free(bmap);
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Check whether the overapproximation of the power of "map" is exactly
+ * the power of "map".  Let R be "map" and A_k the overapproximation.
+ * The approximation is exact if
+ *
+ *	A_1 = R
+ *	A_k = A_{k-1} \circ R			k >= 2
+ *
+ * Since A_k is known to be an overapproximation, we only need to check
+ *
+ *	A_1 \subset R
+ *	A_k \subset A_{k-1} \circ R		k >= 2
+ *
+ * In practice, "app" has an extra input and output coordinate
+ * to encode the length of the path.  So, we first need to add
+ * this coordinate to "map" and set the length of the path to
+ * one.
+ */
+static int check_power_exactness(__isl_take isl_map *map,
+	__isl_take isl_map *app)
+{
+	int exact;
+	isl_map *app_1;
+	isl_map *app_2;
+
+	map = isl_map_add_dims(map, isl_dim_in, 1);
+	map = isl_map_add_dims(map, isl_dim_out, 1);
+	map = set_path_length(map, 1, 1);
+
+	app_1 = set_path_length(isl_map_copy(app), 1, 1);
+
+	exact = isl_map_is_subset(app_1, map);
+	isl_map_free(app_1);
+
+	if (!exact || exact < 0) {
+		isl_map_free(app);
+		isl_map_free(map);
+		return exact;
+	}
+
+	app_1 = set_path_length(isl_map_copy(app), 0, 1);
+	app_2 = set_path_length(app, 0, 2);
+	app_1 = isl_map_apply_range(map, app_1);
+
+	exact = isl_map_is_subset(app_2, app_1);
+
+	isl_map_free(app_1);
+	isl_map_free(app_2);
+
+	return exact;
+}
+
+/* Check whether the overapproximation of the power of "map" is exactly
+ * the power of "map", possibly after projecting out the power (if "project"
+ * is set).
+ *
+ * If "project" is set and if "steps" can only result in acyclic paths,
+ * then we check
+ *
+ *	A = R \cup (A \circ R)
+ *
+ * where A is the overapproximation with the power projected out, i.e.,
+ * an overapproximation of the transitive closure.
+ * More specifically, since A is known to be an overapproximation, we check
+ *
+ *	A \subset R \cup (A \circ R)
+ *
+ * Otherwise, we check if the power is exact.
+ *
+ * Note that "app" has an extra input and output coordinate to encode
+ * the length of the part.  If we are only interested in the transitive
+ * closure, then we can simply project out these coordinates first.
+ */
+static int check_exactness(__isl_take isl_map *map, __isl_take isl_map *app,
+	int project)
+{
+	isl_map *test;
+	int exact;
+	unsigned d;
+
+	if (!project)
+		return check_power_exactness(map, app);
+
+	d = isl_map_dim(map, isl_dim_in);
+	app = set_path_length(app, 0, 1);
+	app = isl_map_project_out(app, isl_dim_in, d, 1);
+	app = isl_map_project_out(app, isl_dim_out, d, 1);
+
+	app = isl_map_reset_space(app, isl_map_get_space(map));
+
+	test = isl_map_apply_range(isl_map_copy(map), isl_map_copy(app));
+	test = isl_map_union(test, isl_map_copy(map));
+
+	exact = isl_map_is_subset(app, test);
+
+	isl_map_free(app);
+	isl_map_free(test);
+
+	isl_map_free(map);
+
+	return exact;
+}
+
+/*
+ * The transitive closure implementation is based on the paper
+ * "Computing the Transitive Closure of a Union of Affine Integer
+ * Tuple Relations" by Anna Beletska, Denis Barthou, Wlodzimierz Bielecki and
+ * Albert Cohen.
+ */
+
+/* Given a set of n offsets v_i (the rows of "steps"), construct a relation
+ * of the given dimension specification (Z^{n+1} -> Z^{n+1})
+ * that maps an element x to any element that can be reached
+ * by taking a non-negative number of steps along any of
+ * the extended offsets v'_i = [v_i 1].
+ * That is, construct
+ *
+ * { [x] -> [y] : exists k_i >= 0, y = x + \sum_i k_i v'_i }
+ *
+ * For any element in this relation, the number of steps taken
+ * is equal to the difference in the final coordinates.
+ */
+static __isl_give isl_map *path_along_steps(__isl_take isl_space *dim,
+	__isl_keep isl_mat *steps)
+{
+	int i, j, k;
+	struct isl_basic_map *path = NULL;
+	unsigned d;
+	unsigned n;
+	unsigned nparam;
+
+	if (!dim || !steps)
+		goto error;
+
+	d = isl_space_dim(dim, isl_dim_in);
+	n = steps->n_row;
+	nparam = isl_space_dim(dim, isl_dim_param);
+
+	path = isl_basic_map_alloc_space(isl_space_copy(dim), n, d, n);
+
+	for (i = 0; i < n; ++i) {
+		k = isl_basic_map_alloc_div(path);
+		if (k < 0)
+			goto error;
+		isl_assert(steps->ctx, i == k, goto error);
+		isl_int_set_si(path->div[k][0], 0);
+	}
+
+	for (i = 0; i < d; ++i) {
+		k = isl_basic_map_alloc_equality(path);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(path->eq[k], 1 + isl_basic_map_total_dim(path));
+		isl_int_set_si(path->eq[k][1 + nparam + i], 1);
+		isl_int_set_si(path->eq[k][1 + nparam + d + i], -1);
+		if (i == d - 1)
+			for (j = 0; j < n; ++j)
+				isl_int_set_si(path->eq[k][1 + nparam + 2 * d + j], 1);
+		else
+			for (j = 0; j < n; ++j)
+				isl_int_set(path->eq[k][1 + nparam + 2 * d + j],
+					    steps->row[j][i]);
+	}
+
+	for (i = 0; i < n; ++i) {
+		k = isl_basic_map_alloc_inequality(path);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(path->ineq[k], 1 + isl_basic_map_total_dim(path));
+		isl_int_set_si(path->ineq[k][1 + nparam + 2 * d + i], 1);
+	}
+
+	isl_space_free(dim);
+
+	path = isl_basic_map_simplify(path);
+	path = isl_basic_map_finalize(path);
+	return isl_map_from_basic_map(path);
+error:
+	isl_space_free(dim);
+	isl_basic_map_free(path);
+	return NULL;
+}
+
+#define IMPURE		0
+#define PURE_PARAM	1
+#define PURE_VAR	2
+#define MIXED		3
+
+/* Check whether the parametric constant term of constraint c is never
+ * positive in "bset".
+ */
+static int parametric_constant_never_positive(__isl_keep isl_basic_set *bset,
+	isl_int *c, int *div_purity)
+{
+	unsigned d;
+	unsigned n_div;
+	unsigned nparam;
+	int i;
+	int k;
+	int empty;
+
+	n_div = isl_basic_set_dim(bset, isl_dim_div);
+	d = isl_basic_set_dim(bset, isl_dim_set);
+	nparam = isl_basic_set_dim(bset, isl_dim_param);
+
+	bset = isl_basic_set_copy(bset);
+	bset = isl_basic_set_cow(bset);
+	bset = isl_basic_set_extend_constraints(bset, 0, 1);
+	k = isl_basic_set_alloc_inequality(bset);
+	if (k < 0)
+		goto error;
+	isl_seq_clr(bset->ineq[k], 1 + isl_basic_set_total_dim(bset));
+	isl_seq_cpy(bset->ineq[k], c, 1 + nparam);
+	for (i = 0; i < n_div; ++i) {
+		if (div_purity[i] != PURE_PARAM)
+			continue;
+		isl_int_set(bset->ineq[k][1 + nparam + d + i],
+			    c[1 + nparam + d + i]);
+	}
+	isl_int_sub_ui(bset->ineq[k][0], bset->ineq[k][0], 1);
+	empty = isl_basic_set_is_empty(bset);
+	isl_basic_set_free(bset);
+
+	return empty;
+error:
+	isl_basic_set_free(bset);
+	return -1;
+}
+
+/* Return PURE_PARAM if only the coefficients of the parameters are non-zero.
+ * Return PURE_VAR if only the coefficients of the set variables are non-zero.
+ * Return MIXED if only the coefficients of the parameters and the set
+ * 	variables are non-zero and if moreover the parametric constant
+ * 	can never attain positive values.
+ * Return IMPURE otherwise.
+ */
+static int purity(__isl_keep isl_basic_set *bset, isl_int *c, int *div_purity,
+	int eq)
+{
+	unsigned d;
+	unsigned n_div;
+	unsigned nparam;
+	int empty;
+	int i;
+	int p = 0, v = 0;
+
+	n_div = isl_basic_set_dim(bset, isl_dim_div);
+	d = isl_basic_set_dim(bset, isl_dim_set);
+	nparam = isl_basic_set_dim(bset, isl_dim_param);
+
+	for (i = 0; i < n_div; ++i) {
+		if (isl_int_is_zero(c[1 + nparam + d + i]))
+			continue;
+		switch (div_purity[i]) {
+		case PURE_PARAM: p = 1; break;
+		case PURE_VAR: v = 1; break;
+		default: return IMPURE;
+		}
+	}
+	if (!p && isl_seq_first_non_zero(c + 1, nparam) == -1)
+		return PURE_VAR;
+	if (!v && isl_seq_first_non_zero(c + 1 + nparam, d) == -1)
+		return PURE_PARAM;
+
+	empty = parametric_constant_never_positive(bset, c, div_purity);
+	if (eq && empty >= 0 && !empty) {
+		isl_seq_neg(c, c, 1 + nparam + d + n_div);
+		empty = parametric_constant_never_positive(bset, c, div_purity);
+	}
+
+	return empty < 0 ? -1 : empty ? MIXED : IMPURE;
+}
+
+/* Return an array of integers indicating the type of each div in bset.
+ * If the div is (recursively) defined in terms of only the parameters,
+ * then the type is PURE_PARAM.
+ * If the div is (recursively) defined in terms of only the set variables,
+ * then the type is PURE_VAR.
+ * Otherwise, the type is IMPURE.
+ */
+static __isl_give int *get_div_purity(__isl_keep isl_basic_set *bset)
+{
+	int i, j;
+	int *div_purity;
+	unsigned d;
+	unsigned n_div;
+	unsigned nparam;
+
+	if (!bset)
+		return NULL;
+
+	n_div = isl_basic_set_dim(bset, isl_dim_div);
+	d = isl_basic_set_dim(bset, isl_dim_set);
+	nparam = isl_basic_set_dim(bset, isl_dim_param);
+
+	div_purity = isl_alloc_array(bset->ctx, int, n_div);
+	if (n_div && !div_purity)
+		return NULL;
+
+	for (i = 0; i < bset->n_div; ++i) {
+		int p = 0, v = 0;
+		if (isl_int_is_zero(bset->div[i][0])) {
+			div_purity[i] = IMPURE;
+			continue;
+		}
+		if (isl_seq_first_non_zero(bset->div[i] + 2, nparam) != -1)
+			p = 1;
+		if (isl_seq_first_non_zero(bset->div[i] + 2 + nparam, d) != -1)
+			v = 1;
+		for (j = 0; j < i; ++j) {
+			if (isl_int_is_zero(bset->div[i][2 + nparam + d + j]))
+				continue;
+			switch (div_purity[j]) {
+			case PURE_PARAM: p = 1; break;
+			case PURE_VAR: v = 1; break;
+			default: p = v = 1; break;
+			}
+		}
+		div_purity[i] = v ? p ? IMPURE : PURE_VAR : PURE_PARAM;
+	}
+
+	return div_purity;
+}
+
+/* Given a path with the as yet unconstrained length at position "pos",
+ * check if setting the length to zero results in only the identity
+ * mapping.
+ */
+static int empty_path_is_identity(__isl_keep isl_basic_map *path, unsigned pos)
+{
+	isl_basic_map *test = NULL;
+	isl_basic_map *id = NULL;
+	int k;
+	int is_id;
+
+	test = isl_basic_map_copy(path);
+	test = isl_basic_map_extend_constraints(test, 1, 0);
+	k = isl_basic_map_alloc_equality(test);
+	if (k < 0)
+		goto error;
+	isl_seq_clr(test->eq[k], 1 + isl_basic_map_total_dim(test));
+	isl_int_set_si(test->eq[k][pos], 1);
+	id = isl_basic_map_identity(isl_basic_map_get_space(path));
+	is_id = isl_basic_map_is_equal(test, id);
+	isl_basic_map_free(test);
+	isl_basic_map_free(id);
+	return is_id;
+error:
+	isl_basic_map_free(test);
+	return -1;
+}
+
+/* If any of the constraints is found to be impure then this function
+ * sets *impurity to 1.
+ *
+ * If impurity is NULL then we are dealing with a non-parametric set
+ * and so the constraints are obviously PURE_VAR.
+ */
+static __isl_give isl_basic_map *add_delta_constraints(
+	__isl_take isl_basic_map *path,
+	__isl_keep isl_basic_set *delta, unsigned off, unsigned nparam,
+	unsigned d, int *div_purity, int eq, int *impurity)
+{
+	int i, k;
+	int n = eq ? delta->n_eq : delta->n_ineq;
+	isl_int **delta_c = eq ? delta->eq : delta->ineq;
+	unsigned n_div;
+
+	n_div = isl_basic_set_dim(delta, isl_dim_div);
+
+	for (i = 0; i < n; ++i) {
+		isl_int *path_c;
+		int p = PURE_VAR;
+		if (impurity)
+			p = purity(delta, delta_c[i], div_purity, eq);
+		if (p < 0)
+			goto error;
+		if (p != PURE_VAR && p != PURE_PARAM && !*impurity)
+			*impurity = 1;
+		if (p == IMPURE)
+			continue;
+		if (eq && p != MIXED) {
+			k = isl_basic_map_alloc_equality(path);
+			path_c = path->eq[k];
+		} else {
+			k = isl_basic_map_alloc_inequality(path);
+			path_c = path->ineq[k];
+		}
+		if (k < 0)
+			goto error;
+		isl_seq_clr(path_c, 1 + isl_basic_map_total_dim(path));
+		if (p == PURE_VAR) {
+			isl_seq_cpy(path_c + off,
+				    delta_c[i] + 1 + nparam, d);
+			isl_int_set(path_c[off + d], delta_c[i][0]);
+		} else if (p == PURE_PARAM) {
+			isl_seq_cpy(path_c, delta_c[i], 1 + nparam);
+		} else {
+			isl_seq_cpy(path_c + off,
+				    delta_c[i] + 1 + nparam, d);
+			isl_seq_cpy(path_c, delta_c[i], 1 + nparam);
+		}
+		isl_seq_cpy(path_c + off - n_div,
+			    delta_c[i] + 1 + nparam + d, n_div);
+	}
+
+	return path;
+error:
+	isl_basic_map_free(path);
+	return NULL;
+}
+
+/* Given a set of offsets "delta", construct a relation of the
+ * given dimension specification (Z^{n+1} -> Z^{n+1}) that
+ * is an overapproximation of the relations that
+ * maps an element x to any element that can be reached
+ * by taking a non-negative number of steps along any of
+ * the elements in "delta".
+ * That is, construct an approximation of
+ *
+ *	{ [x] -> [y] : exists f \in \delta, k \in Z :
+ *					y = x + k [f, 1] and k >= 0 }
+ *
+ * For any element in this relation, the number of steps taken
+ * is equal to the difference in the final coordinates.
+ *
+ * In particular, let delta be defined as
+ *
+ *	\delta = [p] -> { [x] : A x + a >= 0 and B p + b >= 0 and
+ *				C x + C'p + c >= 0 and
+ *				D x + D'p + d >= 0 }
+ *
+ * where the constraints C x + C'p + c >= 0 are such that the parametric
+ * constant term of each constraint j, "C_j x + C'_j p + c_j",
+ * can never attain positive values, then the relation is constructed as
+ *
+ *	{ [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
+ *			A f + k a >= 0 and B p + b >= 0 and
+ *			C f + C'p + c >= 0 and k >= 1 }
+ *	union { [x] -> [x] }
+ *
+ * If the zero-length paths happen to correspond exactly to the identity
+ * mapping, then we return
+ *
+ *	{ [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
+ *			A f + k a >= 0 and B p + b >= 0 and
+ *			C f + C'p + c >= 0 and k >= 0 }
+ *
+ * instead.
+ *
+ * Existentially quantified variables in \delta are handled by
+ * classifying them as independent of the parameters, purely
+ * parameter dependent and others.  Constraints containing
+ * any of the other existentially quantified variables are removed.
+ * This is safe, but leads to an additional overapproximation.
+ *
+ * If there are any impure constraints, then we also eliminate
+ * the parameters from \delta, resulting in a set
+ *
+ *	\delta' = { [x] : E x + e >= 0 }
+ *
+ * and add the constraints
+ *
+ *			E f + k e >= 0
+ *
+ * to the constructed relation.
+ */
+static __isl_give isl_map *path_along_delta(__isl_take isl_space *dim,
+	__isl_take isl_basic_set *delta)
+{
+	isl_basic_map *path = NULL;
+	unsigned d;
+	unsigned n_div;
+	unsigned nparam;
+	unsigned off;
+	int i, k;
+	int is_id;
+	int *div_purity = NULL;
+	int impurity = 0;
+
+	if (!delta)
+		goto error;
+	n_div = isl_basic_set_dim(delta, isl_dim_div);
+	d = isl_basic_set_dim(delta, isl_dim_set);
+	nparam = isl_basic_set_dim(delta, isl_dim_param);
+	path = isl_basic_map_alloc_space(isl_space_copy(dim), n_div + d + 1,
+			d + 1 + delta->n_eq, delta->n_eq + delta->n_ineq + 1);
+	off = 1 + nparam + 2 * (d + 1) + n_div;
+
+	for (i = 0; i < n_div + d + 1; ++i) {
+		k = isl_basic_map_alloc_div(path);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(path->div[k][0], 0);
+	}
+
+	for (i = 0; i < d + 1; ++i) {
+		k = isl_basic_map_alloc_equality(path);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(path->eq[k], 1 + isl_basic_map_total_dim(path));
+		isl_int_set_si(path->eq[k][1 + nparam + i], 1);
+		isl_int_set_si(path->eq[k][1 + nparam + d + 1 + i], -1);
+		isl_int_set_si(path->eq[k][off + i], 1);
+	}
+
+	div_purity = get_div_purity(delta);
+	if (n_div && !div_purity)
+		goto error;
+
+	path = add_delta_constraints(path, delta, off, nparam, d,
+				     div_purity, 1, &impurity);
+	path = add_delta_constraints(path, delta, off, nparam, d,
+				     div_purity, 0, &impurity);
+	if (impurity) {
+		isl_space *dim = isl_basic_set_get_space(delta);
+		delta = isl_basic_set_project_out(delta,
+						  isl_dim_param, 0, nparam);
+		delta = isl_basic_set_add_dims(delta, isl_dim_param, nparam);
+		delta = isl_basic_set_reset_space(delta, dim);
+		if (!delta)
+			goto error;
+		path = isl_basic_map_extend_constraints(path, delta->n_eq,
+							delta->n_ineq + 1);
+		path = add_delta_constraints(path, delta, off, nparam, d,
+					     NULL, 1, NULL);
+		path = add_delta_constraints(path, delta, off, nparam, d,
+					     NULL, 0, NULL);
+		path = isl_basic_map_gauss(path, NULL);
+	}
+
+	is_id = empty_path_is_identity(path, off + d);
+	if (is_id < 0)
+		goto error;
+
+	k = isl_basic_map_alloc_inequality(path);
+	if (k < 0)
+		goto error;
+	isl_seq_clr(path->ineq[k], 1 + isl_basic_map_total_dim(path));
+	if (!is_id)
+		isl_int_set_si(path->ineq[k][0], -1);
+	isl_int_set_si(path->ineq[k][off + d], 1);
+			
+	free(div_purity);
+	isl_basic_set_free(delta);
+	path = isl_basic_map_finalize(path);
+	if (is_id) {
+		isl_space_free(dim);
+		return isl_map_from_basic_map(path);
+	}
+	return isl_basic_map_union(path, isl_basic_map_identity(dim));
+error:
+	free(div_purity);
+	isl_space_free(dim);
+	isl_basic_set_free(delta);
+	isl_basic_map_free(path);
+	return NULL;
+}
+
+/* Given a dimension specification Z^{n+1} -> Z^{n+1} and a parameter "param",
+ * construct a map that equates the parameter to the difference
+ * in the final coordinates and imposes that this difference is positive.
+ * That is, construct
+ *
+ *	{ [x,x_s] -> [y,y_s] : k = y_s - x_s > 0 }
+ */
+static __isl_give isl_map *equate_parameter_to_length(__isl_take isl_space *dim,
+	unsigned param)
+{
+	struct isl_basic_map *bmap;
+	unsigned d;
+	unsigned nparam;
+	int k;
+
+	d = isl_space_dim(dim, isl_dim_in);
+	nparam = isl_space_dim(dim, isl_dim_param);
+	bmap = isl_basic_map_alloc_space(dim, 0, 1, 1);
+	k = isl_basic_map_alloc_equality(bmap);
+	if (k < 0)
+		goto error;
+	isl_seq_clr(bmap->eq[k], 1 + isl_basic_map_total_dim(bmap));
+	isl_int_set_si(bmap->eq[k][1 + param], -1);
+	isl_int_set_si(bmap->eq[k][1 + nparam + d - 1], -1);
+	isl_int_set_si(bmap->eq[k][1 + nparam + d + d - 1], 1);
+
+	k = isl_basic_map_alloc_inequality(bmap);
+	if (k < 0)
+		goto error;
+	isl_seq_clr(bmap->ineq[k], 1 + isl_basic_map_total_dim(bmap));
+	isl_int_set_si(bmap->ineq[k][1 + param], 1);
+	isl_int_set_si(bmap->ineq[k][0], -1);
+
+	bmap = isl_basic_map_finalize(bmap);
+	return isl_map_from_basic_map(bmap);
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Check whether "path" is acyclic, where the last coordinates of domain
+ * and range of path encode the number of steps taken.
+ * That is, check whether
+ *
+ *	{ d | d = y - x and (x,y) in path }
+ *
+ * does not contain any element with positive last coordinate (positive length)
+ * and zero remaining coordinates (cycle).
+ */
+static int is_acyclic(__isl_take isl_map *path)
+{
+	int i;
+	int acyclic;
+	unsigned dim;
+	struct isl_set *delta;
+
+	delta = isl_map_deltas(path);
+	dim = isl_set_dim(delta, isl_dim_set);
+	for (i = 0; i < dim; ++i) {
+		if (i == dim -1)
+			delta = isl_set_lower_bound_si(delta, isl_dim_set, i, 1);
+		else
+			delta = isl_set_fix_si(delta, isl_dim_set, i, 0);
+	}
+
+	acyclic = isl_set_is_empty(delta);
+	isl_set_free(delta);
+
+	return acyclic;
+}
+
+/* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
+ * and a dimension specification (Z^{n+1} -> Z^{n+1}),
+ * construct a map that is an overapproximation of the map
+ * that takes an element from the space D \times Z to another
+ * element from the same space, such that the first n coordinates of the
+ * difference between them is a sum of differences between images
+ * and pre-images in one of the R_i and such that the last coordinate
+ * is equal to the number of steps taken.
+ * That is, let
+ *
+ *	\Delta_i = { y - x | (x, y) in R_i }
+ *
+ * then the constructed map is an overapproximation of
+ *
+ *	{ (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
+ *				d = (\sum_i k_i \delta_i, \sum_i k_i) }
+ *
+ * The elements of the singleton \Delta_i's are collected as the
+ * rows of the steps matrix.  For all these \Delta_i's together,
+ * a single path is constructed.
+ * For each of the other \Delta_i's, we compute an overapproximation
+ * of the paths along elements of \Delta_i.
+ * Since each of these paths performs an addition, composition is
+ * symmetric and we can simply compose all resulting paths in any order.
+ */
+static __isl_give isl_map *construct_extended_path(__isl_take isl_space *dim,
+	__isl_keep isl_map *map, int *project)
+{
+	struct isl_mat *steps = NULL;
+	struct isl_map *path = NULL;
+	unsigned d;
+	int i, j, n;
+
+	d = isl_map_dim(map, isl_dim_in);
+
+	path = isl_map_identity(isl_space_copy(dim));
+
+	steps = isl_mat_alloc(map->ctx, map->n, d);
+	if (!steps)
+		goto error;
+
+	n = 0;
+	for (i = 0; i < map->n; ++i) {
+		struct isl_basic_set *delta;
+
+		delta = isl_basic_map_deltas(isl_basic_map_copy(map->p[i]));
+
+		for (j = 0; j < d; ++j) {
+			int fixed;
+
+			fixed = isl_basic_set_plain_dim_is_fixed(delta, j,
+							    &steps->row[n][j]);
+			if (fixed < 0) {
+				isl_basic_set_free(delta);
+				goto error;
+			}
+			if (!fixed)
+				break;
+		}
+
+
+		if (j < d) {
+			path = isl_map_apply_range(path,
+				path_along_delta(isl_space_copy(dim), delta));
+			path = isl_map_coalesce(path);
+		} else {
+			isl_basic_set_free(delta);
+			++n;
+		}
+	}
+
+	if (n > 0) {
+		steps->n_row = n;
+		path = isl_map_apply_range(path,
+				path_along_steps(isl_space_copy(dim), steps));
+	}
+
+	if (project && *project) {
+		*project = is_acyclic(isl_map_copy(path));
+		if (*project < 0)
+			goto error;
+	}
+
+	isl_space_free(dim);
+	isl_mat_free(steps);
+	return path;
+error:
+	isl_space_free(dim);
+	isl_mat_free(steps);
+	isl_map_free(path);
+	return NULL;
+}
+
+static int isl_set_overlaps(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
+{
+	isl_set *i;
+	int no_overlap;
+
+	if (!isl_space_tuple_is_equal(set1->dim, isl_dim_set,
+					set2->dim, isl_dim_set))
+		return 0;
+
+	i = isl_set_intersect(isl_set_copy(set1), isl_set_copy(set2));
+	no_overlap = isl_set_is_empty(i);
+	isl_set_free(i);
+
+	return no_overlap < 0 ? -1 : !no_overlap;
+}
+
+/* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
+ * and a dimension specification (Z^{n+1} -> Z^{n+1}),
+ * construct a map that is an overapproximation of the map
+ * that takes an element from the dom R \times Z to an
+ * element from ran R \times Z, such that the first n coordinates of the
+ * difference between them is a sum of differences between images
+ * and pre-images in one of the R_i and such that the last coordinate
+ * is equal to the number of steps taken.
+ * That is, let
+ *
+ *	\Delta_i = { y - x | (x, y) in R_i }
+ *
+ * then the constructed map is an overapproximation of
+ *
+ *	{ (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
+ *				d = (\sum_i k_i \delta_i, \sum_i k_i) and
+ *				x in dom R and x + d in ran R and
+ *				\sum_i k_i >= 1 }
+ */
+static __isl_give isl_map *construct_component(__isl_take isl_space *dim,
+	__isl_keep isl_map *map, int *exact, int project)
+{
+	struct isl_set *domain = NULL;
+	struct isl_set *range = NULL;
+	struct isl_map *app = NULL;
+	struct isl_map *path = NULL;
+
+	domain = isl_map_domain(isl_map_copy(map));
+	domain = isl_set_coalesce(domain);
+	range = isl_map_range(isl_map_copy(map));
+	range = isl_set_coalesce(range);
+	if (!isl_set_overlaps(domain, range)) {
+		isl_set_free(domain);
+		isl_set_free(range);
+		isl_space_free(dim);
+
+		map = isl_map_copy(map);
+		map = isl_map_add_dims(map, isl_dim_in, 1);
+		map = isl_map_add_dims(map, isl_dim_out, 1);
+		map = set_path_length(map, 1, 1);
+		return map;
+	}
+	app = isl_map_from_domain_and_range(domain, range);
+	app = isl_map_add_dims(app, isl_dim_in, 1);
+	app = isl_map_add_dims(app, isl_dim_out, 1);
+
+	path = construct_extended_path(isl_space_copy(dim), map,
+					exact && *exact ? &project : NULL);
+	app = isl_map_intersect(app, path);
+
+	if (exact && *exact &&
+	    (*exact = check_exactness(isl_map_copy(map), isl_map_copy(app),
+				      project)) < 0)
+		goto error;
+
+	isl_space_free(dim);
+	app = set_path_length(app, 0, 1);
+	return app;
+error:
+	isl_space_free(dim);
+	isl_map_free(app);
+	return NULL;
+}
+
+/* Call construct_component and, if "project" is set, project out
+ * the final coordinates.
+ */
+static __isl_give isl_map *construct_projected_component(
+	__isl_take isl_space *dim,
+	__isl_keep isl_map *map, int *exact, int project)
+{
+	isl_map *app;
+	unsigned d;
+
+	if (!dim)
+		return NULL;
+	d = isl_space_dim(dim, isl_dim_in);
+
+	app = construct_component(dim, map, exact, project);
+	if (project) {
+		app = isl_map_project_out(app, isl_dim_in, d - 1, 1);
+		app = isl_map_project_out(app, isl_dim_out, d - 1, 1);
+	}
+	return app;
+}
+
+/* Compute an extended version, i.e., with path lengths, of
+ * an overapproximation of the transitive closure of "bmap"
+ * with path lengths greater than or equal to zero and with
+ * domain and range equal to "dom".
+ */
+static __isl_give isl_map *q_closure(__isl_take isl_space *dim,
+	__isl_take isl_set *dom, __isl_keep isl_basic_map *bmap, int *exact)
+{
+	int project = 1;
+	isl_map *path;
+	isl_map *map;
+	isl_map *app;
+
+	dom = isl_set_add_dims(dom, isl_dim_set, 1);
+	app = isl_map_from_domain_and_range(dom, isl_set_copy(dom));
+	map = isl_map_from_basic_map(isl_basic_map_copy(bmap));
+	path = construct_extended_path(dim, map, &project);
+	app = isl_map_intersect(app, path);
+
+	if ((*exact = check_exactness(map, isl_map_copy(app), project)) < 0)
+		goto error;
+
+	return app;
+error:
+	isl_map_free(app);
+	return NULL;
+}
+
+/* Check whether qc has any elements of length at least one
+ * with domain and/or range outside of dom and ran.
+ */
+static int has_spurious_elements(__isl_keep isl_map *qc,
+	__isl_keep isl_set *dom, __isl_keep isl_set *ran)
+{
+	isl_set *s;
+	int subset;
+	unsigned d;
+
+	if (!qc || !dom || !ran)
+		return -1;
+
+	d = isl_map_dim(qc, isl_dim_in);
+
+	qc = isl_map_copy(qc);
+	qc = set_path_length(qc, 0, 1);
+	qc = isl_map_project_out(qc, isl_dim_in, d - 1, 1);
+	qc = isl_map_project_out(qc, isl_dim_out, d - 1, 1);
+
+	s = isl_map_domain(isl_map_copy(qc));
+	subset = isl_set_is_subset(s, dom);
+	isl_set_free(s);
+	if (subset < 0)
+		goto error;
+	if (!subset) {
+		isl_map_free(qc);
+		return 1;
+	}
+
+	s = isl_map_range(qc);
+	subset = isl_set_is_subset(s, ran);
+	isl_set_free(s);
+
+	return subset < 0 ? -1 : !subset;
+error:
+	isl_map_free(qc);
+	return -1;
+}
+
+#define LEFT	2
+#define RIGHT	1
+
+/* For each basic map in "map", except i, check whether it combines
+ * with the transitive closure that is reflexive on C combines
+ * to the left and to the right.
+ *
+ * In particular, if
+ *
+ *	dom map_j \subseteq C
+ *
+ * then right[j] is set to 1.  Otherwise, if
+ *
+ *	ran map_i \cap dom map_j = \emptyset
+ *
+ * then right[j] is set to 0.  Otherwise, composing to the right
+ * is impossible.
+ *
+ * Similar, for composing to the left, we have if
+ *
+ *	ran map_j \subseteq C
+ *
+ * then left[j] is set to 1.  Otherwise, if
+ *
+ *	dom map_i \cap ran map_j = \emptyset
+ *
+ * then left[j] is set to 0.  Otherwise, composing to the left
+ * is impossible.
+ *
+ * The return value is or'd with LEFT if composing to the left
+ * is possible and with RIGHT if composing to the right is possible.
+ */
+static int composability(__isl_keep isl_set *C, int i,
+	isl_set **dom, isl_set **ran, int *left, int *right,
+	__isl_keep isl_map *map)
+{
+	int j;
+	int ok;
+
+	ok = LEFT | RIGHT;
+	for (j = 0; j < map->n && ok; ++j) {
+		int overlaps, subset;
+		if (j == i)
+			continue;
+
+		if (ok & RIGHT) {
+			if (!dom[j])
+				dom[j] = isl_set_from_basic_set(
+					isl_basic_map_domain(
+						isl_basic_map_copy(map->p[j])));
+			if (!dom[j])
+				return -1;
+			overlaps = isl_set_overlaps(ran[i], dom[j]);
+			if (overlaps < 0)
+				return -1;
+			if (!overlaps)
+				right[j] = 0;
+			else {
+				subset = isl_set_is_subset(dom[j], C);
+				if (subset < 0)
+					return -1;
+				if (subset)
+					right[j] = 1;
+				else
+					ok &= ~RIGHT;
+			}
+		}
+
+		if (ok & LEFT) {
+			if (!ran[j])
+				ran[j] = isl_set_from_basic_set(
+					isl_basic_map_range(
+						isl_basic_map_copy(map->p[j])));
+			if (!ran[j])
+				return -1;
+			overlaps = isl_set_overlaps(dom[i], ran[j]);
+			if (overlaps < 0)
+				return -1;
+			if (!overlaps)
+				left[j] = 0;
+			else {
+				subset = isl_set_is_subset(ran[j], C);
+				if (subset < 0)
+					return -1;
+				if (subset)
+					left[j] = 1;
+				else
+					ok &= ~LEFT;
+			}
+		}
+	}
+
+	return ok;
+}
+
+static __isl_give isl_map *anonymize(__isl_take isl_map *map)
+{
+	map = isl_map_reset(map, isl_dim_in);
+	map = isl_map_reset(map, isl_dim_out);
+	return map;
+}
+
+/* Return a map that is a union of the basic maps in "map", except i,
+ * composed to left and right with qc based on the entries of "left"
+ * and "right".
+ */
+static __isl_give isl_map *compose(__isl_keep isl_map *map, int i,
+	__isl_take isl_map *qc, int *left, int *right)
+{
+	int j;
+	isl_map *comp;
+
+	comp = isl_map_empty(isl_map_get_space(map));
+	for (j = 0; j < map->n; ++j) {
+		isl_map *map_j;
+
+		if (j == i)
+			continue;
+
+		map_j = isl_map_from_basic_map(isl_basic_map_copy(map->p[j]));
+		map_j = anonymize(map_j);
+		if (left && left[j])
+			map_j = isl_map_apply_range(map_j, isl_map_copy(qc));
+		if (right && right[j])
+			map_j = isl_map_apply_range(isl_map_copy(qc), map_j);
+		comp = isl_map_union(comp, map_j);
+	}
+
+	comp = isl_map_compute_divs(comp);
+	comp = isl_map_coalesce(comp);
+
+	isl_map_free(qc);
+
+	return comp;
+}
+
+/* Compute the transitive closure of "map" incrementally by
+ * computing
+ *
+ *	map_i^+ \cup qc^+
+ *
+ * or
+ *
+ *	map_i^+ \cup ((id \cup map_i^) \circ qc^+)
+ *
+ * or
+ *
+ *	map_i^+ \cup (qc^+ \circ (id \cup map_i^))
+ *
+ * depending on whether left or right are NULL.
+ */
+static __isl_give isl_map *compute_incremental(
+	__isl_take isl_space *dim, __isl_keep isl_map *map,
+	int i, __isl_take isl_map *qc, int *left, int *right, int *exact)
+{
+	isl_map *map_i;
+	isl_map *tc;
+	isl_map *rtc = NULL;
+
+	if (!map)
+		goto error;
+	isl_assert(map->ctx, left || right, goto error);
+
+	map_i = isl_map_from_basic_map(isl_basic_map_copy(map->p[i]));
+	tc = construct_projected_component(isl_space_copy(dim), map_i,
+						exact, 1);
+	isl_map_free(map_i);
+
+	if (*exact)
+		qc = isl_map_transitive_closure(qc, exact);
+
+	if (!*exact) {
+		isl_space_free(dim);
+		isl_map_free(tc);
+		isl_map_free(qc);
+		return isl_map_universe(isl_map_get_space(map));
+	}
+
+	if (!left || !right)
+		rtc = isl_map_union(isl_map_copy(tc),
+				    isl_map_identity(isl_map_get_space(tc)));
+	if (!right)
+		qc = isl_map_apply_range(rtc, qc);
+	if (!left)
+		qc = isl_map_apply_range(qc, rtc);
+	qc = isl_map_union(tc, qc);
+
+	isl_space_free(dim);
+
+	return qc;
+error:
+	isl_space_free(dim);
+	isl_map_free(qc);
+	return NULL;
+}
+
+/* Given a map "map", try to find a basic map such that
+ * map^+ can be computed as
+ *
+ * map^+ = map_i^+ \cup
+ *    \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
+ *
+ * with C the simple hull of the domain and range of the input map.
+ * map_i^ \cup Id_C is computed by allowing the path lengths to be zero
+ * and by intersecting domain and range with C.
+ * Of course, we need to check that this is actually equal to map_i^ \cup Id_C.
+ * Also, we only use the incremental computation if all the transitive
+ * closures are exact and if the number of basic maps in the union,
+ * after computing the integer divisions, is smaller than the number
+ * of basic maps in the input map.
+ */
+static int incemental_on_entire_domain(__isl_keep isl_space *dim,
+	__isl_keep isl_map *map,
+	isl_set **dom, isl_set **ran, int *left, int *right,
+	__isl_give isl_map **res)
+{
+	int i;
+	isl_set *C;
+	unsigned d;
+
+	*res = NULL;
+
+	C = isl_set_union(isl_map_domain(isl_map_copy(map)),
+			  isl_map_range(isl_map_copy(map)));
+	C = isl_set_from_basic_set(isl_set_simple_hull(C));
+	if (!C)
+		return -1;
+	if (C->n != 1) {
+		isl_set_free(C);
+		return 0;
+	}
+
+	d = isl_map_dim(map, isl_dim_in);
+
+	for (i = 0; i < map->n; ++i) {
+		isl_map *qc;
+		int exact_i, spurious;
+		int j;
+		dom[i] = isl_set_from_basic_set(isl_basic_map_domain(
+					isl_basic_map_copy(map->p[i])));
+		ran[i] = isl_set_from_basic_set(isl_basic_map_range(
+					isl_basic_map_copy(map->p[i])));
+		qc = q_closure(isl_space_copy(dim), isl_set_copy(C),
+				map->p[i], &exact_i);
+		if (!qc)
+			goto error;
+		if (!exact_i) {
+			isl_map_free(qc);
+			continue;
+		}
+		spurious = has_spurious_elements(qc, dom[i], ran[i]);
+		if (spurious) {
+			isl_map_free(qc);
+			if (spurious < 0)
+				goto error;
+			continue;
+		}
+		qc = isl_map_project_out(qc, isl_dim_in, d, 1);
+		qc = isl_map_project_out(qc, isl_dim_out, d, 1);
+		qc = isl_map_compute_divs(qc);
+		for (j = 0; j < map->n; ++j)
+			left[j] = right[j] = 1;
+		qc = compose(map, i, qc, left, right);
+		if (!qc)
+			goto error;
+		if (qc->n >= map->n) {
+			isl_map_free(qc);
+			continue;
+		}
+		*res = compute_incremental(isl_space_copy(dim), map, i, qc,
+				left, right, &exact_i);
+		if (!*res)
+			goto error;
+		if (exact_i)
+			break;
+		isl_map_free(*res);
+		*res = NULL;
+	}
+
+	isl_set_free(C);
+
+	return *res != NULL;
+error:
+	isl_set_free(C);
+	return -1;
+}
+
+/* Try and compute the transitive closure of "map" as
+ *
+ * map^+ = map_i^+ \cup
+ *    \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
+ *
+ * with C either the simple hull of the domain and range of the entire
+ * map or the simple hull of domain and range of map_i.
+ */
+static __isl_give isl_map *incremental_closure(__isl_take isl_space *dim,
+	__isl_keep isl_map *map, int *exact, int project)
+{
+	int i;
+	isl_set **dom = NULL;
+	isl_set **ran = NULL;
+	int *left = NULL;
+	int *right = NULL;
+	isl_set *C;
+	unsigned d;
+	isl_map *res = NULL;
+
+	if (!project)
+		return construct_projected_component(dim, map, exact, project);
+
+	if (!map)
+		goto error;
+	if (map->n <= 1)
+		return construct_projected_component(dim, map, exact, project);
+
+	d = isl_map_dim(map, isl_dim_in);
+
+	dom = isl_calloc_array(map->ctx, isl_set *, map->n);
+	ran = isl_calloc_array(map->ctx, isl_set *, map->n);
+	left = isl_calloc_array(map->ctx, int, map->n);
+	right = isl_calloc_array(map->ctx, int, map->n);
+	if (!ran || !dom || !left || !right)
+		goto error;
+
+	if (incemental_on_entire_domain(dim, map, dom, ran, left, right, &res) < 0)
+		goto error;
+
+	for (i = 0; !res && i < map->n; ++i) {
+		isl_map *qc;
+		int exact_i, spurious, comp;
+		if (!dom[i])
+			dom[i] = isl_set_from_basic_set(
+					isl_basic_map_domain(
+						isl_basic_map_copy(map->p[i])));
+		if (!dom[i])
+			goto error;
+		if (!ran[i])
+			ran[i] = isl_set_from_basic_set(
+					isl_basic_map_range(
+						isl_basic_map_copy(map->p[i])));
+		if (!ran[i])
+			goto error;
+		C = isl_set_union(isl_set_copy(dom[i]),
+				      isl_set_copy(ran[i]));
+		C = isl_set_from_basic_set(isl_set_simple_hull(C));
+		if (!C)
+			goto error;
+		if (C->n != 1) {
+			isl_set_free(C);
+			continue;
+		}
+		comp = composability(C, i, dom, ran, left, right, map);
+		if (!comp || comp < 0) {
+			isl_set_free(C);
+			if (comp < 0)
+				goto error;
+			continue;
+		}
+		qc = q_closure(isl_space_copy(dim), C, map->p[i], &exact_i);
+		if (!qc)
+			goto error;
+		if (!exact_i) {
+			isl_map_free(qc);
+			continue;
+		}
+		spurious = has_spurious_elements(qc, dom[i], ran[i]);
+		if (spurious) {
+			isl_map_free(qc);
+			if (spurious < 0)
+				goto error;
+			continue;
+		}
+		qc = isl_map_project_out(qc, isl_dim_in, d, 1);
+		qc = isl_map_project_out(qc, isl_dim_out, d, 1);
+		qc = isl_map_compute_divs(qc);
+		qc = compose(map, i, qc, (comp & LEFT) ? left : NULL,
+				(comp & RIGHT) ? right : NULL);
+		if (!qc)
+			goto error;
+		if (qc->n >= map->n) {
+			isl_map_free(qc);
+			continue;
+		}
+		res = compute_incremental(isl_space_copy(dim), map, i, qc,
+				(comp & LEFT) ? left : NULL,
+				(comp & RIGHT) ? right : NULL, &exact_i);
+		if (!res)
+			goto error;
+		if (exact_i)
+			break;
+		isl_map_free(res);
+		res = NULL;
+	}
+
+	for (i = 0; i < map->n; ++i) {
+		isl_set_free(dom[i]);
+		isl_set_free(ran[i]);
+	}
+	free(dom);
+	free(ran);
+	free(left);
+	free(right);
+
+	if (res) {
+		isl_space_free(dim);
+		return res;
+	}
+
+	return construct_projected_component(dim, map, exact, project);
+error:
+	if (dom)
+		for (i = 0; i < map->n; ++i)
+			isl_set_free(dom[i]);
+	free(dom);
+	if (ran)
+		for (i = 0; i < map->n; ++i)
+			isl_set_free(ran[i]);
+	free(ran);
+	free(left);
+	free(right);
+	isl_space_free(dim);
+	return NULL;
+}
+
+/* Given an array of sets "set", add "dom" at position "pos"
+ * and search for elements at earlier positions that overlap with "dom".
+ * If any can be found, then merge all of them, together with "dom", into
+ * a single set and assign the union to the first in the array,
+ * which becomes the new group leader for all groups involved in the merge.
+ * During the search, we only consider group leaders, i.e., those with
+ * group[i] = i, as the other sets have already been combined
+ * with one of the group leaders.
+ */
+static int merge(isl_set **set, int *group, __isl_take isl_set *dom, int pos)
+{
+	int i;
+
+	group[pos] = pos;
+	set[pos] = isl_set_copy(dom);
+
+	for (i = pos - 1; i >= 0; --i) {
+		int o;
+
+		if (group[i] != i)
+			continue;
+
+		o = isl_set_overlaps(set[i], dom);
+		if (o < 0)
+			goto error;
+		if (!o)
+			continue;
+
+		set[i] = isl_set_union(set[i], set[group[pos]]);
+		set[group[pos]] = NULL;
+		if (!set[i])
+			goto error;
+		group[group[pos]] = i;
+		group[pos] = i;
+	}
+
+	isl_set_free(dom);
+	return 0;
+error:
+	isl_set_free(dom);
+	return -1;
+}
+
+/* Replace each entry in the n by n grid of maps by the cross product
+ * with the relation { [i] -> [i + 1] }.
+ */
+static int add_length(__isl_keep isl_map *map, isl_map ***grid, int n)
+{
+	int i, j, k;
+	isl_space *dim;
+	isl_basic_map *bstep;
+	isl_map *step;
+	unsigned nparam;
+
+	if (!map)
+		return -1;
+
+	dim = isl_map_get_space(map);
+	nparam = isl_space_dim(dim, isl_dim_param);
+	dim = isl_space_drop_dims(dim, isl_dim_in, 0, isl_space_dim(dim, isl_dim_in));
+	dim = isl_space_drop_dims(dim, isl_dim_out, 0, isl_space_dim(dim, isl_dim_out));
+	dim = isl_space_add_dims(dim, isl_dim_in, 1);
+	dim = isl_space_add_dims(dim, isl_dim_out, 1);
+	bstep = isl_basic_map_alloc_space(dim, 0, 1, 0);
+	k = isl_basic_map_alloc_equality(bstep);
+	if (k < 0) {
+		isl_basic_map_free(bstep);
+		return -1;
+	}
+	isl_seq_clr(bstep->eq[k], 1 + isl_basic_map_total_dim(bstep));
+	isl_int_set_si(bstep->eq[k][0], 1);
+	isl_int_set_si(bstep->eq[k][1 + nparam], 1);
+	isl_int_set_si(bstep->eq[k][1 + nparam + 1], -1);
+	bstep = isl_basic_map_finalize(bstep);
+	step = isl_map_from_basic_map(bstep);
+
+	for (i = 0; i < n; ++i)
+		for (j = 0; j < n; ++j)
+			grid[i][j] = isl_map_product(grid[i][j],
+						     isl_map_copy(step));
+
+	isl_map_free(step);
+
+	return 0;
+}
+
+/* The core of the Floyd-Warshall algorithm.
+ * Updates the given n x x matrix of relations in place.
+ *
+ * The algorithm iterates over all vertices.  In each step, the whole
+ * matrix is updated to include all paths that go to the current vertex,
+ * possibly stay there a while (including passing through earlier vertices)
+ * and then come back.  At the start of each iteration, the diagonal
+ * element corresponding to the current vertex is replaced by its
+ * transitive closure to account for all indirect paths that stay
+ * in the current vertex.
+ */
+static void floyd_warshall_iterate(isl_map ***grid, int n, int *exact)
+{
+	int r, p, q;
+
+	for (r = 0; r < n; ++r) {
+		int r_exact;
+		grid[r][r] = isl_map_transitive_closure(grid[r][r],
+				(exact && *exact) ? &r_exact : NULL);
+		if (exact && *exact && !r_exact)
+			*exact = 0;
+
+		for (p = 0; p < n; ++p)
+			for (q = 0; q < n; ++q) {
+				isl_map *loop;
+				if (p == r && q == r)
+					continue;
+				loop = isl_map_apply_range(
+						isl_map_copy(grid[p][r]),
+						isl_map_copy(grid[r][q]));
+				grid[p][q] = isl_map_union(grid[p][q], loop);
+				loop = isl_map_apply_range(
+						isl_map_copy(grid[p][r]),
+					isl_map_apply_range(
+						isl_map_copy(grid[r][r]),
+						isl_map_copy(grid[r][q])));
+				grid[p][q] = isl_map_union(grid[p][q], loop);
+				grid[p][q] = isl_map_coalesce(grid[p][q]);
+			}
+	}
+}
+
+/* Given a partition of the domains and ranges of the basic maps in "map",
+ * apply the Floyd-Warshall algorithm with the elements in the partition
+ * as vertices.
+ *
+ * In particular, there are "n" elements in the partition and "group" is
+ * an array of length 2 * map->n with entries in [0,n-1].
+ *
+ * We first construct a matrix of relations based on the partition information,
+ * apply Floyd-Warshall on this matrix of relations and then take the
+ * union of all entries in the matrix as the final result.
+ *
+ * If we are actually computing the power instead of the transitive closure,
+ * i.e., when "project" is not set, then the result should have the
+ * path lengths encoded as the difference between an extra pair of
+ * coordinates.  We therefore apply the nested transitive closures
+ * to relations that include these lengths.  In particular, we replace
+ * the input relation by the cross product with the unit length relation
+ * { [i] -> [i + 1] }.
+ */
+static __isl_give isl_map *floyd_warshall_with_groups(__isl_take isl_space *dim,
+	__isl_keep isl_map *map, int *exact, int project, int *group, int n)
+{
+	int i, j, k;
+	isl_map ***grid = NULL;
+	isl_map *app;
+
+	if (!map)
+		goto error;
+
+	if (n == 1) {
+		free(group);
+		return incremental_closure(dim, map, exact, project);
+	}
+
+	grid = isl_calloc_array(map->ctx, isl_map **, n);
+	if (!grid)
+		goto error;
+	for (i = 0; i < n; ++i) {
+		grid[i] = isl_calloc_array(map->ctx, isl_map *, n);
+		if (!grid[i])
+			goto error;
+		for (j = 0; j < n; ++j)
+			grid[i][j] = isl_map_empty(isl_map_get_space(map));
+	}
+
+	for (k = 0; k < map->n; ++k) {
+		i = group[2 * k];
+		j = group[2 * k + 1];
+		grid[i][j] = isl_map_union(grid[i][j],
+				isl_map_from_basic_map(
+					isl_basic_map_copy(map->p[k])));
+	}
+
+	if (!project && add_length(map, grid, n) < 0)
+		goto error;
+
+	floyd_warshall_iterate(grid, n, exact);
+
+	app = isl_map_empty(isl_map_get_space(map));
+
+	for (i = 0; i < n; ++i) {
+		for (j = 0; j < n; ++j)
+			app = isl_map_union(app, grid[i][j]);
+		free(grid[i]);
+	}
+	free(grid);
+
+	free(group);
+	isl_space_free(dim);
+
+	return app;
+error:
+	if (grid)
+		for (i = 0; i < n; ++i) {
+			if (!grid[i])
+				continue;
+			for (j = 0; j < n; ++j)
+				isl_map_free(grid[i][j]);
+			free(grid[i]);
+		}
+	free(grid);
+	free(group);
+	isl_space_free(dim);
+	return NULL;
+}
+
+/* Partition the domains and ranges of the n basic relations in list
+ * into disjoint cells.
+ *
+ * To find the partition, we simply consider all of the domains
+ * and ranges in turn and combine those that overlap.
+ * "set" contains the partition elements and "group" indicates
+ * to which partition element a given domain or range belongs.
+ * The domain of basic map i corresponds to element 2 * i in these arrays,
+ * while the domain corresponds to element 2 * i + 1.
+ * During the construction group[k] is either equal to k,
+ * in which case set[k] contains the union of all the domains and
+ * ranges in the corresponding group, or is equal to some l < k,
+ * with l another domain or range in the same group.
+ */
+static int *setup_groups(isl_ctx *ctx, __isl_keep isl_basic_map **list, int n,
+	isl_set ***set, int *n_group)
+{
+	int i;
+	int *group = NULL;
+	int g;
+
+	*set = isl_calloc_array(ctx, isl_set *, 2 * n);
+	group = isl_alloc_array(ctx, int, 2 * n);
+
+	if (!*set || !group)
+		goto error;
+
+	for (i = 0; i < n; ++i) {
+		isl_set *dom;
+		dom = isl_set_from_basic_set(isl_basic_map_domain(
+				isl_basic_map_copy(list[i])));
+		if (merge(*set, group, dom, 2 * i) < 0)
+			goto error;
+		dom = isl_set_from_basic_set(isl_basic_map_range(
+				isl_basic_map_copy(list[i])));
+		if (merge(*set, group, dom, 2 * i + 1) < 0)
+			goto error;
+	}
+
+	g = 0;
+	for (i = 0; i < 2 * n; ++i)
+		if (group[i] == i) {
+			if (g != i) {
+				(*set)[g] = (*set)[i];
+				(*set)[i] = NULL;
+			}
+			group[i] = g++;
+		} else
+			group[i] = group[group[i]];
+
+	*n_group = g;
+
+	return group;
+error:
+	if (*set) {
+		for (i = 0; i < 2 * n; ++i)
+			isl_set_free((*set)[i]);
+		free(*set);
+		*set = NULL;
+	}
+	free(group);
+	return NULL;
+}
+
+/* Check if the domains and ranges of the basic maps in "map" can
+ * be partitioned, and if so, apply Floyd-Warshall on the elements
+ * of the partition.  Note that we also apply this algorithm
+ * if we want to compute the power, i.e., when "project" is not set.
+ * However, the results are unlikely to be exact since the recursive
+ * calls inside the Floyd-Warshall algorithm typically result in
+ * non-linear path lengths quite quickly.
+ */
+static __isl_give isl_map *floyd_warshall(__isl_take isl_space *dim,
+	__isl_keep isl_map *map, int *exact, int project)
+{
+	int i;
+	isl_set **set = NULL;
+	int *group = NULL;
+	int n;
+
+	if (!map)
+		goto error;
+	if (map->n <= 1)
+		return incremental_closure(dim, map, exact, project);
+
+	group = setup_groups(map->ctx, map->p, map->n, &set, &n);
+	if (!group)
+		goto error;
+
+	for (i = 0; i < 2 * map->n; ++i)
+		isl_set_free(set[i]);
+
+	free(set);
+
+	return floyd_warshall_with_groups(dim, map, exact, project, group, n);
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+/* Structure for representing the nodes of the graph of which
+ * strongly connected components are being computed.
+ *
+ * list contains the actual nodes
+ * check_closed is set if we may have used the fact that
+ * a pair of basic maps can be interchanged
+ */
+struct isl_tc_follows_data {
+	isl_basic_map **list;
+	int check_closed;
+};
+
+/* Check whether in the computation of the transitive closure
+ * "list[i]" (R_1) should follow (or be part of the same component as)
+ * "list[j]" (R_2).
+ *
+ * That is check whether
+ *
+ *	R_1 \circ R_2
+ *
+ * is a subset of
+ *
+ *	R_2 \circ R_1
+ *
+ * If so, then there is no reason for R_1 to immediately follow R_2
+ * in any path.
+ *
+ * *check_closed is set if the subset relation holds while
+ * R_1 \circ R_2 is not empty.
+ */
+static int basic_map_follows(int i, int j, void *user)
+{
+	struct isl_tc_follows_data *data = user;
+	struct isl_map *map12 = NULL;
+	struct isl_map *map21 = NULL;
+	int subset;
+
+	if (!isl_space_tuple_is_equal(data->list[i]->dim, isl_dim_in,
+				    data->list[j]->dim, isl_dim_out))
+		return 0;
+
+	map21 = isl_map_from_basic_map(
+			isl_basic_map_apply_range(
+				isl_basic_map_copy(data->list[j]),
+				isl_basic_map_copy(data->list[i])));
+	subset = isl_map_is_empty(map21);
+	if (subset < 0)
+		goto error;
+	if (subset) {
+		isl_map_free(map21);
+		return 0;
+	}
+
+	if (!isl_space_tuple_is_equal(data->list[i]->dim, isl_dim_in,
+				    data->list[i]->dim, isl_dim_out) ||
+	    !isl_space_tuple_is_equal(data->list[j]->dim, isl_dim_in,
+				    data->list[j]->dim, isl_dim_out)) {
+		isl_map_free(map21);
+		return 1;
+	}
+
+	map12 = isl_map_from_basic_map(
+			isl_basic_map_apply_range(
+				isl_basic_map_copy(data->list[i]),
+				isl_basic_map_copy(data->list[j])));
+
+	subset = isl_map_is_subset(map21, map12);
+
+	isl_map_free(map12);
+	isl_map_free(map21);
+
+	if (subset)
+		data->check_closed = 1;
+
+	return subset < 0 ? -1 : !subset;
+error:
+	isl_map_free(map21);
+	return -1;
+}
+
+/* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
+ * and a dimension specification (Z^{n+1} -> Z^{n+1}),
+ * construct a map that is an overapproximation of the map
+ * that takes an element from the dom R \times Z to an
+ * element from ran R \times Z, such that the first n coordinates of the
+ * difference between them is a sum of differences between images
+ * and pre-images in one of the R_i and such that the last coordinate
+ * is equal to the number of steps taken.
+ * If "project" is set, then these final coordinates are not included,
+ * i.e., a relation of type Z^n -> Z^n is returned.
+ * That is, let
+ *
+ *	\Delta_i = { y - x | (x, y) in R_i }
+ *
+ * then the constructed map is an overapproximation of
+ *
+ *	{ (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
+ *				d = (\sum_i k_i \delta_i, \sum_i k_i) and
+ *				x in dom R and x + d in ran R }
+ *
+ * or
+ *
+ *	{ (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
+ *				d = (\sum_i k_i \delta_i) and
+ *				x in dom R and x + d in ran R }
+ *
+ * if "project" is set.
+ *
+ * We first split the map into strongly connected components, perform
+ * the above on each component and then join the results in the correct
+ * order, at each join also taking in the union of both arguments
+ * to allow for paths that do not go through one of the two arguments.
+ */
+static __isl_give isl_map *construct_power_components(__isl_take isl_space *dim,
+	__isl_keep isl_map *map, int *exact, int project)
+{
+	int i, n, c;
+	struct isl_map *path = NULL;
+	struct isl_tc_follows_data data;
+	struct isl_tarjan_graph *g = NULL;
+	int *orig_exact;
+	int local_exact;
+
+	if (!map)
+		goto error;
+	if (map->n <= 1)
+		return floyd_warshall(dim, map, exact, project);
+
+	data.list = map->p;
+	data.check_closed = 0;
+	g = isl_tarjan_graph_init(map->ctx, map->n, &basic_map_follows, &data);
+	if (!g)
+		goto error;
+
+	orig_exact = exact;
+	if (data.check_closed && !exact)
+		exact = &local_exact;
+
+	c = 0;
+	i = 0;
+	n = map->n;
+	if (project)
+		path = isl_map_empty(isl_map_get_space(map));
+	else
+		path = isl_map_empty(isl_space_copy(dim));
+	path = anonymize(path);
+	while (n) {
+		struct isl_map *comp;
+		isl_map *path_comp, *path_comb;
+		comp = isl_map_alloc_space(isl_map_get_space(map), n, 0);
+		while (g->order[i] != -1) {
+			comp = isl_map_add_basic_map(comp,
+				    isl_basic_map_copy(map->p[g->order[i]]));
+			--n;
+			++i;
+		}
+		path_comp = floyd_warshall(isl_space_copy(dim),
+						comp, exact, project);
+		path_comp = anonymize(path_comp);
+		path_comb = isl_map_apply_range(isl_map_copy(path),
+						isl_map_copy(path_comp));
+		path = isl_map_union(path, path_comp);
+		path = isl_map_union(path, path_comb);
+		isl_map_free(comp);
+		++i;
+		++c;
+	}
+
+	if (c > 1 && data.check_closed && !*exact) {
+		int closed;
+
+		closed = isl_map_is_transitively_closed(path);
+		if (closed < 0)
+			goto error;
+		if (!closed) {
+			isl_tarjan_graph_free(g);
+			isl_map_free(path);
+			return floyd_warshall(dim, map, orig_exact, project);
+		}
+	}
+
+	isl_tarjan_graph_free(g);
+	isl_space_free(dim);
+
+	return path;
+error:
+	isl_tarjan_graph_free(g);
+	isl_space_free(dim);
+	isl_map_free(path);
+	return NULL;
+}
+
+/* Given a union of basic maps R = \cup_i R_i \subseteq D \times D,
+ * construct a map that is an overapproximation of the map
+ * that takes an element from the space D to another
+ * element from the same space, such that the difference between
+ * them is a strictly positive sum of differences between images
+ * and pre-images in one of the R_i.
+ * The number of differences in the sum is equated to parameter "param".
+ * That is, let
+ *
+ *	\Delta_i = { y - x | (x, y) in R_i }
+ *
+ * then the constructed map is an overapproximation of
+ *
+ *	{ (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
+ *				d = \sum_i k_i \delta_i and k = \sum_i k_i > 0 }
+ * or
+ *
+ *	{ (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
+ *				d = \sum_i k_i \delta_i and \sum_i k_i > 0 }
+ *
+ * if "project" is set.
+ *
+ * If "project" is not set, then
+ * we construct an extended mapping with an extra coordinate
+ * that indicates the number of steps taken.  In particular,
+ * the difference in the last coordinate is equal to the number
+ * of steps taken to move from a domain element to the corresponding
+ * image element(s).
+ */
+static __isl_give isl_map *construct_power(__isl_keep isl_map *map,
+	int *exact, int project)
+{
+	struct isl_map *app = NULL;
+	isl_space *dim = NULL;
+	unsigned d;
+
+	if (!map)
+		return NULL;
+
+	dim = isl_map_get_space(map);
+
+	d = isl_space_dim(dim, isl_dim_in);
+	dim = isl_space_add_dims(dim, isl_dim_in, 1);
+	dim = isl_space_add_dims(dim, isl_dim_out, 1);
+
+	app = construct_power_components(isl_space_copy(dim), map,
+					exact, project);
+
+	isl_space_free(dim);
+
+	return app;
+}
+
+/* Compute the positive powers of "map", or an overapproximation.
+ * If the result is exact, then *exact is set to 1.
+ *
+ * If project is set, then we are actually interested in the transitive
+ * closure, so we can use a more relaxed exactness check.
+ * The lengths of the paths are also projected out instead of being
+ * encoded as the difference between an extra pair of final coordinates.
+ */
+static __isl_give isl_map *map_power(__isl_take isl_map *map,
+	int *exact, int project)
+{
+	struct isl_map *app = NULL;
+
+	if (exact)
+		*exact = 1;
+
+	if (!map)
+		return NULL;
+
+	isl_assert(map->ctx,
+		isl_map_dim(map, isl_dim_in) == isl_map_dim(map, isl_dim_out),
+		goto error);
+
+	app = construct_power(map, exact, project);
+
+	isl_map_free(map);
+	return app;
+error:
+	isl_map_free(map);
+	isl_map_free(app);
+	return NULL;
+}
+
+/* Compute the positive powers of "map", or an overapproximation.
+ * The result maps the exponent to a nested copy of the corresponding power.
+ * If the result is exact, then *exact is set to 1.
+ * map_power constructs an extended relation with the path lengths
+ * encoded as the difference between the final coordinates.
+ * In the final step, this difference is equated to an extra parameter
+ * and made positive.  The extra coordinates are subsequently projected out
+ * and the parameter is turned into the domain of the result.
+ */
+__isl_give isl_map *isl_map_power(__isl_take isl_map *map, int *exact)
+{
+	isl_space *target_dim;
+	isl_space *dim;
+	isl_map *diff;
+	unsigned d;
+	unsigned param;
+
+	if (!map)
+		return NULL;
+
+	d = isl_map_dim(map, isl_dim_in);
+	param = isl_map_dim(map, isl_dim_param);
+
+	map = isl_map_compute_divs(map);
+	map = isl_map_coalesce(map);
+
+	if (isl_map_plain_is_empty(map)) {
+		map = isl_map_from_range(isl_map_wrap(map));
+		map = isl_map_add_dims(map, isl_dim_in, 1);
+		map = isl_map_set_dim_name(map, isl_dim_in, 0, "k");
+		return map;
+	}
+
+	target_dim = isl_map_get_space(map);
+	target_dim = isl_space_from_range(isl_space_wrap(target_dim));
+	target_dim = isl_space_add_dims(target_dim, isl_dim_in, 1);
+	target_dim = isl_space_set_dim_name(target_dim, isl_dim_in, 0, "k");
+
+	map = map_power(map, exact, 0);
+
+	map = isl_map_add_dims(map, isl_dim_param, 1);
+	dim = isl_map_get_space(map);
+	diff = equate_parameter_to_length(dim, param);
+	map = isl_map_intersect(map, diff);
+	map = isl_map_project_out(map, isl_dim_in, d, 1);
+	map = isl_map_project_out(map, isl_dim_out, d, 1);
+	map = isl_map_from_range(isl_map_wrap(map));
+	map = isl_map_move_dims(map, isl_dim_in, 0, isl_dim_param, param, 1);
+
+	map = isl_map_reset_space(map, target_dim);
+
+	return map;
+}
+
+/* Compute a relation that maps each element in the range of the input
+ * relation to the lengths of all paths composed of edges in the input
+ * relation that end up in the given range element.
+ * The result may be an overapproximation, in which case *exact is set to 0.
+ * The resulting relation is very similar to the power relation.
+ * The difference are that the domain has been projected out, the
+ * range has become the domain and the exponent is the range instead
+ * of a parameter.
+ */
+__isl_give isl_map *isl_map_reaching_path_lengths(__isl_take isl_map *map,
+	int *exact)
+{
+	isl_space *dim;
+	isl_map *diff;
+	unsigned d;
+	unsigned param;
+
+	if (!map)
+		return NULL;
+
+	d = isl_map_dim(map, isl_dim_in);
+	param = isl_map_dim(map, isl_dim_param);
+
+	map = isl_map_compute_divs(map);
+	map = isl_map_coalesce(map);
+
+	if (isl_map_plain_is_empty(map)) {
+		if (exact)
+			*exact = 1;
+		map = isl_map_project_out(map, isl_dim_out, 0, d);
+		map = isl_map_add_dims(map, isl_dim_out, 1);
+		return map;
+	}
+
+	map = map_power(map, exact, 0);
+
+	map = isl_map_add_dims(map, isl_dim_param, 1);
+	dim = isl_map_get_space(map);
+	diff = equate_parameter_to_length(dim, param);
+	map = isl_map_intersect(map, diff);
+	map = isl_map_project_out(map, isl_dim_in, 0, d + 1);
+	map = isl_map_project_out(map, isl_dim_out, d, 1);
+	map = isl_map_reverse(map);
+	map = isl_map_move_dims(map, isl_dim_out, 0, isl_dim_param, param, 1);
+
+	return map;
+}
+
+/* Check whether equality i of bset is a pure stride constraint
+ * on a single dimensions, i.e., of the form
+ *
+ *	v = k e
+ *
+ * with k a constant and e an existentially quantified variable.
+ */
+static int is_eq_stride(__isl_keep isl_basic_set *bset, int i)
+{
+	unsigned nparam;
+	unsigned d;
+	unsigned n_div;
+	int pos1;
+	int pos2;
+
+	if (!bset)
+		return -1;
+
+	if (!isl_int_is_zero(bset->eq[i][0]))
+		return 0;
+
+	nparam = isl_basic_set_dim(bset, isl_dim_param);
+	d = isl_basic_set_dim(bset, isl_dim_set);
+	n_div = isl_basic_set_dim(bset, isl_dim_div);
+
+	if (isl_seq_first_non_zero(bset->eq[i] + 1, nparam) != -1)
+		return 0;
+	pos1 = isl_seq_first_non_zero(bset->eq[i] + 1 + nparam, d);
+	if (pos1 == -1)
+		return 0;
+	if (isl_seq_first_non_zero(bset->eq[i] + 1 + nparam + pos1 + 1, 
+					d - pos1 - 1) != -1)
+		return 0;
+
+	pos2 = isl_seq_first_non_zero(bset->eq[i] + 1 + nparam + d, n_div);
+	if (pos2 == -1)
+		return 0;
+	if (isl_seq_first_non_zero(bset->eq[i] + 1 + nparam + d  + pos2 + 1,
+				   n_div - pos2 - 1) != -1)
+		return 0;
+	if (!isl_int_is_one(bset->eq[i][1 + nparam + pos1]) &&
+	    !isl_int_is_negone(bset->eq[i][1 + nparam + pos1]))
+		return 0;
+
+	return 1;
+}
+
+/* Given a map, compute the smallest superset of this map that is of the form
+ *
+ *	{ i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
+ *
+ * (where p ranges over the (non-parametric) dimensions),
+ * compute the transitive closure of this map, i.e.,
+ *
+ *	{ i -> j : exists k > 0:
+ *		k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
+ *
+ * and intersect domain and range of this transitive closure with
+ * the given domain and range.
+ *
+ * If with_id is set, then try to include as much of the identity mapping
+ * as possible, by computing
+ *
+ *	{ i -> j : exists k >= 0:
+ *		k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
+ *
+ * instead (i.e., allow k = 0).
+ *
+ * In practice, we compute the difference set
+ *
+ *	delta  = { j - i | i -> j in map },
+ *
+ * look for stride constraint on the individual dimensions and compute
+ * (constant) lower and upper bounds for each individual dimension,
+ * adding a constraint for each bound not equal to infinity.
+ */
+static __isl_give isl_map *box_closure_on_domain(__isl_take isl_map *map,
+	__isl_take isl_set *dom, __isl_take isl_set *ran, int with_id)
+{
+	int i;
+	int k;
+	unsigned d;
+	unsigned nparam;
+	unsigned total;
+	isl_space *dim;
+	isl_set *delta;
+	isl_map *app = NULL;
+	isl_basic_set *aff = NULL;
+	isl_basic_map *bmap = NULL;
+	isl_vec *obj = NULL;
+	isl_int opt;
+
+	isl_int_init(opt);
+
+	delta = isl_map_deltas(isl_map_copy(map));
+
+	aff = isl_set_affine_hull(isl_set_copy(delta));
+	if (!aff)
+		goto error;
+	dim = isl_map_get_space(map);
+	d = isl_space_dim(dim, isl_dim_in);
+	nparam = isl_space_dim(dim, isl_dim_param);
+	total = isl_space_dim(dim, isl_dim_all);
+	bmap = isl_basic_map_alloc_space(dim,
+					aff->n_div + 1, aff->n_div, 2 * d + 1);
+	for (i = 0; i < aff->n_div + 1; ++i) {
+		k = isl_basic_map_alloc_div(bmap);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(bmap->div[k][0], 0);
+	}
+	for (i = 0; i < aff->n_eq; ++i) {
+		if (!is_eq_stride(aff, i))
+			continue;
+		k = isl_basic_map_alloc_equality(bmap);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(bmap->eq[k], 1 + nparam);
+		isl_seq_cpy(bmap->eq[k] + 1 + nparam + d,
+				aff->eq[i] + 1 + nparam, d);
+		isl_seq_neg(bmap->eq[k] + 1 + nparam,
+				aff->eq[i] + 1 + nparam, d);
+		isl_seq_cpy(bmap->eq[k] + 1 + nparam + 2 * d,
+				aff->eq[i] + 1 + nparam + d, aff->n_div);
+		isl_int_set_si(bmap->eq[k][1 + total + aff->n_div], 0);
+	}
+	obj = isl_vec_alloc(map->ctx, 1 + nparam + d);
+	if (!obj)
+		goto error;
+	isl_seq_clr(obj->el, 1 + nparam + d);
+	for (i = 0; i < d; ++ i) {
+		enum isl_lp_result res;
+
+		isl_int_set_si(obj->el[1 + nparam + i], 1);
+
+		res = isl_set_solve_lp(delta, 0, obj->el, map->ctx->one, &opt,
+					NULL, NULL);
+		if (res == isl_lp_error)
+			goto error;
+		if (res == isl_lp_ok) {
+			k = isl_basic_map_alloc_inequality(bmap);
+			if (k < 0)
+				goto error;
+			isl_seq_clr(bmap->ineq[k],
+					1 + nparam + 2 * d + bmap->n_div);
+			isl_int_set_si(bmap->ineq[k][1 + nparam + i], -1);
+			isl_int_set_si(bmap->ineq[k][1 + nparam + d + i], 1);
+			isl_int_neg(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], opt);
+		}
+
+		res = isl_set_solve_lp(delta, 1, obj->el, map->ctx->one, &opt,
+					NULL, NULL);
+		if (res == isl_lp_error)
+			goto error;
+		if (res == isl_lp_ok) {
+			k = isl_basic_map_alloc_inequality(bmap);
+			if (k < 0)
+				goto error;
+			isl_seq_clr(bmap->ineq[k],
+					1 + nparam + 2 * d + bmap->n_div);
+			isl_int_set_si(bmap->ineq[k][1 + nparam + i], 1);
+			isl_int_set_si(bmap->ineq[k][1 + nparam + d + i], -1);
+			isl_int_set(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], opt);
+		}
+
+		isl_int_set_si(obj->el[1 + nparam + i], 0);
+	}
+	k = isl_basic_map_alloc_inequality(bmap);
+	if (k < 0)
+		goto error;
+	isl_seq_clr(bmap->ineq[k],
+			1 + nparam + 2 * d + bmap->n_div);
+	if (!with_id)
+		isl_int_set_si(bmap->ineq[k][0], -1);
+	isl_int_set_si(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], 1);
+
+	app = isl_map_from_domain_and_range(dom, ran);
+
+	isl_vec_free(obj);
+	isl_basic_set_free(aff);
+	isl_map_free(map);
+	bmap = isl_basic_map_finalize(bmap);
+	isl_set_free(delta);
+	isl_int_clear(opt);
+
+	map = isl_map_from_basic_map(bmap);
+	map = isl_map_intersect(map, app);
+
+	return map;
+error:
+	isl_vec_free(obj);
+	isl_basic_map_free(bmap);
+	isl_basic_set_free(aff);
+	isl_set_free(dom);
+	isl_set_free(ran);
+	isl_map_free(map);
+	isl_set_free(delta);
+	isl_int_clear(opt);
+	return NULL;
+}
+
+/* Given a map, compute the smallest superset of this map that is of the form
+ *
+ *	{ i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
+ *
+ * (where p ranges over the (non-parametric) dimensions),
+ * compute the transitive closure of this map, i.e.,
+ *
+ *	{ i -> j : exists k > 0:
+ *		k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
+ *
+ * and intersect domain and range of this transitive closure with
+ * domain and range of the original map.
+ */
+static __isl_give isl_map *box_closure(__isl_take isl_map *map)
+{
+	isl_set *domain;
+	isl_set *range;
+
+	domain = isl_map_domain(isl_map_copy(map));
+	domain = isl_set_coalesce(domain);
+	range = isl_map_range(isl_map_copy(map));
+	range = isl_set_coalesce(range);
+
+	return box_closure_on_domain(map, domain, range, 0);
+}
+
+/* Given a map, compute the smallest superset of this map that is of the form
+ *
+ *	{ i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
+ *
+ * (where p ranges over the (non-parametric) dimensions),
+ * compute the transitive and partially reflexive closure of this map, i.e.,
+ *
+ *	{ i -> j : exists k >= 0:
+ *		k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
+ *
+ * and intersect domain and range of this transitive closure with
+ * the given domain.
+ */
+static __isl_give isl_map *box_closure_with_identity(__isl_take isl_map *map,
+	__isl_take isl_set *dom)
+{
+	return box_closure_on_domain(map, dom, isl_set_copy(dom), 1);
+}
+
+/* Check whether app is the transitive closure of map.
+ * In particular, check that app is acyclic and, if so,
+ * check that
+ *
+ *	app \subset (map \cup (map \circ app))
+ */
+static int check_exactness_omega(__isl_keep isl_map *map,
+	__isl_keep isl_map *app)
+{
+	isl_set *delta;
+	int i;
+	int is_empty, is_exact;
+	unsigned d;
+	isl_map *test;
+
+	delta = isl_map_deltas(isl_map_copy(app));
+	d = isl_set_dim(delta, isl_dim_set);
+	for (i = 0; i < d; ++i)
+		delta = isl_set_fix_si(delta, isl_dim_set, i, 0);
+	is_empty = isl_set_is_empty(delta);
+	isl_set_free(delta);
+	if (is_empty < 0)
+		return -1;
+	if (!is_empty)
+		return 0;
+
+	test = isl_map_apply_range(isl_map_copy(app), isl_map_copy(map));
+	test = isl_map_union(test, isl_map_copy(map));
+	is_exact = isl_map_is_subset(app, test);
+	isl_map_free(test);
+
+	return is_exact;
+}
+
+/* Check if basic map M_i can be combined with all the other
+ * basic maps such that
+ *
+ *	(\cup_j M_j)^+
+ *
+ * can be computed as
+ *
+ *	M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
+ *
+ * In particular, check if we can compute a compact representation
+ * of
+ *
+ *		M_i^* \circ M_j \circ M_i^*
+ *
+ * for each j != i.
+ * Let M_i^? be an extension of M_i^+ that allows paths
+ * of length zero, i.e., the result of box_closure(., 1).
+ * The criterion, as proposed by Kelly et al., is that
+ * id = M_i^? - M_i^+ can be represented as a basic map
+ * and that
+ *
+ *	id \circ M_j \circ id = M_j
+ *
+ * for each j != i.
+ *
+ * If this function returns 1, then tc and qc are set to
+ * M_i^+ and M_i^?, respectively.
+ */
+static int can_be_split_off(__isl_keep isl_map *map, int i,
+	__isl_give isl_map **tc, __isl_give isl_map **qc)
+{
+	isl_map *map_i, *id = NULL;
+	int j = -1;
+	isl_set *C;
+
+	*tc = NULL;
+	*qc = NULL;
+
+	C = isl_set_union(isl_map_domain(isl_map_copy(map)),
+			  isl_map_range(isl_map_copy(map)));
+	C = isl_set_from_basic_set(isl_set_simple_hull(C));
+	if (!C)
+		goto error;
+
+	map_i = isl_map_from_basic_map(isl_basic_map_copy(map->p[i]));
+	*tc = box_closure(isl_map_copy(map_i));
+	*qc = box_closure_with_identity(map_i, C);
+	id = isl_map_subtract(isl_map_copy(*qc), isl_map_copy(*tc));
+
+	if (!id || !*qc)
+		goto error;
+	if (id->n != 1 || (*qc)->n != 1)
+		goto done;
+
+	for (j = 0; j < map->n; ++j) {
+		isl_map *map_j, *test;
+		int is_ok;
+
+		if (i == j)
+			continue;
+		map_j = isl_map_from_basic_map(
+					isl_basic_map_copy(map->p[j]));
+		test = isl_map_apply_range(isl_map_copy(id),
+						isl_map_copy(map_j));
+		test = isl_map_apply_range(test, isl_map_copy(id));
+		is_ok = isl_map_is_equal(test, map_j);
+		isl_map_free(map_j);
+		isl_map_free(test);
+		if (is_ok < 0)
+			goto error;
+		if (!is_ok)
+			break;
+	}
+
+done:
+	isl_map_free(id);
+	if (j == map->n)
+		return 1;
+
+	isl_map_free(*qc);
+	isl_map_free(*tc);
+	*qc = NULL;
+	*tc = NULL;
+
+	return 0;
+error:
+	isl_map_free(id);
+	isl_map_free(*qc);
+	isl_map_free(*tc);
+	*qc = NULL;
+	*tc = NULL;
+	return -1;
+}
+
+static __isl_give isl_map *box_closure_with_check(__isl_take isl_map *map,
+	int *exact)
+{
+	isl_map *app;
+
+	app = box_closure(isl_map_copy(map));
+	if (exact)
+		*exact = check_exactness_omega(map, app);
+
+	isl_map_free(map);
+	return app;
+}
+
+/* Compute an overapproximation of the transitive closure of "map"
+ * using a variation of the algorithm from
+ * "Transitive Closure of Infinite Graphs and its Applications"
+ * by Kelly et al.
+ *
+ * We first check whether we can can split of any basic map M_i and
+ * compute
+ *
+ *	(\cup_j M_j)^+
+ *
+ * as
+ *
+ *	M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
+ *
+ * using a recursive call on the remaining map.
+ *
+ * If not, we simply call box_closure on the whole map.
+ */
+static __isl_give isl_map *transitive_closure_omega(__isl_take isl_map *map,
+	int *exact)
+{
+	int i, j;
+	int exact_i;
+	isl_map *app;
+
+	if (!map)
+		return NULL;
+	if (map->n == 1)
+		return box_closure_with_check(map, exact);
+
+	for (i = 0; i < map->n; ++i) {
+		int ok;
+		isl_map *qc, *tc;
+		ok = can_be_split_off(map, i, &tc, &qc);
+		if (ok < 0)
+			goto error;
+		if (!ok)
+			continue;
+
+		app = isl_map_alloc_space(isl_map_get_space(map), map->n - 1, 0);
+
+		for (j = 0; j < map->n; ++j) {
+			if (j == i)
+				continue;
+			app = isl_map_add_basic_map(app,
+						isl_basic_map_copy(map->p[j]));
+		}
+
+		app = isl_map_apply_range(isl_map_copy(qc), app);
+		app = isl_map_apply_range(app, qc);
+
+		app = isl_map_union(tc, transitive_closure_omega(app, NULL));
+		exact_i = check_exactness_omega(map, app);
+		if (exact_i == 1) {
+			if (exact)
+				*exact = exact_i;
+			isl_map_free(map);
+			return app;
+		}
+		isl_map_free(app);
+		if (exact_i < 0)
+			goto error;
+	}
+
+	return box_closure_with_check(map, exact);
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+/* Compute the transitive closure  of "map", or an overapproximation.
+ * If the result is exact, then *exact is set to 1.
+ * Simply use map_power to compute the powers of map, but tell
+ * it to project out the lengths of the paths instead of equating
+ * the length to a parameter.
+ */
+__isl_give isl_map *isl_map_transitive_closure(__isl_take isl_map *map,
+	int *exact)
+{
+	isl_space *target_dim;
+	int closed;
+
+	if (!map)
+		goto error;
+
+	if (map->ctx->opt->closure == ISL_CLOSURE_BOX)
+		return transitive_closure_omega(map, exact);
+
+	map = isl_map_compute_divs(map);
+	map = isl_map_coalesce(map);
+	closed = isl_map_is_transitively_closed(map);
+	if (closed < 0)
+		goto error;
+	if (closed) {
+		if (exact)
+			*exact = 1;
+		return map;
+	}
+
+	target_dim = isl_map_get_space(map);
+	map = map_power(map, exact, 1);
+	map = isl_map_reset_space(map, target_dim);
+
+	return map;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+static int inc_count(__isl_take isl_map *map, void *user)
+{
+	int *n = user;
+
+	*n += map->n;
+
+	isl_map_free(map);
+
+	return 0;
+}
+
+static int collect_basic_map(__isl_take isl_map *map, void *user)
+{
+	int i;
+	isl_basic_map ***next = user;
+
+	for (i = 0; i < map->n; ++i) {
+		**next = isl_basic_map_copy(map->p[i]);
+		if (!**next)
+			goto error;
+		(*next)++;
+	}
+
+	isl_map_free(map);
+	return 0;
+error:
+	isl_map_free(map);
+	return -1;
+}
+
+/* Perform Floyd-Warshall on the given list of basic relations.
+ * The basic relations may live in different dimensions,
+ * but basic relations that get assigned to the diagonal of the
+ * grid have domains and ranges of the same dimension and so
+ * the standard algorithm can be used because the nested transitive
+ * closures are only applied to diagonal elements and because all
+ * compositions are peformed on relations with compatible domains and ranges.
+ */
+static __isl_give isl_union_map *union_floyd_warshall_on_list(isl_ctx *ctx,
+	__isl_keep isl_basic_map **list, int n, int *exact)
+{
+	int i, j, k;
+	int n_group;
+	int *group = NULL;
+	isl_set **set = NULL;
+	isl_map ***grid = NULL;
+	isl_union_map *app;
+
+	group = setup_groups(ctx, list, n, &set, &n_group);
+	if (!group)
+		goto error;
+
+	grid = isl_calloc_array(ctx, isl_map **, n_group);
+	if (!grid)
+		goto error;
+	for (i = 0; i < n_group; ++i) {
+		grid[i] = isl_calloc_array(ctx, isl_map *, n_group);
+		if (!grid[i])
+			goto error;
+		for (j = 0; j < n_group; ++j) {
+			isl_space *dim1, *dim2, *dim;
+			dim1 = isl_space_reverse(isl_set_get_space(set[i]));
+			dim2 = isl_set_get_space(set[j]);
+			dim = isl_space_join(dim1, dim2);
+			grid[i][j] = isl_map_empty(dim);
+		}
+	}
+
+	for (k = 0; k < n; ++k) {
+		i = group[2 * k];
+		j = group[2 * k + 1];
+		grid[i][j] = isl_map_union(grid[i][j],
+				isl_map_from_basic_map(
+					isl_basic_map_copy(list[k])));
+	}
+	
+	floyd_warshall_iterate(grid, n_group, exact);
+
+	app = isl_union_map_empty(isl_map_get_space(grid[0][0]));
+
+	for (i = 0; i < n_group; ++i) {
+		for (j = 0; j < n_group; ++j)
+			app = isl_union_map_add_map(app, grid[i][j]);
+		free(grid[i]);
+	}
+	free(grid);
+
+	for (i = 0; i < 2 * n; ++i)
+		isl_set_free(set[i]);
+	free(set);
+
+	free(group);
+	return app;
+error:
+	if (grid)
+		for (i = 0; i < n_group; ++i) {
+			if (!grid[i])
+				continue;
+			for (j = 0; j < n_group; ++j)
+				isl_map_free(grid[i][j]);
+			free(grid[i]);
+		}
+	free(grid);
+	if (set) {
+		for (i = 0; i < 2 * n; ++i)
+			isl_set_free(set[i]);
+		free(set);
+	}
+	free(group);
+	return NULL;
+}
+
+/* Perform Floyd-Warshall on the given union relation.
+ * The implementation is very similar to that for non-unions.
+ * The main difference is that it is applied unconditionally.
+ * We first extract a list of basic maps from the union map
+ * and then perform the algorithm on this list.
+ */
+static __isl_give isl_union_map *union_floyd_warshall(
+	__isl_take isl_union_map *umap, int *exact)
+{
+	int i, n;
+	isl_ctx *ctx;
+	isl_basic_map **list = NULL;
+	isl_basic_map **next;
+	isl_union_map *res;
+
+	n = 0;
+	if (isl_union_map_foreach_map(umap, inc_count, &n) < 0)
+		goto error;
+
+	ctx = isl_union_map_get_ctx(umap);
+	list = isl_calloc_array(ctx, isl_basic_map *, n);
+	if (!list)
+		goto error;
+
+	next = list;
+	if (isl_union_map_foreach_map(umap, collect_basic_map, &next) < 0)
+		goto error;
+
+	res = union_floyd_warshall_on_list(ctx, list, n, exact);
+
+	if (list) {
+		for (i = 0; i < n; ++i)
+			isl_basic_map_free(list[i]);
+		free(list);
+	}
+
+	isl_union_map_free(umap);
+	return res;
+error:
+	if (list) {
+		for (i = 0; i < n; ++i)
+			isl_basic_map_free(list[i]);
+		free(list);
+	}
+	isl_union_map_free(umap);
+	return NULL;
+}
+
+/* Decompose the give union relation into strongly connected components.
+ * The implementation is essentially the same as that of
+ * construct_power_components with the major difference that all
+ * operations are performed on union maps.
+ */
+static __isl_give isl_union_map *union_components(
+	__isl_take isl_union_map *umap, int *exact)
+{
+	int i;
+	int n;
+	isl_ctx *ctx;
+	isl_basic_map **list = NULL;
+	isl_basic_map **next;
+	isl_union_map *path = NULL;
+	struct isl_tc_follows_data data;
+	struct isl_tarjan_graph *g = NULL;
+	int c, l;
+	int recheck = 0;
+
+	n = 0;
+	if (isl_union_map_foreach_map(umap, inc_count, &n) < 0)
+		goto error;
+
+	if (n == 0)
+		return umap;
+	if (n <= 1)
+		return union_floyd_warshall(umap, exact);
+
+	ctx = isl_union_map_get_ctx(umap);
+	list = isl_calloc_array(ctx, isl_basic_map *, n);
+	if (!list)
+		goto error;
+
+	next = list;
+	if (isl_union_map_foreach_map(umap, collect_basic_map, &next) < 0)
+		goto error;
+
+	data.list = list;
+	data.check_closed = 0;
+	g = isl_tarjan_graph_init(ctx, n, &basic_map_follows, &data);
+	if (!g)
+		goto error;
+
+	c = 0;
+	i = 0;
+	l = n;
+	path = isl_union_map_empty(isl_union_map_get_space(umap));
+	while (l) {
+		isl_union_map *comp;
+		isl_union_map *path_comp, *path_comb;
+		comp = isl_union_map_empty(isl_union_map_get_space(umap));
+		while (g->order[i] != -1) {
+			comp = isl_union_map_add_map(comp,
+				    isl_map_from_basic_map(
+					isl_basic_map_copy(list[g->order[i]])));
+			--l;
+			++i;
+		}
+		path_comp = union_floyd_warshall(comp, exact);
+		path_comb = isl_union_map_apply_range(isl_union_map_copy(path),
+						isl_union_map_copy(path_comp));
+		path = isl_union_map_union(path, path_comp);
+		path = isl_union_map_union(path, path_comb);
+		++i;
+		++c;
+	}
+
+	if (c > 1 && data.check_closed && !*exact) {
+		int closed;
+
+		closed = isl_union_map_is_transitively_closed(path);
+		if (closed < 0)
+			goto error;
+		recheck = !closed;
+	}
+
+	isl_tarjan_graph_free(g);
+
+	for (i = 0; i < n; ++i)
+		isl_basic_map_free(list[i]);
+	free(list);
+
+	if (recheck) {
+		isl_union_map_free(path);
+		return union_floyd_warshall(umap, exact);
+	}
+
+	isl_union_map_free(umap);
+
+	return path;
+error:
+	isl_tarjan_graph_free(g);
+	if (list) {
+		for (i = 0; i < n; ++i)
+			isl_basic_map_free(list[i]);
+		free(list);
+	}
+	isl_union_map_free(umap);
+	isl_union_map_free(path);
+	return NULL;
+}
+
+/* Compute the transitive closure  of "umap", or an overapproximation.
+ * If the result is exact, then *exact is set to 1.
+ */
+__isl_give isl_union_map *isl_union_map_transitive_closure(
+	__isl_take isl_union_map *umap, int *exact)
+{
+	int closed;
+
+	if (!umap)
+		return NULL;
+
+	if (exact)
+		*exact = 1;
+
+	umap = isl_union_map_compute_divs(umap);
+	umap = isl_union_map_coalesce(umap);
+	closed = isl_union_map_is_transitively_closed(umap);
+	if (closed < 0)
+		goto error;
+	if (closed)
+		return umap;
+	umap = union_components(umap, exact);
+	return umap;
+error:
+	isl_union_map_free(umap);
+	return NULL;
+}
+
+struct isl_union_power {
+	isl_union_map *pow;
+	int *exact;
+};
+
+static int power(__isl_take isl_map *map, void *user)
+{
+	struct isl_union_power *up = user;
+
+	map = isl_map_power(map, up->exact);
+	up->pow = isl_union_map_from_map(map);
+
+	return -1;
+}
+
+/* Construct a map [x] -> [x+1], with parameters prescribed by "dim".
+ */
+static __isl_give isl_union_map *increment(__isl_take isl_space *dim)
+{
+	int k;
+	isl_basic_map *bmap;
+
+	dim = isl_space_add_dims(dim, isl_dim_in, 1);
+	dim = isl_space_add_dims(dim, isl_dim_out, 1);
+	bmap = isl_basic_map_alloc_space(dim, 0, 1, 0);
+	k = isl_basic_map_alloc_equality(bmap);
+	if (k < 0)
+		goto error;
+	isl_seq_clr(bmap->eq[k], isl_basic_map_total_dim(bmap));
+	isl_int_set_si(bmap->eq[k][0], 1);
+	isl_int_set_si(bmap->eq[k][isl_basic_map_offset(bmap, isl_dim_in)], 1);
+	isl_int_set_si(bmap->eq[k][isl_basic_map_offset(bmap, isl_dim_out)], -1);
+	return isl_union_map_from_map(isl_map_from_basic_map(bmap));
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Construct a map [[x]->[y]] -> [y-x], with parameters prescribed by "dim".
+ */
+static __isl_give isl_union_map *deltas_map(__isl_take isl_space *dim)
+{
+	isl_basic_map *bmap;
+
+	dim = isl_space_add_dims(dim, isl_dim_in, 1);
+	dim = isl_space_add_dims(dim, isl_dim_out, 1);
+	bmap = isl_basic_map_universe(dim);
+	bmap = isl_basic_map_deltas_map(bmap);
+
+	return isl_union_map_from_map(isl_map_from_basic_map(bmap));
+}
+
+/* Compute the positive powers of "map", or an overapproximation.
+ * The result maps the exponent to a nested copy of the corresponding power.
+ * If the result is exact, then *exact is set to 1.
+ */
+__isl_give isl_union_map *isl_union_map_power(__isl_take isl_union_map *umap,
+	int *exact)
+{
+	int n;
+	isl_union_map *inc;
+	isl_union_map *dm;
+
+	if (!umap)
+		return NULL;
+	n = isl_union_map_n_map(umap);
+	if (n == 0)
+		return umap;
+	if (n == 1) {
+		struct isl_union_power up = { NULL, exact };
+		isl_union_map_foreach_map(umap, &power, &up);
+		isl_union_map_free(umap);
+		return up.pow;
+	}
+	inc = increment(isl_union_map_get_space(umap));
+	umap = isl_union_map_product(inc, umap);
+	umap = isl_union_map_transitive_closure(umap, exact);
+	umap = isl_union_map_zip(umap);
+	dm = deltas_map(isl_union_map_get_space(umap));
+	umap = isl_union_map_apply_domain(umap, dm);
+	
+	return umap;
+}
+
+#undef TYPE
+#define TYPE isl_map
+#include "isl_power_templ.c"
+
+#undef TYPE
+#define TYPE isl_union_map
+#include "isl_power_templ.c"
diff --git a/polly/lib/External/isl/isl_union_map.c b/polly/lib/External/isl/isl_union_map.c
new file mode 100644
index 00000000000..be39088c8a9
--- /dev/null
+++ b/polly/lib/External/isl/isl_union_map.c
@@ -0,0 +1,3373 @@
+/*
+ * Copyright 2010-2011 INRIA Saclay
+ * Copyright 2013-2014 Ecole Normale Superieure
+ * Copyright 2014      INRIA Rocquencourt
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France 
+ * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
+ * B.P. 105 - 78153 Le Chesnay, France
+ */
+
+#define ISL_DIM_H
+#include <isl_map_private.h>
+#include <isl/ctx.h>
+#include <isl/hash.h>
+#include <isl/aff.h>
+#include <isl/map.h>
+#include <isl/set.h>
+#include <isl_space_private.h>
+#include <isl_union_map_private.h>
+#include <isl/union_set.h>
+#include <isl/deprecated/union_map_int.h>
+
+/* Return the number of parameters of "umap", where "type"
+ * is required to be set to isl_dim_param.
+ */
+unsigned isl_union_map_dim(__isl_keep isl_union_map *umap,
+	enum isl_dim_type type)
+{
+	if (!umap)
+		return 0;
+
+	if (type != isl_dim_param)
+		isl_die(isl_union_map_get_ctx(umap), isl_error_invalid,
+			"can only reference parameters", return 0);
+
+	return isl_space_dim(umap->dim, type);
+}
+
+/* Return the number of parameters of "uset", where "type"
+ * is required to be set to isl_dim_param.
+ */
+unsigned isl_union_set_dim(__isl_keep isl_union_set *uset,
+	enum isl_dim_type type)
+{
+	return isl_union_map_dim(uset, type);
+}
+
+/* Return the id of the specified dimension.
+ */
+__isl_give isl_id *isl_union_map_get_dim_id(__isl_keep isl_union_map *umap,
+	enum isl_dim_type type, unsigned pos)
+{
+	if (!umap)
+		return NULL;
+
+	if (type != isl_dim_param)
+		isl_die(isl_union_map_get_ctx(umap), isl_error_invalid,
+			"can only reference parameters", return NULL);
+
+	return isl_space_get_dim_id(umap->dim, type, pos);
+}
+
+/* Is this union set a parameter domain?
+ */
+int isl_union_set_is_params(__isl_keep isl_union_set *uset)
+{
+	isl_set *set;
+	int params;
+
+	if (!uset)
+		return -1;
+	if (uset->table.n != 1)
+		return 0;
+
+	set = isl_set_from_union_set(isl_union_set_copy(uset));
+	params = isl_set_is_params(set);
+	isl_set_free(set);
+	return params;
+}
+
+static __isl_give isl_union_map *isl_union_map_alloc(
+	__isl_take isl_space *space, int size)
+{
+	isl_union_map *umap;
+
+	space = isl_space_params(space);
+	if (!space)
+		return NULL;
+
+	umap = isl_calloc_type(space->ctx, isl_union_map);
+	if (!umap) {
+		isl_space_free(space);
+		return NULL;
+	}
+
+	umap->ref = 1;
+	umap->dim = space;
+	if (isl_hash_table_init(space->ctx, &umap->table, size) < 0)
+		return isl_union_map_free(umap);
+
+	return umap;
+}
+
+__isl_give isl_union_map *isl_union_map_empty(__isl_take isl_space *dim)
+{
+	return isl_union_map_alloc(dim, 16);
+}
+
+__isl_give isl_union_set *isl_union_set_empty(__isl_take isl_space *dim)
+{
+	return isl_union_map_empty(dim);
+}
+
+isl_ctx *isl_union_map_get_ctx(__isl_keep isl_union_map *umap)
+{
+	return umap ? umap->dim->ctx : NULL;
+}
+
+isl_ctx *isl_union_set_get_ctx(__isl_keep isl_union_set *uset)
+{
+	return uset ? uset->dim->ctx : NULL;
+}
+
+__isl_give isl_space *isl_union_map_get_space(__isl_keep isl_union_map *umap)
+{
+	if (!umap)
+		return NULL;
+	return isl_space_copy(umap->dim);
+}
+
+/* Return the position of the parameter with the given name
+ * in "umap".
+ * Return -1 if no such dimension can be found.
+ */
+int isl_union_map_find_dim_by_name(__isl_keep isl_union_map *umap,
+	enum isl_dim_type type, const char *name)
+{
+	if (!umap)
+		return -1;
+	return isl_space_find_dim_by_name(umap->dim, type, name);
+}
+
+__isl_give isl_space *isl_union_set_get_space(__isl_keep isl_union_set *uset)
+{
+	return isl_union_map_get_space(uset);
+}
+
+static int free_umap_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_map_free(map);
+	return 0;
+}
+
+static int add_map(__isl_take isl_map *map, void *user)
+{
+	isl_union_map **umap = (isl_union_map **)user;
+
+	*umap = isl_union_map_add_map(*umap, map);
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_map_dup(__isl_keep isl_union_map *umap)
+{
+	isl_union_map *dup;
+
+	if (!umap)
+		return NULL;
+
+	dup = isl_union_map_empty(isl_space_copy(umap->dim));
+	if (isl_union_map_foreach_map(umap, &add_map, &dup) < 0)
+		goto error;
+	return dup;
+error:
+	isl_union_map_free(dup);
+	return NULL;
+}
+
+__isl_give isl_union_map *isl_union_map_cow(__isl_take isl_union_map *umap)
+{
+	if (!umap)
+		return NULL;
+
+	if (umap->ref == 1)
+		return umap;
+	umap->ref--;
+	return isl_union_map_dup(umap);
+}
+
+struct isl_union_align {
+	isl_reordering *exp;
+	isl_union_map *res;
+};
+
+static int align_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_reordering *exp;
+	struct isl_union_align *data = user;
+
+	exp = isl_reordering_extend_space(isl_reordering_copy(data->exp),
+				    isl_map_get_space(map));
+
+	data->res = isl_union_map_add_map(data->res,
+					isl_map_realign(isl_map_copy(map), exp));
+
+	return 0;
+}
+
+/* Align the parameters of umap along those of model.
+ * The result has the parameters of model first, in the same order
+ * as they appear in model, followed by any remaining parameters of
+ * umap that do not appear in model.
+ */
+__isl_give isl_union_map *isl_union_map_align_params(
+	__isl_take isl_union_map *umap, __isl_take isl_space *model)
+{
+	struct isl_union_align data = { NULL, NULL };
+
+	if (!umap || !model)
+		goto error;
+
+	if (isl_space_match(umap->dim, isl_dim_param, model, isl_dim_param)) {
+		isl_space_free(model);
+		return umap;
+	}
+
+	model = isl_space_params(model);
+	data.exp = isl_parameter_alignment_reordering(umap->dim, model);
+	if (!data.exp)
+		goto error;
+
+	data.res = isl_union_map_alloc(isl_space_copy(data.exp->dim),
+					umap->table.n);
+	if (isl_hash_table_foreach(umap->dim->ctx, &umap->table,
+					&align_entry, &data) < 0)
+		goto error;
+
+	isl_reordering_free(data.exp);
+	isl_union_map_free(umap);
+	isl_space_free(model);
+	return data.res;
+error:
+	isl_reordering_free(data.exp);
+	isl_union_map_free(umap);
+	isl_union_map_free(data.res);
+	isl_space_free(model);
+	return NULL;
+}
+
+__isl_give isl_union_set *isl_union_set_align_params(
+	__isl_take isl_union_set *uset, __isl_take isl_space *model)
+{
+	return isl_union_map_align_params(uset, model);
+}
+
+__isl_give isl_union_map *isl_union_map_union(__isl_take isl_union_map *umap1,
+	__isl_take isl_union_map *umap2)
+{
+	umap1 = isl_union_map_align_params(umap1, isl_union_map_get_space(umap2));
+	umap2 = isl_union_map_align_params(umap2, isl_union_map_get_space(umap1));
+
+	umap1 = isl_union_map_cow(umap1);
+
+	if (!umap1 || !umap2)
+		goto error;
+
+	if (isl_union_map_foreach_map(umap2, &add_map, &umap1) < 0)
+		goto error;
+
+	isl_union_map_free(umap2);
+
+	return umap1;
+error:
+	isl_union_map_free(umap1);
+	isl_union_map_free(umap2);
+	return NULL;
+}
+
+__isl_give isl_union_set *isl_union_set_union(__isl_take isl_union_set *uset1,
+	__isl_take isl_union_set *uset2)
+{
+	return isl_union_map_union(uset1, uset2);
+}
+
+__isl_give isl_union_map *isl_union_map_copy(__isl_keep isl_union_map *umap)
+{
+	if (!umap)
+		return NULL;
+
+	umap->ref++;
+	return umap;
+}
+
+__isl_give isl_union_set *isl_union_set_copy(__isl_keep isl_union_set *uset)
+{
+	return isl_union_map_copy(uset);
+}
+
+__isl_null isl_union_map *isl_union_map_free(__isl_take isl_union_map *umap)
+{
+	if (!umap)
+		return NULL;
+
+	if (--umap->ref > 0)
+		return NULL;
+
+	isl_hash_table_foreach(umap->dim->ctx, &umap->table,
+			       &free_umap_entry, NULL);
+	isl_hash_table_clear(&umap->table);
+	isl_space_free(umap->dim);
+	free(umap);
+	return NULL;
+}
+
+__isl_null isl_union_set *isl_union_set_free(__isl_take isl_union_set *uset)
+{
+	return isl_union_map_free(uset);
+}
+
+static int has_dim(const void *entry, const void *val)
+{
+	isl_map *map = (isl_map *)entry;
+	isl_space *dim = (isl_space *)val;
+
+	return isl_space_is_equal(map->dim, dim);
+}
+
+__isl_give isl_union_map *isl_union_map_add_map(__isl_take isl_union_map *umap,
+	__isl_take isl_map *map)
+{
+	uint32_t hash;
+	struct isl_hash_table_entry *entry;
+
+	if (!map || !umap)
+		goto error;
+
+	if (isl_map_plain_is_empty(map)) {
+		isl_map_free(map);
+		return umap;
+	}
+
+	if (!isl_space_match(map->dim, isl_dim_param, umap->dim, isl_dim_param)) {
+		umap = isl_union_map_align_params(umap, isl_map_get_space(map));
+		map = isl_map_align_params(map, isl_union_map_get_space(umap));
+	}
+
+	umap = isl_union_map_cow(umap);
+
+	if (!map || !umap)
+		goto error;
+
+	hash = isl_space_get_hash(map->dim);
+	entry = isl_hash_table_find(umap->dim->ctx, &umap->table, hash,
+				    &has_dim, map->dim, 1);
+	if (!entry)
+		goto error;
+
+	if (!entry->data)
+		entry->data = map;
+	else {
+		entry->data = isl_map_union(entry->data, isl_map_copy(map));
+		if (!entry->data)
+			goto error;
+		isl_map_free(map);
+	}
+
+	return umap;
+error:
+	isl_map_free(map);
+	isl_union_map_free(umap);
+	return NULL;
+}
+
+__isl_give isl_union_set *isl_union_set_add_set(__isl_take isl_union_set *uset,
+	__isl_take isl_set *set)
+{
+	return isl_union_map_add_map(uset, (isl_map *)set);
+}
+
+__isl_give isl_union_map *isl_union_map_from_map(__isl_take isl_map *map)
+{
+	isl_space *dim;
+	isl_union_map *umap;
+
+	if (!map)
+		return NULL;
+
+	dim = isl_map_get_space(map);
+	dim = isl_space_params(dim);
+	umap = isl_union_map_empty(dim);
+	umap = isl_union_map_add_map(umap, map);
+
+	return umap;
+}
+
+__isl_give isl_union_set *isl_union_set_from_set(__isl_take isl_set *set)
+{
+	return isl_union_map_from_map((isl_map *)set);
+}
+
+__isl_give isl_union_map *isl_union_map_from_basic_map(
+	__isl_take isl_basic_map *bmap)
+{
+	return isl_union_map_from_map(isl_map_from_basic_map(bmap));
+}
+
+__isl_give isl_union_set *isl_union_set_from_basic_set(
+	__isl_take isl_basic_set *bset)
+{
+	return isl_union_map_from_basic_map(bset);
+}
+
+struct isl_union_map_foreach_data
+{
+	int (*fn)(__isl_take isl_map *map, void *user);
+	void *user;
+};
+
+static int call_on_copy(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	struct isl_union_map_foreach_data *data;
+	data = (struct isl_union_map_foreach_data *)user;
+
+	return data->fn(isl_map_copy(map), data->user);
+}
+
+int isl_union_map_n_map(__isl_keep isl_union_map *umap)
+{
+	return umap ? umap->table.n : 0;
+}
+
+int isl_union_set_n_set(__isl_keep isl_union_set *uset)
+{
+	return uset ? uset->table.n : 0;
+}
+
+int isl_union_map_foreach_map(__isl_keep isl_union_map *umap,
+	int (*fn)(__isl_take isl_map *map, void *user), void *user)
+{
+	struct isl_union_map_foreach_data data = { fn, user };
+
+	if (!umap)
+		return -1;
+
+	return isl_hash_table_foreach(umap->dim->ctx, &umap->table,
+				      &call_on_copy, &data);
+}
+
+static int copy_map(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_map **map_p = user;
+
+	*map_p = isl_map_copy(map);
+
+	return -1;
+}
+
+__isl_give isl_map *isl_map_from_union_map(__isl_take isl_union_map *umap)
+{
+	isl_ctx *ctx;
+	isl_map *map = NULL;
+
+	if (!umap)
+		return NULL;
+	ctx = isl_union_map_get_ctx(umap);
+	if (umap->table.n != 1)
+		isl_die(ctx, isl_error_invalid,
+			"union map needs to contain elements in exactly "
+			"one space", goto error);
+
+	isl_hash_table_foreach(ctx, &umap->table, &copy_map, &map);
+
+	isl_union_map_free(umap);
+
+	return map;
+error:
+	isl_union_map_free(umap);
+	return NULL;
+}
+
+__isl_give isl_set *isl_set_from_union_set(__isl_take isl_union_set *uset)
+{
+	return isl_map_from_union_map(uset);
+}
+
+/* Extract the map in "umap" that lives in the given space (ignoring
+ * parameters).
+ */
+__isl_give isl_map *isl_union_map_extract_map(__isl_keep isl_union_map *umap,
+	__isl_take isl_space *space)
+{
+	uint32_t hash;
+	struct isl_hash_table_entry *entry;
+
+	space = isl_space_drop_dims(space, isl_dim_param,
+					0, isl_space_dim(space, isl_dim_param));
+	space = isl_space_align_params(space, isl_union_map_get_space(umap));
+	if (!umap || !space)
+		goto error;
+
+	hash = isl_space_get_hash(space);
+	entry = isl_hash_table_find(umap->dim->ctx, &umap->table, hash,
+				    &has_dim, space, 0);
+	if (!entry)
+		return isl_map_empty(space);
+	isl_space_free(space);
+	return isl_map_copy(entry->data);
+error:
+	isl_space_free(space);
+	return NULL;
+}
+
+__isl_give isl_set *isl_union_set_extract_set(__isl_keep isl_union_set *uset,
+	__isl_take isl_space *dim)
+{
+	return (isl_set *)isl_union_map_extract_map(uset, dim);
+}
+
+/* Check if umap contains a map in the given space.
+ */
+__isl_give int isl_union_map_contains(__isl_keep isl_union_map *umap,
+	__isl_keep isl_space *dim)
+{
+	uint32_t hash;
+	struct isl_hash_table_entry *entry;
+
+	if (!umap || !dim)
+		return -1;
+
+	hash = isl_space_get_hash(dim);
+	entry = isl_hash_table_find(umap->dim->ctx, &umap->table, hash,
+				    &has_dim, dim, 0);
+	return !!entry;
+}
+
+__isl_give int isl_union_set_contains(__isl_keep isl_union_set *uset,
+	__isl_keep isl_space *dim)
+{
+	return isl_union_map_contains(uset, dim);
+}
+
+int isl_union_set_foreach_set(__isl_keep isl_union_set *uset,
+	int (*fn)(__isl_take isl_set *set, void *user), void *user)
+{
+	return isl_union_map_foreach_map(uset,
+		(int(*)(__isl_take isl_map *, void*))fn, user);
+}
+
+struct isl_union_set_foreach_point_data {
+	int (*fn)(__isl_take isl_point *pnt, void *user);
+	void *user;
+};
+
+static int foreach_point(__isl_take isl_set *set, void *user)
+{
+	struct isl_union_set_foreach_point_data *data = user;
+	int r;
+
+	r = isl_set_foreach_point(set, data->fn, data->user);
+	isl_set_free(set);
+
+	return r;
+}
+
+int isl_union_set_foreach_point(__isl_keep isl_union_set *uset,
+	int (*fn)(__isl_take isl_point *pnt, void *user), void *user)
+{
+	struct isl_union_set_foreach_point_data data = { fn, user };
+	return isl_union_set_foreach_set(uset, &foreach_point, &data);
+}
+
+struct isl_union_map_gen_bin_data {
+	isl_union_map *umap2;
+	isl_union_map *res;
+};
+
+static int subtract_entry(void **entry, void *user)
+{
+	struct isl_union_map_gen_bin_data *data = user;
+	uint32_t hash;
+	struct isl_hash_table_entry *entry2;
+	isl_map *map = *entry;
+
+	hash = isl_space_get_hash(map->dim);
+	entry2 = isl_hash_table_find(data->umap2->dim->ctx, &data->umap2->table,
+				     hash, &has_dim, map->dim, 0);
+	map = isl_map_copy(map);
+	if (entry2) {
+		int empty;
+		map = isl_map_subtract(map, isl_map_copy(entry2->data));
+
+		empty = isl_map_is_empty(map);
+		if (empty < 0) {
+			isl_map_free(map);
+			return -1;
+		}
+		if (empty) {
+			isl_map_free(map);
+			return 0;
+		}
+	}
+	data->res = isl_union_map_add_map(data->res, map);
+
+	return 0;
+}
+
+static __isl_give isl_union_map *gen_bin_op(__isl_take isl_union_map *umap1,
+	__isl_take isl_union_map *umap2, int (*fn)(void **, void *))
+{
+	struct isl_union_map_gen_bin_data data = { NULL, NULL };
+
+	umap1 = isl_union_map_align_params(umap1, isl_union_map_get_space(umap2));
+	umap2 = isl_union_map_align_params(umap2, isl_union_map_get_space(umap1));
+
+	if (!umap1 || !umap2)
+		goto error;
+
+	data.umap2 = umap2;
+	data.res = isl_union_map_alloc(isl_space_copy(umap1->dim),
+				       umap1->table.n);
+	if (isl_hash_table_foreach(umap1->dim->ctx, &umap1->table,
+				   fn, &data) < 0)
+		goto error;
+
+	isl_union_map_free(umap1);
+	isl_union_map_free(umap2);
+	return data.res;
+error:
+	isl_union_map_free(umap1);
+	isl_union_map_free(umap2);
+	isl_union_map_free(data.res);
+	return NULL;
+}
+
+__isl_give isl_union_map *isl_union_map_subtract(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2)
+{
+	return gen_bin_op(umap1, umap2, &subtract_entry);
+}
+
+__isl_give isl_union_set *isl_union_set_subtract(
+	__isl_take isl_union_set *uset1, __isl_take isl_union_set *uset2)
+{
+	return isl_union_map_subtract(uset1, uset2);
+}
+
+struct isl_union_map_gen_bin_set_data {
+	isl_set *set;
+	isl_union_map *res;
+};
+
+static int intersect_params_entry(void **entry, void *user)
+{
+	struct isl_union_map_gen_bin_set_data *data = user;
+	isl_map *map = *entry;
+	int empty;
+
+	map = isl_map_copy(map);
+	map = isl_map_intersect_params(map, isl_set_copy(data->set));
+
+	empty = isl_map_is_empty(map);
+	if (empty < 0) {
+		isl_map_free(map);
+		return -1;
+	}
+
+	data->res = isl_union_map_add_map(data->res, map);
+
+	return 0;
+}
+
+static __isl_give isl_union_map *gen_bin_set_op(__isl_take isl_union_map *umap,
+	__isl_take isl_set *set, int (*fn)(void **, void *))
+{
+	struct isl_union_map_gen_bin_set_data data = { NULL, NULL };
+
+	umap = isl_union_map_align_params(umap, isl_set_get_space(set));
+	set = isl_set_align_params(set, isl_union_map_get_space(umap));
+
+	if (!umap || !set)
+		goto error;
+
+	data.set = set;
+	data.res = isl_union_map_alloc(isl_space_copy(umap->dim),
+				       umap->table.n);
+	if (isl_hash_table_foreach(umap->dim->ctx, &umap->table,
+				   fn, &data) < 0)
+		goto error;
+
+	isl_union_map_free(umap);
+	isl_set_free(set);
+	return data.res;
+error:
+	isl_union_map_free(umap);
+	isl_set_free(set);
+	isl_union_map_free(data.res);
+	return NULL;
+}
+
+__isl_give isl_union_map *isl_union_map_intersect_params(
+	__isl_take isl_union_map *umap, __isl_take isl_set *set)
+{
+	return gen_bin_set_op(umap, set, &intersect_params_entry);
+}
+
+__isl_give isl_union_set *isl_union_set_intersect_params(
+	__isl_take isl_union_set *uset, __isl_take isl_set *set)
+{
+	return isl_union_map_intersect_params(uset, set);
+}
+
+static __isl_give isl_union_map *union_map_intersect_params(
+	__isl_take isl_union_map *umap, __isl_take isl_union_set *uset)
+{
+	return isl_union_map_intersect_params(umap,
+						isl_set_from_union_set(uset));
+}
+
+static __isl_give isl_union_map *union_map_gist_params(
+	__isl_take isl_union_map *umap, __isl_take isl_union_set *uset)
+{
+	return isl_union_map_gist_params(umap, isl_set_from_union_set(uset));
+}
+
+struct isl_union_map_match_bin_data {
+	isl_union_map *umap2;
+	isl_union_map *res;
+	__isl_give isl_map *(*fn)(__isl_take isl_map*, __isl_take isl_map*);
+};
+
+static int match_bin_entry(void **entry, void *user)
+{
+	struct isl_union_map_match_bin_data *data = user;
+	uint32_t hash;
+	struct isl_hash_table_entry *entry2;
+	isl_map *map = *entry;
+	int empty;
+
+	hash = isl_space_get_hash(map->dim);
+	entry2 = isl_hash_table_find(data->umap2->dim->ctx, &data->umap2->table,
+				     hash, &has_dim, map->dim, 0);
+	if (!entry2)
+		return 0;
+
+	map = isl_map_copy(map);
+	map = data->fn(map, isl_map_copy(entry2->data));
+
+	empty = isl_map_is_empty(map);
+	if (empty < 0) {
+		isl_map_free(map);
+		return -1;
+	}
+	if (empty) {
+		isl_map_free(map);
+		return 0;
+	}
+
+	data->res = isl_union_map_add_map(data->res, map);
+
+	return 0;
+}
+
+static __isl_give isl_union_map *match_bin_op(__isl_take isl_union_map *umap1,
+	__isl_take isl_union_map *umap2,
+	__isl_give isl_map *(*fn)(__isl_take isl_map*, __isl_take isl_map*))
+{
+	struct isl_union_map_match_bin_data data = { NULL, NULL, fn };
+
+	umap1 = isl_union_map_align_params(umap1, isl_union_map_get_space(umap2));
+	umap2 = isl_union_map_align_params(umap2, isl_union_map_get_space(umap1));
+
+	if (!umap1 || !umap2)
+		goto error;
+
+	data.umap2 = umap2;
+	data.res = isl_union_map_alloc(isl_space_copy(umap1->dim),
+				       umap1->table.n);
+	if (isl_hash_table_foreach(umap1->dim->ctx, &umap1->table,
+				   &match_bin_entry, &data) < 0)
+		goto error;
+
+	isl_union_map_free(umap1);
+	isl_union_map_free(umap2);
+	return data.res;
+error:
+	isl_union_map_free(umap1);
+	isl_union_map_free(umap2);
+	isl_union_map_free(data.res);
+	return NULL;
+}
+
+__isl_give isl_union_map *isl_union_map_intersect(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2)
+{
+	return match_bin_op(umap1, umap2, &isl_map_intersect);
+}
+
+/* Compute the intersection of the two union_sets.
+ * As a special case, if exactly one of the two union_sets
+ * is a parameter domain, then intersect the parameter domain
+ * of the other one with this set.
+ */
+__isl_give isl_union_set *isl_union_set_intersect(
+	__isl_take isl_union_set *uset1, __isl_take isl_union_set *uset2)
+{
+	int p1, p2;
+
+	p1 = isl_union_set_is_params(uset1);
+	p2 = isl_union_set_is_params(uset2);
+	if (p1 < 0 || p2 < 0)
+		goto error;
+	if (!p1 && p2)
+		return union_map_intersect_params(uset1, uset2);
+	if (p1 && !p2)
+		return union_map_intersect_params(uset2, uset1);
+	return isl_union_map_intersect(uset1, uset2);
+error:
+	isl_union_set_free(uset1);
+	isl_union_set_free(uset2);
+	return NULL;
+}
+
+static int gist_params_entry(void **entry, void *user)
+{
+	struct isl_union_map_gen_bin_set_data *data = user;
+	isl_map *map = *entry;
+	int empty;
+
+	map = isl_map_copy(map);
+	map = isl_map_gist_params(map, isl_set_copy(data->set));
+
+	empty = isl_map_is_empty(map);
+	if (empty < 0) {
+		isl_map_free(map);
+		return -1;
+	}
+
+	data->res = isl_union_map_add_map(data->res, map);
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_map_gist_params(
+	__isl_take isl_union_map *umap, __isl_take isl_set *set)
+{
+	return gen_bin_set_op(umap, set, &gist_params_entry);
+}
+
+__isl_give isl_union_set *isl_union_set_gist_params(
+	__isl_take isl_union_set *uset, __isl_take isl_set *set)
+{
+	return isl_union_map_gist_params(uset, set);
+}
+
+__isl_give isl_union_map *isl_union_map_gist(__isl_take isl_union_map *umap,
+	__isl_take isl_union_map *context)
+{
+	return match_bin_op(umap, context, &isl_map_gist);
+}
+
+__isl_give isl_union_set *isl_union_set_gist(__isl_take isl_union_set *uset,
+	__isl_take isl_union_set *context)
+{
+	if (isl_union_set_is_params(context))
+		return union_map_gist_params(uset, context);
+	return isl_union_map_gist(uset, context);
+}
+
+static __isl_give isl_map *lex_le_set(__isl_take isl_map *set1,
+	__isl_take isl_map *set2)
+{
+	return isl_set_lex_le_set((isl_set *)set1, (isl_set *)set2);
+}
+
+static __isl_give isl_map *lex_lt_set(__isl_take isl_map *set1,
+	__isl_take isl_map *set2)
+{
+	return isl_set_lex_lt_set((isl_set *)set1, (isl_set *)set2);
+}
+
+__isl_give isl_union_map *isl_union_set_lex_lt_union_set(
+	__isl_take isl_union_set *uset1, __isl_take isl_union_set *uset2)
+{
+	return match_bin_op(uset1, uset2, &lex_lt_set);
+}
+
+__isl_give isl_union_map *isl_union_set_lex_le_union_set(
+	__isl_take isl_union_set *uset1, __isl_take isl_union_set *uset2)
+{
+	return match_bin_op(uset1, uset2, &lex_le_set);
+}
+
+__isl_give isl_union_map *isl_union_set_lex_gt_union_set(
+	__isl_take isl_union_set *uset1, __isl_take isl_union_set *uset2)
+{
+	return isl_union_map_reverse(isl_union_set_lex_lt_union_set(uset2, uset1));
+}
+
+__isl_give isl_union_map *isl_union_set_lex_ge_union_set(
+	__isl_take isl_union_set *uset1, __isl_take isl_union_set *uset2)
+{
+	return isl_union_map_reverse(isl_union_set_lex_le_union_set(uset2, uset1));
+}
+
+__isl_give isl_union_map *isl_union_map_lex_gt_union_map(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2)
+{
+	return isl_union_map_reverse(isl_union_map_lex_lt_union_map(umap2, umap1));
+}
+
+__isl_give isl_union_map *isl_union_map_lex_ge_union_map(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2)
+{
+	return isl_union_map_reverse(isl_union_map_lex_le_union_map(umap2, umap1));
+}
+
+static int intersect_domain_entry(void **entry, void *user)
+{
+	struct isl_union_map_gen_bin_data *data = user;
+	uint32_t hash;
+	struct isl_hash_table_entry *entry2;
+	isl_space *dim;
+	isl_map *map = *entry;
+	int empty;
+
+	dim = isl_map_get_space(map);
+	dim = isl_space_domain(dim);
+	hash = isl_space_get_hash(dim);
+	entry2 = isl_hash_table_find(data->umap2->dim->ctx, &data->umap2->table,
+				     hash, &has_dim, dim, 0);
+	isl_space_free(dim);
+	if (!entry2)
+		return 0;
+
+	map = isl_map_copy(map);
+	map = isl_map_intersect_domain(map, isl_set_copy(entry2->data));
+
+	empty = isl_map_is_empty(map);
+	if (empty < 0) {
+		isl_map_free(map);
+		return -1;
+	}
+	if (empty) {
+		isl_map_free(map);
+		return 0;
+	}
+
+	data->res = isl_union_map_add_map(data->res, map);
+
+	return 0;
+}
+
+/* Intersect the domain of "umap" with "uset".
+ * If "uset" is a parameters domain, then intersect the parameter
+ * domain of "umap" with this set.
+ */
+__isl_give isl_union_map *isl_union_map_intersect_domain(
+	__isl_take isl_union_map *umap, __isl_take isl_union_set *uset)
+{
+	if (isl_union_set_is_params(uset))
+		return union_map_intersect_params(umap, uset);
+	return gen_bin_op(umap, uset, &intersect_domain_entry);
+}
+
+/* Remove the elements of data->umap2 from the domain of *entry
+ * and add the result to data->res.
+ */
+static int subtract_domain_entry(void **entry, void *user)
+{
+	struct isl_union_map_gen_bin_data *data = user;
+	uint32_t hash;
+	struct isl_hash_table_entry *entry2;
+	isl_space *dim;
+	isl_map *map = *entry;
+	int empty;
+
+	dim = isl_map_get_space(map);
+	dim = isl_space_domain(dim);
+	hash = isl_space_get_hash(dim);
+	entry2 = isl_hash_table_find(data->umap2->dim->ctx, &data->umap2->table,
+				     hash, &has_dim, dim, 0);
+	isl_space_free(dim);
+
+	map = isl_map_copy(map);
+
+	if (!entry2) {
+		data->res = isl_union_map_add_map(data->res, map);
+		return 0;
+	}
+
+	map = isl_map_subtract_domain(map, isl_set_copy(entry2->data));
+
+	empty = isl_map_is_empty(map);
+	if (empty < 0) {
+		isl_map_free(map);
+		return -1;
+	}
+	if (empty) {
+		isl_map_free(map);
+		return 0;
+	}
+
+	data->res = isl_union_map_add_map(data->res, map);
+
+	return 0;
+}
+
+/* Remove the elements of "uset" from the domain of "umap".
+ */
+__isl_give isl_union_map *isl_union_map_subtract_domain(
+	__isl_take isl_union_map *umap, __isl_take isl_union_set *dom)
+{
+	return gen_bin_op(umap, dom, &subtract_domain_entry);
+}
+
+/* Remove the elements of data->umap2 from the range of *entry
+ * and add the result to data->res.
+ */
+static int subtract_range_entry(void **entry, void *user)
+{
+	struct isl_union_map_gen_bin_data *data = user;
+	uint32_t hash;
+	struct isl_hash_table_entry *entry2;
+	isl_space *space;
+	isl_map *map = *entry;
+	int empty;
+
+	space = isl_map_get_space(map);
+	space = isl_space_range(space);
+	hash = isl_space_get_hash(space);
+	entry2 = isl_hash_table_find(data->umap2->dim->ctx, &data->umap2->table,
+				     hash, &has_dim, space, 0);
+	isl_space_free(space);
+
+	map = isl_map_copy(map);
+
+	if (!entry2) {
+		data->res = isl_union_map_add_map(data->res, map);
+		return 0;
+	}
+
+	map = isl_map_subtract_range(map, isl_set_copy(entry2->data));
+
+	empty = isl_map_is_empty(map);
+	if (empty < 0) {
+		isl_map_free(map);
+		return -1;
+	}
+	if (empty) {
+		isl_map_free(map);
+		return 0;
+	}
+
+	data->res = isl_union_map_add_map(data->res, map);
+
+	return 0;
+}
+
+/* Remove the elements of "uset" from the range of "umap".
+ */
+__isl_give isl_union_map *isl_union_map_subtract_range(
+	__isl_take isl_union_map *umap, __isl_take isl_union_set *dom)
+{
+	return gen_bin_op(umap, dom, &subtract_range_entry);
+}
+
+static int gist_domain_entry(void **entry, void *user)
+{
+	struct isl_union_map_gen_bin_data *data = user;
+	uint32_t hash;
+	struct isl_hash_table_entry *entry2;
+	isl_space *dim;
+	isl_map *map = *entry;
+	int empty;
+
+	dim = isl_map_get_space(map);
+	dim = isl_space_domain(dim);
+	hash = isl_space_get_hash(dim);
+	entry2 = isl_hash_table_find(data->umap2->dim->ctx, &data->umap2->table,
+				     hash, &has_dim, dim, 0);
+	isl_space_free(dim);
+	if (!entry2)
+		return 0;
+
+	map = isl_map_copy(map);
+	map = isl_map_gist_domain(map, isl_set_copy(entry2->data));
+
+	empty = isl_map_is_empty(map);
+	if (empty < 0) {
+		isl_map_free(map);
+		return -1;
+	}
+
+	data->res = isl_union_map_add_map(data->res, map);
+
+	return 0;
+}
+
+/* Compute the gist of "umap" with respect to the domain "uset".
+ * If "uset" is a parameters domain, then compute the gist
+ * with respect to this parameter domain.
+ */
+__isl_give isl_union_map *isl_union_map_gist_domain(
+	__isl_take isl_union_map *umap, __isl_take isl_union_set *uset)
+{
+	if (isl_union_set_is_params(uset))
+		return union_map_gist_params(umap, uset);
+	return gen_bin_op(umap, uset, &gist_domain_entry);
+}
+
+static int gist_range_entry(void **entry, void *user)
+{
+	struct isl_union_map_gen_bin_data *data = user;
+	uint32_t hash;
+	struct isl_hash_table_entry *entry2;
+	isl_space *space;
+	isl_map *map = *entry;
+	int empty;
+
+	space = isl_map_get_space(map);
+	space = isl_space_range(space);
+	hash = isl_space_get_hash(space);
+	entry2 = isl_hash_table_find(data->umap2->dim->ctx, &data->umap2->table,
+				     hash, &has_dim, space, 0);
+	isl_space_free(space);
+	if (!entry2)
+		return 0;
+
+	map = isl_map_copy(map);
+	map = isl_map_gist_range(map, isl_set_copy(entry2->data));
+
+	empty = isl_map_is_empty(map);
+	if (empty < 0) {
+		isl_map_free(map);
+		return -1;
+	}
+
+	data->res = isl_union_map_add_map(data->res, map);
+
+	return 0;
+}
+
+/* Compute the gist of "umap" with respect to the range "uset".
+ */
+__isl_give isl_union_map *isl_union_map_gist_range(
+	__isl_take isl_union_map *umap, __isl_take isl_union_set *uset)
+{
+	return gen_bin_op(umap, uset, &gist_range_entry);
+}
+
+static int intersect_range_entry(void **entry, void *user)
+{
+	struct isl_union_map_gen_bin_data *data = user;
+	uint32_t hash;
+	struct isl_hash_table_entry *entry2;
+	isl_space *dim;
+	isl_map *map = *entry;
+	int empty;
+
+	dim = isl_map_get_space(map);
+	dim = isl_space_range(dim);
+	hash = isl_space_get_hash(dim);
+	entry2 = isl_hash_table_find(data->umap2->dim->ctx, &data->umap2->table,
+				     hash, &has_dim, dim, 0);
+	isl_space_free(dim);
+	if (!entry2)
+		return 0;
+
+	map = isl_map_copy(map);
+	map = isl_map_intersect_range(map, isl_set_copy(entry2->data));
+
+	empty = isl_map_is_empty(map);
+	if (empty < 0) {
+		isl_map_free(map);
+		return -1;
+	}
+	if (empty) {
+		isl_map_free(map);
+		return 0;
+	}
+
+	data->res = isl_union_map_add_map(data->res, map);
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_map_intersect_range(
+	__isl_take isl_union_map *umap, __isl_take isl_union_set *uset)
+{
+	return gen_bin_op(umap, uset, &intersect_range_entry);
+}
+
+struct isl_union_map_bin_data {
+	isl_union_map *umap2;
+	isl_union_map *res;
+	isl_map *map;
+	int (*fn)(void **entry, void *user);
+};
+
+static int apply_range_entry(void **entry, void *user)
+{
+	struct isl_union_map_bin_data *data = user;
+	isl_map *map2 = *entry;
+	int empty;
+
+	if (!isl_space_tuple_is_equal(data->map->dim, isl_dim_out,
+				 map2->dim, isl_dim_in))
+		return 0;
+
+	map2 = isl_map_apply_range(isl_map_copy(data->map), isl_map_copy(map2));
+
+	empty = isl_map_is_empty(map2);
+	if (empty < 0) {
+		isl_map_free(map2);
+		return -1;
+	}
+	if (empty) {
+		isl_map_free(map2);
+		return 0;
+	}
+
+	data->res = isl_union_map_add_map(data->res, map2);
+
+	return 0;
+}
+
+static int bin_entry(void **entry, void *user)
+{
+	struct isl_union_map_bin_data *data = user;
+	isl_map *map = *entry;
+
+	data->map = map;
+	if (isl_hash_table_foreach(data->umap2->dim->ctx, &data->umap2->table,
+				   data->fn, data) < 0)
+		return -1;
+
+	return 0;
+}
+
+static __isl_give isl_union_map *bin_op(__isl_take isl_union_map *umap1,
+	__isl_take isl_union_map *umap2, int (*fn)(void **entry, void *user))
+{
+	struct isl_union_map_bin_data data = { NULL, NULL, NULL, fn };
+
+	umap1 = isl_union_map_align_params(umap1, isl_union_map_get_space(umap2));
+	umap2 = isl_union_map_align_params(umap2, isl_union_map_get_space(umap1));
+
+	if (!umap1 || !umap2)
+		goto error;
+
+	data.umap2 = umap2;
+	data.res = isl_union_map_alloc(isl_space_copy(umap1->dim),
+				       umap1->table.n);
+	if (isl_hash_table_foreach(umap1->dim->ctx, &umap1->table,
+				   &bin_entry, &data) < 0)
+		goto error;
+
+	isl_union_map_free(umap1);
+	isl_union_map_free(umap2);
+	return data.res;
+error:
+	isl_union_map_free(umap1);
+	isl_union_map_free(umap2);
+	isl_union_map_free(data.res);
+	return NULL;
+}
+
+__isl_give isl_union_map *isl_union_map_apply_range(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2)
+{
+	return bin_op(umap1, umap2, &apply_range_entry);
+}
+
+__isl_give isl_union_map *isl_union_map_apply_domain(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2)
+{
+	umap1 = isl_union_map_reverse(umap1);
+	umap1 = isl_union_map_apply_range(umap1, umap2);
+	return isl_union_map_reverse(umap1);
+}
+
+__isl_give isl_union_set *isl_union_set_apply(
+	__isl_take isl_union_set *uset, __isl_take isl_union_map *umap)
+{
+	return isl_union_map_apply_range(uset, umap);
+}
+
+static int map_lex_lt_entry(void **entry, void *user)
+{
+	struct isl_union_map_bin_data *data = user;
+	isl_map *map2 = *entry;
+
+	if (!isl_space_tuple_is_equal(data->map->dim, isl_dim_out,
+				 map2->dim, isl_dim_out))
+		return 0;
+
+	map2 = isl_map_lex_lt_map(isl_map_copy(data->map), isl_map_copy(map2));
+
+	data->res = isl_union_map_add_map(data->res, map2);
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_map_lex_lt_union_map(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2)
+{
+	return bin_op(umap1, umap2, &map_lex_lt_entry);
+}
+
+static int map_lex_le_entry(void **entry, void *user)
+{
+	struct isl_union_map_bin_data *data = user;
+	isl_map *map2 = *entry;
+
+	if (!isl_space_tuple_is_equal(data->map->dim, isl_dim_out,
+				 map2->dim, isl_dim_out))
+		return 0;
+
+	map2 = isl_map_lex_le_map(isl_map_copy(data->map), isl_map_copy(map2));
+
+	data->res = isl_union_map_add_map(data->res, map2);
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_map_lex_le_union_map(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2)
+{
+	return bin_op(umap1, umap2, &map_lex_le_entry);
+}
+
+static int product_entry(void **entry, void *user)
+{
+	struct isl_union_map_bin_data *data = user;
+	isl_map *map2 = *entry;
+
+	map2 = isl_map_product(isl_map_copy(data->map), isl_map_copy(map2));
+
+	data->res = isl_union_map_add_map(data->res, map2);
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_map_product(__isl_take isl_union_map *umap1,
+	__isl_take isl_union_map *umap2)
+{
+	return bin_op(umap1, umap2, &product_entry);
+}
+
+static int set_product_entry(void **entry, void *user)
+{
+	struct isl_union_map_bin_data *data = user;
+	isl_set *set2 = *entry;
+
+	set2 = isl_set_product(isl_set_copy(data->map), isl_set_copy(set2));
+
+	data->res = isl_union_set_add_set(data->res, set2);
+
+	return 0;
+}
+
+__isl_give isl_union_set *isl_union_set_product(__isl_take isl_union_set *uset1,
+	__isl_take isl_union_set *uset2)
+{
+	return bin_op(uset1, uset2, &set_product_entry);
+}
+
+static int domain_product_entry(void **entry, void *user)
+{
+	struct isl_union_map_bin_data *data = user;
+	isl_map *map2 = *entry;
+
+	if (!isl_space_tuple_is_equal(data->map->dim, isl_dim_out,
+				 map2->dim, isl_dim_out))
+		return 0;
+
+	map2 = isl_map_domain_product(isl_map_copy(data->map),
+				     isl_map_copy(map2));
+
+	data->res = isl_union_map_add_map(data->res, map2);
+
+	return 0;
+}
+
+/* Given two maps A -> B and C -> D, construct a map [A -> C] -> (B * D)
+ */
+__isl_give isl_union_map *isl_union_map_domain_product(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2)
+{
+	return bin_op(umap1, umap2, &domain_product_entry);
+}
+
+static int range_product_entry(void **entry, void *user)
+{
+	struct isl_union_map_bin_data *data = user;
+	isl_map *map2 = *entry;
+
+	if (!isl_space_tuple_is_equal(data->map->dim, isl_dim_in,
+				 map2->dim, isl_dim_in))
+		return 0;
+
+	map2 = isl_map_range_product(isl_map_copy(data->map),
+				     isl_map_copy(map2));
+
+	data->res = isl_union_map_add_map(data->res, map2);
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_map_range_product(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2)
+{
+	return bin_op(umap1, umap2, &range_product_entry);
+}
+
+static int flat_range_product_entry(void **entry, void *user)
+{
+	struct isl_union_map_bin_data *data = user;
+	isl_map *map2 = *entry;
+
+	if (!isl_space_tuple_is_equal(data->map->dim, isl_dim_in,
+				 map2->dim, isl_dim_in))
+		return 0;
+
+	map2 = isl_map_flat_range_product(isl_map_copy(data->map),
+					  isl_map_copy(map2));
+
+	data->res = isl_union_map_add_map(data->res, map2);
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_map_flat_range_product(
+	__isl_take isl_union_map *umap1, __isl_take isl_union_map *umap2)
+{
+	return bin_op(umap1, umap2, &flat_range_product_entry);
+}
+
+static __isl_give isl_union_set *cond_un_op(__isl_take isl_union_map *umap,
+	int (*fn)(void **, void *))
+{
+	isl_union_set *res;
+
+	if (!umap)
+		return NULL;
+
+	res = isl_union_map_alloc(isl_space_copy(umap->dim), umap->table.n);
+	if (isl_hash_table_foreach(umap->dim->ctx, &umap->table, fn, &res) < 0)
+		goto error;
+
+	isl_union_map_free(umap);
+	return res;
+error:
+	isl_union_map_free(umap);
+	isl_union_set_free(res);
+	return NULL;
+}
+
+static int from_range_entry(void **entry, void *user)
+{
+	isl_map *set = *entry;
+	isl_union_set **res = user;
+
+	*res = isl_union_map_add_map(*res,
+					isl_map_from_range(isl_set_copy(set)));
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_map_from_range(
+	__isl_take isl_union_set *uset)
+{
+	return cond_un_op(uset, &from_range_entry);
+}
+
+__isl_give isl_union_map *isl_union_map_from_domain(
+	__isl_take isl_union_set *uset)
+{
+	return isl_union_map_reverse(isl_union_map_from_range(uset));
+}
+
+__isl_give isl_union_map *isl_union_map_from_domain_and_range(
+	__isl_take isl_union_set *domain, __isl_take isl_union_set *range)
+{
+	return isl_union_map_apply_range(isl_union_map_from_domain(domain),
+				         isl_union_map_from_range(range));
+}
+
+static __isl_give isl_union_map *un_op(__isl_take isl_union_map *umap,
+	int (*fn)(void **, void *))
+{
+	umap = isl_union_map_cow(umap);
+	if (!umap)
+		return NULL;
+
+	if (isl_hash_table_foreach(umap->dim->ctx, &umap->table, fn, NULL) < 0)
+		goto error;
+
+	return umap;
+error:
+	isl_union_map_free(umap);
+	return NULL;
+}
+
+static int affine_entry(void **entry, void *user)
+{
+	isl_map **map = (isl_map **)entry;
+
+	*map = isl_map_from_basic_map(isl_map_affine_hull(*map));
+
+	return *map ? 0 : -1;
+}
+
+__isl_give isl_union_map *isl_union_map_affine_hull(
+	__isl_take isl_union_map *umap)
+{
+	return un_op(umap, &affine_entry);
+}
+
+__isl_give isl_union_set *isl_union_set_affine_hull(
+	__isl_take isl_union_set *uset)
+{
+	return isl_union_map_affine_hull(uset);
+}
+
+static int polyhedral_entry(void **entry, void *user)
+{
+	isl_map **map = (isl_map **)entry;
+
+	*map = isl_map_from_basic_map(isl_map_polyhedral_hull(*map));
+
+	return *map ? 0 : -1;
+}
+
+__isl_give isl_union_map *isl_union_map_polyhedral_hull(
+	__isl_take isl_union_map *umap)
+{
+	return un_op(umap, &polyhedral_entry);
+}
+
+__isl_give isl_union_set *isl_union_set_polyhedral_hull(
+	__isl_take isl_union_set *uset)
+{
+	return isl_union_map_polyhedral_hull(uset);
+}
+
+static int simple_entry(void **entry, void *user)
+{
+	isl_map **map = (isl_map **)entry;
+
+	*map = isl_map_from_basic_map(isl_map_simple_hull(*map));
+
+	return *map ? 0 : -1;
+}
+
+__isl_give isl_union_map *isl_union_map_simple_hull(
+	__isl_take isl_union_map *umap)
+{
+	return un_op(umap, &simple_entry);
+}
+
+__isl_give isl_union_set *isl_union_set_simple_hull(
+	__isl_take isl_union_set *uset)
+{
+	return isl_union_map_simple_hull(uset);
+}
+
+static int inplace_entry(void **entry, void *user)
+{
+	__isl_give isl_map *(*fn)(__isl_take isl_map *);
+	isl_map **map = (isl_map **)entry;
+	isl_map *copy;
+
+	fn = *(__isl_give isl_map *(**)(__isl_take isl_map *)) user;
+	copy = fn(isl_map_copy(*map));
+	if (!copy)
+		return -1;
+
+	isl_map_free(*map);
+	*map = copy;
+
+	return 0;
+}
+
+static __isl_give isl_union_map *inplace(__isl_take isl_union_map *umap,
+	__isl_give isl_map *(*fn)(__isl_take isl_map *))
+{
+	if (!umap)
+		return NULL;
+
+	if (isl_hash_table_foreach(umap->dim->ctx, &umap->table,
+				    &inplace_entry, &fn) < 0)
+		goto error;
+
+	return umap;
+error:
+	isl_union_map_free(umap);
+	return NULL;
+}
+
+__isl_give isl_union_map *isl_union_map_coalesce(
+	__isl_take isl_union_map *umap)
+{
+	return inplace(umap, &isl_map_coalesce);
+}
+
+__isl_give isl_union_set *isl_union_set_coalesce(
+	__isl_take isl_union_set *uset)
+{
+	return isl_union_map_coalesce(uset);
+}
+
+__isl_give isl_union_map *isl_union_map_detect_equalities(
+	__isl_take isl_union_map *umap)
+{
+	return inplace(umap, &isl_map_detect_equalities);
+}
+
+__isl_give isl_union_set *isl_union_set_detect_equalities(
+	__isl_take isl_union_set *uset)
+{
+	return isl_union_map_detect_equalities(uset);
+}
+
+__isl_give isl_union_map *isl_union_map_compute_divs(
+	__isl_take isl_union_map *umap)
+{
+	return inplace(umap, &isl_map_compute_divs);
+}
+
+__isl_give isl_union_set *isl_union_set_compute_divs(
+	__isl_take isl_union_set *uset)
+{
+	return isl_union_map_compute_divs(uset);
+}
+
+static int lexmin_entry(void **entry, void *user)
+{
+	isl_map **map = (isl_map **)entry;
+
+	*map = isl_map_lexmin(*map);
+
+	return *map ? 0 : -1;
+}
+
+__isl_give isl_union_map *isl_union_map_lexmin(
+	__isl_take isl_union_map *umap)
+{
+	return un_op(umap, &lexmin_entry);
+}
+
+__isl_give isl_union_set *isl_union_set_lexmin(
+	__isl_take isl_union_set *uset)
+{
+	return isl_union_map_lexmin(uset);
+}
+
+static int lexmax_entry(void **entry, void *user)
+{
+	isl_map **map = (isl_map **)entry;
+
+	*map = isl_map_lexmax(*map);
+
+	return *map ? 0 : -1;
+}
+
+__isl_give isl_union_map *isl_union_map_lexmax(
+	__isl_take isl_union_map *umap)
+{
+	return un_op(umap, &lexmax_entry);
+}
+
+__isl_give isl_union_set *isl_union_set_lexmax(
+	__isl_take isl_union_set *uset)
+{
+	return isl_union_map_lexmax(uset);
+}
+
+static int universe_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_map **res = user;
+
+	map = isl_map_universe(isl_map_get_space(map));
+	*res = isl_union_map_add_map(*res, map);
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_map_universe(__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &universe_entry);
+}
+
+__isl_give isl_union_set *isl_union_set_universe(__isl_take isl_union_set *uset)
+{
+	return isl_union_map_universe(uset);
+}
+
+static int reverse_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_map **res = user;
+
+	*res = isl_union_map_add_map(*res, isl_map_reverse(isl_map_copy(map)));
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_map_reverse(__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &reverse_entry);
+}
+
+static int params_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_set **res = user;
+
+	*res = isl_union_set_add_set(*res, isl_map_params(isl_map_copy(map)));
+
+	return 0;
+}
+
+/* Compute the parameter domain of the given union map.
+ */
+__isl_give isl_set *isl_union_map_params(__isl_take isl_union_map *umap)
+{
+	int empty;
+
+	empty = isl_union_map_is_empty(umap);
+	if (empty < 0)
+		goto error;
+	if (empty) {
+		isl_space *space;
+		space = isl_union_map_get_space(umap);
+		isl_union_map_free(umap);
+		return isl_set_empty(space);
+	}
+	return isl_set_from_union_set(cond_un_op(umap, &params_entry));
+error:
+	isl_union_map_free(umap);
+	return NULL;
+}
+
+/* Compute the parameter domain of the given union set.
+ */
+__isl_give isl_set *isl_union_set_params(__isl_take isl_union_set *uset)
+{
+	return isl_union_map_params(uset);
+}
+
+static int domain_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_set **res = user;
+
+	*res = isl_union_set_add_set(*res, isl_map_domain(isl_map_copy(map)));
+
+	return 0;
+}
+
+__isl_give isl_union_set *isl_union_map_domain(__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &domain_entry);
+}
+
+static int range_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_set **res = user;
+
+	*res = isl_union_set_add_set(*res, isl_map_range(isl_map_copy(map)));
+
+	return 0;
+}
+
+__isl_give isl_union_set *isl_union_map_range(__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &range_entry);
+}
+
+static int domain_map_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_set **res = user;
+
+	*res = isl_union_map_add_map(*res,
+					isl_map_domain_map(isl_map_copy(map)));
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_map_domain_map(
+	__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &domain_map_entry);
+}
+
+static int range_map_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_set **res = user;
+
+	*res = isl_union_map_add_map(*res,
+					isl_map_range_map(isl_map_copy(map)));
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_map_range_map(
+	__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &range_map_entry);
+}
+
+/* Check if "set" is of the form A[B -> C].
+ * If so, add A[B -> C] -> B to "res".
+ */
+static int wrapped_domain_map_entry(void **entry, void *user)
+{
+	isl_set *set = *entry;
+	isl_union_set **res = user;
+	int wrapping;
+
+	wrapping = isl_set_is_wrapping(set);
+	if (wrapping < 0)
+		return -1;
+	if (!wrapping)
+		return 0;
+
+	*res = isl_union_map_add_map(*res,
+				isl_set_wrapped_domain_map(isl_set_copy(set)));
+
+	return 0;
+}
+
+/* Given a collection of wrapped maps of the form A[B -> C],
+ * return the collection of maps A[B -> C] -> B.
+ */
+__isl_give isl_union_map *isl_union_set_wrapped_domain_map(
+	__isl_take isl_union_set *uset)
+{
+	return cond_un_op(uset, &wrapped_domain_map_entry);
+}
+
+static int deltas_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_set **res = user;
+
+	if (!isl_space_tuple_is_equal(map->dim, isl_dim_in,
+					map->dim, isl_dim_out))
+		return 0;
+
+	*res = isl_union_set_add_set(*res, isl_map_deltas(isl_map_copy(map)));
+
+	return 0;
+}
+
+__isl_give isl_union_set *isl_union_map_deltas(__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &deltas_entry);
+}
+
+static int deltas_map_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_map **res = user;
+
+	if (!isl_space_tuple_is_equal(map->dim, isl_dim_in,
+					map->dim, isl_dim_out))
+		return 0;
+
+	*res = isl_union_map_add_map(*res,
+				     isl_map_deltas_map(isl_map_copy(map)));
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_map_deltas_map(
+	__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &deltas_map_entry);
+}
+
+static int identity_entry(void **entry, void *user)
+{
+	isl_set *set = *entry;
+	isl_union_map **res = user;
+
+	*res = isl_union_map_add_map(*res, isl_set_identity(isl_set_copy(set)));
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_set_identity(__isl_take isl_union_set *uset)
+{
+	return cond_un_op(uset, &identity_entry);
+}
+
+/* If "map" is of the form [A -> B] -> C, then add A -> C to "res".
+ */
+static int domain_factor_domain_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_map **res = user;
+
+	if (!isl_map_domain_is_wrapping(map))
+		return 0;
+
+	*res = isl_union_map_add_map(*res,
+			    isl_map_domain_factor_domain(isl_map_copy(map)));
+
+	return *res ? 0 : -1;
+}
+
+/* For each map in "umap" of the form [A -> B] -> C,
+ * construct the map A -> C and collect the results.
+ */
+__isl_give isl_union_map *isl_union_map_domain_factor_domain(
+	__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &domain_factor_domain_entry);
+}
+
+/* If "map" is of the form [A -> B] -> C, then add B -> C to "res".
+ */
+static int domain_factor_range_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_map **res = user;
+
+	if (!isl_map_domain_is_wrapping(map))
+		return 0;
+
+	*res = isl_union_map_add_map(*res,
+				isl_map_domain_factor_range(isl_map_copy(map)));
+
+	return *res ? 0 : -1;
+}
+
+/* For each map in "umap" of the form [A -> B] -> C,
+ * construct the map B -> C and collect the results.
+ */
+__isl_give isl_union_map *isl_union_map_domain_factor_range(
+	__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &domain_factor_range_entry);
+}
+
+/* If "map" is of the form A -> [B -> C], then add A -> C to "res".
+ */
+static int range_factor_range_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_map **res = user;
+
+	if (!isl_map_range_is_wrapping(map))
+		return 0;
+
+	*res = isl_union_map_add_map(*res,
+				isl_map_range_factor_range(isl_map_copy(map)));
+
+	return *res ? 0 : -1;
+}
+
+/* For each map in "umap" of the form A -> [B -> C],
+ * construct the map A -> C and collect the results.
+ */
+__isl_give isl_union_map *isl_union_map_range_factor_range(
+	__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &range_factor_range_entry);
+}
+
+/* If "map" is of the form [A -> B] -> [C -> D], then add A -> C to "res".
+ */
+static int factor_domain_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_map **res = user;
+
+	if (!isl_map_domain_is_wrapping(map) || !isl_map_range_is_wrapping(map))
+		return 0;
+
+	*res = isl_union_map_add_map(*res,
+				isl_map_factor_domain(isl_map_copy(map)));
+
+	return *res ? 0 : -1;
+}
+
+/* For each map in "umap" of the form [A -> B] -> [C -> D],
+ * construct the map A -> C and collect the results.
+ */
+__isl_give isl_union_map *isl_union_map_factor_domain(
+	__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &factor_domain_entry);
+}
+
+/* If "map" is of the form [A -> B] -> [C -> D], then add B -> D to "res".
+ */
+static int factor_range_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_map **res = user;
+
+	if (!isl_map_domain_is_wrapping(map) || !isl_map_range_is_wrapping(map))
+		return 0;
+
+	*res = isl_union_map_add_map(*res,
+				isl_map_factor_range(isl_map_copy(map)));
+
+	return *res ? 0 : -1;
+}
+
+/* For each map in "umap" of the form [A -> B] -> [C -> D],
+ * construct the map B -> D and collect the results.
+ */
+__isl_give isl_union_map *isl_union_map_factor_range(
+	__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &factor_range_entry);
+}
+
+static int unwrap_entry(void **entry, void *user)
+{
+	isl_set *set = *entry;
+	isl_union_set **res = user;
+
+	if (!isl_set_is_wrapping(set))
+		return 0;
+
+	*res = isl_union_map_add_map(*res, isl_set_unwrap(isl_set_copy(set)));
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_set_unwrap(__isl_take isl_union_set *uset)
+{
+	return cond_un_op(uset, &unwrap_entry);
+}
+
+static int wrap_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_set **res = user;
+
+	*res = isl_union_set_add_set(*res, isl_map_wrap(isl_map_copy(map)));
+
+	return 0;
+}
+
+__isl_give isl_union_set *isl_union_map_wrap(__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &wrap_entry);
+}
+
+struct isl_union_map_is_subset_data {
+	isl_union_map *umap2;
+	int is_subset;
+};
+
+static int is_subset_entry(void **entry, void *user)
+{
+	struct isl_union_map_is_subset_data *data = user;
+	uint32_t hash;
+	struct isl_hash_table_entry *entry2;
+	isl_map *map = *entry;
+
+	hash = isl_space_get_hash(map->dim);
+	entry2 = isl_hash_table_find(data->umap2->dim->ctx, &data->umap2->table,
+				     hash, &has_dim, map->dim, 0);
+	if (!entry2) {
+		int empty = isl_map_is_empty(map);
+		if (empty < 0)
+			return -1;
+		if (empty)
+			return 0;
+		data->is_subset = 0;
+		return -1;
+	}
+
+	data->is_subset = isl_map_is_subset(map, entry2->data);
+	if (data->is_subset < 0 || !data->is_subset)
+		return -1;
+
+	return 0;
+}
+
+int isl_union_map_is_subset(__isl_keep isl_union_map *umap1,
+	__isl_keep isl_union_map *umap2)
+{
+	struct isl_union_map_is_subset_data data = { NULL, 1 };
+
+	umap1 = isl_union_map_copy(umap1);
+	umap2 = isl_union_map_copy(umap2);
+	umap1 = isl_union_map_align_params(umap1, isl_union_map_get_space(umap2));
+	umap2 = isl_union_map_align_params(umap2, isl_union_map_get_space(umap1));
+
+	if (!umap1 || !umap2)
+		goto error;
+
+	data.umap2 = umap2;
+	if (isl_hash_table_foreach(umap1->dim->ctx, &umap1->table,
+				   &is_subset_entry, &data) < 0 &&
+	    data.is_subset)
+		goto error;
+
+	isl_union_map_free(umap1);
+	isl_union_map_free(umap2);
+
+	return data.is_subset;
+error:
+	isl_union_map_free(umap1);
+	isl_union_map_free(umap2);
+	return -1;
+}
+
+int isl_union_set_is_subset(__isl_keep isl_union_set *uset1,
+	__isl_keep isl_union_set *uset2)
+{
+	return isl_union_map_is_subset(uset1, uset2);
+}
+
+int isl_union_map_is_equal(__isl_keep isl_union_map *umap1,
+	__isl_keep isl_union_map *umap2)
+{
+	int is_subset;
+
+	if (!umap1 || !umap2)
+		return -1;
+	is_subset = isl_union_map_is_subset(umap1, umap2);
+	if (is_subset != 1)
+		return is_subset;
+	is_subset = isl_union_map_is_subset(umap2, umap1);
+	return is_subset;
+}
+
+int isl_union_set_is_equal(__isl_keep isl_union_set *uset1,
+	__isl_keep isl_union_set *uset2)
+{
+	return isl_union_map_is_equal(uset1, uset2);
+}
+
+int isl_union_map_is_strict_subset(__isl_keep isl_union_map *umap1,
+	__isl_keep isl_union_map *umap2)
+{
+	int is_subset;
+
+	if (!umap1 || !umap2)
+		return -1;
+	is_subset = isl_union_map_is_subset(umap1, umap2);
+	if (is_subset != 1)
+		return is_subset;
+	is_subset = isl_union_map_is_subset(umap2, umap1);
+	if (is_subset == -1)
+		return is_subset;
+	return !is_subset;
+}
+
+int isl_union_set_is_strict_subset(__isl_keep isl_union_set *uset1,
+	__isl_keep isl_union_set *uset2)
+{
+	return isl_union_map_is_strict_subset(uset1, uset2);
+}
+
+static int sample_entry(void **entry, void *user)
+{
+	isl_basic_map **sample = (isl_basic_map **)user;
+	isl_map *map = *entry;
+
+	*sample = isl_map_sample(isl_map_copy(map));
+	if (!*sample)
+		return -1;
+	if (!isl_basic_map_plain_is_empty(*sample))
+		return -1;
+	return 0;
+}
+
+__isl_give isl_basic_map *isl_union_map_sample(__isl_take isl_union_map *umap)
+{
+	isl_basic_map *sample = NULL;
+
+	if (!umap)
+		return NULL;
+
+	if (isl_hash_table_foreach(umap->dim->ctx, &umap->table,
+				   &sample_entry, &sample) < 0 &&
+	    !sample)
+		goto error;
+
+	if (!sample)
+		sample = isl_basic_map_empty(isl_union_map_get_space(umap));
+
+	isl_union_map_free(umap);
+
+	return sample;
+error:
+	isl_union_map_free(umap);
+	return NULL;
+}
+
+__isl_give isl_basic_set *isl_union_set_sample(__isl_take isl_union_set *uset)
+{
+	return (isl_basic_set *)isl_union_map_sample(uset);
+}
+
+struct isl_forall_data {
+	int res;
+	int (*fn)(__isl_keep isl_map *map);
+};
+
+static int forall_entry(void **entry, void *user)
+{
+	struct isl_forall_data *data = user;
+	isl_map *map = *entry;
+
+	data->res = data->fn(map);
+	if (data->res < 0)
+		return -1;
+
+	if (!data->res)
+		return -1;
+
+	return 0;
+}
+
+static int union_map_forall(__isl_keep isl_union_map *umap,
+	int (*fn)(__isl_keep isl_map *map))
+{
+	struct isl_forall_data data = { 1, fn };
+
+	if (!umap)
+		return -1;
+
+	if (isl_hash_table_foreach(umap->dim->ctx, &umap->table,
+				   &forall_entry, &data) < 0 && data.res)
+		return -1;
+
+	return data.res;
+}
+
+struct isl_forall_user_data {
+	int res;
+	int (*fn)(__isl_keep isl_map *map, void *user);
+	void *user;
+};
+
+static int forall_user_entry(void **entry, void *user)
+{
+	struct isl_forall_user_data *data = user;
+	isl_map *map = *entry;
+
+	data->res = data->fn(map, data->user);
+	if (data->res < 0)
+		return -1;
+
+	if (!data->res)
+		return -1;
+
+	return 0;
+}
+
+/* Check if fn(map, user) returns true for all maps "map" in umap.
+ */
+static int union_map_forall_user(__isl_keep isl_union_map *umap,
+	int (*fn)(__isl_keep isl_map *map, void *user), void *user)
+{
+	struct isl_forall_user_data data = { 1, fn, user };
+
+	if (!umap)
+		return -1;
+
+	if (isl_hash_table_foreach(umap->dim->ctx, &umap->table,
+				   &forall_user_entry, &data) < 0 && data.res)
+		return -1;
+
+	return data.res;
+}
+
+int isl_union_map_is_empty(__isl_keep isl_union_map *umap)
+{
+	return union_map_forall(umap, &isl_map_is_empty);
+}
+
+int isl_union_set_is_empty(__isl_keep isl_union_set *uset)
+{
+	return isl_union_map_is_empty(uset);
+}
+
+static int is_subset_of_identity(__isl_keep isl_map *map)
+{
+	int is_subset;
+	isl_space *dim;
+	isl_map *id;
+
+	if (!map)
+		return -1;
+
+	if (!isl_space_tuple_is_equal(map->dim, isl_dim_in,
+					map->dim, isl_dim_out))
+		return 0;
+
+	dim = isl_map_get_space(map);
+	id = isl_map_identity(dim);
+
+	is_subset = isl_map_is_subset(map, id);
+
+	isl_map_free(id);
+
+	return is_subset;
+}
+
+/* Given an isl_union_map that consists of a single map, check
+ * if it is single-valued.
+ */
+static int single_map_is_single_valued(__isl_keep isl_union_map *umap)
+{
+	isl_map *map;
+	int sv;
+
+	umap = isl_union_map_copy(umap);
+	map = isl_map_from_union_map(umap);
+	sv = isl_map_is_single_valued(map);
+	isl_map_free(map);
+
+	return sv;
+}
+
+/* Internal data structure for single_valued_on_domain.
+ *
+ * "umap" is the union map to be tested.
+ * "sv" is set to 1 as long as "umap" may still be single-valued.
+ */
+struct isl_union_map_is_sv_data {
+	isl_union_map *umap;
+	int sv;
+};
+
+/* Check if the data->umap is single-valued on "set".
+ *
+ * If data->umap consists of a single map on "set", then test it
+ * as an isl_map.
+ *
+ * Otherwise, compute
+ *
+ *	M \circ M^-1
+ *
+ * check if the result is a subset of the identity mapping and
+ * store the result in data->sv.
+ *
+ * Terminate as soon as data->umap has been determined not to
+ * be single-valued.
+ */
+static int single_valued_on_domain(__isl_take isl_set *set, void *user)
+{
+	struct isl_union_map_is_sv_data *data = user;
+	isl_union_map *umap, *test;
+
+	umap = isl_union_map_copy(data->umap);
+	umap = isl_union_map_intersect_domain(umap,
+						isl_union_set_from_set(set));
+
+	if (isl_union_map_n_map(umap) == 1) {
+		data->sv = single_map_is_single_valued(umap);
+		isl_union_map_free(umap);
+	} else {
+		test = isl_union_map_reverse(isl_union_map_copy(umap));
+		test = isl_union_map_apply_range(test, umap);
+
+		data->sv = union_map_forall(test, &is_subset_of_identity);
+
+		isl_union_map_free(test);
+	}
+
+	if (data->sv < 0 || !data->sv)
+		return -1;
+	return 0;
+}
+
+/* Check if the given map is single-valued.
+ *
+ * If the union map consists of a single map, then test it as an isl_map.
+ * Otherwise, check if the union map is single-valued on each of its
+ * domain spaces.
+ */
+int isl_union_map_is_single_valued(__isl_keep isl_union_map *umap)
+{
+	isl_union_map *universe;
+	isl_union_set *domain;
+	struct isl_union_map_is_sv_data data;
+
+	if (isl_union_map_n_map(umap) == 1)
+		return single_map_is_single_valued(umap);
+
+	universe = isl_union_map_universe(isl_union_map_copy(umap));
+	domain = isl_union_map_domain(universe);
+
+	data.sv = 1;
+	data.umap = umap;
+	if (isl_union_set_foreach_set(domain,
+			    &single_valued_on_domain, &data) < 0 && data.sv)
+		data.sv = -1;
+	isl_union_set_free(domain);
+
+	return data.sv;
+}
+
+int isl_union_map_is_injective(__isl_keep isl_union_map *umap)
+{
+	int in;
+
+	umap = isl_union_map_copy(umap);
+	umap = isl_union_map_reverse(umap);
+	in = isl_union_map_is_single_valued(umap);
+	isl_union_map_free(umap);
+
+	return in;
+}
+
+/* Represents a map that has a fixed value (v) for one of its
+ * range dimensions.
+ * The map in this structure is not reference counted, so it
+ * is only valid while the isl_union_map from which it was
+ * obtained is still alive.
+ */
+struct isl_fixed_map {
+	isl_int v;
+	isl_map *map;
+};
+
+static struct isl_fixed_map *alloc_isl_fixed_map_array(isl_ctx *ctx,
+	int n)
+{
+	int i;
+	struct isl_fixed_map *v;
+
+	v = isl_calloc_array(ctx, struct isl_fixed_map, n);
+	if (!v)
+		return NULL;
+	for (i = 0; i < n; ++i)
+		isl_int_init(v[i].v);
+	return v;
+}
+
+static void free_isl_fixed_map_array(struct isl_fixed_map *v, int n)
+{
+	int i;
+
+	if (!v)
+		return;
+	for (i = 0; i < n; ++i)
+		isl_int_clear(v[i].v);
+	free(v);
+}
+
+/* Compare the "v" field of two isl_fixed_map structs.
+ */
+static int qsort_fixed_map_cmp(const void *p1, const void *p2)
+{
+	const struct isl_fixed_map *e1 = (const struct isl_fixed_map *) p1;
+	const struct isl_fixed_map *e2 = (const struct isl_fixed_map *) p2;
+
+	return isl_int_cmp(e1->v, e2->v);
+}
+
+/* Internal data structure used while checking whether all maps
+ * in a union_map have a fixed value for a given output dimension.
+ * v is the list of maps, with the fixed value for the dimension
+ * n is the number of maps considered so far
+ * pos is the output dimension under investigation
+ */
+struct isl_fixed_dim_data {
+	struct isl_fixed_map *v;
+	int n;
+	int pos;
+};
+
+static int fixed_at_pos(__isl_keep isl_map *map, void *user)
+{
+	struct isl_fixed_dim_data *data = user;
+
+	data->v[data->n].map = map;
+	return isl_map_plain_is_fixed(map, isl_dim_out, data->pos,
+				      &data->v[data->n++].v);
+}
+
+static int plain_injective_on_range(__isl_take isl_union_map *umap,
+	int first, int n_range);
+
+/* Given a list of the maps, with their fixed values at output dimension "pos",
+ * check whether the ranges of the maps form an obvious partition.
+ *
+ * We first sort the maps according to their fixed values.
+ * If all maps have a different value, then we know the ranges form
+ * a partition.
+ * Otherwise, we collect the maps with the same fixed value and
+ * check whether each such collection is obviously injective
+ * based on later dimensions.
+ */
+static int separates(struct isl_fixed_map *v, int n,
+	__isl_take isl_space *dim, int pos, int n_range)
+{
+	int i;
+
+	if (!v)
+		goto error;
+
+	qsort(v, n, sizeof(*v), &qsort_fixed_map_cmp);
+
+	for (i = 0; i + 1 < n; ++i) {
+		int j, k;
+		isl_union_map *part;
+		int injective;
+
+		for (j = i + 1; j < n; ++j)
+			if (isl_int_ne(v[i].v, v[j].v))
+				break;
+
+		if (j == i + 1)
+			continue;
+
+		part = isl_union_map_alloc(isl_space_copy(dim), j - i);
+		for (k = i; k < j; ++k)
+			part = isl_union_map_add_map(part,
+						     isl_map_copy(v[k].map));
+
+		injective = plain_injective_on_range(part, pos + 1, n_range);
+		if (injective < 0)
+			goto error;
+		if (!injective)
+			break;
+
+		i = j - 1;
+	}
+
+	isl_space_free(dim);
+	free_isl_fixed_map_array(v, n);
+	return i + 1 >= n;
+error:
+	isl_space_free(dim);
+	free_isl_fixed_map_array(v, n);
+	return -1;
+}
+
+/* Check whether the maps in umap have obviously distinct ranges.
+ * In particular, check for an output dimension in the range
+ * [first,n_range) for which all maps have a fixed value
+ * and then check if these values, possibly along with fixed values
+ * at later dimensions, entail distinct ranges.
+ */
+static int plain_injective_on_range(__isl_take isl_union_map *umap,
+	int first, int n_range)
+{
+	isl_ctx *ctx;
+	int n;
+	struct isl_fixed_dim_data data = { NULL };
+
+	ctx = isl_union_map_get_ctx(umap);
+
+	n = isl_union_map_n_map(umap);
+	if (!umap)
+		goto error;
+
+	if (n <= 1) {
+		isl_union_map_free(umap);
+		return 1;
+	}
+
+	if (first >= n_range) {
+		isl_union_map_free(umap);
+		return 0;
+	}
+
+	data.v = alloc_isl_fixed_map_array(ctx, n);
+	if (!data.v)
+		goto error;
+
+	for (data.pos = first; data.pos < n_range; ++data.pos) {
+		int fixed;
+		int injective;
+		isl_space *dim;
+
+		data.n = 0;
+		fixed = union_map_forall_user(umap, &fixed_at_pos, &data);
+		if (fixed < 0)
+			goto error;
+		if (!fixed)
+			continue;
+		dim = isl_union_map_get_space(umap);
+		injective = separates(data.v, n, dim, data.pos, n_range);
+		isl_union_map_free(umap);
+		return injective;
+	}
+
+	free_isl_fixed_map_array(data.v, n);
+	isl_union_map_free(umap);
+
+	return 0;
+error:
+	free_isl_fixed_map_array(data.v, n);
+	isl_union_map_free(umap);
+	return -1;
+}
+
+/* Check whether the maps in umap that map to subsets of "ran"
+ * have obviously distinct ranges.
+ */
+static int plain_injective_on_range_wrap(__isl_keep isl_set *ran, void *user)
+{
+	isl_union_map *umap = user;
+
+	umap = isl_union_map_copy(umap);
+	umap = isl_union_map_intersect_range(umap,
+			isl_union_set_from_set(isl_set_copy(ran)));
+	return plain_injective_on_range(umap, 0, isl_set_dim(ran, isl_dim_set));
+}
+
+/* Check if the given union_map is obviously injective.
+ *
+ * In particular, we first check if all individual maps are obviously
+ * injective and then check if all the ranges of these maps are
+ * obviously disjoint.
+ */
+int isl_union_map_plain_is_injective(__isl_keep isl_union_map *umap)
+{
+	int in;
+	isl_union_map *univ;
+	isl_union_set *ran;
+
+	in = union_map_forall(umap, &isl_map_plain_is_injective);
+	if (in < 0)
+		return -1;
+	if (!in)
+		return 0;
+
+	univ = isl_union_map_universe(isl_union_map_copy(umap));
+	ran = isl_union_map_range(univ);
+
+	in = union_map_forall_user(ran, &plain_injective_on_range_wrap, umap);
+
+	isl_union_set_free(ran);
+
+	return in;
+}
+
+int isl_union_map_is_bijective(__isl_keep isl_union_map *umap)
+{
+	int sv;
+
+	sv = isl_union_map_is_single_valued(umap);
+	if (sv < 0 || !sv)
+		return sv;
+
+	return isl_union_map_is_injective(umap);
+}
+
+static int zip_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_map **res = user;
+
+	if (!isl_map_can_zip(map))
+		return 0;
+
+	*res = isl_union_map_add_map(*res, isl_map_zip(isl_map_copy(map)));
+
+	return 0;
+}
+
+__isl_give isl_union_map *isl_union_map_zip(__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &zip_entry);
+}
+
+static int uncurry_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_map **res = user;
+
+	if (!isl_map_can_uncurry(map))
+		return 0;
+
+	*res = isl_union_map_add_map(*res, isl_map_uncurry(isl_map_copy(map)));
+
+	return 0;
+}
+
+/* Given a union map, take the maps of the form A -> (B -> C) and
+ * return the union of the corresponding maps (A -> B) -> C.
+ */
+__isl_give isl_union_map *isl_union_map_uncurry(__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &uncurry_entry);
+}
+
+static int curry_entry(void **entry, void *user)
+{
+	isl_map *map = *entry;
+	isl_union_map **res = user;
+
+	if (!isl_map_can_curry(map))
+		return 0;
+
+	*res = isl_union_map_add_map(*res, isl_map_curry(isl_map_copy(map)));
+
+	return 0;
+}
+
+/* Given a union map, take the maps of the form (A -> B) -> C and
+ * return the union of the corresponding maps A -> (B -> C).
+ */
+__isl_give isl_union_map *isl_union_map_curry(__isl_take isl_union_map *umap)
+{
+	return cond_un_op(umap, &curry_entry);
+}
+
+static int lift_entry(void **entry, void *user)
+{
+	isl_set *set = *entry;
+	isl_union_set **res = user;
+
+	*res = isl_union_set_add_set(*res, isl_set_lift(isl_set_copy(set)));
+
+	return 0;
+}
+
+__isl_give isl_union_set *isl_union_set_lift(__isl_take isl_union_set *uset)
+{
+	return cond_un_op(uset, &lift_entry);
+}
+
+static int coefficients_entry(void **entry, void *user)
+{
+	isl_set *set = *entry;
+	isl_union_set **res = user;
+
+	set = isl_set_copy(set);
+	set = isl_set_from_basic_set(isl_set_coefficients(set));
+	*res = isl_union_set_add_set(*res, set);
+
+	return 0;
+}
+
+__isl_give isl_union_set *isl_union_set_coefficients(
+	__isl_take isl_union_set *uset)
+{
+	isl_ctx *ctx;
+	isl_space *dim;
+	isl_union_set *res;
+
+	if (!uset)
+		return NULL;
+
+	ctx = isl_union_set_get_ctx(uset);
+	dim = isl_space_set_alloc(ctx, 0, 0);
+	res = isl_union_map_alloc(dim, uset->table.n);
+	if (isl_hash_table_foreach(uset->dim->ctx, &uset->table,
+				   &coefficients_entry, &res) < 0)
+		goto error;
+
+	isl_union_set_free(uset);
+	return res;
+error:
+	isl_union_set_free(uset);
+	isl_union_set_free(res);
+	return NULL;
+}
+
+static int solutions_entry(void **entry, void *user)
+{
+	isl_set *set = *entry;
+	isl_union_set **res = user;
+
+	set = isl_set_copy(set);
+	set = isl_set_from_basic_set(isl_set_solutions(set));
+	if (!*res)
+		*res = isl_union_set_from_set(set);
+	else
+		*res = isl_union_set_add_set(*res, set);
+
+	if (!*res)
+		return -1;
+
+	return 0;
+}
+
+__isl_give isl_union_set *isl_union_set_solutions(
+	__isl_take isl_union_set *uset)
+{
+	isl_union_set *res = NULL;
+
+	if (!uset)
+		return NULL;
+
+	if (uset->table.n == 0) {
+		res = isl_union_set_empty(isl_union_set_get_space(uset));
+		isl_union_set_free(uset);
+		return res;
+	}
+
+	if (isl_hash_table_foreach(uset->dim->ctx, &uset->table,
+				   &solutions_entry, &res) < 0)
+		goto error;
+
+	isl_union_set_free(uset);
+	return res;
+error:
+	isl_union_set_free(uset);
+	isl_union_set_free(res);
+	return NULL;
+}
+
+/* Is the domain space of "map" equal to "space"?
+ */
+static int domain_match(__isl_keep isl_map *map, __isl_keep isl_space *space)
+{
+	return isl_space_tuple_is_equal(map->dim, isl_dim_in,
+					space, isl_dim_out);
+}
+
+/* Is the range space of "map" equal to "space"?
+ */
+static int range_match(__isl_keep isl_map *map, __isl_keep isl_space *space)
+{
+	return isl_space_tuple_is_equal(map->dim, isl_dim_out,
+					space, isl_dim_out);
+}
+
+/* Is the set space of "map" equal to "space"?
+ */
+static int set_match(__isl_keep isl_map *map, __isl_keep isl_space *space)
+{
+	return isl_space_tuple_is_equal(map->dim, isl_dim_set,
+					space, isl_dim_out);
+}
+
+/* Internal data structure for preimage_pw_multi_aff.
+ *
+ * "pma" is the function under which the preimage should be taken.
+ * "space" is the space of "pma".
+ * "res" collects the results.
+ * "fn" computes the preimage for a given map.
+ * "match" returns true if "fn" can be called.
+ */
+struct isl_union_map_preimage_data {
+	isl_space *space;
+	isl_pw_multi_aff *pma;
+	isl_union_map *res;
+	int (*match)(__isl_keep isl_map *map, __isl_keep isl_space *space);
+	__isl_give isl_map *(*fn)(__isl_take isl_map *map,
+		__isl_take isl_pw_multi_aff *pma);
+};
+
+/* Call data->fn to compute the preimage of the domain or range of *entry
+ * under the function represented by data->pma, provided the domain/range
+ * space of *entry matches the target space of data->pma
+ * (as given by data->match), and add the result to data->res.
+ */
+static int preimage_entry(void **entry, void *user)
+{
+	int m;
+	isl_map *map = *entry;
+	struct isl_union_map_preimage_data *data = user;
+	int empty;
+
+	m = data->match(map, data->space);
+	if (m < 0)
+		return -1;
+	if (!m)
+		return 0;
+
+	map = isl_map_copy(map);
+	map = data->fn(map, isl_pw_multi_aff_copy(data->pma));
+
+	empty = isl_map_is_empty(map);
+	if (empty < 0 || empty) {
+		isl_map_free(map);
+		return empty < 0 ? -1 : 0;
+	}
+
+	data->res = isl_union_map_add_map(data->res, map);
+
+	return 0;
+}
+
+/* Compute the preimage of the domain or range of "umap" under the function
+ * represented by "pma".
+ * In other words, plug in "pma" in the domain or range of "umap".
+ * The function "fn" performs the actual preimage computation on a map,
+ * while "match" determines to which maps the function should be applied.
+ */
+static __isl_give isl_union_map *preimage_pw_multi_aff(
+	__isl_take isl_union_map *umap, __isl_take isl_pw_multi_aff *pma,
+	int (*match)(__isl_keep isl_map *map, __isl_keep isl_space *space),
+	__isl_give isl_map *(*fn)(__isl_take isl_map *map,
+		__isl_take isl_pw_multi_aff *pma))
+{
+	isl_ctx *ctx;
+	isl_space *space;
+	struct isl_union_map_preimage_data data;
+
+	umap = isl_union_map_align_params(umap,
+					    isl_pw_multi_aff_get_space(pma));
+	pma = isl_pw_multi_aff_align_params(pma, isl_union_map_get_space(umap));
+
+	if (!umap || !pma)
+		goto error;
+
+	ctx = isl_union_map_get_ctx(umap);
+	space = isl_union_map_get_space(umap);
+	data.space = isl_pw_multi_aff_get_space(pma);
+	data.pma = pma;
+	data.res = isl_union_map_alloc(space, umap->table.n);
+	data.match = match;
+	data.fn = fn;
+	if (isl_hash_table_foreach(ctx, &umap->table, &preimage_entry,
+					&data) < 0)
+		data.res = isl_union_map_free(data.res);
+
+	isl_space_free(data.space);
+	isl_union_map_free(umap);
+	isl_pw_multi_aff_free(pma);
+	return data.res;
+error:
+	isl_union_map_free(umap);
+	isl_pw_multi_aff_free(pma);
+	return NULL;
+}
+
+/* Compute the preimage of the domain of "umap" under the function
+ * represented by "pma".
+ * In other words, plug in "pma" in the domain of "umap".
+ * The result contains maps that live in the same spaces as the maps of "umap"
+ * with domain space equal to the target space of "pma",
+ * except that the domain has been replaced by the domain space of "pma".
+ */
+__isl_give isl_union_map *isl_union_map_preimage_domain_pw_multi_aff(
+	__isl_take isl_union_map *umap, __isl_take isl_pw_multi_aff *pma)
+{
+	return preimage_pw_multi_aff(umap, pma, &domain_match,
+					&isl_map_preimage_domain_pw_multi_aff);
+}
+
+/* Compute the preimage of the range of "umap" under the function
+ * represented by "pma".
+ * In other words, plug in "pma" in the range of "umap".
+ * The result contains maps that live in the same spaces as the maps of "umap"
+ * with range space equal to the target space of "pma",
+ * except that the range has been replaced by the domain space of "pma".
+ */
+__isl_give isl_union_map *isl_union_map_preimage_range_pw_multi_aff(
+	__isl_take isl_union_map *umap, __isl_take isl_pw_multi_aff *pma)
+{
+	return preimage_pw_multi_aff(umap, pma, &range_match,
+					&isl_map_preimage_range_pw_multi_aff);
+}
+
+/* Compute the preimage of "uset" under the function represented by "pma".
+ * In other words, plug in "pma" in "uset".
+ * The result contains sets that live in the same spaces as the sets of "uset"
+ * with space equal to the target space of "pma",
+ * except that the space has been replaced by the domain space of "pma".
+ */
+__isl_give isl_union_set *isl_union_set_preimage_pw_multi_aff(
+	__isl_take isl_union_set *uset, __isl_take isl_pw_multi_aff *pma)
+{
+	return preimage_pw_multi_aff(uset, pma, &set_match,
+					&isl_set_preimage_pw_multi_aff);
+}
+
+/* Compute the preimage of the domain of "umap" under the function
+ * represented by "ma".
+ * In other words, plug in "ma" in the domain of "umap".
+ * The result contains maps that live in the same spaces as the maps of "umap"
+ * with domain space equal to the target space of "ma",
+ * except that the domain has been replaced by the domain space of "ma".
+ */
+__isl_give isl_union_map *isl_union_map_preimage_domain_multi_aff(
+	__isl_take isl_union_map *umap, __isl_take isl_multi_aff *ma)
+{
+	return isl_union_map_preimage_domain_pw_multi_aff(umap,
+					isl_pw_multi_aff_from_multi_aff(ma));
+}
+
+/* Compute the preimage of the range of "umap" under the function
+ * represented by "ma".
+ * In other words, plug in "ma" in the range of "umap".
+ * The result contains maps that live in the same spaces as the maps of "umap"
+ * with range space equal to the target space of "ma",
+ * except that the range has been replaced by the domain space of "ma".
+ */
+__isl_give isl_union_map *isl_union_map_preimage_range_multi_aff(
+	__isl_take isl_union_map *umap, __isl_take isl_multi_aff *ma)
+{
+	return isl_union_map_preimage_range_pw_multi_aff(umap,
+					isl_pw_multi_aff_from_multi_aff(ma));
+}
+
+/* Compute the preimage of "uset" under the function represented by "ma".
+ * In other words, plug in "ma" in "uset".
+ * The result contains sets that live in the same spaces as the sets of "uset"
+ * with space equal to the target space of "ma",
+ * except that the space has been replaced by the domain space of "ma".
+ */
+__isl_give isl_union_map *isl_union_set_preimage_multi_aff(
+	__isl_take isl_union_set *uset, __isl_take isl_multi_aff *ma)
+{
+	return isl_union_set_preimage_pw_multi_aff(uset,
+					isl_pw_multi_aff_from_multi_aff(ma));
+}
+
+/* Internal data structure for preimage_multi_pw_aff.
+ *
+ * "mpa" is the function under which the preimage should be taken.
+ * "space" is the space of "mpa".
+ * "res" collects the results.
+ * "fn" computes the preimage for a given map.
+ * "match" returns true if "fn" can be called.
+ */
+struct isl_union_map_preimage_mpa_data {
+	isl_space *space;
+	isl_multi_pw_aff *mpa;
+	isl_union_map *res;
+	int (*match)(__isl_keep isl_map *map, __isl_keep isl_space *space);
+	__isl_give isl_map *(*fn)(__isl_take isl_map *map,
+		__isl_take isl_multi_pw_aff *mpa);
+};
+
+/* Call data->fn to compute the preimage of the domain or range of *entry
+ * under the function represented by data->mpa, provided the domain/range
+ * space of *entry matches the target space of data->mpa
+ * (as given by data->match), and add the result to data->res.
+ */
+static int preimage_mpa_entry(void **entry, void *user)
+{
+	int m;
+	isl_map *map = *entry;
+	struct isl_union_map_preimage_mpa_data *data = user;
+	int empty;
+
+	m = data->match(map, data->space);
+	if (m < 0)
+		return -1;
+	if (!m)
+		return 0;
+
+	map = isl_map_copy(map);
+	map = data->fn(map, isl_multi_pw_aff_copy(data->mpa));
+
+	empty = isl_map_is_empty(map);
+	if (empty < 0 || empty) {
+		isl_map_free(map);
+		return empty < 0 ? -1 : 0;
+	}
+
+	data->res = isl_union_map_add_map(data->res, map);
+
+	return 0;
+}
+
+/* Compute the preimage of the domain or range of "umap" under the function
+ * represented by "mpa".
+ * In other words, plug in "mpa" in the domain or range of "umap".
+ * The function "fn" performs the actual preimage computation on a map,
+ * while "match" determines to which maps the function should be applied.
+ */
+static __isl_give isl_union_map *preimage_multi_pw_aff(
+	__isl_take isl_union_map *umap, __isl_take isl_multi_pw_aff *mpa,
+	int (*match)(__isl_keep isl_map *map, __isl_keep isl_space *space),
+	__isl_give isl_map *(*fn)(__isl_take isl_map *map,
+		__isl_take isl_multi_pw_aff *mpa))
+{
+	isl_ctx *ctx;
+	isl_space *space;
+	struct isl_union_map_preimage_mpa_data data;
+
+	umap = isl_union_map_align_params(umap,
+					    isl_multi_pw_aff_get_space(mpa));
+	mpa = isl_multi_pw_aff_align_params(mpa, isl_union_map_get_space(umap));
+
+	if (!umap || !mpa)
+		goto error;
+
+	ctx = isl_union_map_get_ctx(umap);
+	space = isl_union_map_get_space(umap);
+	data.space = isl_multi_pw_aff_get_space(mpa);
+	data.mpa = mpa;
+	data.res = isl_union_map_alloc(space, umap->table.n);
+	data.match = match;
+	data.fn = fn;
+	if (isl_hash_table_foreach(ctx, &umap->table, &preimage_mpa_entry,
+					&data) < 0)
+		data.res = isl_union_map_free(data.res);
+
+	isl_space_free(data.space);
+	isl_union_map_free(umap);
+	isl_multi_pw_aff_free(mpa);
+	return data.res;
+error:
+	isl_union_map_free(umap);
+	isl_multi_pw_aff_free(mpa);
+	return NULL;
+}
+
+/* Compute the preimage of the domain of "umap" under the function
+ * represented by "mpa".
+ * In other words, plug in "mpa" in the domain of "umap".
+ * The result contains maps that live in the same spaces as the maps of "umap"
+ * with domain space equal to the target space of "mpa",
+ * except that the domain has been replaced by the domain space of "mpa".
+ */
+__isl_give isl_union_map *isl_union_map_preimage_domain_multi_pw_aff(
+	__isl_take isl_union_map *umap, __isl_take isl_multi_pw_aff *mpa)
+{
+	return preimage_multi_pw_aff(umap, mpa, &domain_match,
+					&isl_map_preimage_domain_multi_pw_aff);
+}
+
+/* Internal data structure for preimage_upma.
+ *
+ * "umap" is the map of which the preimage should be computed.
+ * "res" collects the results.
+ * "fn" computes the preimage for a given piecewise multi-affine function.
+ */
+struct isl_union_map_preimage_upma_data {
+	isl_union_map *umap;
+	isl_union_map *res;
+	__isl_give isl_union_map *(*fn)(__isl_take isl_union_map *umap,
+		__isl_take isl_pw_multi_aff *pma);
+};
+
+/* Call data->fn to compute the preimage of the domain or range of data->umap
+ * under the function represented by pma and add the result to data->res.
+ */
+static int preimage_upma(__isl_take isl_pw_multi_aff *pma, void *user)
+{
+	struct isl_union_map_preimage_upma_data *data = user;
+	isl_union_map *umap;
+
+	umap = isl_union_map_copy(data->umap);
+	umap = data->fn(umap, pma);
+	data->res = isl_union_map_union(data->res, umap);
+
+	return data->res ? 0 : -1;
+}
+
+/* Compute the preimage of the domain or range of "umap" under the function
+ * represented by "upma".
+ * In other words, plug in "upma" in the domain or range of "umap".
+ * The function "fn" performs the actual preimage computation
+ * on a piecewise multi-affine function.
+ */
+static __isl_give isl_union_map *preimage_union_pw_multi_aff(
+	__isl_take isl_union_map *umap,
+	__isl_take isl_union_pw_multi_aff *upma,
+	__isl_give isl_union_map *(*fn)(__isl_take isl_union_map *umap,
+		__isl_take isl_pw_multi_aff *pma))
+{
+	struct isl_union_map_preimage_upma_data data;
+
+	data.umap = umap;
+	data.res = isl_union_map_empty(isl_union_map_get_space(umap));
+	data.fn = fn;
+	if (isl_union_pw_multi_aff_foreach_pw_multi_aff(upma,
+						    &preimage_upma, &data) < 0)
+		data.res = isl_union_map_free(data.res);
+
+	isl_union_map_free(umap);
+	isl_union_pw_multi_aff_free(upma);
+
+	return data.res;
+}
+
+/* Compute the preimage of the domain of "umap" under the function
+ * represented by "upma".
+ * In other words, plug in "upma" in the domain of "umap".
+ * The result contains maps that live in the same spaces as the maps of "umap"
+ * with domain space equal to one of the target spaces of "upma",
+ * except that the domain has been replaced by one of the the domain spaces that
+ * corresponds to that target space of "upma".
+ */
+__isl_give isl_union_map *isl_union_map_preimage_domain_union_pw_multi_aff(
+	__isl_take isl_union_map *umap,
+	__isl_take isl_union_pw_multi_aff *upma)
+{
+	return preimage_union_pw_multi_aff(umap, upma,
+				&isl_union_map_preimage_domain_pw_multi_aff);
+}
+
+/* Compute the preimage of the range of "umap" under the function
+ * represented by "upma".
+ * In other words, plug in "upma" in the range of "umap".
+ * The result contains maps that live in the same spaces as the maps of "umap"
+ * with range space equal to one of the target spaces of "upma",
+ * except that the range has been replaced by one of the the domain spaces that
+ * corresponds to that target space of "upma".
+ */
+__isl_give isl_union_map *isl_union_map_preimage_range_union_pw_multi_aff(
+	__isl_take isl_union_map *umap,
+	__isl_take isl_union_pw_multi_aff *upma)
+{
+	return preimage_union_pw_multi_aff(umap, upma,
+				&isl_union_map_preimage_range_pw_multi_aff);
+}
+
+/* Compute the preimage of "uset" under the function represented by "upma".
+ * In other words, plug in "upma" in the range of "uset".
+ * The result contains sets that live in the same spaces as the sets of "uset"
+ * with space equal to one of the target spaces of "upma",
+ * except that the space has been replaced by one of the the domain spaces that
+ * corresponds to that target space of "upma".
+ */
+__isl_give isl_union_set *isl_union_set_preimage_union_pw_multi_aff(
+	__isl_take isl_union_set *uset,
+	__isl_take isl_union_pw_multi_aff *upma)
+{
+	return preimage_union_pw_multi_aff(uset, upma,
+					&isl_union_set_preimage_pw_multi_aff);
+}
+
+/* Reset the user pointer on all identifiers of parameters and tuples
+ * of the space of *entry.
+ */
+static int reset_user(void **entry, void *user)
+{
+	isl_map **map = (isl_map **)entry;
+
+	*map = isl_map_reset_user(*map);
+
+	return *map ? 0 : -1;
+}
+
+/* Reset the user pointer on all identifiers of parameters and tuples
+ * of the spaces of "umap".
+ */
+__isl_give isl_union_map *isl_union_map_reset_user(
+	__isl_take isl_union_map *umap)
+{
+	umap = isl_union_map_cow(umap);
+	if (!umap)
+		return NULL;
+	umap->dim = isl_space_reset_user(umap->dim);
+	if (!umap->dim)
+		return isl_union_map_free(umap);
+	umap = un_op(umap, &reset_user);
+
+	return umap;
+}
+
+/* Reset the user pointer on all identifiers of parameters and tuples
+ * of the spaces of "uset".
+ */
+__isl_give isl_union_set *isl_union_set_reset_user(
+	__isl_take isl_union_set *uset)
+{
+	return isl_union_map_reset_user(uset);
+}
+
+/* Internal data structure for isl_union_map_project_out.
+ * "type", "first" and "n" are the arguments for the isl_map_project_out
+ * call.
+ * "res" collects the results.
+ */
+struct isl_union_map_project_out_data {
+	enum isl_dim_type type;
+	unsigned first;
+	unsigned n;
+
+	isl_union_map *res;
+};
+
+/* Turn the data->n dimensions of type data->type, starting at data->first
+ * into existentially quantified variables and add the result to data->res.
+ */
+static int project_out(__isl_take isl_map *map, void *user)
+{
+	struct isl_union_map_project_out_data *data = user;
+
+	map = isl_map_project_out(map, data->type, data->first, data->n);
+	data->res = isl_union_map_add_map(data->res, map);
+
+	return 0;
+}
+
+/* Turn the "n" dimensions of type "type", starting at "first"
+ * into existentially quantified variables.
+ * Since the space of an isl_union_map only contains parameters,
+ * type is required to be equal to isl_dim_param.
+ */
+__isl_give isl_union_map *isl_union_map_project_out(
+	__isl_take isl_union_map *umap,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	isl_space *space;
+	struct isl_union_map_project_out_data data = { type, first, n };
+
+	if (!umap)
+		return NULL;
+
+	if (type != isl_dim_param)
+		isl_die(isl_union_map_get_ctx(umap), isl_error_invalid,
+			"can only project out parameters",
+			return isl_union_map_free(umap));
+
+	space = isl_union_map_get_space(umap);
+	space = isl_space_drop_dims(space, type, first, n);
+	data.res = isl_union_map_empty(space);
+	if (isl_union_map_foreach_map(umap, &project_out, &data) < 0)
+		data.res = isl_union_map_free(data.res);
+
+	isl_union_map_free(umap);
+
+	return data.res;
+}
+
+/* Internal data structure for isl_union_map_involves_dims.
+ * "first" and "n" are the arguments for the isl_map_involves_dims calls.
+ */
+struct isl_union_map_involves_dims_data {
+	unsigned first;
+	unsigned n;
+};
+
+/* Does "map" _not_ involve the data->n parameters starting at data->first?
+ */
+static int map_excludes(__isl_keep isl_map *map, void *user)
+{
+	struct isl_union_map_involves_dims_data *data = user;
+	int involves;
+
+	involves = isl_map_involves_dims(map,
+					isl_dim_param, data->first, data->n);
+	if (involves < 0)
+		return -1;
+	return !involves;
+}
+
+/* Does "umap" involve any of the n parameters starting at first?
+ * "type" is required to be set to isl_dim_param.
+ *
+ * "umap" involves any of those parameters if any of its maps
+ * involve the parameters.  In other words, "umap" does not
+ * involve any of the parameters if all its maps to not
+ * involve the parameters.
+ */
+int isl_union_map_involves_dims(__isl_keep isl_union_map *umap,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	struct isl_union_map_involves_dims_data data = { first, n };
+	int excludes;
+
+	if (type != isl_dim_param)
+		isl_die(isl_union_map_get_ctx(umap), isl_error_invalid,
+			"can only reference parameters", return 0);
+
+	excludes = union_map_forall_user(umap, &map_excludes, &data);
+
+	if (excludes < 0)
+		return -1;
+
+	return !excludes;
+}
diff --git a/polly/lib/External/isl/isl_union_map_private.h b/polly/lib/External/isl/isl_union_map_private.h
new file mode 100644
index 00000000000..7a502e7bdd5
--- /dev/null
+++ b/polly/lib/External/isl/isl_union_map_private.h
@@ -0,0 +1,10 @@
+#define isl_union_set	isl_union_map
+#include <isl/union_map.h>
+#include <isl/union_set.h>
+
+struct isl_union_map {
+	int ref;
+	isl_space *dim;
+
+	struct isl_hash_table	table;
+};
diff --git a/polly/lib/External/isl/isl_union_templ.c b/polly/lib/External/isl/isl_union_templ.c
new file mode 100644
index 00000000000..4cd1d9dd183
--- /dev/null
+++ b/polly/lib/External/isl/isl_union_templ.c
@@ -0,0 +1,1107 @@
+/*
+ * Copyright 2010      INRIA Saclay
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France 
+ */
+
+#define xFN(TYPE,NAME) TYPE ## _ ## NAME
+#define FN(TYPE,NAME) xFN(TYPE,NAME)
+#define xS(TYPE,NAME) struct TYPE ## _ ## NAME
+#define S(TYPE,NAME) xS(TYPE,NAME)
+
+struct UNION {
+	int ref;
+#ifdef HAS_TYPE
+	enum isl_fold type;
+#endif
+	isl_space *space;
+
+	struct isl_hash_table	table;
+};
+
+__isl_give UNION *FN(UNION,cow)(__isl_take UNION *u);
+
+isl_ctx *FN(UNION,get_ctx)(__isl_keep UNION *u)
+{
+	return u ? u->space->ctx : NULL;
+}
+
+__isl_give isl_space *FN(UNION,get_space)(__isl_keep UNION *u)
+{
+	if (!u)
+		return NULL;
+	return isl_space_copy(u->space);
+}
+
+/* Return the number of parameters of "u", where "type"
+ * is required to be set to isl_dim_param.
+ */
+unsigned FN(UNION,dim)(__isl_keep UNION *u, enum isl_dim_type type)
+{
+	if (!u)
+		return 0;
+
+	if (type != isl_dim_param)
+		isl_die(FN(UNION,get_ctx)(u), isl_error_invalid,
+			"can only reference parameters", return 0);
+
+	return isl_space_dim(u->space, type);
+}
+
+/* Return the position of the parameter with the given name
+ * in "u".
+ * Return -1 if no such dimension can be found.
+ */
+int FN(UNION,find_dim_by_name)(__isl_keep UNION *u, enum isl_dim_type type,
+	const char *name)
+{
+	if (!u)
+		return -1;
+	return isl_space_find_dim_by_name(u->space, type, name);
+}
+
+#ifdef HAS_TYPE
+static __isl_give UNION *FN(UNION,alloc)(__isl_take isl_space *dim,
+	enum isl_fold type, int size)
+#else
+static __isl_give UNION *FN(UNION,alloc)(__isl_take isl_space *dim, int size)
+#endif
+{
+	UNION *u;
+
+	dim = isl_space_params(dim);
+	if (!dim)
+		return NULL;
+
+	u = isl_calloc_type(dim->ctx, UNION);
+	if (!u)
+		goto error;
+
+	u->ref = 1;
+#ifdef HAS_TYPE
+	u->type = type;
+#endif
+	u->space = dim;
+	if (isl_hash_table_init(dim->ctx, &u->table, size) < 0)
+		return FN(UNION,free)(u);
+
+	return u;
+error:
+	isl_space_free(dim);
+	return NULL;
+}
+
+#ifdef HAS_TYPE
+__isl_give UNION *FN(UNION,ZERO)(__isl_take isl_space *dim, enum isl_fold type)
+{
+	return FN(UNION,alloc)(dim, type, 16);
+}
+#else
+__isl_give UNION *FN(UNION,ZERO)(__isl_take isl_space *dim)
+{
+	return FN(UNION,alloc)(dim, 16);
+}
+#endif
+
+__isl_give UNION *FN(UNION,copy)(__isl_keep UNION *u)
+{
+	if (!u)
+		return NULL;
+
+	u->ref++;
+	return u;
+}
+
+S(UNION,foreach_data)
+{
+	int (*fn)(__isl_take PART *part, void *user);
+	void *user;
+};
+
+static int call_on_copy(void **entry, void *user)
+{
+	PART *part = *entry;
+	S(UNION,foreach_data) *data = (S(UNION,foreach_data) *)user;
+
+	return data->fn(FN(PART,copy)(part), data->user);
+}
+
+int FN(FN(UNION,foreach),PARTS)(__isl_keep UNION *u,
+	int (*fn)(__isl_take PART *part, void *user), void *user)
+{
+	S(UNION,foreach_data) data = { fn, user };
+
+	if (!u)
+		return -1;
+
+	return isl_hash_table_foreach(u->space->ctx, &u->table,
+				      &call_on_copy, &data);
+}
+
+/* Is the space of "entry" equal to "space"?
+ */
+static int has_space(const void *entry, const void *val)
+{
+	PART *part = (PART *)entry;
+	isl_space *space = (isl_space *) val;
+
+	return isl_space_is_equal(part->dim, space);
+}
+
+/* This function is not currently used by isl_aff.c.
+ */
+static int has_domain_space(const void *entry, const void *val)
+	__attribute__ ((unused));
+
+/* Is the domain space of "entry" equal to "space"?
+ */
+static int has_domain_space(const void *entry, const void *val)
+{
+	PART *part = (PART *)entry;
+	isl_space *space = (isl_space *) val;
+
+	if (isl_space_is_params(space))
+		return isl_space_is_set(part->dim);
+
+	return isl_space_tuple_is_equal(part->dim, isl_dim_in,
+					space, isl_dim_set);
+}
+
+/* Is the domain space of "entry" equal to the domain of "space"?
+ */
+static int has_same_domain_space(const void *entry, const void *val)
+{
+	PART *part = (PART *)entry;
+	isl_space *space = (isl_space *) val;
+
+	if (isl_space_is_set(space))
+		return isl_space_is_set(part->dim);
+
+	return isl_space_tuple_is_equal(part->dim, isl_dim_in,
+					space, isl_dim_in);
+}
+
+/* Extract the element of "u" living in "space" (ignoring parameters).
+ *
+ * Return the ZERO element if "u" does not contain any element
+ * living in "space".
+ */
+__isl_give PART *FN(FN(UNION,extract),PARTS)(__isl_keep UNION *u,
+	__isl_take isl_space *space)
+{
+	uint32_t hash;
+	struct isl_hash_table_entry *entry;
+
+	if (!u || !space)
+		goto error;
+	if (!isl_space_match(u->space, isl_dim_param, space, isl_dim_param)) {
+		space = isl_space_drop_dims(space, isl_dim_param,
+					0, isl_space_dim(space, isl_dim_param));
+		space = isl_space_align_params(space,
+					FN(UNION,get_space)(u));
+		if (!space)
+			goto error;
+	}
+
+	hash = isl_space_get_hash(space);
+	entry = isl_hash_table_find(u->space->ctx, &u->table, hash,
+				    &has_space, space, 0);
+	if (!entry)
+#ifdef HAS_TYPE
+		return FN(PART,ZERO)(space, u->type);
+#else
+		return FN(PART,ZERO)(space);
+#endif
+	isl_space_free(space);
+	return FN(PART,copy)(entry->data);
+error:
+	isl_space_free(space);
+	return NULL;
+}
+
+/* Add "part" to "u".
+ * If "disjoint" is set, then "u" is not allowed to already have
+ * a part that is defined on the same space as "part".
+ * Otherwise, compute the union sum of "part" and the part in "u"
+ * defined on the same space.
+ */
+static __isl_give UNION *FN(UNION,add_part_generic)(__isl_take UNION *u,
+	__isl_take PART *part, int disjoint)
+{
+	int empty;
+	uint32_t hash;
+	struct isl_hash_table_entry *entry;
+
+	if (!part)
+		goto error;
+
+	empty = FN(PART,IS_ZERO)(part);
+	if (empty < 0)
+		goto error;
+	if (empty) {
+		FN(PART,free)(part);
+		return u;
+	}
+
+	u = FN(UNION,align_params)(u, FN(PART,get_space)(part));
+	part = FN(PART,align_params)(part, FN(UNION,get_space)(u));
+
+	u = FN(UNION,cow)(u);
+
+	if (!u)
+		goto error;
+
+	hash = isl_space_get_hash(part->dim);
+	entry = isl_hash_table_find(u->space->ctx, &u->table, hash,
+				    &has_same_domain_space, part->dim, 1);
+	if (!entry)
+		goto error;
+
+	if (!entry->data)
+		entry->data = part;
+	else {
+		PART *entry_part = entry->data;
+		if (disjoint)
+			isl_die(FN(UNION,get_ctx)(u), isl_error_invalid,
+				"additional part should live on separate "
+				"space", goto error);
+		if (!isl_space_tuple_is_equal(entry_part->dim, isl_dim_out,
+						part->dim, isl_dim_out))
+			isl_die(FN(UNION,get_ctx)(u), isl_error_invalid,
+				"union expression can only contain a single "
+				"expression over a given domain", goto error);
+		entry->data = FN(PART,union_add_)(entry->data,
+						FN(PART,copy)(part));
+		if (!entry->data)
+			goto error;
+		empty = FN(PART,IS_ZERO)(part);
+		if (empty < 0)
+			goto error;
+		if (empty) {
+			FN(PART,free)(entry->data);
+			isl_hash_table_remove(u->space->ctx, &u->table, entry);
+		}
+		FN(PART,free)(part);
+	}
+
+	return u;
+error:
+	FN(PART,free)(part);
+	FN(UNION,free)(u);
+	return NULL;
+}
+
+/* Add "part" to "u", where "u" is assumed not to already have
+ * a part that is defined on the same space as "part".
+ */
+__isl_give UNION *FN(FN(UNION,add),PARTS)(__isl_take UNION *u,
+	__isl_take PART *part)
+{
+	return FN(UNION,add_part_generic)(u, part, 1);
+}
+
+static int add_part(__isl_take PART *part, void *user)
+{
+	UNION **u = (UNION **)user;
+
+	*u = FN(FN(UNION,add),PARTS)(*u, part);
+
+	return 0;
+}
+
+__isl_give UNION *FN(UNION,dup)(__isl_keep UNION *u)
+{
+	UNION *dup;
+
+	if (!u)
+		return NULL;
+
+#ifdef HAS_TYPE
+	dup = FN(UNION,ZERO)(isl_space_copy(u->space), u->type);
+#else
+	dup = FN(UNION,ZERO)(isl_space_copy(u->space));
+#endif
+	if (FN(FN(UNION,foreach),PARTS)(u, &add_part, &dup) < 0)
+		goto error;
+	return dup;
+error:
+	FN(UNION,free)(dup);
+	return NULL;
+}
+
+__isl_give UNION *FN(UNION,cow)(__isl_take UNION *u)
+{
+	if (!u)
+		return NULL;
+
+	if (u->ref == 1)
+		return u;
+	u->ref--;
+	return FN(UNION,dup)(u);
+}
+
+static int free_u_entry(void **entry, void *user)
+{
+	PART *part = *entry;
+	FN(PART,free)(part);
+	return 0;
+}
+
+__isl_null UNION *FN(UNION,free)(__isl_take UNION *u)
+{
+	if (!u)
+		return NULL;
+
+	if (--u->ref > 0)
+		return NULL;
+
+	isl_hash_table_foreach(u->space->ctx, &u->table, &free_u_entry, NULL);
+	isl_hash_table_clear(&u->table);
+	isl_space_free(u->space);
+	free(u);
+	return NULL;
+}
+
+S(UNION,align) {
+	isl_reordering *exp;
+	UNION *res;
+};
+
+#ifdef ALIGN_DOMAIN
+static int align_entry(__isl_take PART *part, void *user)
+{
+	isl_reordering *exp;
+	S(UNION,align) *data = user;
+
+	exp = isl_reordering_extend_space(isl_reordering_copy(data->exp),
+				    FN(PART,get_domain_space)(part));
+
+	data->res = FN(FN(UNION,add),PARTS)(data->res,
+					    FN(PART,realign_domain)(part, exp));
+
+	return 0;
+}
+#else
+static int align_entry(__isl_take PART *part, void *user)
+{
+	isl_reordering *exp;
+	S(UNION,align) *data = user;
+
+	exp = isl_reordering_extend_space(isl_reordering_copy(data->exp),
+				    FN(PART,get_space)(part));
+
+	data->res = FN(FN(UNION,add),PARTS)(data->res,
+					    FN(PART,realign)(part, exp));
+
+	return 0;
+}
+#endif
+
+__isl_give UNION *FN(UNION,align_params)(__isl_take UNION *u,
+	__isl_take isl_space *model)
+{
+	S(UNION,align) data = { NULL, NULL };
+
+	if (!u || !model)
+		goto error;
+
+	if (isl_space_match(u->space, isl_dim_param, model, isl_dim_param)) {
+		isl_space_free(model);
+		return u;
+	}
+
+	model = isl_space_params(model);
+	data.exp = isl_parameter_alignment_reordering(u->space, model);
+	if (!data.exp)
+		goto error;
+
+#ifdef HAS_TYPE
+	data.res = FN(UNION,alloc)(isl_space_copy(data.exp->dim),
+						u->type, u->table.n);
+#else
+	data.res = FN(UNION,alloc)(isl_space_copy(data.exp->dim), u->table.n);
+#endif
+	if (FN(FN(UNION,foreach),PARTS)(u, &align_entry, &data) < 0)
+		goto error;
+
+	isl_reordering_free(data.exp);
+	FN(UNION,free)(u);
+	isl_space_free(model);
+	return data.res;
+error:
+	isl_reordering_free(data.exp);
+	FN(UNION,free)(u);
+	FN(UNION,free)(data.res);
+	isl_space_free(model);
+	return NULL;
+}
+
+/* Add "part" to *u, taking the union sum if "u" already has
+ * a part defined on the same space as "part".
+ */
+static int union_add_part(__isl_take PART *part, void *user)
+{
+	UNION **u = (UNION **)user;
+
+	*u = FN(UNION,add_part_generic)(*u, part, 0);
+
+	return 0;
+}
+
+/* Compute the sum of "u1" and "u2" on the union of their domains,
+ * with the actual sum on the shared domain and
+ * the defined expression on the symmetric difference of the domains.
+ *
+ * This is an internal function that is exposed under different
+ * names depending on whether the base expressions have a zero default
+ * value.
+ * If they do, then this function is called "add".
+ * Otherwise, it is called "union_add".
+ */
+static __isl_give UNION *FN(UNION,union_add_)(__isl_take UNION *u1,
+	__isl_take UNION *u2)
+{
+	u1 = FN(UNION,align_params)(u1, FN(UNION,get_space)(u2));
+	u2 = FN(UNION,align_params)(u2, FN(UNION,get_space)(u1));
+
+	u1 = FN(UNION,cow)(u1);
+
+	if (!u1 || !u2)
+		goto error;
+
+	if (FN(FN(UNION,foreach),PARTS)(u2, &union_add_part, &u1) < 0)
+		goto error;
+
+	FN(UNION,free)(u2);
+
+	return u1;
+error:
+	FN(UNION,free)(u1);
+	FN(UNION,free)(u2);
+	return NULL;
+}
+
+__isl_give UNION *FN(FN(UNION,from),PARTS)(__isl_take PART *part)
+{
+	isl_space *dim;
+	UNION *u;
+
+	if (!part)
+		return NULL;
+
+	dim = FN(PART,get_space)(part);
+	dim = isl_space_drop_dims(dim, isl_dim_in, 0, isl_space_dim(dim, isl_dim_in));
+	dim = isl_space_drop_dims(dim, isl_dim_out, 0, isl_space_dim(dim, isl_dim_out));
+#ifdef HAS_TYPE
+	u = FN(UNION,ZERO)(dim, part->type);
+#else
+	u = FN(UNION,ZERO)(dim);
+#endif
+	u = FN(FN(UNION,add),PARTS)(u, part);
+
+	return u;
+}
+
+S(UNION,match_bin_data) {
+	UNION *u2;
+	UNION *res;
+	__isl_give PART *(*fn)(__isl_take PART *, __isl_take PART *);
+};
+
+/* Check if data->u2 has an element living in the same space as *entry.
+ * If so, call data->fn on the two elements and add the result to
+ * data->res.
+ */
+static int match_bin_entry(void **entry, void *user)
+{
+	S(UNION,match_bin_data) *data = user;
+	uint32_t hash;
+	struct isl_hash_table_entry *entry2;
+	isl_space *space;
+	PART *part = *entry;
+	PART *part2;
+
+	space = FN(PART,get_space)(part);
+	hash = isl_space_get_hash(space);
+	entry2 = isl_hash_table_find(data->u2->space->ctx, &data->u2->table,
+				     hash, &has_same_domain_space, space, 0);
+	isl_space_free(space);
+	if (!entry2)
+		return 0;
+
+	part2 = entry2->data;
+	if (!isl_space_tuple_is_equal(part->dim, isl_dim_out,
+					part2->dim, isl_dim_out))
+		isl_die(FN(UNION,get_ctx)(data->u2), isl_error_invalid,
+			"entries should have the same range space",
+			return -1);
+
+	part = FN(PART, copy)(part);
+	part = data->fn(part, FN(PART, copy)(entry2->data));
+
+	data->res = FN(FN(UNION,add),PARTS)(data->res, part);
+	if (!data->res)
+		return -1;
+
+	return 0;
+}
+
+/* This function is currently only used from isl_polynomial.c
+ * and not from isl_fold.c.
+ */
+static __isl_give UNION *match_bin_op(__isl_take UNION *u1,
+	__isl_take UNION *u2,
+	__isl_give PART *(*fn)(__isl_take PART *, __isl_take PART *))
+	__attribute__ ((unused));
+/* For each pair of elements in "u1" and "u2" living in the same space,
+ * call "fn" and collect the results.
+ */
+static __isl_give UNION *match_bin_op(__isl_take UNION *u1,
+	__isl_take UNION *u2,
+	__isl_give PART *(*fn)(__isl_take PART *, __isl_take PART *))
+{
+	S(UNION,match_bin_data) data = { NULL, NULL, fn };
+
+	u1 = FN(UNION,align_params)(u1, FN(UNION,get_space)(u2));
+	u2 = FN(UNION,align_params)(u2, FN(UNION,get_space)(u1));
+
+	if (!u1 || !u2)
+		goto error;
+
+	data.u2 = u2;
+#ifdef HAS_TYPE
+	data.res = FN(UNION,alloc)(isl_space_copy(u1->space), u1->type,
+				    u1->table.n);
+#else
+	data.res = FN(UNION,alloc)(isl_space_copy(u1->space), u1->table.n);
+#endif
+	if (isl_hash_table_foreach(u1->space->ctx, &u1->table,
+				    &match_bin_entry, &data) < 0)
+		goto error;
+
+	FN(UNION,free)(u1);
+	FN(UNION,free)(u2);
+	return data.res;
+error:
+	FN(UNION,free)(u1);
+	FN(UNION,free)(u2);
+	FN(UNION,free)(data.res);
+	return NULL;
+}
+
+/* Compute the sum of "u1" and "u2".
+ *
+ * If the base expressions have a default zero value, then the sum
+ * is computed on the union of the domains of "u1" and "u2".
+ * Otherwise, it is computed on their shared domains.
+ */
+__isl_give UNION *FN(UNION,add)(__isl_take UNION *u1, __isl_take UNION *u2)
+{
+#if DEFAULT_IS_ZERO
+	return FN(UNION,union_add_)(u1, u2);
+#else
+	return match_bin_op(u1, u2, &FN(PART,add));
+#endif
+}
+
+#ifndef NO_SUB
+/* Subtract "u2" from "u1" and return the result.
+ */
+__isl_give UNION *FN(UNION,sub)(__isl_take UNION *u1, __isl_take UNION *u2)
+{
+	return match_bin_op(u1, u2, &FN(PART,sub));
+}
+#endif
+
+S(UNION,any_set_data) {
+	isl_set *set;
+	UNION *res;
+	__isl_give PW *(*fn)(__isl_take PW*, __isl_take isl_set*);
+};
+
+static int any_set_entry(void **entry, void *user)
+{
+	S(UNION,any_set_data) *data = user;
+	PW *pw = *entry;
+
+	pw = FN(PW,copy)(pw);
+	pw = data->fn(pw, isl_set_copy(data->set));
+
+	data->res = FN(FN(UNION,add),PARTS)(data->res, pw);
+	if (!data->res)
+		return -1;
+
+	return 0;
+}
+
+/* Update each element of "u" by calling "fn" on the element and "set".
+ */
+static __isl_give UNION *any_set_op(__isl_take UNION *u,
+	__isl_take isl_set *set,
+	__isl_give PW *(*fn)(__isl_take PW*, __isl_take isl_set*))
+{
+	S(UNION,any_set_data) data = { NULL, NULL, fn };
+
+	u = FN(UNION,align_params)(u, isl_set_get_space(set));
+	set = isl_set_align_params(set, FN(UNION,get_space)(u));
+
+	if (!u || !set)
+		goto error;
+
+	data.set = set;
+#ifdef HAS_TYPE
+	data.res = FN(UNION,alloc)(isl_space_copy(u->space), u->type,
+					u->table.n);
+#else
+	data.res = FN(UNION,alloc)(isl_space_copy(u->space), u->table.n);
+#endif
+	if (isl_hash_table_foreach(u->space->ctx, &u->table,
+				   &any_set_entry, &data) < 0)
+		goto error;
+
+	FN(UNION,free)(u);
+	isl_set_free(set);
+	return data.res;
+error:
+	FN(UNION,free)(u);
+	isl_set_free(set);
+	FN(UNION,free)(data.res);
+	return NULL;
+}
+
+/* Intersect the domain of "u" with the parameter domain "context".
+ */
+__isl_give UNION *FN(UNION,intersect_params)(__isl_take UNION *u,
+	__isl_take isl_set *set)
+{
+	return any_set_op(u, set, &FN(PW,intersect_params));
+}
+
+/* Compute the gist of the domain of "u" with respect to
+ * the parameter domain "context".
+ */
+__isl_give UNION *FN(UNION,gist_params)(__isl_take UNION *u,
+	__isl_take isl_set *set)
+{
+	return any_set_op(u, set, &FN(PW,gist_params));
+}
+
+S(UNION,match_domain_data) {
+	isl_union_set *uset;
+	UNION *res;
+	__isl_give PW *(*fn)(__isl_take PW*, __isl_take isl_set*);
+};
+
+static int set_has_dim(const void *entry, const void *val)
+{
+	isl_set *set = (isl_set *)entry;
+	isl_space *dim = (isl_space *)val;
+
+	return isl_space_is_equal(set->dim, dim);
+}
+
+/* Find the set in data->uset that lives in the same space as the domain
+ * of *entry, apply data->fn to *entry and this set (if any), and add
+ * the result to data->res.
+ */
+static int match_domain_entry(void **entry, void *user)
+{
+	S(UNION,match_domain_data) *data = user;
+	uint32_t hash;
+	struct isl_hash_table_entry *entry2;
+	PW *pw = *entry;
+	isl_space *space;
+
+	space = FN(PW,get_domain_space)(pw);
+	hash = isl_space_get_hash(space);
+	entry2 = isl_hash_table_find(data->uset->dim->ctx, &data->uset->table,
+				     hash, &set_has_dim, space, 0);
+	isl_space_free(space);
+	if (!entry2)
+		return 0;
+
+	pw = FN(PW,copy)(pw);
+	pw = data->fn(pw, isl_set_copy(entry2->data));
+
+	data->res = FN(FN(UNION,add),PARTS)(data->res, pw);
+	if (!data->res)
+		return -1;
+
+	return 0;
+}
+
+/* Apply fn to each pair of PW in u and set in uset such that
+ * the set lives in the same space as the domain of PW
+ * and collect the results.
+ */
+static __isl_give UNION *match_domain_op(__isl_take UNION *u,
+	__isl_take isl_union_set *uset,
+	__isl_give PW *(*fn)(__isl_take PW*, __isl_take isl_set*))
+{
+	S(UNION,match_domain_data) data = { NULL, NULL, fn };
+
+	u = FN(UNION,align_params)(u, isl_union_set_get_space(uset));
+	uset = isl_union_set_align_params(uset, FN(UNION,get_space)(u));
+
+	if (!u || !uset)
+		goto error;
+
+	data.uset = uset;
+#ifdef HAS_TYPE
+	data.res = FN(UNION,alloc)(isl_space_copy(u->space), u->type,
+					u->table.n);
+#else
+	data.res = FN(UNION,alloc)(isl_space_copy(u->space), u->table.n);
+#endif
+	if (isl_hash_table_foreach(u->space->ctx, &u->table,
+				   &match_domain_entry, &data) < 0)
+		goto error;
+
+	FN(UNION,free)(u);
+	isl_union_set_free(uset);
+	return data.res;
+error:
+	FN(UNION,free)(u);
+	isl_union_set_free(uset);
+	FN(UNION,free)(data.res);
+	return NULL;
+}
+
+/* Intersect the domain of "u" with "uset".
+ * If "uset" is a parameters domain, then intersect the parameter
+ * domain of "u" with this set.
+ */
+__isl_give UNION *FN(UNION,intersect_domain)(__isl_take UNION *u,
+	__isl_take isl_union_set *uset)
+{
+	if (isl_union_set_is_params(uset))
+		return FN(UNION,intersect_params)(u,
+						isl_set_from_union_set(uset));
+	return match_domain_op(u, uset, &FN(PW,intersect_domain));
+}
+
+__isl_give UNION *FN(UNION,gist)(__isl_take UNION *u,
+	__isl_take isl_union_set *uset)
+{
+	if (isl_union_set_is_params(uset))
+		return FN(UNION,gist_params)(u, isl_set_from_union_set(uset));
+	return match_domain_op(u, uset, &FN(PW,gist));
+}
+
+#ifndef NO_EVAL
+__isl_give isl_val *FN(UNION,eval)(__isl_take UNION *u,
+	__isl_take isl_point *pnt)
+{
+	uint32_t hash;
+	struct isl_hash_table_entry *entry;
+	isl_space *space;
+	isl_val *v;
+
+	if (!u || !pnt)
+		goto error;
+
+	space = isl_space_copy(pnt->dim);
+	if (!space)
+		goto error;
+	hash = isl_space_get_hash(space);
+	entry = isl_hash_table_find(u->space->ctx, &u->table,
+				    hash, &has_domain_space, space, 0);
+	isl_space_free(space);
+	if (!entry) {
+		v = isl_val_zero(isl_point_get_ctx(pnt));
+		isl_point_free(pnt);
+	} else {
+		v = FN(PART,eval)(FN(PART,copy)(entry->data), pnt);
+	}
+	FN(UNION,free)(u);
+	return v;
+error:
+	FN(UNION,free)(u);
+	isl_point_free(pnt);
+	return NULL;
+}
+#endif
+
+static int coalesce_entry(void **entry, void *user)
+{
+	PW **pw = (PW **)entry;
+
+	*pw = FN(PW,coalesce)(*pw);
+	if (!*pw)
+		return -1;
+
+	return 0;
+}
+
+__isl_give UNION *FN(UNION,coalesce)(__isl_take UNION *u)
+{
+	if (!u)
+		return NULL;
+
+	if (isl_hash_table_foreach(u->space->ctx, &u->table,
+				   &coalesce_entry, NULL) < 0)
+		goto error;
+
+	return u;
+error:
+	FN(UNION,free)(u);
+	return NULL;
+}
+
+static int domain(__isl_take PART *part, void *user)
+{
+	isl_union_set **uset = (isl_union_set **)user;
+
+	*uset = isl_union_set_add_set(*uset, FN(PART,domain)(part));
+
+	return 0;
+}
+
+__isl_give isl_union_set *FN(UNION,domain)(__isl_take UNION *u)
+{
+	isl_union_set *uset;
+
+	uset = isl_union_set_empty(FN(UNION,get_space)(u));
+	if (FN(FN(UNION,foreach),PARTS)(u, &domain, &uset) < 0)
+		goto error;
+
+	FN(UNION,free)(u);
+	
+	return uset;
+error:
+	isl_union_set_free(uset);
+	FN(UNION,free)(u);
+	return NULL;
+}
+
+static int mul_isl_int(void **entry, void *user)
+{
+	PW **pw = (PW **)entry;
+	isl_int *v = user;
+
+	*pw = FN(PW,mul_isl_int)(*pw, *v);
+	if (!*pw)
+		return -1;
+
+	return 0;
+}
+
+__isl_give UNION *FN(UNION,mul_isl_int)(__isl_take UNION *u, isl_int v)
+{
+	if (isl_int_is_one(v))
+		return u;
+
+	if (DEFAULT_IS_ZERO && u && isl_int_is_zero(v)) {
+		UNION *zero;
+		isl_space *dim = FN(UNION,get_space)(u);
+#ifdef HAS_TYPE
+		zero = FN(UNION,ZERO)(dim, u->type);
+#else
+		zero = FN(UNION,ZERO)(dim);
+#endif
+		FN(UNION,free)(u);
+		return zero;
+	}
+
+	u = FN(UNION,cow)(u);
+	if (!u)
+		return NULL;
+
+#ifdef HAS_TYPE
+	if (isl_int_is_neg(v))
+		u->type = isl_fold_type_negate(u->type);
+#endif
+	if (isl_hash_table_foreach(u->space->ctx, &u->table,
+				    &mul_isl_int, &v) < 0)
+		goto error;
+
+	return u;
+error:
+	FN(UNION,free)(u);
+	return NULL;
+}
+
+/* Multiply *entry by the isl_val "user".
+ *
+ * Return 0 on success and -1 on error.
+ */
+static int scale_val(void **entry, void *user)
+{
+	PW **pw = (PW **)entry;
+	isl_val *v = user;
+
+	*pw = FN(PW,scale_val)(*pw, isl_val_copy(v));
+	if (!*pw)
+		return -1;
+
+	return 0;
+}
+
+/* Multiply "u" by "v" and return the result.
+ */
+__isl_give UNION *FN(UNION,scale_val)(__isl_take UNION *u,
+	__isl_take isl_val *v)
+{
+	if (!u || !v)
+		goto error;
+	if (isl_val_is_one(v)) {
+		isl_val_free(v);
+		return u;
+	}
+
+	if (DEFAULT_IS_ZERO && u && isl_val_is_zero(v)) {
+		UNION *zero;
+		isl_space *space = FN(UNION,get_space)(u);
+#ifdef HAS_TYPE
+		zero = FN(UNION,ZERO)(space, u->type);
+#else
+		zero = FN(UNION,ZERO)(space);
+#endif
+		FN(UNION,free)(u);
+		isl_val_free(v);
+		return zero;
+	}
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting rational factor", goto error);
+
+	u = FN(UNION,cow)(u);
+	if (!u)
+		return NULL;
+
+#ifdef HAS_TYPE
+	if (isl_val_is_neg(v))
+		u->type = isl_fold_type_negate(u->type);
+#endif
+	if (isl_hash_table_foreach(u->space->ctx, &u->table, &scale_val, v) < 0)
+		goto error;
+
+	isl_val_free(v);
+	return u;
+error:
+	isl_val_free(v);
+	FN(UNION,free)(u);
+	return NULL;
+}
+
+/* Divide *entry by the isl_val "user".
+ *
+ * Return 0 on success and -1 on error.
+ */
+static int FN(UNION,scale_down_val_entry)(void **entry, void *user)
+{
+	PW **pw = (PW **)entry;
+	isl_val *v = user;
+
+	*pw = FN(PW,scale_down_val)(*pw, isl_val_copy(v));
+	if (!*pw)
+		return -1;
+
+	return 0;
+}
+
+/* Divide "u" by "v" and return the result.
+ */
+__isl_give UNION *FN(UNION,scale_down_val)(__isl_take UNION *u,
+	__isl_take isl_val *v)
+{
+	if (!u || !v)
+		goto error;
+	if (isl_val_is_one(v)) {
+		isl_val_free(v);
+		return u;
+	}
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting rational factor", goto error);
+	if (isl_val_is_zero(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"cannot scale down by zero", goto error);
+
+	u = FN(UNION,cow)(u);
+	if (!u)
+		return NULL;
+
+#ifdef HAS_TYPE
+	if (isl_val_is_neg(v))
+		u->type = isl_fold_type_negate(u->type);
+#endif
+	if (isl_hash_table_foreach(FN(UNION,get_ctx)(u), &u->table,
+				    &FN(UNION,scale_down_val_entry), v) < 0)
+		goto error;
+
+	isl_val_free(v);
+	return u;
+error:
+	isl_val_free(v);
+	FN(UNION,free)(u);
+	return NULL;
+}
+
+S(UNION,plain_is_equal_data)
+{
+	UNION *u2;
+	int is_equal;
+};
+
+static int plain_is_equal_entry(void **entry, void *user)
+{
+	S(UNION,plain_is_equal_data) *data = user;
+	uint32_t hash;
+	struct isl_hash_table_entry *entry2;
+	PW *pw = *entry;
+
+	hash = isl_space_get_hash(pw->dim);
+	entry2 = isl_hash_table_find(data->u2->space->ctx, &data->u2->table,
+				     hash, &has_same_domain_space, pw->dim, 0);
+	if (!entry2) {
+		data->is_equal = 0;
+		return -1;
+	}
+
+	data->is_equal = FN(PW,plain_is_equal)(pw, entry2->data);
+	if (data->is_equal < 0 || !data->is_equal)
+		return -1;
+
+	return 0;
+}
+
+int FN(UNION,plain_is_equal)(__isl_keep UNION *u1, __isl_keep UNION *u2)
+{
+	S(UNION,plain_is_equal_data) data = { NULL, 1 };
+
+	if (!u1 || !u2)
+		return -1;
+	if (u1 == u2)
+		return 1;
+	if (u1->table.n != u2->table.n)
+		return 0;
+
+	u1 = FN(UNION,copy)(u1);
+	u2 = FN(UNION,copy)(u2);
+	u1 = FN(UNION,align_params)(u1, FN(UNION,get_space)(u2));
+	u2 = FN(UNION,align_params)(u2, FN(UNION,get_space)(u1));
+	if (!u1 || !u2)
+		goto error;
+
+	data.u2 = u2;
+	if (isl_hash_table_foreach(u1->space->ctx, &u1->table,
+				   &plain_is_equal_entry, &data) < 0 &&
+	    data.is_equal)
+		goto error;
+
+	FN(UNION,free)(u1);
+	FN(UNION,free)(u2);
+
+	return data.is_equal;
+error:
+	FN(UNION,free)(u1);
+	FN(UNION,free)(u2);
+	return -1;
+}
diff --git a/polly/lib/External/isl/isl_val.c b/polly/lib/External/isl/isl_val.c
new file mode 100644
index 00000000000..4938c4c3823
--- /dev/null
+++ b/polly/lib/External/isl/isl_val.c
@@ -0,0 +1,1624 @@
+/*
+ * Copyright 2013      Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege,
+ * Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France
+ */
+
+#include <isl_int.h>
+#include <isl_ctx_private.h>
+#include <isl_val_private.h>
+
+#undef BASE
+#define BASE val
+
+#include <isl_list_templ.c>
+
+/* Allocate an isl_val object with indeterminate value.
+ */
+__isl_give isl_val *isl_val_alloc(isl_ctx *ctx)
+{
+	isl_val *v;
+
+	v = isl_alloc_type(ctx, struct isl_val);
+	if (!v)
+		return NULL;
+
+	v->ctx = ctx;
+	isl_ctx_ref(ctx);
+	v->ref = 1;
+	isl_int_init(v->n);
+	isl_int_init(v->d);
+
+	return v;
+}
+
+/* Return a reference to an isl_val representing zero.
+ */
+__isl_give isl_val *isl_val_zero(isl_ctx *ctx)
+{
+	return isl_val_int_from_si(ctx, 0);
+}
+
+/* Return a reference to an isl_val representing one.
+ */
+__isl_give isl_val *isl_val_one(isl_ctx *ctx)
+{
+	return isl_val_int_from_si(ctx, 1);
+}
+
+/* Return a reference to an isl_val representing negative one.
+ */
+__isl_give isl_val *isl_val_negone(isl_ctx *ctx)
+{
+	return isl_val_int_from_si(ctx, -1);
+}
+
+/* Return a reference to an isl_val representing NaN.
+ */
+__isl_give isl_val *isl_val_nan(isl_ctx *ctx)
+{
+	isl_val *v;
+
+	v = isl_val_alloc(ctx);
+	if (!v)
+		return NULL;
+
+	isl_int_set_si(v->n, 0);
+	isl_int_set_si(v->d, 0);
+
+	return v;
+}
+
+/* Change "v" into a NaN.
+ */
+__isl_give isl_val *isl_val_set_nan(__isl_take isl_val *v)
+{
+	if (!v)
+		return NULL;
+	if (isl_val_is_nan(v))
+		return v;
+	v = isl_val_cow(v);
+	if (!v)
+		return NULL;
+
+	isl_int_set_si(v->n, 0);
+	isl_int_set_si(v->d, 0);
+
+	return v;
+}
+
+/* Return a reference to an isl_val representing +infinity.
+ */
+__isl_give isl_val *isl_val_infty(isl_ctx *ctx)
+{
+	isl_val *v;
+
+	v = isl_val_alloc(ctx);
+	if (!v)
+		return NULL;
+
+	isl_int_set_si(v->n, 1);
+	isl_int_set_si(v->d, 0);
+
+	return v;
+}
+
+/* Return a reference to an isl_val representing -infinity.
+ */
+__isl_give isl_val *isl_val_neginfty(isl_ctx *ctx)
+{
+	isl_val *v;
+
+	v = isl_val_alloc(ctx);
+	if (!v)
+		return NULL;
+
+	isl_int_set_si(v->n, -1);
+	isl_int_set_si(v->d, 0);
+
+	return v;
+}
+
+/* Return a reference to an isl_val representing the integer "i".
+ */
+__isl_give isl_val *isl_val_int_from_si(isl_ctx *ctx, long i)
+{
+	isl_val *v;
+
+	v = isl_val_alloc(ctx);
+	if (!v)
+		return NULL;
+
+	isl_int_set_si(v->n, i);
+	isl_int_set_si(v->d, 1);
+
+	return v;
+}
+
+/* Change the value of "v" to be equal to the integer "i".
+ */
+__isl_give isl_val *isl_val_set_si(__isl_take isl_val *v, long i)
+{
+	if (!v)
+		return NULL;
+	if (isl_val_is_int(v) && isl_int_cmp_si(v->n, i) == 0)
+		return v;
+	v = isl_val_cow(v);
+	if (!v)
+		return NULL;
+
+	isl_int_set_si(v->n, i);
+	isl_int_set_si(v->d, 1);
+
+	return v;
+}
+
+/* Change the value of "v" to be equal to zero.
+ */
+__isl_give isl_val *isl_val_set_zero(__isl_take isl_val *v)
+{
+	return isl_val_set_si(v, 0);
+}
+
+/* Return a reference to an isl_val representing the unsigned integer "u".
+ */
+__isl_give isl_val *isl_val_int_from_ui(isl_ctx *ctx, unsigned long u)
+{
+	isl_val *v;
+
+	v = isl_val_alloc(ctx);
+	if (!v)
+		return NULL;
+
+	isl_int_set_ui(v->n, u);
+	isl_int_set_si(v->d, 1);
+
+	return v;
+}
+
+/* Return a reference to an isl_val representing the integer "n".
+ */
+__isl_give isl_val *isl_val_int_from_isl_int(isl_ctx *ctx, isl_int n)
+{
+	isl_val *v;
+
+	v = isl_val_alloc(ctx);
+	if (!v)
+		return NULL;
+
+	isl_int_set(v->n, n);
+	isl_int_set_si(v->d, 1);
+
+	return v;
+}
+
+/* Return a reference to an isl_val representing the rational value "n"/"d".
+ * Normalizing the isl_val (if needed) is left to the caller.
+ */
+__isl_give isl_val *isl_val_rat_from_isl_int(isl_ctx *ctx,
+	isl_int n, isl_int d)
+{
+	isl_val *v;
+
+	v = isl_val_alloc(ctx);
+	if (!v)
+		return NULL;
+
+	isl_int_set(v->n, n);
+	isl_int_set(v->d, d);
+
+	return v;
+}
+
+/* Return a new reference to "v".
+ */
+__isl_give isl_val *isl_val_copy(__isl_keep isl_val *v)
+{
+	if (!v)
+		return NULL;
+
+	v->ref++;
+	return v;
+}
+
+/* Return a fresh copy of "val".
+ */
+__isl_give isl_val *isl_val_dup(__isl_keep isl_val *val)
+{
+	isl_val *dup;
+
+	if (!val)
+		return NULL;
+
+	dup = isl_val_alloc(isl_val_get_ctx(val));
+	if (!dup)
+		return NULL;
+
+	isl_int_set(dup->n, val->n);
+	isl_int_set(dup->d, val->d);
+
+	return dup;
+}
+
+/* Return an isl_val that is equal to "val" and that has only
+ * a single reference.
+ */
+__isl_give isl_val *isl_val_cow(__isl_take isl_val *val)
+{
+	if (!val)
+		return NULL;
+
+	if (val->ref == 1)
+		return val;
+	val->ref--;
+	return isl_val_dup(val);
+}
+
+/* Free "v" and return NULL.
+ */
+__isl_null isl_val *isl_val_free(__isl_take isl_val *v)
+{
+	if (!v)
+		return NULL;
+
+	if (--v->ref > 0)
+		return NULL;
+
+	isl_ctx_deref(v->ctx);
+	isl_int_clear(v->n);
+	isl_int_clear(v->d);
+	free(v);
+	return NULL;
+}
+
+/* Extract the numerator of a rational value "v" as an integer.
+ *
+ * If "v" is not a rational value, then the result is undefined.
+ */
+long isl_val_get_num_si(__isl_keep isl_val *v)
+{
+	if (!v)
+		return 0;
+	if (!isl_val_is_rat(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting rational value", return 0);
+	if (!isl_int_fits_slong(v->n))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"numerator too large", return 0);
+	return isl_int_get_si(v->n);
+}
+
+/* Extract the numerator of a rational value "v" as an isl_int.
+ *
+ * If "v" is not a rational value, then the result is undefined.
+ */
+int isl_val_get_num_isl_int(__isl_keep isl_val *v, isl_int *n)
+{
+	if (!v)
+		return -1;
+	if (!isl_val_is_rat(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting rational value", return -1);
+	isl_int_set(*n, v->n);
+	return 0;
+}
+
+/* Extract the denominator of a rational value "v" as an integer.
+ *
+ * If "v" is not a rational value, then the result is undefined.
+ */
+long isl_val_get_den_si(__isl_keep isl_val *v)
+{
+	if (!v)
+		return 0;
+	if (!isl_val_is_rat(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting rational value", return 0);
+	if (!isl_int_fits_slong(v->d))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"denominator too large", return 0);
+	return isl_int_get_si(v->d);
+}
+
+/* Return an approximation of "v" as a double.
+ */
+double isl_val_get_d(__isl_keep isl_val *v)
+{
+	if (!v)
+		return 0;
+	if (!isl_val_is_rat(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting rational value", return 0);
+	return isl_int_get_d(v->n) / isl_int_get_d(v->d);
+}
+
+/* Return the isl_ctx to which "val" belongs.
+ */
+isl_ctx *isl_val_get_ctx(__isl_keep isl_val *val)
+{
+	return val ? val->ctx : NULL;
+}
+
+/* Normalize "v".
+ *
+ * In particular, make sure that the denominator of a rational value
+ * is positive and the numerator and denominator do not have any
+ * common divisors.
+ *
+ * This function should not be called by an external user
+ * since it will only be given normalized values.
+ */
+__isl_give isl_val *isl_val_normalize(__isl_take isl_val *v)
+{
+	isl_ctx *ctx;
+
+	if (!v)
+		return NULL;
+	if (isl_val_is_int(v))
+		return v;
+	if (!isl_val_is_rat(v))
+		return v;
+	if (isl_int_is_neg(v->d)) {
+		isl_int_neg(v->d, v->d);
+		isl_int_neg(v->n, v->n);
+	}
+	ctx = isl_val_get_ctx(v);
+	isl_int_gcd(ctx->normalize_gcd, v->n, v->d);
+	if (isl_int_is_one(ctx->normalize_gcd))
+		return v;
+	isl_int_divexact(v->n, v->n, ctx->normalize_gcd);
+	isl_int_divexact(v->d, v->d, ctx->normalize_gcd);
+	return v;
+}
+
+/* Return the opposite of "v".
+ */
+__isl_give isl_val *isl_val_neg(__isl_take isl_val *v)
+{
+	if (!v)
+		return NULL;
+	if (isl_val_is_nan(v))
+		return v;
+	if (isl_val_is_zero(v))
+		return v;
+
+	v = isl_val_cow(v);
+	if (!v)
+		return NULL;
+	isl_int_neg(v->n, v->n);
+
+	return v;
+}
+
+/* Return the inverse of "v".
+ */
+__isl_give isl_val *isl_val_inv(__isl_take isl_val *v)
+{
+	if (!v)
+		return NULL;
+	if (isl_val_is_nan(v))
+		return v;
+	if (isl_val_is_zero(v)) {
+		isl_ctx *ctx = isl_val_get_ctx(v);
+		isl_val_free(v);
+		return isl_val_nan(ctx);
+	}
+	if (isl_val_is_infty(v) || isl_val_is_neginfty(v)) {
+		isl_ctx *ctx = isl_val_get_ctx(v);
+		isl_val_free(v);
+		return isl_val_zero(ctx);
+	}
+
+	v = isl_val_cow(v);
+	if (!v)
+		return NULL;
+	isl_int_swap(v->n, v->d);
+
+	return isl_val_normalize(v);
+}
+
+/* Return the absolute value of "v".
+ */
+__isl_give isl_val *isl_val_abs(__isl_take isl_val *v)
+{
+	if (!v)
+		return NULL;
+	if (isl_val_is_nan(v))
+		return v;
+	if (isl_val_is_nonneg(v))
+		return v;
+	return isl_val_neg(v);
+}
+
+/* Return the "floor" (greatest integer part) of "v".
+ * That is, return the result of rounding towards -infinity.
+ */
+__isl_give isl_val *isl_val_floor(__isl_take isl_val *v)
+{
+	if (!v)
+		return NULL;
+	if (isl_val_is_int(v))
+		return v;
+	if (!isl_val_is_rat(v))
+		return v;
+
+	v = isl_val_cow(v);
+	if (!v)
+		return NULL;
+	isl_int_fdiv_q(v->n, v->n, v->d);
+	isl_int_set_si(v->d, 1);
+
+	return v;
+}
+
+/* Return the "ceiling" of "v".
+ * That is, return the result of rounding towards +infinity.
+ */
+__isl_give isl_val *isl_val_ceil(__isl_take isl_val *v)
+{
+	if (!v)
+		return NULL;
+	if (isl_val_is_int(v))
+		return v;
+	if (!isl_val_is_rat(v))
+		return v;
+
+	v = isl_val_cow(v);
+	if (!v)
+		return NULL;
+	isl_int_cdiv_q(v->n, v->n, v->d);
+	isl_int_set_si(v->d, 1);
+
+	return v;
+}
+
+/* Truncate "v".
+ * That is, return the result of rounding towards zero.
+ */
+__isl_give isl_val *isl_val_trunc(__isl_take isl_val *v)
+{
+	if (!v)
+		return NULL;
+	if (isl_val_is_int(v))
+		return v;
+	if (!isl_val_is_rat(v))
+		return v;
+
+	v = isl_val_cow(v);
+	if (!v)
+		return NULL;
+	isl_int_tdiv_q(v->n, v->n, v->d);
+	isl_int_set_si(v->d, 1);
+
+	return v;
+}
+
+/* Return 2^v, where v is an integer (that is not too large).
+ */
+__isl_give isl_val *isl_val_2exp(__isl_take isl_val *v)
+{
+	unsigned long exp;
+	int neg;
+
+	v = isl_val_cow(v);
+	if (!v)
+		return NULL;
+	if (!isl_val_is_int(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"can only compute integer powers",
+			return isl_val_free(v));
+	neg = isl_val_is_neg(v);
+	if (neg)
+		isl_int_neg(v->n, v->n);
+	if (!isl_int_fits_ulong(v->n))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"exponent too large", return isl_val_free(v));
+	exp = isl_int_get_ui(v->n);
+	if (neg) {
+		isl_int_mul_2exp(v->d, v->d, exp);
+		isl_int_set_si(v->n, 1);
+	} else {
+		isl_int_mul_2exp(v->n, v->d, exp);
+	}
+
+	return v;
+}
+
+/* Return the minimum of "v1" and "v2".
+ */
+__isl_give isl_val *isl_val_min(__isl_take isl_val *v1, __isl_take isl_val *v2)
+{
+	if (!v1 || !v2)
+		goto error;
+
+	if (isl_val_is_nan(v1)) {
+		isl_val_free(v2);
+		return v1;
+	}
+	if (isl_val_is_nan(v2)) {
+		isl_val_free(v1);
+		return v2;
+	}
+	if (isl_val_le(v1, v2)) {
+		isl_val_free(v2);
+		return v1;
+	} else {
+		isl_val_free(v1);
+		return v2;
+	}
+error:
+	isl_val_free(v1);
+	isl_val_free(v2);
+	return NULL;
+}
+
+/* Return the maximum of "v1" and "v2".
+ */
+__isl_give isl_val *isl_val_max(__isl_take isl_val *v1, __isl_take isl_val *v2)
+{
+	if (!v1 || !v2)
+		goto error;
+
+	if (isl_val_is_nan(v1)) {
+		isl_val_free(v2);
+		return v1;
+	}
+	if (isl_val_is_nan(v2)) {
+		isl_val_free(v1);
+		return v2;
+	}
+	if (isl_val_ge(v1, v2)) {
+		isl_val_free(v2);
+		return v1;
+	} else {
+		isl_val_free(v1);
+		return v2;
+	}
+error:
+	isl_val_free(v1);
+	isl_val_free(v2);
+	return NULL;
+}
+
+/* Return the sum of "v1" and "v2".
+ */
+__isl_give isl_val *isl_val_add(__isl_take isl_val *v1, __isl_take isl_val *v2)
+{
+	if (!v1 || !v2)
+		goto error;
+	if (isl_val_is_nan(v1)) {
+		isl_val_free(v2);
+		return v1;
+	}
+	if (isl_val_is_nan(v2)) {
+		isl_val_free(v1);
+		return v2;
+	}
+	if ((isl_val_is_infty(v1) && isl_val_is_neginfty(v2)) ||
+	    (isl_val_is_neginfty(v1) && isl_val_is_infty(v2))) {
+		isl_val_free(v2);
+		return isl_val_set_nan(v1);
+	}
+	if (isl_val_is_infty(v1) || isl_val_is_neginfty(v1)) {
+		isl_val_free(v2);
+		return v1;
+	}
+	if (isl_val_is_infty(v2) || isl_val_is_neginfty(v2)) {
+		isl_val_free(v1);
+		return v2;
+	}
+	if (isl_val_is_zero(v1)) {
+		isl_val_free(v1);
+		return v2;
+	}
+	if (isl_val_is_zero(v2)) {
+		isl_val_free(v2);
+		return v1;
+	}
+
+	v1 = isl_val_cow(v1);
+	if (!v1)
+		goto error;
+	if (isl_val_is_int(v1) && isl_val_is_int(v2))
+		isl_int_add(v1->n, v1->n, v2->n);
+	else {
+		if (isl_int_eq(v1->d, v2->d))
+			isl_int_add(v1->n, v1->n, v2->n);
+		else {
+			isl_int_mul(v1->n, v1->n, v2->d);
+			isl_int_addmul(v1->n, v2->n, v1->d);
+			isl_int_mul(v1->d, v1->d, v2->d);
+		}
+		v1 = isl_val_normalize(v1);
+	}
+	isl_val_free(v2);
+	return v1;
+error:
+	isl_val_free(v1);
+	isl_val_free(v2);
+	return NULL;
+}
+
+/* Return the sum of "v1" and "v2".
+ */
+__isl_give isl_val *isl_val_add_ui(__isl_take isl_val *v1, unsigned long v2)
+{
+	if (!v1)
+		return NULL;
+	if (!isl_val_is_rat(v1))
+		return v1;
+	if (v2 == 0)
+		return v1;
+	v1 = isl_val_cow(v1);
+	if (!v1)
+		return NULL;
+
+	isl_int_addmul_ui(v1->n, v1->d, v2);
+
+	return v1;
+}
+
+/* Subtract "v2" from "v1".
+ */
+__isl_give isl_val *isl_val_sub(__isl_take isl_val *v1, __isl_take isl_val *v2)
+{
+	if (!v1 || !v2)
+		goto error;
+	if (isl_val_is_nan(v1)) {
+		isl_val_free(v2);
+		return v1;
+	}
+	if (isl_val_is_nan(v2)) {
+		isl_val_free(v1);
+		return v2;
+	}
+	if ((isl_val_is_infty(v1) && isl_val_is_infty(v2)) ||
+	    (isl_val_is_neginfty(v1) && isl_val_is_neginfty(v2))) {
+		isl_val_free(v2);
+		return isl_val_set_nan(v1);
+	}
+	if (isl_val_is_infty(v1) || isl_val_is_neginfty(v1)) {
+		isl_val_free(v2);
+		return v1;
+	}
+	if (isl_val_is_infty(v2) || isl_val_is_neginfty(v2)) {
+		isl_val_free(v1);
+		return isl_val_neg(v2);
+	}
+	if (isl_val_is_zero(v2)) {
+		isl_val_free(v2);
+		return v1;
+	}
+	if (isl_val_is_zero(v1)) {
+		isl_val_free(v1);
+		return isl_val_neg(v2);
+	}
+
+	v1 = isl_val_cow(v1);
+	if (!v1)
+		goto error;
+	if (isl_val_is_int(v1) && isl_val_is_int(v2))
+		isl_int_sub(v1->n, v1->n, v2->n);
+	else {
+		if (isl_int_eq(v1->d, v2->d))
+			isl_int_sub(v1->n, v1->n, v2->n);
+		else {
+			isl_int_mul(v1->n, v1->n, v2->d);
+			isl_int_submul(v1->n, v2->n, v1->d);
+			isl_int_mul(v1->d, v1->d, v2->d);
+		}
+		v1 = isl_val_normalize(v1);
+	}
+	isl_val_free(v2);
+	return v1;
+error:
+	isl_val_free(v1);
+	isl_val_free(v2);
+	return NULL;
+}
+
+/* Subtract "v2" from "v1".
+ */
+__isl_give isl_val *isl_val_sub_ui(__isl_take isl_val *v1, unsigned long v2)
+{
+	if (!v1)
+		return NULL;
+	if (!isl_val_is_rat(v1))
+		return v1;
+	if (v2 == 0)
+		return v1;
+	v1 = isl_val_cow(v1);
+	if (!v1)
+		return NULL;
+
+	isl_int_submul_ui(v1->n, v1->d, v2);
+
+	return v1;
+}
+
+/* Return the product of "v1" and "v2".
+ */
+__isl_give isl_val *isl_val_mul(__isl_take isl_val *v1, __isl_take isl_val *v2)
+{
+	if (!v1 || !v2)
+		goto error;
+	if (isl_val_is_nan(v1)) {
+		isl_val_free(v2);
+		return v1;
+	}
+	if (isl_val_is_nan(v2)) {
+		isl_val_free(v1);
+		return v2;
+	}
+	if ((!isl_val_is_rat(v1) && isl_val_is_zero(v2)) ||
+	    (isl_val_is_zero(v1) && !isl_val_is_rat(v2))) {
+		isl_val_free(v2);
+		return isl_val_set_nan(v1);
+	}
+	if (isl_val_is_zero(v1)) {
+		isl_val_free(v2);
+		return v1;
+	}
+	if (isl_val_is_zero(v2)) {
+		isl_val_free(v1);
+		return v2;
+	}
+	if (isl_val_is_infty(v1) || isl_val_is_neginfty(v1)) {
+		if (isl_val_is_neg(v2))
+			v1 = isl_val_neg(v1);
+		isl_val_free(v2);
+		return v1;
+	}
+	if (isl_val_is_infty(v2) || isl_val_is_neginfty(v2)) {
+		if (isl_val_is_neg(v1))
+			v2 = isl_val_neg(v2);
+		isl_val_free(v1);
+		return v2;
+	}
+
+	v1 = isl_val_cow(v1);
+	if (!v1)
+		goto error;
+	if (isl_val_is_int(v1) && isl_val_is_int(v2))
+		isl_int_mul(v1->n, v1->n, v2->n);
+	else {
+		isl_int_mul(v1->n, v1->n, v2->n);
+		isl_int_mul(v1->d, v1->d, v2->d);
+		v1 = isl_val_normalize(v1);
+	}
+	isl_val_free(v2);
+	return v1;
+error:
+	isl_val_free(v1);
+	isl_val_free(v2);
+	return NULL;
+}
+
+/* Return the product of "v1" and "v2".
+ *
+ * This is a private copy of isl_val_mul for use in the generic
+ * isl_multi_*_scale_val instantiated for isl_val.
+ */
+__isl_give isl_val *isl_val_scale_val(__isl_take isl_val *v1,
+	__isl_take isl_val *v2)
+{
+	return isl_val_mul(v1, v2);
+}
+
+/* Return the product of "v1" and "v2".
+ */
+__isl_give isl_val *isl_val_mul_ui(__isl_take isl_val *v1, unsigned long v2)
+{
+	if (!v1)
+		return NULL;
+	if (isl_val_is_nan(v1))
+		return v1;
+	if (!isl_val_is_rat(v1)) {
+		if (v2 == 0)
+			v1 = isl_val_set_nan(v1);
+		return v1;
+	}
+	if (v2 == 1)
+		return v1;
+	v1 = isl_val_cow(v1);
+	if (!v1)
+		return NULL;
+
+	isl_int_mul_ui(v1->n, v1->n, v2);
+
+	return isl_val_normalize(v1);
+}
+
+/* Divide "v1" by "v2".
+ */
+__isl_give isl_val *isl_val_div(__isl_take isl_val *v1, __isl_take isl_val *v2)
+{
+	if (!v1 || !v2)
+		goto error;
+	if (isl_val_is_nan(v1)) {
+		isl_val_free(v2);
+		return v1;
+	}
+	if (isl_val_is_nan(v2)) {
+		isl_val_free(v1);
+		return v2;
+	}
+	if (isl_val_is_zero(v2) ||
+	    (!isl_val_is_rat(v1) && !isl_val_is_rat(v2))) {
+		isl_val_free(v2);
+		return isl_val_set_nan(v1);
+	}
+	if (isl_val_is_zero(v1)) {
+		isl_val_free(v2);
+		return v1;
+	}
+	if (isl_val_is_infty(v1) || isl_val_is_neginfty(v1)) {
+		if (isl_val_is_neg(v2))
+			v1 = isl_val_neg(v1);
+		isl_val_free(v2);
+		return v1;
+	}
+	if (isl_val_is_infty(v2) || isl_val_is_neginfty(v2)) {
+		isl_val_free(v2);
+		return isl_val_set_zero(v1);
+	}
+
+	v1 = isl_val_cow(v1);
+	if (!v1)
+		goto error;
+	if (isl_val_is_int(v2)) {
+		isl_int_mul(v1->d, v1->d, v2->n);
+		v1 = isl_val_normalize(v1);
+	} else {
+		isl_int_mul(v1->d, v1->d, v2->n);
+		isl_int_mul(v1->n, v1->n, v2->d);
+		v1 = isl_val_normalize(v1);
+	}
+	isl_val_free(v2);
+	return v1;
+error:
+	isl_val_free(v1);
+	isl_val_free(v2);
+	return NULL;
+}
+
+/* Divide "v1" by "v2".
+ *
+ * This is a private copy of isl_val_div for use in the generic
+ * isl_multi_*_scale_down_val instantiated for isl_val.
+ */
+__isl_give isl_val *isl_val_scale_down_val(__isl_take isl_val *v1,
+	__isl_take isl_val *v2)
+{
+	return isl_val_div(v1, v2);
+}
+
+/* Given two integer values "v1" and "v2", check if "v1" is divisible by "v2".
+ */
+int isl_val_is_divisible_by(__isl_keep isl_val *v1, __isl_keep isl_val *v2)
+{
+	if (!v1 || !v2)
+		return -1;
+
+	if (!isl_val_is_int(v1) || !isl_val_is_int(v2))
+		isl_die(isl_val_get_ctx(v1), isl_error_invalid,
+			"expecting two integers", return -1);
+
+	return isl_int_is_divisible_by(v1->n, v2->n);
+}
+
+/* Given two integer values "v1" and "v2", return the residue of "v1"
+ * modulo "v2".
+ */
+__isl_give isl_val *isl_val_mod(__isl_take isl_val *v1, __isl_take isl_val *v2)
+{
+	if (!v1 || !v2)
+		goto error;
+	if (!isl_val_is_int(v1) || !isl_val_is_int(v2))
+		isl_die(isl_val_get_ctx(v1), isl_error_invalid,
+			"expecting two integers", goto error);
+	if (isl_val_is_nonneg(v1) && isl_val_lt(v1, v2)) {
+		isl_val_free(v2);
+		return v1;
+	}
+	v1 = isl_val_cow(v1);
+	if (!v1)
+		goto error;
+	isl_int_fdiv_r(v1->n, v1->n, v2->n);
+	isl_val_free(v2);
+	return v1;
+error:
+	isl_val_free(v1);
+	isl_val_free(v2);
+	return NULL;
+}
+
+/* Given two integer values, return their greatest common divisor.
+ */
+__isl_give isl_val *isl_val_gcd(__isl_take isl_val *v1, __isl_take isl_val *v2)
+{
+	if (!v1 || !v2)
+		goto error;
+	if (!isl_val_is_int(v1) || !isl_val_is_int(v2))
+		isl_die(isl_val_get_ctx(v1), isl_error_invalid,
+			"expecting two integers", goto error);
+	if (isl_val_eq(v1, v2)) {
+		isl_val_free(v2);
+		return v1;
+	}
+	if (isl_val_is_one(v1)) {
+		isl_val_free(v2);
+		return v1;
+	}
+	if (isl_val_is_one(v2)) {
+		isl_val_free(v1);
+		return v2;
+	}
+	v1 = isl_val_cow(v1);
+	if (!v1)
+		goto error;
+	isl_int_gcd(v1->n, v1->n, v2->n);
+	isl_val_free(v2);
+	return v1;
+error:
+	isl_val_free(v1);
+	isl_val_free(v2);
+	return NULL;
+}
+
+/* Compute x, y and g such that g = gcd(a,b) and a*x+b*y = g.
+ */
+static void isl_int_gcdext(isl_int g, isl_int x, isl_int y,
+	isl_int a, isl_int b)
+{
+	isl_int d, tmp;
+	isl_int a_copy, b_copy;
+
+	isl_int_init(a_copy);
+	isl_int_init(b_copy);
+	isl_int_init(d);
+	isl_int_init(tmp);
+	isl_int_set(a_copy, a);
+	isl_int_set(b_copy, b);
+	isl_int_abs(g, a_copy);
+	isl_int_abs(d, b_copy);
+	isl_int_set_si(x, 1);
+	isl_int_set_si(y, 0);
+	while (isl_int_is_pos(d)) {
+		isl_int_fdiv_q(tmp, g, d);
+		isl_int_submul(x, tmp, y);
+		isl_int_submul(g, tmp, d);
+		isl_int_swap(g, d);
+		isl_int_swap(x, y);
+	}
+	if (isl_int_is_zero(a_copy))
+		isl_int_set_si(x, 0);
+	else if (isl_int_is_neg(a_copy))
+		isl_int_neg(x, x);
+	if (isl_int_is_zero(b_copy))
+		isl_int_set_si(y, 0);
+	else {
+		isl_int_mul(tmp, a_copy, x);
+		isl_int_sub(tmp, g, tmp);
+		isl_int_divexact(y, tmp, b_copy);
+	}
+	isl_int_clear(d);
+	isl_int_clear(tmp);
+	isl_int_clear(a_copy);
+	isl_int_clear(b_copy);
+}
+
+/* Given two integer values v1 and v2, return their greatest common divisor g,
+ * as well as two integers x and y such that x * v1 + y * v2 = g.
+ */
+__isl_give isl_val *isl_val_gcdext(__isl_take isl_val *v1,
+	__isl_take isl_val *v2, __isl_give isl_val **x, __isl_give isl_val **y)
+{
+	isl_ctx *ctx;
+	isl_val *a = NULL, *b = NULL;
+
+	if (!x && !y)
+		return isl_val_gcd(v1, v2);
+
+	if (!v1 || !v2)
+		goto error;
+
+	ctx = isl_val_get_ctx(v1);
+	if (!isl_val_is_int(v1) || !isl_val_is_int(v2))
+		isl_die(ctx, isl_error_invalid,
+			"expecting two integers", goto error);
+
+	v1 = isl_val_cow(v1);
+	a = isl_val_alloc(ctx);
+	b = isl_val_alloc(ctx);
+	if (!v1 || !a || !b)
+		goto error;
+	isl_int_gcdext(v1->n, a->n, b->n, v1->n, v2->n);
+	if (x) {
+		isl_int_set_si(a->d, 1);
+		*x = a;
+	} else
+		isl_val_free(a);
+	if (y) {
+		isl_int_set_si(b->d, 1);
+		*y = b;
+	} else
+		isl_val_free(b);
+	isl_val_free(v2);
+	return v1;
+error:
+	isl_val_free(v1);
+	isl_val_free(v2);
+	isl_val_free(a);
+	isl_val_free(b);
+	if (x)
+		*x = NULL;
+	if (y)
+		*y = NULL;
+	return NULL;
+}
+
+/* Does "v" represent an integer value?
+ */
+int isl_val_is_int(__isl_keep isl_val *v)
+{
+	if (!v)
+		return -1;
+
+	return isl_int_is_one(v->d);
+}
+
+/* Does "v" represent a rational value?
+ */
+int isl_val_is_rat(__isl_keep isl_val *v)
+{
+	if (!v)
+		return -1;
+
+	return !isl_int_is_zero(v->d);
+}
+
+/* Does "v" represent NaN?
+ */
+int isl_val_is_nan(__isl_keep isl_val *v)
+{
+	if (!v)
+		return -1;
+
+	return isl_int_is_zero(v->n) && isl_int_is_zero(v->d);
+}
+
+/* Does "v" represent +infinity?
+ */
+int isl_val_is_infty(__isl_keep isl_val *v)
+{
+	if (!v)
+		return -1;
+
+	return isl_int_is_pos(v->n) && isl_int_is_zero(v->d);
+}
+
+/* Does "v" represent -infinity?
+ */
+int isl_val_is_neginfty(__isl_keep isl_val *v)
+{
+	if (!v)
+		return -1;
+
+	return isl_int_is_neg(v->n) && isl_int_is_zero(v->d);
+}
+
+/* Does "v" represent the integer zero?
+ */
+int isl_val_is_zero(__isl_keep isl_val *v)
+{
+	if (!v)
+		return -1;
+
+	return isl_int_is_zero(v->n) && !isl_int_is_zero(v->d);
+}
+
+/* Does "v" represent the integer one?
+ */
+int isl_val_is_one(__isl_keep isl_val *v)
+{
+	if (!v)
+		return -1;
+
+	return isl_int_eq(v->n, v->d);
+}
+
+/* Does "v" represent the integer negative one?
+ */
+int isl_val_is_negone(__isl_keep isl_val *v)
+{
+	if (!v)
+		return -1;
+
+	return isl_int_is_neg(v->n) && isl_int_abs_eq(v->n, v->d);
+}
+
+/* Is "v" (strictly) positive?
+ */
+int isl_val_is_pos(__isl_keep isl_val *v)
+{
+	if (!v)
+		return -1;
+
+	return isl_int_is_pos(v->n);
+}
+
+/* Is "v" (strictly) negative?
+ */
+int isl_val_is_neg(__isl_keep isl_val *v)
+{
+	if (!v)
+		return -1;
+
+	return isl_int_is_neg(v->n);
+}
+
+/* Is "v" non-negative?
+ */
+int isl_val_is_nonneg(__isl_keep isl_val *v)
+{
+	if (!v)
+		return -1;
+
+	if (isl_val_is_nan(v))
+		return 0;
+
+	return isl_int_is_nonneg(v->n);
+}
+
+/* Is "v" non-positive?
+ */
+int isl_val_is_nonpos(__isl_keep isl_val *v)
+{
+	if (!v)
+		return -1;
+
+	if (isl_val_is_nan(v))
+		return 0;
+
+	return isl_int_is_nonpos(v->n);
+}
+
+/* Return the sign of "v".
+ *
+ * The sign of NaN is undefined.
+ */
+int isl_val_sgn(__isl_keep isl_val *v)
+{
+	if (!v)
+		return 0;
+	if (isl_val_is_zero(v))
+		return 0;
+	if (isl_val_is_pos(v))
+		return 1;
+	return -1;
+}
+
+/* Is "v1" (strictly) less than "v2"?
+ */
+int isl_val_lt(__isl_keep isl_val *v1, __isl_keep isl_val *v2)
+{
+	isl_int t;
+	int lt;
+
+	if (!v1 || !v2)
+		return -1;
+	if (isl_val_is_int(v1) && isl_val_is_int(v2))
+		return isl_int_lt(v1->n, v2->n);
+	if (isl_val_is_nan(v1) || isl_val_is_nan(v2))
+		return 0;
+	if (isl_val_eq(v1, v2))
+		return 0;
+	if (isl_val_is_infty(v2))
+		return 1;
+	if (isl_val_is_infty(v1))
+		return 0;
+	if (isl_val_is_neginfty(v1))
+		return 1;
+	if (isl_val_is_neginfty(v2))
+		return 0;
+
+	isl_int_init(t);
+	isl_int_mul(t, v1->n, v2->d);
+	isl_int_submul(t, v2->n, v1->d);
+	lt = isl_int_is_neg(t);
+	isl_int_clear(t);
+
+	return lt;
+}
+
+/* Is "v1" (strictly) greater than "v2"?
+ */
+int isl_val_gt(__isl_keep isl_val *v1, __isl_keep isl_val *v2)
+{
+	return isl_val_lt(v2, v1);
+}
+
+/* Is "v1" less than or equal to "v2"?
+ */
+int isl_val_le(__isl_keep isl_val *v1, __isl_keep isl_val *v2)
+{
+	isl_int t;
+	int le;
+
+	if (!v1 || !v2)
+		return -1;
+	if (isl_val_is_int(v1) && isl_val_is_int(v2))
+		return isl_int_le(v1->n, v2->n);
+	if (isl_val_is_nan(v1) || isl_val_is_nan(v2))
+		return 0;
+	if (isl_val_eq(v1, v2))
+		return 1;
+	if (isl_val_is_infty(v2))
+		return 1;
+	if (isl_val_is_infty(v1))
+		return 0;
+	if (isl_val_is_neginfty(v1))
+		return 1;
+	if (isl_val_is_neginfty(v2))
+		return 0;
+
+	isl_int_init(t);
+	isl_int_mul(t, v1->n, v2->d);
+	isl_int_submul(t, v2->n, v1->d);
+	le = isl_int_is_nonpos(t);
+	isl_int_clear(t);
+
+	return le;
+}
+
+/* Is "v1" greater than or equal to "v2"?
+ */
+int isl_val_ge(__isl_keep isl_val *v1, __isl_keep isl_val *v2)
+{
+	return isl_val_le(v2, v1);
+}
+
+/* How does "v" compare to "i"?
+ *
+ * Return 1 if v is greater, -1 if v is smaller and 0 if v is equal to i.
+ *
+ * If v is NaN (or NULL), then the result is undefined.
+ */
+int isl_val_cmp_si(__isl_keep isl_val *v, long i)
+{
+	isl_int t;
+	int cmp;
+
+	if (!v)
+		return 0;
+	if (isl_val_is_int(v))
+		return isl_int_cmp_si(v->n, i);
+	if (isl_val_is_nan(v))
+		return 0;
+	if (isl_val_is_infty(v))
+		return 1;
+	if (isl_val_is_neginfty(v))
+		return -1;
+
+	isl_int_init(t);
+	isl_int_mul_si(t, v->d, i);
+	isl_int_sub(t, v->n, t);
+	cmp = isl_int_sgn(t);
+	isl_int_clear(t);
+
+	return cmp;
+}
+
+/* Is "v1" equal to "v2"?
+ */
+int isl_val_eq(__isl_keep isl_val *v1, __isl_keep isl_val *v2)
+{
+	if (!v1 || !v2)
+		return -1;
+	if (isl_val_is_nan(v1) || isl_val_is_nan(v2))
+		return 0;
+
+	return isl_int_eq(v1->n, v2->n) && isl_int_eq(v1->d, v2->d);
+}
+
+/* Is "v1" equal to "v2" in absolute value?
+ */
+int isl_val_abs_eq(__isl_keep isl_val *v1, __isl_keep isl_val *v2)
+{
+	if (!v1 || !v2)
+		return -1;
+	if (isl_val_is_nan(v1) || isl_val_is_nan(v2))
+		return 0;
+
+	return isl_int_abs_eq(v1->n, v2->n) && isl_int_eq(v1->d, v2->d);
+}
+
+/* Is "v1" different from "v2"?
+ */
+int isl_val_ne(__isl_keep isl_val *v1, __isl_keep isl_val *v2)
+{
+	if (!v1 || !v2)
+		return -1;
+	if (isl_val_is_nan(v1) || isl_val_is_nan(v2))
+		return 0;
+
+	return isl_int_ne(v1->n, v2->n) || isl_int_ne(v1->d, v2->d);
+}
+
+/* Print a textual representation of "v" onto "p".
+ */
+__isl_give isl_printer *isl_printer_print_val(__isl_take isl_printer *p,
+	__isl_keep isl_val *v)
+{
+	int neg;
+
+	if (!p || !v)
+		return isl_printer_free(p);
+
+	neg = isl_int_is_neg(v->n);
+	if (neg) {
+		p = isl_printer_print_str(p, "-");
+		isl_int_neg(v->n, v->n);
+	}
+	if (isl_int_is_zero(v->d)) {
+		int sgn = isl_int_sgn(v->n);
+		p = isl_printer_print_str(p, sgn < 0 ? "-infty" :
+					    sgn == 0 ? "NaN" : "infty");
+	} else
+		p = isl_printer_print_isl_int(p, v->n);
+	if (neg)
+		isl_int_neg(v->n, v->n);
+	if (!isl_int_is_zero(v->d) && !isl_int_is_one(v->d)) {
+		p = isl_printer_print_str(p, "/");
+		p = isl_printer_print_isl_int(p, v->d);
+	}
+
+	return p;
+}
+
+/* Is "val1" (obviously) equal to "val2"?
+ *
+ * This is a private copy of isl_val_eq for use in the generic
+ * isl_multi_*_plain_is_equal instantiated for isl_val.
+ */
+int isl_val_plain_is_equal(__isl_keep isl_val *val1, __isl_keep isl_val *val2)
+{
+	return isl_val_eq(val1, val2);
+}
+
+/* Does "v" have any non-zero coefficients
+ * for any dimension in the given range?
+ *
+ * This function is only meant to be used in the generic isl_multi_*
+ * functions which have to deal with base objects that have an associated
+ * space.  Since an isl_val does not have any coefficients, this function
+ * always return 0.
+ */
+int isl_val_involves_dims(__isl_keep isl_val *v, enum isl_dim_type type,
+	unsigned first, unsigned n)
+{
+	if (!v)
+		return -1;
+
+	return 0;
+}
+
+/* Insert "n" dimensions of type "type" at position "first".
+ *
+ * This function is only meant to be used in the generic isl_multi_*
+ * functions which have to deal with base objects that have an associated
+ * space.  Since an isl_val does not have an associated space, this function
+ * does not do anything.
+ */
+__isl_give isl_val *isl_val_insert_dims(__isl_take isl_val *v,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return v;
+}
+
+/* Drop the the "n" first dimensions of type "type" at position "first".
+ *
+ * This function is only meant to be used in the generic isl_multi_*
+ * functions which have to deal with base objects that have an associated
+ * space.  Since an isl_val does not have an associated space, this function
+ * does not do anything.
+ */
+__isl_give isl_val *isl_val_drop_dims(__isl_take isl_val *v,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	return v;
+}
+
+/* Change the name of the dimension of type "type" at position "pos" to "s".
+ *
+ * This function is only meant to be used in the generic isl_multi_*
+ * functions which have to deal with base objects that have an associated
+ * space.  Since an isl_val does not have an associated space, this function
+ * does not do anything.
+ */
+__isl_give isl_val *isl_val_set_dim_name(__isl_take isl_val *v,
+	enum isl_dim_type type, unsigned pos, const char *s)
+{
+	return v;
+}
+
+/* Return the space of "v".
+ *
+ * This function is only meant to be used in the generic isl_multi_*
+ * functions which have to deal with base objects that have an associated
+ * space.  The conditions surrounding the call to this function make sure
+ * that this function will never actually get called.  We return a valid
+ * space anyway, just in case.
+ */
+__isl_give isl_space *isl_val_get_space(__isl_keep isl_val *v)
+{
+	if (!v)
+		return NULL;
+
+	return isl_space_params_alloc(isl_val_get_ctx(v), 0);
+}
+
+/* Reset the domain space of "v" to "space".
+ *
+ * This function is only meant to be used in the generic isl_multi_*
+ * functions which have to deal with base objects that have an associated
+ * space.  Since an isl_val does not have an associated space, this function
+ * does not do anything, apart from error handling and cleaning up memory.
+ */
+__isl_give isl_val *isl_val_reset_domain_space(__isl_take isl_val *v,
+	__isl_take isl_space *space)
+{
+	if (!space)
+		return isl_val_free(v);
+	isl_space_free(space);
+	return v;
+}
+
+/* Align the parameters of "v" to those of "space".
+ *
+ * This function is only meant to be used in the generic isl_multi_*
+ * functions which have to deal with base objects that have an associated
+ * space.  Since an isl_val does not have an associated space, this function
+ * does not do anything, apart from error handling and cleaning up memory.
+ * Note that the conditions surrounding the call to this function make sure
+ * that this function will never actually get called.
+ */
+__isl_give isl_val *isl_val_align_params(__isl_take isl_val *v,
+	__isl_take isl_space *space)
+{
+	if (!space)
+		return isl_val_free(v);
+	isl_space_free(space);
+	return v;
+}
+
+/* Reorder the dimensions of the domain of "v" according
+ * to the given reordering.
+ *
+ * This function is only meant to be used in the generic isl_multi_*
+ * functions which have to deal with base objects that have an associated
+ * space.  Since an isl_val does not have an associated space, this function
+ * does not do anything, apart from error handling and cleaning up memory.
+ */
+__isl_give isl_val *isl_val_realign_domain(__isl_take isl_val *v,
+	__isl_take isl_reordering *r)
+{
+	if (!r)
+		return isl_val_free(v);
+	isl_reordering_free(r);
+	return v;
+}
+
+/* Return an isl_val that is zero on "ls".
+ *
+ * This function is only meant to be used in the generic isl_multi_*
+ * functions which have to deal with base objects that have an associated
+ * space.  Since an isl_val does not have an associated space, this function
+ * simply returns a zero isl_val in the same context as "ls".
+ */
+__isl_give isl_val *isl_val_zero_on_domain(__isl_take isl_local_space *ls)
+{
+	isl_ctx *ctx;
+
+	if (!ls)
+		return NULL;
+	ctx = isl_local_space_get_ctx(ls);
+	isl_local_space_free(ls);
+	return isl_val_zero(ctx);
+}
+
+/* Do the parameters of "v" match those of "space"?
+ *
+ * This function is only meant to be used in the generic isl_multi_*
+ * functions which have to deal with base objects that have an associated
+ * space.  Since an isl_val does not have an associated space, this function
+ * simply returns 1, except if "v" or "space" are NULL.
+ */
+int isl_val_matching_params(__isl_keep isl_val *v, __isl_keep isl_space *space)
+{
+	if (!v || !space)
+		return -1;
+	return 1;
+}
+
+/* Check that the domain space of "v" matches "space".
+ *
+ * Return 0 on success and -1 on error.
+ *
+ * This function is only meant to be used in the generic isl_multi_*
+ * functions which have to deal with base objects that have an associated
+ * space.  Since an isl_val does not have an associated space, this function
+ * simply returns 0, except if "v" or "space" are NULL.
+ */
+int isl_val_check_match_domain_space(__isl_keep isl_val *v,
+	__isl_keep isl_space *space)
+{
+	if (!v || !space)
+		return -1;
+	return 0;
+}
+
+#undef BASE
+#define BASE val
+
+#define NO_DOMAIN
+#define NO_INTERSECT_DOMAIN
+#define NO_GIST
+#define NO_IDENTITY
+#define NO_FROM_BASE
+#define NO_MOVE_DIMS
+#include <isl_multi_templ.c>
+
+/* Apply "fn" to each of the elements of "mv" with as second argument "v".
+ */
+static __isl_give isl_multi_val *isl_multi_val_fn_val(
+	__isl_take isl_multi_val *mv,
+	__isl_give isl_val *(*fn)(__isl_take isl_val *v1,
+					__isl_take isl_val *v2),
+	__isl_take isl_val *v)
+{
+	int i;
+
+	mv = isl_multi_val_cow(mv);
+	if (!mv || !v)
+		goto error;
+
+	for (i = 0; i < mv->n; ++i) {
+		mv->p[i] = fn(mv->p[i], isl_val_copy(v));
+		if (!mv->p[i])
+			goto error;
+	}
+
+	isl_val_free(v);
+	return mv;
+error:
+	isl_val_free(v);
+	isl_multi_val_free(mv);
+	return NULL;
+}
+
+/* Add "v" to each of the elements of "mv".
+ */
+__isl_give isl_multi_val *isl_multi_val_add_val(__isl_take isl_multi_val *mv,
+	__isl_take isl_val *v)
+{
+	if (!v)
+		return isl_multi_val_free(mv);
+	if (isl_val_is_zero(v)) {
+		isl_val_free(v);
+		return mv;
+	}
+	return isl_multi_val_fn_val(mv, &isl_val_add, v);
+}
+
+/* Reduce the elements of "mv" modulo "v".
+ */
+__isl_give isl_multi_val *isl_multi_val_mod_val(__isl_take isl_multi_val *mv,
+	__isl_take isl_val *v)
+{
+	return isl_multi_val_fn_val(mv, &isl_val_mod, v);
+}
diff --git a/polly/lib/External/isl/isl_val_gmp.c b/polly/lib/External/isl/isl_val_gmp.c
new file mode 100644
index 00000000000..42e8d444671
--- /dev/null
+++ b/polly/lib/External/isl/isl_val_gmp.c
@@ -0,0 +1,128 @@
+#include <string.h>
+#include <isl/val_gmp.h>
+#include <isl_val_private.h>
+
+/* Return a reference to an isl_val representing the integer "z".
+ */
+__isl_give isl_val *isl_val_int_from_gmp(isl_ctx *ctx, mpz_t z)
+{
+	isl_val *v;
+
+	v = isl_val_alloc(ctx);
+	if (!v)
+		return NULL;
+
+	isl_int_set(v->n, z);
+	isl_int_set_si(v->d, 1);
+
+	return v;
+}
+
+/* Return a reference to an isl_val representing the rational value "n"/"d".
+ */
+__isl_give isl_val *isl_val_from_gmp(isl_ctx *ctx, const mpz_t n, const mpz_t d)
+{
+	isl_val *v;
+
+	v = isl_val_alloc(ctx);
+	if (!v)
+		return NULL;
+
+	isl_int_set(v->n, n);
+	isl_int_set(v->d, d);
+
+	return isl_val_normalize(v);
+}
+
+/* Extract the numerator of a rational value "v" in "z".
+ *
+ * If "v" is not a rational value, then the result is undefined.
+ */
+int isl_val_get_num_gmp(__isl_keep isl_val *v, mpz_t z)
+{
+	if (!v)
+		return -1;
+	if (!isl_val_is_rat(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting rational value", return -1);
+	mpz_set(z, v->n);
+	return 0;
+}
+
+/* Extract the denominator of a rational value "v" in "z".
+ *
+ * If "v" is not a rational value, then the result is undefined.
+ */
+int isl_val_get_den_gmp(__isl_keep isl_val *v, mpz_t z)
+{
+	if (!v)
+		return -1;
+	if (!isl_val_is_rat(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting rational value", return -1);
+	mpz_set(z, v->d);
+	return 0;
+}
+
+/* Return a reference to an isl_val representing the unsigned
+ * integer value stored in the "n" chunks of size "size" at "chunks".
+ * The least significant chunk is assumed to be stored first.
+ */
+__isl_give isl_val *isl_val_int_from_chunks(isl_ctx *ctx, size_t n,
+	size_t size, const void *chunks)
+{
+	isl_val *v;
+
+	v = isl_val_alloc(ctx);
+	if (!v)
+		return NULL;
+
+	mpz_import(v->n, n, -1, size, 0, 0, chunks);
+	isl_int_set_si(v->d, 1);
+
+	return v;
+}
+
+/* Return the number of chunks of size "size" required to
+ * store the absolute value of the numerator of "v".
+ */
+size_t isl_val_n_abs_num_chunks(__isl_keep isl_val *v, size_t size)
+{
+	if (!v)
+		return 0;
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting rational value", return 0);
+
+	size *= 8;
+	return (mpz_sizeinbase(v->n, 2) + size - 1) / size;
+}
+
+/* Store a representation of the absolute value of the numerator of "v"
+ * in terms of chunks of size "size" at "chunks".
+ * The least significant chunk is stored first.
+ * The number of chunks in the result can be obtained by calling
+ * isl_val_n_abs_num_chunks.  The user is responsible for allocating
+ * enough memory to store the results.
+ *
+ * In the special case of a zero value, isl_val_n_abs_num_chunks will
+ * return one, while mpz_export will not fill in any chunks.  We therefore
+ * do it ourselves.
+ */
+int isl_val_get_abs_num_chunks(__isl_keep isl_val *v, size_t size,
+	void *chunks)
+{
+	if (!v || !chunks)
+		return -1;
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting rational value", return -1);
+
+	mpz_export(chunks, NULL, -1, size, 0, 0, v->n);
+	if (isl_val_is_zero(v))
+		memset(chunks, 0, size);
+
+	return 0;
+}
diff --git a/polly/lib/External/isl/isl_val_imath.c b/polly/lib/External/isl/isl_val_imath.c
new file mode 100644
index 00000000000..8f91700a895
--- /dev/null
+++ b/polly/lib/External/isl/isl_val_imath.c
@@ -0,0 +1,64 @@
+#include <isl_val_private.h>
+
+/* Return a reference to an isl_val representing the unsigned
+ * integer value stored in the "n" chunks of size "size" at "chunks".
+ * The least significant chunk is assumed to be stored first.
+ */
+__isl_give isl_val *isl_val_int_from_chunks(isl_ctx *ctx, size_t n,
+	size_t size, const void *chunks)
+{
+	isl_val *v;
+
+	v = isl_val_alloc(ctx);
+	if (!v)
+		return NULL;
+
+	impz_import(v->n, n, -1, size, 0, 0, chunks);
+	isl_int_set_si(v->d, 1);
+
+	return v;
+}
+
+/* Store a representation of the absolute value of the numerator of "v"
+ * in terms of chunks of size "size" at "chunks".
+ * The least significant chunk is stored first.
+ * The number of chunks in the result can be obtained by calling
+ * isl_val_n_abs_num_chunks.  The user is responsible for allocating
+ * enough memory to store the results.
+ *
+ * In the special case of a zero value, isl_val_n_abs_num_chunks will
+ * return one, while impz_export will not fill in any chunks.  We therefore
+ * do it ourselves.
+ */
+int isl_val_get_abs_num_chunks(__isl_keep isl_val *v, size_t size,
+	void *chunks)
+{
+	if (!v || !chunks)
+		return -1;
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting rational value", return -1);
+
+	impz_export(chunks, NULL, -1, size, 0, 0, v->n);
+	if (isl_val_is_zero(v))
+		memset(chunks, 0, size);
+
+	return 0;
+}
+
+/* Return the number of chunks of size "size" required to
+ * store the absolute value of the numerator of "v".
+ */
+size_t isl_val_n_abs_num_chunks(__isl_keep isl_val *v, size_t size)
+{
+	if (!v)
+		return 0;
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting rational value", return 0);
+
+	size *= 8;
+	return (impz_sizeinbase(v->n, 2) + size - 1) / size;
+}
diff --git a/polly/lib/External/isl/isl_val_private.h b/polly/lib/External/isl/isl_val_private.h
new file mode 100644
index 00000000000..722fcf07639
--- /dev/null
+++ b/polly/lib/External/isl/isl_val_private.h
@@ -0,0 +1,70 @@
+#ifndef ISL_VAL_PRIVATE_H
+#define ISL_VAL_PRIVATE_H
+
+#include <isl_int.h>
+#include <isl/val.h>
+#include <isl/local_space.h>
+#include <isl_reordering.h>
+
+/* Represents a "value", which may be an integer value, a rational value,
+ * plus or minus infinity or "not a number".
+ *
+ * Internally, +infinity is represented as 1/0,
+ * -infinity as -1/0 and NaN as 0/0.
+ *
+ * A rational value is always normalized before it is passed to the user.
+ */
+struct isl_val {
+	int ref;
+	isl_ctx *ctx;
+
+	isl_int n;
+	isl_int d;
+};
+
+#undef EL
+#define EL isl_val
+
+#include <isl_list_templ.h>
+
+__isl_give isl_val *isl_val_alloc(isl_ctx *ctx);
+__isl_give isl_val *isl_val_normalize(__isl_take isl_val *v);
+__isl_give isl_val *isl_val_int_from_isl_int(isl_ctx *ctx, isl_int n);
+__isl_give isl_val *isl_val_rat_from_isl_int(isl_ctx *ctx,
+	isl_int n, isl_int d);
+__isl_give isl_val *isl_val_cow(__isl_take isl_val *val);
+
+int isl_val_involves_dims(__isl_keep isl_val *v, enum isl_dim_type type,
+	unsigned first, unsigned n);
+__isl_give isl_val *isl_val_insert_dims(__isl_take isl_val *v,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_val *isl_val_drop_dims(__isl_take isl_val *v,
+	enum isl_dim_type type, unsigned first, unsigned n);
+__isl_give isl_val *isl_val_set_dim_name(__isl_take isl_val *v,
+	enum isl_dim_type type, unsigned pos, const char *s);
+__isl_give isl_space *isl_val_get_space(__isl_keep isl_val *v);
+__isl_give isl_val *isl_val_reset_domain_space(__isl_take isl_val *v,
+	__isl_take isl_space *space);
+__isl_give isl_val *isl_val_align_params(__isl_take isl_val *v,
+	__isl_take isl_space *space);
+__isl_give isl_val *isl_val_realign_domain(__isl_take isl_val *v,
+	__isl_take isl_reordering *r);
+__isl_give isl_val *isl_val_zero_on_domain(__isl_take isl_local_space *ls);
+
+__isl_give isl_val *isl_val_scale_val(__isl_take isl_val *v1,
+	__isl_take isl_val *v2);
+__isl_give isl_val *isl_val_scale_down_val(__isl_take isl_val *v1,
+	__isl_take isl_val *v2);
+
+int isl_val_plain_is_equal(__isl_keep isl_val *val1, __isl_keep isl_val *val2);
+
+int isl_val_matching_params(__isl_keep isl_val *v, __isl_keep isl_space *space);
+int isl_val_check_match_domain_space(__isl_keep isl_val *v,
+	__isl_keep isl_space *space);
+
+#undef BASE
+#define BASE val
+
+#include <isl_multi_templ.h>
+
+#endif
diff --git a/polly/lib/External/isl/isl_vec.c b/polly/lib/External/isl/isl_vec.c
new file mode 100644
index 00000000000..a8582be3996
--- /dev/null
+++ b/polly/lib/External/isl/isl_vec.c
@@ -0,0 +1,579 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ * Copyright 2013      Ecole Normale Superieure
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
+ */
+
+#include <isl_ctx_private.h>
+#include <isl_seq.h>
+#include <isl_val_private.h>
+#include <isl_vec_private.h>
+#include <isl/deprecated/vec_int.h>
+
+isl_ctx *isl_vec_get_ctx(__isl_keep isl_vec *vec)
+{
+	return vec ? vec->ctx : NULL;
+}
+
+struct isl_vec *isl_vec_alloc(struct isl_ctx *ctx, unsigned size)
+{
+	struct isl_vec *vec;
+
+	vec = isl_alloc_type(ctx, struct isl_vec);
+	if (!vec)
+		return NULL;
+
+	vec->block = isl_blk_alloc(ctx, size);
+	if (isl_blk_is_error(vec->block))
+		goto error;
+
+	vec->ctx = ctx;
+	isl_ctx_ref(ctx);
+	vec->ref = 1;
+	vec->size = size;
+	vec->el = vec->block.data;
+
+	return vec;
+error:
+	isl_blk_free(ctx, vec->block);
+	return NULL;
+}
+
+__isl_give isl_vec *isl_vec_extend(__isl_take isl_vec *vec, unsigned size)
+{
+	if (!vec)
+		return NULL;
+	if (size <= vec->size)
+		return vec;
+
+	vec = isl_vec_cow(vec);
+	if (!vec)
+		return NULL;
+
+	vec->block = isl_blk_extend(vec->ctx, vec->block, size);
+	if (!vec->block.data)
+		goto error;
+
+	vec->size = size;
+	vec->el = vec->block.data;
+
+	return vec;
+error:
+	isl_vec_free(vec);
+	return NULL;
+}
+
+__isl_give isl_vec *isl_vec_zero_extend(__isl_take isl_vec *vec, unsigned size)
+{
+	int extra;
+
+	if (!vec)
+		return NULL;
+	if (size <= vec->size)
+		return vec;
+
+	vec = isl_vec_cow(vec);
+	if (!vec)
+		return NULL;
+
+	extra = size - vec->size;
+	vec = isl_vec_extend(vec, size);
+	if (!vec)
+		return NULL;
+
+	isl_seq_clr(vec->el + size - extra, extra);
+
+	return vec;
+}
+
+/* Return a vector containing the elements of "vec1" followed by
+ * those of "vec2".
+ */
+__isl_give isl_vec *isl_vec_concat(__isl_take isl_vec *vec1,
+	__isl_take isl_vec *vec2)
+{
+	if (!vec1 || !vec2)
+		goto error;
+
+	if (vec2->size == 0) {
+		isl_vec_free(vec2);
+		return vec1;
+	}
+
+	if (vec1->size == 0) {
+		isl_vec_free(vec1);
+		return vec2;
+	}
+
+	vec1 = isl_vec_extend(vec1, vec1->size + vec2->size);
+	if (!vec1)
+		goto error;
+
+	isl_seq_cpy(vec1->el + vec1->size - vec2->size, vec2->el, vec2->size);
+
+	isl_vec_free(vec2);
+	return vec1;
+error:
+	isl_vec_free(vec1);
+	isl_vec_free(vec2);
+	return NULL;
+}
+
+struct isl_vec *isl_vec_copy(struct isl_vec *vec)
+{
+	if (!vec)
+		return NULL;
+
+	vec->ref++;
+	return vec;
+}
+
+struct isl_vec *isl_vec_dup(struct isl_vec *vec)
+{
+	struct isl_vec *vec2;
+
+	if (!vec)
+		return NULL;
+	vec2 = isl_vec_alloc(vec->ctx, vec->size);
+	if (!vec2)
+		return NULL;
+	isl_seq_cpy(vec2->el, vec->el, vec->size);
+	return vec2;
+}
+
+struct isl_vec *isl_vec_cow(struct isl_vec *vec)
+{
+	struct isl_vec *vec2;
+	if (!vec)
+		return NULL;
+
+	if (vec->ref == 1)
+		return vec;
+
+	vec2 = isl_vec_dup(vec);
+	isl_vec_free(vec);
+	return vec2;
+}
+
+__isl_null isl_vec *isl_vec_free(__isl_take isl_vec *vec)
+{
+	if (!vec)
+		return NULL;
+
+	if (--vec->ref > 0)
+		return NULL;
+
+	isl_ctx_deref(vec->ctx);
+	isl_blk_free(vec->ctx, vec->block);
+	free(vec);
+
+	return NULL;
+}
+
+int isl_vec_size(__isl_keep isl_vec *vec)
+{
+	return vec ? vec->size : -1;
+}
+
+int isl_vec_get_element(__isl_keep isl_vec *vec, int pos, isl_int *v)
+{
+	if (!vec)
+		return -1;
+
+	if (pos < 0 || pos >= vec->size)
+		isl_die(vec->ctx, isl_error_invalid, "position out of range",
+			return -1);
+	isl_int_set(*v, vec->el[pos]);
+	return 0;
+}
+
+/* Extract the element at position "pos" of "vec".
+ */
+__isl_give isl_val *isl_vec_get_element_val(__isl_keep isl_vec *vec, int pos)
+{
+	isl_ctx *ctx;
+
+	if (!vec)
+		return NULL;
+	ctx = isl_vec_get_ctx(vec);
+	if (pos < 0 || pos >= vec->size)
+		isl_die(ctx, isl_error_invalid, "position out of range",
+			return NULL);
+	return isl_val_int_from_isl_int(ctx, vec->el[pos]);
+}
+
+__isl_give isl_vec *isl_vec_set_element(__isl_take isl_vec *vec,
+	int pos, isl_int v)
+{
+	vec = isl_vec_cow(vec);
+	if (!vec)
+		return NULL;
+	if (pos < 0 || pos >= vec->size)
+		isl_die(vec->ctx, isl_error_invalid, "position out of range",
+			goto error);
+	isl_int_set(vec->el[pos], v);
+	return vec;
+error:
+	isl_vec_free(vec);
+	return NULL;
+}
+
+__isl_give isl_vec *isl_vec_set_element_si(__isl_take isl_vec *vec,
+	int pos, int v)
+{
+	vec = isl_vec_cow(vec);
+	if (!vec)
+		return NULL;
+	if (pos < 0 || pos >= vec->size)
+		isl_die(vec->ctx, isl_error_invalid, "position out of range",
+			goto error);
+	isl_int_set_si(vec->el[pos], v);
+	return vec;
+error:
+	isl_vec_free(vec);
+	return NULL;
+}
+
+/* Replace the element at position "pos" of "vec" by "v".
+ */
+__isl_give isl_vec *isl_vec_set_element_val(__isl_take isl_vec *vec,
+	int pos, __isl_take isl_val *v)
+{
+	if (!v)
+		return isl_vec_free(vec);
+	if (!isl_val_is_int(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting integer value", goto error);
+	vec = isl_vec_set_element(vec, pos, v->n);
+	isl_val_free(v);
+	return vec;
+error:
+	isl_val_free(v);
+	return isl_vec_free(vec);
+}
+
+/* Compare the elements of "vec1" and "vec2" at position "pos".
+ */
+int isl_vec_cmp_element(__isl_keep isl_vec *vec1, __isl_keep isl_vec *vec2,
+	int pos)
+{
+	if (!vec1 || !vec2)
+		return 0;
+	if (pos < 0 || pos >= vec1->size || pos >= vec2->size)
+		isl_die(isl_vec_get_ctx(vec1), isl_error_invalid,
+			"position out of range", return 0);
+	return isl_int_cmp(vec1->el[pos], vec2->el[pos]);
+}
+
+int isl_vec_is_equal(__isl_keep isl_vec *vec1, __isl_keep isl_vec *vec2)
+{
+	if (!vec1 || !vec2)
+		return -1;
+
+	if (vec1->size != vec2->size)
+		return 0;
+
+	return isl_seq_eq(vec1->el, vec2->el, vec1->size);
+}
+
+__isl_give isl_printer *isl_printer_print_vec(__isl_take isl_printer *printer,
+	__isl_keep isl_vec *vec)
+{
+	int i;
+
+	if (!printer || !vec)
+		goto error;
+
+	printer = isl_printer_print_str(printer, "[");
+	for (i = 0; i < vec->size; ++i) {
+		if (i)
+			printer = isl_printer_print_str(printer, ",");
+		printer = isl_printer_print_isl_int(printer, vec->el[i]);
+	}
+	printer = isl_printer_print_str(printer, "]");
+
+	return printer;
+error:
+	isl_printer_free(printer);
+	return NULL;
+}
+
+void isl_vec_dump(struct isl_vec *vec)
+{
+	isl_printer *printer;
+
+	if (!vec)
+		return;
+
+	printer = isl_printer_to_file(vec->ctx, stderr);
+	printer = isl_printer_print_vec(printer, vec);
+	printer = isl_printer_end_line(printer);
+
+	isl_printer_free(printer);
+}
+
+__isl_give isl_vec *isl_vec_set(__isl_take isl_vec *vec, isl_int v)
+{
+	vec = isl_vec_cow(vec);
+	if (!vec)
+		return NULL;
+	isl_seq_set(vec->el, v, vec->size);
+	return vec;
+}
+
+__isl_give isl_vec *isl_vec_set_si(__isl_take isl_vec *vec, int v)
+{
+	vec = isl_vec_cow(vec);
+	if (!vec)
+		return NULL;
+	isl_seq_set_si(vec->el, v, vec->size);
+	return vec;
+}
+
+/* Replace all elements of "vec" by "v".
+ */
+__isl_give isl_vec *isl_vec_set_val(__isl_take isl_vec *vec,
+	__isl_take isl_val *v)
+{
+	vec = isl_vec_cow(vec);
+	if (!vec || !v)
+		goto error;
+	if (!isl_val_is_int(v))
+		isl_die(isl_val_get_ctx(v), isl_error_invalid,
+			"expecting integer value", goto error);
+	isl_seq_set(vec->el, v->n, vec->size);
+	isl_val_free(v);
+	return vec;
+error:
+	isl_vec_free(vec);
+	isl_val_free(v);
+	return NULL;
+}
+
+__isl_give isl_vec *isl_vec_clr(__isl_take isl_vec *vec)
+{
+	vec = isl_vec_cow(vec);
+	if (!vec)
+		return NULL;
+	isl_seq_clr(vec->el, vec->size);
+	return vec;
+}
+
+void isl_vec_lcm(struct isl_vec *vec, isl_int *lcm)
+{
+	isl_seq_lcm(vec->block.data, vec->size, lcm);
+}
+
+/* Given a rational vector, with the denominator in the first element
+ * of the vector, round up all coordinates.
+ */
+struct isl_vec *isl_vec_ceil(struct isl_vec *vec)
+{
+	vec = isl_vec_cow(vec);
+	if (!vec)
+		return NULL;
+
+	isl_seq_cdiv_q(vec->el + 1, vec->el + 1, vec->el[0], vec->size - 1);
+
+	isl_int_set_si(vec->el[0], 1);
+
+	return vec;
+}
+
+struct isl_vec *isl_vec_normalize(struct isl_vec *vec)
+{
+	if (!vec)
+		return NULL;
+	isl_seq_normalize(vec->ctx, vec->el, vec->size);
+	return vec;
+}
+
+__isl_give isl_vec *isl_vec_neg(__isl_take isl_vec *vec)
+{
+	vec = isl_vec_cow(vec);
+	if (!vec)
+		return NULL;
+	isl_seq_neg(vec->el, vec->el, vec->size);
+	return vec;
+}
+
+__isl_give isl_vec *isl_vec_scale(__isl_take isl_vec *vec, isl_int m)
+{
+	if (isl_int_is_one(m))
+		return vec;
+	vec = isl_vec_cow(vec);
+	if (!vec)
+		return NULL;
+	isl_seq_scale(vec->el, vec->el, m, vec->size);
+	return vec;
+}
+
+/* Reduce the elements of "vec" modulo "m".
+ */
+__isl_give isl_vec *isl_vec_fdiv_r(__isl_take isl_vec *vec, isl_int m)
+{
+	vec = isl_vec_cow(vec);
+	if (!vec)
+		return NULL;
+
+	isl_seq_fdiv_r(vec->el, vec->el, m, vec->size);
+
+	return vec;
+}
+
+__isl_give isl_vec *isl_vec_add(__isl_take isl_vec *vec1,
+	__isl_take isl_vec *vec2)
+{
+	vec1 = isl_vec_cow(vec1);
+	if (!vec1 || !vec2)
+		goto error;
+
+	isl_assert(vec1->ctx, vec1->size == vec2->size, goto error);
+
+	isl_seq_combine(vec1->el, vec1->ctx->one, vec1->el,
+			vec1->ctx->one, vec2->el, vec1->size);
+	
+	isl_vec_free(vec2);
+	return vec1;
+error:
+	isl_vec_free(vec1);
+	isl_vec_free(vec2);
+	return NULL;
+}
+
+static int qsort_int_cmp(const void *p1, const void *p2)
+{
+	const isl_int *i1 = (const isl_int *) p1;
+	const isl_int *i2 = (const isl_int *) p2;
+
+	return isl_int_cmp(*i1, *i2);
+}
+
+__isl_give isl_vec *isl_vec_sort(__isl_take isl_vec *vec)
+{
+	if (!vec)
+		return NULL;
+	
+	qsort(vec->el, vec->size, sizeof(*vec->el), &qsort_int_cmp);
+
+	return vec;
+}
+
+__isl_give isl_vec *isl_vec_drop_els(__isl_take isl_vec *vec,
+	unsigned pos, unsigned n)
+{
+	if (n == 0)
+		return vec;
+	vec = isl_vec_cow(vec);
+	if (!vec)
+		return NULL;
+
+	if (pos + n > vec->size)
+		isl_die(vec->ctx, isl_error_invalid,
+			"range out of bounds", goto error);
+
+	if (pos + n != vec->size)
+		isl_seq_cpy(vec->el + pos, vec->el + pos + n,
+			    vec->size - pos - n);
+
+	vec->size -= n;
+	
+	return vec;
+error:
+	isl_vec_free(vec);
+	return NULL;
+}
+
+__isl_give isl_vec *isl_vec_insert_els(__isl_take isl_vec *vec,
+	unsigned pos, unsigned n)
+{
+	isl_vec *ext = NULL;
+
+	if (n == 0)
+		return vec;
+	if (!vec)
+		return NULL;
+
+	if (pos > vec->size)
+		isl_die(vec->ctx, isl_error_invalid,
+			"position out of bounds", goto error);
+
+	ext =  isl_vec_alloc(vec->ctx, vec->size + n);
+	if (!ext)
+		goto error;
+
+	isl_seq_cpy(ext->el, vec->el, pos);
+	isl_seq_cpy(ext->el + pos + n, vec->el + pos, vec->size - pos);
+
+	isl_vec_free(vec);
+	return ext;
+error:
+	isl_vec_free(vec);
+	isl_vec_free(ext);
+	return NULL;
+}
+
+__isl_give isl_vec *isl_vec_insert_zero_els(__isl_take isl_vec *vec,
+	unsigned pos, unsigned n)
+{
+	vec = isl_vec_insert_els(vec, pos, n);
+	if (!vec)
+		return NULL;
+
+	isl_seq_clr(vec->el + pos, n);
+
+	return vec;
+}
+
+/* Move the "n" elements starting as "src_pos" of "vec"
+ * to "dst_pos".  The elements originally at "dst_pos" are moved
+ * up or down depending on whether "dst_pos" is smaller or greater
+ * than "src_pos".
+ */
+__isl_give isl_vec *isl_vec_move_els(__isl_take isl_vec *vec,
+	unsigned dst_pos, unsigned src_pos, unsigned n)
+{
+	isl_vec *res;
+
+	if (!vec)
+		return NULL;
+
+	if (src_pos + n > vec->size)
+		isl_die(vec->ctx, isl_error_invalid,
+			"source range out of bounds", return isl_vec_free(vec));
+	if (dst_pos + n > vec->size)
+		isl_die(vec->ctx, isl_error_invalid,
+			"destination range out of bounds",
+			return isl_vec_free(vec));
+
+	if (n == 0 || dst_pos == src_pos)
+		return vec;
+
+	res = isl_vec_alloc(vec->ctx, vec->size);
+	if (!res)
+		return isl_vec_free(vec);
+
+	if (dst_pos < src_pos) {
+		isl_seq_cpy(res->el, vec->el, dst_pos);
+		isl_seq_cpy(res->el + dst_pos, vec->el + src_pos, n);
+		isl_seq_cpy(res->el + dst_pos + n,
+			    vec->el + dst_pos, src_pos - dst_pos);
+		isl_seq_cpy(res->el + src_pos + n,
+			    vec->el + src_pos + n, res->size - src_pos - n);
+	} else {
+		isl_seq_cpy(res->el, vec->el, src_pos);
+		isl_seq_cpy(res->el + src_pos,
+			    vec->el + src_pos + n, dst_pos - src_pos);
+		isl_seq_cpy(res->el + dst_pos, vec->el + src_pos, n);
+		isl_seq_cpy(res->el + dst_pos + n,
+			    vec->el + dst_pos + n, res->size - dst_pos - n);
+	}
+
+	isl_vec_free(vec);
+	return res;
+}
diff --git a/polly/lib/External/isl/isl_vec_private.h b/polly/lib/External/isl/isl_vec_private.h
new file mode 100644
index 00000000000..c3fcd9227a2
--- /dev/null
+++ b/polly/lib/External/isl/isl_vec_private.h
@@ -0,0 +1,24 @@
+#ifndef ISL_VEC_PRIVATE_H
+#define ISL_VEC_PRIVATE_H
+
+#include <isl_blk.h>
+#include <isl/vec.h>
+
+struct isl_vec {
+	int ref;
+
+	struct isl_ctx *ctx;
+
+	unsigned size;
+	isl_int *el;
+
+	struct isl_blk block;
+};
+
+__isl_give isl_vec *isl_vec_cow(__isl_take isl_vec *vec);
+
+void isl_vec_lcm(struct isl_vec *vec, isl_int *lcm);
+int isl_vec_get_element(__isl_keep isl_vec *vec, int pos, isl_int *v);
+__isl_give isl_vec *isl_vec_set(__isl_take isl_vec *vec, isl_int v);
+
+#endif
diff --git a/polly/lib/External/isl/isl_version.c b/polly/lib/External/isl/isl_version.c
new file mode 100644
index 00000000000..a7ddd9dc8f9
--- /dev/null
+++ b/polly/lib/External/isl/isl_version.c
@@ -0,0 +1,14 @@
+#include "isl_config.h"
+#include "gitversion.h"
+
+const char *isl_version(void)
+{
+	return GIT_HEAD_ID
+#ifdef USE_GMP_FOR_MP
+	"-GMP"
+#endif
+#ifdef USE_IMATH_FOR_MP
+	"-IMath"
+#endif
+	"\n";
+}
diff --git a/polly/lib/External/isl/isl_vertices.c b/polly/lib/External/isl/isl_vertices.c
new file mode 100644
index 00000000000..c17224d08bb
--- /dev/null
+++ b/polly/lib/External/isl/isl_vertices.c
@@ -0,0 +1,1578 @@
+/*
+ * Copyright 2010      INRIA Saclay
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
+ * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
+ * 91893 Orsay, France 
+ */
+
+#include <isl_map_private.h>
+#include <isl_aff_private.h>
+#include <isl/set.h>
+#include <isl_seq.h>
+#include <isl_tab.h>
+#include <isl_space_private.h>
+#include <isl_morph.h>
+#include <isl_vertices_private.h>
+#include <isl_mat_private.h>
+#include <isl_vec_private.h>
+
+#define SELECTED	1
+#define DESELECTED	-1
+#define UNSELECTED	0
+
+static __isl_give isl_vertices *compute_chambers(__isl_take isl_basic_set *bset,
+	__isl_take isl_vertices *vertices);
+
+__isl_give isl_vertices *isl_vertices_copy(__isl_keep isl_vertices *vertices)
+{
+	if (!vertices)
+		return NULL;
+
+	vertices->ref++;
+	return vertices;
+}
+
+void isl_vertices_free(__isl_take isl_vertices *vertices)
+{
+	int i;
+
+	if (!vertices)
+		return;
+
+	if (--vertices->ref > 0)
+		return;
+
+	for (i = 0; i < vertices->n_vertices; ++i) {
+		isl_basic_set_free(vertices->v[i].vertex);
+		isl_basic_set_free(vertices->v[i].dom);
+	}
+	free(vertices->v);
+
+	for (i = 0; i < vertices->n_chambers; ++i) {
+		free(vertices->c[i].vertices);
+		isl_basic_set_free(vertices->c[i].dom);
+	}
+	free(vertices->c);
+
+	isl_basic_set_free(vertices->bset);
+	free(vertices);
+}
+
+struct isl_vertex_list {
+	struct isl_vertex v;
+	struct isl_vertex_list *next;
+};
+
+static void free_vertex_list(struct isl_vertex_list *list)
+{
+	struct isl_vertex_list *next;
+
+	for (; list; list = next) {
+		next = list->next;
+		isl_basic_set_free(list->v.vertex);
+		isl_basic_set_free(list->v.dom);
+		free(list);
+	}
+}
+
+static __isl_give isl_vertices *vertices_from_list(__isl_keep isl_basic_set *bset,
+	int n_vertices, struct isl_vertex_list *list)
+{
+	int i;
+	struct isl_vertex_list *next;
+	isl_vertices *vertices;
+
+	vertices = isl_calloc_type(bset->ctx, isl_vertices);
+	if (!vertices)
+		goto error;
+	vertices->ref = 1;
+	vertices->bset = isl_basic_set_copy(bset);
+	vertices->v = isl_alloc_array(bset->ctx, struct isl_vertex, n_vertices);
+	if (n_vertices && !vertices->v)
+		goto error;
+	vertices->n_vertices = n_vertices;
+
+	for (i = 0; list; list = next, i++) {
+		next = list->next;
+		vertices->v[i] = list->v;
+		free(list);
+	}
+
+	return vertices;
+error:
+	isl_vertices_free(vertices);
+	free_vertex_list(list);
+	return NULL;
+}
+
+/* Prepend a vertex to the linked list "list" based on the equalities in "tab".
+ */
+static int add_vertex(struct isl_vertex_list **list,
+	__isl_keep isl_basic_set *bset, struct isl_tab *tab)
+{
+	unsigned nvar;
+	unsigned nparam;
+	struct isl_vertex_list *v = NULL;
+
+	if (isl_tab_detect_implicit_equalities(tab) < 0)
+		return -1;
+
+	nvar = isl_basic_set_dim(bset, isl_dim_set);
+	nparam = isl_basic_set_dim(bset, isl_dim_param);
+
+	v = isl_calloc_type(tab->mat->ctx, struct isl_vertex_list);
+	if (!v)
+		goto error;
+
+	v->v.vertex = isl_basic_set_copy(bset);
+	v->v.vertex = isl_basic_set_cow(v->v.vertex);
+	v->v.vertex = isl_basic_set_update_from_tab(v->v.vertex, tab);
+	v->v.vertex = isl_basic_set_simplify(v->v.vertex);
+	v->v.vertex = isl_basic_set_finalize(v->v.vertex);
+	if (!v->v.vertex)
+		goto error;
+	isl_assert(bset->ctx, v->v.vertex->n_eq >= nvar, goto error);
+	v->v.dom = isl_basic_set_copy(v->v.vertex);
+	v->v.dom = isl_basic_set_params(v->v.dom);
+	if (!v->v.dom)
+		goto error;
+
+	v->next = *list;
+	*list = v;
+
+	return 0;
+error:
+	free_vertex_list(v);
+	return -1;
+}
+
+/* Compute the parametric vertices and the chamber decomposition
+ * of an empty parametric polytope.
+ */
+static __isl_give isl_vertices *vertices_empty(__isl_keep isl_basic_set *bset)
+{
+	isl_vertices *vertices;
+	unsigned nparam;
+
+	if (!bset)
+		return NULL;
+
+	nparam = isl_basic_set_dim(bset, isl_dim_param);
+
+	vertices = isl_calloc_type(bset->ctx, isl_vertices);
+	if (!vertices)
+		return NULL;
+	vertices->bset = isl_basic_set_copy(bset);
+	vertices->ref = 1;
+
+	vertices->n_vertices = 0;
+	vertices->n_chambers = 0;
+
+	return vertices;
+}
+
+/* Compute the parametric vertices and the chamber decomposition
+ * of the parametric polytope defined using the same constraints
+ * as "bset" in the 0D case.
+ * There is exactly one 0D vertex and a single chamber containing
+ * the vertex.
+ */
+static __isl_give isl_vertices *vertices_0D(__isl_keep isl_basic_set *bset)
+{
+	isl_vertices *vertices;
+	unsigned nparam;
+
+	if (!bset)
+		return NULL;
+
+	nparam = isl_basic_set_dim(bset, isl_dim_param);
+
+	vertices = isl_calloc_type(bset->ctx, isl_vertices);
+	if (!vertices)
+		return NULL;
+	vertices->ref = 1;
+	vertices->bset = isl_basic_set_copy(bset);
+
+	vertices->v = isl_calloc_array(bset->ctx, struct isl_vertex, 1);
+	if (!vertices->v)
+		goto error;
+	vertices->n_vertices = 1;
+	vertices->v[0].vertex = isl_basic_set_copy(bset);
+	vertices->v[0].dom = isl_basic_set_params(isl_basic_set_copy(bset));
+	if (!vertices->v[0].vertex || !vertices->v[0].dom)
+		goto error;
+
+	vertices->c = isl_calloc_array(bset->ctx, struct isl_chamber, 1);
+	if (!vertices->c)
+		goto error;
+	vertices->n_chambers = 1;
+	vertices->c[0].n_vertices = 1;
+	vertices->c[0].vertices = isl_calloc_array(bset->ctx, int, 1);
+	if (!vertices->c[0].vertices)
+		goto error;
+	vertices->c[0].dom = isl_basic_set_copy(vertices->v[0].dom);
+	if (!vertices->c[0].dom)
+		goto error;
+
+	return vertices;
+error:
+	isl_vertices_free(vertices);
+	return NULL;
+}
+
+static int isl_mat_rank(__isl_keep isl_mat *mat)
+{
+	int row, col;
+	isl_mat *H;
+
+	H = isl_mat_left_hermite(isl_mat_copy(mat), 0, NULL, NULL);
+	if (!H)
+		return -1;
+
+	for (col = 0; col < H->n_col; ++col) {
+		for (row = 0; row < H->n_row; ++row)
+			if (!isl_int_is_zero(H->row[row][col]))
+				break;
+		if (row == H->n_row)
+			break;
+	}
+
+	isl_mat_free(H);
+
+	return col;
+}
+
+/* Is the row pointed to by "f" linearly independent of the "n" first
+ * rows in "facets"?
+ */
+static int is_independent(__isl_keep isl_mat *facets, int n, isl_int *f)
+{
+	int rank;
+
+	if (isl_seq_first_non_zero(f, facets->n_col) < 0)
+		return 0;
+
+	isl_seq_cpy(facets->row[n], f, facets->n_col);
+	facets->n_row = n + 1;
+	rank = isl_mat_rank(facets);
+	if (rank < 0)
+		return -1;
+
+	return rank == n + 1;
+}
+
+/* Check whether we can select constraint "level", given the current selection
+ * reflected by facets in "tab", the rows of "facets" and the earlier
+ * "selected" elements of "selection".
+ *
+ * If the constraint is (strictly) redundant in the tableau, selecting it would
+ * result in an empty tableau, so it can't be selected.
+ * If the set variable part of the constraint is not linearly indepedent
+ * of the set variable parts of the already selected constraints,
+ * the constraint cannot be selected.
+ * If selecting the constraint results in an empty tableau, the constraint
+ * cannot be selected.
+ * Finally, if selecting the constraint results in some explicitly
+ * deselected constraints turning into equalities, then the corresponding
+ * vertices have already been generated, so the constraint cannot be selected.
+ */
+static int can_select(__isl_keep isl_basic_set *bset, int level,
+	struct isl_tab *tab, __isl_keep isl_mat *facets, int selected,
+	int *selection)
+{
+	int i;
+	int indep;
+	unsigned ovar;
+	struct isl_tab_undo *snap;
+
+	if (isl_tab_is_redundant(tab, level))
+		return 0;
+
+	ovar = isl_space_offset(bset->dim, isl_dim_set);
+
+	indep = is_independent(facets, selected, bset->ineq[level] + 1 + ovar);
+	if (indep < 0)
+		return -1;
+	if (!indep)
+		return 0;
+
+	snap = isl_tab_snap(tab);
+	if (isl_tab_select_facet(tab, level) < 0)
+		return -1;
+
+	if (tab->empty) {
+		if (isl_tab_rollback(tab, snap) < 0)
+			return -1;
+		return 0;
+	}
+
+	for (i = 0; i < level; ++i) {
+		int sgn;
+
+		if (selection[i] != DESELECTED)
+			continue;
+
+		if (isl_tab_is_equality(tab, i))
+			sgn = 0;
+		else if (isl_tab_is_redundant(tab, i))
+			sgn = 1;
+		else
+			sgn = isl_tab_sign_of_max(tab, i);
+		if (sgn < -1)
+			return -1;
+		if (sgn <= 0) {
+			if (isl_tab_rollback(tab, snap) < 0)
+				return -1;
+			return 0;
+		}
+	}
+
+	return 1;
+}
+
+/* Compute the parametric vertices and the chamber decomposition
+ * of a parametric polytope that is not full-dimensional.
+ *
+ * Simply map the parametric polytope to a lower dimensional space
+ * and map the resulting vertices back.
+ */
+static __isl_give isl_vertices *lower_dim_vertices(
+	__isl_keep isl_basic_set *bset)
+{
+	isl_morph *morph;
+	isl_vertices *vertices;
+
+	bset = isl_basic_set_copy(bset);
+	morph = isl_basic_set_full_compression(bset);
+	bset = isl_morph_basic_set(isl_morph_copy(morph), bset);
+
+	vertices = isl_basic_set_compute_vertices(bset);
+	isl_basic_set_free(bset);
+
+	morph = isl_morph_inverse(morph);
+
+	vertices = isl_morph_vertices(morph, vertices);
+
+	return vertices;
+}
+
+/* Compute the parametric vertices and the chamber decomposition
+ * of the parametric polytope defined using the same constraints
+ * as "bset".  "bset" is assumed to have no existentially quantified
+ * variables.
+ *
+ * The vertices themselves are computed in a fairly simplistic way.
+ * We simply run through all combinations of d constraints,
+ * with d the number of set variables, and check if those d constraints
+ * define a vertex.  To avoid the generation of duplicate vertices,
+ * which we may happen if a vertex is defined by more that d constraints,
+ * we make sure we only generate the vertex for the d constraints with
+ * smallest index.
+ *
+ * We set up a tableau and keep track of which facets have been
+ * selected.  The tableau is marked strict_redundant so that we can be
+ * sure that any constraint that is marked redundant (and that is not
+ * also marked zero) is not an equality.
+ * If a constraint is marked DESELECTED, it means the constraint was
+ * SELECTED before (in combination with the same selection of earlier
+ * constraints).  If such a deselected constraint turns out to be an
+ * equality, then any vertex that may still be found with the current
+ * selection has already been generated when the constraint was selected.
+ * A constraint is marked UNSELECTED when there is no way selecting
+ * the constraint could lead to a vertex (in combination with the current
+ * selection of earlier constraints).
+ *
+ * The set variable coefficients of the selected constraints are stored
+ * in the facets matrix.
+ */
+__isl_give isl_vertices *isl_basic_set_compute_vertices(
+	__isl_keep isl_basic_set *bset)
+{
+	struct isl_tab *tab;
+	int level;
+	int init;
+	unsigned nvar;
+	int *selection = NULL;
+	int selected;
+	struct isl_tab_undo **snap = NULL;
+	isl_mat *facets = NULL;
+	struct isl_vertex_list *list = NULL;
+	int n_vertices = 0;
+	isl_vertices *vertices;
+
+	if (!bset)
+		return NULL;
+
+	if (isl_basic_set_plain_is_empty(bset))
+		return vertices_empty(bset);
+
+	if (bset->n_eq != 0)
+		return lower_dim_vertices(bset);
+
+	isl_assert(bset->ctx, isl_basic_set_dim(bset, isl_dim_div) == 0,
+		return NULL);
+
+	if (isl_basic_set_dim(bset, isl_dim_set) == 0)
+		return vertices_0D(bset);
+
+	nvar = isl_basic_set_dim(bset, isl_dim_set);
+
+	bset = isl_basic_set_copy(bset);
+	bset = isl_basic_set_set_rational(bset);
+	if (!bset)
+		return NULL;
+
+	tab = isl_tab_from_basic_set(bset, 0);
+	if (!tab)
+		goto error;
+	tab->strict_redundant = 1;
+
+	if (tab->empty)	{
+		vertices = vertices_empty(bset);
+		isl_basic_set_free(bset);
+		isl_tab_free(tab);
+		return vertices;
+	}
+
+	selection = isl_alloc_array(bset->ctx, int, bset->n_ineq);
+	snap = isl_alloc_array(bset->ctx, struct isl_tab_undo *, bset->n_ineq);
+	facets = isl_mat_alloc(bset->ctx, nvar, nvar);
+	if ((bset->n_ineq && (!selection || !snap)) || !facets)
+		goto error;
+
+	level = 0;
+	init = 1;
+	selected = 0;
+
+	while (level >= 0) {
+		if (level >= bset->n_ineq ||
+		    (!init && selection[level] != SELECTED)) {
+			--level;
+			init = 0;
+			continue;
+		}
+		if (init) {
+			int ok;
+			snap[level] = isl_tab_snap(tab);
+			ok = can_select(bset, level, tab, facets, selected,
+					selection);
+			if (ok < 0)
+				goto error;
+			if (ok) {
+				selection[level] = SELECTED;
+				selected++;
+			} else
+				selection[level] = UNSELECTED;
+		} else {
+			selection[level] = DESELECTED;
+			selected--;
+			if (isl_tab_rollback(tab, snap[level]) < 0)
+				goto error;
+		}
+		if (selected == nvar) {
+			if (tab->n_dead == nvar) {
+				if (add_vertex(&list, bset, tab) < 0)
+					goto error;
+				n_vertices++;
+			}
+			init = 0;
+			continue;
+		}
+		++level;
+		init = 1;
+	}
+
+	isl_mat_free(facets);
+	free(selection);
+	free(snap);
+
+	isl_tab_free(tab);
+
+	vertices = vertices_from_list(bset, n_vertices, list);
+
+	vertices = compute_chambers(bset, vertices);
+
+	return vertices;
+error:
+	free_vertex_list(list);
+	isl_mat_free(facets);
+	free(selection);
+	free(snap);
+	isl_tab_free(tab);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+struct isl_chamber_list {
+	struct isl_chamber c;
+	struct isl_chamber_list *next;
+};
+
+static void free_chamber_list(struct isl_chamber_list *list)
+{
+	struct isl_chamber_list *next;
+
+	for (; list; list = next) {
+		next = list->next;
+		isl_basic_set_free(list->c.dom);
+		free(list->c.vertices);
+		free(list);
+	}
+}
+
+/* Check whether the basic set "bset" is a superset of the basic set described
+ * by "tab", i.e., check whether all constraints of "bset" are redundant.
+ */
+static int bset_covers_tab(__isl_keep isl_basic_set *bset, struct isl_tab *tab)
+{
+	int i;
+
+	if (!bset || !tab)
+		return -1;
+
+	for (i = 0; i < bset->n_ineq; ++i) {
+		enum isl_ineq_type type = isl_tab_ineq_type(tab, bset->ineq[i]);
+		switch (type) {
+		case isl_ineq_error:		return -1;
+		case isl_ineq_redundant:	continue;
+		default:			return 0;
+		}
+	}
+
+	return 1;
+}
+
+static __isl_give isl_vertices *vertices_add_chambers(
+	__isl_take isl_vertices *vertices, int n_chambers,
+	struct isl_chamber_list *list)
+{
+	int i;
+	isl_ctx *ctx;
+	struct isl_chamber_list *next;
+
+	ctx = isl_vertices_get_ctx(vertices);
+	vertices->c = isl_alloc_array(ctx, struct isl_chamber, n_chambers);
+	if (!vertices->c)
+		goto error;
+	vertices->n_chambers = n_chambers;
+
+	for (i = 0; list; list = next, i++) {
+		next = list->next;
+		vertices->c[i] = list->c;
+		free(list);
+	}
+
+	return vertices;
+error:
+	isl_vertices_free(vertices);
+	free_chamber_list(list);
+	return NULL;
+}
+
+/* Can "tab" be intersected with "bset" without resulting in
+ * a lower-dimensional set.
+ */
+static int can_intersect(struct isl_tab *tab, __isl_keep isl_basic_set *bset)
+{
+	int i;
+	struct isl_tab_undo *snap;
+
+	if (isl_tab_extend_cons(tab, bset->n_ineq) < 0)
+		return -1;
+
+	snap = isl_tab_snap(tab);
+
+	for (i = 0; i < bset->n_ineq; ++i) {
+		if (isl_tab_ineq_type(tab, bset->ineq[i]) == isl_ineq_redundant)
+			continue;
+		if (isl_tab_add_ineq(tab, bset->ineq[i]) < 0)
+			return -1;
+	}
+
+	if (isl_tab_detect_implicit_equalities(tab) < 0)
+		return -1;
+	if (tab->n_dead) {
+		if (isl_tab_rollback(tab, snap) < 0)
+			return -1;
+		return 0;
+	}
+
+	return 1;
+}
+
+static int add_chamber(struct isl_chamber_list **list,
+	__isl_keep isl_vertices *vertices, struct isl_tab *tab, int *selection)
+{
+	int n_frozen;
+	int i, j;
+	int n_vertices = 0;
+	struct isl_tab_undo *snap;
+	struct isl_chamber_list *c = NULL;
+
+	for (i = 0; i < vertices->n_vertices; ++i)
+		if (selection[i])
+			n_vertices++;
+
+	snap = isl_tab_snap(tab);
+
+	for (i = 0; i < tab->n_con && tab->con[i].frozen; ++i)
+		tab->con[i].frozen = 0;
+	n_frozen = i;
+
+	if (isl_tab_detect_redundant(tab) < 0)
+		return -1;
+
+	c = isl_calloc_type(tab->mat->ctx, struct isl_chamber_list);
+	if (!c)
+		goto error;
+	c->c.vertices = isl_alloc_array(tab->mat->ctx, int, n_vertices);
+	if (n_vertices && !c->c.vertices)
+		goto error;
+	c->c.dom = isl_basic_set_from_basic_map(isl_basic_map_copy(tab->bmap));
+	c->c.dom = isl_basic_set_set_rational(c->c.dom);
+	c->c.dom = isl_basic_set_cow(c->c.dom);
+	c->c.dom = isl_basic_set_update_from_tab(c->c.dom, tab);
+	c->c.dom = isl_basic_set_simplify(c->c.dom);
+	c->c.dom = isl_basic_set_finalize(c->c.dom);
+	if (!c->c.dom)
+		goto error;
+
+	c->c.n_vertices = n_vertices;
+
+	for (i = 0, j = 0; i < vertices->n_vertices; ++i)
+		if (selection[i]) {
+			c->c.vertices[j] = i;
+			j++;
+		}
+
+	c->next = *list;
+	*list = c;
+
+	for (i = 0; i < n_frozen; ++i)
+		tab->con[i].frozen = 1;
+
+	if (isl_tab_rollback(tab, snap) < 0)
+		return -1;
+
+	return 0;
+error:
+	free_chamber_list(c);
+	return -1;
+}
+
+struct isl_facet_todo {
+	struct isl_tab *tab;	/* A tableau representation of the facet */
+	isl_basic_set *bset;    /* A normalized basic set representation */
+	isl_vec *constraint;	/* Constraint pointing to the other side */
+	struct isl_facet_todo *next;
+};
+
+static void free_todo(struct isl_facet_todo *todo)
+{
+	while (todo) {
+		struct isl_facet_todo *next = todo->next;
+
+		isl_tab_free(todo->tab);
+		isl_basic_set_free(todo->bset);
+		isl_vec_free(todo->constraint);
+		free(todo);
+
+		todo = next;
+	}
+}
+
+static struct isl_facet_todo *create_todo(struct isl_tab *tab, int con)
+{
+	int i;
+	int n_frozen;
+	struct isl_tab_undo *snap;
+	struct isl_facet_todo *todo;
+
+	snap = isl_tab_snap(tab);
+
+	for (i = 0; i < tab->n_con && tab->con[i].frozen; ++i)
+		tab->con[i].frozen = 0;
+	n_frozen = i;
+
+	if (isl_tab_detect_redundant(tab) < 0)
+		return NULL;
+
+	todo = isl_calloc_type(tab->mat->ctx, struct isl_facet_todo);
+	if (!todo)
+		return NULL;
+
+	todo->constraint = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_var);
+	if (!todo->constraint)
+		goto error;
+	isl_seq_neg(todo->constraint->el, tab->bmap->ineq[con], 1 + tab->n_var);
+	todo->bset = isl_basic_set_from_basic_map(isl_basic_map_copy(tab->bmap));
+	todo->bset = isl_basic_set_set_rational(todo->bset);
+	todo->bset = isl_basic_set_cow(todo->bset);
+	todo->bset = isl_basic_set_update_from_tab(todo->bset, tab);
+	todo->bset = isl_basic_set_simplify(todo->bset);
+	todo->bset = isl_basic_set_sort_constraints(todo->bset);
+	if (!todo->bset)
+		goto error;
+	ISL_F_SET(todo->bset, ISL_BASIC_SET_NORMALIZED);
+	todo->tab = isl_tab_dup(tab);
+	if (!todo->tab)
+		goto error;
+
+	for (i = 0; i < n_frozen; ++i)
+		tab->con[i].frozen = 1;
+
+	if (isl_tab_rollback(tab, snap) < 0)
+		goto error;
+
+	return todo;
+error:
+	free_todo(todo);
+	return NULL;
+}
+
+/* Create todo items for all interior facets of the chamber represented
+ * by "tab" and collect them in "next".
+ */
+static int init_todo(struct isl_facet_todo **next, struct isl_tab *tab)
+{
+	int i;
+	struct isl_tab_undo *snap;
+	struct isl_facet_todo *todo;
+
+	snap = isl_tab_snap(tab);
+
+	for (i = 0; i < tab->n_con; ++i) {
+		if (tab->con[i].frozen)
+			continue;
+		if (tab->con[i].is_redundant)
+			continue;
+
+		if (isl_tab_select_facet(tab, i) < 0)
+			return -1;
+
+		todo = create_todo(tab, i);
+		if (!todo)
+			return -1;
+
+		todo->next = *next;
+		*next = todo;
+
+		if (isl_tab_rollback(tab, snap) < 0)
+			return -1;
+	}
+
+	return 0;
+}
+
+/* Does the linked list contain a todo item that is the opposite of "todo".
+ * If so, return 1 and remove the opposite todo item.
+ */
+static int has_opposite(struct isl_facet_todo *todo,
+	struct isl_facet_todo **list)
+{
+	for (; *list; list = &(*list)->next) {
+		int eq;
+		eq = isl_basic_set_plain_is_equal(todo->bset, (*list)->bset);
+		if (eq < 0)
+			return -1;
+		if (!eq)
+			continue;
+		todo = *list;
+		*list = todo->next;
+		todo->next = NULL;
+		free_todo(todo);
+		return 1;
+	}
+
+	return 0;
+}
+
+/* Create todo items for all interior facets of the chamber represented
+ * by "tab" and collect them in first->next, taking care to cancel
+ * opposite todo items.
+ */
+static int update_todo(struct isl_facet_todo *first, struct isl_tab *tab)
+{
+	int i;
+	struct isl_tab_undo *snap;
+	struct isl_facet_todo *todo;
+
+	snap = isl_tab_snap(tab);
+
+	for (i = 0; i < tab->n_con; ++i) {
+		int drop;
+
+		if (tab->con[i].frozen)
+			continue;
+		if (tab->con[i].is_redundant)
+			continue;
+
+		if (isl_tab_select_facet(tab, i) < 0)
+			return -1;
+
+		todo = create_todo(tab, i);
+		if (!todo)
+			return -1;
+
+		drop = has_opposite(todo, &first->next);
+		if (drop < 0)
+			return -1;
+
+		if (drop)
+			free_todo(todo);
+		else {
+			todo->next = first->next;
+			first->next = todo;
+		}
+
+		if (isl_tab_rollback(tab, snap) < 0)
+			return -1;
+	}
+
+	return 0;
+}
+
+/* Compute the chamber decomposition of the parametric polytope respresented
+ * by "bset" given the parametric vertices and their activity domains.
+ *
+ * We are only interested in full-dimensional chambers.
+ * Each of these chambers is the intersection of the activity domains of
+ * one or more vertices and the union of all chambers is equal to the
+ * projection of the entire parametric polytope onto the parameter space.
+ *
+ * We first create an initial chamber by intersecting as many activity
+ * domains as possible without ending up with an empty or lower-dimensional
+ * set.  As a minor optimization, we only consider those activity domains
+ * that contain some arbitrary point.
+ *
+ * For each of interior facets of the chamber, we construct a todo item,
+ * containing the facet and a constraint containing the other side of the facet,
+ * for constructing the chamber on the other side.
+ * While their are any todo items left, we pick a todo item and
+ * create the required chamber by intersecting all activity domains
+ * that contain the facet and have a full-dimensional intersection with
+ * the other side of the facet.  For each of the interior facets, we
+ * again create todo items, taking care to cancel opposite todo items.
+ */
+static __isl_give isl_vertices *compute_chambers(__isl_take isl_basic_set *bset,
+	__isl_take isl_vertices *vertices)
+{
+	int i;
+	isl_ctx *ctx;
+	isl_vec *sample = NULL;
+	struct isl_tab *tab = NULL;
+	struct isl_tab_undo *snap;
+	int *selection = NULL;
+	int n_chambers = 0;
+	struct isl_chamber_list *list = NULL;
+	struct isl_facet_todo *todo = NULL;
+
+	if (!bset || !vertices)
+		goto error;
+
+	ctx = isl_vertices_get_ctx(vertices);
+	selection = isl_alloc_array(ctx, int, vertices->n_vertices);
+	if (vertices->n_vertices && !selection)
+		goto error;
+
+	bset = isl_basic_set_params(bset);
+
+	tab = isl_tab_from_basic_set(bset, 1);
+	if (!tab)
+		goto error;
+	for (i = 0; i < bset->n_ineq; ++i)
+		if (isl_tab_freeze_constraint(tab, i) < 0)
+			goto error;
+	isl_basic_set_free(bset);
+
+	snap = isl_tab_snap(tab);
+
+	sample = isl_tab_get_sample_value(tab);
+
+	for (i = 0; i < vertices->n_vertices; ++i) {
+		selection[i] = isl_basic_set_contains(vertices->v[i].dom, sample);
+		if (selection[i] < 0)
+			goto error;
+		if (!selection[i])
+			continue;
+		selection[i] = can_intersect(tab, vertices->v[i].dom);
+		if (selection[i] < 0)
+			goto error;
+	}
+
+	if (isl_tab_detect_redundant(tab) < 0)
+		goto error;
+
+	if (add_chamber(&list, vertices, tab, selection) < 0)
+		goto error;
+	n_chambers++;
+
+	if (init_todo(&todo, tab) < 0)
+		goto error;
+
+	while (todo) {
+		struct isl_facet_todo *next;
+
+		if (isl_tab_rollback(tab, snap) < 0)
+			goto error;
+
+		if (isl_tab_add_ineq(tab, todo->constraint->el) < 0)
+			goto error;
+		if (isl_tab_freeze_constraint(tab, tab->n_con - 1) < 0)
+			goto error;
+
+		for (i = 0; i < vertices->n_vertices; ++i) {
+			selection[i] = bset_covers_tab(vertices->v[i].dom,
+							todo->tab);
+			if (selection[i] < 0)
+				goto error;
+			if (!selection[i])
+				continue;
+			selection[i] = can_intersect(tab, vertices->v[i].dom);
+			if (selection[i] < 0)
+				goto error;
+		}
+
+		if (isl_tab_detect_redundant(tab) < 0)
+			goto error;
+
+		if (add_chamber(&list, vertices, tab, selection) < 0)
+			goto error;
+		n_chambers++;
+
+		if (update_todo(todo, tab) < 0)
+			goto error;
+
+		next = todo->next;
+		todo->next = NULL;
+		free_todo(todo);
+		todo = next;
+	}
+
+	isl_vec_free(sample);
+
+	isl_tab_free(tab);
+	free(selection);
+
+	vertices = vertices_add_chambers(vertices, n_chambers, list);
+
+	for (i = 0; vertices && i < vertices->n_vertices; ++i) {
+		isl_basic_set_free(vertices->v[i].dom);
+		vertices->v[i].dom = NULL;
+	}
+
+	return vertices;
+error:
+	free_chamber_list(list);
+	free_todo(todo);
+	isl_vec_free(sample);
+	isl_tab_free(tab);
+	free(selection);
+	if (!tab)
+		isl_basic_set_free(bset);
+	isl_vertices_free(vertices);
+	return NULL;
+}
+
+isl_ctx *isl_vertex_get_ctx(__isl_keep isl_vertex *vertex)
+{
+	return vertex ? isl_vertices_get_ctx(vertex->vertices) : NULL;
+}
+
+int isl_vertex_get_id(__isl_keep isl_vertex *vertex)
+{
+	return vertex ? vertex->id : -1;
+}
+
+__isl_give isl_basic_set *isl_basic_set_set_integral(__isl_take isl_basic_set *bset)
+{
+	if (!bset)
+		return NULL;
+
+	if (!ISL_F_ISSET(bset, ISL_BASIC_MAP_RATIONAL))
+		return bset;
+
+	bset = isl_basic_set_cow(bset);
+	if (!bset)
+		return NULL;
+
+	ISL_F_CLR(bset, ISL_BASIC_MAP_RATIONAL);
+
+	return isl_basic_set_finalize(bset);
+}
+
+/* Return the activity domain of the vertex "vertex".
+ */
+__isl_give isl_basic_set *isl_vertex_get_domain(__isl_keep isl_vertex *vertex)
+{
+	struct isl_vertex *v;
+
+	if (!vertex)
+		return NULL;
+
+	v = &vertex->vertices->v[vertex->id];
+	if (!v->dom) {
+		v->dom = isl_basic_set_copy(v->vertex);
+		v->dom = isl_basic_set_params(v->dom);
+		v->dom = isl_basic_set_set_integral(v->dom);
+	}
+
+	return isl_basic_set_copy(v->dom);
+}
+
+/* Return a multiple quasi-affine expression describing the vertex "vertex"
+ * in terms of the parameters,
+ */
+__isl_give isl_multi_aff *isl_vertex_get_expr(__isl_keep isl_vertex *vertex)
+{
+	struct isl_vertex *v;
+	isl_basic_set *bset;
+
+	if (!vertex)
+		return NULL;
+
+	v = &vertex->vertices->v[vertex->id];
+
+	bset = isl_basic_set_copy(v->vertex);
+	return isl_multi_aff_from_basic_set_equalities(bset);
+}
+
+static __isl_give isl_vertex *isl_vertex_alloc(__isl_take isl_vertices *vertices,
+	int id)
+{
+	isl_ctx *ctx;
+	isl_vertex *vertex;
+
+	if (!vertices)
+		return NULL;
+
+	ctx = isl_vertices_get_ctx(vertices);
+	vertex = isl_alloc_type(ctx, isl_vertex);
+	if (!vertex)
+		goto error;
+
+	vertex->vertices = vertices;
+	vertex->id = id;
+
+	return vertex;
+error:
+	isl_vertices_free(vertices);
+	return NULL;
+}
+
+void isl_vertex_free(__isl_take isl_vertex *vertex)
+{
+	if (!vertex)
+		return;
+	isl_vertices_free(vertex->vertices);
+	free(vertex);
+}
+
+isl_ctx *isl_cell_get_ctx(__isl_keep isl_cell *cell)
+{
+	return cell ? cell->dom->ctx : NULL;
+}
+
+__isl_give isl_basic_set *isl_cell_get_domain(__isl_keep isl_cell *cell)
+{
+	return cell ? isl_basic_set_copy(cell->dom) : NULL;
+}
+
+static __isl_give isl_cell *isl_cell_alloc(__isl_take isl_vertices *vertices,
+	__isl_take isl_basic_set *dom, int id)
+{
+	int i;
+	isl_cell *cell = NULL;
+
+	if (!vertices || !dom)
+		goto error;
+
+	cell = isl_calloc_type(dom->ctx, isl_cell);
+	if (!cell)
+		goto error;
+
+	cell->n_vertices = vertices->c[id].n_vertices;
+	cell->ids = isl_alloc_array(dom->ctx, int, cell->n_vertices);
+	if (cell->n_vertices && !cell->ids)
+		goto error;
+	for (i = 0; i < cell->n_vertices; ++i)
+		cell->ids[i] = vertices->c[id].vertices[i];
+	cell->vertices = vertices;
+	cell->dom = dom;
+
+	return cell;
+error:
+	isl_cell_free(cell);
+	isl_vertices_free(vertices);
+	isl_basic_set_free(dom);
+	return NULL;
+}
+
+void isl_cell_free(__isl_take isl_cell *cell)
+{
+	if (!cell)
+		return;
+
+	isl_vertices_free(cell->vertices);
+	free(cell->ids);
+	isl_basic_set_free(cell->dom);
+	free(cell);
+}
+
+/* Create a tableau of the cone obtained by first homogenizing the given
+ * polytope and then making all inequalities strict by setting the
+ * constant term to -1.
+ */
+static struct isl_tab *tab_for_shifted_cone(__isl_keep isl_basic_set *bset)
+{
+	int i;
+	isl_vec *c = NULL;
+	struct isl_tab *tab;
+
+	if (!bset)
+		return NULL;
+	tab = isl_tab_alloc(bset->ctx, bset->n_ineq + 1,
+			    1 + isl_basic_set_total_dim(bset), 0);
+	if (!tab)
+		return NULL;
+	tab->rational = ISL_F_ISSET(bset, ISL_BASIC_SET_RATIONAL);
+	if (ISL_F_ISSET(bset, ISL_BASIC_MAP_EMPTY)) {
+		if (isl_tab_mark_empty(tab) < 0)
+			goto error;
+		return tab;
+	}
+
+	c = isl_vec_alloc(bset->ctx, 1 + 1 + isl_basic_set_total_dim(bset));
+	if (!c)
+		goto error;
+
+	isl_int_set_si(c->el[0], 0);
+	for (i = 0; i < bset->n_eq; ++i) {
+		isl_seq_cpy(c->el + 1, bset->eq[i], c->size - 1);
+		if (isl_tab_add_eq(tab, c->el) < 0)
+			goto error;
+	}
+
+	isl_int_set_si(c->el[0], -1);
+	for (i = 0; i < bset->n_ineq; ++i) {
+		isl_seq_cpy(c->el + 1, bset->ineq[i], c->size - 1);
+		if (isl_tab_add_ineq(tab, c->el) < 0)
+			goto error;
+		if (tab->empty) {
+			isl_vec_free(c);
+			return tab;
+		}
+	}
+
+	isl_seq_clr(c->el + 1, c->size - 1);
+	isl_int_set_si(c->el[1], 1);
+	if (isl_tab_add_ineq(tab, c->el) < 0)
+		goto error;
+
+	isl_vec_free(c);
+	return tab;
+error:
+	isl_vec_free(c);
+	isl_tab_free(tab);
+	return NULL;
+}
+
+/* Compute an interior point of "bset" by selecting an interior
+ * point in homogeneous space and projecting the point back down.
+ */
+static __isl_give isl_vec *isl_basic_set_interior_point(
+	__isl_keep isl_basic_set *bset)
+{
+	isl_vec *vec;
+	struct isl_tab *tab;
+
+	tab = tab_for_shifted_cone(bset);
+	vec = isl_tab_get_sample_value(tab);
+	isl_tab_free(tab);
+	if (!vec)
+		return NULL;
+
+	isl_seq_cpy(vec->el, vec->el + 1, vec->size - 1);
+	vec->size--;
+
+	return vec;
+}
+
+/* Call "fn" on all chambers of the parametric polytope with the shared
+ * facets of neighboring chambers only appearing in one of the chambers.
+ *
+ * We pick an interior point from one of the chambers and then make
+ * all constraints that do not satisfy this point strict.
+ */
+int isl_vertices_foreach_disjoint_cell(__isl_keep isl_vertices *vertices,
+	int (*fn)(__isl_take isl_cell *cell, void *user), void *user)
+{
+	int i, j;
+	isl_vec *vec;
+	isl_int v;
+	isl_cell *cell;
+
+	if (!vertices)
+		return -1;
+
+	if (vertices->n_chambers == 0)
+		return 0;
+
+	if (vertices->n_chambers == 1) {
+		isl_basic_set *dom = isl_basic_set_copy(vertices->c[0].dom);
+		dom = isl_basic_set_set_integral(dom);
+		cell = isl_cell_alloc(isl_vertices_copy(vertices), dom, 0);
+		if (!cell)
+			return -1;
+		return fn(cell, user);
+	}
+
+	vec = isl_basic_set_interior_point(vertices->c[0].dom);
+	if (!vec)
+		return -1;
+
+	isl_int_init(v);
+
+	for (i = 0; i < vertices->n_chambers; ++i) {
+		int r;
+		isl_basic_set *dom = isl_basic_set_copy(vertices->c[i].dom);
+		dom = isl_basic_set_cow(dom);
+		if (!dom)
+			goto error;
+		for (j = 0; i && j < dom->n_ineq; ++j) {
+			isl_seq_inner_product(vec->el, dom->ineq[j], vec->size,
+						&v);
+			if (!isl_int_is_neg(v))
+				continue;
+			isl_int_sub_ui(dom->ineq[j][0], dom->ineq[j][0], 1);
+		}
+		dom = isl_basic_set_set_integral(dom);
+		cell = isl_cell_alloc(isl_vertices_copy(vertices), dom, i);
+		if (!cell)
+			goto error;
+		r = fn(cell, user);
+		if (r < 0)
+			goto error;
+	}
+
+	isl_int_clear(v);
+	isl_vec_free(vec);
+
+	return 0;
+error:
+	isl_int_clear(v);
+	isl_vec_free(vec);
+	return -1;
+}
+
+int isl_vertices_foreach_cell(__isl_keep isl_vertices *vertices,
+	int (*fn)(__isl_take isl_cell *cell, void *user), void *user)
+{
+	int i;
+	isl_cell *cell;
+
+	if (!vertices)
+		return -1;
+
+	if (vertices->n_chambers == 0)
+		return 0;
+
+	for (i = 0; i < vertices->n_chambers; ++i) {
+		int r;
+		isl_basic_set *dom = isl_basic_set_copy(vertices->c[i].dom);
+
+		cell = isl_cell_alloc(isl_vertices_copy(vertices), dom, i);
+		if (!cell)
+			return -1;
+
+		r = fn(cell, user);
+		if (r < 0)
+			return -1;
+	}
+
+	return 0;
+}
+
+int isl_vertices_foreach_vertex(__isl_keep isl_vertices *vertices,
+	int (*fn)(__isl_take isl_vertex *vertex, void *user), void *user)
+{
+	int i;
+	isl_vertex *vertex;
+
+	if (!vertices)
+		return -1;
+
+	if (vertices->n_vertices == 0)
+		return 0;
+
+	for (i = 0; i < vertices->n_vertices; ++i) {
+		int r;
+
+		vertex = isl_vertex_alloc(isl_vertices_copy(vertices), i);
+		if (!vertex)
+			return -1;
+
+		r = fn(vertex, user);
+		if (r < 0)
+			return -1;
+	}
+
+	return 0;
+}
+
+int isl_cell_foreach_vertex(__isl_keep isl_cell *cell,
+	int (*fn)(__isl_take isl_vertex *vertex, void *user), void *user)
+{
+	int i;
+	isl_vertex *vertex;
+
+	if (!cell)
+		return -1;
+
+	if (cell->n_vertices == 0)
+		return 0;
+
+	for (i = 0; i < cell->n_vertices; ++i) {
+		int r;
+
+		vertex = isl_vertex_alloc(isl_vertices_copy(cell->vertices),
+					  cell->ids[i]);
+		if (!vertex)
+			return -1;
+
+		r = fn(vertex, user);
+		if (r < 0)
+			return -1;
+	}
+
+	return 0;
+}
+
+isl_ctx *isl_vertices_get_ctx(__isl_keep isl_vertices *vertices)
+{
+	return vertices ? vertices->bset->ctx : NULL;
+}
+
+int isl_vertices_get_n_vertices(__isl_keep isl_vertices *vertices)
+{
+	return vertices ? vertices->n_vertices : -1;
+}
+
+__isl_give isl_vertices *isl_morph_vertices(__isl_take isl_morph *morph,
+	__isl_take isl_vertices *vertices)
+{
+	int i;
+	isl_morph *param_morph = NULL;
+
+	if (!morph || !vertices)
+		goto error;
+
+	isl_assert(vertices->bset->ctx, vertices->ref == 1, goto error);
+
+	param_morph = isl_morph_copy(morph);
+	param_morph = isl_morph_dom_params(param_morph);
+	param_morph = isl_morph_ran_params(param_morph);
+
+	for (i = 0; i < vertices->n_vertices; ++i) {
+		vertices->v[i].dom = isl_morph_basic_set(
+			isl_morph_copy(param_morph), vertices->v[i].dom);
+		vertices->v[i].vertex = isl_morph_basic_set(
+			isl_morph_copy(morph), vertices->v[i].vertex);
+		if (!vertices->v[i].vertex)
+			goto error;
+	}
+
+	for (i = 0; i < vertices->n_chambers; ++i) {
+		vertices->c[i].dom = isl_morph_basic_set(
+			isl_morph_copy(param_morph), vertices->c[i].dom);
+		if (!vertices->c[i].dom)
+			goto error;
+	}
+
+	isl_morph_free(param_morph);
+	isl_morph_free(morph);
+	return vertices;
+error:
+	isl_morph_free(param_morph);
+	isl_morph_free(morph);
+	isl_vertices_free(vertices);
+	return NULL;
+}
+
+/* Construct a simplex isl_cell spanned by the vertices with indices in
+ * "simplex_ids" and "other_ids" and call "fn" on this isl_cell.
+ */
+static int call_on_simplex(__isl_keep isl_cell *cell,
+	int *simplex_ids, int n_simplex, int *other_ids, int n_other,
+	int (*fn)(__isl_take isl_cell *simplex, void *user), void *user)
+{
+	int i;
+	isl_ctx *ctx;
+	struct isl_cell *simplex;
+
+	ctx = isl_cell_get_ctx(cell);
+
+	simplex = isl_calloc_type(ctx, struct isl_cell);
+	if (!simplex)
+		return -1;
+	simplex->vertices = isl_vertices_copy(cell->vertices);
+	if (!simplex->vertices)
+		goto error;
+	simplex->dom = isl_basic_set_copy(cell->dom);
+	if (!simplex->dom)
+		goto error;
+	simplex->n_vertices = n_simplex + n_other;
+	simplex->ids = isl_alloc_array(ctx, int, simplex->n_vertices);
+	if (!simplex->ids)
+		goto error;
+
+	for (i = 0; i < n_simplex; ++i)
+		simplex->ids[i] = simplex_ids[i];
+	for (i = 0; i < n_other; ++i)
+		simplex->ids[n_simplex + i] = other_ids[i];
+
+	return fn(simplex, user);
+error:
+	isl_cell_free(simplex);
+	return -1;
+}
+
+/* Check whether the parametric vertex described by "vertex"
+ * lies on the facet corresponding to constraint "facet" of "bset".
+ * The isl_vec "v" is a temporary vector than can be used by this function.
+ *
+ * We eliminate the variables from the facet constraint using the
+ * equalities defining the vertex and check if the result is identical
+ * to zero.
+ *
+ * It would probably be better to keep track of the constraints defining
+ * a vertex during the vertex construction so that we could simply look
+ * it up here.
+ */
+static int vertex_on_facet(__isl_keep isl_basic_set *vertex,
+	__isl_keep isl_basic_set *bset, int facet, __isl_keep isl_vec *v)
+{
+	int i;
+	isl_int m;
+
+	isl_seq_cpy(v->el, bset->ineq[facet], v->size);
+
+	isl_int_init(m);
+	for (i = 0; i < vertex->n_eq; ++i) {
+		int k = isl_seq_last_non_zero(vertex->eq[i], v->size);
+		isl_seq_elim(v->el, vertex->eq[i], k, v->size, &m);
+	}
+	isl_int_clear(m);
+
+	return isl_seq_first_non_zero(v->el, v->size) == -1;
+}
+
+/* Triangulate the polytope spanned by the vertices with ids
+ * in "simplex_ids" and "other_ids" and call "fn" on each of
+ * the resulting simplices.
+ * If the input polytope is already a simplex, we simply call "fn".
+ * Otherwise, we pick a point from "other_ids" and add it to "simplex_ids".
+ * Then we consider each facet of "bset" that does not contain the point
+ * we just picked, but does contain some of the other points in "other_ids"
+ * and call ourselves recursively on the polytope spanned by the new
+ * "simplex_ids" and those points in "other_ids" that lie on the facet.
+ */
+static int triangulate(__isl_keep isl_cell *cell, __isl_keep isl_vec *v,
+	int *simplex_ids, int n_simplex, int *other_ids, int n_other,
+	int (*fn)(__isl_take isl_cell *simplex, void *user), void *user)
+{
+	int i, j, k;
+	int d, nparam;
+	int *ids;
+	isl_ctx *ctx;
+	isl_basic_set *vertex;
+	isl_basic_set *bset;
+
+	ctx = isl_cell_get_ctx(cell);
+	d = isl_basic_set_dim(cell->vertices->bset, isl_dim_set);
+	nparam = isl_basic_set_dim(cell->vertices->bset, isl_dim_param);
+
+	if (n_simplex + n_other == d + 1)
+		return call_on_simplex(cell, simplex_ids, n_simplex,
+				       other_ids, n_other, fn, user);
+
+	simplex_ids[n_simplex] = other_ids[0];
+	vertex = cell->vertices->v[other_ids[0]].vertex;
+	bset = cell->vertices->bset;
+
+	ids = isl_alloc_array(ctx, int, n_other - 1);
+	for (i = 0; i < bset->n_ineq; ++i) {
+		if (isl_seq_first_non_zero(bset->ineq[i] + 1 + nparam, d) == -1)
+			continue;
+		if (vertex_on_facet(vertex, bset, i, v))
+			continue;
+
+		for (j = 1, k = 0; j < n_other; ++j) {
+			isl_basic_set *ov;
+			ov = cell->vertices->v[other_ids[j]].vertex;
+			if (vertex_on_facet(ov, bset, i, v))
+				ids[k++] = other_ids[j];
+		}
+		if (k == 0)
+			continue;
+
+		if (triangulate(cell, v, simplex_ids, n_simplex + 1,
+				ids, k, fn, user) < 0)
+			goto error;
+	}
+	free(ids);
+
+	return 0;
+error:
+	free(ids);
+	return -1;
+}
+
+/* Triangulate the given cell and call "fn" on each of the resulting
+ * simplices.
+ */
+int isl_cell_foreach_simplex(__isl_take isl_cell *cell,
+	int (*fn)(__isl_take isl_cell *simplex, void *user), void *user)
+{
+	int d, total;
+	int r;
+	isl_ctx *ctx;
+	isl_vec *v = NULL;
+	int *simplex_ids = NULL;
+
+	if (!cell)
+		return -1;
+
+	d = isl_basic_set_dim(cell->vertices->bset, isl_dim_set);
+	total = isl_basic_set_total_dim(cell->vertices->bset);
+
+	if (cell->n_vertices == d + 1)
+		return fn(cell, user);
+
+	ctx = isl_cell_get_ctx(cell);
+	simplex_ids = isl_alloc_array(ctx, int, d + 1);
+	if (!simplex_ids)
+		goto error;
+
+	v = isl_vec_alloc(ctx, 1 + total);
+	if (!v)
+		goto error;
+
+	r = triangulate(cell, v, simplex_ids, 0,
+			cell->ids, cell->n_vertices, fn, user);
+
+	isl_vec_free(v);
+	free(simplex_ids);
+
+	isl_cell_free(cell);
+
+	return r;
+error:
+	free(simplex_ids);
+	isl_vec_free(v);
+	isl_cell_free(cell);
+	return -1;
+}
diff --git a/polly/lib/External/isl/isl_vertices_private.h b/polly/lib/External/isl/isl_vertices_private.h
new file mode 100644
index 00000000000..7c707fd5492
--- /dev/null
+++ b/polly/lib/External/isl/isl_vertices_private.h
@@ -0,0 +1,64 @@
+#include <isl/set.h>
+#include <isl/vertices.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+struct isl_morph;
+
+/* A parametric vertex.  "vertex" contains the actual description
+ * of the vertex as a singleton parametric set.  "dom" is the projection
+ * of "vertex" onto the parameter space, i.e., the activity domain
+ * of the vertex.
+ */
+struct isl_vertex {
+	isl_basic_set *dom;
+	isl_basic_set *vertex;
+};
+
+/* A chamber in the chamber decomposition.  The indices of the "n_vertices"
+ * active vertices are stored in "vertices".
+ */
+struct isl_chamber {
+	int n_vertices;
+	int *vertices;
+	isl_basic_set *dom;
+};
+
+struct isl_vertices {
+	int ref;
+
+	/* The rational basic set spanned by the vertices. */
+	isl_basic_set *bset;
+
+	int n_vertices;
+	struct isl_vertex *v;
+
+	int n_chambers;
+	struct isl_chamber *c;
+};
+
+struct isl_cell {
+	int n_vertices;
+	int *ids;
+	isl_vertices *vertices;
+	isl_basic_set *dom;
+};
+
+struct isl_external_vertex {
+	isl_vertices *vertices;
+	int id;
+};
+
+int isl_vertices_foreach_disjoint_cell(__isl_keep isl_vertices *vertices,
+	int (*fn)(__isl_take isl_cell *cell, void *user), void *user);
+int isl_cell_foreach_simplex(__isl_take isl_cell *cell,
+	int (*fn)(__isl_take isl_cell *simplex, void *user), void *user);
+
+__isl_give isl_vertices *isl_morph_vertices(__isl_take struct isl_morph *morph,
+	__isl_take isl_vertices *vertices);
+
+#if defined(__cplusplus)
+}
+#endif
diff --git a/polly/lib/External/isl/m4/ax_c___attribute__.m4 b/polly/lib/External/isl/m4/ax_c___attribute__.m4
new file mode 100644
index 00000000000..cf3d62bbdf4
--- /dev/null
+++ b/polly/lib/External/isl/m4/ax_c___attribute__.m4
@@ -0,0 +1,66 @@
+# ===========================================================================
+#    http://www.gnu.org/software/autoconf-archive/ax_c___attribute__.html
+# ===========================================================================
+#
+# SYNOPSIS
+#
+#   AX_C___ATTRIBUTE__
+#
+# DESCRIPTION
+#
+#   Provides a test for the compiler support of __attribute__ extensions.
+#   Defines HAVE___ATTRIBUTE__ if it is found.
+#
+# LICENSE
+#
+#   Copyright (c) 2008 Stepan Kasal <skasal@redhat.com>
+#   Copyright (c) 2008 Christian Haggstrom
+#   Copyright (c) 2008 Ryan McCabe <ryan@numb.org>
+#
+#   This program is free software; you can redistribute it and/or modify it
+#   under the terms of the GNU General Public License as published by the
+#   Free Software Foundation; either version 2 of the License, or (at your
+#   option) any later version.
+#
+#   This program is distributed in the hope that it will be useful, but
+#   WITHOUT ANY WARRANTY; without even the implied warranty of
+#   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
+#   Public License for more details.
+#
+#   You should have received a copy of the GNU General Public License along
+#   with this program. If not, see <http://www.gnu.org/licenses/>.
+#
+#   As a special exception, the respective Autoconf Macro's copyright owner
+#   gives unlimited permission to copy, distribute and modify the configure
+#   scripts that are the output of Autoconf when processing the Macro. You
+#   need not follow the terms of the GNU General Public License when using
+#   or distributing such scripts, even though portions of the text of the
+#   Macro appear in them. The GNU General Public License (GPL) does govern
+#   all other use of the material that constitutes the Autoconf Macro.
+#
+#   This special exception to the GPL applies to versions of the Autoconf
+#   Macro released by the Autoconf Archive. When you make and distribute a
+#   modified version of the Autoconf Macro, you may extend this special
+#   exception to the GPL to apply to your modified version as well.
+
+#serial 8
+
+AC_DEFUN([AX_C___ATTRIBUTE__], [
+  AC_CACHE_CHECK([for __attribute__], [ax_cv___attribute__],
+    [AC_COMPILE_IFELSE(
+      [AC_LANG_PROGRAM(
+	[[#include <stdlib.h>
+	  static void foo(void) __attribute__ ((unused));
+	  static void
+	  foo(void) {
+	      exit(1);
+	  }
+        ]], [])],
+      [ax_cv___attribute__=yes],
+      [ax_cv___attribute__=no]
+    )
+  ])
+  if test "$ax_cv___attribute__" = "yes"; then
+    AC_DEFINE([HAVE___ATTRIBUTE__], 1, [define if your compiler has __attribute__])
+  fi
+])
diff --git a/polly/lib/External/isl/m4/ax_cc_maxopt.m4 b/polly/lib/External/isl/m4/ax_cc_maxopt.m4
new file mode 100644
index 00000000000..92d522d8603
--- /dev/null
+++ b/polly/lib/External/isl/m4/ax_cc_maxopt.m4
@@ -0,0 +1,188 @@
+# ===========================================================================
+#          http://www.nongnu.org/autoconf-archive/ax_cc_maxopt.html
+# ===========================================================================
+#
+# SYNOPSIS
+#
+#   AX_CC_MAXOPT
+#
+# DESCRIPTION
+#
+#   Try to turn on "good" C optimization flags for various compilers and
+#   architectures, for some definition of "good". (In our case, good for
+#   FFTW and hopefully for other scientific codes. Modify as needed.)
+#
+#   The user can override the flags by setting the CFLAGS environment
+#   variable. The user can also specify --enable-portable-binary in order to
+#   disable any optimization flags that might result in a binary that only
+#   runs on the host architecture.
+#
+#   Note also that the flags assume that ANSI C aliasing rules are followed
+#   by the code (e.g. for gcc's -fstrict-aliasing), and that floating-point
+#   computations can be re-ordered as needed.
+#
+#   Requires macros: AX_CHECK_COMPILER_FLAGS, AX_COMPILER_VENDOR,
+#   AX_GCC_ARCHFLAG, AX_GCC_X86_CPUID.
+#
+# LICENSE
+#
+#   Copyright (c) 2008 Steven G. Johnson <stevenj@alum.mit.edu>
+#   Copyright (c) 2008 Matteo Frigo
+#
+#   This program is free software: you can redistribute it and/or modify it
+#   under the terms of the GNU General Public License as published by the
+#   Free Software Foundation, either version 3 of the License, or (at your
+#   option) any later version.
+#
+#   This program is distributed in the hope that it will be useful, but
+#   WITHOUT ANY WARRANTY; without even the implied warranty of
+#   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
+#   Public License for more details.
+#
+#   You should have received a copy of the GNU General Public License along
+#   with this program. If not, see <http://www.gnu.org/licenses/>.
+#
+#   As a special exception, the respective Autoconf Macro's copyright owner
+#   gives unlimited permission to copy, distribute and modify the configure
+#   scripts that are the output of Autoconf when processing the Macro. You
+#   need not follow the terms of the GNU General Public License when using
+#   or distributing such scripts, even though portions of the text of the
+#   Macro appear in them. The GNU General Public License (GPL) does govern
+#   all other use of the material that constitutes the Autoconf Macro.
+#
+#   This special exception to the GPL applies to versions of the Autoconf
+#   Macro released by the Autoconf Archive. When you make and distribute a
+#   modified version of the Autoconf Macro, you may extend this special
+#   exception to the GPL to apply to your modified version as well.
+
+AC_DEFUN([AX_CC_MAXOPT],
+[
+AC_REQUIRE([AC_PROG_CC])
+AC_REQUIRE([AX_COMPILER_VENDOR])
+AC_REQUIRE([AC_CANONICAL_HOST])
+
+AC_ARG_ENABLE(portable-binary, [AC_HELP_STRING([--enable-portable-binary], [disable compiler optimizations that would produce unportable binaries])],
+	acx_maxopt_portable=$withval, acx_maxopt_portable=no)
+
+# Try to determine "good" native compiler flags if none specified via CFLAGS
+if test "$ac_test_CFLAGS" != "set"; then
+  CFLAGS=""
+  case $ax_cv_c_compiler_vendor in
+    dec) CFLAGS="-newc -w0 -O5 -ansi_alias -ansi_args -fp_reorder -tune host"
+	 if test "x$acx_maxopt_portable" = xno; then
+           CFLAGS="$CFLAGS -arch host"
+         fi;;
+
+    sun) CFLAGS="-native -fast -xO5 -dalign"
+	 if test "x$acx_maxopt_portable" = xyes; then
+	   CFLAGS="$CFLAGS -xarch=generic"
+         fi;;
+
+    hp)  CFLAGS="+Oall +Optrs_ansi +DSnative"
+	 if test "x$acx_maxopt_portable" = xyes; then
+	   CFLAGS="$CFLAGS +DAportable"
+	 fi;;
+
+    ibm) if test "x$acx_maxopt_portable" = xno; then
+           xlc_opt="-qarch=auto -qtune=auto"
+	 else
+           xlc_opt="-qtune=auto"
+	 fi
+         AX_CHECK_COMPILER_FLAGS($xlc_opt,
+         	CFLAGS="-O3 -qansialias -w $xlc_opt",
+               [CFLAGS="-O3 -qansialias -w"
+                echo "******************************************************"
+                echo "*  You seem to have the IBM  C compiler.  It is      *"
+                echo "*  recommended for best performance that you use:    *"
+                echo "*                                                    *"
+                echo "*    CFLAGS=-O3 -qarch=xxx -qtune=xxx -qansialias -w *"
+                echo "*                      ^^^        ^^^                *"
+                echo "*  where xxx is pwr2, pwr3, 604, or whatever kind of *"
+                echo "*  CPU you have.  (Set the CFLAGS environment var.   *"
+                echo "*  and re-run configure.)  For more info, man cc.    *"
+                echo "******************************************************"])
+         ;;
+
+    intel) CFLAGS="-O3 -ansi_alias"
+	if test "x$acx_maxopt_portable" = xno; then
+	  icc_archflag=unknown
+	  icc_flags=""
+	  case $host_cpu in
+	    i686*|x86_64*)
+              # icc accepts gcc assembly syntax, so these should work:
+	      AX_GCC_X86_CPUID(0)
+              AX_GCC_X86_CPUID(1)
+	      case $ax_cv_gcc_x86_cpuid_0 in # see AX_GCC_ARCHFLAG
+                *:756e6547:*:*) # Intel
+                  case $ax_cv_gcc_x86_cpuid_1 in
+                    *6a?:*[[234]]:*:*|*6[[789b]]?:*:*:*) icc_flags="-xK";;
+                    *f3[[347]]:*:*:*|*f4[1347]:*:*:*) icc_flags="-xP -xN -xW -xK";;
+                    *f??:*:*:*) icc_flags="-xN -xW -xK";;
+                  esac ;;
+              esac ;;
+          esac
+          if test "x$icc_flags" != x; then
+            for flag in $icc_flags; do
+              AX_CHECK_COMPILER_FLAGS($flag, [icc_archflag=$flag; break])
+            done
+          fi
+          AC_MSG_CHECKING([for icc architecture flag])
+	  AC_MSG_RESULT($icc_archflag)
+          if test "x$icc_archflag" != xunknown; then
+            CFLAGS="$CFLAGS $icc_archflag"
+          fi
+        fi
+	;;
+
+    gnu)
+     # default optimization flags for gcc on all systems
+     CFLAGS="-O3 -fomit-frame-pointer"
+
+     # -malign-double for x86 systems
+     AX_CHECK_COMPILER_FLAGS(-malign-double, CFLAGS="$CFLAGS -malign-double")
+
+     #  -fstrict-aliasing for gcc-2.95+
+     AX_CHECK_COMPILER_FLAGS(-fstrict-aliasing,
+	CFLAGS="$CFLAGS -fstrict-aliasing")
+
+     # note that we enable "unsafe" fp optimization with other compilers, too
+     AX_CHECK_COMPILER_FLAGS(-ffast-math, CFLAGS="$CFLAGS -ffast-math")
+
+     AX_GCC_ARCHFLAG($acx_maxopt_portable)
+
+     # drop to -O1 for gcc 4.2
+     $CC --version |
+	sed -e 's/.* \(@<:@0-9@:>@@<:@0-9@:>@*\)\.\(@<:@0-9@:>@@<:@0-9@:>@*\).*/\1 \2/' |
+	(read major minor
+	    if test $major -eq 4 -a $minor -eq 2; then
+				exit 0
+	    fi
+	    exit 1
+	) && CFLAGS="-O1"
+     ;;
+  esac
+
+  if test -z "$CFLAGS"; then
+	echo ""
+	echo "********************************************************"
+        echo "* WARNING: Don't know the best CFLAGS for this system  *"
+        echo "* Use ./configure CFLAGS=... to specify your own flags *"
+	echo "* (otherwise, a default of CFLAGS=-O3 will be used)    *"
+	echo "********************************************************"
+	echo ""
+        CFLAGS="-O3"
+  fi
+
+  AX_CHECK_COMPILER_FLAGS($CFLAGS, [], [
+	echo ""
+        echo "********************************************************"
+        echo "* WARNING: The guessed CFLAGS don't seem to work with  *"
+        echo "* your compiler.                                       *"
+        echo "* Use ./configure CFLAGS=... to specify your own flags *"
+        echo "********************************************************"
+        echo ""
+        CFLAGS=""
+  ])
+
+fi
+])
diff --git a/polly/lib/External/isl/m4/ax_cflags_warn_all.m4 b/polly/lib/External/isl/m4/ax_cflags_warn_all.m4
new file mode 100644
index 00000000000..026c6e98415
--- /dev/null
+++ b/polly/lib/External/isl/m4/ax_cflags_warn_all.m4
@@ -0,0 +1,149 @@
+# ===========================================================================
+#       http://www.nongnu.org/autoconf-archive/ax_cflags_warn_all.html
+# ===========================================================================
+#
+# SYNOPSIS
+#
+#   AX_CFLAGS_WARN_ALL [(shellvar [,default, [A/NA]])]
+#
+# DESCRIPTION
+#
+#   Try to find a compiler option that enables most reasonable warnings.
+#
+#   For the GNU CC compiler it will be -Wall (and -ansi -pedantic) The
+#   result is added to the shellvar being CFLAGS by default.
+#
+#   Currently this macro knows about GCC, Solaris C compiler, Digital Unix C
+#   compiler, C for AIX Compiler, HP-UX C compiler, IRIX C compiler, NEC
+#   SX-5 (Super-UX 10) C compiler, and Cray J90 (Unicos 10.0.0.8) C
+#   compiler.
+#
+#    - $1 shell-variable-to-add-to : CFLAGS
+#    - $2 add-value-if-not-found : nothing
+#    - $3 action-if-found : add value to shellvariable
+#    - $4 action-if-not-found : nothing
+#
+# LICENSE
+#
+#   Copyright (c) 2008 Guido U. Draheim <guidod@gmx.de>
+#
+#   This program is free software; you can redistribute it and/or modify it
+#   under the terms of the GNU General Public License as published by the
+#   Free Software Foundation; either version 2 of the License, or (at your
+#   option) any later version.
+#
+#   This program is distributed in the hope that it will be useful, but
+#   WITHOUT ANY WARRANTY; without even the implied warranty of
+#   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
+#   Public License for more details.
+#
+#   You should have received a copy of the GNU General Public License along
+#   with this program. If not, see <http://www.gnu.org/licenses/>.
+#
+#   As a special exception, the respective Autoconf Macro's copyright owner
+#   gives unlimited permission to copy, distribute and modify the configure
+#   scripts that are the output of Autoconf when processing the Macro. You
+#   need not follow the terms of the GNU General Public License when using
+#   or distributing such scripts, even though portions of the text of the
+#   Macro appear in them. The GNU General Public License (GPL) does govern
+#   all other use of the material that constitutes the Autoconf Macro.
+#
+#   This special exception to the GPL applies to versions of the Autoconf
+#   Macro released by the Autoconf Archive. When you make and distribute a
+#   modified version of the Autoconf Macro, you may extend this special
+#   exception to the GPL to apply to your modified version as well.
+
+AC_DEFUN([AX_CFLAGS_WARN_ALL],[dnl
+AS_VAR_PUSHDEF([FLAGS],[CFLAGS])dnl
+AS_VAR_PUSHDEF([VAR],[ac_cv_cflags_warn_all])dnl
+AC_CACHE_CHECK([m4_ifval($1,$1,FLAGS) for maximum warnings],
+VAR,[VAR="no, unknown"
+ AC_LANG_SAVE
+ AC_LANG_C
+ ac_save_[]FLAGS="$[]FLAGS"
+for ac_arg dnl
+in "-pedantic  % -Wall"       dnl   GCC
+   "-xstrconst % -v"          dnl Solaris C
+   "-std1      % -verbose -w0 -warnprotos" dnl Digital Unix
+   "-qlanglvl=ansi % -qsrcmsg -qinfo=all:noppt:noppc:noobs:nocnd" dnl AIX
+   "-ansi -ansiE % -fullwarn" dnl IRIX
+   "+ESlit     % +w1"         dnl HP-UX C
+   "-Xc        % -pvctl[,]fullmsg" dnl NEC SX-5 (Super-UX 10)
+   "-h conform % -h msglevel 2" dnl Cray C (Unicos)
+   #
+do FLAGS="$ac_save_[]FLAGS "`echo $ac_arg | sed -e 's,%%.*,,' -e 's,%,,'`
+   AC_TRY_COMPILE([],[return 0;],
+   [VAR=`echo $ac_arg | sed -e 's,.*% *,,'` ; break])
+done
+ FLAGS="$ac_save_[]FLAGS"
+ AC_LANG_RESTORE
+])
+case ".$VAR" in
+     .ok|.ok,*) m4_ifvaln($3,$3) ;;
+   .|.no|.no,*) m4_ifvaln($4,$4,[m4_ifval($2,[
+        AC_RUN_LOG([: m4_ifval($1,$1,FLAGS)="$m4_ifval($1,$1,FLAGS) $2"])
+                      m4_ifval($1,$1,FLAGS)="$m4_ifval($1,$1,FLAGS) $2"])]) ;;
+   *) m4_ifvaln($3,$3,[
+   if echo " $[]m4_ifval($1,$1,FLAGS) " | grep " $VAR " 2>&1 >/dev/null
+   then AC_RUN_LOG([: m4_ifval($1,$1,FLAGS) does contain $VAR])
+   else AC_RUN_LOG([: m4_ifval($1,$1,FLAGS)="$m4_ifval($1,$1,FLAGS) $VAR"])
+                      m4_ifval($1,$1,FLAGS)="$m4_ifval($1,$1,FLAGS) $VAR"
+   fi ]) ;;
+esac
+AS_VAR_POPDEF([VAR])dnl
+AS_VAR_POPDEF([FLAGS])dnl
+])
+
+dnl the only difference - the LANG selection... and the default FLAGS
+
+AC_DEFUN([AX_CXXFLAGS_WARN_ALL],[dnl
+AS_VAR_PUSHDEF([FLAGS],[CXXFLAGS])dnl
+AS_VAR_PUSHDEF([VAR],[ax_cv_cxxflags_warn_all])dnl
+AC_CACHE_CHECK([m4_ifval($1,$1,FLAGS) for maximum warnings],
+VAR,[VAR="no, unknown"
+ AC_LANG_SAVE
+ AC_LANG_CPLUSPLUS
+ ac_save_[]FLAGS="$[]FLAGS"
+for ac_arg dnl
+in "-pedantic  % -Wall"       dnl   GCC
+   "-xstrconst % -v"          dnl Solaris C
+   "-std1      % -verbose -w0 -warnprotos" dnl Digital Unix
+   "-qlanglvl=ansi % -qsrcmsg -qinfo=all:noppt:noppc:noobs:nocnd" dnl AIX
+   "-ansi -ansiE % -fullwarn" dnl IRIX
+   "+ESlit     % +w1"         dnl HP-UX C
+   "-Xc        % -pvctl[,]fullmsg" dnl NEC SX-5 (Super-UX 10)
+   "-h conform % -h msglevel 2" dnl Cray C (Unicos)
+   #
+do FLAGS="$ac_save_[]FLAGS "`echo $ac_arg | sed -e 's,%%.*,,' -e 's,%,,'`
+   AC_TRY_COMPILE([],[return 0;],
+   [VAR=`echo $ac_arg | sed -e 's,.*% *,,'` ; break])
+done
+ FLAGS="$ac_save_[]FLAGS"
+ AC_LANG_RESTORE
+])
+case ".$VAR" in
+     .ok|.ok,*) m4_ifvaln($3,$3) ;;
+   .|.no|.no,*) m4_ifvaln($4,$4,[m4_ifval($2,[
+        AC_RUN_LOG([: m4_ifval($1,$1,FLAGS)="$m4_ifval($1,$1,FLAGS) $2"])
+                      m4_ifval($1,$1,FLAGS)="$m4_ifval($1,$1,FLAGS) $2"])]) ;;
+   *) m4_ifvaln($3,$3,[
+   if echo " $[]m4_ifval($1,$1,FLAGS) " | grep " $VAR " 2>&1 >/dev/null
+   then AC_RUN_LOG([: m4_ifval($1,$1,FLAGS) does contain $VAR])
+   else AC_RUN_LOG([: m4_ifval($1,$1,FLAGS)="$m4_ifval($1,$1,FLAGS) $VAR"])
+                      m4_ifval($1,$1,FLAGS)="$m4_ifval($1,$1,FLAGS) $VAR"
+   fi ]) ;;
+esac
+AS_VAR_POPDEF([VAR])dnl
+AS_VAR_POPDEF([FLAGS])dnl
+])
+
+dnl  implementation tactics:
+dnl   the for-argument contains a list of options. The first part of
+dnl   these does only exist to detect the compiler - usually it is
+dnl   a global option to enable -ansi or -extrawarnings. All other
+dnl   compilers will fail about it. That was needed since a lot of
+dnl   compilers will give false positives for some option-syntax
+dnl   like -Woption or -Xoption as they think of it is a pass-through
+dnl   to later compile stages or something. The "%" is used as a
+dnl   delimimiter. A non-option comment can be given after "%%" marks
+dnl   which will be shown but not added to the respective C/CXXFLAGS.
diff --git a/polly/lib/External/isl/m4/ax_check_compiler_flags.m4 b/polly/lib/External/isl/m4/ax_check_compiler_flags.m4
new file mode 100644
index 00000000000..7da8324b70f
--- /dev/null
+++ b/polly/lib/External/isl/m4/ax_check_compiler_flags.m4
@@ -0,0 +1,74 @@
+# ===========================================================================
+#     http://www.nongnu.org/autoconf-archive/ax_check_compiler_flags.html
+# ===========================================================================
+#
+# SYNOPSIS
+#
+#   AX_CHECK_COMPILER_FLAGS(FLAGS, [ACTION-SUCCESS], [ACTION-FAILURE])
+#
+# DESCRIPTION
+#
+#   Check whether the given compiler FLAGS work with the current language's
+#   compiler, or whether they give an error. (Warnings, however, are
+#   ignored.)
+#
+#   ACTION-SUCCESS/ACTION-FAILURE are shell commands to execute on
+#   success/failure.
+#
+# LICENSE
+#
+#   Copyright (c) 2009 Steven G. Johnson <stevenj@alum.mit.edu>
+#   Copyright (c) 2009 Matteo Frigo
+#
+#   This program is free software: you can redistribute it and/or modify it
+#   under the terms of the GNU General Public License as published by the
+#   Free Software Foundation, either version 3 of the License, or (at your
+#   option) any later version.
+#
+#   This program is distributed in the hope that it will be useful, but
+#   WITHOUT ANY WARRANTY; without even the implied warranty of
+#   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
+#   Public License for more details.
+#
+#   You should have received a copy of the GNU General Public License along
+#   with this program. If not, see <http://www.gnu.org/licenses/>.
+#
+#   As a special exception, the respective Autoconf Macro's copyright owner
+#   gives unlimited permission to copy, distribute and modify the configure
+#   scripts that are the output of Autoconf when processing the Macro. You
+#   need not follow the terms of the GNU General Public License when using
+#   or distributing such scripts, even though portions of the text of the
+#   Macro appear in them. The GNU General Public License (GPL) does govern
+#   all other use of the material that constitutes the Autoconf Macro.
+#
+#   This special exception to the GPL applies to versions of the Autoconf
+#   Macro released by the Autoconf Archive. When you make and distribute a
+#   modified version of the Autoconf Macro, you may extend this special
+#   exception to the GPL to apply to your modified version as well.
+
+AC_DEFUN([AX_CHECK_COMPILER_FLAGS],
+[AC_PREREQ(2.59) dnl for _AC_LANG_PREFIX
+AC_MSG_CHECKING([whether _AC_LANG compiler accepts $1])
+dnl Some hackery here since AC_CACHE_VAL can't handle a non-literal varname:
+AS_LITERAL_IF([$1],
+  [AC_CACHE_VAL(AS_TR_SH(ax_cv_[]_AC_LANG_ABBREV[]_flags_[$1]), [
+      ax_save_FLAGS=$[]_AC_LANG_PREFIX[]FLAGS
+      _AC_LANG_PREFIX[]FLAGS="$1"
+      AC_COMPILE_IFELSE([AC_LANG_PROGRAM()],
+        AS_TR_SH(ax_cv_[]_AC_LANG_ABBREV[]_flags_[$1])=yes,
+        AS_TR_SH(ax_cv_[]_AC_LANG_ABBREV[]_flags_[$1])=no)
+      _AC_LANG_PREFIX[]FLAGS=$ax_save_FLAGS])],
+  [ax_save_FLAGS=$[]_AC_LANG_PREFIX[]FLAGS
+   _AC_LANG_PREFIX[]FLAGS="$1"
+   AC_COMPILE_IFELSE([AC_LANG_PROGRAM()],
+     eval AS_TR_SH(ax_cv_[]_AC_LANG_ABBREV[]_flags_[$1])=yes,
+     eval AS_TR_SH(ax_cv_[]_AC_LANG_ABBREV[]_flags_[$1])=no)
+   _AC_LANG_PREFIX[]FLAGS=$ax_save_FLAGS])
+eval ax_check_compiler_flags=$AS_TR_SH(ax_cv_[]_AC_LANG_ABBREV[]_flags_[$1])
+AC_MSG_RESULT($ax_check_compiler_flags)
+if test "x$ax_check_compiler_flags" = xyes; then
+	m4_default([$2], :)
+else
+	m4_default([$3], :)
+fi
+])dnl AX_CHECK_COMPILER_FLAGS
diff --git a/polly/lib/External/isl/m4/ax_compiler_vendor.m4 b/polly/lib/External/isl/m4/ax_compiler_vendor.m4
new file mode 100644
index 00000000000..321470697e3
--- /dev/null
+++ b/polly/lib/External/isl/m4/ax_compiler_vendor.m4
@@ -0,0 +1,63 @@
+# ===========================================================================
+#    http://www.gnu.org/software/autoconf-archive/ax_compiler_vendor.html
+# ===========================================================================
+#
+# SYNOPSIS
+#
+#   AX_COMPILER_VENDOR
+#
+# DESCRIPTION
+#
+#   Determine the vendor of the C/C++ compiler, e.g., gnu, intel, ibm, sun,
+#   hp, borland, comeau, dec, cray, kai, lcc, metrowerks, sgi, microsoft,
+#   watcom, etc. The vendor is returned in the cache variable
+#   $ax_cv_c_compiler_vendor for C and $ax_cv_cxx_compiler_vendor for C++.
+#
+# LICENSE
+#
+#   Copyright (c) 2008 Steven G. Johnson <stevenj@alum.mit.edu>
+#   Copyright (c) 2008 Matteo Frigo
+#
+#   This program is free software: you can redistribute it and/or modify it
+#   under the terms of the GNU General Public License as published by the
+#   Free Software Foundation, either version 3 of the License, or (at your
+#   option) any later version.
+#
+#   This program is distributed in the hope that it will be useful, but
+#   WITHOUT ANY WARRANTY; without even the implied warranty of
+#   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
+#   Public License for more details.
+#
+#   You should have received a copy of the GNU General Public License along
+#   with this program. If not, see <http://www.gnu.org/licenses/>.
+#
+#   As a special exception, the respective Autoconf Macro's copyright owner
+#   gives unlimited permission to copy, distribute and modify the configure
+#   scripts that are the output of Autoconf when processing the Macro. You
+#   need not follow the terms of the GNU General Public License when using
+#   or distributing such scripts, even though portions of the text of the
+#   Macro appear in them. The GNU General Public License (GPL) does govern
+#   all other use of the material that constitutes the Autoconf Macro.
+#
+#   This special exception to the GPL applies to versions of the Autoconf
+#   Macro released by the Autoconf Archive. When you make and distribute a
+#   modified version of the Autoconf Macro, you may extend this special
+#   exception to the GPL to apply to your modified version as well.
+
+#serial 9
+
+AC_DEFUN([AX_COMPILER_VENDOR],
+[
+AC_CACHE_CHECK([for _AC_LANG compiler vendor], ax_cv_[]_AC_LANG_ABBREV[]_compiler_vendor,
+ [ax_cv_[]_AC_LANG_ABBREV[]_compiler_vendor=unknown
+  # note: don't check for gcc first since some other compilers define __GNUC__
+  for ventest in intel:__ICC,__ECC,__INTEL_COMPILER ibm:__xlc__,__xlC__,__IBMC__,__IBMCPP__ pathscale:__PATHCC__,__PATHSCALE__ clang:__clang__ gnu:__GNUC__ sun:__SUNPRO_C,__SUNPRO_CC hp:__HP_cc,__HP_aCC dec:__DECC,__DECCXX,__DECC_VER,__DECCXX_VER borland:__BORLANDC__,__TURBOC__ comeau:__COMO__ cray:_CRAYC kai:__KCC lcc:__LCC__ metrowerks:__MWERKS__ sgi:__sgi,sgi microsoft:_MSC_VER watcom:__WATCOMC__ portland:__PGI; do
+    vencpp="defined("`echo $ventest | cut -d: -f2 | sed 's/,/) || defined(/g'`")"
+    AC_COMPILE_IFELSE([AC_LANG_PROGRAM(,[
+#if !($vencpp)
+      thisisanerror;
+#endif
+])], [ax_cv_]_AC_LANG_ABBREV[_compiler_vendor=`echo $ventest | cut -d: -f1`; break])
+  done
+ ])
+])
diff --git a/polly/lib/External/isl/m4/ax_create_pkgconfig_info.m4 b/polly/lib/External/isl/m4/ax_create_pkgconfig_info.m4
new file mode 100644
index 00000000000..308e64ff854
--- /dev/null
+++ b/polly/lib/External/isl/m4/ax_create_pkgconfig_info.m4
@@ -0,0 +1,351 @@
+# ============================================================================
+#  http://www.gnu.org/software/autoconf-archive/ax_create_pkgconfig_info.html
+# ============================================================================
+#
+# SYNOPSIS
+#
+#   AX_CREATE_PKGCONFIG_INFO [(outputfile, [requires [,libs [,summary [,cflags [, ldflags]]]]])]
+#
+# DESCRIPTION
+#
+#   Defaults:
+#
+#     $1 = $PACKAGE_NAME.pc
+#     $2 = (empty)
+#     $3 = $PACKAGE_LIBS $LIBS (as set at that point in configure.ac)
+#     $4 = $PACKAGE_SUMMARY (or $1 Library)
+#     $5 = $PACKAGE_CFLAGS (as set at the point in configure.ac)
+#     $6 = $PACKAGE_LDFLAGS (as set at the point in configure.ac)
+#
+#     PACKAGE_NAME defaults to $PACKAGE if not set.
+#     PACKAGE_LIBS defaults to -l$PACKAGE_NAME if not set.
+#
+#   The resulting file is called $PACKAGE.pc.in / $PACKAGE.pc
+#
+#   You will find this macro most useful in conjunction with
+#   ax_spec_defaults that can read good initializers from the .spec file. In
+#   consequencd, most of the generatable installable stuff can be made from
+#   information being updated in a single place for the whole project.
+#
+# LICENSE
+#
+#   Copyright (c) 2008 Guido U. Draheim <guidod@gmx.de>
+#   Copyright (c) 2008 Sven Verdoolaege <skimo@kotnet.org>
+#
+#   This program is free software; you can redistribute it and/or modify it
+#   under the terms of the GNU General Public License as published by the
+#   Free Software Foundation; either version 3 of the License, or (at your
+#   option) any later version.
+#
+#   This program is distributed in the hope that it will be useful, but
+#   WITHOUT ANY WARRANTY; without even the implied warranty of
+#   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
+#   Public License for more details.
+#
+#   You should have received a copy of the GNU General Public License along
+#   with this program. If not, see <http://www.gnu.org/licenses/>.
+#
+#   As a special exception, the respective Autoconf Macro's copyright owner
+#   gives unlimited permission to copy, distribute and modify the configure
+#   scripts that are the output of Autoconf when processing the Macro. You
+#   need not follow the terms of the GNU General Public License when using
+#   or distributing such scripts, even though portions of the text of the
+#   Macro appear in them. The GNU General Public License (GPL) does govern
+#   all other use of the material that constitutes the Autoconf Macro.
+#
+#   This special exception to the GPL applies to versions of the Autoconf
+#   Macro released by the Autoconf Archive. When you make and distribute a
+#   modified version of the Autoconf Macro, you may extend this special
+#   exception to the GPL to apply to your modified version as well.
+
+#serial 12
+
+AC_DEFUN([AX_CREATE_PKGCONFIG_INFO],[dnl
+AS_VAR_PUSHDEF([PKGCONFIG_suffix],[ax_create_pkgconfig_suffix])dnl
+AS_VAR_PUSHDEF([PKGCONFIG_libdir],[ax_create_pkgconfig_libdir])dnl
+AS_VAR_PUSHDEF([PKGCONFIG_libfile],[ax_create_pkgconfig_libfile])dnl
+AS_VAR_PUSHDEF([PKGCONFIG_libname],[ax_create_pkgconfig_libname])dnl
+AS_VAR_PUSHDEF([PKGCONFIG_version],[ax_create_pkgconfig_version])dnl
+AS_VAR_PUSHDEF([PKGCONFIG_description],[ax_create_pkgconfig_description])dnl
+AS_VAR_PUSHDEF([PKGCONFIG_requires],[ax_create_pkgconfig_requires])dnl
+AS_VAR_PUSHDEF([PKGCONFIG_pkglibs],[ax_create_pkgconfig_pkglibs])dnl
+AS_VAR_PUSHDEF([PKGCONFIG_libs],[ax_create_pkgconfig_libs])dnl
+AS_VAR_PUSHDEF([PKGCONFIG_ldflags],[ax_create_pkgconfig_ldflags])dnl
+AS_VAR_PUSHDEF([PKGCONFIG_cppflags],[ax_create_pkgconfig_cppflags])dnl
+AS_VAR_PUSHDEF([PKGCONFIG_generate],[ax_create_pkgconfig_generate])dnl
+AS_VAR_PUSHDEF([PKGCONFIG_src_libdir],[ax_create_pkgconfig_src_libdir])dnl
+AS_VAR_PUSHDEF([PKGCONFIG_src_headers],[ax_create_pkgconfig_src_headers])dnl
+
+# we need the expanded forms...
+test "x$prefix" = xNONE && prefix=$ac_default_prefix
+test "x$exec_prefix" = xNONE && exec_prefix='${prefix}'
+
+AC_MSG_CHECKING(our pkgconfig libname)
+test ".$PKGCONFIG_libname" != "." || \
+PKGCONFIG_libname="ifelse($1,,${PACKAGE_NAME},`basename $1 .pc`)"
+test ".$PKGCONFIG_libname" != "." || \
+PKGCONFIG_libname="$PACKAGE"
+PKGCONFIG_libname=`eval echo "$PKGCONFIG_libname"`
+PKGCONFIG_libname=`eval echo "$PKGCONFIG_libname"`
+AC_MSG_RESULT($PKGCONFIG_libname)
+
+AC_MSG_CHECKING(our pkgconfig version)
+test ".$PKGCONFIG_version" != "." || \
+PKGCONFIG_version="${PACKAGE_VERSION}"
+test ".$PKGCONFIG_version" != "." || \
+PKGCONFIG_version="$VERSION"
+PKGCONFIG_version=`eval echo "$PKGCONFIG_version"`
+PKGCONFIG_version=`eval echo "$PKGCONFIG_version"`
+AC_MSG_RESULT($PKGCONFIG_version)
+
+AC_MSG_CHECKING(our pkgconfig_libdir)
+test ".$pkgconfig_libdir" = "." && \
+pkgconfig_libdir='${libdir}/pkgconfig'
+PKGCONFIG_libdir=`eval echo "$pkgconfig_libdir"`
+PKGCONFIG_libdir=`eval echo "$PKGCONFIG_libdir"`
+PKGCONFIG_libdir=`eval echo "$PKGCONFIG_libdir"`
+AC_MSG_RESULT($pkgconfig_libdir)
+test "$pkgconfig_libdir" != "$PKGCONFIG_libdir" && (
+AC_MSG_RESULT(expanded our pkgconfig_libdir... $PKGCONFIG_libdir))
+AC_SUBST([pkgconfig_libdir])
+
+AC_MSG_CHECKING(our pkgconfig_libfile)
+test ".$pkgconfig_libfile" != "." || \
+pkgconfig_libfile="ifelse($1,,$PKGCONFIG_libname.pc,`basename $1`)"
+PKGCONFIG_libfile=`eval echo "$pkgconfig_libfile"`
+PKGCONFIG_libfile=`eval echo "$PKGCONFIG_libfile"`
+AC_MSG_RESULT($pkgconfig_libfile)
+test "$pkgconfig_libfile" != "$PKGCONFIG_libfile" && (
+AC_MSG_RESULT(expanded our pkgconfig_libfile... $PKGCONFIG_libfile))
+AC_SUBST([pkgconfig_libfile])
+
+AC_MSG_CHECKING(our package / suffix)
+PKGCONFIG_suffix="$program_suffix"
+test ".$PKGCONFIG_suffix" != .NONE || PKGCONFIG_suffix=""
+AC_MSG_RESULT(${PACKAGE_NAME} / ${PKGCONFIG_suffix})
+
+AC_MSG_CHECKING(our pkgconfig description)
+PKGCONFIG_description="ifelse($4,,$PACKAGE_SUMMARY,$4)"
+test ".$PKGCONFIG_description" != "." || \
+PKGCONFIG_description="$PKGCONFIG_libname Library"
+PKGCONFIG_description=`eval echo "$PKGCONFIG_description"`
+PKGCONFIG_description=`eval echo "$PKGCONFIG_description"`
+AC_MSG_RESULT($PKGCONFIG_description)
+
+AC_MSG_CHECKING(our pkgconfig requires)
+PKGCONFIG_requires="ifelse($2,,$PACKAGE_REQUIRES,$2)"
+PKGCONFIG_requires=`eval echo "$PKGCONFIG_requires"`
+PKGCONFIG_requires=`eval echo "$PKGCONFIG_requires"`
+AC_MSG_RESULT($PKGCONFIG_requires)
+
+AC_MSG_CHECKING(our pkgconfig ext libs)
+PKGCONFIG_pkglibs="$PACKAGE_LIBS"
+test ".$PKGCONFIG_pkglibs" != "." || PKGCONFIG_pkglibs="-l$PKGCONFIG_libname"
+PKGCONFIG_libs="ifelse($3,,$PKGCONFIG_pkglibs $LIBS,$3)"
+PKGCONFIG_libs=`eval echo "$PKGCONFIG_libs"`
+PKGCONFIG_libs=`eval echo "$PKGCONFIG_libs"`
+AC_MSG_RESULT($PKGCONFIG_libs)
+
+AC_MSG_CHECKING(our pkgconfig cppflags)
+PKGCONFIG_cppflags="ifelse($5,,$PACKAGE_CFLAGS,$5)"
+PKGCONFIG_cppflags=`eval echo "$PKGCONFIG_cppflags"`
+PKGCONFIG_cppflags=`eval echo "$PKGCONFIG_cppflags"`
+AC_MSG_RESULT($PKGCONFIG_cppflags)
+
+AC_MSG_CHECKING(our pkgconfig ldflags)
+PKGCONFIG_ldflags="ifelse($6,,$PACKAGE_LDFLAGS,$5)"
+PKGCONFIG_ldflags=`eval echo "$PKGCONFIG_ldflags"`
+PKGCONFIG_ldflags=`eval echo "$PKGCONFIG_ldflags"`
+AC_MSG_RESULT($PKGCONFIG_ldflags)
+
+test ".$PKGCONFIG_generate" != "." || \
+PKGCONFIG_generate="ifelse($1,,$PKGCONFIG_libname.pc,$1)"
+PKGCONFIG_generate=`eval echo "$PKGCONFIG_generate"`
+PKGCONFIG_generate=`eval echo "$PKGCONFIG_generate"`
+test "$pkgconfig_libfile" != "$PKGCONFIG_generate" && (
+AC_MSG_RESULT(generate the pkgconfig later... $PKGCONFIG_generate))
+
+if test ".$PKGCONFIG_src_libdir" = "." ; then
+PKGCONFIG_src_libdir=`pwd`
+PKGCONFIG_src_libdir=`AS_DIRNAME("$PKGCONFIG_src_libdir/$PKGCONFIG_generate")`
+test ! -d $PKGCONFIG_src_libdir/src || \
+PKGCONFIG_src_libdir="$PKGCONFIG_src_libdir/src"
+case ".$objdir" in
+*libs) PKGCONFIG_src_libdir="$PKGCONFIG_src_libdir/$objdir" ;; esac
+AC_MSG_RESULT(noninstalled pkgconfig -L $PKGCONFIG_src_libdir)
+fi
+
+if test ".$PKGCONFIG_src_headers" = "." ; then
+PKGCONFIG_src_headers=`pwd`
+v="$ac_top_srcdir" ;
+test ".$v" != "." || v="$ax_spec_dir"
+test ".$v" != "." || v="$srcdir"
+case "$v" in /*) PKGCONFIG_src_headers="" ;; esac
+PKGCONFIG_src_headers=`AS_DIRNAME("$PKGCONFIG_src_headers/$v/x")`
+test ! -d $PKGCONFIG_src_headers/incl[]ude || \
+PKGCONFIG_src_headers="$PKGCONFIG_src_headers/incl[]ude"
+AC_MSG_RESULT(noninstalled pkgconfig -I $PKGCONFIG_src_headers)
+fi
+
+
+dnl AC_CONFIG_COMMANDS crap disallows to use $PKGCONFIG_libfile here...
+AC_CONFIG_COMMANDS([$ax_create_pkgconfig_generate],[
+pkgconfig_generate="$ax_create_pkgconfig_generate"
+if test ! -f "$pkgconfig_generate.in"
+then generate="true"
+elif grep ' generated by configure ' $pkgconfig_generate.in >/dev/null
+then generate="true"
+else generate="false";
+fi
+if $generate ; then
+AC_MSG_NOTICE(creating $pkgconfig_generate.in)
+cat > $pkgconfig_generate.in <<AXEOF
+# generated by configure / remove this line to disable regeneration
+prefix=@prefix@
+exec_prefix=@exec_prefix@
+bindir=@bindir@
+libdir=@libdir@
+datarootdir=@datarootdir@
+datadir=@datadir@
+sysconfdir=@sysconfdir@
+includedir=@includedir@
+package=@PACKAGE@
+suffix=@suffix@
+
+Name: @PACKAGE_NAME@
+Description: @PACKAGE_DESCRIPTION@
+Version: @PACKAGE_VERSION@
+Requires: @PACKAGE_REQUIRES@
+Libs: -L\${libdir} @LDFLAGS@ @LIBS@
+Cflags: -I\${includedir} @CPPFLAGS@
+AXEOF
+fi # DONE generate $pkgconfig_generate.in
+AC_MSG_NOTICE(creating $pkgconfig_generate)
+cat >conftest.sed <<AXEOF
+s|@prefix@|${pkgconfig_prefix}|
+s|@exec_prefix@|${pkgconfig_execprefix}|
+s|@bindir@|${pkgconfig_bindir}|
+s|@libdir@|${pkgconfig_libdir}|
+s|@datarootdir@|${pkgconfig_datarootdir}|
+s|@datadir@|${pkgconfig_datadir}|
+s|@sysconfdir@|${pkgconfig_sysconfdir}|
+s|@includedir@|${pkgconfig_includedir}|
+s|@suffix@|${pkgconfig_suffix}|
+s|@PACKAGE@|${pkgconfig_package}|
+s|@PACKAGE_NAME@|${pkgconfig_libname}|
+s|@PACKAGE_DESCRIPTION@|${pkgconfig_description}|
+s|@PACKAGE_VERSION@|${pkgconfig_version}|
+s|@PACKAGE_REQUIRES@|${pkgconfig_requires}|
+s|@LIBS@|${pkgconfig_libs}|
+s|@LDFLAGS@|${pkgconfig_ldflags}|
+s|@CPPFLAGS@|${pkgconfig_cppflags}|
+AXEOF
+sed -f conftest.sed  $pkgconfig_generate.in > $pkgconfig_generate
+if test ! -s $pkgconfig_generate ; then
+    AC_MSG_ERROR([$pkgconfig_generate is empty])
+fi ; rm conftest.sed # DONE generate $pkgconfig_generate
+pkgconfig_uninstalled=`echo $pkgconfig_generate |sed 's/.pc$/-uninstalled.pc/'`
+AC_MSG_NOTICE(creating $pkgconfig_uninstalled)
+cat >conftest.sed <<AXEOF
+s|@prefix@|${pkgconfig_prefix}|
+s|@exec_prefix@|${pkgconfig_execprefix}|
+s|@bindir@|${pkgconfig_bindir}|
+s|@libdir@|${pkgconfig_src_libdir}|
+s|@datarootdir@|${pkgconfig_datarootdir}|
+s|@datadir@|${pkgconfig_datadir}|
+s|@sysconfdir@|${pkgconfig_sysconfdir}|
+s|@includedir@|${pkgconfig_src_headers}|
+s|@suffix@|${pkgconfig_suffix}|
+s|@PACKAGE@|${pkgconfig_package}|
+s|@PACKAGE_NAME@|${pkgconfig_libname}|
+s|@PACKAGE_DESCRIPTION@|${pkgconfig_description}|
+s|@PACKAGE_VERSION@|${pkgconfig_version}|
+s|@PACKAGE_REQUIRES@|${pkgconfig_requires}|
+s|@LIBS@|${pkgconfig_libs}|
+s|@LDFLAGS@|${pkgconfig_ldflags}|
+s|@CPPFLAGS@|${pkgconfig_cppflags}|
+AXEOF
+sed -f conftest.sed $pkgconfig_generate.in > $pkgconfig_uninstalled
+if test ! -s $pkgconfig_uninstalled ; then
+    AC_MSG_ERROR([$pkgconfig_uninstalled is empty])
+fi ; rm conftest.sed # DONE generate $pkgconfig_uninstalled
+           pkgconfig_requires_add=`echo ${pkgconfig_requires}`
+if test ".$pkgconfig_requires_add" != "." ; then
+           pkgconfig_requires_add="pkg-config $pkgconfig_requires_add"
+    else   pkgconfig_requires_add=":" ; fi
+pkgconfig_uninstalled=`echo $pkgconfig_generate |sed 's/.pc$/-uninstalled.sh/'`
+AC_MSG_NOTICE(creating $pkgconfig_uninstalled)
+cat >conftest.sed <<AXEOF
+s|@prefix@|\"${pkgconfig_prefix}\"|
+s|@exec_prefix@|\"${pkgconfig_execprefix}\"|
+s|@bindir@|\"${pkgconfig_bindir}\"|
+s|@libdir@|\"${pkgconfig_src_libdir}\"|
+s|@datarootdir@|\"${pkgconfig_datarootdir}\"|
+s|@datadir@|\"${pkgconfig_datadir}\"|
+s|@sysconfdir@|\"${pkgconfig_sysconfdir}\"|
+s|@includedir@|\"${pkgconfig_src_headers}\"|
+s|@suffix@|\"${pkgconfig_suffix}\"|
+s|@PACKAGE@|\"${pkgconfig_package}\"|
+s|@PACKAGE_NAME@|\"${pkgconfig_libname}\"|
+s|@PACKAGE_DESCRIPTION@|\"${pkgconfig_description}\"|
+s|@PACKAGE_VERSION@|\"${pkgconfig_version}\"|
+s|@PACKAGE_REQUIRES@|\"${pkgconfig_requires}\"|
+s|@LIBS@|\"${pkgconfig_libs}\"|
+s|@LDFLAGS@|\"${pkgconfig_ldflags}\"|
+s|@CPPFLAGS@|\"${pkgconfig_cppflags}\"|
+s>Name:>for option\\; do case \"\$option\" in --list-all|--name) echo >
+s>Description: *>\\;\\; --help) pkg-config --help \\; echo Buildscript Of >
+s>Version: *>\\;\\; --modversion|--version) echo >
+s>Requires:>\\;\\; --requires) echo $pkgconfig_requires_add>
+s>Libs: *>\\;\\; --libs) echo >
+s>Cflags: *>\\;\\; --cflags) echo >
+/--libs)/a\\
+       $pkgconfig_requires_add
+/--cflags)/a\\
+       $pkgconfig_requires_add\\
+;; --variable=*) eval echo '\$'\`echo \$option | sed -e 's/.*=//'\`\\
+;; --uninstalled) exit 0 \\
+;; *) ;; esac done
+AXEOF
+sed -f conftest.sed  $pkgconfig_generate.in > $pkgconfig_uninstalled
+if test ! -s $pkgconfig_uninstalled ; then
+    AC_MSG_ERROR([$pkgconfig_uninstalled is empty])
+fi ; rm conftest.sed # DONE generate $pkgconfig_uninstalled
+],[
+dnl AC_CONFIG_COMMANDS crap, the AS_PUSHVAR defines are invalid here...
+ax_create_pkgconfig_generate="$ax_create_pkgconfig_generate"
+pkgconfig_prefix='$prefix'
+pkgconfig_execprefix='$exec_prefix'
+pkgconfig_bindir='$bindir'
+pkgconfig_libdir='$libdir'
+pkgconfig_includedir='$includedir'
+pkgconfig_datarootdir='$datarootdir'
+pkgconfig_datadir='$datadir'
+pkgconfig_sysconfdir='$sysconfdir'
+pkgconfig_suffix='$ax_create_pkgconfig_suffix'
+pkgconfig_package='$PACKAGE_NAME'
+pkgconfig_libname='$ax_create_pkgconfig_libname'
+pkgconfig_description='$ax_create_pkgconfig_description'
+pkgconfig_version='$ax_create_pkgconfig_version'
+pkgconfig_requires='$ax_create_pkgconfig_requires'
+pkgconfig_libs='$ax_create_pkgconfig_libs'
+pkgconfig_ldflags='$ax_create_pkgconfig_ldflags'
+pkgconfig_cppflags='$ax_create_pkgconfig_cppflags'
+pkgconfig_src_libdir='$ax_create_pkgconfig_src_libdir'
+pkgconfig_src_headers='$ax_create_pkgconfig_src_headers'
+])dnl
+AS_VAR_POPDEF([PKGCONFIG_suffix])dnl
+AS_VAR_POPDEF([PKGCONFIG_libdir])dnl
+AS_VAR_POPDEF([PKGCONFIG_libfile])dnl
+AS_VAR_POPDEF([PKGCONFIG_libname])dnl
+AS_VAR_POPDEF([PKGCONFIG_version])dnl
+AS_VAR_POPDEF([PKGCONFIG_description])dnl
+AS_VAR_POPDEF([PKGCONFIG_requires])dnl
+AS_VAR_POPDEF([PKGCONFIG_pkglibs])dnl
+AS_VAR_POPDEF([PKGCONFIG_libs])dnl
+AS_VAR_POPDEF([PKGCONFIG_ldflags])dnl
+AS_VAR_POPDEF([PKGCONFIG_cppflags])dnl
+AS_VAR_POPDEF([PKGCONFIG_generate])dnl
+AS_VAR_POPDEF([PKGCONFIG_src_libdir])dnl
+AS_VAR_POPDEF([PKGCONFIG_src_headers])dnl
+])
diff --git a/polly/lib/External/isl/m4/ax_create_stdint_h.m4 b/polly/lib/External/isl/m4/ax_create_stdint_h.m4
new file mode 100644
index 00000000000..7b8216599e7
--- /dev/null
+++ b/polly/lib/External/isl/m4/ax_create_stdint_h.m4
@@ -0,0 +1,739 @@
+# ===========================================================================
+#           http://autoconf-archive.cryp.to/ax_create_stdint_h.html
+# ===========================================================================
+#
+# SYNOPSIS
+#
+#   AX_CREATE_STDINT_H [( HEADER-TO-GENERATE [, HEDERS-TO-CHECK])]
+#
+# DESCRIPTION
+#
+#   the "ISO C9X: 7.18 Integer types <stdint.h>" section requires the
+#   existence of an include file <stdint.h> that defines a set of typedefs,
+#   especially uint8_t,int32_t,uintptr_t. Many older installations will not
+#   provide this file, but some will have the very same definitions in
+#   <inttypes.h>. In other enviroments we can use the inet-types in
+#   <sys/types.h> which would define the typedefs int8_t and u_int8_t
+#   respectivly.
+#
+#   This macros will create a local "_stdint.h" or the headerfile given as
+#   an argument. In many cases that file will just "#include <stdint.h>" or
+#   "#include <inttypes.h>", while in other environments it will provide the
+#   set of basic 'stdint's definitions/typedefs:
+#
+#     int8_t,uint8_t,int16_t,uint16_t,int32_t,uint32_t,intptr_t,uintptr_t
+#     int_least32_t.. int_fast32_t.. intmax_t
+#
+#   which may or may not rely on the definitions of other files, or using
+#   the AC_CHECK_SIZEOF macro to determine the actual sizeof each type.
+#
+#   if your header files require the stdint-types you will want to create an
+#   installable file mylib-int.h that all your other installable header may
+#   include. So if you have a library package named "mylib", just use
+#
+#        AX_CREATE_STDINT_H(mylib-int.h)
+#
+#   in configure.ac and go to install that very header file in Makefile.am
+#   along with the other headers (mylib.h) - and the mylib-specific headers
+#   can simply use "#include <mylib-int.h>" to obtain the stdint-types.
+#
+#   Remember, if the system already had a valid <stdint.h>, the generated
+#   file will include it directly. No need for fuzzy HAVE_STDINT_H things...
+#   (oops, GCC 4.2.x has deliberatly disabled its stdint.h for non-c99
+#   compilation and the c99-mode is not the default. Therefore this macro
+#   will not use the compiler's stdint.h - please complain to the GCC
+#   developers).
+#
+# LAST MODIFICATION
+#
+#   2008-04-12
+#
+# COPYLEFT
+#
+#   Copyright (c) 2008 Guido U. Draheim <guidod@gmx.de>
+#
+#   This program is free software; you can redistribute it and/or modify it
+#   under the terms of the GNU General Public License as published by the
+#   Free Software Foundation; either version 2 of the License, or (at your
+#   option) any later version.
+#
+#   This program is distributed in the hope that it will be useful, but
+#   WITHOUT ANY WARRANTY; without even the implied warranty of
+#   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
+#   Public License for more details.
+#
+#   You should have received a copy of the GNU General Public License along
+#   with this program. If not, see <http://www.gnu.org/licenses/>.
+#
+#   As a special exception, the respective Autoconf Macro's copyright owner
+#   gives unlimited permission to copy, distribute and modify the configure
+#   scripts that are the output of Autoconf when processing the Macro. You
+#   need not follow the terms of the GNU General Public License when using
+#   or distributing such scripts, even though portions of the text of the
+#   Macro appear in them. The GNU General Public License (GPL) does govern
+#   all other use of the material that constitutes the Autoconf Macro.
+#
+#   This special exception to the GPL applies to versions of the Autoconf
+#   Macro released by the Autoconf Macro Archive. When you make and
+#   distribute a modified version of the Autoconf Macro, you may extend this
+#   special exception to the GPL to apply to your modified version as well.
+
+AC_DEFUN([AX_CHECK_DATA_MODEL],[
+   AC_CHECK_SIZEOF(char)
+   AC_CHECK_SIZEOF(short)
+   AC_CHECK_SIZEOF(int)
+   AC_CHECK_SIZEOF(long)
+   AC_CHECK_SIZEOF(void*)
+   ac_cv_char_data_model=""
+   ac_cv_char_data_model="$ac_cv_char_data_model$ac_cv_sizeof_char"
+   ac_cv_char_data_model="$ac_cv_char_data_model$ac_cv_sizeof_short"
+   ac_cv_char_data_model="$ac_cv_char_data_model$ac_cv_sizeof_int"
+   ac_cv_long_data_model=""
+   ac_cv_long_data_model="$ac_cv_long_data_model$ac_cv_sizeof_int"
+   ac_cv_long_data_model="$ac_cv_long_data_model$ac_cv_sizeof_long"
+   ac_cv_long_data_model="$ac_cv_long_data_model$ac_cv_sizeof_voidp"
+   AC_MSG_CHECKING([data model])
+   case "$ac_cv_char_data_model/$ac_cv_long_data_model" in
+    122/242)     ac_cv_data_model="IP16"  ; n="standard 16bit machine" ;;
+    122/244)     ac_cv_data_model="LP32"  ; n="standard 32bit machine" ;;
+    122/*)       ac_cv_data_model="i16"   ; n="unusual int16 model" ;;
+    124/444)     ac_cv_data_model="ILP32" ; n="standard 32bit unixish" ;;
+    124/488)     ac_cv_data_model="LP64"  ; n="standard 64bit unixish" ;;
+    124/448)     ac_cv_data_model="LLP64" ; n="unusual 64bit unixish" ;;
+    124/*)       ac_cv_data_model="i32"   ; n="unusual int32 model" ;;
+    128/888)     ac_cv_data_model="ILP64" ; n="unusual 64bit numeric" ;;
+    128/*)       ac_cv_data_model="i64"   ; n="unusual int64 model" ;;
+    222/*2)      ac_cv_data_model="DSP16" ; n="strict 16bit dsptype" ;;
+    333/*3)      ac_cv_data_model="DSP24" ; n="strict 24bit dsptype" ;;
+    444/*4)      ac_cv_data_model="DSP32" ; n="strict 32bit dsptype" ;;
+    666/*6)      ac_cv_data_model="DSP48" ; n="strict 48bit dsptype" ;;
+    888/*8)      ac_cv_data_model="DSP64" ; n="strict 64bit dsptype" ;;
+    222/*|333/*|444/*|666/*|888/*) :
+                 ac_cv_data_model="iDSP"  ; n="unusual dsptype" ;;
+     *)          ac_cv_data_model="none"  ; n="very unusual model" ;;
+   esac
+   AC_MSG_RESULT([$ac_cv_data_model ($ac_cv_long_data_model, $n)])
+])
+
+dnl AX_CHECK_HEADER_STDINT_X([HEADERLIST][,ACTION-IF])
+AC_DEFUN([AX_CHECK_HEADER_STDINT_X],[
+AC_CACHE_CHECK([for stdint uintptr_t], [ac_cv_header_stdint_x],[
+ ac_cv_header_stdint_x="" # the 1997 typedefs (inttypes.h)
+  AC_MSG_RESULT([(..)])
+  for i in m4_ifval([$1],[$1],[stdint.h inttypes.h sys/inttypes.h sys/types.h])
+  do
+   unset ac_cv_type_uintptr_t
+   unset ac_cv_type_uint64_t
+   AC_CHECK_TYPE(uintptr_t,[ac_cv_header_stdint_x=$i],continue,[#include <$i>])
+   AC_CHECK_TYPE(uint64_t,[and64="/uint64_t"],[and64=""],[#include<$i>])
+   m4_ifvaln([$2],[$2]) break
+  done
+  AC_MSG_CHECKING([for stdint uintptr_t])
+ ])
+])
+
+AC_DEFUN([AX_CHECK_HEADER_STDINT_O],[
+AC_CACHE_CHECK([for stdint uint32_t], [ac_cv_header_stdint_o],[
+ ac_cv_header_stdint_o="" # the 1995 typedefs (sys/inttypes.h)
+  AC_MSG_RESULT([(..)])
+  for i in m4_ifval([$1],[$1],[inttypes.h sys/inttypes.h sys/types.h stdint.h])
+  do
+   unset ac_cv_type_uint32_t
+   unset ac_cv_type_uint64_t
+   AC_CHECK_TYPE(uint32_t,[ac_cv_header_stdint_o=$i],continue,[#include <$i>])
+   AC_CHECK_TYPE(uint64_t,[and64="/uint64_t"],[and64=""],[#include<$i>])
+   m4_ifvaln([$2],[$2]) break
+   break;
+  done
+  AC_MSG_CHECKING([for stdint uint32_t])
+ ])
+])
+
+AC_DEFUN([AX_CHECK_HEADER_STDINT_U],[
+AC_CACHE_CHECK([for stdint u_int32_t], [ac_cv_header_stdint_u],[
+ ac_cv_header_stdint_u="" # the BSD typedefs (sys/types.h)
+  AC_MSG_RESULT([(..)])
+  for i in m4_ifval([$1],[$1],[sys/types.h inttypes.h sys/inttypes.h]) ; do
+   unset ac_cv_type_u_int32_t
+   unset ac_cv_type_u_int64_t
+   AC_CHECK_TYPE(u_int32_t,[ac_cv_header_stdint_u=$i],continue,[#include <$i>])
+   AC_CHECK_TYPE(u_int64_t,[and64="/u_int64_t"],[and64=""],[#include<$i>])
+   m4_ifvaln([$2],[$2]) break
+   break;
+  done
+  AC_MSG_CHECKING([for stdint u_int32_t])
+ ])
+])
+
+AC_DEFUN([AX_CREATE_STDINT_H],
+[# ------ AX CREATE STDINT H -------------------------------------
+AC_MSG_CHECKING([for stdint types])
+ac_stdint_h=`echo ifelse($1, , _stdint.h, $1)`
+# try to shortcircuit - if the default include path of the compiler
+# can find a "stdint.h" header then we assume that all compilers can.
+AC_CACHE_VAL([ac_cv_header_stdint_t],[
+old_CXXFLAGS="$CXXFLAGS" ; CXXFLAGS=""
+old_CPPFLAGS="$CPPFLAGS" ; CPPFLAGS=""
+old_CFLAGS="$CFLAGS"     ; CFLAGS=""
+AC_TRY_COMPILE([#include <stdint.h>],[int_least32_t v = 0;],
+[ac_cv_stdint_result="(assuming C99 compatible system)"
+ ac_cv_header_stdint_t="stdint.h"; ],
+[ac_cv_header_stdint_t=""])
+if test "$GCC" = "yes" && test ".$ac_cv_header_stdint_t" = "."; then
+CFLAGS="-std=c99"
+AC_TRY_COMPILE([#include <stdint.h>],[int_least32_t v = 0;],
+[AC_MSG_WARN(your GCC compiler has a defunct stdint.h for its default-mode)])
+fi
+CXXFLAGS="$old_CXXFLAGS"
+CPPFLAGS="$old_CPPFLAGS"
+CFLAGS="$old_CFLAGS" ])
+
+v="... $ac_cv_header_stdint_h"
+if test "$ac_stdint_h" = "stdint.h" ; then
+ AC_MSG_RESULT([(are you sure you want them in ./stdint.h?)])
+elif test "$ac_stdint_h" = "inttypes.h" ; then
+ AC_MSG_RESULT([(are you sure you want them in ./inttypes.h?)])
+elif test "_$ac_cv_header_stdint_t" = "_" ; then
+ AC_MSG_RESULT([(putting them into $ac_stdint_h)$v])
+else
+ ac_cv_header_stdint="$ac_cv_header_stdint_t"
+ AC_MSG_RESULT([$ac_cv_header_stdint (shortcircuit)])
+fi
+
+if test "_$ac_cv_header_stdint_t" = "_" ; then # can not shortcircuit..
+
+dnl .....intro message done, now do a few system checks.....
+dnl btw, all old CHECK_TYPE macros do automatically "DEFINE" a type,
+dnl therefore we use the autoconf implementation detail CHECK_TYPE_NEW
+dnl instead that is triggered with 3 or more arguments (see types.m4)
+
+inttype_headers=`echo $2 | sed -e 's/,/ /g'`
+
+ac_cv_stdint_result="(no helpful system typedefs seen)"
+AX_CHECK_HEADER_STDINT_X(dnl
+   stdint.h inttypes.h sys/inttypes.h $inttype_headers,
+   ac_cv_stdint_result="(seen uintptr_t$and64 in $i)")
+
+if test "_$ac_cv_header_stdint_x" = "_" ; then
+AX_CHECK_HEADER_STDINT_O(dnl,
+   inttypes.h sys/inttypes.h stdint.h $inttype_headers,
+   ac_cv_stdint_result="(seen uint32_t$and64 in $i)")
+fi
+
+if test "_$ac_cv_header_stdint_x" = "_" ; then
+if test "_$ac_cv_header_stdint_o" = "_" ; then
+AX_CHECK_HEADER_STDINT_U(dnl,
+   sys/types.h inttypes.h sys/inttypes.h $inttype_headers,
+   ac_cv_stdint_result="(seen u_int32_t$and64 in $i)")
+fi fi
+
+dnl if there was no good C99 header file, do some typedef checks...
+if test "_$ac_cv_header_stdint_x" = "_" ; then
+   AC_MSG_CHECKING([for stdint datatype model])
+   AC_MSG_RESULT([(..)])
+   AX_CHECK_DATA_MODEL
+fi
+
+if test "_$ac_cv_header_stdint_x" != "_" ; then
+   ac_cv_header_stdint="$ac_cv_header_stdint_x"
+elif  test "_$ac_cv_header_stdint_o" != "_" ; then
+   ac_cv_header_stdint="$ac_cv_header_stdint_o"
+elif  test "_$ac_cv_header_stdint_u" != "_" ; then
+   ac_cv_header_stdint="$ac_cv_header_stdint_u"
+else
+   ac_cv_header_stdint="stddef.h"
+fi
+
+AC_MSG_CHECKING([for extra inttypes in chosen header])
+AC_MSG_RESULT([($ac_cv_header_stdint)])
+dnl see if int_least and int_fast types are present in _this_ header.
+unset ac_cv_type_int_least32_t
+unset ac_cv_type_int_fast32_t
+AC_CHECK_TYPE(int_least32_t,,,[#include <$ac_cv_header_stdint>])
+AC_CHECK_TYPE(int_fast32_t,,,[#include<$ac_cv_header_stdint>])
+AC_CHECK_TYPE(intmax_t,,,[#include <$ac_cv_header_stdint>])
+
+fi # shortcircut to system "stdint.h"
+# ------------------ PREPARE VARIABLES ------------------------------
+if test "$GCC" = "yes" ; then
+ac_cv_stdint_message="using gnu compiler "`$CC --version | head -1`
+else
+ac_cv_stdint_message="using $CC"
+fi
+
+AC_MSG_RESULT([make use of $ac_cv_header_stdint in $ac_stdint_h dnl
+$ac_cv_stdint_result])
+
+dnl -----------------------------------------------------------------
+# ----------------- DONE inttypes.h checks START header -------------
+AC_CONFIG_COMMANDS([$ac_stdint_h],[
+AC_MSG_NOTICE(creating $ac_stdint_h : $_ac_stdint_h)
+ac_stdint=$tmp/_stdint.h
+
+echo "#ifndef" $_ac_stdint_h >$ac_stdint
+echo "#define" $_ac_stdint_h "1" >>$ac_stdint
+echo "#ifndef" _GENERATED_STDINT_H >>$ac_stdint
+echo "#define" _GENERATED_STDINT_H '"'$PACKAGE $VERSION'"' >>$ac_stdint
+echo "/* generated $ac_cv_stdint_message */" >>$ac_stdint
+if test "_$ac_cv_header_stdint_t" != "_" ; then
+echo "#define _STDINT_HAVE_STDINT_H" "1" >>$ac_stdint
+echo "#include <stdint.h>" >>$ac_stdint
+echo "#endif" >>$ac_stdint
+echo "#endif" >>$ac_stdint
+else
+
+cat >>$ac_stdint <<STDINT_EOF
+
+/* ................... shortcircuit part ........................... */
+
+#if defined HAVE_STDINT_H || defined _STDINT_HAVE_STDINT_H
+#include <stdint.h>
+#else
+#include <stddef.h>
+
+/* .................... configured part ............................ */
+
+STDINT_EOF
+
+echo "/* whether we have a C99 compatible stdint header file */" >>$ac_stdint
+if test "_$ac_cv_header_stdint_x" != "_" ; then
+  ac_header="$ac_cv_header_stdint_x"
+  echo "#define _STDINT_HEADER_INTPTR" '"'"$ac_header"'"' >>$ac_stdint
+else
+  echo "/* #undef _STDINT_HEADER_INTPTR */" >>$ac_stdint
+fi
+
+echo "/* whether we have a C96 compatible inttypes header file */" >>$ac_stdint
+if  test "_$ac_cv_header_stdint_o" != "_" ; then
+  ac_header="$ac_cv_header_stdint_o"
+  echo "#define _STDINT_HEADER_UINT32" '"'"$ac_header"'"' >>$ac_stdint
+else
+  echo "/* #undef _STDINT_HEADER_UINT32 */" >>$ac_stdint
+fi
+
+echo "/* whether we have a BSD compatible inet types header */" >>$ac_stdint
+if  test "_$ac_cv_header_stdint_u" != "_" ; then
+  ac_header="$ac_cv_header_stdint_u"
+  echo "#define _STDINT_HEADER_U_INT32" '"'"$ac_header"'"' >>$ac_stdint
+else
+  echo "/* #undef _STDINT_HEADER_U_INT32 */" >>$ac_stdint
+fi
+
+echo "" >>$ac_stdint
+
+if test "_$ac_header" != "_" ; then if test "$ac_header" != "stddef.h" ; then
+  echo "#include <$ac_header>" >>$ac_stdint
+  echo "" >>$ac_stdint
+fi fi
+
+echo "/* which 64bit typedef has been found */" >>$ac_stdint
+if test "$ac_cv_type_uint64_t" = "yes" ; then
+echo "#define   _STDINT_HAVE_UINT64_T" "1"  >>$ac_stdint
+else
+echo "/* #undef _STDINT_HAVE_UINT64_T */" >>$ac_stdint
+fi
+if test "$ac_cv_type_u_int64_t" = "yes" ; then
+echo "#define   _STDINT_HAVE_U_INT64_T" "1"  >>$ac_stdint
+else
+echo "/* #undef _STDINT_HAVE_U_INT64_T */" >>$ac_stdint
+fi
+echo "" >>$ac_stdint
+
+echo "/* which type model has been detected */" >>$ac_stdint
+if test "_$ac_cv_char_data_model" != "_" ; then
+echo "#define   _STDINT_CHAR_MODEL" "$ac_cv_char_data_model" >>$ac_stdint
+echo "#define   _STDINT_LONG_MODEL" "$ac_cv_long_data_model" >>$ac_stdint
+else
+echo "/* #undef _STDINT_CHAR_MODEL // skipped */" >>$ac_stdint
+echo "/* #undef _STDINT_LONG_MODEL // skipped */" >>$ac_stdint
+fi
+echo "" >>$ac_stdint
+
+echo "/* whether int_least types were detected */" >>$ac_stdint
+if test "$ac_cv_type_int_least32_t" = "yes"; then
+echo "#define   _STDINT_HAVE_INT_LEAST32_T" "1"  >>$ac_stdint
+else
+echo "/* #undef _STDINT_HAVE_INT_LEAST32_T */" >>$ac_stdint
+fi
+echo "/* whether int_fast types were detected */" >>$ac_stdint
+if test "$ac_cv_type_int_fast32_t" = "yes"; then
+echo "#define   _STDINT_HAVE_INT_FAST32_T" "1" >>$ac_stdint
+else
+echo "/* #undef _STDINT_HAVE_INT_FAST32_T */" >>$ac_stdint
+fi
+echo "/* whether intmax_t type was detected */" >>$ac_stdint
+if test "$ac_cv_type_intmax_t" = "yes"; then
+echo "#define   _STDINT_HAVE_INTMAX_T" "1" >>$ac_stdint
+else
+echo "/* #undef _STDINT_HAVE_INTMAX_T */" >>$ac_stdint
+fi
+echo "" >>$ac_stdint
+
+  cat >>$ac_stdint <<STDINT_EOF
+/* .................... detections part ............................ */
+
+/* whether we need to define bitspecific types from compiler base types */
+#ifndef _STDINT_HEADER_INTPTR
+#ifndef _STDINT_HEADER_UINT32
+#ifndef _STDINT_HEADER_U_INT32
+#define _STDINT_NEED_INT_MODEL_T
+#else
+#define _STDINT_HAVE_U_INT_TYPES
+#endif
+#endif
+#endif
+
+#ifdef _STDINT_HAVE_U_INT_TYPES
+#undef _STDINT_NEED_INT_MODEL_T
+#endif
+
+#ifdef  _STDINT_CHAR_MODEL
+#if     _STDINT_CHAR_MODEL+0 == 122 || _STDINT_CHAR_MODEL+0 == 124
+#ifndef _STDINT_BYTE_MODEL
+#define _STDINT_BYTE_MODEL 12
+#endif
+#endif
+#endif
+
+#ifndef _STDINT_HAVE_INT_LEAST32_T
+#define _STDINT_NEED_INT_LEAST_T
+#endif
+
+#ifndef _STDINT_HAVE_INT_FAST32_T
+#define _STDINT_NEED_INT_FAST_T
+#endif
+
+#ifndef _STDINT_HEADER_INTPTR
+#define _STDINT_NEED_INTPTR_T
+#ifndef _STDINT_HAVE_INTMAX_T
+#define _STDINT_NEED_INTMAX_T
+#endif
+#endif
+
+
+/* .................... definition part ............................ */
+
+/* some system headers have good uint64_t */
+#ifndef _HAVE_UINT64_T
+#if     defined _STDINT_HAVE_UINT64_T  || defined HAVE_UINT64_T
+#define _HAVE_UINT64_T
+#elif   defined _STDINT_HAVE_U_INT64_T || defined HAVE_U_INT64_T
+#define _HAVE_UINT64_T
+typedef u_int64_t uint64_t;
+#endif
+#endif
+
+#ifndef _HAVE_UINT64_T
+/* .. here are some common heuristics using compiler runtime specifics */
+#if defined __STDC_VERSION__ && defined __STDC_VERSION__ >= 199901L
+#define _HAVE_UINT64_T
+#define _HAVE_LONGLONG_UINT64_T
+typedef long long int64_t;
+typedef unsigned long long uint64_t;
+
+#elif !defined __STRICT_ANSI__
+#if defined _MSC_VER || defined __WATCOMC__ || defined __BORLANDC__
+#define _HAVE_UINT64_T
+typedef __int64 int64_t;
+typedef unsigned __int64 uint64_t;
+
+#elif defined __GNUC__ || defined __MWERKS__ || defined __ELF__
+/* note: all ELF-systems seem to have loff-support which needs 64-bit */
+#if !defined _NO_LONGLONG
+#define _HAVE_UINT64_T
+#define _HAVE_LONGLONG_UINT64_T
+typedef long long int64_t;
+typedef unsigned long long uint64_t;
+#endif
+
+#elif defined __alpha || (defined __mips && defined _ABIN32)
+#if !defined _NO_LONGLONG
+typedef long int64_t;
+typedef unsigned long uint64_t;
+#endif
+  /* compiler/cpu type to define int64_t */
+#endif
+#endif
+#endif
+
+#if defined _STDINT_HAVE_U_INT_TYPES
+/* int8_t int16_t int32_t defined by inet code, redeclare the u_intXX types */
+typedef u_int8_t uint8_t;
+typedef u_int16_t uint16_t;
+typedef u_int32_t uint32_t;
+
+/* glibc compatibility */
+#ifndef __int8_t_defined
+#define __int8_t_defined
+#endif
+#endif
+
+#ifdef _STDINT_NEED_INT_MODEL_T
+/* we must guess all the basic types. Apart from byte-adressable system, */
+/* there a few 32-bit-only dsp-systems that we guard with BYTE_MODEL 8-} */
+/* (btw, those nibble-addressable systems are way off, or so we assume) */
+
+dnl   /* have a look at "64bit and data size neutrality" at */
+dnl   /* http://unix.org/version2/whatsnew/login_64bit.html */
+dnl   /* (the shorthand "ILP" types always have a "P" part) */
+
+#if defined _STDINT_BYTE_MODEL
+#if _STDINT_LONG_MODEL+0 == 242
+/* 2:4:2 =  IP16 = a normal 16-bit system                */
+typedef unsigned char   uint8_t;
+typedef unsigned short  uint16_t;
+typedef unsigned long   uint32_t;
+#ifndef __int8_t_defined
+#define __int8_t_defined
+typedef          char    int8_t;
+typedef          short   int16_t;
+typedef          long    int32_t;
+#endif
+#elif _STDINT_LONG_MODEL+0 == 244 || _STDINT_LONG_MODEL == 444
+/* 2:4:4 =  LP32 = a 32-bit system derived from a 16-bit */
+/* 4:4:4 = ILP32 = a normal 32-bit system                */
+typedef unsigned char   uint8_t;
+typedef unsigned short  uint16_t;
+typedef unsigned int    uint32_t;
+#ifndef __int8_t_defined
+#define __int8_t_defined
+typedef          char    int8_t;
+typedef          short   int16_t;
+typedef          int     int32_t;
+#endif
+#elif _STDINT_LONG_MODEL+0 == 484 || _STDINT_LONG_MODEL+0 == 488
+/* 4:8:4 =  IP32 = a 32-bit system prepared for 64-bit    */
+/* 4:8:8 =  LP64 = a normal 64-bit system                 */
+typedef unsigned char   uint8_t;
+typedef unsigned short  uint16_t;
+typedef unsigned int    uint32_t;
+#ifndef __int8_t_defined
+#define __int8_t_defined
+typedef          char    int8_t;
+typedef          short   int16_t;
+typedef          int     int32_t;
+#endif
+/* this system has a "long" of 64bit */
+#ifndef _HAVE_UINT64_T
+#define _HAVE_UINT64_T
+typedef unsigned long   uint64_t;
+typedef          long    int64_t;
+#endif
+#elif _STDINT_LONG_MODEL+0 == 448
+/*      LLP64   a 64-bit system derived from a 32-bit system */
+typedef unsigned char   uint8_t;
+typedef unsigned short  uint16_t;
+typedef unsigned int    uint32_t;
+#ifndef __int8_t_defined
+#define __int8_t_defined
+typedef          char    int8_t;
+typedef          short   int16_t;
+typedef          int     int32_t;
+#endif
+/* assuming the system has a "long long" */
+#ifndef _HAVE_UINT64_T
+#define _HAVE_UINT64_T
+#define _HAVE_LONGLONG_UINT64_T
+typedef unsigned long long uint64_t;
+typedef          long long  int64_t;
+#endif
+#else
+#define _STDINT_NO_INT32_T
+#endif
+#else
+#define _STDINT_NO_INT8_T
+#define _STDINT_NO_INT32_T
+#endif
+#endif
+
+/*
+ * quote from SunOS-5.8 sys/inttypes.h:
+ * Use at your own risk.  As of February 1996, the committee is squarely
+ * behind the fixed sized types; the "least" and "fast" types are still being
+ * discussed.  The probability that the "fast" types may be removed before
+ * the standard is finalized is high enough that they are not currently
+ * implemented.
+ */
+
+#if defined _STDINT_NEED_INT_LEAST_T
+typedef  int8_t    int_least8_t;
+typedef  int16_t   int_least16_t;
+typedef  int32_t   int_least32_t;
+#ifdef _HAVE_UINT64_T
+typedef  int64_t   int_least64_t;
+#endif
+
+typedef uint8_t   uint_least8_t;
+typedef uint16_t  uint_least16_t;
+typedef uint32_t  uint_least32_t;
+#ifdef _HAVE_UINT64_T
+typedef uint64_t  uint_least64_t;
+#endif
+  /* least types */
+#endif
+
+#if defined _STDINT_NEED_INT_FAST_T
+typedef  int8_t    int_fast8_t;
+typedef  int       int_fast16_t;
+typedef  int32_t   int_fast32_t;
+#ifdef _HAVE_UINT64_T
+typedef  int64_t   int_fast64_t;
+#endif
+
+typedef uint8_t   uint_fast8_t;
+typedef unsigned  uint_fast16_t;
+typedef uint32_t  uint_fast32_t;
+#ifdef _HAVE_UINT64_T
+typedef uint64_t  uint_fast64_t;
+#endif
+  /* fast types */
+#endif
+
+#ifdef _STDINT_NEED_INTMAX_T
+#ifdef _HAVE_UINT64_T
+typedef  int64_t       intmax_t;
+typedef uint64_t      uintmax_t;
+#else
+typedef          long  intmax_t;
+typedef unsigned long uintmax_t;
+#endif
+#endif
+
+#ifdef _STDINT_NEED_INTPTR_T
+#ifndef __intptr_t_defined
+#define __intptr_t_defined
+/* we encourage using "long" to store pointer values, never use "int" ! */
+#if   _STDINT_LONG_MODEL+0 == 242 || _STDINT_LONG_MODEL+0 == 484
+typedef  unsigned int   uintptr_t;
+typedef           int    intptr_t;
+#elif _STDINT_LONG_MODEL+0 == 244 || _STDINT_LONG_MODEL+0 == 444
+typedef  unsigned long  uintptr_t;
+typedef           long   intptr_t;
+#elif _STDINT_LONG_MODEL+0 == 448 && defined _HAVE_UINT64_T
+typedef        uint64_t uintptr_t;
+typedef         int64_t  intptr_t;
+#else /* matches typical system types ILP32 and LP64 - but not IP16 or LLP64 */
+typedef  unsigned long  uintptr_t;
+typedef           long   intptr_t;
+#endif
+#endif
+#endif
+
+/* The ISO C99 standard specifies that in C++ implementations these
+   should only be defined if explicitly requested.  */
+#if !defined __cplusplus || defined __STDC_CONSTANT_MACROS
+#ifndef UINT32_C
+
+/* Signed.  */
+# define INT8_C(c)      c
+# define INT16_C(c)     c
+# define INT32_C(c)     c
+# ifdef _HAVE_LONGLONG_UINT64_T
+#  define INT64_C(c)    c ## L
+# else
+#  define INT64_C(c)    c ## LL
+# endif
+
+/* Unsigned.  */
+# define UINT8_C(c)     c ## U
+# define UINT16_C(c)    c ## U
+# define UINT32_C(c)    c ## U
+# ifdef _HAVE_LONGLONG_UINT64_T
+#  define UINT64_C(c)   c ## UL
+# else
+#  define UINT64_C(c)   c ## ULL
+# endif
+
+/* Maximal type.  */
+# ifdef _HAVE_LONGLONG_UINT64_T
+#  define INTMAX_C(c)   c ## L
+#  define UINTMAX_C(c)  c ## UL
+# else
+#  define INTMAX_C(c)   c ## LL
+#  define UINTMAX_C(c)  c ## ULL
+# endif
+
+  /* literalnumbers */
+#endif
+#endif
+
+/* These limits are merily those of a two complement byte-oriented system */
+
+/* Minimum of signed integral types.  */
+# define INT8_MIN               (-128)
+# define INT16_MIN              (-32767-1)
+# define INT32_MIN              (-2147483647-1)
+#ifndef INT64_MIN
+# define INT64_MIN              (-__INT64_C(9223372036854775807)-1)
+#endif
+/* Maximum of signed integral types.  */
+# define INT8_MAX               (127)
+# define INT16_MAX              (32767)
+# define INT32_MAX              (2147483647)
+#ifndef INT64_MAX
+# define INT64_MAX              (__INT64_C(9223372036854775807))
+#endif
+
+/* Maximum of unsigned integral types.  */
+#ifndef UINT8_MAX
+# define UINT8_MAX              (255)
+#endif
+#ifndef UINT16_MAX
+# define UINT16_MAX             (65535)
+#endif
+# define UINT32_MAX             (4294967295U)
+#ifndef UINT64_MAX
+# define UINT64_MAX             (__UINT64_C(18446744073709551615))
+#endif
+
+/* Minimum of signed integral types having a minimum size.  */
+# define INT_LEAST8_MIN         INT8_MIN
+# define INT_LEAST16_MIN        INT16_MIN
+# define INT_LEAST32_MIN        INT32_MIN
+# define INT_LEAST64_MIN        INT64_MIN
+/* Maximum of signed integral types having a minimum size.  */
+# define INT_LEAST8_MAX         INT8_MAX
+# define INT_LEAST16_MAX        INT16_MAX
+# define INT_LEAST32_MAX        INT32_MAX
+# define INT_LEAST64_MAX        INT64_MAX
+
+/* Maximum of unsigned integral types having a minimum size.  */
+# define UINT_LEAST8_MAX        UINT8_MAX
+# define UINT_LEAST16_MAX       UINT16_MAX
+# define UINT_LEAST32_MAX       UINT32_MAX
+# define UINT_LEAST64_MAX       UINT64_MAX
+
+  /* shortcircuit*/
+#endif
+  /* once */
+#endif
+#endif
+STDINT_EOF
+fi
+    if cmp -s $ac_stdint_h $ac_stdint 2>/dev/null; then
+      AC_MSG_NOTICE([$ac_stdint_h is unchanged])
+    else
+      ac_dir=`AS_DIRNAME(["$ac_stdint_h"])`
+      AS_MKDIR_P(["$ac_dir"])
+      rm -f $ac_stdint_h
+      mv $ac_stdint $ac_stdint_h
+    fi
+],[# variables for create stdint.h replacement
+PACKAGE="$PACKAGE"
+VERSION="$VERSION"
+ac_stdint_h="$ac_stdint_h"
+_ac_stdint_h=AS_TR_CPP(_$PACKAGE-$ac_stdint_h)
+ac_cv_stdint_message="$ac_cv_stdint_message"
+ac_cv_header_stdint_t="$ac_cv_header_stdint_t"
+ac_cv_header_stdint_x="$ac_cv_header_stdint_x"
+ac_cv_header_stdint_o="$ac_cv_header_stdint_o"
+ac_cv_header_stdint_u="$ac_cv_header_stdint_u"
+ac_cv_type_uint64_t="$ac_cv_type_uint64_t"
+ac_cv_type_u_int64_t="$ac_cv_type_u_int64_t"
+ac_cv_char_data_model="$ac_cv_char_data_model"
+ac_cv_long_data_model="$ac_cv_long_data_model"
+ac_cv_type_int_least32_t="$ac_cv_type_int_least32_t"
+ac_cv_type_int_fast32_t="$ac_cv_type_int_fast32_t"
+ac_cv_type_intmax_t="$ac_cv_type_intmax_t"
+])
+])
diff --git a/polly/lib/External/isl/m4/ax_detect_git_head.m4 b/polly/lib/External/isl/m4/ax_detect_git_head.m4
new file mode 100644
index 00000000000..31a6fe32945
--- /dev/null
+++ b/polly/lib/External/isl/m4/ax_detect_git_head.m4
@@ -0,0 +1,32 @@
+AC_DEFUN([AX_DETECT_GIT_HEAD], [
+	AC_SUBST(GIT_HEAD_ID)
+	AC_SUBST(GIT_HEAD)
+	AC_SUBST(GIT_HEAD_VERSION)
+	if test -f $srcdir/.git; then
+		gitdir=`GIT_DIR=$srcdir/.git git rev-parse --git-dir`
+		GIT_HEAD="$gitdir/index"
+		GIT_REPO="$gitdir"
+		GIT_HEAD_ID=`GIT_DIR=$GIT_REPO git describe --always`
+	elif test -f $srcdir/.git/HEAD; then
+		GIT_HEAD="$srcdir/.git/index"
+		GIT_REPO="$srcdir/.git"
+		GIT_HEAD_ID=`GIT_DIR=$GIT_REPO git describe --always`
+	elif test -f $srcdir/GIT_HEAD_ID; then
+		GIT_HEAD_ID=`cat $srcdir/GIT_HEAD_ID`
+	else
+		mysrcdir=`(cd $srcdir; pwd)`
+		head=`basename $mysrcdir | sed -e 's/.*-//'`
+		head2=`echo $head | sed -e 's/[^0-9a-f]//'`
+		head3=`echo $head2 | sed -e 's/........................................//'`
+		if test "x$head3" = "x" -a "x$head" = "x$head2"; then
+			GIT_HEAD_ID="$head"
+		else
+			GIT_HEAD_ID="UNKNOWN"
+		fi
+	fi
+	if test -z "$GIT_REPO" ; then
+		GIT_HEAD_VERSION="$GIT_HEAD_ID"
+	else
+	    GIT_HEAD_VERSION="\`GIT_DIR=$GIT_REPO git describe --always\`"
+	fi
+])
diff --git a/polly/lib/External/isl/m4/ax_detect_gmp.m4 b/polly/lib/External/isl/m4/ax_detect_gmp.m4
new file mode 100644
index 00000000000..43fa20dc531
--- /dev/null
+++ b/polly/lib/External/isl/m4/ax_detect_gmp.m4
@@ -0,0 +1,48 @@
+AC_DEFUN([AX_DETECT_GMP], [
+AC_DEFINE([USE_GMP_FOR_MP], [], [use gmp to implement isl_int])
+AX_SUBMODULE(gmp,system|build,system)
+case "$with_gmp" in
+system)
+	if test "x$with_gmp_prefix" != "x"; then
+		isl_configure_args="$isl_configure_args --with-gmp=$with_gmp_prefix"
+		MP_CPPFLAGS="-I$with_gmp_prefix/include"
+		MP_LDFLAGS="-L$with_gmp_prefix/lib"
+	fi
+	MP_LIBS=-lgmp
+	SAVE_CPPFLAGS="$CPPFLAGS"
+	SAVE_LDFLAGS="$LDFLAGS"
+	SAVE_LIBS="$LIBS"
+	CPPFLAGS="$MP_CPPFLAGS $CPPFLAGS"
+	LDFLAGS="$MP_LDFLAGS $LDFLAGS"
+	LIBS="$MP_LIBS $LIBS"
+	AC_CHECK_HEADER([gmp.h], [], [AC_ERROR([gmp.h header not found])])
+	AC_CHECK_LIB([gmp], [main], [], [AC_ERROR([gmp library not found])])
+	AC_LINK_IFELSE([AC_LANG_PROGRAM([[#include <gmp.h>]], [[
+		mpz_t n, d;
+		if (mpz_divisible_p(n, d))
+			mpz_divexact_ui(n, n, 4);
+	]])], [], [AC_ERROR([gmp library too old])])
+	CPPFLAGS="$SAVE_CPPFLAGS"
+	LDFLAGS="$SAVE_LDFLAGS"
+	LIBS="$SAVE_LIBS"
+	;;
+build)
+	MP_CPPFLAGS="-I$gmp_srcdir -I$with_gmp_builddir"
+	MP_LIBS="$with_gmp_builddir/libgmp.la"
+	;;
+esac
+SAVE_CPPFLAGS="$CPPFLAGS"
+SAVE_LDFLAGS="$LDFLAGS"
+SAVE_LIBS="$LIBS"
+CPPFLAGS="$MP_CPPFLAGS $CPPFLAGS"
+LDFLAGS="$MP_LDFLAGS $LDFLAGS"
+LIBS="$MP_LIBS $LIBS"
+need_get_memory_functions=false
+AC_CHECK_DECLS(mp_get_memory_functions,[],[
+	need_get_memory_functions=true
+],[#include <gmp.h>])
+CPPFLAGS="$SAVE_CPPFLAGS"
+LDFLAGS="$SAVE_LDFLAGS"
+LIBS="$SAVE_LIBS"
+AM_CONDITIONAL(NEED_GET_MEMORY_FUNCTIONS, test x$need_get_memory_functions = xtrue)
+])
diff --git a/polly/lib/External/isl/m4/ax_detect_imath.m4 b/polly/lib/External/isl/m4/ax_detect_imath.m4
new file mode 100644
index 00000000000..9c30a614e3f
--- /dev/null
+++ b/polly/lib/External/isl/m4/ax_detect_imath.m4
@@ -0,0 +1,15 @@
+AC_DEFUN([AX_DETECT_IMATH], [
+AC_DEFINE([USE_IMATH_FOR_MP], [], [use imath to implement isl_int])
+
+MP_CPPFLAGS="-I$srcdir/imath_wrap"
+MP_LDFLAGS=""
+MP_LIBS=""
+
+SAVE_CPPFLAGS="$CPPFLAGS"
+CPPFLAGS="$MP_CPPFLAGS $CPPFLAGS"
+AC_CHECK_HEADER([imath.h], [], [AC_ERROR([imath.h header not found])])
+AC_CHECK_HEADER([gmp_compat.h], [], [AC_ERROR([gmp_compat.h header not found])])
+CPPFLAGS="$SAVE_CPPFLAGS"
+
+AM_CONDITIONAL(NEED_GET_MEMORY_FUNCTIONS, test x = xfalse)
+])
diff --git a/polly/lib/External/isl/m4/ax_gcc_archflag.m4 b/polly/lib/External/isl/m4/ax_gcc_archflag.m4
new file mode 100644
index 00000000000..dedeef424e9
--- /dev/null
+++ b/polly/lib/External/isl/m4/ax_gcc_archflag.m4
@@ -0,0 +1,213 @@
+# ===========================================================================
+#         http://www.nongnu.org/autoconf-archive/ax_gcc_archflag.html
+# ===========================================================================
+#
+# SYNOPSIS
+#
+#   AX_GCC_ARCHFLAG([PORTABLE?], [ACTION-SUCCESS], [ACTION-FAILURE])
+#
+# DESCRIPTION
+#
+#   This macro tries to guess the "native" arch corresponding to the target
+#   architecture for use with gcc's -march=arch or -mtune=arch flags. If
+#   found, the cache variable $ax_cv_gcc_archflag is set to this flag and
+#   ACTION-SUCCESS is executed; otherwise $ax_cv_gcc_archflag is is set to
+#   "unknown" and ACTION-FAILURE is executed. The default ACTION-SUCCESS is
+#   to add $ax_cv_gcc_archflag to the end of $CFLAGS.
+#
+#   PORTABLE? should be either [yes] (default) or [no]. In the former case,
+#   the flag is set to -mtune (or equivalent) so that the architecture is
+#   only used for tuning, but the instruction set used is still portable. In
+#   the latter case, the flag is set to -march (or equivalent) so that
+#   architecture-specific instructions are enabled.
+#
+#   The user can specify --with-gcc-arch=<arch> in order to override the
+#   macro's choice of architecture, or --without-gcc-arch to disable this.
+#
+#   When cross-compiling, or if $CC is not gcc, then ACTION-FAILURE is
+#   called unless the user specified --with-gcc-arch manually.
+#
+#   Requires macros: AX_CHECK_COMPILER_FLAGS, AX_GCC_X86_CPUID
+#
+#   (The main emphasis here is on recent CPUs, on the principle that doing
+#   high-performance computing on old hardware is uncommon.)
+#
+# LICENSE
+#
+#   Copyright (c) 2008 Steven G. Johnson <stevenj@alum.mit.edu>
+#   Copyright (c) 2008 Matteo Frigo
+#
+#   This program is free software: you can redistribute it and/or modify it
+#   under the terms of the GNU General Public License as published by the
+#   Free Software Foundation, either version 3 of the License, or (at your
+#   option) any later version.
+#
+#   This program is distributed in the hope that it will be useful, but
+#   WITHOUT ANY WARRANTY; without even the implied warranty of
+#   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
+#   Public License for more details.
+#
+#   You should have received a copy of the GNU General Public License along
+#   with this program. If not, see <http://www.gnu.org/licenses/>.
+#
+#   As a special exception, the respective Autoconf Macro's copyright owner
+#   gives unlimited permission to copy, distribute and modify the configure
+#   scripts that are the output of Autoconf when processing the Macro. You
+#   need not follow the terms of the GNU General Public License when using
+#   or distributing such scripts, even though portions of the text of the
+#   Macro appear in them. The GNU General Public License (GPL) does govern
+#   all other use of the material that constitutes the Autoconf Macro.
+#
+#   This special exception to the GPL applies to versions of the Autoconf
+#   Macro released by the Autoconf Archive. When you make and distribute a
+#   modified version of the Autoconf Macro, you may extend this special
+#   exception to the GPL to apply to your modified version as well.
+
+AC_DEFUN([AX_GCC_ARCHFLAG],
+[AC_REQUIRE([AC_PROG_CC])
+AC_REQUIRE([AC_CANONICAL_HOST])
+
+AC_ARG_WITH(gcc-arch, [AC_HELP_STRING([--with-gcc-arch=<arch>], [use architecture <arch> for gcc -march/-mtune, instead of guessing])],
+	ax_gcc_arch=$withval, ax_gcc_arch=yes)
+
+AC_MSG_CHECKING([for gcc architecture flag])
+AC_MSG_RESULT([])
+AC_CACHE_VAL(ax_cv_gcc_archflag,
+[
+ax_cv_gcc_archflag="unknown"
+
+if test "$GCC" = yes; then
+
+if test "x$ax_gcc_arch" = xyes; then
+ax_gcc_arch=""
+if test "$cross_compiling" = no; then
+case $host_cpu in
+  i[[3456]]86*|x86_64*) # use cpuid codes, in part from x86info-1.7 by D. Jones
+     AX_GCC_X86_CPUID(0)
+     AX_GCC_X86_CPUID(1)
+     case $ax_cv_gcc_x86_cpuid_0 in
+       *:756e6547:*:*) # Intel
+          case $ax_cv_gcc_x86_cpuid_1 in
+	    *5[[48]]?:*:*:*) ax_gcc_arch="pentium-mmx pentium" ;;
+	    *5??:*:*:*) ax_gcc_arch=pentium ;;
+	    *6[[3456]]?:*:*:*) ax_gcc_arch="pentium2 pentiumpro" ;;
+	    *6a?:*[[01]]:*:*) ax_gcc_arch="pentium2 pentiumpro" ;;
+	    *6a?:*[[234]]:*:*) ax_gcc_arch="pentium3 pentiumpro" ;;
+	    *6[[9d]]?:*:*:*) ax_gcc_arch="pentium-m pentium3 pentiumpro" ;;
+	    *6[[78b]]?:*:*:*) ax_gcc_arch="pentium3 pentiumpro" ;;
+	    *6??:*:*:*) ax_gcc_arch=pentiumpro ;;
+            *f3[[347]]:*:*:*|*f4[1347]:*:*:*)
+		case $host_cpu in
+                  x86_64*) ax_gcc_arch="nocona pentium4 pentiumpro" ;;
+                  *) ax_gcc_arch="prescott pentium4 pentiumpro" ;;
+                esac ;;
+            *f??:*:*:*) ax_gcc_arch="pentium4 pentiumpro";;
+          esac ;;
+       *:68747541:*:*) # AMD
+          case $ax_cv_gcc_x86_cpuid_1 in
+	    *5[[67]]?:*:*:*) ax_gcc_arch=k6 ;;
+	    *5[[8d]]?:*:*:*) ax_gcc_arch="k6-2 k6" ;;
+	    *5[[9]]?:*:*:*) ax_gcc_arch="k6-3 k6" ;;
+	    *60?:*:*:*) ax_gcc_arch=k7 ;;
+	    *6[[12]]?:*:*:*) ax_gcc_arch="athlon k7" ;;
+	    *6[[34]]?:*:*:*) ax_gcc_arch="athlon-tbird k7" ;;
+	    *67?:*:*:*) ax_gcc_arch="athlon-4 athlon k7" ;;
+	    *6[[68a]]?:*:*:*)
+	       AX_GCC_X86_CPUID(0x80000006) # L2 cache size
+	       case $ax_cv_gcc_x86_cpuid_0x80000006 in
+                 *:*:*[[1-9a-f]]??????:*) # (L2 = ecx >> 16) >= 256
+			ax_gcc_arch="athlon-xp athlon-4 athlon k7" ;;
+                 *) ax_gcc_arch="athlon-4 athlon k7" ;;
+	       esac ;;
+	    *f[[4cef8b]]?:*:*:*) ax_gcc_arch="athlon64 k8" ;;
+	    *f5?:*:*:*) ax_gcc_arch="opteron k8" ;;
+	    *f7?:*:*:*) ax_gcc_arch="athlon-fx opteron k8" ;;
+	    *f??:*:*:*) ax_gcc_arch="k8" ;;
+          esac ;;
+	*:746e6543:*:*) # IDT
+	   case $ax_cv_gcc_x86_cpuid_1 in
+	     *54?:*:*:*) ax_gcc_arch=winchip-c6 ;;
+	     *58?:*:*:*) ax_gcc_arch=winchip2 ;;
+	     *6[[78]]?:*:*:*) ax_gcc_arch=c3 ;;
+	     *69?:*:*:*) ax_gcc_arch="c3-2 c3" ;;
+	   esac ;;
+     esac
+     if test x"$ax_gcc_arch" = x; then # fallback
+	case $host_cpu in
+	  i586*) ax_gcc_arch=pentium ;;
+	  i686*) ax_gcc_arch=pentiumpro ;;
+        esac
+     fi
+     ;;
+
+  sparc*)
+     AC_PATH_PROG([PRTDIAG], [prtdiag], [prtdiag], [$PATH:/usr/platform/`uname -i`/sbin/:/usr/platform/`uname -m`/sbin/])
+     cputype=`(((grep cpu /proc/cpuinfo | cut -d: -f2) ; ($PRTDIAG -v |grep -i sparc) ; grep -i cpu /var/run/dmesg.boot ) | head -n 1) 2> /dev/null`
+     cputype=`echo "$cputype" | tr -d ' -' |tr $as_cr_LETTERS $as_cr_letters`
+     case $cputype in
+         *ultrasparciv*) ax_gcc_arch="ultrasparc4 ultrasparc3 ultrasparc v9" ;;
+         *ultrasparciii*) ax_gcc_arch="ultrasparc3 ultrasparc v9" ;;
+         *ultrasparc*) ax_gcc_arch="ultrasparc v9" ;;
+         *supersparc*|*tms390z5[[05]]*) ax_gcc_arch="supersparc v8" ;;
+         *hypersparc*|*rt62[[056]]*) ax_gcc_arch="hypersparc v8" ;;
+         *cypress*) ax_gcc_arch=cypress ;;
+     esac ;;
+
+  alphaev5) ax_gcc_arch=ev5 ;;
+  alphaev56) ax_gcc_arch=ev56 ;;
+  alphapca56) ax_gcc_arch="pca56 ev56" ;;
+  alphapca57) ax_gcc_arch="pca57 pca56 ev56" ;;
+  alphaev6) ax_gcc_arch=ev6 ;;
+  alphaev67) ax_gcc_arch=ev67 ;;
+  alphaev68) ax_gcc_arch="ev68 ev67" ;;
+  alphaev69) ax_gcc_arch="ev69 ev68 ev67" ;;
+  alphaev7) ax_gcc_arch="ev7 ev69 ev68 ev67" ;;
+  alphaev79) ax_gcc_arch="ev79 ev7 ev69 ev68 ev67" ;;
+
+  powerpc*)
+     cputype=`((grep cpu /proc/cpuinfo | head -n 1 | cut -d: -f2 | cut -d, -f1 | sed 's/ //g') ; /usr/bin/machine ; /bin/machine; grep CPU /var/run/dmesg.boot | head -n 1 | cut -d" " -f2) 2> /dev/null`
+     cputype=`echo $cputype | sed -e 's/ppc//g;s/ *//g'`
+     case $cputype in
+       *750*) ax_gcc_arch="750 G3" ;;
+       *740[[0-9]]*) ax_gcc_arch="$cputype 7400 G4" ;;
+       *74[[4-5]][[0-9]]*) ax_gcc_arch="$cputype 7450 G4" ;;
+       *74[[0-9]][[0-9]]*) ax_gcc_arch="$cputype G4" ;;
+       *970*) ax_gcc_arch="970 G5 power4";;
+       *POWER4*|*power4*|*gq*) ax_gcc_arch="power4 970";;
+       *POWER5*|*power5*|*gr*|*gs*) ax_gcc_arch="power5 power4 970";;
+       603ev|8240) ax_gcc_arch="$cputype 603e 603";;
+       *) ax_gcc_arch=$cputype ;;
+     esac
+     ax_gcc_arch="$ax_gcc_arch powerpc"
+     ;;
+esac
+fi # not cross-compiling
+fi # guess arch
+
+if test "x$ax_gcc_arch" != x -a "x$ax_gcc_arch" != xno; then
+for arch in $ax_gcc_arch; do
+  if test "x[]m4_default([$1],yes)" = xyes; then # if we require portable code
+    flags="-mtune=$arch"
+    # -mcpu=$arch and m$arch generate nonportable code on every arch except
+    # x86.  And some other arches (e.g. Alpha) don't accept -mtune.  Grrr.
+    case $host_cpu in i*86|x86_64*) flags="$flags -mcpu=$arch -m$arch";; esac
+  else
+    flags="-march=$arch -mcpu=$arch -m$arch"
+  fi
+  for flag in $flags; do
+    AX_CHECK_COMPILER_FLAGS($flag, [ax_cv_gcc_archflag=$flag; break])
+  done
+  test "x$ax_cv_gcc_archflag" = xunknown || break
+done
+fi
+
+fi # $GCC=yes
+])
+AC_MSG_CHECKING([for gcc architecture flag])
+AC_MSG_RESULT($ax_cv_gcc_archflag)
+if test "x$ax_cv_gcc_archflag" = xunknown; then
+  m4_default([$3],:)
+else
+  m4_default([$2], [CFLAGS="$CFLAGS $ax_cv_gcc_archflag"])
+fi
+])
diff --git a/polly/lib/External/isl/m4/ax_gcc_warn_unused_result.m4 b/polly/lib/External/isl/m4/ax_gcc_warn_unused_result.m4
new file mode 100644
index 00000000000..a957f8f9eaf
--- /dev/null
+++ b/polly/lib/External/isl/m4/ax_gcc_warn_unused_result.m4
@@ -0,0 +1,56 @@
+# ===========================================================================
+#    http://www.nongnu.org/autoconf-archive/ax_gcc_warn_unused_result.html
+# ===========================================================================
+#
+# SYNOPSIS
+#
+#   AX_GCC_WARN_UNUSED_RESULT
+#
+# DESCRIPTION
+#
+#   The macro will compile a test program to see whether the compiler does
+#   understand the per-function postfix pragma.
+#
+# LICENSE
+#
+#   Copyright (c) 2008 Guido U. Draheim <guidod@gmx.de>
+#
+#   This program is free software; you can redistribute it and/or modify it
+#   under the terms of the GNU General Public License as published by the
+#   Free Software Foundation; either version 2 of the License, or (at your
+#   option) any later version.
+#
+#   This program is distributed in the hope that it will be useful, but
+#   WITHOUT ANY WARRANTY; without even the implied warranty of
+#   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
+#   Public License for more details.
+#
+#   You should have received a copy of the GNU General Public License along
+#   with this program. If not, see <http://www.gnu.org/licenses/>.
+#
+#   As a special exception, the respective Autoconf Macro's copyright owner
+#   gives unlimited permission to copy, distribute and modify the configure
+#   scripts that are the output of Autoconf when processing the Macro. You
+#   need not follow the terms of the GNU General Public License when using
+#   or distributing such scripts, even though portions of the text of the
+#   Macro appear in them. The GNU General Public License (GPL) does govern
+#   all other use of the material that constitutes the Autoconf Macro.
+#
+#   This special exception to the GPL applies to versions of the Autoconf
+#   Macro released by the Autoconf Archive. When you make and distribute a
+#   modified version of the Autoconf Macro, you may extend this special
+#   exception to the GPL to apply to your modified version as well.
+
+AC_DEFUN([AX_GCC_WARN_UNUSED_RESULT],[dnl
+AC_CACHE_CHECK(
+ [whether the compiler supports function __attribute__((__warn_unused_result__))],
+ ax_cv_gcc_warn_unused_result,[
+ AC_TRY_COMPILE([__attribute__((__warn_unused_result__))
+ int f(int i) { return i; }],
+ [],
+ ax_cv_gcc_warn_unused_result=yes, ax_cv_gcc_warn_unused_result=no)])
+ if test "$ax_cv_gcc_warn_unused_result" = yes; then
+   AC_DEFINE([GCC_WARN_UNUSED_RESULT],[__attribute__((__warn_unused_result__))],
+    [most gcc compilers know a function __attribute__((__warn_unused_result__))])
+ fi
+])
diff --git a/polly/lib/External/isl/m4/ax_gcc_x86_cpuid.m4 b/polly/lib/External/isl/m4/ax_gcc_x86_cpuid.m4
new file mode 100644
index 00000000000..5420b095900
--- /dev/null
+++ b/polly/lib/External/isl/m4/ax_gcc_x86_cpuid.m4
@@ -0,0 +1,77 @@
+# ===========================================================================
+#        http://www.nongnu.org/autoconf-archive/ax_gcc_x86_cpuid.html
+# ===========================================================================
+#
+# SYNOPSIS
+#
+#   AX_GCC_X86_CPUID(OP)
+#
+# DESCRIPTION
+#
+#   On Pentium and later x86 processors, with gcc or a compiler that has a
+#   compatible syntax for inline assembly instructions, run a small program
+#   that executes the cpuid instruction with input OP. This can be used to
+#   detect the CPU type.
+#
+#   On output, the values of the eax, ebx, ecx, and edx registers are stored
+#   as hexadecimal strings as "eax:ebx:ecx:edx" in the cache variable
+#   ax_cv_gcc_x86_cpuid_OP.
+#
+#   If the cpuid instruction fails (because you are running a
+#   cross-compiler, or because you are not using gcc, or because you are on
+#   a processor that doesn't have this instruction), ax_cv_gcc_x86_cpuid_OP
+#   is set to the string "unknown".
+#
+#   This macro mainly exists to be used in AX_GCC_ARCHFLAG.
+#
+# LICENSE
+#
+#   Copyright (c) 2008 Steven G. Johnson <stevenj@alum.mit.edu>
+#   Copyright (c) 2008 Matteo Frigo
+#
+#   This program is free software: you can redistribute it and/or modify it
+#   under the terms of the GNU General Public License as published by the
+#   Free Software Foundation, either version 3 of the License, or (at your
+#   option) any later version.
+#
+#   This program is distributed in the hope that it will be useful, but
+#   WITHOUT ANY WARRANTY; without even the implied warranty of
+#   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
+#   Public License for more details.
+#
+#   You should have received a copy of the GNU General Public License along
+#   with this program. If not, see <http://www.gnu.org/licenses/>.
+#
+#   As a special exception, the respective Autoconf Macro's copyright owner
+#   gives unlimited permission to copy, distribute and modify the configure
+#   scripts that are the output of Autoconf when processing the Macro. You
+#   need not follow the terms of the GNU General Public License when using
+#   or distributing such scripts, even though portions of the text of the
+#   Macro appear in them. The GNU General Public License (GPL) does govern
+#   all other use of the material that constitutes the Autoconf Macro.
+#
+#   This special exception to the GPL applies to versions of the Autoconf
+#   Macro released by the Autoconf Archive. When you make and distribute a
+#   modified version of the Autoconf Macro, you may extend this special
+#   exception to the GPL to apply to your modified version as well.
+
+AC_DEFUN([AX_GCC_X86_CPUID],
+[AC_REQUIRE([AC_PROG_CC])
+AC_LANG_PUSH([C])
+AC_CACHE_CHECK(for x86 cpuid $1 output, ax_cv_gcc_x86_cpuid_$1,
+ [AC_RUN_IFELSE([AC_LANG_PROGRAM([#include <stdio.h>], [
+     int op = $1, eax, ebx, ecx, edx;
+     FILE *f;
+      __asm__("cpuid"
+        : "=a" (eax), "=b" (ebx), "=c" (ecx), "=d" (edx)
+        : "a" (op));
+     f = fopen("conftest_cpuid", "w"); if (!f) return 1;
+     fprintf(f, "%x:%x:%x:%x\n", eax, ebx, ecx, edx);
+     fclose(f);
+     return 0;
+])],
+     [ax_cv_gcc_x86_cpuid_$1=`cat conftest_cpuid`; rm -f conftest_cpuid],
+     [ax_cv_gcc_x86_cpuid_$1=unknown; rm -f conftest_cpuid],
+     [ax_cv_gcc_x86_cpuid_$1=unknown])])
+AC_LANG_POP([C])
+])
diff --git a/polly/lib/External/isl/m4/ax_set_warning_flags.m4 b/polly/lib/External/isl/m4/ax_set_warning_flags.m4
new file mode 100644
index 00000000000..c64ad7d8478
--- /dev/null
+++ b/polly/lib/External/isl/m4/ax_set_warning_flags.m4
@@ -0,0 +1,17 @@
+dnl Add a set of flags to WARNING_FLAGS, that enable compiler warnings for
+dnl isl. The warnings that are enabled vary with the compiler and only include
+dnl warnings that did not trigger at the time of adding these flags.
+AC_DEFUN([AX_SET_WARNING_FLAGS],[dnl
+	AX_COMPILER_VENDOR
+
+	WARNING_FLAGS=""
+
+	if test "${ax_cv_c_compiler_vendor}" = "clang"; then
+		dnl isl is at the moment clean of -Wall warnings.  If clang adds
+		dnl new warnings to -Wall which cause false positives, the
+		dnl specific warning types will be disabled explicitally (by
+		dnl adding for example -Wno-return-type). To temporarily disable
+		dnl all warnings run configure with CFLAGS=-Wno-all.
+		WARNING_FLAGS="-Wall"
+	fi
+])
diff --git a/polly/lib/External/isl/m4/ax_submodule.m4 b/polly/lib/External/isl/m4/ax_submodule.m4
new file mode 100644
index 00000000000..7cf899576d7
--- /dev/null
+++ b/polly/lib/External/isl/m4/ax_submodule.m4
@@ -0,0 +1,71 @@
+AC_DEFUN([AX_SUBMODULE],
+[
+
+m4_if(m4_bregexp($2,|,choice),choice,
+	[AC_ARG_WITH($1,
+		[AS_HELP_STRING([--with-$1=$2],
+				[Which $1 to use [default=$3]])])])
+case "system" in
+$2)
+	AC_ARG_WITH($1_prefix,
+		    [AS_HELP_STRING([--with-$1-prefix=DIR],
+				    [Prefix of $1 installation])])
+	AC_ARG_WITH($1_exec_prefix,
+		    [AS_HELP_STRING([--with-$1-exec-prefix=DIR],
+				    [Exec prefix of $1 installation])])
+esac
+m4_if(m4_bregexp($2,build,build),build,
+	[AC_ARG_WITH($1_builddir,
+		[AS_HELP_STRING([--with-$1-builddir=DIR],
+				[Location of $1 builddir])])])
+if test "x$with_$1_prefix" != "x" -a "x$with_$1_exec_prefix" = "x"; then
+	with_$1_exec_prefix=$with_$1_prefix
+fi
+if test "x$with_$1_prefix" != "x" -o "x$with_$1_exec_prefix" != "x"; then
+	if test "x$with_$1" != "x" -a "x$with_$1" != "xyes" -a "x$with_$1" != "xsystem"; then
+		AC_MSG_ERROR([Setting $with_$1_prefix implies use of system $1])
+	fi
+	with_$1="system"
+fi
+if test "x$with_$1_builddir" != "x"; then
+	if test "x$with_$1" != "x" -a "x$with_$1" != "xyes" -a "x$with_$1" != "xbuild"; then
+		AC_MSG_ERROR([Setting $with_$1_builddir implies use of build $1])
+	fi
+	with_$1="build"
+	$1_srcdir=`echo @abs_srcdir@ | $with_$1_builddir/config.status --file=-`
+	AC_MSG_NOTICE($1 sources in $$1_srcdir)
+fi
+if test "x$with_$1_exec_prefix" != "x"; then
+	export PKG_CONFIG_PATH="$with_$1_exec_prefix/lib/pkgconfig${PKG_CONFIG_PATH+:$PKG_CONFIG_PATH}"
+fi
+case "$with_$1" in
+$2)
+	;;
+*)
+	case "$3" in
+	bundled)
+		if test -d $srcdir/.git -a \
+			-d $srcdir/$1 -a \
+			! -d $srcdir/$1/.git; then
+			AC_MSG_WARN([git repo detected, but submodule $1 not initialized])
+			AC_MSG_WARN([You may want to run])
+			AC_MSG_WARN([	git submodule init])
+			AC_MSG_WARN([	git submodule update])
+			AC_MSG_WARN([	sh autogen.sh])
+		fi
+		if test -f $srcdir/$1/configure; then
+			with_$1="bundled"
+		else
+			with_$1="no"
+		fi
+		;;
+	*)
+		with_$1="$3"
+		;;
+	esac
+	;;
+esac
+AC_MSG_CHECKING([which $1 to use])
+AC_MSG_RESULT($with_$1)
+
+])
diff --git a/polly/lib/External/isl/mp_get_memory_functions.c b/polly/lib/External/isl/mp_get_memory_functions.c
new file mode 100644
index 00000000000..e14e336ceb6
--- /dev/null
+++ b/polly/lib/External/isl/mp_get_memory_functions.c
@@ -0,0 +1,14 @@
+#include <gmp.h>
+
+void mp_get_memory_functions(
+		void *(**alloc_func_ptr) (size_t),
+		void *(**realloc_func_ptr) (void *, size_t, size_t),
+		void (**free_func_ptr) (void *, size_t))
+{
+	if (alloc_func_ptr)
+		*alloc_func_ptr = __gmp_allocate_func;
+	if (realloc_func_ptr)
+		*realloc_func_ptr = __gmp_reallocate_func;
+	if (free_func_ptr)
+		*free_func_ptr = __gmp_free_func;
+}
diff --git a/polly/lib/External/isl/pip.c b/polly/lib/External/isl/pip.c
new file mode 100644
index 00000000000..707da1c4993
--- /dev/null
+++ b/polly/lib/External/isl/pip.c
@@ -0,0 +1,382 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <assert.h>
+#include <string.h>
+#include <strings.h>
+#include <isl_map_private.h>
+#include <isl/aff.h>
+#include <isl/set.h>
+#include "isl_tab.h"
+#include "isl_sample.h"
+#include "isl_scan.h"
+#include <isl_seq.h>
+#include <isl_ilp_private.h>
+#include <isl/printer.h>
+#include <isl_point_private.h>
+#include <isl_vec_private.h>
+#include <isl/options.h>
+
+/* The input of this program is the same as that of the "example" program
+ * from the PipLib distribution, except that the "big parameter column"
+ * should always be -1.
+ *
+ * Context constraints in PolyLib format
+ * -1
+ * Problem constraints in PolyLib format
+ * Optional list of options
+ *
+ * The options are
+ *	Maximize	compute maximum instead of minimum
+ *	Rational	compute rational optimum instead of integer optimum
+ *	Urs_parms	don't assume parameters are non-negative
+ *	Urs_unknowns	don't assume unknowns are non-negative
+ */
+
+struct options {
+	struct isl_options	*isl;
+	unsigned		 verify;
+	unsigned		 format;
+};
+
+#define FORMAT_SET	0
+#define FORMAT_AFF	1
+
+struct isl_arg_choice pip_format[] = {
+	{"set",		FORMAT_SET},
+	{"affine",	FORMAT_AFF},
+	{0}
+};
+
+ISL_ARGS_START(struct options, options_args)
+ISL_ARG_CHILD(struct options, isl, "isl", &isl_options_args, "isl options")
+ISL_ARG_BOOL(struct options, verify, 'T', "verify", 0, NULL)
+ISL_ARG_CHOICE(struct options, format, 0, "format",
+	pip_format, FORMAT_SET, "output format")
+ISL_ARGS_END
+
+ISL_ARG_DEF(options, struct options, options_args)
+
+static __isl_give isl_basic_set *set_bounds(__isl_take isl_basic_set *bset)
+{
+	unsigned nparam;
+	int i, r;
+	isl_point *pt, *pt2;
+	isl_basic_set *box;
+
+	nparam = isl_basic_set_dim(bset, isl_dim_param);
+	r = nparam >= 8 ? 4 : nparam >= 5 ? 6 : 30;
+
+	pt = isl_basic_set_sample_point(isl_basic_set_copy(bset));
+	pt2 = isl_point_copy(pt);
+
+	for (i = 0; i < nparam; ++i) {
+		pt = isl_point_add_ui(pt, isl_dim_param, i, r);
+		pt2 = isl_point_sub_ui(pt2, isl_dim_param, i, r);
+	}
+
+	box = isl_basic_set_box_from_points(pt, pt2);
+
+	return isl_basic_set_intersect(bset, box);
+}
+
+static struct isl_basic_set *to_parameter_domain(struct isl_basic_set *context)
+{
+	context = isl_basic_set_move_dims(context, isl_dim_param, 0,
+		    isl_dim_set, 0, isl_basic_set_dim(context, isl_dim_set));
+	context = isl_basic_set_params(context);
+	return context;
+}
+
+isl_basic_set *plug_in_parameters(isl_basic_set *bset, struct isl_vec *params)
+{
+	int i;
+
+	for (i = 0; i < params->size - 1; ++i)
+		bset = isl_basic_set_fix(bset,
+					 isl_dim_param, i, params->el[1 + i]);
+
+	bset = isl_basic_set_remove_dims(bset,
+					 isl_dim_param, 0, params->size - 1);
+
+	isl_vec_free(params);
+
+	return bset;
+}
+
+isl_set *set_plug_in_parameters(isl_set *set, struct isl_vec *params)
+{
+	int i;
+
+	for (i = 0; i < params->size - 1; ++i)
+		set = isl_set_fix(set, isl_dim_param, i, params->el[1 + i]);
+
+	set = isl_set_remove_dims(set, isl_dim_param, 0, params->size - 1);
+
+	isl_vec_free(params);
+
+	return set;
+}
+
+/* Compute the lexicographically minimal (or maximal if max is set)
+ * element of bset for the given values of the parameters, by
+ * successively solving an ilp problem in each direction.
+ */
+struct isl_vec *opt_at(struct isl_basic_set *bset,
+	struct isl_vec *params, int max)
+{
+	unsigned dim;
+	struct isl_vec *opt;
+	struct isl_vec *obj;
+	int i;
+
+	dim = isl_basic_set_dim(bset, isl_dim_set);
+
+	bset = plug_in_parameters(bset, params);
+
+	if (isl_basic_set_plain_is_empty(bset)) {
+		opt = isl_vec_alloc(bset->ctx, 0);
+		isl_basic_set_free(bset);
+		return opt;
+	}
+
+	opt = isl_vec_alloc(bset->ctx, 1 + dim);
+	assert(opt);
+
+	obj = isl_vec_alloc(bset->ctx, 1 + dim);
+	assert(obj);
+
+	isl_int_set_si(opt->el[0], 1);
+	isl_int_set_si(obj->el[0], 0);
+
+	for (i = 0; i < dim; ++i) {
+		enum isl_lp_result res;
+
+		isl_seq_clr(obj->el + 1, dim);
+		isl_int_set_si(obj->el[1 + i], 1);
+		res = isl_basic_set_solve_ilp(bset, max, obj->el,
+						&opt->el[1 + i], NULL);
+		if (res == isl_lp_empty)
+			goto empty;
+		assert(res == isl_lp_ok);
+		bset = isl_basic_set_fix(bset, isl_dim_set, i, opt->el[1 + i]);
+	}
+
+	isl_basic_set_free(bset);
+	isl_vec_free(obj);
+
+	return opt;
+empty:
+	isl_vec_free(opt);
+	opt = isl_vec_alloc(bset->ctx, 0);
+	isl_basic_set_free(bset);
+	isl_vec_free(obj);
+
+	return opt;
+}
+
+struct isl_scan_pip {
+	struct isl_scan_callback callback;
+	isl_basic_set *bset;
+	isl_set *sol;
+	isl_set *empty;
+	int stride;
+	int n;
+	int max;
+};
+
+/* Check if the "manually" computed optimum of bset at the "sample"
+ * values of the parameters agrees with the solution of pilp problem
+ * represented by the pair (sol, empty).
+ * In particular, if there is no solution for this value of the parameters,
+ * then it should be an element of the parameter domain "empty".
+ * Otherwise, the optimal solution, should be equal to the result of
+ * plugging in the value of the parameters in "sol".
+ */
+static int scan_one(struct isl_scan_callback *callback,
+	__isl_take isl_vec *sample)
+{
+	struct isl_scan_pip *sp = (struct isl_scan_pip *)callback;
+	struct isl_vec *opt;
+
+	sp->n--;
+
+	opt = opt_at(isl_basic_set_copy(sp->bset), isl_vec_copy(sample), sp->max);
+	assert(opt);
+
+	if (opt->size == 0) {
+		isl_point *sample_pnt;
+		sample_pnt = isl_point_alloc(isl_set_get_space(sp->empty), sample);
+		assert(isl_set_contains_point(sp->empty, sample_pnt));
+		isl_point_free(sample_pnt);
+		isl_vec_free(opt);
+	} else {
+		isl_set *sol;
+		isl_set *opt_set;
+		opt_set = isl_set_from_basic_set(isl_basic_set_from_vec(opt));
+		sol = set_plug_in_parameters(isl_set_copy(sp->sol), sample);
+		assert(isl_set_is_equal(opt_set, sol));
+		isl_set_free(sol);
+		isl_set_free(opt_set);
+	}
+
+	if (!(sp->n % sp->stride)) {
+		printf("o");
+		fflush(stdout);
+	}
+
+	return sp->n >= 1 ? 0 : -1;
+}
+
+static void check_solution(isl_basic_set *bset, isl_basic_set *context,
+	isl_set *sol, isl_set *empty, int max)
+{
+	struct isl_scan_pip sp;
+	isl_int count, count_max;
+	int i, n;
+	int r;
+
+	context = set_bounds(context);
+	context = isl_basic_set_underlying_set(context);
+
+	isl_int_init(count);
+	isl_int_init(count_max);
+
+	isl_int_set_si(count_max, 2000);
+	r = isl_basic_set_count_upto(context, count_max, &count);
+	assert(r >= 0);
+	n = isl_int_get_si(count);
+
+	isl_int_clear(count_max);
+	isl_int_clear(count);
+
+	sp.callback.add = scan_one;
+	sp.bset = bset;
+	sp.sol = sol;
+	sp.empty = empty;
+	sp.n = n;
+	sp.stride = n > 70 ? 1 + (n + 1)/70 : 1;
+	sp.max = max;
+
+	for (i = 0; i < n; i += sp.stride)
+		printf(".");
+	printf("\r");
+	fflush(stdout);
+
+	isl_basic_set_scan(context, &sp.callback);
+
+	printf("\n");
+
+	isl_basic_set_free(bset);
+}
+
+int main(int argc, char **argv)
+{
+	struct isl_ctx *ctx;
+	struct isl_basic_set *context, *bset, *copy, *context_copy;
+	struct isl_set *set = NULL;
+	struct isl_set *empty;
+	isl_pw_multi_aff *pma = NULL;
+	int neg_one;
+	char s[1024];
+	int urs_parms = 0;
+	int urs_unknowns = 0;
+	int max = 0;
+	int rational = 0;
+	int n;
+	int nparam;
+	struct options *options;
+
+	options = options_new_with_defaults();
+	assert(options);
+	argc = options_parse(options, argc, argv, ISL_ARG_ALL);
+
+	ctx = isl_ctx_alloc_with_options(&options_args, options);
+
+	context = isl_basic_set_read_from_file(ctx, stdin);
+	assert(context);
+	n = fscanf(stdin, "%d", &neg_one);
+	assert(n == 1);
+	assert(neg_one == -1);
+	bset = isl_basic_set_read_from_file(ctx, stdin);
+
+	while (fgets(s, sizeof(s), stdin)) {
+		if (strncasecmp(s, "Maximize", 8) == 0)
+			max = 1;
+		if (strncasecmp(s, "Rational", 8) == 0) {
+			rational = 1;
+			bset = isl_basic_set_set_rational(bset);
+		}
+		if (strncasecmp(s, "Urs_parms", 9) == 0)
+			urs_parms = 1;
+		if (strncasecmp(s, "Urs_unknowns", 12) == 0)
+			urs_unknowns = 1;
+	}
+	if (!urs_parms)
+		context = isl_basic_set_intersect(context,
+		isl_basic_set_positive_orthant(isl_basic_set_get_space(context)));
+	context = to_parameter_domain(context);
+	nparam = isl_basic_set_dim(context, isl_dim_param);
+	if (nparam != isl_basic_set_dim(bset, isl_dim_param)) {
+		int dim = isl_basic_set_dim(bset, isl_dim_set);
+		bset = isl_basic_set_move_dims(bset, isl_dim_param, 0,
+					    isl_dim_set, dim - nparam, nparam);
+	}
+	if (!urs_unknowns)
+		bset = isl_basic_set_intersect(bset,
+		isl_basic_set_positive_orthant(isl_basic_set_get_space(bset)));
+
+	if (options->verify) {
+		copy = isl_basic_set_copy(bset);
+		context_copy = isl_basic_set_copy(context);
+	}
+
+	if (options->format == FORMAT_AFF) {
+		if (max)
+			pma = isl_basic_set_partial_lexmax_pw_multi_aff(bset,
+								context, &empty);
+		else
+			pma = isl_basic_set_partial_lexmin_pw_multi_aff(bset,
+								context, &empty);
+	} else {
+		if (max)
+			set = isl_basic_set_partial_lexmax(bset,
+								context, &empty);
+		else
+			set = isl_basic_set_partial_lexmin(bset,
+								context, &empty);
+	}
+
+	if (options->verify) {
+		assert(!rational);
+		if (options->format == FORMAT_AFF)
+			set = isl_set_from_pw_multi_aff(pma);
+		check_solution(copy, context_copy, set, empty, max);
+		isl_set_free(set);
+	} else {
+		isl_printer *p;
+		p = isl_printer_to_file(ctx, stdout);
+		if (options->format == FORMAT_AFF)
+			p = isl_printer_print_pw_multi_aff(p, pma);
+		else
+			p = isl_printer_print_set(p, set);
+		p = isl_printer_end_line(p);
+		p = isl_printer_print_str(p, "no solution: ");
+		p = isl_printer_print_set(p, empty);
+		p = isl_printer_end_line(p);
+		isl_printer_free(p);
+		isl_set_free(set);
+		isl_pw_multi_aff_free(pma);
+	}
+
+	isl_set_free(empty);
+	isl_ctx_free(ctx);
+
+	return 0;
+}
diff --git a/polly/lib/External/isl/pip_test.sh.in b/polly/lib/External/isl/pip_test.sh.in
new file mode 100755
index 00000000000..28ad1ef4245
--- /dev/null
+++ b/polly/lib/External/isl/pip_test.sh.in
@@ -0,0 +1,28 @@
+#!/bin/sh
+
+EXEEXT=@EXEEXT@
+
+PIP_TESTS="\
+	boulet.pip \
+	brisebarre.pip \
+	cg1.pip \
+	esced.pip \
+	ex2.pip \
+	ex.pip \
+	fimmel.pip \
+	max.pip \
+	negative.pip \
+	seghir-vd.pip \
+	small.pip \
+	sor1d.pip \
+	square.pip \
+	sven.pip \
+	tobi.pip"
+
+for i in $PIP_TESTS; do
+	echo $i;
+	./isl_pip$EXEEXT --format=set --context=gbr -T < $srcdir/test_inputs/$i || exit
+	./isl_pip$EXEEXT --format=set --context=lexmin -T < $srcdir/test_inputs/$i || exit
+	./isl_pip$EXEEXT --format=affine --context=gbr -T < $srcdir/test_inputs/$i || exit
+	./isl_pip$EXEEXT --format=affine --context=lexmin -T < $srcdir/test_inputs/$i || exit
+done
diff --git a/polly/lib/External/isl/polyhedron_detect_equalities.c b/polly/lib/External/isl/polyhedron_detect_equalities.c
new file mode 100644
index 00000000000..567acec71df
--- /dev/null
+++ b/polly/lib/External/isl/polyhedron_detect_equalities.c
@@ -0,0 +1,24 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <isl/set.h>
+
+int main(int argc, char **argv)
+{
+	struct isl_ctx *ctx = isl_ctx_alloc();
+	struct isl_basic_set *bset;
+
+	bset = isl_basic_set_read_from_file(ctx, stdin);
+	bset = isl_basic_set_detect_equalities(bset);
+	isl_basic_set_print(bset, stdout, 0, "", "", ISL_FORMAT_POLYLIB);
+	isl_basic_set_free(bset);
+	isl_ctx_free(ctx);
+
+	return 0;
+}
diff --git a/polly/lib/External/isl/polyhedron_minimize.c b/polly/lib/External/isl/polyhedron_minimize.c
new file mode 100644
index 00000000000..526b37368f2
--- /dev/null
+++ b/polly/lib/External/isl/polyhedron_minimize.c
@@ -0,0 +1,105 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <assert.h>
+#include <isl/set.h>
+#include <isl/vec.h>
+#include <isl_ilp_private.h>
+#include <isl_seq.h>
+#include <isl_vec_private.h>
+
+/* The input of this program is the same as that of the "polytope_minimize"
+ * program from the barvinok distribution.
+ *
+ * Constraints of set is PolyLib format.
+ * Linear or affine objective function in PolyLib format.
+ */
+
+static struct isl_vec *isl_vec_lin_to_aff(struct isl_vec *vec)
+{
+	struct isl_vec *aff;
+
+	if (!vec)
+		return NULL;
+	aff = isl_vec_alloc(vec->ctx, 1 + vec->size);
+	if (!aff)
+		goto error;
+	isl_int_set_si(aff->el[0], 0);
+	isl_seq_cpy(aff->el + 1, vec->el, vec->size);
+	isl_vec_free(vec);
+	return aff;
+error:
+	isl_vec_free(vec);
+	return NULL;
+}
+
+/* Rotate elements of vector right.
+ * In particular, move the constant term from the end of the
+ * vector to the start of the vector.
+ */
+static struct isl_vec *vec_ror(struct isl_vec *vec)
+{
+	int i;
+
+	if (!vec)
+		return NULL;
+	for (i = vec->size - 2; i >= 0; --i)
+		isl_int_swap(vec->el[i], vec->el[i + 1]);
+	return vec;
+}
+
+int main(int argc, char **argv)
+{
+	struct isl_ctx *ctx = isl_ctx_alloc();
+	struct isl_basic_set *bset;
+	struct isl_vec *obj;
+	struct isl_vec *sol;
+	isl_int opt;
+	unsigned dim;
+	enum isl_lp_result res;
+	isl_printer *p;
+
+	isl_int_init(opt);
+	bset = isl_basic_set_read_from_file(ctx, stdin);
+	assert(bset);
+	obj = isl_vec_read_from_file(ctx, stdin);
+	assert(obj);
+	dim = isl_basic_set_total_dim(bset);
+	assert(obj->size >= dim && obj->size <= dim + 1);
+	if (obj->size != dim + 1)
+		obj = isl_vec_lin_to_aff(obj);
+	else
+		obj = vec_ror(obj);
+	res = isl_basic_set_solve_ilp(bset, 0, obj->el, &opt, &sol);
+	switch (res) {
+	case isl_lp_error:
+		fprintf(stderr, "error\n");
+		return -1;
+	case isl_lp_empty:
+		fprintf(stdout, "empty\n");
+		break;
+	case isl_lp_unbounded:
+		fprintf(stdout, "unbounded\n");
+		break;
+	case isl_lp_ok:
+		p = isl_printer_to_file(ctx, stdout);
+		p = isl_printer_print_vec(p, sol);
+		p = isl_printer_end_line(p);
+		p = isl_printer_print_isl_int(p, opt);
+		p = isl_printer_end_line(p);
+		isl_printer_free(p);
+	}
+	isl_basic_set_free(bset);
+	isl_vec_free(obj);
+	isl_vec_free(sol);
+	isl_ctx_free(ctx);
+	isl_int_clear(opt);
+
+	return 0;
+}
diff --git a/polly/lib/External/isl/polyhedron_sample.c b/polly/lib/External/isl/polyhedron_sample.c
new file mode 100644
index 00000000000..3fc442f2999
--- /dev/null
+++ b/polly/lib/External/isl/polyhedron_sample.c
@@ -0,0 +1,36 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <assert.h>
+#include <isl_map_private.h>
+#include "isl_sample.h"
+#include <isl/vec.h>
+
+int main(int argc, char **argv)
+{
+	struct isl_ctx *ctx = isl_ctx_alloc();
+	struct isl_basic_set *bset;
+	struct isl_vec *sample;
+	isl_printer *p;
+
+	bset = isl_basic_set_read_from_file(ctx, stdin);
+	sample = isl_basic_set_sample_vec(isl_basic_set_copy(bset));
+	p = isl_printer_to_file(ctx, stdout);
+	p = isl_printer_print_vec(p, sample);
+	p = isl_printer_end_line(p);
+	isl_printer_free(p);
+	assert(sample);
+	if (isl_vec_size(sample) > 0)
+		assert(isl_basic_set_contains(bset, sample));
+	isl_basic_set_free(bset);
+	isl_vec_free(sample);
+	isl_ctx_free(ctx);
+
+	return 0;
+}
diff --git a/polly/lib/External/isl/polytope_scan.c b/polly/lib/External/isl/polytope_scan.c
new file mode 100644
index 00000000000..3fb3a4da460
--- /dev/null
+++ b/polly/lib/External/isl/polytope_scan.c
@@ -0,0 +1,104 @@
+/*
+ * Copyright 2008-2009 Katholieke Universiteit Leuven
+ *
+ * Use of this software is governed by the MIT license
+ *
+ * Written by Sven Verdoolaege, K.U.Leuven, Departement
+ * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
+ */
+
+#include <assert.h>
+#include <isl_map_private.h>
+#include "isl_equalities.h"
+#include <isl_seq.h>
+#include "isl_scan.h"
+#include <isl_mat_private.h>
+#include <isl_vec_private.h>
+
+/* The input of this program is the same as that of the "polytope_scan"
+ * program from the barvinok distribution.
+ *
+ * Constraints of set is PolyLib format.
+ *
+ * The input set is assumed to be bounded.
+ */
+
+struct scan_samples {
+	struct isl_scan_callback callback;
+	struct isl_mat *samples;
+};
+
+static int scan_samples_add_sample(struct isl_scan_callback *cb,
+	__isl_take isl_vec *sample)
+{
+	struct scan_samples *ss = (struct scan_samples *)cb;
+
+	ss->samples = isl_mat_extend(ss->samples, ss->samples->n_row + 1,
+						  ss->samples->n_col);
+	if (!ss->samples)
+		goto error;
+
+	isl_seq_cpy(ss->samples->row[ss->samples->n_row - 1],
+		    sample->el, sample->size);
+
+	isl_vec_free(sample);
+	return 0;
+error:
+	isl_vec_free(sample);
+	return -1;
+}
+
+static struct isl_mat *isl_basic_set_scan_samples(struct isl_basic_set *bset)
+{
+	isl_ctx *ctx;
+	unsigned dim;
+	struct scan_samples ss;
+
+	ctx = isl_basic_set_get_ctx(bset);
+	dim = isl_basic_set_total_dim(bset);
+	ss.callback.add = scan_samples_add_sample;
+	ss.samples = isl_mat_alloc(ctx, 0, 1 + dim);
+	if (!ss.samples)
+		goto error;
+
+	if (isl_basic_set_scan(bset, &ss.callback) < 0) {
+		isl_mat_free(ss.samples);
+		return NULL;
+	}
+
+	return ss.samples;
+error:
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+static struct isl_mat *isl_basic_set_samples(struct isl_basic_set *bset)
+{
+	struct isl_mat *T;
+	struct isl_mat *samples;
+
+	if (!bset)
+		return NULL;
+
+	if (bset->n_eq == 0)
+		return isl_basic_set_scan_samples(bset);
+
+	bset = isl_basic_set_remove_equalities(bset, &T, NULL);
+	samples = isl_basic_set_scan_samples(bset);
+	return isl_mat_product(samples, isl_mat_transpose(T));
+}
+
+int main(int argc, char **argv)
+{
+	struct isl_ctx *ctx = isl_ctx_alloc();
+	struct isl_basic_set *bset;
+	struct isl_mat *samples;
+
+	bset = isl_basic_set_read_from_file(ctx, stdin);
+	samples = isl_basic_set_samples(bset);
+	isl_mat_print_internal(samples, stdout, 0);
+	isl_mat_free(samples);
+	isl_ctx_free(ctx);
+
+	return 0;
+}
diff --git a/polly/lib/External/isl/print.c b/polly/lib/External/isl/print.c
new file mode 100644
index 00000000000..5e3a5319636
--- /dev/null
+++ b/polly/lib/External/isl/print.c
@@ -0,0 +1,100 @@
+#include <isl/ctx.h>
+#include <isl/id.h>
+#include <isl/space.h>
+#include <isl/local_space.h>
+#include <isl/set.h>
+#include <isl/map.h>
+#include <isl/union_set.h>
+#include <isl/union_map.h>
+#include <isl/polynomial.h>
+#include <isl/band.h>
+#include <isl/constraint.h>
+#include <isl/aff.h>
+#include <isl/ast.h>
+#include <isl/printer.h>
+#include <isl/val.h>
+
+#undef BASE
+#define BASE id
+#include <print_templ.c>
+#undef BASE
+#define BASE val
+#include <print_templ.c>
+#undef BASE
+#define BASE multi_val
+#include <print_templ.c>
+#undef BASE
+#define BASE space
+#include <print_templ.c>
+#undef BASE
+#define BASE local_space
+#include <print_templ.c>
+#undef BASE
+#define BASE basic_set
+#include <print_templ.c>
+#undef BASE
+#define BASE basic_map
+#include <print_templ.c>
+#undef BASE
+#define BASE set
+#include <print_templ.c>
+#undef BASE
+#define BASE map
+#include <print_templ.c>
+#undef BASE
+#define BASE union_set
+#include <print_templ.c>
+#undef BASE
+#define BASE union_map
+#include <print_templ.c>
+#undef BASE
+#define BASE qpolynomial
+#include <print_templ.c>
+#undef BASE
+#define BASE qpolynomial_fold
+#include <print_templ.c>
+#undef BASE
+#define BASE pw_qpolynomial
+#include <print_templ.c>
+#undef BASE
+#define BASE pw_qpolynomial_fold
+#include <print_templ.c>
+#undef BASE
+#define BASE union_pw_qpolynomial
+#include <print_templ.c>
+#undef BASE
+#define BASE union_pw_qpolynomial_fold
+#include <print_templ.c>
+#undef BASE
+#define BASE band
+#include <print_templ.c>
+#undef BASE
+#define BASE constraint
+#include <print_templ.c>
+#undef BASE
+#define BASE aff
+#include <print_templ.c>
+#undef BASE
+#define BASE pw_aff
+#include <print_templ.c>
+#undef BASE
+#define BASE multi_aff
+#include <print_templ.c>
+#undef BASE
+#define BASE pw_multi_aff
+#include <print_templ.c>
+#undef BASE
+#define BASE union_pw_multi_aff
+#include <print_templ.c>
+#undef BASE
+#define BASE multi_pw_aff
+#include <print_templ.c>
+#undef BASE
+#define BASE point
+#include <print_templ.c>
+#undef BASE
+#define BASE ast_expr
+#include <print_templ.c>
+#undef BASE
+#define BASE ast_node
+#include <print_templ.c>
diff --git a/polly/lib/External/isl/print_templ.c b/polly/lib/External/isl/print_templ.c
new file mode 100644
index 00000000000..f147fdc5480
--- /dev/null
+++ b/polly/lib/External/isl/print_templ.c
@@ -0,0 +1,33 @@
+#define xCAT(A,B) A ## B
+#define CAT(A,B) xCAT(A,B)
+#undef TYPE
+#define TYPE CAT(isl_,BASE)
+#define xFN(TYPE,NAME) TYPE ## _ ## NAME
+#define FN(TYPE,NAME) xFN(TYPE,NAME)
+
+void FN(TYPE,dump)(__isl_keep TYPE *obj)
+{
+	isl_printer *p;
+
+	if (!obj)
+		return;
+	p = isl_printer_to_file(FN(TYPE,get_ctx)(obj), stderr);
+	p = FN(isl_printer_print,BASE)(p, obj);
+	p = isl_printer_end_line(p);
+	isl_printer_free(p);
+}
+
+__isl_give char *FN(TYPE,to_str)(__isl_keep TYPE *obj)
+{
+	isl_printer *p;
+	char *s;
+
+	if (!obj)
+		return NULL;
+	p = isl_printer_to_str(FN(TYPE,get_ctx)(obj));
+	p = FN(isl_printer_print,BASE)(p, obj);
+	s = isl_printer_get_str(p);
+	isl_printer_free(p);
+
+	return s;
+}
diff --git a/polly/lib/External/isl/test_inputs/affine.polylib b/polly/lib/External/isl/test_inputs/affine.polylib
new file mode 100644
index 00000000000..f14720c0a0a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/affine.polylib
@@ -0,0 +1,9 @@
+# the affine hull of {[a,b] : a=b && 1 <= a <= 163} ...
+3 4
+0 1 -1 0
+1 1  0 -1
+1 -1 0 163
+
+# ... is {[a,b] : a=b} (and not {[In_1,In_2]}, as Omega 1.2 claims)
+1 4
+0 1 -1 0
diff --git a/polly/lib/External/isl/test_inputs/affine2.polylib b/polly/lib/External/isl/test_inputs/affine2.polylib
new file mode 100644
index 00000000000..c67db77bb6e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/affine2.polylib
@@ -0,0 +1,9 @@
+5 5
+1 -2 0 1 0
+1 2 0 -1 1
+1 0 -2 1 0
+1 0 2 -1 1
+1 0 0 1 -1
+
+1 5
+0 1 -1 0 0
diff --git a/polly/lib/External/isl/test_inputs/affine3.polylib b/polly/lib/External/isl/test_inputs/affine3.polylib
new file mode 100644
index 00000000000..f2bc9a2db2f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/affine3.polylib
@@ -0,0 +1,7 @@
+3 4
+1 1 0 0
+1 -7 4 2
+1 5 -4 2
+
+1 4
+0 3 -2 0
diff --git a/polly/lib/External/isl/test_inputs/application.omega b/polly/lib/External/isl/test_inputs/application.omega
new file mode 100644
index 00000000000..8f4fd1db39a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/application.omega
@@ -0,0 +1,3 @@
+{[x]}
+{[x] -> [y] : y = 2x}
+{[y]: Exists ( alpha : 2alpha = y)}
diff --git a/polly/lib/External/isl/test_inputs/application2.omega b/polly/lib/External/isl/test_inputs/application2.omega
new file mode 100644
index 00000000000..f2af1e86958
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/application2.omega
@@ -0,0 +1,3 @@
+{[x] : x >= 0 && x <= 20 }
+{[x] -> [y] : y = 2x}
+{[y]: Exists ( alpha : 2alpha = y && 0 <= y && y <= 40)}
diff --git a/polly/lib/External/isl/test_inputs/basicLinear.pwqp b/polly/lib/External/isl/test_inputs/basicLinear.pwqp
new file mode 100644
index 00000000000..0af7fabb53f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/basicLinear.pwqp
@@ -0,0 +1 @@
+[P, Q] -> { [n, m] -> n : n >= 1 and m >= n and m <= P and m <= Q }
diff --git a/polly/lib/External/isl/test_inputs/basicLinear2.pwqp b/polly/lib/External/isl/test_inputs/basicLinear2.pwqp
new file mode 100644
index 00000000000..d411a369158
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/basicLinear2.pwqp
@@ -0,0 +1 @@
+[P, Q] -> { [n, m] -> n : n >= 1 and m >= n and m <= P and n >= -1 + Q }
diff --git a/polly/lib/External/isl/test_inputs/basicTest.pwqp b/polly/lib/External/isl/test_inputs/basicTest.pwqp
new file mode 100644
index 00000000000..52e7fc8e697
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/basicTest.pwqp
@@ -0,0 +1 @@
+[p] -> { [n, m] -> (n + n^2) : n >= 1 and m >= n and m <= p }
diff --git a/polly/lib/External/isl/test_inputs/basicTestParameterPosNeg.pwqp b/polly/lib/External/isl/test_inputs/basicTestParameterPosNeg.pwqp
new file mode 100644
index 00000000000..6cb4490215c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/basicTestParameterPosNeg.pwqp
@@ -0,0 +1 @@
+[p] -> { [n, m] -> (n + n^3) : n >= -1 and m >= n and m <= p }
diff --git a/polly/lib/External/isl/test_inputs/boulet.pip b/polly/lib/External/isl/test_inputs/boulet.pip
new file mode 100644
index 00000000000..78e90ddf58b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/boulet.pip
@@ -0,0 +1,13 @@
+0 3
+
+-1
+
+5 6
+1 1 -1  2 0 0
+1 0  1  1 4 20
+1 0 -1 -1 0 0
+1 0  1 -1 2 10
+1 0 -1  1 2 10
+
+Urs_parms
+Urs_unknowns
diff --git a/polly/lib/External/isl/test_inputs/brisebarre.pip b/polly/lib/External/isl/test_inputs/brisebarre.pip
new file mode 100644
index 00000000000..f3decadfd27
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/brisebarre.pip
@@ -0,0 +1,34 @@
+# ---------------------- CONTEXT ----------------------
+1 2
+1    0
+
+-1
+
+# ----------------------- DOMAIN ----------------------
+26 6
+1    3    0    0     0    -98300
+1   -3    0    0     0     98308
+1  432   36    6     1 -14757611
+1 -432  -36   -6    -1  14758510
+1   54    9    3     1  -1923190
+1  -54   -9   -3    -1   1923303
+1   48   12    6     3  -1782238
+1  -48  -12   -6    -3   1782339
+1   27    9    6     4  -1045164
+1  -27   -9   -6    -4   1045221
+1  432  180  150   125 -17434139
+1 -432 -180 -150  -125  17435038
+1    6    3    3     3   -252443
+1   -6   -3   -3    -3    252456
+1  432  252  294   343 -18949275
+1 -432 -252 -294  -343  18950174
+1   27   18   24    32  -1234720
+1  -27  -18  -24   -32   1234777
+1   48   36   54    81  -2288453
+1  -48  -36  -54   -81   2288554
+1   54   45   75   125  -2684050
+1  -54  -45  -75  -125   2684163
+1  432  396  726  1331 -22386005
+1 -432 -396 -726 -1331  22386904
+1    3    3    6    12   -162072
+1   -3   -3   -6   -12    162080
diff --git a/polly/lib/External/isl/test_inputs/cg1.pip b/polly/lib/External/isl/test_inputs/cg1.pip
new file mode 100644
index 00000000000..78e31f187b1
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/cg1.pip
@@ -0,0 +1,15 @@
+2 4
+  1  1  0 -1
+  1 -1  1  0
+
+-1
+
+8 7
+  1  0  1  0 -1  0  0
+  1  0 -1  0  1  0  0
+  1  1  0  0  0 -1  0
+  1 -1  0  0  0  1  0
+  1  0  1  0  0  0 -1
+  1  0 -1  0  0  1  0
+  1  0 -1  1  0  0 -1
+  1  0  0 -1  0  1  0
diff --git a/polly/lib/External/isl/test_inputs/codegen/atomic.c b/polly/lib/External/isl/test_inputs/codegen/atomic.c
new file mode 100644
index 00000000000..73731c0c22b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/atomic.c
@@ -0,0 +1,6 @@
+for (int c0 = 0; c0 <= 10; c0 += 1) {
+  if (c0 >= 1)
+    b(c0 - 1);
+  if (c0 <= 9)
+    a(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/atomic.in b/polly/lib/External/isl/test_inputs/codegen/atomic.in
new file mode 100644
index 00000000000..754b3d0cf2f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/atomic.in
@@ -0,0 +1,3 @@
+{ a[i] -> [i] : 0 <= i < 10; b[i] -> [i+1] : 0 <= i < 10 }
+{ : }
+{ [i] -> atomic[x] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/atomic2.c b/polly/lib/External/isl/test_inputs/codegen/atomic2.c
new file mode 100644
index 00000000000..9c4a57e3e5a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/atomic2.c
@@ -0,0 +1,2 @@
+for (int c0 = ((b0 + 32767) % 32768) + 1; c0 <= 65534; c0 += 32768)
+  A(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/atomic2.in b/polly/lib/External/isl/test_inputs/codegen/atomic2.in
new file mode 100644
index 00000000000..99901d0e191
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/atomic2.in
@@ -0,0 +1,4 @@
+# Check that isl properly handles atomic domains that are unions.
+[nn, b0] -> { A[a] -> [a, 0, b0] : exists (e0 = [(b0 - a)/32768]: 32768e0 = b0 - a and a >= 1 and b0 >= 0 and b0 <= 32767 and a <= 65534) }
+[nn, b0] -> { : b0 >= 0 and b0 <= 32767 }
+[nn, b0] -> { [a, b, c] -> atomic[2] : c >= 1; [a, 0, c] -> atomic[2] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/atomic3.c b/polly/lib/External/isl/test_inputs/codegen/atomic3.c
new file mode 100644
index 00000000000..e63e09201a1
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/atomic3.c
@@ -0,0 +1,8 @@
+for (int c0 = 0; c0 <= 64; c0 += 1) {
+  if (c0 >= 63) {
+    sync();
+  } else if (c0 >= 1) {
+    sync();
+  } else
+    sync();
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/atomic3.in b/polly/lib/External/isl/test_inputs/codegen/atomic3.in
new file mode 100644
index 00000000000..4d0c4953107
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/atomic3.in
@@ -0,0 +1,5 @@
+# Check that isl is not confused by inconsistent
+# separation_class and atomic options.
+{ sync[] -> [i, 0] : 0 <= i <= 64 }
+{ : }
+{ [i, 0] -> separation_class[[1] -> [0]] : 1 <= i <= 62; [i, 0] -> atomic[1]}
diff --git a/polly/lib/External/isl/test_inputs/codegen/atomic4.c b/polly/lib/External/isl/test_inputs/codegen/atomic4.c
new file mode 100644
index 00000000000..624c2af1d25
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/atomic4.c
@@ -0,0 +1,2 @@
+for (int c0 = 0; c0 <= 64; c0 += 1)
+  sync();
diff --git a/polly/lib/External/isl/test_inputs/codegen/atomic4.in b/polly/lib/External/isl/test_inputs/codegen/atomic4.in
new file mode 100644
index 00000000000..c5dab102c6a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/atomic4.in
@@ -0,0 +1,4 @@
+# Check that isl is not confused by inconsistent separate and atomic options.
+{ sync[] -> [i, 0] : 0 <= i <= 64 }
+{ : }
+{ [i, 0] -> separate[1] : 1 <= i <= 62; [i, 0] -> atomic[1] : i <= 10 or i >= 20 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/0D-1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/0D-1.c
new file mode 100644
index 00000000000..7f22e2e417e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/0D-1.c
@@ -0,0 +1 @@
+S1();
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/0D-1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/0D-1.in
new file mode 100644
index 00000000000..10e3e408a6d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/0D-1.in
@@ -0,0 +1,3 @@
+{ S1[] -> [0] }
+{  :  }
+{ [i] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/0D-2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/0D-2.c
new file mode 100644
index 00000000000..b87b57eacfb
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/0D-2.c
@@ -0,0 +1,2 @@
+if (M >= 0)
+  S1();
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/0D-2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/0D-2.in
new file mode 100644
index 00000000000..3bbfd35cc2b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/0D-2.in
@@ -0,0 +1,3 @@
+[M] -> { S1[] -> [0] : M >= 0 }
+[M] -> {  :  }
+[M] -> { [i] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/0D-3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/0D-3.c
new file mode 100644
index 00000000000..7f22e2e417e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/0D-3.c
@@ -0,0 +1 @@
+S1();
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/0D-3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/0D-3.in
new file mode 100644
index 00000000000..f99dd344e46
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/0D-3.in
@@ -0,0 +1,3 @@
+[M] -> { S1[] -> [0] : M >= 0 }
+[M] -> {  : M >= 0 }
+[M] -> { [i] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/1point-1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/1point-1.c
new file mode 100644
index 00000000000..53dd8931e6f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/1point-1.c
@@ -0,0 +1 @@
+S1(2 * M, M);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/1point-1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/1point-1.in
new file mode 100644
index 00000000000..efd0a7ff827
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/1point-1.in
@@ -0,0 +1,3 @@
+[M] -> { S1[2M, M] -> [2M, M, 0] }
+[M] -> {  :  }
+[M] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/1point-2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/1point-2.c
new file mode 100644
index 00000000000..0a8cbe37d2f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/1point-2.c
@@ -0,0 +1 @@
+S1(2 * M, N + 2);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/1point-2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/1point-2.in
new file mode 100644
index 00000000000..de5157ceac6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/1point-2.in
@@ -0,0 +1,3 @@
+[M, N] -> { S1[2M, 2 + N] -> [2M, 2 + N, 0] }
+[M, N] -> {  :  }
+[M, N] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/4-param.c b/polly/lib/External/isl/test_inputs/codegen/cloog/4-param.c
new file mode 100644
index 00000000000..56659a06779
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/4-param.c
@@ -0,0 +1,14 @@
+{
+  for (int c0 = m; c0 <= min(n, p - 1); c0 += 1)
+    S1(c0);
+  for (int c0 = p; c0 <= min(m - 1, q); c0 += 1)
+    S2(c0);
+  for (int c0 = max(m, p); c0 <= min(n, q); c0 += 1) {
+    S1(c0);
+    S2(c0);
+  }
+  for (int c0 = max(max(m, p), q + 1); c0 <= n; c0 += 1)
+    S1(c0);
+  for (int c0 = max(max(m, n + 1), p); c0 <= q; c0 += 1)
+    S2(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/4-param.in b/polly/lib/External/isl/test_inputs/codegen/cloog/4-param.in
new file mode 100644
index 00000000000..7814076053b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/4-param.in
@@ -0,0 +1,3 @@
+[m, n, p, q] -> { S1[i0] -> [i0, 0] : i0 >= m and i0 <= n; S2[i0] -> [i0, 1] : i0 >= p and i0 <= q }
+[m, n, p, q] -> {  :  }
+[m, n, p, q] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/README b/polly/lib/External/isl/test_inputs/codegen/cloog/README
new file mode 100644
index 00000000000..9250f11e161
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/README
@@ -0,0 +1,2 @@
+The tests in this directory have been adapted from the corresponding CLooG
+test cases.
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/backtrack.c b/polly/lib/External/isl/test_inputs/codegen/cloog/backtrack.c
new file mode 100644
index 00000000000..df0407a437e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/backtrack.c
@@ -0,0 +1 @@
+S1(0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/backtrack.in b/polly/lib/External/isl/test_inputs/codegen/cloog/backtrack.in
new file mode 100644
index 00000000000..b8ec466f559
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/backtrack.in
@@ -0,0 +1,3 @@
+{ S1[0] -> [0, 0] }
+{  :  }
+{ [i, j] -> atomic[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-1.c
new file mode 100644
index 00000000000..95eb5f716e3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-1.c
@@ -0,0 +1,2 @@
+for (int c0 = 0; c0 <= 2; c0 += 1)
+  S1(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-1.in
new file mode 100644
index 00000000000..e5f8bfafd15
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-1.in
@@ -0,0 +1,3 @@
+{ S1[i0] -> [i0, 0] : i0 >= 0 and i0 <= 2 }
+{  :  }
+{ [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-2.c
new file mode 100644
index 00000000000..df0407a437e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-2.c
@@ -0,0 +1 @@
+S1(0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-2.in
new file mode 100644
index 00000000000..7eb48f2da74
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-2.in
@@ -0,0 +1,3 @@
+{ S1[0] -> [0, 0] }
+{  :  }
+{ [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-3.c
new file mode 100644
index 00000000000..d0dd54bd089
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-3.c
@@ -0,0 +1,2 @@
+for (int c0 = 0; c0 <= M; c0 += 1)
+  S1(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-3.in
new file mode 100644
index 00000000000..59ce2c9249b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-3.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0] -> [i0, 0] : i0 >= 0 and i0 <= M }
+[M] -> {  : M >= 0 }
+[M] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-4.c b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-4.c
new file mode 100644
index 00000000000..0ebb7ccbb43
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-4.c
@@ -0,0 +1,2 @@
+for (int c0 = 0; c0 <= M + 1; c0 += 1)
+  S1(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-4.in b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-4.in
new file mode 100644
index 00000000000..5f79acf698c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-4.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0] -> [i0, 0] : i0 >= 0 and i0 <= 1 + M }
+[M] -> {  : M >= 0 }
+[M] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-5.c b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-5.c
new file mode 100644
index 00000000000..09e8c422168
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-5.c
@@ -0,0 +1 @@
+S1(1, floord(M + 1, 2));
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-5.in b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-5.in
new file mode 100644
index 00000000000..3a6f9085514
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/basic-bounds-5.in
@@ -0,0 +1,3 @@
+[M] -> { S1[1, i1] -> [1, i1, 0] : 2i1 >= M and 2i1 <= 1 + M }
+[M] -> {  :  }
+[M] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/block.c b/polly/lib/External/isl/test_inputs/codegen/cloog/block.c
new file mode 100644
index 00000000000..e24ef30ade7
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/block.c
@@ -0,0 +1,6 @@
+{
+  S1();
+  S3(0);
+  S2();
+  S3(1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/block.in b/polly/lib/External/isl/test_inputs/codegen/cloog/block.in
new file mode 100644
index 00000000000..faf2af5f466
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/block.in
@@ -0,0 +1,3 @@
+{ S3[i0] -> [i0, 1] : i0 >= 0 and i0 <= 1; S1[] -> [0, 0]; S2[] -> [1, 0] }
+{  :  }
+{ [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/block2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/block2.c
new file mode 100644
index 00000000000..d3fbbd7802a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/block2.c
@@ -0,0 +1,5 @@
+for (int c0 = 0; c0 <= 9; c0 += 1) {
+  S1(c0, 1);
+  S3(c0, 1);
+  S2(c0, 1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/block2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/block2.in
new file mode 100644
index 00000000000..d2b5db44039
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/block2.in
@@ -0,0 +1,3 @@
+{ S1[i0, 1] -> [i0, 1, 6] : i0 >= 0 and i0 <= 9; S2[i0, 1] -> [i0, 1, 11] : i0 >= 0 and i0 <= 9; S3[i0, 1] -> [i0, 1, 8] : i0 >= 0 and i0 <= 9 }
+{  :  }
+{ [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/block3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/block3.c
new file mode 100644
index 00000000000..ff4d5539178
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/block3.c
@@ -0,0 +1,6 @@
+{
+  S1();
+  for (int c0 = 0; c0 <= 1; c0 += 1)
+    S3(c0);
+  S2();
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/block3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/block3.in
new file mode 100644
index 00000000000..39c0abb790d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/block3.in
@@ -0,0 +1,3 @@
+{ S2[] -> [1]; S3[i0] -> [i0] : i0 >= 0 and i0 <= 1; S1[] -> [0] }
+{  :  }
+{ [i] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/byu98-1-2-3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/byu98-1-2-3.c
new file mode 100644
index 00000000000..8e87cba33db
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/byu98-1-2-3.c
@@ -0,0 +1,18 @@
+{
+  for (int c0 = 2; c0 <= 3; c0 += 1)
+    for (int c1 = -c0 + 6; c1 <= 6; c1 += 1)
+      S1(c0, c1);
+  for (int c0 = 4; c0 <= 8; c0 += 1) {
+    if (c0 >= 6) {
+      S2(c0, -c0 + 9);
+    } else {
+      if (c0 == 4)
+        for (int c1 = 3; c1 <= 4; c1 += 1)
+          S1(4, c1);
+      S1(c0, -c0 + 9);
+      S2(c0, -c0 + 9);
+    }
+    for (int c1 = max(c0 - 1, -c0 + 10); c1 <= 6; c1 += 1)
+      S1(c0, c1);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/byu98-1-2-3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/byu98-1-2-3.in
new file mode 100644
index 00000000000..9f98fb85990
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/byu98-1-2-3.in
@@ -0,0 +1,3 @@
+{ S1[i0, i1] -> [i0, i1, 0] : i1 >= 6 - i0 and i0 >= 2 and i1 >= 3 and i1 <= 6 and i1 >= -1 + i0; S2[i0, 9 - i0] -> [i0, 9 - i0, 1] : i0 <= 8 and i0 >= 4 }
+{  :  }
+{ [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/cholesky.c b/polly/lib/External/isl/test_inputs/codegen/cloog/cholesky.c
new file mode 100644
index 00000000000..20d4e4d0b9e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/cholesky.c
@@ -0,0 +1,12 @@
+for (int c0 = 1; c0 <= n; c0 += 1) {
+  S1(c0);
+  for (int c2 = 1; c2 < c0; c2 += 1)
+    S2(c0, c2);
+  S3(c0);
+  for (int c2 = c0 + 1; c2 <= n; c2 += 1) {
+    S4(c0, c2);
+    for (int c4 = 1; c4 < c0; c4 += 1)
+      S5(c0, c2, c4);
+    S6(c0, c2);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/cholesky.in b/polly/lib/External/isl/test_inputs/codegen/cloog/cholesky.in
new file mode 100644
index 00000000000..90b56c638b3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/cholesky.in
@@ -0,0 +1,3 @@
+[n] -> { S1[i0] -> [i0, 1, 0, 0, 0, 0] : i0 >= 1 and i0 <= n; S2[i0, i1] -> [i0, 2, i1, 1, 0, 0] : i0 >= 1 and i0 <= n and i1 >= 1 and i1 <= -1 + i0; S6[i0, i1] -> [i0, 4, i1, 3, 0, 0] : i0 >= 1 and i0 <= n and i1 >= 1 + i0 and i1 <= n; S3[i0] -> [i0, 3, 0, 0, 0, 0] : i0 >= 1 and i0 <= n; S4[i0, i1] -> [i0, 4, i1, 1, 0, 0] : i0 >= 1 and i0 <= n and i1 >= 1 + i0 and i1 <= n; S5[i0, i1, i2] -> [i0, 4, i1, 2, i2, 1] : i0 >= 1 and i0 <= n and i1 >= 1 + i0 and i1 <= n and i2 >= 1 and i2 <= -1 + i0 }
+[n] -> {  :  }
+[n] -> { [i, j, k, l, m, n'] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/cholesky2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/cholesky2.c
new file mode 100644
index 00000000000..c746460b4fa
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/cholesky2.c
@@ -0,0 +1,20 @@
+{
+  for (int c1 = 1; c1 <= M; c1 += 1) {
+    S1(c1);
+    for (int c2 = c1 + 1; c2 <= M; c2 += 1)
+      S4(c1, c2);
+  }
+  for (int c0 = 1; c0 < 3 * M - 1; c0 += 3) {
+    S3((c0 + 2) / 3);
+    if (3 * M >= c0 + 8) {
+      for (int c1 = (c0 + 5) / 3; c1 <= M; c1 += 1) {
+        S6((c0 + 2) / 3, c1);
+        for (int c4 = (c0 + 5) / 3; c4 < c1; c4 += 1)
+          S5(c4, c1, (c0 + 2) / 3);
+      }
+    } else if (c0 + 5 == 3 * M)
+      S6(M - 1, M);
+    for (int c1 = (c0 + 5) / 3; c1 <= M; c1 += 1)
+      S2(c1, (c0 + 2) / 3);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/cholesky2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/cholesky2.in
new file mode 100644
index 00000000000..b44b4568f71
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/cholesky2.in
@@ -0,0 +1,3 @@
+[M] -> { S2[i0, i1] -> [3i1, i0, 0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= -1 + i0; S4[i0, i1] -> [0, i0, i1] : i0 >= 1 and i0 <= M and i1 >= 1 + i0 and i1 <= M; S1[i0] -> [0, i0, 0] : i0 >= 1 and i0 <= M; S6[i0, i1] -> [-1 + 3i0, i1, 0] : i0 >= 1 and i0 <= M and i1 >= 1 + i0 and i1 <= M; S3[i0] -> [-2 + 3i0, 0, 0] : i0 >= 1 and i0 <= M; S5[i0, i1, i2] -> [-1 + 3i2, i1, i2] : i0 >= 1 and i0 <= M and i1 >= 1 + i0 and i1 <= M and i2 >= 1 and i2 <= -1 + i0 }
+[M] -> {  :  }
+[M] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/christian.c b/polly/lib/External/isl/test_inputs/codegen/cloog/christian.c
new file mode 100644
index 00000000000..98b52d8d0fe
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/christian.c
@@ -0,0 +1,6 @@
+for (int c0 = -N + 1; c0 <= N; c0 += 1) {
+  for (int c1 = max(0, c0 - 1); c1 < min(N, N + c0 - 1); c1 += 1)
+    S2(c1, -c0 + c1 + 1);
+  for (int c1 = max(0, c0); c1 < min(N, N + c0); c1 += 1)
+    S1(c1, -c0 + c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/christian.in b/polly/lib/External/isl/test_inputs/codegen/cloog/christian.in
new file mode 100644
index 00000000000..af3df07967f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/christian.in
@@ -0,0 +1,3 @@
+[N] -> { S1[i0, i1] -> [i0 - i1] : i0 >= 0 and i0 <= -1 + N and i1 >= 0 and i1 <= -1 + N; S2[i0, i1] -> [1 + i0 - i1] : i0 >= 0 and i0 <= -1 + N and i1 >= 0 and i1 <= -1 + N }
+[N] -> {  :  }
+[N] -> { [i] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/classen.c b/polly/lib/External/isl/test_inputs/codegen/cloog/classen.c
new file mode 100644
index 00000000000..ad7be335451
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/classen.c
@@ -0,0 +1,86 @@
+{
+  if (m == 1) {
+    S1(0, 1, 1, 1);
+    S8(0, 1);
+  } else if (m >= 2) {
+    S1(0, 1, 1, 1);
+    S3(0, 1, 1, 2, 1, 1, 1, 2);
+    S4(0, 1, 2, 2, 1, 1, 2, 2);
+    S2(0, 1, 1, 1, 1, 1, 2, 1);
+    S8(0, 1);
+  }
+  for (int c0 = 1; c0 < 2 * m - 3; c0 += 1) {
+    if (c0 + 1 == m) {
+      S5(m - 2, 1, m - 1, 1, m - 1, 1, m, 1);
+      S1(m - 1, 1, m, 1);
+      S3(m - 1, 1, m, 2, m, 1, m, 2);
+    } else if (m >= c0 + 2) {
+      S5(c0 - 1, 1, c0, 1, c0, 1, c0 + 1, 1);
+      S1(c0, 1, c0 + 1, 1);
+      S3(c0, 1, c0 + 1, 2, c0 + 1, 1, c0 + 1, 2);
+      S4(c0, 1, c0 + 2, 2, c0 + 1, 1, c0 + 2, 2);
+      S2(c0, 1, c0 + 1, 1, c0 + 1, 1, c0 + 2, 1);
+    } else {
+      S5(c0 - 1, -m + c0 + 2, c0, -m + c0 + 2, m - 1, -m + c0 + 2, m, -m + c0 + 2);
+      S6(c0 - 1, -m + c0 + 1, c0, -m + c0 + 2, m, -m + c0 + 1, m, -m + c0 + 2);
+      S1(c0, -m + c0 + 2, m, -m + c0 + 2);
+      S3(c0, -m + c0 + 2, c0 + 1, -m + c0 + 3, m, -m + c0 + 2, m, -m + c0 + 3);
+    }
+    for (int c2 = max(2, -m + c0 + 3); c2 <= min(m - 1, c0); c2 += 1) {
+      S5(c0 - 1, c2, c0, c2, c0 - c2 + 1, c2, c0 - c2 + 2, c2);
+      S7(c0 - 1, c2 - 1, c0 + 1, c2, c0 - c2 + 2, c2 - 1, c0 - c2 + 3, c2);
+      S6(c0 - 1, c2 - 1, c0, c2, c0 - c2 + 2, c2 - 1, c0 - c2 + 2, c2);
+      S1(c0, c2, c0 - c2 + 2, c2);
+      S3(c0, c2, c0 + 1, c2 + 1, c0 - c2 + 2, c2, c0 - c2 + 2, c2 + 1);
+      S4(c0, c2, c0 + 2, c2 + 1, c0 - c2 + 2, c2, c0 - c2 + 3, c2 + 1);
+      S2(c0, c2, c0 + 1, c2, c0 - c2 + 2, c2, c0 - c2 + 3, c2);
+    }
+    if (c0 + 1 == m) {
+      S7(m - 2, m - 1, m, m, 1, m - 1, 2, m);
+      S6(m - 2, m - 1, m - 1, m, 1, m - 1, 1, m);
+      S1(m - 1, m, 1, m);
+      S2(m - 1, m, m, m, 1, m, 2, m);
+    } else if (c0 >= m) {
+      S5(c0 - 1, m, c0, m, -m + c0 + 1, m, -m + c0 + 2, m);
+      S7(c0 - 1, m - 1, c0 + 1, m, -m + c0 + 2, m - 1, -m + c0 + 3, m);
+      S6(c0 - 1, m - 1, c0, m, -m + c0 + 2, m - 1, -m + c0 + 2, m);
+      S1(c0, m, -m + c0 + 2, m);
+      S2(c0, m, c0 + 1, m, -m + c0 + 2, m, -m + c0 + 3, m);
+    } else {
+      S7(c0 - 1, c0, c0 + 1, c0 + 1, 1, c0, 2, c0 + 1);
+      S6(c0 - 1, c0, c0, c0 + 1, 1, c0, 1, c0 + 1);
+      S1(c0, c0 + 1, 1, c0 + 1);
+      S3(c0, c0 + 1, c0 + 1, c0 + 2, 1, c0 + 1, 1, c0 + 2);
+      S4(c0, c0 + 1, c0 + 2, c0 + 2, 1, c0 + 1, 2, c0 + 2);
+      S2(c0, c0 + 1, c0 + 1, c0 + 1, 1, c0 + 1, 2, c0 + 1);
+    }
+    for (int c8 = max(1, -m + c0 + 2); c8 <= min(m, c0 + 1); c8 += 1)
+      S8(c0, c8);
+  }
+  if (m >= 2) {
+    if (m >= 3) {
+      S5(2 * m - 4, m - 1, 2 * m - 3, m - 1, m - 1, m - 1, m, m - 1);
+      S6(2 * m - 4, m - 2, 2 * m - 3, m - 1, m, m - 2, m, m - 1);
+      S1(2 * m - 3, m - 1, m, m - 1);
+      S3(2 * m - 3, m - 1, 2 * m - 2, m, m, m - 1, m, m);
+      S5(2 * m - 4, m, 2 * m - 3, m, m - 2, m, m - 1, m);
+      S7(2 * m - 4, m - 1, 2 * m - 2, m, m - 1, m - 1, m, m);
+      S6(2 * m - 4, m - 1, 2 * m - 3, m, m - 1, m - 1, m - 1, m);
+      S1(2 * m - 3, m, m - 1, m);
+    } else {
+      S5(0, 1, 1, 1, 1, 1, 2, 1);
+      S1(1, 1, 2, 1);
+      S3(1, 1, 2, 2, 2, 1, 2, 2);
+      S7(0, 1, 2, 2, 1, 1, 2, 2);
+      S6(0, 1, 1, 2, 1, 1, 1, 2);
+      S1(1, 2, 1, 2);
+    }
+    S2(2 * m - 3, m, 2 * m - 2, m, m - 1, m, m, m);
+    for (int c8 = m - 1; c8 <= m; c8 += 1)
+      S8(2 * m - 3, c8);
+    S5(2 * m - 3, m, 2 * m - 2, m, m - 1, m, m, m);
+    S6(2 * m - 3, m - 1, 2 * m - 2, m, m, m - 1, m, m);
+    S1(2 * m - 2, m, m, m);
+    S8(2 * m - 2, m);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/classen.in b/polly/lib/External/isl/test_inputs/codegen/cloog/classen.in
new file mode 100644
index 00000000000..8dc0b76a505
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/classen.in
@@ -0,0 +1,3 @@
+[m] -> { S2[i0, i1, 1 + i0, i1, 2 + i0 - i1, i1, 3 + i0 - i1, i1] -> [i0, 0, i1, 2, 2 + i0 - i1, i1, 1] : m >= 1 and i0 <= -3 + 2m and i0 >= 0 and i1 <= 1 + i0 and i1 <= m and i1 >= 3 - m + i0 and i1 >= 1; S4[i0, i1, 2 + i0, 1 + i1, 2 + i0 - i1, i1, 3 + i0 - i1, 1 + i1] -> [i0, 0, i1, 2, 2 + i0 - i1, i1, 1] : m >= 1 and i0 <= -4 + 2m and i0 >= 0 and i1 <= 1 + i0 and i1 <= -1 + m and i1 >= 3 - m + i0 and i1 >= 1; S5[i0, i1, 1 + i0, i1, 2 + i0 - i1, i1, 3 + i0 - i1, i1] -> [1 + i0, 0, i1, 0, 2 + i0 - i1, i1, 1] : m >= 1 and i0 <= -3 + 2m and i0 >= 0 and i1 <= 1 + i0 and i1 <= m and i1 >= 3 - m + i0 and i1 >= 1; S7[i0, i1, 2 + i0, 1 + i1, 2 + i0 - i1, i1, 3 + i0 - i1, 1 + i1] -> [1 + i0, 0, 1 + i1, 0, 2 + i0 - i1, i1, 1] : m >= 1 and i0 <= -4 + 2m and i0 >= 0 and i1 <= 1 + i0 and i1 <= -1 + m and i1 >= 3 - m + i0 and i1 >= 1; S6[i0, i1, 1 + i0, 1 + i1, 2 + i0 - i1, i1, 2 + i0 - i1, 1 + i1] -> [1 + i0, 0, 1 + i1, 0, 2 + i0 - i1, i1, 1] : m >= 1 and i0 <= -3 + 2m and i0 >= 0 and i1 <= 1 + i0 and i1 <= -1 + m and i1 >= 2 - m + i0 and i1 >= 1; S3[i0, i1, 1 + i0, 1 + i1, 2 + i0 - i1, i1, 2 + i0 - i1, 1 + i1] -> [i0, 0, i1, 2, 2 + i0 - i1, i1, 1] : m >= 1 and i0 <= -3 + 2m and i0 >= 0 and i1 <= 1 + i0 and i1 <= -1 + m and i1 >= 2 - m + i0 and i1 >= 1; S8[i0, i1] -> [i0, 1, 0, 0, 0, 0, 0] : i0 <= -2 + 2m and i0 >= 0 and i1 <= 1 + i0 and i1 <= m and i1 >= 2 - m + i0 and i1 >= 1; S1[i0, i1, 2 + i0 - i1, i1] -> [i0, 0, i1, 1, 0, 0, 0] : m >= 1 and i1 >= 2 - m + i0 and i1 <= 1 + i0 and i1 <= m and i1 >= 1 }
+[m] -> {  : m >= 0 }
+[m] -> { [i, j, k, l, m', n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/classen2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/classen2.c
new file mode 100644
index 00000000000..a737d2d4b1a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/classen2.c
@@ -0,0 +1,4 @@
+for (int c0 = max(max(max(max(max(max(4, 5 * outerTimeTileScatter), 5 * outerProcTileScatter1), 5 * outerProcTileScatter2 + 1), 5 * outerProcTileScatter1 + 5 * outerProcTileScatter2 - N), 10 * outerProcTileScatter2 - N + 1), 10 * outerProcTileScatter1 - 2 * N + 2); c0 <= min(min(min(min(min(min(5 * outerTimeTileScatter + 4, 10 * outerProcTileScatter1 + 4), 5 * outerProcTileScatter1 + 5 * outerProcTileScatter2 + 5), 5 * outerProcTileScatter1 + M + 2), 2 * M + 2 * N - 6), 5 * outerProcTileScatter2 + M + N), 10 * outerProcTileScatter2 + N + 3); c0 += 1)
+  for (int c1 = max(max(max(max(5 * outerProcTileScatter1, 5 * outerProcTileScatter2 + 1), -5 * outerProcTileScatter2 + c0 - 1), -M + c0 + 2), (c0 + 1) / 2 + 2); c1 <= min(min(min(min(5 * outerProcTileScatter1 + 4, 5 * outerProcTileScatter2 + N + 2), -5 * outerProcTileScatter2 + N + c0), c0), N + c0 / 2 - 1); c1 += 1)
+    for (int c2 = max(max(5 * outerProcTileScatter2, -N + c1 + 2), c0 - c1 + 3); c2 <= min(min(5 * outerProcTileScatter2 + 4, c1 - 1), N + c0 - c1); c2 += 1)
+      S1(c0 - c1 + 1, -c0 + c1 + c2 - 2, c1 - c2, c0, c1, c2);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/classen2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/classen2.in
new file mode 100644
index 00000000000..8a74eaa9728
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/classen2.in
@@ -0,0 +1,3 @@
+[outerTimeTileScatter, outerProcTileScatter1, outerProcTileScatter2, M, N] -> { S1[i0, i1, i2, 2i0 + i1 + i2, 1 + i0 + i1 + i2, 1 + i0 + i1] -> [2i0 + i1 + i2, 1 + i0 + i1 + i2, 1 + i0 + i1] : N >= 3 and i2 <= 3 + 5outerProcTileScatter1 - i0 - i1 and i1 >= -1 + 5outerProcTileScatter2 - i0 and M >= 2 and i2 <= 4 + 5outerTimeTileScatter - 2i0 - i1 and i1 <= 3 + 5outerProcTileScatter2 - i0 and i2 >= 1 and i2 <= -2 + N and i1 >= 1 and i1 <= -2 + N and i0 >= 1 and i0 <= -1 + M and i2 >= 5outerTimeTileScatter - 2i0 - i1 and i2 >= -1 + 5outerProcTileScatter1 - i0 - i1 }
+[outerTimeTileScatter, outerProcTileScatter1, outerProcTileScatter2, M, N] -> {  :  }
+[outerTimeTileScatter, outerProcTileScatter1, outerProcTileScatter2, M, N] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/constant.c b/polly/lib/External/isl/test_inputs/codegen/cloog/constant.c
new file mode 100644
index 00000000000..f39a3a2bde0
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/constant.c
@@ -0,0 +1,18 @@
+{
+  for (int c1 = 0; c1 <= min(1023, M + 1024); c1 += 1) {
+    S1(c1);
+    S3(c1);
+  }
+  for (int c1 = max(0, M + 1025); c1 <= 1023; c1 += 1) {
+    S2(c1);
+    S3(c1);
+  }
+  for (int c0 = 0; c0 <= min(1023, M + 1024); c0 += 1) {
+    S4(c0);
+    S6(c0);
+  }
+  for (int c0 = max(0, M + 1025); c0 <= 1023; c0 += 1) {
+    S5(c0);
+    S6(c0);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/constant.in b/polly/lib/External/isl/test_inputs/codegen/cloog/constant.in
new file mode 100644
index 00000000000..7cd365a8967
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/constant.in
@@ -0,0 +1,3 @@
+[M] -> { S5[i0] -> [i0, 0, 1] : i0 >= 0 and i0 <= 1023 and i0 >= 1025 + M; S1[i0] -> [-1, i0, 0] : i0 >= 0 and i0 <= 1023 and i0 <= 1024 + M; S3[i0] -> [-1, i0, 2] : i0 >= 0 and i0 <= 1023; S2[i0] -> [-1, i0, 1] : i0 >= 0 and i0 <= 1023 and i0 >= 1025 + M; S4[i0] -> [i0, 0, 0] : i0 >= 0 and i0 <= 1023 and i0 <= 1024 + M; S6[i0] -> [i0, 0, 2] : i0 >= 0 and i0 <= 1023 }
+[M] -> {  :  }
+[M] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/constbound.c b/polly/lib/External/isl/test_inputs/codegen/cloog/constbound.c
new file mode 100644
index 00000000000..219165dc8c9
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/constbound.c
@@ -0,0 +1,8 @@
+for (int c0 = 0; c0 <= 199; c0 += 1) {
+  for (int c2 = 50 * c0; c2 <= 50 * c0 + 24; c2 += 1)
+    for (int c3 = 0; c3 <= c2; c3 += 1)
+      S1(c0, c2, c3);
+  for (int c2 = 50 * c0 + 25; c2 <= 50 * c0 + 49; c2 += 1)
+    for (int c3 = 0; c3 <= c2; c3 += 1)
+      S2(c0, c2, c3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/constbound.in b/polly/lib/External/isl/test_inputs/codegen/cloog/constbound.in
new file mode 100644
index 00000000000..24727a8737c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/constbound.in
@@ -0,0 +1,3 @@
+{ S2[i0, i1, i2] -> [i0, 1, i1, i2] : i1 >= 0 and i1 <= 9999 and i2 >= 0 and i2 <= i1 and i1 >= 25 + 50i0 and i1 <= 49 + 50i0; S1[i0, i1, i2] -> [i0, 0, i1, i2] : i1 >= 0 and i1 <= 9999 and i2 >= 0 and i2 <= i1 and i1 >= 50i0 and i1 <= 24 + 50i0 }
+{  :  }
+{ [i, j, k, l] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/darte.c b/polly/lib/External/isl/test_inputs/codegen/cloog/darte.c
new file mode 100644
index 00000000000..acb63714ed2
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/darte.c
@@ -0,0 +1,14 @@
+for (int c0 = -n + 1; c0 <= n; c0 += 1) {
+  if (c0 <= 0)
+    for (int c2 = -c0 + 4; c2 <= 2 * n - c0 + 2; c2 += 2)
+      S1(1, -c0 + 1, ((c0 + c2) / 2) - 1);
+  for (int c1 = max(c0 + 2, -c0 + 4); c1 <= min(2 * n - c0, 2 * n + c0); c1 += 2) {
+    for (int c2 = c1 + 2; c2 <= 2 * n + c1; c2 += 2)
+      S1((c0 + c1) / 2, (-c0 + c1) / 2, (-c1 + c2) / 2);
+    for (int c2 = 1; c2 <= n; c2 += 1)
+      S2(((c0 + c1) / 2) - 1, (-c0 + c1) / 2, c2);
+  }
+  if (c0 >= 1)
+    for (int c2 = 1; c2 <= n; c2 += 1)
+      S2(n, n - c0 + 1, c2);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/darte.in b/polly/lib/External/isl/test_inputs/codegen/cloog/darte.in
new file mode 100644
index 00000000000..55897a5a7e4
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/darte.in
@@ -0,0 +1,3 @@
+[n] -> { S2[i0, i1, i2] -> [1 + i0 - i1, 2 + i0 + i1, i2] : i0 >= 1 and i0 <= n and i1 >= 1 and i1 <= n and i2 >= 1 and i2 <= n; S1[i0, i1, i2] -> [i0 - i1, i0 + i1, i0 + i1 + 2i2] : i0 >= 1 and i0 <= n and i1 >= 1 and i1 <= n and i2 >= 1 and i2 <= n }
+[n] -> {  :  }
+[n] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/dealII.c b/polly/lib/External/isl/test_inputs/codegen/cloog/dealII.c
new file mode 100644
index 00000000000..7db605d5d5b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/dealII.c
@@ -0,0 +1,12 @@
+{
+  for (int c0 = 0; c0 <= min(T_2 - 1, T_66); c0 += 1) {
+    S1(c0);
+    S2(c0);
+  }
+  for (int c0 = T_2; c0 <= min(T_67 - 1, T_66); c0 += 1)
+    S2(c0);
+  for (int c0 = max(0, T_66 + 1); c0 < T_2; c0 += 1)
+    S1(c0);
+  if (T_2 == 0 && T_67 == 0)
+    S1(0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/dealII.in b/polly/lib/External/isl/test_inputs/codegen/cloog/dealII.in
new file mode 100644
index 00000000000..fc0310b638f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/dealII.in
@@ -0,0 +1,3 @@
+[T_2, T_67, T_66] -> { S1[scat_0] -> [scat_0, 0] : (scat_0 <= -1 + T_2 and scat_0 >= 0) or (scat_0 <= -T_67 and scat_0 >= 0); S2[scat_0] -> [scat_0, 1] : (scat_0 <= -1 + T_2 and scat_0 >= 0 and scat_0 <= T_66) or (scat_0 <= -1 + T_67 and scat_0 >= 0 and scat_0 <= T_66) }
+[T_2, T_67, T_66] -> {  : T_2 <= 4 and T_2 >= 0 and T_67 <= 4 and T_67 >= 0 }
+[T_2, T_67, T_66] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/donotsimp.c b/polly/lib/External/isl/test_inputs/codegen/cloog/donotsimp.c
new file mode 100644
index 00000000000..4c482e2997e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/donotsimp.c
@@ -0,0 +1,6 @@
+for (int c1 = 1; c1 <= 10; c1 += 1) {
+  for (int c3 = 1; c3 <= c1; c3 += 1)
+    S1(c1, c3);
+  for (int c3 = 11; c3 <= M; c3 += 1)
+    S2(c1, c3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/donotsimp.in b/polly/lib/External/isl/test_inputs/codegen/cloog/donotsimp.in
new file mode 100644
index 00000000000..c1541ccff7e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/donotsimp.in
@@ -0,0 +1,3 @@
+[M] -> { S2[i0, i1] -> [0, i0, 0, i1, 0] : i0 >= 1 and i0 <= 10 and i1 >= 11 and i1 <= M; S1[i0, i1] -> [0, i0, 0, i1, 0] : i0 >= 1 and i0 <= 10 and i1 >= 1 and i1 <= i0 }
+[M] -> {  : M >= 20 }
+[M] -> { [i, j, k, l, m] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/dot.c b/polly/lib/External/isl/test_inputs/codegen/cloog/dot.c
new file mode 100644
index 00000000000..b08de331226
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/dot.c
@@ -0,0 +1,7 @@
+{
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    S1(0, c1);
+  for (int c0 = 1; c0 <= N; c0 += 1)
+    for (int c1 = 1; c1 <= M; c1 += 1)
+      S2(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/dot.in b/polly/lib/External/isl/test_inputs/codegen/cloog/dot.in
new file mode 100644
index 00000000000..37e1f14bad2
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/dot.in
@@ -0,0 +1,3 @@
+[M, N] -> { S1[0, i1] -> [0, i1, 0] : i1 <= M and N >= 0 and i1 >= 1; S2[i0, i1] -> [i0, i1, 1] : i0 >= 1 and i1 <= M and i0 <= N and i1 >= 1 }
+[M, N] -> {  : M >= 1 and N >= 1 }
+[M, N] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/dot2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/dot2.c
new file mode 100644
index 00000000000..a1b10692c2a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/dot2.c
@@ -0,0 +1,12 @@
+{
+  for (int c0 = 1; c0 <= min(M, N); c0 += 1) {
+    S1(c0);
+    for (int c1 = 1; c1 <= M; c1 += 1)
+      S2(c0, c1);
+  }
+  for (int c0 = N + 1; c0 <= M; c0 += 1)
+    S1(c0);
+  for (int c0 = M + 1; c0 <= N; c0 += 1)
+    for (int c1 = 1; c1 <= M; c1 += 1)
+      S2(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/dot2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/dot2.in
new file mode 100644
index 00000000000..8fa8381e20b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/dot2.in
@@ -0,0 +1,3 @@
+[M, N] -> { S2[i0, i1] -> [i0, i1, 1] : i0 >= 1 and i0 <= N and i1 >= 1 and i1 <= M; S1[i0] -> [i0, 0, 0] : i0 >= 1 and i0 <= M }
+[M, N] -> {  : M >= 1 and N >= 1 }
+[M, N] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/durbin_e_s.c b/polly/lib/External/isl/test_inputs/codegen/cloog/durbin_e_s.c
new file mode 100644
index 00000000000..25590998d5f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/durbin_e_s.c
@@ -0,0 +1,23 @@
+{
+  S4(1, 0, 0);
+  S7(1, 0, 0);
+  S8(1, 0, 3);
+  for (int c0 = 2; c0 <= 9; c0 += 1) {
+    S2(c0, -7, 0);
+    for (int c1 = -7; c1 < c0 - 8; c1 += 1)
+      S3(c0, c1, 1);
+    S6(c0, c0 - 9, 2);
+    S8(c0, 0, 3);
+    for (int c1 = 1; c1 < c0; c1 += 1)
+      S5(c0, c1, 3);
+  }
+  S2(10, -7, 0);
+  for (int c1 = -7; c1 <= 1; c1 += 1)
+    S3(10, c1, 1);
+  S6(10, 1, 2);
+  for (int c1 = 1; c1 <= 9; c1 += 1) {
+    S5(10, c1, 3);
+    S1(10, c1, 4);
+  }
+  S1(10, 10, 4);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/durbin_e_s.in b/polly/lib/External/isl/test_inputs/codegen/cloog/durbin_e_s.in
new file mode 100644
index 00000000000..77d32ad73f9
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/durbin_e_s.in
@@ -0,0 +1,3 @@
+{ S5[i0, i1, 3] -> [i0, i1, 3, 4] : i1 <= -1 + i0 and i0 <= 10 and i1 >= 1; S8[i0, 0, 3] -> [i0, 0, 3, 7] : i0 >= 1 and i0 <= 9; S2[i0, -7, 0] -> [i0, -7, 0, 1] : i0 >= 2 and i0 <= 10; S3[i0, i1, 1] -> [i0, i1, 1, 2] : i1 >= -7 and i0 <= 10 and i1 <= -9 + i0; S1[10, i1, 4] -> [10, i1, 4, 0] : i1 >= 1 and i1 <= 10; S7[1, 0, 0] -> [1, 0, 0, 6]; S4[1, 0, 0] -> [1, 0, 0, 3]; S6[i0, -9 + i0, 2] -> [i0, -9 + i0, 2, 5] : i0 >= 2 and i0 <= 10 }
+{  :  }
+{ [i, j, k, l] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/emploi.c b/polly/lib/External/isl/test_inputs/codegen/cloog/emploi.c
new file mode 100644
index 00000000000..80ee37d5921
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/emploi.c
@@ -0,0 +1,5 @@
+for (int c0 = 1; c0 <= n; c0 += 1) {
+  S1(c0);
+  for (int c1 = 1; c1 <= m; c1 += 1)
+    S2(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/emploi.in b/polly/lib/External/isl/test_inputs/codegen/cloog/emploi.in
new file mode 100644
index 00000000000..57016623a25
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/emploi.in
@@ -0,0 +1,3 @@
+[m, n] -> { S1[i0] -> [i0, 0, 0] : (i0 >= 1 and i0 <= n and i0 <= 2m) or (i0 >= 1 and i0 <= n and i0 >= m); S2[i0, i1] -> [i0, i1, 1] : i0 >= 1 and i0 <= n and i1 >= 1 and i1 <= m }
+[m, n] -> {  :  }
+[m, n] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/equality.c b/polly/lib/External/isl/test_inputs/codegen/cloog/equality.c
new file mode 100644
index 00000000000..2535b8b975e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/equality.c
@@ -0,0 +1,7 @@
+for (int c0 = 0; c0 <= 5; c0 += 1)
+  for (int c1 = c0 <= 2 ? 2 * c0 : 4; c1 <= (c0 >= 2 ? 2 * c0 : 4); c1 += 1) {
+    if (c1 == 2 * c0)
+      S1(c0, 2 * c0);
+    if (c1 == 4)
+      S2(c0, 4);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/equality.in b/polly/lib/External/isl/test_inputs/codegen/cloog/equality.in
new file mode 100644
index 00000000000..677fa901324
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/equality.in
@@ -0,0 +1,3 @@
+{ S1[i0, 2i0] -> [i0, 2i0, 0] : i0 >= 0 and i0 <= 5; S2[i0, 4] -> [i0, 4, 1] : i0 >= 0 and i0 <= 5 }
+{  :  }
+{ [i, j, k] -> atomic[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/equality2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/equality2.c
new file mode 100644
index 00000000000..be22da28be4
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/equality2.c
@@ -0,0 +1,8 @@
+for (int c0 = 1; c0 <= 10000; c0 += 1)
+  for (int c1 = 1000; c1 <= 1016; c1 += 1)
+    for (int c2 = 1; c2 < 2 * c1 - 1998; c2 += 1) {
+      if (c1 <= 1008 && c2 + 1999 == 2 * c1)
+        S2(c0, c1, 2 * c1 - 1999, 1, c0, 2 * c1 - 1000, 1, 2, c0, c1 - 499, 2 * c1 - 1999, c0, 2 * c1 - 1999, c1 - 999, c1 - 999);
+      if (c2 == 1 && c1 % 2 == 0)
+        S1(c0, c1, 1, 2, c0, (c1 / 2) + 1, c1 - 999, c0, c1 - 999, (c1 / 2) - 499, (c1 / 2) - 499);
+    }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/equality2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/equality2.in
new file mode 100644
index 00000000000..95b3f793407
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/equality2.in
@@ -0,0 +1,3 @@
+{ S1[i0, i1, 1, 2, i0, i5, -999 + i1, i0, -999 + i1, i9, i10] -> [i0, i1, 1, 2, i0, n, -999 + i1, i0, -999 + i1, r, s, 0, 0, 0, 0, 0] : 2s = -998 + i1 and 2n = 2 + i1 and 2i10 = -998 + i1 and 2i5 = 2 + i1 and 2i9 = -998 + i1 and 2r = -998 + i1 and i0 >= 1 and i0 <= 10000 and i1 >= 1000 and i1 <= 1016; S2[i0, i1, -1999 + 2i1, 1, i0, -1000 + 2i1, 1, 2, i0, -499 + i1, -1999 + 2i1, i0, -1999 + 2i1, -999 + i1, -999 + i1] -> [i0, i1, -1999 + 2i1, 1, i0, -1000 + 2i1, 1, 2, i0, -499 + i1, -1999 + 2i1, i0, -1999 + 2i1, -999 + i1, -999 + i1, 1] : i0 >= 1 and i0 <= 10000 and i1 >= 1000 and i1 <= 1008 }
+{  :  }
+{ [i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x] -> atomic[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/esced.c b/polly/lib/External/isl/test_inputs/codegen/cloog/esced.c
new file mode 100644
index 00000000000..f7c7ee0678c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/esced.c
@@ -0,0 +1,5 @@
+for (int c0 = 1; c0 <= m; c0 += 1) {
+  S1(c0);
+  for (int c1 = 1; c1 <= n; c1 += 1)
+    S2(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/esced.in b/polly/lib/External/isl/test_inputs/codegen/cloog/esced.in
new file mode 100644
index 00000000000..3bef6fd90f0
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/esced.in
@@ -0,0 +1,3 @@
+[n, m] -> { S1[i0] -> [i0, 0, 0] : i0 >= 1 and i0 <= m; S2[i0, i1] -> [i0, i1, 1] : i0 >= 1 and i0 <= m and i1 >= 1 and i1 <= n }
+[n, m] -> {  :  }
+[n, m] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/ex1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/ex1.c
new file mode 100644
index 00000000000..2627f23ddde
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/ex1.c
@@ -0,0 +1,15 @@
+{
+  for (int c0 = 0; c0 <= 14; c0 += 1)
+    for (int c1 = 0; c1 < n - 14; c1 += 1)
+      S1(c0, c1);
+  for (int c0 = 15; c0 <= n; c0 += 1) {
+    for (int c1 = 0; c1 <= 9; c1 += 1)
+      S1(c0, c1);
+    for (int c1 = 10; c1 < n - 14; c1 += 1) {
+      S1(c0, c1);
+      S2(c0, c1);
+    }
+    for (int c1 = n - 14; c1 <= n; c1 += 1)
+      S2(c0, c1);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/ex1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/ex1.in
new file mode 100644
index 00000000000..457b834afe2
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/ex1.in
@@ -0,0 +1,3 @@
+[n] -> { S2[i0, i1] -> [i0, i1, 1] : i0 >= 15 and i0 <= n and i1 >= 10 and i1 <= n; S1[i0, i1] -> [i0, i1, 0] : i0 >= 0 and i0 <= n and i1 >= 0 and i1 <= -15 + n }
+[n] -> {  : n >= 25 }
+[n] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/faber.c b/polly/lib/External/isl/test_inputs/codegen/cloog/faber.c
new file mode 100644
index 00000000000..b398206b260
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/faber.c
@@ -0,0 +1,155 @@
+{
+  for (int c0 = 0; c0 <= 36; c0 += 1) {
+    for (int c1 = -6; c1 < c0 / 14 - 5; c1 += 1) {
+      for (int c2 = -((-2 * c1 + 3) / 5) + 9; c2 <= c1 + 12; c2 += 1)
+        S6(c0, c1, c2);
+      for (int c2 = c1 + 24; c2 <= -2 * c1 + 24; c2 += 1)
+        S2(c0, c1, c2);
+      for (int c2 = -2 * c1 + 30; c2 <= c1 + 48; c2 += 1)
+        S1(c0, c1, c2);
+    }
+    for (int c1 = c0 / 14 - 5; c1 < 0; c1 += 1) {
+      if (c1 >= -3 && 2 * c0 >= 7 * c1 + 42)
+        S7(c0, c1, 6);
+      for (int c2 = max(c1 - (6 * c0 + 77) / 77 + 13, -((-2 * c1 + 3) / 5) + 9); c2 <= c1 + 12; c2 += 1)
+        S6(c0, c1, c2);
+      for (int c2 = c1 - (3 * c0 + 14) / 14 + 49; c2 <= c1 + 48; c2 += 1)
+        S1(c0, c1, c2);
+    }
+    S3(c0, 0, 0);
+    S10(c0, 0, 0);
+    for (int c2 = 1; c2 <= 5; c2 += 1)
+      S3(c0, 0, c2);
+    for (int c2 = 6; c2 <= 2 * c0 / 21 + 4; c2 += 1) {
+      S3(c0, 0, c2);
+      S7(c0, 0, c2);
+    }
+    for (int c2 = max(6, 2 * c0 / 21 + 5); c2 <= -((6 * c0 + 77) / 77) + 12; c2 += 1)
+      S3(c0, 0, c2);
+    for (int c2 = -((6 * c0 + 77) / 77) + 13; c2 <= 12; c2 += 1) {
+      S3(c0, 0, c2);
+      S6(c0, 0, c2);
+    }
+    for (int c2 = 13; c2 <= 24; c2 += 1)
+      S3(c0, 0, c2);
+    for (int c2 = -((3 * c0 + 14) / 14) + 49; c2 <= 48; c2 += 1)
+      S1(c0, 0, c2);
+    for (int c1 = 1; c1 <= 18; c1 += 1) {
+      for (int c2 = -8 * c1; c2 <= min(6, -8 * c1 + 24); c2 += 1)
+        S3(c0, c1, c2);
+      if (c1 == 2) {
+        S3(c0, 2, 7);
+      } else if (c0 >= 35 && c1 == 1) {
+        S3(c0, 1, 7);
+        S7(c0, 1, 7);
+      } else if (c0 <= 34 && c1 == 1)
+        S3(c0, 1, 7);
+      for (int c2 = 8; c2 <= min(-8 * c1 + 24, c1 - (6 * c0 + 77) / 77 + 12); c2 += 1)
+        S3(c0, c1, c2);
+      for (int c2 = max(-8 * c1 + 25, c1 - (6 * c0 + 77) / 77 + 13); c2 <= c1 + 12; c2 += 1)
+        S6(c0, c1, c2);
+      if (c1 == 1) {
+        for (int c2 = -((6 * c0 + 77) / 77) + 14; c2 <= 13; c2 += 1) {
+          S3(c0, 1, c2);
+          S6(c0, 1, c2);
+        }
+        for (int c2 = 14; c2 <= 16; c2 += 1)
+          S3(c0, 1, c2);
+      }
+      for (int c2 = c1 - (3 * c0 + 14) / 14 + 49; c2 <= c1 + 48; c2 += 1)
+        S1(c0, c1, c2);
+    }
+    for (int c1 = 19; c1 <= 24; c1 += 1) {
+      for (int c2 = -8 * c1; c2 <= -8 * c1 + 24; c2 += 1)
+        S3(c0, c1, c2);
+      for (int c2 = c1 - (6 * c0 + 77) / 77 + 13; c2 <= 30; c2 += 1)
+        S6(c0, c1, c2);
+    }
+  }
+  for (int c0 = 37; c0 <= 218; c0 += 1) {
+    for (int c1 = (c0 + 5) / 14 - 8; c1 < min(0, c0 / 14 - 5); c1 += 1) {
+      if (c0 <= 46 && c1 == -3)
+        S7(c0, -3, 6);
+      if (-77 * ((-3 * c1 + 1) / 5) + 447 >= 6 * c0)
+        S6(c0, c1, -((-2 * c1 + 3) / 5) + 9);
+      for (int c2 = c1 + 24; c2 <= -2 * c1 + 24; c2 += 1)
+        S2(c0, c1, c2);
+      for (int c2 = -2 * c1 + 30; c2 <= c1 - (3 * c0 + 17) / 14 + 56; c2 += 1)
+        S1(c0, c1, c2);
+    }
+    if (c0 <= 148)
+      for (int c1 = max(0, (c0 + 5) / 14 - 8); c1 < c0 / 14 - 5; c1 += 1) {
+        if (c1 == 0)
+          S2(c0, 0, 24);
+        for (int c2 = max(c1 + 24, -2 * c1 + 30); c2 <= c1 - (3 * c0 + 17) / 14 + 56; c2 += 1)
+          S1(c0, c1, c2);
+      }
+    if (c0 >= 79 && c0 % 14 >= 9) {
+      for (int c2 = max((c0 - 70) / 14 + 24, (c0 - 70) / 14 - (3 * c0 + 14) / 14 + 49); c2 <= (c0 - 70) / 14 - (3 * c0 + 17) / 14 + 56; c2 += 1)
+        S1(c0, c0 / 14 - 5, c2);
+    } else if (c0 <= 69 && c0 % 14 >= 9) {
+      if (c0 <= 41)
+        S7(c0, -3, 6);
+      S6(c0, c0 / 14 - 5, 8);
+      for (int c2 = -((-c0 + 83) / 14) - (3 * c0 + 14) / 14 + 49; c2 <= -((-c0 + 83) / 14) - (3 * c0 + 17) / 14 + 56; c2 += 1)
+        S1(c0, c0 / 14 - 5, c2);
+    }
+    for (int c1 = (c0 + 5) / 14 - 5; c1 < 0; c1 += 1) {
+      if (7 * c1 + 114 >= 2 * c0)
+        S7(c0, c1, 6);
+      for (int c2 = max(8, c1 - (6 * c0 + 77) / 77 + 13); c2 <= c1 - (6 * c0 + 91) / 77 + 15; c2 += 1)
+        S6(c0, c1, c2);
+      for (int c2 = c1 - (3 * c0 + 14) / 14 + 49; c2 <= c1 - (3 * c0 + 17) / 14 + 56; c2 += 1)
+        S1(c0, c1, c2);
+    }
+    for (int c1 = max(0, (c0 + 5) / 14 - 5); c1 < c0 / 14 - 2; c1 += 1) {
+      for (int c2 = max(c1, -2 * c1 + 6); c2 <= min(-2 * c1 + 24, (2 * c0 - 7 * c1 - 10) / 21 + 1); c2 += 1)
+        S9(c0, c1, c2);
+      if (c1 >= 1 && c1 <= 6 && 14 * c1 + 46 >= c0)
+        S9(c0, c1, c1 + 5);
+      for (int c2 = max(c1 + 6, (2 * c0 - 7 * c1 - 10) / 21 + 2); c2 <= (2 * c1 + 1) / 5 + 7; c2 += 1) {
+        S7(c0, c1, c2);
+        S9(c0, c1, c2);
+      }
+      if (c1 <= 3 && 7 * c1 + 21 * ((2 * c1 + 41) / 5) >= 2 * c0 + 12)
+        S9(c0, c1, (2 * c1 + 1) / 5 + 8);
+      for (int c2 = (2 * c1 + 1) / 5 + 9; c2 <= c1 - (6 * c0 + 91) / 77 + 15; c2 += 1) {
+        S6(c0, c1, c2);
+        S9(c0, c1, c2);
+      }
+      for (int c2 = max(max(c1 + 6, c1 - (6 * c0 + 91) / 77 + 16), (2 * c0 - 7 * c1 - 10) / 21 + 2); c2 <= -2 * c1 + 24; c2 += 1)
+        S9(c0, c1, c2);
+      for (int c2 = max(c1, -2 * c1 + 30); c2 <= min(c1 + 24, c1 - (3 * c0 + 17) / 14 + 47); c2 += 1)
+        S8(c0, c1, c2);
+      for (int c2 = max(c1 + 24, c1 - (3 * c0 + 14) / 14 + 49); c2 <= c1 - (3 * c0 + 17) / 14 + 56; c2 += 1)
+        S1(c0, c1, c2);
+    }
+    for (int c1 = c0 / 14 - 2; c1 <= 18; c1 += 1) {
+      for (int c2 = c1 + 6; c2 <= min((2 * c1 + 1) / 5 + 7, floord(2 * c0 - 7 * c1, 21) + 4); c2 += 1)
+        S7(c0, c1, c2);
+      for (int c2 = max(6, (c0 + 5) / 14 + 1); c2 <= min(min(c1, c0 / 14 + 3), -c1 + c1 / 2 + 18); c2 += 1)
+        S5(c0, c1, c2);
+      for (int c2 = max(max(c1 + 6, c1 - (6 * c0 + 77) / 77 + 13), (2 * c1 + 1) / 5 + 9); c2 <= c1 - (6 * c0 + 91) / 77 + 15; c2 += 1)
+        S6(c0, c1, c2);
+      for (int c2 = max(c1 + (3 * c0 + 3) / 14 - 40, -c1 + (c1 + 1) / 2 + 21); c2 <= min(c1, c1 + 3 * c0 / 14 - 33); c2 += 1)
+        S4(c0, c1, c2);
+      for (int c2 = max(c1, c1 - (3 * c0 + 14) / 14 + 40); c2 <= min(c1 + 24, c1 - (3 * c0 + 17) / 14 + 47); c2 += 1)
+        S8(c0, c1, c2);
+      for (int c2 = max(c1 + 24, c1 - (3 * c0 + 14) / 14 + 49); c2 <= c1 - (3 * c0 + 17) / 14 + 56; c2 += 1)
+        S1(c0, c1, c2);
+    }
+    for (int c1 = 19; c1 <= 24; c1 += 1) {
+      for (int c2 = max(c1 - 12, (c0 + 5) / 14 + 1); c2 <= min(c0 / 14 + 3, -c1 + c1 / 2 + 18); c2 += 1)
+        S5(c0, c1, c2);
+      for (int c2 = max(max(c1 - 12, c1 + (3 * c0 + 3) / 14 - 40), -c1 + (c1 + 1) / 2 + 21); c2 <= min(c1, c1 + 3 * c0 / 14 - 33); c2 += 1)
+        S4(c0, c1, c2);
+      for (int c2 = max(c1 + 6, c1 - (6 * c0 + 77) / 77 + 13); c2 <= min(30, c1 - (6 * c0 + 91) / 77 + 15); c2 += 1)
+        S6(c0, c1, c2);
+      for (int c2 = max(c1, c1 - (3 * c0 + 14) / 14 + 40); c2 <= min(c1 + 24, c1 - (3 * c0 + 17) / 14 + 47); c2 += 1)
+        S8(c0, c1, c2);
+    }
+    for (int c1 = 25; c1 <= min(42, -((3 * c0 + 17) / 14) + 71); c1 += 1)
+      for (int c2 = max(c1 - 12, c1 + (3 * c0 + 3) / 14 - 40); c2 <= min(min(30, c1), c1 + 3 * c0 / 14 - 33); c2 += 1)
+        S4(c0, c1, c2);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/faber.in b/polly/lib/External/isl/test_inputs/codegen/cloog/faber.in
new file mode 100644
index 00000000000..4f78a5561d0
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/faber.in
@@ -0,0 +1,3 @@
+{ S5[idx4, idx5, idx6] -> [idx4, idx5, idx6, 4] : 14idx6 <= 42 + idx4 and 14idx6 >= 6 + idx4 and idx6 <= idx5 and idx6 >= -12 + idx5 and idx6 >= 6 and idx6 <= 30 and 2idx6 <= 36 - idx5; S8[idx4, idx5, idx6] -> [idx4, idx5, idx6, 7] : 14idx6 >= 546 - 3idx4 + 14idx5 and 14idx6 <= 654 - 3idx4 + 14idx5 and idx6 >= idx5 and idx6 <= 24 + idx5 and idx5 >= 0 and idx5 <= 24 and idx6 >= 30 - 2idx5; S2[idx4, idx5, idx6] -> [idx4, idx5, idx6, 1] : 14idx5 <= -84 + idx4 and 14idx5 >= -120 + idx4 and idx6 >= 24 + idx5 and idx6 <= 48 + idx5 and idx5 >= -6 and idx5 <= 18 and idx6 <= 24 - 2idx5; S10[idx4, idx5, idx6] -> [idx4, idx5, idx6, 9] : 7idx6 <= idx4 - 28idx5 and 7idx6 >= -36 + idx4 - 28idx5 and idx6 >= idx5 and idx6 <= 24 + idx5 and idx5 >= 0 and idx5 <= 24 and idx6 <= -2idx5; S3[idx4, idx5, idx6] -> [idx4, idx5, idx6, 2] : idx4 >= 0 and idx4 <= 36 and idx6 >= -8idx5 and idx6 <= 24 - 8idx5 and idx5 >= 0 and idx5 <= 24; S1[idx4, idx5, idx6] -> [idx4, idx5, idx6, 0] : 14idx6 >= 672 - 3idx4 + 14idx5 and 14idx6 <= 780 - 3idx4 + 14idx5 and idx6 >= 24 + idx5 and idx6 <= 48 + idx5 and idx5 >= -6 and idx5 <= 18 and idx6 >= 30 - 2idx5; S7[idx4, idx5, idx6] -> [idx4, idx5, idx6, 6] : 21idx6 <= 84 + 2idx4 - 7idx5 and 21idx6 >= 12 + 2idx4 - 7idx5 and idx6 <= 12 + idx5 and idx6 >= 6 + idx5 and idx6 >= 6 and idx6 <= 30 and 5idx6 <= 36 + 2idx5; S9[idx4, idx5, idx6] -> [idx4, idx5, idx6, 8] : 14idx5 <= -42 + idx4 and 14idx5 >= -78 + idx4 and idx6 >= idx5 and idx6 <= 24 + idx5 and idx5 >= 0 and idx5 <= 24 and idx6 <= 24 - 2idx5 and idx6 >= 6 - 2idx5; S4[idx4, idx5, idx6] -> [idx4, idx5, idx6, 3] : 14idx6 <= -462 + 3idx4 + 14idx5 and 14idx6 >= -570 + 3idx4 + 14idx5 and idx6 <= idx5 and idx6 >= -12 + idx5 and idx6 >= 6 and idx6 <= 30 and 2idx6 >= 42 - idx5; S6[idx4, idx5, idx6] -> [idx4, idx5, idx6, 5] : 77idx6 >= 924 - 6idx4 + 77idx5 and 77idx6 <= 1140 - 6idx4 + 77idx5 and idx6 <= 12 + idx5 and idx6 >= 6 + idx5 and idx6 >= 6 and idx6 <= 30 and 5idx6 >= 42 + 2idx5 }
+{  :  }
+{ [i, j, k, l] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-1-1-2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-1-1-2.c
new file mode 100644
index 00000000000..45fe75ef372
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-1-1-2.c
@@ -0,0 +1,9 @@
+{
+  S3(1, 1);
+  for (int c0 = 2; c0 <= M; c0 += 1) {
+    S1(c0, 1);
+    for (int c1 = 2; c1 < c0; c1 += 1)
+      S2(c0, c1);
+    S4(c0, c0);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-1-1-2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-1-1-2.in
new file mode 100644
index 00000000000..ede8856d545
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-1-1-2.in
@@ -0,0 +1,3 @@
+[M] -> { S2[i0, i1] -> [i0, i1, 1] : i1 <= -1 + i0 and i1 >= 2 and i0 <= M; S4[i0, i0] -> [i0, i0, 3] : M >= 3 and i0 <= M and i0 >= 2; S1[i0, 1] -> [i0, 1, 0] : M >= 3 and i0 <= M and i0 >= 2; S3[1, 1] -> [1, 1, 2] : M >= 3 }
+[M] -> {  : M >= 3 }
+[M] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-2-1-2-3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-2-1-2-3.c
new file mode 100644
index 00000000000..cfc3e7d7bea
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-2-1-2-3.c
@@ -0,0 +1,10 @@
+{
+  S3(1, 0);
+  for (int c2 = 2; c2 <= M; c2 += 1)
+    S1(1, 1, c2);
+  for (int c0 = 2; c0 <= M; c0 += 1) {
+    S4(c0, 0);
+    for (int c2 = c0 + 1; c2 <= M; c2 += 1)
+      S2(c0, 1, c2);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-2-1-2-3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-2-1-2-3.in
new file mode 100644
index 00000000000..57863cc5727
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-2-1-2-3.in
@@ -0,0 +1,3 @@
+[M] -> { S1[1, 1, i2] -> [1, 1, i2, 0] : M >= 3 and i2 <= M and i2 >= 2; S2[i0, 1, i2] -> [i0, 1, i2, 1] : i2 >= 1 + i0 and i0 >= 2 and i2 <= M; S4[i0, 0] -> [i0, 0, 0, 3] : i0 >= 2 and M >= 3 and i0 <= M; S3[1, 0] -> [1, 0, 0, 2] : M >= 3 }
+[M] -> {  : M >= 3 }
+[M] -> { [i, j, k, l] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-3-1-2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-3-1-2.c
new file mode 100644
index 00000000000..9300d189691
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-3-1-2.c
@@ -0,0 +1,17 @@
+{
+  S3(2, 1);
+  S1(3, 1);
+  for (int c0 = 4; c0 <= M + 1; c0 += 1) {
+    S1(c0, 1);
+    for (int c1 = 2; c1 < (c0 + 1) / 2; c1 += 1)
+      S2(c0, c1);
+    if (c0 % 2 == 0)
+      S4(c0, c0 / 2);
+  }
+  for (int c0 = M + 2; c0 <= 2 * M; c0 += 1) {
+    for (int c1 = -M + c0; c1 < (c0 + 1) / 2; c1 += 1)
+      S2(c0, c1);
+    if (c0 % 2 == 0)
+      S4(c0, c0 / 2);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-3-1-2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-3-1-2.in
new file mode 100644
index 00000000000..c40383d8b6e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/forwardsub-3-1-2.in
@@ -0,0 +1,3 @@
+[M] -> { S2[i0, i1] -> [i0, i1, 1] : 2i1 <= -1 + i0 and i1 >= 2 and i1 >= -M + i0; S4[i0, i1] -> [i0, j, 3] : 2j = i0 and 2i1 = i0 and M >= 3 and i0 <= 2M and i0 >= 4; S1[i0, 1] -> [i0, 1, 0] : M >= 3 and i0 <= 1 + M and i0 >= 3; S3[2, 1] -> [2, 1, 2] : M >= 3 }
+[M] -> {  : M >= 3 }
+[M] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/gauss.c b/polly/lib/External/isl/test_inputs/codegen/cloog/gauss.c
new file mode 100644
index 00000000000..13a8e137173
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/gauss.c
@@ -0,0 +1,7 @@
+for (int c0 = 1; c0 < M; c0 += 1)
+  for (int c1 = c0 + 1; c1 <= M; c1 += 1) {
+    for (int c3 = c0 + 1; c3 <= M; c3 += 1)
+      S2(c0, c3, c1);
+    for (int c3 = 1; c3 < c0; c3 += 1)
+      S1(c0, c3, c1);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/gauss.in b/polly/lib/External/isl/test_inputs/codegen/cloog/gauss.in
new file mode 100644
index 00000000000..439e72e7f34
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/gauss.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0, i1, i2] -> [i0, i2] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= -1 + i0 and i2 >= 1 + i0 and i2 <= M; S2[i0, i1, i2] -> [i0, i2] : i0 >= 1 and i0 <= M and i1 >= 1 + i0 and i1 <= M and i2 >= 1 + i0 and i2 <= M }
+[M] -> {  :  }
+[M] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/gesced.c b/polly/lib/External/isl/test_inputs/codegen/cloog/gesced.c
new file mode 100644
index 00000000000..a36715ca3db
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/gesced.c
@@ -0,0 +1,16 @@
+{
+  for (int c0 = 1; c0 <= N; c0 += 1)
+    S1(c0);
+  for (int c0 = N + 1; c0 <= 2 * N; c0 += 1)
+    for (int c1 = 1; c1 <= N; c1 += 1)
+      S2(c1, -N + c0);
+  for (int c0 = 2 * N + 1; c0 <= M + N; c0 += 1) {
+    for (int c1 = 1; c1 <= N; c1 += 1)
+      S3(c1, -2 * N + c0);
+    for (int c1 = 1; c1 <= N; c1 += 1)
+      S2(c1, -N + c0);
+  }
+  for (int c0 = M + N + 1; c0 <= M + 2 * N; c0 += 1)
+    for (int c1 = 1; c1 <= N; c1 += 1)
+      S3(c1, -2 * N + c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/gesced.in b/polly/lib/External/isl/test_inputs/codegen/cloog/gesced.in
new file mode 100644
index 00000000000..6043f4b3fb9
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/gesced.in
@@ -0,0 +1,3 @@
+[M, N] -> { S3[i0, i1] -> [2N + i1] : i0 >= 1 and i0 <= N and i1 >= 1 and i1 <= M; S2[i0, i1] -> [N + i1] : i0 >= 1 and i0 <= N and i1 >= 1 and i1 <= M; S1[i0] -> [i0] : i0 >= 1 and i0 <= N }
+[M, N] -> {  : N <= M and M >= 2 and N >= 2 }
+[M, N] -> { [i] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/gesced2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/gesced2.c
new file mode 100644
index 00000000000..0baf706ed99
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/gesced2.c
@@ -0,0 +1,20 @@
+{
+  for (int c0 = 1; c0 <= 4; c0 += 1)
+    for (int c1 = 5; c1 < M - 9; c1 += 1)
+      S1(c0, c1);
+  for (int c0 = 5; c0 < M - 9; c0 += 1) {
+    for (int c1 = -c0 + 1; c1 <= 4; c1 += 1)
+      S2(c0 + c1, c0);
+    for (int c1 = 5; c1 <= min(M - 10, M - c0); c1 += 1) {
+      S1(c0, c1);
+      S2(c0 + c1, c0);
+    }
+    for (int c1 = M - c0 + 1; c1 < M - 9; c1 += 1)
+      S1(c0, c1);
+    for (int c1 = M - 9; c1 <= M - c0; c1 += 1)
+      S2(c0 + c1, c0);
+  }
+  for (int c0 = M - 9; c0 <= M; c0 += 1)
+    for (int c1 = 5; c1 < M - 9; c1 += 1)
+      S1(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/gesced2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/gesced2.in
new file mode 100644
index 00000000000..a5d5bd27e28
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/gesced2.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0, i1] -> [i0, i1] : i0 >= 1 and i0 <= M and i1 >= 5 and i1 <= -10 + M; S2[i0, i1] -> [i1, i0 - i1] : i0 >= 1 and i0 <= M and i1 >= 5 and i1 <= -10 + M }
+[M] -> {  : M >= 16 }
+[M] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/gesced3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/gesced3.c
new file mode 100644
index 00000000000..c163ed0735e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/gesced3.c
@@ -0,0 +1,10 @@
+{
+  for (int c0 = M + 1; c0 <= 2 * M; c0 += 1)
+    S1(-M + c0);
+  for (int c0 = 2 * M + 1; c0 <= M + N; c0 += 1) {
+    S2(-2 * M + c0);
+    S1(-M + c0);
+  }
+  for (int c0 = M + N + 1; c0 <= 2 * M + N; c0 += 1)
+    S2(-2 * M + c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/gesced3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/gesced3.in
new file mode 100644
index 00000000000..29f57f32176
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/gesced3.in
@@ -0,0 +1,3 @@
+[M, N] -> { S2[i0] -> [2M + i0] : i0 >= 1 and i0 <= N; S1[i0] -> [M + i0] : i0 >= 1 and i0 <= N }
+[M, N] -> {  : N >= M and M >= 2 }
+[M, N] -> { [i] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/guide.c b/polly/lib/External/isl/test_inputs/codegen/cloog/guide.c
new file mode 100644
index 00000000000..bc48f1efe8a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/guide.c
@@ -0,0 +1,6 @@
+{
+  for (int c0 = 1; c0 <= N; c0 += 1)
+    S1(c0);
+  for (int c0 = N + 1; c0 <= 2 * N; c0 += 1)
+    S2(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/guide.in b/polly/lib/External/isl/test_inputs/codegen/cloog/guide.in
new file mode 100644
index 00000000000..38c418c6c65
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/guide.in
@@ -0,0 +1,3 @@
+[M, N] -> { S1[i0] -> [i0, 0] : (i0 >= 1 and i0 <= N and i0 <= 2M) or (i0 >= 1 and i0 <= N and i0 >= M); S2[i0] -> [i0, 1] : i0 >= 1 + N and i0 <= 2N }
+[M, N] -> {  :  }
+[M, N] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/iftest.c b/polly/lib/External/isl/test_inputs/codegen/cloog/iftest.c
new file mode 100644
index 00000000000..0f41fb7902a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/iftest.c
@@ -0,0 +1,2 @@
+for (int c0 = 1; c0 <= n; c0 += 1)
+  S1(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/iftest.in b/polly/lib/External/isl/test_inputs/codegen/cloog/iftest.in
new file mode 100644
index 00000000000..741399d48a0
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/iftest.in
@@ -0,0 +1,3 @@
+[m, n] -> { S1[i0] -> [i0, 0] : (i0 >= 1 and i0 <= n and i0 >= m) or (i0 >= 1 and i0 <= n and i0 <= 2m) }
+[m, n] -> {  :  }
+[m, n] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/iftest2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/iftest2.c
new file mode 100644
index 00000000000..8e3e4c1b90b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/iftest2.c
@@ -0,0 +1,3 @@
+for (int c0 = 1; c0 <= N; c0 += 1)
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    S1(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/iftest2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/iftest2.in
new file mode 100644
index 00000000000..a2e3212ece3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/iftest2.in
@@ -0,0 +1,3 @@
+[M, N] -> { S1[i0, i1] -> [i0, i1, 0] : (i0 >= 1 and i0 <= N and i0 >= M and i1 >= 1 and i1 <= M) or (i0 >= 1 and i0 <= N and i0 <= 2M and i1 >= 1 and i1 <= M) }
+[M, N] -> {  :  }
+[M, N] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/infinite2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/infinite2.c
new file mode 100644
index 00000000000..bbb6d6e750a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/infinite2.c
@@ -0,0 +1,9 @@
+{
+  for (int c0 = 1; c0 <= N; c0 += 1) {
+    S1(c0);
+    for (int c1 = 1; c1 <= M; c1 += 1)
+      S2(c0, c1);
+  }
+  for (int c0 = N + 1; 1; c0 += 1)
+    S1(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/infinite2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/infinite2.in
new file mode 100644
index 00000000000..64fb5d55735
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/infinite2.in
@@ -0,0 +1,3 @@
+[M, N] -> { S2[i0, i1] -> [i0, i1, 1] : i0 >= 1 and i0 <= N and i1 >= 1 and i1 <= M; S1[i0] -> [i0, 0, 0] : i0 >= 1 }
+[M, N] -> {  : M >= 1 and N >= 1 }
+[M, N] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/jacobi-shared.c b/polly/lib/External/isl/test_inputs/codegen/cloog/jacobi-shared.c
new file mode 100644
index 00000000000..50840a86877
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/jacobi-shared.c
@@ -0,0 +1,3 @@
+if (((t1 + 31) % 32) + g2 >= 2 && N >= ((t1 + 31) % 32) + g2 + 2 && (h0 - 1) % 2 == 0)
+  for (int c0 = max(((t0 + 15) % 16) + 1, ((g1 + t0 + 13) % 16) - g1 + 3); c0 <= min(32, N - g1 - 1); c0 += 16)
+    S1(g1 + c0 - 1, ((t1 + 31) % 32) + g2);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/jacobi-shared.in b/polly/lib/External/isl/test_inputs/codegen/cloog/jacobi-shared.in
new file mode 100644
index 00000000000..d6153244284
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/jacobi-shared.in
@@ -0,0 +1,3 @@
+[T, N, h0, b0, b1, g0, g1, g2, g3, g4, t0, t1] -> { S1[i0, i1] -> [1 - g1 + i0, 1 - g2 + i1, t0, t1] : exists (e0 = [(-1 + h0)/2], e1 = [(-2016b0 - g1)/2048], e2 = [(-992b1 - g2)/1024], e3 = [(-1 + t0 - i0)/16], e4 = [(-1 + t1 - i1)/32]: g0 = h0 and 2e0 = -1 + h0 and 2048e1 = -2016b0 - g1 and 1024e2 = -992b1 - g2 and 16e3 = -1 + t0 - i0 and 32e4 = -1 + t1 - i1 and h0 >= 1 and h0 <= -1 + 2T and i0 >= 2 and i0 <= -2 + N and i1 >= 2 and i1 <= -2 + N and b1 <= 31 and b1 >= 0 and b0 <= 63 and b0 >= 0 and i1 <= 31 + g2 and i1 >= g2 and N >= 4 and i0 >= g1 and i0 <= 31 + g1 and g2 <= -2 + N and g2 >= -29 and g1 <= -2 + N and g1 >= -29 and g1 >= 32b0 and g2 >= 32b1 and 32b0 <= -2 + N and 32b1 <= -2 + N and t0 >= 0 and t0 <= 15 and t1 >= 0 and t1 <= 31) }
+[T, N, h0, b0, b1, g0, g1, g2, g3, g4, t0, t1] -> {  : exists (e0 = [(-32b0 + g1)/2048], e1 = [(-32b1 + g2)/1024]: g0 = h0 and 2048e0 = -32b0 + g1 and 1024e1 = -32b1 + g2 and g2 <= -2 + N and g2 >= -29 and g1 <= -2 + N and g1 >= -29 and b1 >= 0 and b1 <= 31 and b0 <= 63 and 32b1 <= -2 + N and 32b0 <= -2 + N and b0 >= 0 and N >= 4 and h0 >= 0 and h0 <= -1 + 2T and g2 >= 32b1 and g1 >= 32b0 and t0 >= 0 and t0 <= 15 and t1 >= 0 and t1 <= 31) }
+[T, N, h0, b0, b1, g0, g1, g2, g3, g4, t0, t1] -> { [i, j, k, l] -> separate[x] : x >= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/largeur.c b/polly/lib/External/isl/test_inputs/codegen/cloog/largeur.c
new file mode 100644
index 00000000000..faced0a5573
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/largeur.c
@@ -0,0 +1,3 @@
+for (int c0 = 1; c0 <= M; c0 += 1)
+  for (int c1 = 1; c1 <= c0; c1 += 1)
+    S1(c1, c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/largeur.in b/polly/lib/External/isl/test_inputs/codegen/cloog/largeur.in
new file mode 100644
index 00000000000..3a25146b10f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/largeur.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0, i1] -> [i1, i0] : i0 >= 1 and i0 <= M and i1 >= i0 and i1 <= M }
+[M] -> {  : M >= 0 }
+[M] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/levenshtein-1-2-3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/levenshtein-1-2-3.c
new file mode 100644
index 00000000000..0e470b4d73b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/levenshtein-1-2-3.c
@@ -0,0 +1,32 @@
+{
+  S1(0, 0);
+  for (int c0 = 1; c0 <= N; c0 += 1) {
+    S2(c0, 0);
+    for (int c1 = 1; c1 < c0; c1 += 1)
+      S6(c0, c1);
+    S3(c0, c0);
+  }
+  S7(N + 1, 0);
+  for (int c1 = 1; c1 <= N; c1 += 1) {
+    S6(N + 1, c1);
+    S8(N + 1, c1);
+  }
+  for (int c0 = N + 2; c0 < 2 * M - N - 1; c0 += 1) {
+    S7(c0, -N + (N + c0 + 1) / 2 - 1);
+    if ((N - c0) % 2 == 0) {
+      S5(c0, (-N + c0) / 2);
+      S8(c0, (-N + c0) / 2);
+    }
+    for (int c1 = -N + (N + c0) / 2 + 1; c1 < (N + c0 + 1) / 2; c1 += 1) {
+      S6(c0, c1);
+      S8(c0, c1);
+    }
+    if ((N - c0) % 2 == 0) {
+      S4(c0, (N + c0) / 2);
+      S8(c0, (N + c0) / 2);
+    }
+  }
+  for (int c0 = 2 * M - N - 1; c0 < 2 * M - 1; c0 += 1)
+    for (int c1 = -M + c0 + 1; c1 < M; c1 += 1)
+      S6(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/levenshtein-1-2-3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/levenshtein-1-2-3.in
new file mode 100644
index 00000000000..dc9446410e6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/levenshtein-1-2-3.in
@@ -0,0 +1,3 @@
+[M, N] -> { S8[i0, i1] -> [i0, i1, 7] : i0 >= 1 + N and 2i1 <= N + i0 and 2i1 >= -N + i0 and i0 <= -2 + 2M - N and N <= -2 + M and N >= 1; S1[0, 0] -> [0, 0, 0] : N <= -2 + M and N >= 1; S5[i0, i1] -> [i0, j, 4] : 2j = -N + i0 and 2i1 = -N + i0 and i0 >= 2 + N and i0 <= -2 + 2M - N and N >= 1; S7[i0, i1] -> [i0, i1, 6] : i0 >= 1 + N and 2i1 <= -1 - N + i0 and i0 <= -2 + 2M - N and 2i1 >= -2 - N + i0 and N <= -2 + M and N >= 1; S2[i0, 0] -> [i0, 0, 1] : i0 >= 1 and i0 <= N and N <= -2 + M; S3[i0, i0] -> [i0, i0, 2] : i0 >= 1 and i0 <= N and N <= -2 + M; S4[i0, i1] -> [i0, j, 3] : 2j = N + i0 and 2i1 = N + i0 and i0 >= 2 + N and i0 <= -2 + 2M - N and N >= 1; S6[i0, i1] -> [i0, i1, 5] : 2i1 <= -1 + N + i0 and i1 <= -1 + i0 and i1 >= 1 - M + i0 and 2i1 >= 1 - N + i0 and i1 >= 1 and i1 <= -1 + M and N <= -2 + M }
+[M, N] -> {  : N <= -2 + M and N >= 1 }
+[M, N] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/lex.c b/polly/lib/External/isl/test_inputs/codegen/cloog/lex.c
new file mode 100644
index 00000000000..e81e97e75a9
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/lex.c
@@ -0,0 +1,4 @@
+for (int c0 = 0; c0 <= 10; c0 += 1) {
+  S2(c0);
+  S1(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/lex.in b/polly/lib/External/isl/test_inputs/codegen/cloog/lex.in
new file mode 100644
index 00000000000..7a268b605f0
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/lex.in
@@ -0,0 +1,3 @@
+{ S2[i0] -> [i0, -1, 0] : i0 >= 0 and i0 <= 10; S1[i0] -> [i0, 0, 0] : i0 >= 0 and i0 <= 10 }
+{  :  }
+{ [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/lineality-1-2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/lineality-1-2.c
new file mode 100644
index 00000000000..bb1e0718719
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/lineality-1-2.c
@@ -0,0 +1,8 @@
+for (int c0 = 1; c0 <= M; c0 += 1) {
+  for (int c1 = 1; c1 < c0; c1 += 1)
+    S1(c0, c1);
+  S1(c0, c0);
+  S2(c0, c0);
+  for (int c1 = c0 + 1; c1 <= M; c1 += 1)
+    S1(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/lineality-1-2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/lineality-1-2.in
new file mode 100644
index 00000000000..b7d06d05953
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/lineality-1-2.in
@@ -0,0 +1,3 @@
+[M] -> { S2[i0, i0] -> [i0, i0, 1] : i0 >= 1 and i0 <= M; S1[i0, i1] -> [i0, i1, 0] : i0 >= 1 and i1 >= 1 and i0 <= M and i1 <= M }
+[M] -> {  : M >= 2 }
+[M] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/lineality-2-1-2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/lineality-2-1-2.c
new file mode 100644
index 00000000000..23f50d24818
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/lineality-2-1-2.c
@@ -0,0 +1,12 @@
+for (int c0 = 1; c0 <= M; c0 += 1) {
+  for (int c1 = 1; c1 <= min(M, c0 + 1); c1 += 1)
+    S1(c0, c1);
+  if (c0 + 1 >= M) {
+    S2(c0, c0 + 2);
+  } else {
+    S1(c0, c0 + 2);
+    S2(c0, c0 + 2);
+  }
+  for (int c1 = c0 + 3; c1 <= M; c1 += 1)
+    S1(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/lineality-2-1-2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/lineality-2-1-2.in
new file mode 100644
index 00000000000..6ed9040ab03
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/lineality-2-1-2.in
@@ -0,0 +1,3 @@
+[M] -> { S2[i0, 2 + i0] -> [i0, 2 + i0, 1] : i0 >= 1 and i0 <= M; S1[i0, i1] -> [i0, i1, 0] : i0 >= 1 and i1 >= 1 and i0 <= M and i1 <= M }
+[M] -> {  : M >= 2 }
+[M] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/logo.c b/polly/lib/External/isl/test_inputs/codegen/cloog/logo.c
new file mode 100644
index 00000000000..abe35b5916f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/logo.c
@@ -0,0 +1,17 @@
+{
+  for (int c1 = 0; c1 <= 7; c1 += 1)
+    S1(1, c1);
+  for (int c0 = 2; c0 <= 6; c0 += 1) {
+    for (int c1 = 0; c1 < c0 - 1; c1 += 1)
+      S2(c0, c1);
+    for (int c1 = c0 - 1; c1 <= 4; c1 += 1) {
+      S1(c0, c1);
+      S2(c0, c1);
+    }
+    for (int c1 = 5; c1 <= 7; c1 += 1)
+      S1(c0, c1);
+  }
+  for (int c0 = 7; c0 <= 8; c0 += 1)
+    for (int c1 = c0 - 1; c1 <= 7; c1 += 1)
+      S1(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/logo.in b/polly/lib/External/isl/test_inputs/codegen/cloog/logo.in
new file mode 100644
index 00000000000..33e2d88b540
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/logo.in
@@ -0,0 +1,3 @@
+[M] -> { S2[i0, i1] -> [i0, i1, 1] : i0 >= 2 and i0 <= 6 and i1 >= 0 and i1 <= 4; S1[i0, i1] -> [i0, i1, 0] : i0 >= 1 and i1 <= 7 and i1 >= -1 + i0 }
+[M] -> {  :  }
+[M] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/logopar.c b/polly/lib/External/isl/test_inputs/codegen/cloog/logopar.c
new file mode 100644
index 00000000000..70f98e8cd89
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/logopar.c
@@ -0,0 +1,17 @@
+{
+  for (int c1 = 0; c1 <= m; c1 += 1)
+    S1(1, c1);
+  for (int c0 = 2; c0 <= n; c0 += 1) {
+    for (int c1 = 0; c1 < c0 - 1; c1 += 1)
+      S2(c0, c1);
+    for (int c1 = c0 - 1; c1 <= n; c1 += 1) {
+      S1(c0, c1);
+      S2(c0, c1);
+    }
+    for (int c1 = n + 1; c1 <= m; c1 += 1)
+      S1(c0, c1);
+  }
+  for (int c0 = n + 1; c0 <= m + 1; c0 += 1)
+    for (int c1 = c0 - 1; c1 <= m; c1 += 1)
+      S1(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/logopar.in b/polly/lib/External/isl/test_inputs/codegen/cloog/logopar.in
new file mode 100644
index 00000000000..48b70fde5b5
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/logopar.in
@@ -0,0 +1,3 @@
+[m, n] -> { S1[i0, i1] -> [i0, i1, 0] : i0 >= 1 and i1 <= m and i1 >= -1 + i0; S2[i0, i1] -> [i0, i1, 1] : i0 >= 2 and i0 <= n and i1 >= 0 and i1 <= n }
+[m, n] -> {  : n <= m and m >= 0 and n >= 2 }
+[m, n] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/lu.c b/polly/lib/External/isl/test_inputs/codegen/cloog/lu.c
new file mode 100644
index 00000000000..a7fe2b36eed
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/lu.c
@@ -0,0 +1,7 @@
+for (int c0 = 1; c0 <= n; c0 += 1) {
+  for (int c1 = 2; c1 <= n; c1 += 1)
+    for (int c2 = 1; c2 < min(c0, c1); c2 += 1)
+      S2(c2, c1, c0);
+  for (int c3 = c0 + 1; c3 <= n; c3 += 1)
+    S1(c0, c3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/lu.in b/polly/lib/External/isl/test_inputs/codegen/cloog/lu.in
new file mode 100644
index 00000000000..49952ccc46e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/lu.in
@@ -0,0 +1,3 @@
+[n] -> { S2[i0, i1, i2] -> [i2, i1] : i0 >= 1 and i0 <= n and i1 >= 1 + i0 and i1 <= n and i2 >= 1 + i0 and i2 <= n; S1[i0, i1] -> [i0, n] : i0 >= 1 and i0 <= n and i1 >= 1 + i0 and i1 <= n }
+[n] -> {  :  }
+[n] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/lu2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/lu2.c
new file mode 100644
index 00000000000..d5cc912359c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/lu2.c
@@ -0,0 +1,7 @@
+for (int c0 = 1; c0 <= n; c0 += 1) {
+  for (int c1 = 2; c1 <= n; c1 += 1)
+    for (int c2 = 1; c2 < min(c0, c1); c2 += 1)
+      S2(c0, c1, c2, c1, c0);
+  for (int c3 = c0 + 1; c3 <= n; c3 += 1)
+    S1(c0, n, c0, c3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/lu2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/lu2.in
new file mode 100644
index 00000000000..5c71fc0aacc
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/lu2.in
@@ -0,0 +1,3 @@
+[n] -> { S1[i0, n, i0, i3] -> [i0, n, i0, i3, 0, 0] : i0 >= 1 and i0 <= n and i3 >= 1 + i0 and i3 <= n; S2[i0, i1, i2, i1, i0] -> [i0, i1, i2, i1, i0, 1] : i2 >= 1 and i2 <= n and i2 <= -1 + i1 and i1 <= n and i2 <= -1 + i0 and i0 <= n }
+[n] -> {  :  }
+[n] -> { [i, j, k, l, m, n'] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/lux.c b/polly/lib/External/isl/test_inputs/codegen/cloog/lux.c
new file mode 100644
index 00000000000..3b33025e227
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/lux.c
@@ -0,0 +1,7 @@
+for (int c0 = 1; c0 <= M; c0 += 1) {
+  for (int c1 = 1; c1 < c0; c1 += 1)
+    for (int c2 = c1 + 1; c2 <= M; c2 += 1)
+      S2(c0, c1, c2, c2, c0);
+  for (int c3 = c0 + 1; c3 <= M; c3 += 1)
+    S1(c0, c0, M, c3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/lux.in b/polly/lib/External/isl/test_inputs/codegen/cloog/lux.in
new file mode 100644
index 00000000000..75ed7e24000
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/lux.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0, i0, M, i3] -> [i0, i0, M, i3, 0, 0] : i0 >= 1 and i0 <= M and i3 >= 1 + i0 and i3 <= M; S2[i0, i1, i2, i2, i0] -> [i0, i1, i2, i2, i0, 1] : i1 >= 1 and i1 <= M and i2 >= 1 + i1 and i2 <= M and i1 <= -1 + i0 and i0 <= M }
+[M] -> {  :  }
+[M] -> { [i, j, k, l, m, n] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/merge.c b/polly/lib/External/isl/test_inputs/codegen/cloog/merge.c
new file mode 100644
index 00000000000..64564ad9987
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/merge.c
@@ -0,0 +1,8 @@
+{
+  S1(0);
+  for (int c0 = 0; c0 <= 10; c0 += 1) {
+    if (c0 >= 2)
+      S2(c0);
+    S3(c0);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/merge.in b/polly/lib/External/isl/test_inputs/codegen/cloog/merge.in
new file mode 100644
index 00000000000..1747c44cf32
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/merge.in
@@ -0,0 +1,3 @@
+{ S3[i0] -> [i0, 2] : i0 >= 0 and i0 <= 10; S2[i0] -> [i0, 1] : i0 >= 2 and i0 <= 10; S1[0] -> [0, 0] }
+{  :  }
+{ [i, j] -> atomic[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/min-1-1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/min-1-1.c
new file mode 100644
index 00000000000..b869e1b8b40
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/min-1-1.c
@@ -0,0 +1,3 @@
+for (int c0 = 1; c0 <= N; c0 += 1)
+  for (int c1 = 0; c1 <= min(min(M, c0), N - c0); c1 += 1)
+    S1(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/min-1-1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/min-1-1.in
new file mode 100644
index 00000000000..76e511ff2fa
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/min-1-1.in
@@ -0,0 +1,3 @@
+[M, N] -> { S1[i0, i1] -> [i0, i1, 0] : i0 >= 1 and i1 >= 0 and i1 <= M and i1 <= i0 and i1 <= N - i0 }
+[M, N] -> {  :  }
+[M, N] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/min-2-1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/min-2-1.c
new file mode 100644
index 00000000000..8e6fabf7d28
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/min-2-1.c
@@ -0,0 +1,4 @@
+for (int c0 = 1; c0 <= N; c0 += 1)
+  for (int c1 = 0; c1 <= min(min(M, c0), N - c0); c1 += 1)
+    for (int c2 = 0; c2 <= min(min(M, c0), N - c0); c2 += 1)
+      S1(c0, c1, c2);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/min-2-1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/min-2-1.in
new file mode 100644
index 00000000000..aa0bcaefd86
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/min-2-1.in
@@ -0,0 +1,3 @@
+[M, N] -> { S1[i0, i1, i2] -> [i0, i1, i2, 0] : i0 >= 1 and i1 >= 0 and i1 <= M and i1 <= i0 and i1 <= N - i0 and i2 >= 0 and i2 <= M and i2 <= i0 and i2 <= N - i0 }
+[M, N] -> {  :  }
+[M, N] -> { [i, j, k, l] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/min-3-1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/min-3-1.c
new file mode 100644
index 00000000000..8d11d76d568
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/min-3-1.c
@@ -0,0 +1,3 @@
+for (int c0 = 0; c0 <= min(10, M); c0 += 1)
+  for (int c1 = 0; c1 <= min(10, M); c1 += 1)
+    S1(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/min-3-1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/min-3-1.in
new file mode 100644
index 00000000000..1676ac9318e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/min-3-1.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0, i1] -> [i0, i1, 0] : i0 >= 0 and i0 <= M and i0 <= 10 and i1 >= 0 and i1 <= M and i1 <= 10 }
+[M] -> {  : M >= 0 }
+[M] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/min-4-1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/min-4-1.c
new file mode 100644
index 00000000000..da272fc291f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/min-4-1.c
@@ -0,0 +1,2 @@
+for (int c0 = max(-M, -N); c0 <= min(N, O); c0 += 1)
+  S1(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/min-4-1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/min-4-1.in
new file mode 100644
index 00000000000..72fd01269be
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/min-4-1.in
@@ -0,0 +1,3 @@
+[M, N, O] -> { S1[i0] -> [i0, 0] : i0 >= -M and i0 >= -N and i0 <= N and i0 <= O }
+[M, N, O] -> {  :  }
+[M, N, O] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/mod.c b/polly/lib/External/isl/test_inputs/codegen/cloog/mod.c
new file mode 100644
index 00000000000..cafe0b85ddb
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/mod.c
@@ -0,0 +1,3 @@
+for (int c0 = 0; c0 <= 3; c0 += 1)
+  if ((c0 + 1) % 3 >= 1)
+    S1(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/mod.in b/polly/lib/External/isl/test_inputs/codegen/cloog/mod.in
new file mode 100644
index 00000000000..2759c0e506e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/mod.in
@@ -0,0 +1,3 @@
+{ S1[i0] -> [i0, 0] : exists (e0 = [(1 + i0)/3]: i0 >= 0 and i0 <= 3 and 3e0 <= i0 and 3e0 >= -1 + i0) }
+{  :  }
+{ [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/mod2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/mod2.c
new file mode 100644
index 00000000000..cafe0b85ddb
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/mod2.c
@@ -0,0 +1,3 @@
+for (int c0 = 0; c0 <= 3; c0 += 1)
+  if ((c0 + 1) % 3 >= 1)
+    S1(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/mod2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/mod2.in
new file mode 100644
index 00000000000..58134bbbfe6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/mod2.in
@@ -0,0 +1,3 @@
+{ S1[i] -> [i, 0] : exists (e0 = [(i)/3]: i >= 0 and i <= 3 and 3e0 <= i and 3e0 >= -1 + i) }
+{  :  }
+{ [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/mod3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/mod3.c
new file mode 100644
index 00000000000..f8d879ce1ae
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/mod3.c
@@ -0,0 +1,4 @@
+for (int c0 = max(0, 32 * h0 - 1991); c0 <= min(999, 32 * h0 + 31); c0 += 1)
+  if ((32 * h0 - c0 + 32) % 64 >= 1)
+    for (int c1 = 0; c1 <= 999; c1 += 1)
+      S1(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/mod3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/mod3.in
new file mode 100644
index 00000000000..bd0713eeba3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/mod3.in
@@ -0,0 +1,3 @@
+[h0] -> { S1[i0, i1] -> [i0, i1, 0] : exists (e0 = [(32 + 32h0 - i0)/64]: i0 >= 0 and i0 <= 999 and i0 >= -2015 + 32h0 and 32e0 >= -999 + 32h0 - i0 and i1 >= 0 and i1 <= 999 and 64e0 >= -31 + 32h0 - i0 and 64e0 <= 31 + 32h0 - i0 and i0 <= 32 + 32h0) }
+[h0] -> {  : h0 <= 93 and h0 >= 0 }
+[h0] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/mod4.c b/polly/lib/External/isl/test_inputs/codegen/cloog/mod4.c
new file mode 100644
index 00000000000..a5dca247b96
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/mod4.c
@@ -0,0 +1,5 @@
+for (int c0 = 2; c0 <= 10; c0 += 3) {
+  S1(c0, (c0 + 1) / 3, (c0 + 1) / 3, 2, (c0 - 2) / 3);
+  S2(c0, (c0 + 1) / 3, (c0 + 1) / 3, 2, (c0 - 2) / 3);
+  S3(c0, (c0 + 1) / 3, (c0 + 1) / 3, 2, (c0 - 2) / 3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/mod4.in b/polly/lib/External/isl/test_inputs/codegen/cloog/mod4.in
new file mode 100644
index 00000000000..2002cef3ed3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/mod4.in
@@ -0,0 +1,3 @@
+{ S2[j, div41, div42, 2, mod6_a] -> [j, div41, k, 2, m, 1] : 3k = 1 + j and 3div42 = 1 + j and 3m = -2 + j and 3mod6_a = -2 + j and 3div41 >= 1 + j and 3div41 <= 2 + j and j >= 1 and j <= 10; S1[j, div41, div42, 2, mod6_a] -> [j, div41, div42, 2, m, 0] : 3m = -2 + j and 3mod6_a = -2 + j and j >= 1 and j <= 10 and 3div41 >= j and 3div42 >= -1 + j and 3div42 <= 1 + j and 3div41 <= 2 + j; S3[j, div41, div42, 2, mod6_a] -> [j, div41, div42, 2, m, 2] : 3m = -2 + j and 3mod6_a = -2 + j and j >= 1 and j <= 10 and 3div41 >= j and 3div42 >= -1 + j and 3div42 <= 1 + j and 3div41 <= 2 + j }
+{  :  }
+{ [i, j, k, l, m, n] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/mode.c b/polly/lib/External/isl/test_inputs/codegen/cloog/mode.c
new file mode 100644
index 00000000000..bcfd31839bb
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/mode.c
@@ -0,0 +1,10 @@
+for (int c0 = 0; c0 <= M; c0 += 1) {
+  for (int c1 = 0; c1 <= min(N, c0); c1 += 1) {
+    S1(c0, c1);
+    S2(c0, c1);
+  }
+  for (int c1 = max(0, N + 1); c1 <= c0; c1 += 1)
+    S1(c0, c1);
+  for (int c1 = c0 + 1; c1 <= N; c1 += 1)
+    S2(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/mode.in b/polly/lib/External/isl/test_inputs/codegen/cloog/mode.in
new file mode 100644
index 00000000000..914c37e642a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/mode.in
@@ -0,0 +1,3 @@
+[M, N] -> { S1[i0, i1] -> [i0, i1, 0] : i0 >= 0 and i0 <= M and i1 >= 0 and i1 <= i0; S2[i0, i1] -> [i0, i1, 1] : i0 >= 0 and i0 <= M and i1 >= 0 and i1 <= N }
+[M, N] -> {  :  }
+[M, N] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/multi-mm-1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/multi-mm-1.c
new file mode 100644
index 00000000000..271d863f008
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/multi-mm-1.c
@@ -0,0 +1,8 @@
+for (int c0 = 0; c0 <= M; c0 += 1) {
+  for (int c1 = 0; c1 <= min(N, c0); c1 += 1) {
+    S1(c0, c1);
+    S2(c0, c1);
+  }
+  for (int c1 = N + 1; c1 <= c0; c1 += 1)
+    S1(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/multi-mm-1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/multi-mm-1.in
new file mode 100644
index 00000000000..22bdd1b4234
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/multi-mm-1.in
@@ -0,0 +1,3 @@
+[M, N] -> { S1[i0, i1] -> [i0, i1, 0] : i1 >= 0 and i1 <= i0 and i0 <= M; S2[i0, i1] -> [i0, i1, 1] : i1 >= 0 and i1 <= i0 and i0 <= M and i1 <= N }
+[M, N] -> {  : N <= M and N >= 1 }
+[M, N] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/multi-stride.c b/polly/lib/External/isl/test_inputs/codegen/cloog/multi-stride.c
new file mode 100644
index 00000000000..2c63c085104
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/multi-stride.c
@@ -0,0 +1,2 @@
+{
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/multi-stride.in b/polly/lib/External/isl/test_inputs/codegen/cloog/multi-stride.in
new file mode 100644
index 00000000000..fdef8f90c2e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/multi-stride.in
@@ -0,0 +1,3 @@
+{ S1[i0, i1, i2] -> [i0, j, k, 0] : 2i1 = -1 + i0 and 2j = -1 + i0 and 6k = -2 + i0 and 6i2 = -2 + i0 and i0 >= 0 and i0 <= 100 }
+{  :  }
+{ [i, j, k, l] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/multi-stride2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/multi-stride2.c
new file mode 100644
index 00000000000..14f80506e0b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/multi-stride2.c
@@ -0,0 +1,2 @@
+for (int c0 = 5; c0 <= 100; c0 += 6)
+  S1(c0, (c0 - 1) / 2, (c0 - 2) / 3);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/multi-stride2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/multi-stride2.in
new file mode 100644
index 00000000000..d2c4bbc965e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/multi-stride2.in
@@ -0,0 +1,3 @@
+{ S1[i0, i1, i2] -> [i0, j, k, 0] : 2i1 = -1 + i0 and 2j = -1 + i0 and 3k = -2 + i0 and 3i2 = -2 + i0 and i0 >= 0 and i0 <= 100 }
+{  :  }
+{ [i, j, k, l] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/mxm-shared.c b/polly/lib/External/isl/test_inputs/codegen/cloog/mxm-shared.c
new file mode 100644
index 00000000000..c2423f01f5c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/mxm-shared.c
@@ -0,0 +1,6 @@
+if (g4 == 0 && N >= g0 + t1 + 1 && t1 <= 7) {
+  for (int c0 = t0; c0 <= min(127, N - g1 - 1); c0 += 16)
+    S1(g0 + t1, g1 + c0);
+} else if (g4 >= 4 && N >= g0 + t1 + 1 && t1 <= 7 && g4 % 4 == 0)
+  for (int c0 = t0; c0 <= min(127, N - g1 - 1); c0 += 16)
+    S1(g0 + t1, g1 + c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/mxm-shared.in b/polly/lib/External/isl/test_inputs/codegen/cloog/mxm-shared.in
new file mode 100644
index 00000000000..a525e9c0dd8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/mxm-shared.in
@@ -0,0 +1,3 @@
+[N, b0, b1, g0, g1, g2, g3, g4, t0, t1] -> { S1[g0 + t1, i1] -> [-g1 + i1, t1, t0, t1] : (exists (e0 = [(-g1)/128], e1 = [(128b1 + 31g1)/4096], e2 = [(t0 - i1)/16], e3 = [(-120b0 - g0)/128]: g4 = 0 and g2 = 8b0 and g3 = 128b1 and 128e0 = -g1 and 4096e1 = 128b1 + 31g1 and 16e2 = t0 - i1 and 128e3 = -120b0 - g0 and g1 >= 128b1 and t1 <= -1 + N - g0 and g0 >= 8b0 and i1 <= -1 + N and b0 <= 15 and b0 >= 0 and b1 <= 31 and b1 >= 0 and i1 <= 127 + g1 and t1 >= 0 and t1 <= 7 and i1 >= g1 and t0 >= 0 and t0 <= 15)) or (exists (e0 = [(-g1)/128], e1 = [(128b1 + 31g1)/4096], e2 = [(t0 - i1)/16], e3 = [(-120b0 - g0)/128]: g4 = 0 and g2 = 8b0 and g3 = 128b1 and 128e0 = -g1 and 4096e1 = 128b1 + 31g1 and 16e2 = t0 - i1 and 128e3 = -120b0 - g0 and g1 >= 128b1 and t1 <= -1 + N - g0 and g0 >= 8b0 and i1 <= -1 + N and b0 <= 15 and b0 >= 0 and b1 <= 31 and b1 >= 0 and i1 <= 127 + g1 and t1 >= 0 and t1 <= 7 and i1 >= g1 and t0 >= 0 and t0 <= 15 and N >= 1)) or (exists (e0 = [(-g1)/128], e1 = [(128b1 + 31g1)/4096], e2 = [(t0 - i1)/16], e3 = [(-120b0 - g0)/128]: g4 = 0 and g2 = 8b0 and g3 = 128b1 and 128e0 = -g1 and 4096e1 = 128b1 + 31g1 and 16e2 = t0 - i1 and 128e3 = -120b0 - g0 and g0 >= 8b0 and t1 <= -1 + N - g0 and g1 >= 128b1 and i1 <= -1 + N and N >= 1 and t1 <= 7 and b1 <= 31 and b1 >= 0 and b0 <= 15 and b0 >= 0 and t0 <= 15 and i1 <= 127 + g1 and i1 >= g1 and t1 >= 0 and t0 >= 0)); S1[g0 + t1, i1] -> [-g1 + i1, t1, t0, t1] : exists (e0 = [(g4)/4], e1 = [(-g1)/128], e2 = [(128b1 + 31g1)/4096], e3 = [(t0 - i1)/16], e4 = [(-120b0 - g0)/128]: g3 = 128b1 and g2 = 8b0 and 4e0 = g4 and 128e1 = -g1 and 4096e2 = 128b1 + 31g1 and 16e3 = t0 - i1 and 128e4 = -120b0 - g0 and g0 >= 8b0 and t1 <= -1 + N - g0 and g1 >= 128b1 and i1 <= -1 + N and g4 <= -1 + N and t1 <= 7 and b1 <= 31 and b1 >= 0 and b0 <= 15 and b0 >= 0 and g4 >= 0 and i1 <= 127 + g1 and i1 >= g1 and t1 >= 0 and t0 >= 0 and t0 <= 15) }
+[N, b0, b1, g0, g1, g2, g3, g4, t0, t1] -> {  : exists (e0 = [(g0)/8], e1 = [(-128b1 + g1)/4096], e2 = [(8b0 - g0)/128]: g2 = 8b0 and g3 = 128b1 and 8e0 = g0 and 4096e1 = -128b1 + g1 and 128e2 = 8b0 - g0 and b0 >= 0 and g4 <= -1 + N and b0 <= 15 and g1 <= -1 + N and g4 >= 0 and b1 <= 31 and g0 <= -1 + N and g1 >= 128b1 and b1 >= 0 and g0 >= 8b0 and t0 >= 0 and t0 <= 15 and t1 >= 0 and t1 <= 15) }
+[N, b0, b1, g0, g1, g2, g3, g4, t0, t1] -> { [i, j, k, l] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/no_lindep.c b/polly/lib/External/isl/test_inputs/codegen/cloog/no_lindep.c
new file mode 100644
index 00000000000..1432b5e4e3c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/no_lindep.c
@@ -0,0 +1 @@
+S1(N + 2);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/no_lindep.in b/polly/lib/External/isl/test_inputs/codegen/cloog/no_lindep.in
new file mode 100644
index 00000000000..078992925f7
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/no_lindep.in
@@ -0,0 +1,3 @@
+[M, N] -> { S1[2 + N] -> [1 + M, N] }
+[M, N] -> {  :  }
+[M, N] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/nul_basic1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_basic1.c
new file mode 100644
index 00000000000..9293aef92c7
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_basic1.c
@@ -0,0 +1,2 @@
+for (int c0 = 0; c0 <= M; c0 += 2)
+  S1(c0, c0 / 2);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/nul_basic1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_basic1.in
new file mode 100644
index 00000000000..e10488f68ef
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_basic1.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0, i1] -> [i0, j, 0] : 2j = i0 and 2i1 = i0 and i0 >= 0 and i0 <= M }
+[M] -> {  :  }
+[M] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/nul_basic2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_basic2.c
new file mode 100644
index 00000000000..2b7bb4d22c6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_basic2.c
@@ -0,0 +1,5 @@
+for (int c0 = 2; c0 <= n; c0 += 2) {
+  if (c0 % 4 == 0)
+    S2(c0, c0 / 4);
+  S1(c0, c0 / 2);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/nul_basic2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_basic2.in
new file mode 100644
index 00000000000..54f9374a5d3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_basic2.in
@@ -0,0 +1,3 @@
+[n] -> { S2[i0, i1] -> [i0, j, 1] : 4j = i0 and 4i1 = i0 and i0 >= 1 and i0 <= n; S1[i0, i1] -> [i0, j, 0] : 2j = i0 and 2i1 = i0 and i0 >= 1 and i0 <= n }
+[n] -> {  : n >= 2 }
+[n] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/nul_complex1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_complex1.c
new file mode 100644
index 00000000000..7053351603e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_complex1.c
@@ -0,0 +1,3 @@
+for (int c0 = 0; c0 <= 5 * n; c0 += 1)
+  for (int c1 = max(-((5 * n - c0 + 1) % 2) - n + c0 + 1, 2 * floord(c0 - 1, 3) + 2); c1 <= min(c0, n + c0 - (n + c0 + 2) / 3); c1 += 2)
+    S1((3 * c1 / 2) - c0, c0 - c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/nul_complex1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_complex1.in
new file mode 100644
index 00000000000..277e62ef171
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_complex1.in
@@ -0,0 +1,3 @@
+[n] -> { S1[i0, i1] -> [2i0 + 3i1, 2i0 + 2i1] : i0 >= 0 and i0 <= n and i1 >= 0 and i1 <= n }
+[n] -> {  :  }
+[n] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/nul_lcpc.c b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_lcpc.c
new file mode 100644
index 00000000000..e6f39682d49
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_lcpc.c
@@ -0,0 +1,13 @@
+{
+  for (int c0 = 1; c0 <= 6; c0 += 2) {
+    for (int c2 = 1; c2 <= c0; c2 += 1) {
+      S1(c0, (c0 - 1) / 2, c2);
+      S2(c0, (c0 - 1) / 2, c2);
+    }
+    for (int c2 = c0 + 1; c2 <= p; c2 += 1)
+      S1(c0, (c0 - 1) / 2, c2);
+  }
+  for (int c0 = 7; c0 <= m; c0 += 2)
+    for (int c2 = 1; c2 <= p; c2 += 1)
+      S1(c0, (c0 - 1) / 2, c2);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/nul_lcpc.in b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_lcpc.in
new file mode 100644
index 00000000000..e3a4e99fcf3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/nul_lcpc.in
@@ -0,0 +1,3 @@
+[m, n, p] -> { S1[i, k, j] -> [i, j', j, 0] : 2k = -1 + i and 2j' = -1 + i and i >= 1 and i <= m and j >= 1 and j <= p; S2[i, k, j] -> [i, j', j, 1] : 2k = -1 + i and 2j' = -1 + i and i >= 1 and i <= n and j >= 1 and j <= i }
+[m, n, p] -> {  : n = 6 and m >= 7 and p >= 7 }
+[m, n, p] -> { [i, j, k, l] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/orc.c b/polly/lib/External/isl/test_inputs/codegen/cloog/orc.c
new file mode 100644
index 00000000000..8ad0162832c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/orc.c
@@ -0,0 +1,16 @@
+{
+  for (int c1 = 0; c1 <= 2; c1 += 1) {
+    S1(c1);
+    for (int c2 = 0; c2 <= -c1 + 11; c2 += 1) {
+      S2(c1, c2);
+      S3(c1, c2);
+    }
+    S4(c1);
+  }
+  for (int c1 = 0; c1 <= 14; c1 += 1) {
+    S5(c1);
+    for (int c2 = 0; c2 <= 9; c2 += 1)
+      S6(c1, c2);
+    S7(c1);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/orc.in b/polly/lib/External/isl/test_inputs/codegen/cloog/orc.in
new file mode 100644
index 00000000000..d81bf4728e9
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/orc.in
@@ -0,0 +1,3 @@
+{ S2[i0, i1] -> [0, 1 + 3i0, 2i1] : i0 >= 0 and i0 <= 2 and i1 >= 0 and i1 <= 11 - i0; S4[i0] -> [0, 2 + 3i0, 0] : i0 >= 0 and i0 <= 2; S5[i0] -> [2, 3i0, 0] : i0 >= 0 and i0 <= 14; S6[i0, i1] -> [2, 1 + 3i0, i1] : i0 >= 0 and i0 <= 14 and i1 >= 0 and i1 <= 9; S1[i0] -> [0, 3i0, 0] : i0 >= 0 and i0 <= 2; S7[i0] -> [2, 2 + 3i0, 0] : i0 >= 0 and i0 <= 14; S3[i0, i1] -> [0, 1 + 3i0, 1 + 2i1] : i0 >= 0 and i0 <= 2 and i1 >= 0 and i1 <= 11 - i0 }
+{  :  }
+{ [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/otl.c b/polly/lib/External/isl/test_inputs/codegen/cloog/otl.c
new file mode 100644
index 00000000000..793733447ee
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/otl.c
@@ -0,0 +1,7 @@
+if (M >= 3 && N >= 4)
+  for (int c0 = 1; c0 < (2 * M + 2 * N - 2) / 5; c0 += 1)
+    for (int c1 = max(c0 - (M + 2) / 5, (c0 + 1) / 2); c1 <= min(min(c0, (M + 2 * N) / 5 - 1), (2 * N + 5 * c0 + 1) / 10); c1 += 1)
+      for (int c2 = max(max(max(max(0, c0 - c1 - 1), c1 - (N + 6) / 5 + 1), c0 - (M + N + 4) / 5 + 1), floord(-N + 5 * c0 - 3, 10) + 1); c2 <= min(min(min(c1, (M + N - 2) / 5), c0 - c1 + (N - 1) / 5 + 1), (N + 5 * c0 + 3) / 10); c2 += 1)
+        for (int c3 = max(max(max(c0, 2 * c1 - (2 * N + 5) / 5 + 1), c1 + c2 - (N + 3) / 5), 2 * c2 - (N + 2) / 5); c3 <= min(min(min(min(min(c0 + 1, c1 + c2 + 1), c1 + (M - 2) / 5 + 1), 2 * c2 + (N - 2) / 5 + 1), (2 * M + 2 * N - 1) / 5 - 1), c2 + (M + N) / 5); c3 += 1)
+          for (int c4 = max(max(max(max(c1, c0 - c2), c0 - (M + 6) / 5 + 1), c3 - (M + 2) / 5), (c3 + 1) / 2); c4 <= min(min(min(min(min(min(min(c0, c1 + 1), -c2 + c3 + (N - 1) / 5 + 1), c0 - c2 + N / 5 + 1), (M + 2 * N + 1) / 5 - 1), c2 + (N + 2) / 5), (2 * N + 5 * c0 + 3) / 10), (2 * N + 5 * c3 + 2) / 10); c4 += 1)
+            S1(c0, c1, c2, c3, c4, c2);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/otl.in b/polly/lib/External/isl/test_inputs/codegen/cloog/otl.in
new file mode 100644
index 00000000000..a6ad5d2fc58
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/otl.in
@@ -0,0 +1,3 @@
+[M, N] -> { S1[i0, i1, i2, i3, i4, i2] -> [i0, i1, i2, i3, i4, i2] : 5i0 <= -7 + 2M + 2N and i4 >= i0 - i2 and 10i2 >= -2 - N + 5i0 and i2 >= -1 + i0 - i1 and 2i4 >= i0 and i0 >= 1 and i1 >= 1 and 2i1 >= i0 and i2 >= 0 and 5i2 >= 1 - M - N + 5i0 and 5i4 >= -1 - M + 5i0 and i3 >= i0 and 5i1 >= -2 - M + 5i0 and i3 >= 1 and 5i3 >= -3 - N + 5i1 + 5i2 and 5i2 <= 4 + N + 5i0 - 5i1 and i4 >= 1 and i2 <= i0 and 5i3 >= -2N + 10i1 and 5i1 <= -5 + M + 2N and 10i1 <= 1 + 2N + 5i0 and 5i2 >= -1 - N + 5i1 and i4 >= i1 and i3 >= i1 and i2 <= i1 and i1 <= i0 and 5i4 <= 4 + N - 5i2 + 5i3 and 5i4 <= 5 + N + 5i0 - 5i2 and 5i3 >= -2 - N + 10i2 and 5i2 <= -2 + M + N and 10i2 <= 3 + N + 5i0 and N >= 4 and i4 >= i2 and i3 >= i2 and M >= 3 and i4 <= i0 and 5i3 <= -6 + 2M + 2N and i4 >= -1 - i2 + i3 and 5i3 <= 3 + N + 10i2 and i3 <= 1 + i1 + i2 and i4 <= 1 + i1 and 2i4 >= i3 and 5i4 <= 2 + N + 5i2 and i3 <= 1 + 2i1 and i4 <= i3 and 5i3 <= M + N + 5i2 and 5i4 >= -2 - M + 5i3 and 10i4 <= 3 + 2N + 5i0 and i3 <= 1 + i0 and 5i3 <= 3 + M + 5i1 and 5i4 <= -4 + M + 2N and 10i4 <= 2 + 2N + 5i3 }
+[M, N] -> {  : M >= 1 and N >= 1 }
+[M, N] -> { [i, j, k, l, m, n] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/param-split.c b/polly/lib/External/isl/test_inputs/codegen/cloog/param-split.c
new file mode 100644
index 00000000000..e7774d55d1e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/param-split.c
@@ -0,0 +1,6 @@
+for (int c0 = 0; c0 <= (M <= 0 ? 0 : M); c0 += 1) {
+  if (M >= c0)
+    S1(c0);
+  if (c0 == 0)
+    S2(0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/param-split.in b/polly/lib/External/isl/test_inputs/codegen/cloog/param-split.in
new file mode 100644
index 00000000000..7b8a59e641e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/param-split.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0] -> [i0, 0] : i0 >= 0 and i0 <= M; S2[0] -> [0, 1] }
+[M] -> {  :  }
+[M] -> { [i, j] -> atomic[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/pouchet.c b/polly/lib/External/isl/test_inputs/codegen/cloog/pouchet.c
new file mode 100644
index 00000000000..dc844442ca5
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/pouchet.c
@@ -0,0 +1,11 @@
+for (int c0 = 1; c0 <= floord(Ny, 2) + 2; c0 += 1)
+  for (int c1 = max(c0 - 1, c0 / 2 + 1); c1 <= min(c0, (Ny + 2 * c0) / 4); c1 += 1) {
+    if (Ny + 2 * c0 >= 4 * c1 + 1) {
+      for (int c2 = 1; c2 <= 2; c2 += 1) {
+        S1(c0 - c1, c1, 2 * c0 - 2 * c1, -2 * c0 + 4 * c1, c2);
+        S2(c0 - c1, c1, 2 * c0 - 2 * c1, -2 * c0 + 4 * c1 - 1, c2);
+      }
+    } else
+      for (int c2 = 1; c2 <= 2; c2 += 1)
+        S2((-Ny + 2 * c0) / 4, (Ny + 2 * c0) / 4, (-Ny / 2) + c0, Ny - 1, c2);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/pouchet.in b/polly/lib/External/isl/test_inputs/codegen/cloog/pouchet.in
new file mode 100644
index 00000000000..95d59dc29da
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/pouchet.in
@@ -0,0 +1,3 @@
+[Ny] -> { S1[i0, i1, 2i0, -2i0 + 2i1, i4] -> [i0 + i1, i1, i4, 2i0, -2i0 + 2i1, i4] : i0 >= 0 and i0 <= 1 and i1 >= 1 + i0 and 2i1 <= -1 + Ny + 2i0 and i4 >= 1 and i4 <= 2; S2[i0, i1, 2i0, -1 - 2i0 + 2i1, i4] -> [i0 + i1, i1, i4, 2i0, -2i0 + 2i1, 1 + i4] : i0 >= 0 and i0 <= 1 and i1 >= 1 + i0 and 2i1 <= Ny + 2i0 and i4 >= 1 and i4 <= 2 }
+[Ny] -> {  :  }
+[Ny] -> { [i, j, k, l, m, n] -> separate[x] : x >= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/rectangle.c b/polly/lib/External/isl/test_inputs/codegen/cloog/rectangle.c
new file mode 100644
index 00000000000..10d1f76c19d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/rectangle.c
@@ -0,0 +1,3 @@
+for (int c0 = 0; c0 <= 2 * n; c0 += 1)
+  for (int c1 = max(0, -n + c0); c1 <= min(n, c0); c1 += 1)
+    S1(c1, c0 - c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/rectangle.in b/polly/lib/External/isl/test_inputs/codegen/cloog/rectangle.in
new file mode 100644
index 00000000000..fab0f945c37
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/rectangle.in
@@ -0,0 +1,3 @@
+[n] -> { S1[i0, i1] -> [i0 + i1] : i0 >= 0 and i0 <= n and i1 >= 0 and i1 <= n }
+[n] -> {  : n >= 0 }
+[n] -> { [i] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-QR.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-QR.c
new file mode 100644
index 00000000000..253791a50b0
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-QR.c
@@ -0,0 +1,54 @@
+if (N >= 1) {
+  S1(0);
+  if (N == 1) {
+    for (int c3 = 0; c3 < M; c3 += 1)
+      S2(0, c3);
+    S3(0);
+    for (int c3 = 0; c3 < M; c3 += 1)
+      S4(0, c3);
+    S10(0);
+    S5(0);
+  } else {
+    for (int c3 = 0; c3 < M; c3 += 1)
+      S2(0, c3);
+    S3(0);
+    for (int c3 = 0; c3 < M; c3 += 1)
+      S4(0, c3);
+    S10(0);
+    S1(1);
+    S5(0);
+  }
+  for (int c1 = 2; c1 < N; c1 += 1) {
+    for (int c3 = c1 - 1; c3 < N; c3 += 1) {
+      S6(c1 - 2, c3);
+      for (int c5 = c1 - 2; c5 < M; c5 += 1)
+        S7(c1 - 2, c3, c5);
+      S8(c1 - 2, c3);
+      for (int c5 = c1 - 2; c5 < M; c5 += 1)
+        S9(c1 - 2, c3, c5);
+    }
+    for (int c3 = c1 - 1; c3 < M; c3 += 1)
+      S2(c1 - 1, c3);
+    S3(c1 - 1);
+    for (int c3 = c1 - 1; c3 < M; c3 += 1)
+      S4(c1 - 1, c3);
+    S10(c1 - 1);
+    S1(c1);
+    S5(c1 - 1);
+  }
+  if (N >= 2) {
+    S6(N - 2, N - 1);
+    for (int c5 = N - 2; c5 < M; c5 += 1)
+      S7(N - 2, N - 1, c5);
+    S8(N - 2, N - 1);
+    for (int c5 = N - 2; c5 < M; c5 += 1)
+      S9(N - 2, N - 1, c5);
+    for (int c3 = N - 1; c3 < M; c3 += 1)
+      S2(N - 1, c3);
+    S3(N - 1);
+    for (int c3 = N - 1; c3 < M; c3 += 1)
+      S4(N - 1, c3);
+    S10(N - 1);
+    S5(N - 1);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-QR.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-QR.in
new file mode 100644
index 00000000000..b81c03d7252
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-QR.in
@@ -0,0 +1,3 @@
+[M, N] -> { S1[i0] -> [0, i0, 5, 0, 0, 0, 0] : i0 >= 0 and i0 <= -1 + N; S9[i0, i1, i2] -> [0, 2 + i0, 0, i1, 3, i2, 0] : i0 >= 0 and i1 >= 1 + i0 and i1 <= -1 + N and i2 >= i0 and i2 <= -1 + M; S10[i0] -> [0, 1 + i0, 4, 0, 0, 0, 0] : i0 >= 0 and i0 <= -1 + N; S3[i0] -> [0, 1 + i0, 2, 0, 0, 0, 0] : i0 >= 0 and i0 <= -1 + N; S6[i0, i1] -> [0, 2 + i0, 0, i1, 0, 0, 0] : i0 >= 0 and i1 >= 1 + i0 and i1 <= -1 + N; S8[i0, i1] -> [0, 2 + i0, 0, i1, 2, 0, 0] : i0 >= 0 and i1 >= 1 + i0 and i1 <= -1 + N; S2[i0, i1] -> [0, 1 + i0, 1, i1, 0, 0, 0] : i0 >= 0 and i0 <= -1 + N and i1 >= i0 and i1 <= -1 + M; S4[i0, i1] -> [0, 1 + i0, 3, i1, 0, 0, 0] : i0 >= 0 and i0 <= -1 + N and i1 >= i0 and i1 <= -1 + M; S7[i0, i1, i2] -> [0, 2 + i0, 0, i1, 1, i2, 0] : i0 >= 0 and i1 >= 1 + i0 and i1 <= -1 + N and i2 >= i0 and i2 <= -1 + M; S5[i0] -> [0, 1 + i0, 6, 0, 0, 0, 0] : i0 >= 0 and i0 <= -1 + N }
+[M, N] -> {  :  }
+[M, N] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-bastoul3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-bastoul3.c
new file mode 100644
index 00000000000..6d82ce4a89b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-bastoul3.c
@@ -0,0 +1,3 @@
+for (int c0 = 3; c0 <= 9; c0 += 1)
+  for (int c1 = max(c0 - 6, -(c0 % 2) + 2); c1 <= min(3, c0 - 2); c1 += 2)
+    S1(c0, c1, (c0 - c1) / 2);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-bastoul3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-bastoul3.in
new file mode 100644
index 00000000000..a6a4ffc9297
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-bastoul3.in
@@ -0,0 +1,3 @@
+{ S1[i0, i1, i2] -> [i0, i1, k, 0] : 2k = i0 - i1 and 2i2 = i0 - i1 and i1 >= 1 and i1 <= 3 and i1 <= -2 + i0 and i1 >= -6 + i0 }
+{  :  }
+{ [i, j, k, l] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-cholesky2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-cholesky2.c
new file mode 100644
index 00000000000..4d1a9118bb9
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-cholesky2.c
@@ -0,0 +1,9 @@
+for (int c1 = 2; c1 < 3 * M; c1 += 1) {
+  if ((c1 - 2) % 3 == 0)
+    S1((c1 + 1) / 3);
+  for (int c3 = (c1 + 1) / 3 + 1; c3 <= min(M, c1 - 2); c3 += 1)
+    for (int c5 = -c3 + (c1 + c3 + 1) / 2 + 1; c5 <= min(c3, c1 - c3); c5 += 1)
+      S3(c1 - c3 - c5 + 1, c3, c5);
+  for (int c3 = -c1 + 2 * ((2 * c1 + 1) / 3) + 2; c3 <= min(M, c1); c3 += 2)
+    S2(((c1 - c3) / 2) + 1, c3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-cholesky2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-cholesky2.in
new file mode 100644
index 00000000000..7f62b1f0264
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-cholesky2.in
@@ -0,0 +1,3 @@
+[M] -> { S3[i0, i1, i2] -> [0, -1 + i0 + i1 + i2, 1, i1, 1, i2, 0] : i0 >= 1 and i1 <= M and i2 >= 1 + i0 and i2 <= i1; S2[i0, i1] -> [0, -2 + 2i0 + i1, 2, i1, 0, 0, 0] : i0 >= 1 and i1 >= 1 + i0 and i1 <= M; S1[i0] -> [0, -1 + 3i0, 0, 0, 0, 0, 0] : i0 >= 1 and i0 <= M }
+[M] -> {  :  }
+[M] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-fusion1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-fusion1.c
new file mode 100644
index 00000000000..201ffee0bc3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-fusion1.c
@@ -0,0 +1,8 @@
+{
+  for (int c1 = 0; c1 <= M; c1 += 1)
+    S1(c1);
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    S2(c1);
+  for (int c1 = 0; c1 <= M; c1 += 1)
+    S3(c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-fusion1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-fusion1.in
new file mode 100644
index 00000000000..ca4282e49b3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-fusion1.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0] -> [0, i0, 0] : i0 >= 0 and i0 <= M; S3[i0] -> [2, i0, 0] : i0 >= 0 and i0 <= M; S2[i0] -> [1, i0, 0] : i0 >= 1 and i0 <= M }
+[M] -> {  : M >= 1 }
+[M] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-fusion2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-fusion2.c
new file mode 100644
index 00000000000..13c30301620
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-fusion2.c
@@ -0,0 +1,12 @@
+if (N >= 1) {
+  for (int c3 = 1; c3 <= M; c3 += 1)
+    S1(1, c3);
+  for (int c1 = 2; c1 <= N; c1 += 1) {
+    for (int c3 = 1; c3 <= M; c3 += 1)
+      S2(c1 - 1, c3);
+    for (int c3 = 1; c3 <= M; c3 += 1)
+      S1(c1, c3);
+  }
+  for (int c3 = 1; c3 <= M; c3 += 1)
+    S2(N, c3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-fusion2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-fusion2.in
new file mode 100644
index 00000000000..53905d8df35
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-fusion2.in
@@ -0,0 +1,3 @@
+[M, N] -> { S2[i0, i1] -> [0, 1 + i0, 0, i1, 0] : i0 >= 1 and i0 <= N and i1 >= 1 and i1 <= M; S1[i0, i1] -> [0, i0, 1, i1, 0] : i0 >= 1 and i0 <= N and i1 >= 1 and i1 <= M }
+[M, N] -> {  :  }
+[M, N] -> { [i, j, k, l, m] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-jacobi2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-jacobi2.c
new file mode 100644
index 00000000000..407fc8b73e1
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-jacobi2.c
@@ -0,0 +1,3 @@
+for (int c1 = 0; c1 < M; c1 += 1)
+  for (int c3 = 0; c3 < M; c3 += 1)
+    S1(c1, c3);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-jacobi2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-jacobi2.in
new file mode 100644
index 00000000000..ea86283a7b7
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-jacobi2.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0, i1] -> [0, i0, 0, i1, 0] : i0 >= 0 and i0 <= -1 + M and i1 >= 0 and i1 <= -1 + M }
+[M] -> {  : M >= 1 }
+[M] -> { [i, j, k, l, m] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-jacobi3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-jacobi3.c
new file mode 100644
index 00000000000..1df9bb90c24
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-jacobi3.c
@@ -0,0 +1,8 @@
+for (int c1 = 1; c1 <= M; c1 += 1) {
+  for (int c3 = 2; c3 < N; c3 += 1)
+    for (int c5 = 2; c5 < N; c5 += 1)
+      S1(c1, c3, c5);
+  for (int c3 = 2; c3 < N; c3 += 1)
+    for (int c5 = 2; c5 < N; c5 += 1)
+      S2(c1, c3, c5);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-jacobi3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-jacobi3.in
new file mode 100644
index 00000000000..4f5cc77904a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-jacobi3.in
@@ -0,0 +1,3 @@
+[M, N] -> { S2[i0, i1, i2] -> [0, 1 + 2i0, 1, i1, 0, i2, 0] : i0 >= 1 and i0 <= M and i1 >= 2 and i1 <= -1 + N and i2 >= 2 and i2 <= -1 + N; S1[i0, i1, i2] -> [0, 2i0, 0, i1, 0, i2, 0] : i0 >= 1 and i0 <= M and i1 >= 2 and i1 <= -1 + N and i2 >= 2 and i2 <= -1 + N }
+[M, N] -> {  :  }
+[M, N] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam1.c
new file mode 100644
index 00000000000..d8a80d66363
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam1.c
@@ -0,0 +1,10 @@
+for (int c1 = -99; c1 <= 100; c1 += 1) {
+  if (c1 <= 0)
+    S1(1, -c1 + 1);
+  for (int c3 = max(1, -2 * c1 + 3); c3 <= min(199, -2 * c1 + 199); c3 += 2) {
+    S2(((c3 - 1) / 2) + c1, (c3 + 1) / 2);
+    S1(((c3 + 1) / 2) + c1, (c3 + 1) / 2);
+  }
+  if (c1 >= 1)
+    S2(100, -c1 + 101);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam1.in
new file mode 100644
index 00000000000..b2338b0321b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam1.in
@@ -0,0 +1,3 @@
+{ S2[i0, i1] -> [0, 1 + i0 - i1, 0, -1 + 2i1, 1] : i0 >= 1 and i0 <= 100 and i1 >= 1 and i1 <= 100; S1[i0, i1] -> [0, i0 - i1, 0, 2i1, 0] : i0 >= 1 and i0 <= 100 and i1 >= 1 and i1 <= 100 }
+{  :  }
+{ [i, j, k, l, m] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam2.c
new file mode 100644
index 00000000000..f3215bf69bd
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam2.c
@@ -0,0 +1,10 @@
+{
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    S1(c1);
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    for (int c3 = 2; c3 <= N; c3 += 1)
+      S2(c1, c3);
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    for (int c3 = 1; c3 < N; c3 += 1)
+      S3(c1, c3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam2.in
new file mode 100644
index 00000000000..eb62bb7d009
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam2.in
@@ -0,0 +1,3 @@
+[M, N] -> { S1[i0] -> [0, i0, 0, 0, 0] : i0 >= 1 and i0 <= M; S2[i0, i1] -> [1, i0, 1, i1, 0] : i0 >= 1 and i0 <= M and i1 >= 2 and i1 <= N; S3[i0, i1] -> [2, i0, 2, i1, 0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= -1 + N }
+[M, N] -> {  : M >= 1 and N >= 1 }
+[M, N] -> { [i, j, k, l, m] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam3.c
new file mode 100644
index 00000000000..c786628d804
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam3.c
@@ -0,0 +1,11 @@
+for (int c1 = 5; c1 <= 5 * M; c1 += 1) {
+  for (int c3 = max(2, floord(-M + c1, 4)); c3 < min(M, (c1 + 1) / 3 - 2); c3 += 1)
+    for (int c5 = max(1, -M - c3 + (M + c1) / 2 - 2); c5 < min(c3, -2 * c3 + (c1 + c3) / 2 - 2); c5 += 1)
+      S1(c1 - 2 * c3 - 2 * c5 - 5, c3, c5);
+  for (int c3 = max(1, floord(-M + c1, 4)); c3 < (c1 + 1) / 5; c3 += 1)
+    S2(c1 - 4 * c3 - 3, c3);
+  if (c1 % 5 == 0)
+    S4(c1 / 5);
+  for (int c3 = max(-3 * M - c1 + 3 * ((M + c1) / 2) + 1, -((c1 - 1) % 3) + 3); c3 < (c1 + 1) / 5; c3 += 3)
+    S3((c1 - 2 * c3 - 1) / 3, c3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam3.in
new file mode 100644
index 00000000000..ebf78ae0b6b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam3.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0, i1, i2] -> [0, 5 + i0 + 2i1 + 2i2, 0, i1, 0, i2, 0] : i0 <= M and i1 <= -1 + i0 and i2 >= 1 and i2 <= -1 + i1; S3[i0, i1] -> [0, 1 + 3i0 + 2i1, 2, i1, 0, 0, 0] : i0 <= M and i1 >= 1 and i1 <= -1 + i0; S2[i0, i1] -> [0, 3 + i0 + 4i1, 1, i1, 1, 0, 0] : i0 <= M and i1 >= 1 and i1 <= -1 + i0; S4[i0] -> [0, 5i0, 2, 0, 0, 0, 0] : i0 >= 1 and i0 <= M }
+[M] -> {  : M >= 1 }
+[M] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam4.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam4.c
new file mode 100644
index 00000000000..6a0b0badab6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam4.c
@@ -0,0 +1,10 @@
+for (int c1 = 1; c1 < 2 * M - 1; c1 += 1) {
+  for (int c3 = max(-M + 1, -c1 + 1); c3 < 0; c3 += 1) {
+    for (int c7 = max(1, -M + c1 + 1); c7 <= min(M - 1, c1 + c3); c7 += 1)
+      S1(c7, c1 + c3 - c7, -c3);
+    for (int c5 = max(-M + c1 + 1, -c3); c5 < min(M, c1); c5 += 1)
+      S2(c1 - c5, c3 + c5, c5);
+  }
+  for (int c7 = max(1, -M + c1 + 1); c7 <= min(M - 1, c1); c7 += 1)
+    S1(c7, c1 - c7, 0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam4.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam4.in
new file mode 100644
index 00000000000..1bb684f8e2b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam4.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0, i1, i2] -> [0, i0 + i1 + i2, 0, -i2, 0, i2, 0] : i0 >= 1 and i0 <= -1 + M and i1 >= 0 and i2 >= 0 and i2 <= -1 + M - i1; S2[i0, i1, i2] -> [0, i0 + i2, 0, i1 - i2, 1, i2, 0] : i0 >= 1 and i0 <= -1 + M and i1 >= 0 and i2 >= 1 + i1 and i2 <= -1 + M }
+[M] -> {  :  }
+[M] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam5.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam5.c
new file mode 100644
index 00000000000..f476185713c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam5.c
@@ -0,0 +1,11 @@
+{
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    for (int c3 = 1; c3 <= M; c3 += 1)
+      S1(c1, c3);
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    for (int c3 = 1; c3 <= M; c3 += 1)
+      S2(c1, c3);
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    for (int c3 = 1; c3 <= M; c3 += 1)
+      S3(c1, c3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam5.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam5.in
new file mode 100644
index 00000000000..2e2046f1db9
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam5.in
@@ -0,0 +1,3 @@
+[M] -> { S2[i0, i1] -> [1, i0, 0, i1, 0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= M; S1[i0, i1] -> [0, i0, 0, i1, 0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= M; S3[i0, i1] -> [2, i0, 0, i1, 0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= M }
+[M] -> {  :  }
+[M] -> { [i, j, k, l, m] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam6.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam6.c
new file mode 100644
index 00000000000..2cc512261d6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam6.c
@@ -0,0 +1,8 @@
+{
+  for (int c1 = 0; c1 <= M; c1 += 1)
+    for (int c3 = 1; c3 <= M; c3 += 1)
+      S1(c1, c3);
+  for (int c1 = 0; c1 <= M; c1 += 1)
+    for (int c3 = 1; c3 <= M; c3 += 1)
+      S2(c3, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam6.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam6.in
new file mode 100644
index 00000000000..2699212b1df
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-lim-lam6.in
@@ -0,0 +1,3 @@
+[M] -> { S2[i0, i1] -> [1, i1, 0, i0, 0] : i0 >= 1 and i0 <= M and i1 >= 0 and i1 <= M; S1[i0, i1] -> [0, i0, 0, i1, 0] : i0 >= 0 and i0 <= M and i1 >= 1 and i1 <= M }
+[M] -> {  :  }
+[M] -> { [i, j, k, l, m] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-liu-zhuge1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-liu-zhuge1.c
new file mode 100644
index 00000000000..dfc7d37d901
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-liu-zhuge1.c
@@ -0,0 +1,17 @@
+if (M >= 0 && N >= 0)
+  for (int c1 = -4; c1 <= 3 * M + N; c1 += 1) {
+    if (3 * M + N >= c1 + 1 && c1 >= 3 * M) {
+      S2(M, -3 * M + c1);
+    } else if (3 * M >= c1 + 4 && (c1 - 2) % 3 == 0)
+      S1((c1 + 4) / 3, 0);
+    for (int c3 = max(-3 * M + c1 + 3, (c1 + 6) % 3); c3 <= min(N - 1, c1); c3 += 3) {
+      S2((c1 - c3) / 3, c3);
+      S1(((c1 - c3) / 3) + 1, c3 + 1);
+    }
+    if (c1 >= N && (N - c1) % 3 == 0) {
+      S2((-N + c1) / 3, N);
+    } else if (N >= c1 + 4 && c1 >= -3)
+      S1(0, c1 + 4);
+    for (int c3 = max(-3 * M + c1, (c1 + 6) % 3); c3 <= min(N, c1); c3 += 3)
+      S3((c1 - c3) / 3, c3);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-liu-zhuge1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-liu-zhuge1.in
new file mode 100644
index 00000000000..fd4dcc884f9
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-liu-zhuge1.in
@@ -0,0 +1,3 @@
+[M, N] -> { S2[i0, i1] -> [0, 3i0 + i1, 0, i1, 0] : i0 >= 0 and i0 <= M and i1 >= 0 and i1 <= N; S3[i0, i1] -> [0, 3i0 + i1, 1, i1, 0] : i0 >= 0 and i0 <= M and i1 >= 0 and i1 <= N; S1[i0, i1] -> [0, -4 + 3i0 + i1, 0, i1, 0] : i0 >= 0 and i0 <= M and i1 >= 0 and i1 <= N }
+[M, N] -> {  :  }
+[M, N] -> { [i, j, k, l, m] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner3.c
new file mode 100644
index 00000000000..d37df98b692
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner3.c
@@ -0,0 +1,4 @@
+for (int c1 = 1; c1 <= M; c1 += 1)
+  for (int c3 = 2; c3 <= M + c1; c3 += 1)
+    for (int c5 = max(1, -c1 + c3); c5 <= min(M, c3 - 1); c5 += 1)
+      S1(c1, c5, c3 - c5);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner3.in
new file mode 100644
index 00000000000..03731713ff7
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner3.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0, i1, i2] -> [0, i0, 0, i1 + i2, 0, i1, 0] : i0 <= M and i1 >= 1 and i1 <= M and i2 >= 1 and i2 <= i0 }
+[M] -> {  :  }
+[M] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner4.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner4.c
new file mode 100644
index 00000000000..7a67212b2bb
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner4.c
@@ -0,0 +1,5 @@
+for (int c1 = 2; c1 <= 2 * M; c1 += 1)
+  for (int c3 = 1; c3 <= M; c3 += 1)
+    for (int c5 = 1; c5 <= M; c5 += 1)
+      for (int c7 = max(1, -M + c1); c7 <= min(M, c1 - 1); c7 += 1)
+        S1(c5, c3, c7, c1 - c7);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner4.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner4.in
new file mode 100644
index 00000000000..1a8070210cd
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner4.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0, i1, i2, i3] -> [0, i2 + i3, 0, i1, 0, i0, 0, i2, 0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= M and i2 >= 1 and i2 <= M and i3 >= 1 and i3 <= M }
+[M] -> {  :  }
+[M] -> { [i, j, k, l, m, n, o, p, q] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner5.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner5.c
new file mode 100644
index 00000000000..c5f8fcc38e8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner5.c
@@ -0,0 +1,5 @@
+for (int c1 = 1; c1 <= M; c1 += 1)
+  for (int c3 = 1; c3 <= M; c3 += 1)
+    for (int c5 = 1; c5 <= M; c5 += 1)
+      for (int c7 = 1; c7 <= M; c7 += 1)
+        S1(c3, c5, c1, c7);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner5.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner5.in
new file mode 100644
index 00000000000..9122fc0a626
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-loechner5.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0, i1, i2, i3] -> [0, i2, 0, i0, 0, i1, 0, i3, 0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= M and i2 >= 1 and i2 <= M and i3 >= 1 and i3 <= M }
+[M] -> {  :  }
+[M] -> { [i, j, k, l, m, n, o, p, q] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-interp.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-interp.c
new file mode 100644
index 00000000000..cff8f1724e3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-interp.c
@@ -0,0 +1,85 @@
+{
+  if (N >= 2)
+    for (int c1 = 1; c1 < O; c1 += 1) {
+      for (int c5 = 1; c5 <= M; c5 += 1)
+        S1(c1, 1, c5);
+      for (int c5 = 1; c5 < M; c5 += 1) {
+        S6(c1, 1, c5);
+        S7(c1, 1, c5);
+      }
+      if (N >= 3) {
+        for (int c5 = 1; c5 <= M; c5 += 1)
+          S3(c1, 1, c5);
+        for (int c5 = 1; c5 <= M; c5 += 1)
+          S1(c1, 2, c5);
+        for (int c5 = 1; c5 < M; c5 += 1) {
+          S6(c1, 2, c5);
+          S7(c1, 2, c5);
+        }
+        for (int c5 = 1; c5 < M; c5 += 1)
+          S11(c1, 1, c5);
+      } else {
+        for (int c5 = 1; c5 <= M; c5 += 1)
+          S3(c1, 1, c5);
+        for (int c5 = 1; c5 < M; c5 += 1)
+          S11(c1, 1, c5);
+      }
+      for (int c3 = 3; c3 < 2 * N - 4; c3 += 2) {
+        for (int c5 = 1; c5 < M; c5 += 1)
+          S10(c1, (c3 - 1) / 2, c5);
+        for (int c5 = 1; c5 <= M; c5 += 1)
+          S3(c1, (c3 + 1) / 2, c5);
+        for (int c5 = 1; c5 <= M; c5 += 1)
+          S1(c1, (c3 + 3) / 2, c5);
+        for (int c5 = 1; c5 < M; c5 += 1) {
+          S6(c1, (c3 + 3) / 2, c5);
+          S7(c1, (c3 + 3) / 2, c5);
+        }
+        for (int c5 = 1; c5 < M; c5 += 1)
+          S11(c1, (c3 + 1) / 2, c5);
+      }
+      if (N >= 3) {
+        for (int c5 = 1; c5 < M; c5 += 1)
+          S10(c1, N - 2, c5);
+        for (int c5 = 1; c5 <= M; c5 += 1)
+          S3(c1, N - 1, c5);
+        for (int c5 = 1; c5 < M; c5 += 1)
+          S11(c1, N - 1, c5);
+      }
+      for (int c5 = 1; c5 < M; c5 += 1)
+        S10(c1, N - 1, c5);
+    }
+  for (int c1 = 1; c1 < O; c1 += 1)
+    for (int c3 = 1; c3 < N; c3 += 1) {
+      for (int c5 = 1; c5 <= M; c5 += 1)
+        S2(c1, c3, c5);
+      for (int c5 = 1; c5 < M; c5 += 1)
+        S8(c1, c3, c5);
+      for (int c5 = 1; c5 < M; c5 += 1)
+        S9(c1, c3, c5);
+    }
+  for (int c1 = 1; c1 < O; c1 += 1)
+    for (int c3 = 1; c3 < N; c3 += 1)
+      for (int c5 = 1; c5 < M; c5 += 1)
+        S4(c1, c3, c5);
+  for (int c1 = 1; c1 < O; c1 += 1)
+    for (int c3 = 1; c3 < N; c3 += 1)
+      for (int c5 = 1; c5 < M; c5 += 1)
+        S5(c1, c3, c5);
+  for (int c1 = R; c1 < O; c1 += 1)
+    for (int c3 = Q; c3 < N; c3 += 1)
+      for (int c5 = P; c5 < M; c5 += 1)
+        S12(c1, c3, c5);
+  for (int c1 = R; c1 < O; c1 += 1)
+    for (int c3 = Q; c3 < N; c3 += 1)
+      for (int c5 = 1; c5 < M; c5 += 1)
+        S13(c1, c3, c5);
+  for (int c1 = R; c1 < O; c1 += 1)
+    for (int c3 = 1; c3 < N; c3 += 1)
+      for (int c5 = P; c5 < M; c5 += 1)
+        S14(c1, c3, c5);
+  for (int c1 = R; c1 < O; c1 += 1)
+    for (int c3 = 1; c3 < N; c3 += 1)
+      for (int c5 = 1; c5 < M; c5 += 1)
+        S15(c1, c3, c5);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-interp.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-interp.in
new file mode 100644
index 00000000000..b7fdb69788c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-interp.in
@@ -0,0 +1,3 @@
+[M, N, O, P, Q, R, S, T, U] -> { S8[i0, i1, i2] -> [1, i0, 0, 2i1, 1, i2, 0] : i0 >= 1 and i0 <= -1 + O and i1 >= 1 and i1 <= -1 + N and i2 >= 1 and i2 <= -1 + M; S11[i0, i1, i2] -> [0, i0, 0, 2i1, 4, i2, 1] : i0 >= 1 and i0 <= -1 + O and i1 >= 1 and i1 <= -1 + N and i2 >= 1 and i2 <= -1 + M; S6[i0, i1, i2] -> [0, i0, 0, -2 + 2i1, 2, i2, 0] : i0 >= 1 and i0 <= -1 + O and i1 >= 1 and i1 <= -1 + N and i2 >= 1 and i2 <= -1 + M; S9[i0, i1, i2] -> [1, i0, 0, 1 + 2i1, 3, i2, 1] : i0 >= 1 and i0 <= -1 + O and i1 >= 1 and i1 <= -1 + N and i2 >= 1 and i2 <= -1 + M; S1[i0, i1, i2] -> [0, i0, 0, -3 + 2i1, 2, i2, 0] : i0 >= 1 and i0 <= -1 + O and i1 >= 1 and i1 <= -1 + N and i2 >= 1 and i2 <= M; S4[i0, i1, i2] -> [2, i0, 0, i1, 1, i2, 0] : i0 >= 1 and i0 <= -1 + O and i1 >= 1 and i1 <= -1 + N and i2 >= 1 and i2 <= -1 + M; S2[i0, i1, i2] -> [1, i0, 0, 2i1, 0, i2, 1] : i0 >= 1 and i0 <= -1 + O and i1 >= 1 and i1 <= -1 + N and i2 >= 1 and i2 <= M; S7[i0, i1, i2] -> [0, i0, 0, -2 + 2i1, 2, i2, 1] : i0 >= 1 and i0 <= -1 + O and i1 >= 1 and i1 <= -1 + N and i2 >= 1 and i2 <= -1 + M; S10[i0, i1, i2] -> [0, i0, 0, 1 + 2i1, 0, i2, 0] : i0 >= 1 and i0 <= -1 + O and i1 >= 1 and i1 <= -1 + N and i2 >= 1 and i2 <= -1 + M; S3[i0, i1, i2] -> [0, i0, 0, -1 + 2i1, 1, i2, 2] : i0 >= 1 and i0 <= -1 + O and i1 >= 1 and i1 <= -1 + N and i2 >= 1 and i2 <= M; S15[i0, i1, i2] -> [7, i0, 1, i1, 1, i2, 0] : i0 >= R and i0 <= -1 + O and i1 >= 1 and i1 <= -1 + N and i2 >= 1 and i2 <= -1 + M; S5[i0, i1, i2] -> [3, i0, 0, i1, 1, i2, 1] : i0 >= 1 and i0 <= -1 + O and i1 >= 1 and i1 <= -1 + N and i2 >= 1 and i2 <= -1 + M; S13[i0, i1, i2] -> [5, i0, 0, i1, 1, i2, 0] : i0 >= R and i0 <= -1 + O and i1 >= Q and i1 <= -1 + N and i2 >= 1 and i2 <= -1 + M; S12[i0, i1, i2] -> [4, i0, 0, i1, 0, i2, 0] : i0 >= R and i0 <= -1 + O and i1 >= Q and i1 <= -1 + N and i2 >= P and i2 <= -1 + M; S14[i0, i1, i2] -> [6, i0, 1, i1, 0, i2, 0] : i0 >= R and i0 <= -1 + O and i1 >= 1 and i1 <= -1 + N and i2 >= P and i2 <= -1 + M }
+[M, N, O, P, Q, R, S, T, U] -> {  :  }
+[M, N, O, P, Q, R, S, T, U] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-interp2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-interp2.c
new file mode 100644
index 00000000000..98cd4b0707a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-interp2.c
@@ -0,0 +1,18 @@
+{
+  for (int c1 = 1; c1 < O; c1 += 1)
+    for (int c3 = Q; c3 < N; c3 += 1)
+      for (int c5 = P; c5 < M; c5 += 1)
+        S1(c1, c3, c5);
+  for (int c1 = 1; c1 < O; c1 += 1)
+    for (int c3 = Q; c3 < N; c3 += 1)
+      for (int c5 = 1; c5 < M; c5 += 1)
+        S2(c1, c3, c5);
+  for (int c1 = 1; c1 < O; c1 += 1)
+    for (int c3 = 1; c3 < N; c3 += 1)
+      for (int c5 = P; c5 < M; c5 += 1)
+        S3(c1, c3, c5);
+  for (int c1 = 1; c1 < O; c1 += 1)
+    for (int c3 = 1; c3 < N; c3 += 1)
+      for (int c5 = 1; c5 < M; c5 += 1)
+        S4(c1, c3, c5);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-interp2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-interp2.in
new file mode 100644
index 00000000000..8a5dfb12d5a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-interp2.in
@@ -0,0 +1,3 @@
+[M, N, O, P, Q, R, S, T, U] -> { S1[i0, i1, i2] -> [0, i0, 0, i1, 0, i2, 0] : i0 >= 1 and i0 <= -1 + O and i1 >= Q and i1 <= -1 + N and i2 >= P and i2 <= -1 + M; S4[i0, i1, i2] -> [3, i0, 1, i1, 1, i2, 0] : i0 >= 1 and i0 <= -1 + O and i1 >= 1 and i1 <= -1 + N and i2 >= 1 and i2 <= -1 + M; S2[i0, i1, i2] -> [1, i0, 0, i1, 1, i2, 0] : i0 >= 1 and i0 <= -1 + O and i1 >= Q and i1 <= -1 + N and i2 >= 1 and i2 <= -1 + M; S3[i0, i1, i2] -> [2, i0, 1, i1, 0, i2, 0] : i0 >= 1 and i0 <= -1 + O and i1 >= 1 and i1 <= -1 + N and i2 >= P and i2 <= -1 + M }
+[M, N, O, P, Q, R, S, T, U] -> {  :  }
+[M, N, O, P, Q, R, S, T, U] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-psinv.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-psinv.c
new file mode 100644
index 00000000000..e3a5deeb1e7
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-psinv.c
@@ -0,0 +1,9 @@
+for (int c1 = 2; c1 < O; c1 += 1)
+  for (int c3 = 3; c3 < 2 * N - 2; c3 += 2) {
+    for (int c5 = 1; c5 <= M; c5 += 1) {
+      S1(c1, (c3 + 1) / 2, c5);
+      S2(c1, (c3 + 1) / 2, c5);
+    }
+    for (int c5 = 2; c5 < M; c5 += 1)
+      S3(c1, (c3 + 1) / 2, c5);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-psinv.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-psinv.in
new file mode 100644
index 00000000000..835f81a4725
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-psinv.in
@@ -0,0 +1,3 @@
+[M, N, O] -> { S1[i0, i1, i2] -> [0, i0, 0, -1 + 2i1, 0, i2, 0] : i0 >= 2 and i0 <= -1 + O and i1 >= 2 and i1 <= -1 + N and i2 >= 1 and i2 <= M; S2[i0, i1, i2] -> [0, i0, 0, -1 + 2i1, 0, i2, 1] : i0 >= 2 and i0 <= -1 + O and i1 >= 2 and i1 <= -1 + N and i2 >= 1 and i2 <= M; S3[i0, i1, i2] -> [0, i0, 0, 2i1, 1, i2, 0] : i0 >= 2 and i0 <= -1 + O and i1 >= 2 and i1 <= -1 + N and i2 >= 2 and i2 <= -1 + M }
+[M, N, O] -> {  :  }
+[M, N, O] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-resid.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-resid.c
new file mode 100644
index 00000000000..e3a5deeb1e7
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-resid.c
@@ -0,0 +1,9 @@
+for (int c1 = 2; c1 < O; c1 += 1)
+  for (int c3 = 3; c3 < 2 * N - 2; c3 += 2) {
+    for (int c5 = 1; c5 <= M; c5 += 1) {
+      S1(c1, (c3 + 1) / 2, c5);
+      S2(c1, (c3 + 1) / 2, c5);
+    }
+    for (int c5 = 2; c5 < M; c5 += 1)
+      S3(c1, (c3 + 1) / 2, c5);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-resid.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-resid.in
new file mode 100644
index 00000000000..835f81a4725
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-resid.in
@@ -0,0 +1,3 @@
+[M, N, O] -> { S1[i0, i1, i2] -> [0, i0, 0, -1 + 2i1, 0, i2, 0] : i0 >= 2 and i0 <= -1 + O and i1 >= 2 and i1 <= -1 + N and i2 >= 1 and i2 <= M; S2[i0, i1, i2] -> [0, i0, 0, -1 + 2i1, 0, i2, 1] : i0 >= 2 and i0 <= -1 + O and i1 >= 2 and i1 <= -1 + N and i2 >= 1 and i2 <= M; S3[i0, i1, i2] -> [0, i0, 0, 2i1, 1, i2, 0] : i0 >= 2 and i0 <= -1 + O and i1 >= 2 and i1 <= -1 + N and i2 >= 2 and i2 <= -1 + M }
+[M, N, O] -> {  :  }
+[M, N, O] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-rprj3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-rprj3.c
new file mode 100644
index 00000000000..992bfd53f02
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-rprj3.c
@@ -0,0 +1,33 @@
+if (M >= 2 && N >= 3)
+  for (int c1 = 2; c1 < O; c1 += 1) {
+    for (int c5 = 2; c5 <= M; c5 += 1)
+      S1(c1, 2, c5);
+    for (int c3 = 3; c3 < N; c3 += 1) {
+      for (int c5 = 2; c5 <= M; c5 += 1)
+        S2(c1, c3 - 1, c5);
+      if (M >= 3) {
+        S4(c1, c3 - 1, 2);
+        for (int c5 = 2; c5 < M - 1; c5 += 1) {
+          S3(c1, c3 - 1, c5);
+          S5(c1, c3 - 1, c5);
+          S4(c1, c3 - 1, c5 + 1);
+        }
+        S3(c1, c3 - 1, M - 1);
+        S5(c1, c3 - 1, M - 1);
+      }
+      for (int c5 = 2; c5 <= M; c5 += 1)
+        S1(c1, c3, c5);
+    }
+    for (int c5 = 2; c5 <= M; c5 += 1)
+      S2(c1, N - 1, c5);
+    if (M >= 3) {
+      S4(c1, N - 1, 2);
+      for (int c5 = 2; c5 < M - 1; c5 += 1) {
+        S3(c1, N - 1, c5);
+        S5(c1, N - 1, c5);
+        S4(c1, N - 1, c5 + 1);
+      }
+      S3(c1, N - 1, M - 1);
+      S5(c1, N - 1, M - 1);
+    }
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-rprj3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-rprj3.in
new file mode 100644
index 00000000000..03f4134661f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-mg-rprj3.in
@@ -0,0 +1,3 @@
+[M, N, O, P, Q, R] -> { S1[i0, i1, i2] -> [0, i0, 0, i1, 2, i2, 0] : i0 >= 2 and i0 <= -1 + O and i1 >= 2 and i1 <= -1 + N and i2 >= 2 and i2 <= M; S4[i0, i1, i2] -> [0, i0, 0, 1 + i1, 1, -1 + i2, 2] : i0 >= 2 and i0 <= -1 + O and i1 >= 2 and i1 <= -1 + N and i2 >= 2 and i2 <= -1 + M; S3[i0, i1, i2] -> [0, i0, 0, 1 + i1, 1, i2, 0] : i0 >= 2 and i0 <= -1 + O and i1 >= 2 and i1 <= -1 + N and i2 >= 2 and i2 <= -1 + M; S5[i0, i1, i2] -> [0, i0, 0, 1 + i1, 1, i2, 1] : i0 >= 2 and i0 <= -1 + O and i1 >= 2 and i1 <= -1 + N and i2 >= 2 and i2 <= -1 + M; S2[i0, i1, i2] -> [0, i0, 0, 1 + i1, 0, i2, 1] : i0 >= 2 and i0 <= -1 + O and i1 >= 2 and i1 <= -1 + N and i2 >= 2 and i2 <= M }
+[M, N, O, P, Q, R] -> {  :  }
+[M, N, O, P, Q, R] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali1.c
new file mode 100644
index 00000000000..7268c23d080
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali1.c
@@ -0,0 +1,7 @@
+for (int c1 = 1; c1 <= M; c1 += 1)
+  for (int c3 = 1; c3 < 2 * N; c3 += 1) {
+    for (int c5 = max(1, -N + c3); c5 < (c3 + 1) / 2; c5 += 1)
+      S1(c1, c3 - c5, c5);
+    if ((c3 - 1) % 2 == 0)
+      S2(c1, (c3 + 1) / 2);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali1.in
new file mode 100644
index 00000000000..ecbb320de01
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali1.in
@@ -0,0 +1,3 @@
+[M, N] -> { S1[i0, i1, i2] -> [0, i0, 0, i1 + i2, 0, i2, 0] : i0 >= 1 and i0 <= M and i1 <= N and i2 >= 1 and i2 <= -1 + i1; S2[i0, i1] -> [0, i0, 0, -1 + 2i1, 1, 0, 0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= N }
+[M, N] -> {  :  }
+[M, N] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali2.c
new file mode 100644
index 00000000000..a407cad52d2
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali2.c
@@ -0,0 +1,8 @@
+{
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    for (int c3 = 1; c3 <= M; c3 += 1)
+      S1(c1, c3);
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    for (int c3 = 1; c3 <= M; c3 += 1)
+      S2(c1, c3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali2.in
new file mode 100644
index 00000000000..9ba248fb0fc
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali2.in
@@ -0,0 +1,3 @@
+[M] -> { S2[i0, i1] -> [1, i0, 1, i1, 0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= M; S1[i0, i1] -> [0, i0, 0, i1, 0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= M }
+[M] -> {  :  }
+[M] -> { [i, j, k, l, m] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali3.c
new file mode 100644
index 00000000000..c281b826e01
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali3.c
@@ -0,0 +1,9 @@
+{
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    for (int c3 = 1; c3 <= M; c3 += 1)
+      S1(c1, c3);
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    for (int c3 = 1; c3 <= M; c3 += 1)
+      for (int c5 = 1; c5 <= M; c5 += 1)
+        S2(c1, c3, c5);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali3.in
new file mode 100644
index 00000000000..70da8ea5c11
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali3.in
@@ -0,0 +1,3 @@
+[M] -> { S2[i0, i1, i2] -> [1, i0, 0, i1, 1, i2, 0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= M and i2 >= 1 and i2 <= M; S1[i0, i1] -> [0, i0, 0, i1, 0, 0, 0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= M }
+[M] -> {  :  }
+[M] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali4.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali4.c
new file mode 100644
index 00000000000..a407cad52d2
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali4.c
@@ -0,0 +1,8 @@
+{
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    for (int c3 = 1; c3 <= M; c3 += 1)
+      S1(c1, c3);
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    for (int c3 = 1; c3 <= M; c3 += 1)
+      S2(c1, c3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali4.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali4.in
new file mode 100644
index 00000000000..e975faef4c7
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali4.in
@@ -0,0 +1,3 @@
+[M] -> { S2[i0, i1] -> [1, i0, 0, i1, 0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= M; S1[i0, i1] -> [0, i0, 0, i1, 0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= M }
+[M] -> {  : M >= 2 }
+[M] -> { [i, j, k, l, m] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali5.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali5.c
new file mode 100644
index 00000000000..51f2f0c154e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali5.c
@@ -0,0 +1,10 @@
+for (int c1 = 3; c1 < 2 * M; c1 += 1) {
+  for (int c3 = c1 / 2 + 2; c3 <= M; c3 += 1)
+    for (int c7 = c1 / 2 + 1; c7 < min(c1, c3); c7 += 1)
+      S1(c7, c1 - c7, c3);
+  for (int c3 = max(1, -M + c1); c3 < (c1 + 1) / 2; c3 += 1)
+    S2(c1 - c3, c3);
+  for (int c3 = c1 / 2 + 2; c3 <= M; c3 += 1)
+    for (int c7 = c1 / 2 + 1; c7 < min(c1, c3); c7 += 1)
+      S3(c7, c1 - c7, c3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali5.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali5.in
new file mode 100644
index 00000000000..1b77acdc8d6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali5.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0, i1, i2] -> [0, i0 + i1, 0, i2, 0, i2, 0] : i1 >= 1 and i1 <= -1 + i0 and i2 >= 1 + i0 and i2 <= M; S3[i0, i1, i2] -> [0, i0 + i1, 2, i2, 2, i2, 0] : i1 >= 1 and i1 <= -1 + i0 and i2 >= 1 + i0 and i2 <= M; S2[i0, i1] -> [0, i0 + i1, 1, i1, 1, 0, 0] : i0 <= M and i1 >= 1 and i1 <= -1 + i0 }
+[M] -> {  :  }
+[M] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali6.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali6.c
new file mode 100644
index 00000000000..1df9bb90c24
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali6.c
@@ -0,0 +1,8 @@
+for (int c1 = 1; c1 <= M; c1 += 1) {
+  for (int c3 = 2; c3 < N; c3 += 1)
+    for (int c5 = 2; c5 < N; c5 += 1)
+      S1(c1, c3, c5);
+  for (int c3 = 2; c3 < N; c3 += 1)
+    for (int c5 = 2; c5 < N; c5 += 1)
+      S2(c1, c3, c5);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali6.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali6.in
new file mode 100644
index 00000000000..7a08ca76e8e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-pingali6.in
@@ -0,0 +1,3 @@
+[M, N] -> { S2[i0, i1, i2] -> [0, 1 + 2i0, 1, i1, 0, i2, 0] : i0 >= 1 and i0 <= M and i1 >= 2 and i1 <= -1 + N and i2 >= 2 and i2 <= -1 + N; S1[i0, i1, i2] -> [0, 2i0, 0, i1, 0, i2, 0] : i0 >= 1 and i0 <= M and i1 >= 2 and i1 <= -1 + N and i2 >= 2 and i2 <= -1 + N }
+[M, N] -> {  : M >= 1 and N >= 1 }
+[M, N] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-stride.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-stride.c
new file mode 100644
index 00000000000..104e2620fdd
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-stride.c
@@ -0,0 +1,2 @@
+for (int c1 = 2; c1 <= M; c1 += 7)
+  S1(c1, (c1 - 2) / 7);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-stride.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-stride.in
new file mode 100644
index 00000000000..46a1c35375e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-stride.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0, i1] -> [0, i0, 0, 0, 0] : 7i1 = -2 + i0 and i0 >= 2 and i0 <= M }
+[M] -> {  :  }
+[M] -> { [i, j, k, l, m] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-stride2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-stride2.c
new file mode 100644
index 00000000000..104e2620fdd
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-stride2.c
@@ -0,0 +1,2 @@
+for (int c1 = 2; c1 <= M; c1 += 7)
+  S1(c1, (c1 - 2) / 7);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-stride2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-stride2.in
new file mode 100644
index 00000000000..be6b1978a4b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-stride2.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0, i1] -> [0, i0, 0, 0, 0] : 7i1 = -2 + i0 and i0 >= 0 and i0 <= M }
+[M] -> {  :  }
+[M] -> { [i, j, k, l, m] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-tang-xue1.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-tang-xue1.c
new file mode 100644
index 00000000000..8874eabfceb
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-tang-xue1.c
@@ -0,0 +1,5 @@
+for (int c1 = 0; c1 <= 9; c1 += 2)
+  for (int c3 = 0; c3 <= min(4, c1 + 3); c3 += 2)
+    for (int c5 = max(1, c1); c5 <= min(c1 + 1, c1 - c3 + 4); c5 += 1)
+      for (int c7 = max(1, -c1 + c3 + c5); c7 <= min(4, -c1 + c3 + c5 + 1); c7 += 1)
+        S1(c1 / 2, (-c1 + c3) / 2, -c1 + c5, -c3 + c7);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-tang-xue1.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-tang-xue1.in
new file mode 100644
index 00000000000..0c9ffc9f77f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-tang-xue1.in
@@ -0,0 +1,3 @@
+{ S1[i0, i1, i2, i3] -> [0, 2i0, 0, 2i0 + 2i1, 0, 2i0 + i2, 0, 2i0 + 2i1 + i3, 0] : i3 <= 4 - 2i0 - 2i1 and i3 >= i2 and i2 <= 9 - 2i0 and i2 >= 0 and i2 >= 1 - 2i0 and i3 <= 1 + i2 and i2 <= 1 and i3 >= 1 - 2i0 - 2i1 }
+{  :  }
+{ [i, j, k, l, m, n, o, p, q] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-two.c b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-two.c
new file mode 100644
index 00000000000..3946c423f15
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-two.c
@@ -0,0 +1 @@
+S1(1, 1, 5);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-two.in b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-two.in
new file mode 100644
index 00000000000..a7eb204753b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/reservoir-two.in
@@ -0,0 +1,3 @@
+{ S1[i0, i1, i2] -> [i0, j, k, 0] : 2i1 = 3 - i0 and 2j = 3 - i0 and 2k = 9 + i0 and 2i2 = 9 + i0 and i0 >= 0 and i0 <= 1 }
+{  :  }
+{ [i, j, k, l] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/singleton.c b/polly/lib/External/isl/test_inputs/codegen/cloog/singleton.c
new file mode 100644
index 00000000000..0023bc6cd9a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/singleton.c
@@ -0,0 +1,4 @@
+{
+  S2();
+  S1();
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/singleton.in b/polly/lib/External/isl/test_inputs/codegen/cloog/singleton.in
new file mode 100644
index 00000000000..a0a5a8a1dde
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/singleton.in
@@ -0,0 +1,3 @@
+{ S2[] -> [-1]; S1[] -> [0] }
+{  :  }
+{ [i] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/square+triangle-1-1-2-3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/square+triangle-1-1-2-3.c
new file mode 100644
index 00000000000..316213f18f1
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/square+triangle-1-1-2-3.c
@@ -0,0 +1,9 @@
+for (int c0 = 1; c0 <= M; c0 += 1) {
+  S1(c0, 1);
+  for (int c1 = 2; c1 <= c0; c1 += 1) {
+    S1(c0, c1);
+    S2(c0, c1);
+  }
+  for (int c1 = c0 + 1; c1 <= M; c1 += 1)
+    S1(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/square+triangle-1-1-2-3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/square+triangle-1-1-2-3.in
new file mode 100644
index 00000000000..028c792b640
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/square+triangle-1-1-2-3.in
@@ -0,0 +1,3 @@
+[M] -> { S2[i0, i1] -> [i0, i1, 1] : i1 >= 2 and i1 <= i0 and i0 <= M; S1[i0, i1] -> [i0, i1, 0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= M }
+[M] -> {  : M >= 1 }
+[M] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/stride.c b/polly/lib/External/isl/test_inputs/codegen/cloog/stride.c
new file mode 100644
index 00000000000..9bd4ddfb307
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/stride.c
@@ -0,0 +1,7 @@
+{
+  for (int c0 = 3; c0 <= 24; c0 += 3)
+    S2(c0, c0 / 3);
+  S1(25);
+  for (int c0 = 27; c0 <= 100; c0 += 3)
+    S2(c0, c0 / 3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/stride.in b/polly/lib/External/isl/test_inputs/codegen/cloog/stride.in
new file mode 100644
index 00000000000..410c6843527
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/stride.in
@@ -0,0 +1,3 @@
+{ S1[25] -> [25, 0]; S2[i0, i1] -> [i0, j] : 3i1 = i0 and 3j = i0 and i0 >= 3 and i0 <= 100 }
+{  :  }
+{ [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/stride2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/stride2.c
new file mode 100644
index 00000000000..c56af8e749e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/stride2.c
@@ -0,0 +1,8 @@
+{
+  for (int c0 = 3; c0 <= 26; c0 += 3)
+    S2(c0, c0 / 3);
+  S1(27);
+  S2(27, 9);
+  for (int c0 = 30; c0 <= 100; c0 += 3)
+    S2(c0, c0 / 3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/stride2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/stride2.in
new file mode 100644
index 00000000000..97e40866adf
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/stride2.in
@@ -0,0 +1,3 @@
+{ S1[27] -> [27, 0]; S2[i0, i1] -> [i0, j] : 3i1 = i0 and 3j = i0 and i0 >= 3 and i0 <= 100 }
+{  :  }
+{ [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/stride3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/stride3.c
new file mode 100644
index 00000000000..d145938b663
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/stride3.c
@@ -0,0 +1,2 @@
+for (int c0 = max(1, m); c0 <= n; c0 += 1)
+  S1(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/stride3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/stride3.in
new file mode 100644
index 00000000000..9b87b899243
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/stride3.in
@@ -0,0 +1,3 @@
+[m, n] -> { S1[i0] -> [50i0, 0] : i0 >= 1 and i0 <= n and i0 >= m }
+[m, n] -> {  :  }
+[m, n] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/stride4.c b/polly/lib/External/isl/test_inputs/codegen/cloog/stride4.c
new file mode 100644
index 00000000000..03908e60c07
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/stride4.c
@@ -0,0 +1,3 @@
+if (t >= 0 && t <= 15)
+  for (int c0 = t; c0 <= 99; c0 += 16)
+    S1(c0, t);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/stride4.in b/polly/lib/External/isl/test_inputs/codegen/cloog/stride4.in
new file mode 100644
index 00000000000..436cad3c5a5
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/stride4.in
@@ -0,0 +1,3 @@
+[t] -> { S1[i0, t] -> [i0, t, 0] : exists (e0 = [(t - i0)/16]: 16e0 = t - i0 and i0 >= 0 and i0 <= 99 and t >= 0 and t <= 15) }
+[t] -> {  :  }
+[t] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/swim.c b/polly/lib/External/isl/test_inputs/codegen/cloog/swim.c
new file mode 100644
index 00000000000..47b4ebe5ff8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/swim.c
@@ -0,0 +1,159 @@
+if (M == 1) {
+  S1();
+  S2();
+  S3();
+  S4();
+  S5();
+  S6();
+  S7();
+  S8();
+  S9();
+  S10();
+  S11();
+  S12();
+  S13();
+  S14();
+  S15();
+  S16();
+  S17();
+  S18();
+  S19();
+  S20();
+  S21();
+  S22();
+  S23();
+  S24();
+  S25();
+  S26();
+  S27();
+  for (int c1 = 1; c1 <= N; c1 += 1) {
+    for (int c3 = 1; c3 <= N; c3 += 1) {
+      S28(c1, c3);
+      S29(c1, c3);
+      S30(c1, c3);
+    }
+    S31(c1);
+  }
+  S32();
+  S33();
+  S34();
+  if (O <= 1)
+    S35();
+  S36();
+  S37();
+  for (int c1 = 2; c1 <= P; c1 += 1) {
+    S38(c1);
+    S39(c1);
+    for (int c3 = 1; c3 <= Q; c3 += 1)
+      for (int c5 = 1; c5 <= R; c5 += 1) {
+        S40(c1, c3, c5);
+        S41(c1, c3, c5);
+        S42(c1, c3, c5);
+        S43(c1, c3, c5);
+      }
+    for (int c3 = 1; c3 <= Q; c3 += 1) {
+      S44(c1, c3);
+      S45(c1, c3);
+      S46(c1, c3);
+      S47(c1, c3);
+    }
+    for (int c3 = 1; c3 <= R; c3 += 1) {
+      S48(c1, c3);
+      S49(c1, c3);
+      S50(c1, c3);
+      S51(c1, c3);
+    }
+    S52(c1);
+    S53(c1);
+    S54(c1);
+    S55(c1);
+    S56(c1);
+    S57(c1);
+    S58(c1);
+    for (int c3 = 1; c3 <= Q; c3 += 1)
+      for (int c5 = 1; c5 <= R; c5 += 1) {
+        S59(c1, c3, c5);
+        S60(c1, c3, c5);
+        S61(c1, c3, c5);
+      }
+    for (int c3 = 1; c3 <= Q; c3 += 1) {
+      S62(c1, c3);
+      S63(c1, c3);
+      S64(c1, c3);
+    }
+    for (int c3 = 1; c3 <= R; c3 += 1) {
+      S65(c1, c3);
+      S66(c1, c3);
+      S67(c1, c3);
+    }
+    S68(c1);
+    S69(c1);
+    S70(c1);
+    S71(c1);
+    S72(c1);
+    S73(c1);
+    S74(c1);
+    S75(c1);
+    S76(c1);
+    S77(c1);
+    S78(c1);
+    S79(c1);
+    S80(c1);
+    S81(c1);
+    S82(c1);
+    S83(c1);
+    S84(c1);
+    S85(c1);
+    S86(c1);
+    S87(c1);
+    S88(c1);
+    S89(c1);
+    S90(c1);
+    S91(c1);
+    S92(c1);
+    S93(c1);
+    S94(c1);
+    for (int c3 = 1; c3 <= N; c3 += 1) {
+      for (int c5 = 1; c5 <= N; c5 += 1) {
+        S95(c1, c3, c5);
+        S96(c1, c3, c5);
+        S97(c1, c3, c5);
+      }
+      S98(c1, c3);
+    }
+    S99(c1);
+    S100(c1);
+    S101(c1);
+    for (int c3 = 1; c3 <= Q; c3 += 1)
+      for (int c5 = 1; c5 <= R; c5 += 1) {
+        S102(c1, c3, c5);
+        S103(c1, c3, c5);
+        S104(c1, c3, c5);
+        S105(c1, c3, c5);
+        S106(c1, c3, c5);
+        S107(c1, c3, c5);
+      }
+    for (int c3 = 1; c3 <= Q; c3 += 1) {
+      S108(c1, c3);
+      S109(c1, c3);
+      S110(c1, c3);
+      S111(c1, c3);
+      S112(c1, c3);
+      S113(c1, c3);
+    }
+    for (int c3 = 1; c3 <= R; c3 += 1) {
+      S114(c1, c3);
+      S115(c1, c3);
+      S116(c1, c3);
+      S117(c1, c3);
+      S118(c1, c3);
+      S119(c1, c3);
+    }
+    S120(c1);
+    S121(c1);
+    S122(c1);
+    S123(c1);
+    S124(c1);
+    S125(c1);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/swim.in b/polly/lib/External/isl/test_inputs/codegen/cloog/swim.in
new file mode 100644
index 00000000000..4661ff0cff2
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/swim.in
@@ -0,0 +1,3 @@
+[M, N, O, P, Q, R] -> { S84[i0] -> [34, i0, 31, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S6[] -> [5, 0, 0, 0, 0, 0, 0] : M = 1; S25[] -> [24, 0, 0, 0, 0, 0, 0] : M = 1; S115[i0, i1] -> [34, i0, 48, i1, 1, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= R; S64[i0, i1] -> [34, i0, 13, i1, 2, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q; S62[i0, i1] -> [34, i0, 13, i1, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q; S40[i0, i1, i2] -> [34, i0, 2, i1, 0, i2, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q and i2 >= 1 and i2 <= R; S123[i0] -> [34, i0, 52, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S5[] -> [4, 0, 0, 0, 0, 0, 0] : M = 1; S15[] -> [14, 0, 0, 0, 0, 0, 0] : M = 1; S76[i0] -> [34, i0, 23, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S39[i0] -> [34, i0, 1, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S13[] -> [12, 0, 0, 0, 0, 0, 0] : M = 1; S28[i0, i1] -> [27, i0, 0, i1, 0, 0, 0] : M = 1 and i0 >= 1 and i0 <= N and i1 >= 1 and i1 <= N; S70[i0] -> [34, i0, 17, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S53[i0] -> [34, i0, 6, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S121[i0] -> [34, i0, 50, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S47[i0, i1] -> [34, i0, 3, i1, 3, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q; S59[i0, i1, i2] -> [34, i0, 12, i1, 0, i2, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q and i2 >= 1 and i2 <= R; S71[i0] -> [34, i0, 18, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S77[i0] -> [34, i0, 24, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S63[i0, i1] -> [34, i0, 13, i1, 1, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q; S51[i0, i1] -> [34, i0, 4, i1, 3, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= R; S100[i0] -> [34, i0, 44, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S22[] -> [21, 0, 0, 0, 0, 0, 0] : M = 1; S95[i0, i1, i2] -> [34, i0, 42, i1, 0, i2, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= N and i2 >= 1 and i2 <= N; S8[] -> [7, 0, 0, 0, 0, 0, 0] : M = 1; S120[i0] -> [34, i0, 49, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S50[i0, i1] -> [34, i0, 4, i1, 2, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= R; S45[i0, i1] -> [34, i0, 3, i1, 1, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q; S72[i0] -> [34, i0, 19, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S52[i0] -> [34, i0, 5, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S7[] -> [6, 0, 0, 0, 0, 0, 0] : M = 1; S78[i0] -> [34, i0, 25, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S110[i0, i1] -> [34, i0, 47, i1, 2, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q; S124[i0] -> [34, i0, 53, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S60[i0, i1, i2] -> [34, i0, 12, i1, 0, i2, 1] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q and i2 >= 1 and i2 <= R; S27[] -> [26, 0, 0, 0, 0, 0, 0] : M = 1; S114[i0, i1] -> [34, i0, 48, i1, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= R; S49[i0, i1] -> [34, i0, 4, i1, 1, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= R; S94[i0] -> [34, i0, 41, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S57[i0] -> [34, i0, 10, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S66[i0, i1] -> [34, i0, 14, i1, 1, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= R; S18[] -> [17, 0, 0, 0, 0, 0, 0] : M = 1; S92[i0] -> [34, i0, 39, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S3[] -> [2, 0, 0, 0, 0, 0, 0] : M = 1; S35[] -> [31, 0, 0, 0, 0, 0, 0] : M = 1 and O <= 1; S36[] -> [32, 0, 0, 0, 0, 0, 0] : M = 1; S10[] -> [9, 0, 0, 0, 0, 0, 0] : M = 1; S2[] -> [1, 0, 0, 0, 0, 0, 0] : M = 1; S48[i0, i1] -> [34, i0, 4, i1, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= R; S30[i0, i1] -> [27, i0, 0, i1, 2, 0, 0] : M = 1 and i0 >= 1 and i0 <= N and i1 >= 1 and i1 <= N; S75[i0] -> [34, i0, 22, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S83[i0] -> [34, i0, 30, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S87[i0] -> [34, i0, 34, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S88[i0] -> [34, i0, 35, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S33[] -> [29, 0, 0, 0, 0, 0, 0] : M = 1; S101[i0] -> [34, i0, 45, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S21[] -> [20, 0, 0, 0, 0, 0, 0] : M = 1; S32[] -> [28, 0, 0, 0, 0, 0, 0] : M = 1; S118[i0, i1] -> [34, i0, 48, i1, 4, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= R; S79[i0] -> [34, i0, 26, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S38[i0] -> [34, i0, 0, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S108[i0, i1] -> [34, i0, 47, i1, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q; S65[i0, i1] -> [34, i0, 14, i1, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= R; S91[i0] -> [34, i0, 38, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S17[] -> [16, 0, 0, 0, 0, 0, 0] : M = 1; S80[i0] -> [34, i0, 27, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S54[i0] -> [34, i0, 7, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S46[i0, i1] -> [34, i0, 3, i1, 2, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q; S68[i0] -> [34, i0, 15, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S116[i0, i1] -> [34, i0, 48, i1, 2, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= R; S43[i0, i1, i2] -> [34, i0, 2, i1, 0, i2, 3] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q and i2 >= 1 and i2 <= R; S26[] -> [25, 0, 0, 0, 0, 0, 0] : M = 1; S31[i0] -> [27, i0, 1, 0, 0, 0, 0] : M = 1 and i0 >= 1 and i0 <= N; S69[i0] -> [34, i0, 16, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S24[] -> [23, 0, 0, 0, 0, 0, 0] : M = 1; S90[i0] -> [34, i0, 37, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S89[i0] -> [34, i0, 36, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S82[i0] -> [34, i0, 29, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S67[i0, i1] -> [34, i0, 14, i1, 2, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= R; S58[i0] -> [34, i0, 11, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S109[i0, i1] -> [34, i0, 47, i1, 1, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q; S122[i0] -> [34, i0, 51, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S61[i0, i1, i2] -> [34, i0, 12, i1, 0, i2, 2] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q and i2 >= 1 and i2 <= R; S1[] -> [0, 0, 0, 0, 0, 0, 0] : M = 1; S4[] -> [3, 0, 0, 0, 0, 0, 0] : M = 1; S86[i0] -> [34, i0, 33, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S14[] -> [13, 0, 0, 0, 0, 0, 0] : M = 1; S93[i0] -> [34, i0, 40, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S56[i0] -> [34, i0, 9, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S37[] -> [33, 0, 0, 0, 0, 0, 0] : M = 1; S125[i0] -> [34, i0, 54, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S96[i0, i1, i2] -> [34, i0, 42, i1, 0, i2, 1] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= N and i2 >= 1 and i2 <= N; S9[] -> [8, 0, 0, 0, 0, 0, 0] : M = 1; S41[i0, i1, i2] -> [34, i0, 2, i1, 0, i2, 1] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q and i2 >= 1 and i2 <= R; S102[i0, i1, i2] -> [34, i0, 46, i1, 0, i2, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q and i2 >= 1 and i2 <= R; S112[i0, i1] -> [34, i0, 47, i1, 4, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q; S16[] -> [15, 0, 0, 0, 0, 0, 0] : M = 1; S85[i0] -> [34, i0, 32, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S42[i0, i1, i2] -> [34, i0, 2, i1, 0, i2, 2] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q and i2 >= 1 and i2 <= R; S12[] -> [11, 0, 0, 0, 0, 0, 0] : M = 1; S55[i0] -> [34, i0, 8, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S11[] -> [10, 0, 0, 0, 0, 0, 0] : M = 1; S19[] -> [18, 0, 0, 0, 0, 0, 0] : M = 1; S107[i0, i1, i2] -> [34, i0, 46, i1, 0, i2, 5] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q and i2 >= 1 and i2 <= R; S98[i0, i1] -> [34, i0, 42, i1, 1, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= N; S104[i0, i1, i2] -> [34, i0, 46, i1, 0, i2, 2] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q and i2 >= 1 and i2 <= R; S29[i0, i1] -> [27, i0, 0, i1, 1, 0, 0] : M = 1 and i0 >= 1 and i0 <= N and i1 >= 1 and i1 <= N; S20[] -> [19, 0, 0, 0, 0, 0, 0] : M = 1; S103[i0, i1, i2] -> [34, i0, 46, i1, 0, i2, 1] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q and i2 >= 1 and i2 <= R; S105[i0, i1, i2] -> [34, i0, 46, i1, 0, i2, 3] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q and i2 >= 1 and i2 <= R; S97[i0, i1, i2] -> [34, i0, 42, i1, 0, i2, 2] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= N and i2 >= 1 and i2 <= N; S44[i0, i1] -> [34, i0, 3, i1, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q; S106[i0, i1, i2] -> [34, i0, 46, i1, 0, i2, 4] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q and i2 >= 1 and i2 <= R; S34[] -> [30, 0, 0, 0, 0, 0, 0] : M = 1; S111[i0, i1] -> [34, i0, 47, i1, 3, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q; S99[i0] -> [34, i0, 43, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S81[i0] -> [34, i0, 28, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S23[] -> [22, 0, 0, 0, 0, 0, 0] : M = 1; S117[i0, i1] -> [34, i0, 48, i1, 3, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= R; S74[i0] -> [34, i0, 21, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P; S113[i0, i1] -> [34, i0, 47, i1, 5, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= Q; S119[i0, i1] -> [34, i0, 48, i1, 5, 0, 0] : M = 1 and i0 >= 2 and i0 <= P and i1 >= 1 and i1 <= R; S73[i0] -> [34, i0, 20, 0, 0, 0, 0] : M = 1 and i0 >= 2 and i0 <= P }
+[M, N, O, P, Q, R] -> {  :  }
+[M, N, O, P, Q, R] -> { [i, j, k, l, m, n, o] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/test.c b/polly/lib/External/isl/test_inputs/codegen/cloog/test.c
new file mode 100644
index 00000000000..84c84931e35
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/test.c
@@ -0,0 +1,17 @@
+{
+  for (int c0 = 1; c0 <= 2; c0 += 1)
+    for (int c1 = 1; c1 <= M; c1 += 1)
+      S1(c0, c1);
+  for (int c0 = 3; c0 <= N; c0 += 1) {
+    for (int c1 = 1; c1 <= min(M, c0 - 1); c1 += 1)
+      S1(c0, c1);
+    if (M >= c0) {
+      S1(c0, c0);
+      S2(c0, c0);
+    }
+    for (int c1 = c0 + 1; c1 <= M; c1 += 1)
+      S1(c0, c1);
+    if (c0 >= M + 1)
+      S2(c0, c0);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/test.in b/polly/lib/External/isl/test_inputs/codegen/cloog/test.in
new file mode 100644
index 00000000000..981d9339f52
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/test.in
@@ -0,0 +1,3 @@
+[M, N] -> { S1[i0, i1] -> [i0, i1, 0] : i0 >= 1 and i0 <= N and i1 >= 1 and i1 <= M; S2[i0, i0] -> [i0, i0, 1] : i0 >= 3 and i0 <= N }
+[M, N] -> {  : N >= M and M >= 4 }
+[M, N] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/thomasset.c b/polly/lib/External/isl/test_inputs/codegen/cloog/thomasset.c
new file mode 100644
index 00000000000..8d6709e7d9b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/thomasset.c
@@ -0,0 +1,9 @@
+{
+  for (int c0 = 0; c0 <= floord(n - 1, 3); c0 += 1)
+    for (int c2 = 3 * c0 + 1; c2 <= min(n, 3 * c0 + 3); c2 += 1)
+      S1(c2, c0);
+  for (int c0 = floord(n, 3); c0 <= 2 * floord(n, 3); c0 += 1)
+    for (int c1 = 0; c1 < n; c1 += 1)
+      for (int c3 = max(1, (n % 3) - n + 3 * c0); c3 <= min(n, (n % 3) - n + 3 * c0 + 2); c3 += 1)
+        S2(c1 + 1, c3, 0, n / 3, c0 - n / 3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/thomasset.in b/polly/lib/External/isl/test_inputs/codegen/cloog/thomasset.in
new file mode 100644
index 00000000000..398e5234603
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/thomasset.in
@@ -0,0 +1,3 @@
+[n] -> { S2[i0, i1, 0, i3, i4] -> [i3 + i4, -1 + i0] : i0 <= n and i1 <= n and i1 >= 1 and i0 >= 1 and 3i4 <= i1 and 3i4 >= -2 + i1 and 3i3 <= n and 3i3 >= -2 + n; S1[i0, i1] -> [i1, 0] : i0 <= n and i0 >= 1 and 3i1 <= -1 + i0 and 3i1 >= -3 + i0 }
+[n] -> {  :  }
+[n] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/tiling.c b/polly/lib/External/isl/test_inputs/codegen/cloog/tiling.c
new file mode 100644
index 00000000000..971b988a578
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/tiling.c
@@ -0,0 +1,3 @@
+for (int c0 = 0; c0 <= n / 10; c0 += 1)
+  for (int c1 = 10 * c0; c1 <= min(n, 10 * c0 + 9); c1 += 1)
+    S1(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/tiling.in b/polly/lib/External/isl/test_inputs/codegen/cloog/tiling.in
new file mode 100644
index 00000000000..6f0530f2ba0
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/tiling.in
@@ -0,0 +1,3 @@
+[n] -> { S1[ii, i] -> [ii, i, 0] : i >= 0 and i <= n and i <= 9 + 10ii and i >= 10ii }
+[n] -> {  : n >= 0 }
+[n] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/uday_scalars.c b/polly/lib/External/isl/test_inputs/codegen/cloog/uday_scalars.c
new file mode 100644
index 00000000000..d8002a86d86
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/uday_scalars.c
@@ -0,0 +1,6 @@
+{
+  for (int c2 = 0; c2 <= n; c2 += 1)
+    S1(c2, 0, 0);
+  for (int c2 = 0; c2 <= n; c2 += 1)
+    S2(0, c2, 0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/uday_scalars.in b/polly/lib/External/isl/test_inputs/codegen/cloog/uday_scalars.in
new file mode 100644
index 00000000000..ae29bb54f89
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/uday_scalars.in
@@ -0,0 +1,3 @@
+[n] -> { S1[i0, 0, 0] -> [0, 1, i0, 0] : i0 >= 0 and i0 <= n; S2[0, i1, 0] -> [1, 0, i1, 1] : i1 >= 0 and i1 <= n }
+[n] -> {  :  }
+[n] -> { [i, j, k, l] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/union.c b/polly/lib/External/isl/test_inputs/codegen/cloog/union.c
new file mode 100644
index 00000000000..6dc0a9b877f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/union.c
@@ -0,0 +1,6 @@
+if (M >= 11) {
+  for (int c0 = -100; c0 <= 0; c0 += 1)
+    S1(-c0);
+} else
+  for (int c0 = 0; c0 <= 100; c0 += 1)
+    S1(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/union.in b/polly/lib/External/isl/test_inputs/codegen/cloog/union.in
new file mode 100644
index 00000000000..23b6cab9a89
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/union.in
@@ -0,0 +1,3 @@
+[M] -> { S1[i0] -> [i0] : M <= 10 and i0 >= 0 and i0 <= 100; S1[i0] -> [-i0] : M >= 11 and i0 >= 0 and i0 <= 100 }
+[M] -> {  : M >= 1 or M <= -1 }
+[M] -> { [i] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/unroll.c b/polly/lib/External/isl/test_inputs/codegen/cloog/unroll.c
new file mode 100644
index 00000000000..83c4577b479
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/unroll.c
@@ -0,0 +1,13 @@
+{
+  S1(0);
+  S1(1);
+  S1(2);
+  S1(3);
+  S1(4);
+  S1(5);
+  S1(6);
+  S1(7);
+  S1(8);
+  S1(9);
+  S1(10);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/unroll.in b/polly/lib/External/isl/test_inputs/codegen/cloog/unroll.in
new file mode 100644
index 00000000000..1e449e0b420
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/unroll.in
@@ -0,0 +1,3 @@
+[n] -> { S1[i] -> [i, 0] : i >= 0 and i <= 10 }
+[n] -> {  :  }
+[n] -> { [i, j] -> unroll[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/unroll2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/unroll2.c
new file mode 100644
index 00000000000..b96ddd73fe2
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/unroll2.c
@@ -0,0 +1,6 @@
+{
+  if (n >= 0 && n <= 9)
+    S1(n);
+  if (n >= -1 && n <= 9)
+    S1(n + 1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/unroll2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/unroll2.in
new file mode 100644
index 00000000000..eb71e075ab3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/unroll2.in
@@ -0,0 +1,3 @@
+[n] -> { S1[i] -> [i, 0] : i >= n and i <= 1 + n and n <= 9 and i >= 0 }
+[n] -> {  :  }
+[n] -> { [i, j] -> unroll[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/usvd_e_t.c b/polly/lib/External/isl/test_inputs/codegen/cloog/usvd_e_t.c
new file mode 100644
index 00000000000..af8c9fe42f0
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/usvd_e_t.c
@@ -0,0 +1,348 @@
+{
+  for (int c0 = 0; c0 <= 2; c0 += 1) {
+    S1(c0, 0, 0);
+    for (int c1 = 0; c1 <= 4; c1 += 1)
+      S2(c0, c1, 0);
+  }
+  S1(3, 0, 0);
+  for (int c1 = 0; c1 <= 4; c1 += 1)
+    S2(3, c1, 0);
+  for (int c1 = 7; c1 <= 11; c1 += 1)
+    S8(3, c1, 0);
+  S1(4, 0, 0);
+  S2(4, 0, 0);
+  S3(4, 0, 0);
+  S5(4, 0, 0);
+  for (int c1 = 1; c1 <= 4; c1 += 1) {
+    S2(4, c1, 0);
+    S5(4, c1, 0);
+  }
+  for (int c1 = 7; c1 <= 11; c1 += 1)
+    S8(4, c1, 0);
+  for (int c0 = 5; c0 <= 6; c0 += 1) {
+    for (int c1 = -4; c1 < c0 - 8; c1 += 1)
+      S6(c0, c1, 0);
+    for (int c1 = c0 - 9; c1 < 0; c1 += 1)
+      S7(c0, c1, 0);
+    S3(c0, 0, 0);
+    S7(c0, 0, 0);
+    for (int c1 = 1; c1 < c0 - 3; c1 += 1)
+      S4(c0, c1, -1);
+    for (int c1 = c0 - 4; c1 <= 4; c1 += 1)
+      S5(c0, c1, 0);
+    for (int c1 = 7; c1 <= 11; c1 += 1)
+      S8(c0, c1, 0);
+  }
+  for (int c1 = -4; c1 < -1; c1 += 1)
+    S6(7, c1, 0);
+  for (int c1 = -2; c1 < 0; c1 += 1)
+    S7(7, c1, 0);
+  S3(7, 0, 0);
+  S7(7, 0, 0);
+  for (int c1 = 1; c1 <= 3; c1 += 1)
+    S4(7, c1, -1);
+  for (int c1 = 3; c1 <= 4; c1 += 1)
+    S5(7, c1, 0);
+  S9(7, 4, 0);
+  S10(7, 4, 0);
+  S11(7, 4, 0);
+  S21(7, 4, 0);
+  S23(7, 4, 0);
+  S11(7, 4, 1);
+  S16(7, 4, 1);
+  S17(7, 4, 1);
+  for (int c2 = 2; c2 <= 4; c2 += 1)
+    S11(7, 4, c2);
+  S12(7, 5, 0);
+  S21(7, 5, 0);
+  S22(7, 5, 0);
+  S23(7, 5, 0);
+  S12(7, 5, 1);
+  S16(7, 5, 1);
+  S17(7, 5, 1);
+  for (int c2 = 2; c2 <= 4; c2 += 1)
+    S12(7, 5, c2);
+  S21(7, 6, 0);
+  S22(7, 6, 0);
+  S23(7, 6, 0);
+  for (int c1 = 7; c1 <= 8; c1 += 1) {
+    S8(7, c1, 0);
+    S21(7, c1, 0);
+    S22(7, c1, 0);
+    S23(7, c1, 0);
+  }
+  S8(7, 9, 0);
+  S22(7, 9, 0);
+  for (int c1 = 10; c1 <= 11; c1 += 1)
+    S8(7, c1, 0);
+  for (int c1 = -4; c1 < 0; c1 += 1)
+    S6(8, c1, 0);
+  S7(8, -1, 0);
+  S3(8, 0, 0);
+  S7(8, 0, 0);
+  S19(8, 1, -2);
+  S4(8, 1, -1);
+  S19(8, 1, -1);
+  S19(8, 1, 0);
+  S15(8, 1, 4);
+  S18(8, 1, 4);
+  for (int c2 = -4; c2 < -2; c2 += 1) {
+    S14(8, 2, c2);
+    S20(8, 2, c2);
+  }
+  S14(8, 2, -2);
+  S19(8, 2, -2);
+  S20(8, 2, -2);
+  S4(8, 2, -1);
+  S14(8, 2, -1);
+  S19(8, 2, -1);
+  S20(8, 2, -1);
+  S14(8, 2, 0);
+  S19(8, 2, 0);
+  S20(8, 2, 0);
+  S15(8, 2, 4);
+  S18(8, 2, 4);
+  for (int c2 = -4; c2 < -1; c2 += 1) {
+    S14(8, 3, c2);
+    S20(8, 3, c2);
+  }
+  S4(8, 3, -1);
+  S14(8, 3, -1);
+  S20(8, 3, -1);
+  S14(8, 3, 0);
+  S20(8, 3, 0);
+  S15(8, 3, 4);
+  S18(8, 3, 4);
+  for (int c2 = -4; c2 < -1; c2 += 1) {
+    S14(8, 4, c2);
+    S20(8, 4, c2);
+  }
+  S4(8, 4, -1);
+  S14(8, 4, -1);
+  S20(8, 4, -1);
+  S5(8, 4, 0);
+  S9(8, 4, 0);
+  S10(8, 4, 0);
+  S14(8, 4, 0);
+  S20(8, 4, 0);
+  S23(8, 4, 0);
+  S13(8, 4, 1);
+  S21(8, 4, 1);
+  S23(8, 4, 1);
+  S24(8, 4, 1);
+  S13(8, 4, 2);
+  S16(8, 4, 2);
+  S17(8, 4, 2);
+  S24(8, 4, 2);
+  S13(8, 4, 3);
+  S24(8, 4, 3);
+  S13(8, 4, 4);
+  S15(8, 4, 4);
+  S23(8, 5, 0);
+  S11(8, 5, 1);
+  S21(8, 5, 1);
+  S22(8, 5, 1);
+  S23(8, 5, 1);
+  S24(8, 5, 1);
+  S11(8, 5, 2);
+  S16(8, 5, 2);
+  S17(8, 5, 2);
+  S24(8, 5, 2);
+  S11(8, 5, 3);
+  S24(8, 5, 3);
+  S11(8, 5, 4);
+  S15(8, 5, 4);
+  S23(8, 6, 0);
+  S12(8, 6, 1);
+  S21(8, 6, 1);
+  S22(8, 6, 1);
+  S23(8, 6, 1);
+  S24(8, 6, 1);
+  S12(8, 6, 2);
+  S16(8, 6, 2);
+  S17(8, 6, 2);
+  S24(8, 6, 2);
+  S12(8, 6, 3);
+  S24(8, 6, 3);
+  S12(8, 6, 4);
+  for (int c1 = 7; c1 <= 8; c1 += 1) {
+    S23(8, c1, 0);
+    S21(8, c1, 1);
+    S22(8, c1, 1);
+    S23(8, c1, 1);
+    for (int c2 = 1; c2 <= 3; c2 += 1)
+      S24(8, c1, c2);
+  }
+  S22(8, 9, 1);
+  S7(9, 0, 0);
+  for (int c1 = 1; c1 <= 2; c1 += 1) {
+    for (int c2 = -1; c2 <= 0; c2 += 1)
+      S19(9, c1, c2);
+    for (int c2 = 4; c2 <= 5; c2 += 1) {
+      S15(9, c1, c2);
+      S18(9, c1, c2);
+    }
+  }
+  S20(9, 3, -4);
+  for (int c2 = -3; c2 < -1; c2 += 1) {
+    S14(9, 3, c2);
+    S20(9, 3, c2);
+  }
+  for (int c2 = -1; c2 <= 0; c2 += 1) {
+    S14(9, 3, c2);
+    S19(9, 3, c2);
+    S20(9, 3, c2);
+  }
+  for (int c2 = 4; c2 <= 5; c2 += 1) {
+    S15(9, 3, c2);
+    S18(9, 3, c2);
+  }
+  S20(9, 4, -4);
+  for (int c2 = -3; c2 < 0; c2 += 1) {
+    S14(9, 4, c2);
+    S20(9, 4, c2);
+  }
+  S9(9, 4, 0);
+  S10(9, 4, 0);
+  S14(9, 4, 0);
+  S20(9, 4, 0);
+  for (int c2 = 0; c2 <= 1; c2 += 1)
+    S23(9, 4, c2);
+  S13(9, 4, 2);
+  S21(9, 4, 2);
+  S23(9, 4, 2);
+  S24(9, 4, 2);
+  S13(9, 4, 3);
+  S16(9, 4, 3);
+  S17(9, 4, 3);
+  S24(9, 4, 3);
+  S13(9, 4, 4);
+  for (int c2 = 4; c2 <= 5; c2 += 1) {
+    S15(9, 4, c2);
+    S18(9, 4, c2);
+  }
+  for (int c2 = 0; c2 <= 1; c2 += 1)
+    S23(9, 5, c2);
+  S13(9, 5, 2);
+  S21(9, 5, 2);
+  S22(9, 5, 2);
+  S23(9, 5, 2);
+  S24(9, 5, 2);
+  S13(9, 5, 3);
+  S16(9, 5, 3);
+  S17(9, 5, 3);
+  S24(9, 5, 3);
+  S13(9, 5, 4);
+  for (int c2 = 4; c2 <= 5; c2 += 1)
+    S15(9, 5, c2);
+  for (int c2 = 0; c2 <= 1; c2 += 1)
+    S23(9, 6, c2);
+  S11(9, 6, 2);
+  S21(9, 6, 2);
+  S22(9, 6, 2);
+  S23(9, 6, 2);
+  S24(9, 6, 2);
+  S11(9, 6, 3);
+  S16(9, 6, 3);
+  S17(9, 6, 3);
+  S24(9, 6, 3);
+  S11(9, 6, 4);
+  for (int c2 = 0; c2 <= 1; c2 += 1)
+    S23(9, 7, c2);
+  S12(9, 7, 2);
+  S21(9, 7, 2);
+  S22(9, 7, 2);
+  S23(9, 7, 2);
+  S24(9, 7, 2);
+  S12(9, 7, 3);
+  S16(9, 7, 3);
+  S17(9, 7, 3);
+  S24(9, 7, 3);
+  S12(9, 7, 4);
+  for (int c2 = 0; c2 <= 1; c2 += 1)
+    S23(9, 8, c2);
+  S21(9, 8, 2);
+  S22(9, 8, 2);
+  S23(9, 8, 2);
+  for (int c2 = 2; c2 <= 3; c2 += 1)
+    S24(9, 8, c2);
+  S22(9, 9, 2);
+  for (int c1 = 1; c1 <= 3; c1 += 1) {
+    S19(10, c1, 0);
+    S26(10, c1, 3);
+    S15(10, c1, 4);
+    S18(10, c1, 4);
+    S25(10, c1, 4);
+    for (int c2 = 5; c2 <= 6; c2 += 1) {
+      S15(10, c1, c2);
+      S18(10, c1, c2);
+    }
+  }
+  for (int c2 = -4; c2 < -2; c2 += 1)
+    S20(10, 4, c2);
+  for (int c2 = -2; c2 < 0; c2 += 1) {
+    S14(10, 4, c2);
+    S20(10, 4, c2);
+  }
+  S9(10, 4, 0);
+  S10(10, 4, 0);
+  S14(10, 4, 0);
+  S19(10, 4, 0);
+  S20(10, 4, 0);
+  S13(10, 4, 3);
+  S21(10, 4, 3);
+  S24(10, 4, 3);
+  S26(10, 4, 3);
+  S13(10, 4, 4);
+  S15(10, 4, 4);
+  S16(10, 4, 4);
+  S17(10, 4, 4);
+  S18(10, 4, 4);
+  S25(10, 4, 4);
+  for (int c2 = 5; c2 <= 6; c2 += 1) {
+    S15(10, 4, c2);
+    S18(10, 4, c2);
+  }
+  S13(10, 5, 3);
+  S21(10, 5, 3);
+  S22(10, 5, 3);
+  S24(10, 5, 3);
+  S26(10, 5, 3);
+  S13(10, 5, 4);
+  S15(10, 5, 4);
+  S16(10, 5, 4);
+  S17(10, 5, 4);
+  S18(10, 5, 4);
+  S25(10, 5, 4);
+  for (int c2 = 5; c2 <= 6; c2 += 1) {
+    S15(10, 5, c2);
+    S18(10, 5, c2);
+  }
+  S13(10, 6, 3);
+  S21(10, 6, 3);
+  S22(10, 6, 3);
+  S24(10, 6, 3);
+  S13(10, 6, 4);
+  S16(10, 6, 4);
+  S17(10, 6, 4);
+  S11(10, 7, 3);
+  S21(10, 7, 3);
+  S22(10, 7, 3);
+  S24(10, 7, 3);
+  S11(10, 7, 4);
+  S16(10, 7, 4);
+  S17(10, 7, 4);
+  S12(10, 8, 3);
+  S21(10, 8, 3);
+  S22(10, 8, 3);
+  S24(10, 8, 3);
+  S12(10, 8, 4);
+  S16(10, 8, 4);
+  S17(10, 8, 4);
+  S22(10, 9, 3);
+  for (int c0 = 11; c0 <= 14; c0 += 1)
+    for (int c1 = 1; c1 <= 5; c1 += 1) {
+      S26(c0, c1, 3);
+      S25(c0, c1, 4);
+    }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/usvd_e_t.in b/polly/lib/External/isl/test_inputs/codegen/cloog/usvd_e_t.in
new file mode 100644
index 00000000000..feb8f0898bb
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/usvd_e_t.in
@@ -0,0 +1,3 @@
+{ S19[i0, i1, i2] -> [i0, i1, i2, 18] : i0 >= 8 and i1 >= 1 and i2 <= 0 and i2 >= -10 + i0 and i1 <= -6 + i0; S21[i0, i1, -7 + i0] -> [i0, i1, -7 + i0, 20] : i0 <= 10 and i1 >= 4 and i1 <= 8 and i0 >= 7; S17[i0, i1, -6 + i0] -> [i0, i1, -6 + i0, 16] : i0 >= 7 and i1 >= 4 and i0 <= 10 and i1 <= -2 + i0; S4[i0, i1, -1] -> [i0, i1, -1, 3] : i0 <= 8 and i1 >= 1 and i1 <= -4 + i0; S1[i0, 0, 0] -> [i0, 0, 0, 0] : i0 >= 0 and i0 <= 4; S10[i0, 4, 0] -> [i0, 4, 0, 9] : i0 >= 7 and i0 <= 10; S16[i0, i1, -6 + i0] -> [i0, i1, -6 + i0, 15] : i0 <= 10 and i1 >= 4 and i0 >= 7 and i1 <= -2 + i0; S24[i0, i1, i2] -> [i0, i1, i2, 23] : i0 >= 8 and i1 >= 4 and i1 <= 8 and i2 <= 3 and i2 >= -7 + i0; S23[i0, i1, i2] -> [i0, i1, i2, 22] : i0 <= 9 and i1 >= 4 and i1 <= 8 and i2 >= 0 and i2 <= -7 + i0; S7[i0, i1, 0] -> [i0, i1, 0, 6] : i0 >= 5 and i1 <= 0 and i1 >= -9 + i0; S25[i0, i1, 4] -> [i0, i1, 4, 24] : i0 >= 10 and i0 <= 14 and i1 >= 1 and i1 <= 5; S18[i0, i1, i2] -> [i0, i1, i2, 17] : i0 <= 10 and i1 >= 1 and i2 >= 4 and i2 <= -4 + i0 and i1 <= -5 + i0; S14[i0, i1, i2] -> [i0, i1, i2, 13] : i0 >= 8 and i1 <= 4 and i2 <= 0 and i2 >= -12 + i0 and i1 >= -6 + i0; S6[i0, i1, 0] -> [i0, i1, 0, 5] : i0 <= 8 and i1 >= -4 and i1 <= -9 + i0; S8[i0, i1, 0] -> [i0, i1, 0, 7] : i0 >= 3 and i0 <= 7 and i1 >= 7 and i1 <= 11; S15[i0, i1, i2] -> [i0, i1, i2, 14] : i0 <= 10 and i1 >= 1 and i1 <= 5 and i2 >= 4 and i2 <= -4 + i0; S11[i0, -3 + i0, i2] -> [i0, -3 + i0, i2, 10] : i0 <= 10 and i0 >= 7 and i2 <= 4 and i2 >= -7 + i0; S20[i0, i1, i2] -> [i0, i1, i2, 19] : i0 >= 8 and i1 <= 4 and i2 >= -4 and i2 <= 0 and i1 >= -6 + i0; S12[i0, -2 + i0, i2] -> [i0, -2 + i0, i2, 11] : i0 >= 7 and i0 <= 10 and i2 <= 4 and i2 >= -7 + i0; S9[i0, 4, 0] -> [i0, 4, 0, 8] : i0 >= 7 and i0 <= 10; S22[i0, i1, -7 + i0] -> [i0, i1, -7 + i0, 21] : i0 >= 7 and i1 >= 5 and i1 <= 9 and i0 <= 10; S13[i0, i1, i2] -> [i0, i1, i2, 12] : i0 <= 10 and i1 >= 4 and i2 <= 4 and i2 >= -7 + i0 and i1 <= -4 + i0; S26[i0, i1, 3] -> [i0, i1, 3, 25] : i0 >= 10 and i0 <= 14 and i1 >= 1 and i1 <= 5; S3[i0, 0, 0] -> [i0, 0, 0, 2] : i0 >= 4 and i0 <= 8; S5[i0, i1, 0] -> [i0, i1, 0, 4] : i0 >= 4 and i1 <= 4 and i1 >= -4 + i0; S2[i0, i1, 0] -> [i0, i1, 0, 1] : i0 >= 0 and i0 <= 4 and i1 >= 0 and i1 <= 4 }
+{  :  }
+{ [i, j, k, l] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/vasilache.c b/polly/lib/External/isl/test_inputs/codegen/cloog/vasilache.c
new file mode 100644
index 00000000000..c6d32ca637b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/vasilache.c
@@ -0,0 +1,23 @@
+{
+  S1();
+  S2();
+  for (int c1 = 0; c1 < N; c1 += 1)
+    for (int c3 = 0; c3 < N; c3 += 1) {
+      S4(c1, c3);
+      S5(c1, c3);
+    }
+  for (int c1 = 0; c1 < N; c1 += 1)
+    for (int c3 = 0; c3 < N; c3 += 1)
+      for (int c5 = 0; c5 <= (N - 1) / 32; c5 += 1) {
+        S7(c1, c3, c5, 32 * c5);
+        for (int c7 = 32 * c5 + 1; c7 <= min(N - 1, 32 * c5 + 31); c7 += 1) {
+          S6(c1, c3, c5, c7 - 1);
+          S7(c1, c3, c5, c7);
+        }
+        if (32 * c5 + 31 >= N) {
+          S6(c1, c3, c5, N - 1);
+        } else
+          S6(c1, c3, c5, 32 * c5 + 31);
+      }
+  S8();
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/vasilache.in b/polly/lib/External/isl/test_inputs/codegen/cloog/vasilache.in
new file mode 100644
index 00000000000..a9b9860fd2b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/vasilache.in
@@ -0,0 +1,3 @@
+[M, N] -> { S4[i0, i1] -> [3, i0, 0, i1, 0, 0, 0, 0, 0] : i0 >= 0 and i0 <= -1 + N and i1 >= 0 and i1 <= -1 + N; S1[] -> [0, 0, 0, 0, 0, 0, 0, 0, 0]; S6[i0, i1, i2, i3] -> [4, i0, 0, i1, 0, i2, 0, 1 + i3, 0] : i0 >= 0 and i0 <= -1 + N and i1 >= 0 and i1 <= -1 + N and i3 >= 0 and i3 <= -1 + N and i3 >= 32i2 and i3 <= 31 + 32i2; S7[i0, i1, i2, i3] -> [4, i0, 0, i1, 0, i2, 0, i3, 1] : i0 >= 0 and i0 <= -1 + N and i1 >= 0 and i1 <= -1 + N and i3 >= 0 and i3 <= -1 + N and i3 >= 32i2 and i3 <= 31 + 32i2; S5[i0, i1] -> [3, i0, 0, i1, 1, 0, 0, 0, 0] : i0 >= 0 and i0 <= -1 + N and i1 >= 0 and i1 <= -1 + N; S2[] -> [1, 0, 0, 0, 0, 0, 0, 0, 0]; S8[] -> [5, 0, 0, 0, 0, 0, 0, 0, 0]; S3[] -> [2, 0, 0, 0, 0, 0, 0, 0, 0] : M >= 79 }
+[M, N] -> {  : M <= 3 and N >= 100 }
+[M, N] -> { [i, j, k, l, m, n, o, p, q] -> separate[x] : x >= 7 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/vivien.c b/polly/lib/External/isl/test_inputs/codegen/cloog/vivien.c
new file mode 100644
index 00000000000..ce38c10e833
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/vivien.c
@@ -0,0 +1,85 @@
+{
+  for (int c0 = -27 * n + 2; c0 <= 1; c0 += 1)
+    S1(c0 - 1);
+  for (int c0 = 2; c0 <= min(2 * n, n + 29); c0 += 1) {
+    if (c0 >= 3) {
+      if (2 * n >= c0 + 1) {
+        S4(c0 - c0 / 2 - 1, c0 / 2 + 1);
+        if (c0 + 2 >= 2 * n) {
+          for (int c2 = 1; c2 < -n + c0; c2 += 1)
+            S5(-n + c0, n, c2);
+        } else if (c0 >= 5) {
+          S4(c0 - c0 / 2 - 2, c0 / 2 + 2);
+          for (int c2 = 1; c2 < c0 - c0 / 2 - 1; c2 += 1)
+            S5(c0 - c0 / 2 - 1, c0 / 2 + 1, c2);
+        }
+      }
+      for (int c1 = -c0 + c0 / 2 + 3; c1 <= min(-1, n - c0); c1 += 1) {
+        S4(-c1, c0 + c1);
+        S6(-c1 + 2, c0 + c1 - 2);
+        for (int c2 = 1; c2 <= -c1; c2 += 1)
+          S5(-c1 + 1, c0 + c1 - 1, c2);
+      }
+      if (2 * n >= c0 + 3 && c0 >= n + 2) {
+        S6(-n + c0 + 1, n - 1);
+        for (int c2 = 1; c2 < -n + c0; c2 += 1)
+          S5(-n + c0, n, c2);
+      }
+      if (c0 >= n + 3 && 2 * n >= c0 + 1) {
+        S6(-n + c0, n);
+      } else {
+        if (c0 >= 5 && n + 1 >= c0) {
+          S6(2, c0 - 2);
+          S1(c0 - 1);
+        } else if (n + 1 >= c0 && c0 <= 4)
+          S1(c0 - 1);
+        if (n + 1 >= c0) {
+          S6(1, c0 - 1);
+        } else if (n >= 3 && c0 == n + 2) {
+          S6(2, n);
+          S1(n + 1);
+        }
+      }
+      if (c0 >= n + 3)
+        S1(c0 - 1);
+      if (n == 2 && c0 == 4)
+        S1(3);
+    } else
+      S1(1);
+    if (c0 % 2 == 0)
+      S3(c0 / 2);
+    for (int c1 = max(1, -n + c0); c1 < (c0 + 1) / 2; c1 += 1)
+      S2(c0 - c1, c1);
+  }
+  for (int c0 = max(2 * n + 1, -27 * n + 2); c0 <= n + 29; c0 += 1)
+    S1(c0 - 1);
+  for (int c0 = n + 30; c0 <= 2 * n; c0 += 1) {
+    if (2 * n >= c0 + 1) {
+      S4(c0 - c0 / 2 - 1, c0 / 2 + 1);
+      if (c0 + 2 >= 2 * n) {
+        for (int c2 = 1; c2 < -n + c0; c2 += 1)
+          S5(-n + c0, n, c2);
+      } else {
+        S4(c0 - c0 / 2 - 2, c0 / 2 + 2);
+        for (int c2 = 1; c2 < c0 - c0 / 2 - 1; c2 += 1)
+          S5(c0 - c0 / 2 - 1, c0 / 2 + 1, c2);
+      }
+      for (int c1 = -c0 + c0 / 2 + 3; c1 <= n - c0; c1 += 1) {
+        S4(-c1, c0 + c1);
+        S6(-c1 + 2, c0 + c1 - 2);
+        for (int c2 = 1; c2 <= -c1; c2 += 1)
+          S5(-c1 + 1, c0 + c1 - 1, c2);
+      }
+      if (2 * n >= c0 + 3) {
+        S6(-n + c0 + 1, n - 1);
+        for (int c2 = 1; c2 < -n + c0; c2 += 1)
+          S5(-n + c0, n, c2);
+      }
+      S6(-n + c0, n);
+    }
+    if (c0 % 2 == 0)
+      S3(c0 / 2);
+    for (int c1 = -n + c0; c1 < (c0 + 1) / 2; c1 += 1)
+      S2(c0 - c1, c1);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/vivien.in b/polly/lib/External/isl/test_inputs/codegen/cloog/vivien.in
new file mode 100644
index 00000000000..b591ef35976
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/vivien.in
@@ -0,0 +1,3 @@
+[n] -> { S6[i0, i1] -> [2i0 + 2i1, 2 - i0, 0] : i0 >= 1 and i0 <= n and i1 >= 1 + i0 and i1 <= n; S2[i0, i1] -> [1 + 2i0 + 2i1, i1, 0] : 29i1 >= 1 - i0 and i0 <= n and i1 >= 1 and i1 <= -1 + i0; S1[i0] -> [2 + 2i0, 0, 0] : i0 >= 1 - 27n and i0 <= 28 + n; S4[i0, i1] -> [2i0 + 2i1, -i0, 0] : i0 >= 1 and i0 <= n and i1 >= 1 + i0 and i1 <= n; S3[i0] -> [1 + 4i0, 0, 0] : i0 >= 1 and i0 <= n; S5[i0, i1, i2] -> [2i0 + 2i1, 1 - i0, i2] : i0 >= 1 and i0 <= n and i1 >= 1 + i0 and i1 <= n and i2 >= 1 and i2 <= -1 + i0 }
+[n] -> {  :  }
+[n] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/vivien2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/vivien2.c
new file mode 100644
index 00000000000..7fe52ff319b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/vivien2.c
@@ -0,0 +1,77 @@
+{
+  for (int c0 = -27 * n + 2; c0 <= 1; c0 += 1)
+    S1(c0 - 1);
+  for (int c0 = 2; c0 <= n + 29; c0 += 1) {
+    if (c0 >= 3) {
+      S4(c0 - c0 / 2 - 1, c0 / 2 + 1);
+      if (c0 >= 5 && 2 * n >= c0 + 3) {
+        S4(c0 - c0 / 2 - 2, c0 / 2 + 2);
+        for (int c2 = 1; c2 < c0 - c0 / 2 - 1; c2 += 1)
+          S5(c0 - c0 / 2 - 1, c0 / 2 + 1, c2);
+      }
+      for (int c1 = -c0 + c0 / 2 + 3; c1 <= min(-1, n - c0); c1 += 1) {
+        S4(-c1, c0 + c1);
+        S6(-c1 + 2, c0 + c1 - 2);
+        for (int c2 = 1; c2 <= -c1; c2 += 1)
+          S5(-c1 + 1, c0 + c1 - 1, c2);
+      }
+      if (2 * n >= c0 + 3 && c0 >= n + 2) {
+        S6(-n + c0 + 1, n - 1);
+        for (int c2 = 1; c2 < -n + c0; c2 += 1)
+          S5(-n + c0, n, c2);
+      } else if (c0 + 2 >= 2 * n)
+        for (int c2 = 1; c2 < -n + c0; c2 += 1)
+          S5(-n + c0, n, c2);
+      if (c0 >= n + 3) {
+        S6(-n + c0, n);
+        S1(c0 - 1);
+      } else {
+        if (c0 >= 5 && n + 1 >= c0) {
+          S6(2, c0 - 2);
+          S1(c0 - 1);
+        } else if (c0 <= 4)
+          S1(c0 - 1);
+        if (n + 1 >= c0) {
+          S6(1, c0 - 1);
+        } else {
+          S6(2, n);
+          S1(n + 1);
+        }
+      }
+    } else
+      S1(1);
+    if (c0 % 2 == 0)
+      S3(c0 / 2);
+    for (int c1 = max(1, -n + c0); c1 < (c0 + 1) / 2; c1 += 1)
+      S2(c0 - c1, c1);
+  }
+  for (int c0 = n + 30; c0 <= 2 * n; c0 += 1) {
+    if (2 * n >= c0 + 1) {
+      S4(c0 - c0 / 2 - 1, c0 / 2 + 1);
+      if (c0 + 2 >= 2 * n) {
+        for (int c2 = 1; c2 < -n + c0; c2 += 1)
+          S5(-n + c0, n, c2);
+      } else {
+        S4(c0 - c0 / 2 - 2, c0 / 2 + 2);
+        for (int c2 = 1; c2 < c0 - c0 / 2 - 1; c2 += 1)
+          S5(c0 - c0 / 2 - 1, c0 / 2 + 1, c2);
+      }
+      for (int c1 = -c0 + c0 / 2 + 3; c1 <= n - c0; c1 += 1) {
+        S4(-c1, c0 + c1);
+        S6(-c1 + 2, c0 + c1 - 2);
+        for (int c2 = 1; c2 <= -c1; c2 += 1)
+          S5(-c1 + 1, c0 + c1 - 1, c2);
+      }
+      if (2 * n >= c0 + 3) {
+        S6(-n + c0 + 1, n - 1);
+        for (int c2 = 1; c2 < -n + c0; c2 += 1)
+          S5(-n + c0, n, c2);
+      }
+      S6(-n + c0, n);
+    }
+    if (c0 % 2 == 0)
+      S3(c0 / 2);
+    for (int c1 = -n + c0; c1 < (c0 + 1) / 2; c1 += 1)
+      S2(c0 - c1, c1);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/vivien2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/vivien2.in
new file mode 100644
index 00000000000..d8b1b1a007b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/vivien2.in
@@ -0,0 +1,3 @@
+[n] -> { S6[i0, i1] -> [2i0 + 2i1, 2 - i0, 0] : i0 >= 1 and i0 <= n and i1 >= 1 + i0 and i1 <= n; S2[i0, i1] -> [1 + 2i0 + 2i1, i1, 0] : 29i1 >= 1 - i0 and i0 <= n and i1 >= 1 and i1 <= -1 + i0; S1[i0] -> [2 + 2i0, 0, 0] : i0 >= 1 - 27n and i0 <= 28 + n; S4[i0, i1] -> [2i0 + 2i1, -i0, 0] : i0 >= 1 and i0 <= n and i1 >= 1 + i0 and i1 <= n; S3[i0] -> [1 + 4i0, 0, 0] : i0 >= 1 and i0 <= n; S5[i0, i1, i2] -> [2i0 + 2i1, 1 - i0, i2] : i0 >= 1 and i0 <= n and i1 >= 1 + i0 and i1 <= n and i2 >= 1 and i2 <= -1 + i0 }
+[n] -> {  : n >= 30 }
+[n] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/walters.c b/polly/lib/External/isl/test_inputs/codegen/cloog/walters.c
new file mode 100644
index 00000000000..2c1bd93858b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/walters.c
@@ -0,0 +1,15 @@
+{
+  S2(1, 0, 1, 0);
+  S4(1, 0, 1, 0);
+  S3(2, 0, 1, 1);
+  S4(2, 0, 1, 1);
+  for (int c0 = 3; c0 <= 10; c0 += 1) {
+    if (c0 % 3 == 0) {
+      S1(c0, c0 / 3, c0 / 3, c0 / 3);
+    } else if ((c0 - 1) % 3 == 0) {
+      S2(c0, (c0 - 1) / 3, (c0 + 2) / 3, (c0 - 1) / 3);
+    } else
+      S3(c0, (c0 - 2) / 3, (c0 + 1) / 3, (c0 + 1) / 3);
+    S4(c0, c0 / 3, (c0 - 1) / 3 + 1, c0 - (c0 - 1) / 3 - c0 / 3 - 1);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/walters.in b/polly/lib/External/isl/test_inputs/codegen/cloog/walters.in
new file mode 100644
index 00000000000..6e2645147e6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/walters.in
@@ -0,0 +1,3 @@
+{ S2[i, div36, div37, div38] -> [i, div36, k, div38, 1] : 3div37 = 2 + i and 3k = 2 + i and i >= 1 and i <= 10 and 3div36 >= -2 + i and 3div38 <= 1 + i and 3div38 >= -1 + i and 3div36 <= i; S1[i, div36, div37, div38] -> [i, j, div37, div38, 0] : 3j = i and 3div36 = i and i >= 3 and i <= 10 and 3div37 >= i and 3div38 <= 1 + i and 3div37 <= 2 + i and 3div38 >= -1 + i; S3[i, div36, div37, div38] -> [i, div36, div37, l, 2] : 3l = 1 + i and 3div38 = 1 + i and i <= 10 and i >= 2 and 3div36 >= -2 + i and 3div37 <= 2 + i and 3div36 <= i and 3div37 >= i; S4[i, div36, div37, div38] -> [i, div36, div37, div38, 3] : i >= 1 and i <= 10 and 3div36 <= i and 3div36 >= -2 + i and 3div37 <= 2 + i and 3div37 >= i and 3div38 <= 1 + i and 3div38 >= -1 + i }
+{  :  }
+{ [i, j, k, l, m] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/walters2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/walters2.c
new file mode 100644
index 00000000000..27565fbde9a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/walters2.c
@@ -0,0 +1,12 @@
+{
+  for (int c1 = 0; c1 <= 51; c1 += 1)
+    S2(0, c1);
+  for (int c0 = 1; c0 <= 24; c0 += 1) {
+    S2(c0, 0);
+    for (int c1 = 1; c1 <= 50; c1 += 1)
+      S1(c0, c1);
+    S2(c0, 51);
+  }
+  for (int c1 = 0; c1 <= 51; c1 += 1)
+    S2(25, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/walters2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/walters2.in
new file mode 100644
index 00000000000..ea01a10a336
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/walters2.in
@@ -0,0 +1,3 @@
+{ S1[j, i] -> [j, i, 0] : j >= 1 and j <= 24 and i >= 1 and i <= 50; S2[j, 51] -> [j, 51, 1] : j >= 1 and j <= 24; S2[25, i] -> [25, i, 1] : i >= 1 and i <= 51; S2[0, i] -> [0, i, 1] : i >= 0 and i <= 51; S2[j, 0] -> [j, 0, 1] : j >= 1 and j <= 25 }
+{  :  }
+{ [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/walters3.c b/polly/lib/External/isl/test_inputs/codegen/cloog/walters3.c
new file mode 100644
index 00000000000..bafb9acbc8e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/walters3.c
@@ -0,0 +1,7 @@
+{
+  for (int c0 = 2; c0 <= 8; c0 += 2) {
+    S1(c0, c0 / 2, c0 / 2);
+    S2(c0, c0 / 2, c0 / 2);
+  }
+  S2(10, 5, 5);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/walters3.in b/polly/lib/External/isl/test_inputs/codegen/cloog/walters3.in
new file mode 100644
index 00000000000..802bd4f0435
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/walters3.in
@@ -0,0 +1,3 @@
+{ S2[j, a, b] -> [j, j', b, 1] : 2a = j and 2j' = j and j >= 1 and j <= 10 and 2b <= j and 2b >= -1 + j; S1[j, a, b] -> [j, j', k, 0] : 2a = j and 2j' = j and 2k = j and 2b = j and j <= 8 and j >= 2 }
+{  :  }
+{ [i, j, k, l] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/wavefront.c b/polly/lib/External/isl/test_inputs/codegen/cloog/wavefront.c
new file mode 100644
index 00000000000..7db0788fc68
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/wavefront.c
@@ -0,0 +1,3 @@
+for (int c0 = 2; c0 <= n + m; c0 += 1)
+  for (int c1 = max(1, -m + c0); c1 <= min(n, c0 - 1); c1 += 1)
+    S1(c1, c0 - c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/wavefront.in b/polly/lib/External/isl/test_inputs/codegen/cloog/wavefront.in
new file mode 100644
index 00000000000..61618c37a45
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/wavefront.in
@@ -0,0 +1,3 @@
+[n, m] -> { S1[i0, i1] -> [i0 + i1, i0] : i0 >= 1 and i0 <= n and i1 >= 1 and i1 <= m }
+[n, m] -> {  :  }
+[n, m] -> { [i, j] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/yosr.c b/polly/lib/External/isl/test_inputs/codegen/cloog/yosr.c
new file mode 100644
index 00000000000..1c20057bd1b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/yosr.c
@@ -0,0 +1,12 @@
+{
+  for (int c0 = 1; c0 < n; c0 += 1) {
+    for (int c2 = c0 + 1; c2 <= n; c2 += 1)
+      S1(c0, c2);
+    for (int c1 = 1; c1 < c0; c1 += 1)
+      for (int c2 = c1 + 1; c2 <= n; c2 += 1)
+        S2(c1, c2, c0);
+  }
+  for (int c1 = 1; c1 < n; c1 += 1)
+    for (int c2 = c1 + 1; c2 <= n; c2 += 1)
+      S2(c1, c2, n);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/yosr.in b/polly/lib/External/isl/test_inputs/codegen/cloog/yosr.in
new file mode 100644
index 00000000000..dcc93e5a69e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/yosr.in
@@ -0,0 +1,3 @@
+[n] -> { S2[i0, i1, i2] -> [i2] : i0 >= 1 and i0 <= -1 + n and i1 >= 1 + i0 and i1 <= n and i2 >= 1 + i0 and i2 <= n; S1[i0, i1] -> [i0] : i0 >= 1 and i0 <= -1 + n and i1 >= 1 + i0 and i1 <= n }
+[n] -> {  :  }
+[n] -> { [i] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/yosr2.c b/polly/lib/External/isl/test_inputs/codegen/cloog/yosr2.c
new file mode 100644
index 00000000000..a52909ccd45
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/yosr2.c
@@ -0,0 +1,13 @@
+{
+  for (int c1 = 1; c1 <= M; c1 += 1)
+    S2(c1);
+  for (int c0 = 2; c0 <= M; c0 += 1) {
+    for (int c2 = c0 + 1; c2 <= M; c2 += 1)
+      for (int c3 = 1; c3 < c0; c3 += 1)
+        S3(c0, c2, c3);
+    for (int c1 = 1; c1 < c0; c1 += 1)
+      S4(c1, c0);
+    for (int c2 = 1; c2 < c0; c2 += 1)
+      S1(c0, c2);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/yosr2.in b/polly/lib/External/isl/test_inputs/codegen/cloog/yosr2.in
new file mode 100644
index 00000000000..a07e1492b1c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/yosr2.in
@@ -0,0 +1,3 @@
+[M] -> { S3[i0, i1, i2] -> [i0] : i0 >= 1 and i0 <= M and i1 >= 1 + i0 and i1 <= M and i2 >= 1 and i2 <= -1 + i0; S4[i0, i1] -> [i1] : i0 >= 1 and i0 <= M and i1 >= 1 + i0 and i1 <= M; S1[i0, i1] -> [i0] : i0 >= 1 and i0 <= M and i1 >= 1 and i1 <= -1 + i0; S2[i0] -> [0] : i0 >= 1 and i0 <= M }
+[M] -> {  : M >= 2 }
+[M] -> { [i] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/youcef.c b/polly/lib/External/isl/test_inputs/codegen/cloog/youcef.c
new file mode 100644
index 00000000000..1bc7ffd0524
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/youcef.c
@@ -0,0 +1,6 @@
+for (int c0 = 0; c0 <= 5; c0 += 1) {
+  S1(c0, c0);
+  for (int c1 = c0; c1 <= 5; c1 += 1)
+    S2(c0, c1);
+  S3(c0, 5);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/youcef.in b/polly/lib/External/isl/test_inputs/codegen/cloog/youcef.in
new file mode 100644
index 00000000000..5164abe2874
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/youcef.in
@@ -0,0 +1,3 @@
+{ S1[i0, i0] -> [i0, i0, 0] : i0 >= 0 and i0 <= 5; S2[i0, i1] -> [i0, i1, 1] : i0 >= 0 and i0 <= 5 and i1 >= i0 and i1 <= 5; S3[i0, 5] -> [i0, 5, 2] : i0 >= 0 and i0 <= 5 }
+{  :  }
+{ [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/youcefn.c b/polly/lib/External/isl/test_inputs/codegen/cloog/youcefn.c
new file mode 100644
index 00000000000..1107790b201
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/youcefn.c
@@ -0,0 +1,10 @@
+{
+  for (int c0 = 1; c0 <= n; c0 += 1) {
+    S1(c0, c0);
+    for (int c1 = c0; c1 <= n; c1 += 1)
+      S2(c0, c1);
+    S3(c0, n);
+  }
+  for (int c0 = n + 1; c0 <= m; c0 += 1)
+    S3(c0, n);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/cloog/youcefn.in b/polly/lib/External/isl/test_inputs/codegen/cloog/youcefn.in
new file mode 100644
index 00000000000..b21e30d7fa7
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/cloog/youcefn.in
@@ -0,0 +1,3 @@
+[n, m] -> { S3[i0, n] -> [i0, n, 2] : i0 >= 1 and i0 <= m; S2[i0, i1] -> [i0, i1, 1] : i0 >= 1 and i0 <= n and i1 >= i0 and i1 <= n; S1[i0, i0] -> [i0, i0, 0] : i0 >= 1 and i0 <= n }
+[n, m] -> {  : n >= 2 and m >= n }
+[n, m] -> { [i, j, k] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/disjuncts.c b/polly/lib/External/isl/test_inputs/codegen/disjuncts.c
new file mode 100644
index 00000000000..207e3377835
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/disjuncts.c
@@ -0,0 +1,10 @@
+for (int c0 = 0; c0 <= n; c0 += 1)
+  for (int c1 = 0; c1 <= n; c1 += 1)
+    if (c1 == n || c0 == n || c1 == 0 || c0 == 0) {
+      for (int c3 = 0; c3 <= n; c3 += 1)
+        for (int c4 = 0; c4 <= n; c4 += 1)
+          a(c0, c1, c3, c4);
+      for (int c3 = 0; c3 <= n; c3 += 1)
+        for (int c4 = 0; c4 <= n; c4 += 1)
+          b(c0, c1, c3, c4);
+    }
diff --git a/polly/lib/External/isl/test_inputs/codegen/disjuncts.in b/polly/lib/External/isl/test_inputs/codegen/disjuncts.in
new file mode 100644
index 00000000000..01386701c69
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/disjuncts.in
@@ -0,0 +1,7 @@
+# Check that conditions are hoisted up from the innermost loop
+[n] -> { a[i,j,k,l] -> [i,j,0,k,l] :
+	    0 <= i,j,k,l <= n and (i = 0 or j = 0 or i = n or j = n);
+	 b[i,j,k,l] -> [i,j,1,k,l] :
+	    0 <= i,j,k,l <= n and (i = 0 or j = 0 or i = n or j = n) }
+{ : }
+{ [i,j,t,k,l] -> atomic[x] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/dwt.c b/polly/lib/External/isl/test_inputs/codegen/dwt.c
new file mode 100644
index 00000000000..52adb5d6e0d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/dwt.c
@@ -0,0 +1,8 @@
+for (int c0 = 0; c0 < Ncl; c0 += 1) {
+  if (Ncl >= c0 + 2 && c0 >= 1) {
+    S(c0, 28);
+  } else if (c0 == 0) {
+    S(0, 26);
+  } else
+    S(Ncl - 1, 27);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/dwt.in b/polly/lib/External/isl/test_inputs/codegen/dwt.in
new file mode 100644
index 00000000000..424b0f3fd09
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/dwt.in
@@ -0,0 +1,3 @@
+[Ncl] -> { S[j, 28] -> [j] : j <= -2 + Ncl and Ncl <= 256 and Ncl >= 40 and j >= 1; S[0, 26] -> [0] : Ncl <= 256 and Ncl >= 40; S[-1 + Ncl, 27] -> [-1 + Ncl] : Ncl <= 256 and Ncl >= 40 }
+[Ncl] -> {  :  Ncl >= 40 and Ncl <= 256 }
+{  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/empty.c b/polly/lib/External/isl/test_inputs/codegen/empty.c
new file mode 100644
index 00000000000..58f7b7b295b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/empty.c
@@ -0,0 +1,6 @@
+for (int c0 = 0; c0 <= 10; c0 += 1) {
+  S0(c0);
+  if (c0 == 5)
+    S2();
+  S1(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/empty.in b/polly/lib/External/isl/test_inputs/codegen/empty.in
new file mode 100644
index 00000000000..17febea4be6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/empty.in
@@ -0,0 +1,5 @@
+# Earlier versions of isl would end up with an empty partial
+# executed relation and fail to detect this emptiness.
+[M] -> { S0[i] -> [i, -M] : 0 <= i <= 10; S1[i] -> [i, 0] : 0 <= i <= 10; S2[] -> [5, 0] }
+[M] -> {  : M >= 1 }
+{ }
diff --git a/polly/lib/External/isl/test_inputs/codegen/hoist.c b/polly/lib/External/isl/test_inputs/codegen/hoist.c
new file mode 100644
index 00000000000..7ba854f8e2f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/hoist.c
@@ -0,0 +1,45 @@
+if (ni >= t0 + 1 && nj >= t1 + 1)
+  for (int c2 = 0; c2 <= min(15, nk - 1); c2 += 1) {
+    S_1(t0, t1, c2);
+    if (nj >= t1 + 17) {
+      S_1(t0, t1 + 16, c2);
+      if (nj >= t1 + 33) {
+        S_1(t0, t1 + 32, c2);
+        if (nj >= t1 + 49)
+          S_1(t0, t1 + 48, c2);
+      }
+    }
+    if (ni >= t0 + 17) {
+      S_1(t0 + 16, t1, c2);
+      if (nj >= t1 + 17) {
+        S_1(t0 + 16, t1 + 16, c2);
+        if (nj >= t1 + 33) {
+          S_1(t0 + 16, t1 + 32, c2);
+          if (nj >= t1 + 49)
+            S_1(t0 + 16, t1 + 48, c2);
+        }
+      }
+      if (ni >= t0 + 33) {
+        S_1(t0 + 32, t1, c2);
+        if (nj >= t1 + 17) {
+          S_1(t0 + 32, t1 + 16, c2);
+          if (nj >= t1 + 33) {
+            S_1(t0 + 32, t1 + 32, c2);
+            if (nj >= t1 + 49)
+              S_1(t0 + 32, t1 + 48, c2);
+          }
+        }
+        if (ni >= t0 + 49) {
+          S_1(t0 + 48, t1, c2);
+          if (nj >= t1 + 17) {
+            S_1(t0 + 48, t1 + 16, c2);
+            if (nj >= t1 + 33) {
+              S_1(t0 + 48, t1 + 32, c2);
+              if (nj >= t1 + 49)
+                S_1(t0 + 48, t1 + 48, c2);
+            }
+          }
+        }
+      }
+    }
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/hoist.in b/polly/lib/External/isl/test_inputs/codegen/hoist.in
new file mode 100644
index 00000000000..80a905200e4
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/hoist.in
@@ -0,0 +1,10 @@
+# check that the shared conditions ni >= t0 + 1 and nj >= t1 + 1
+# are hoisted out of the loop
+[ni, nj, nk, t0, t1] -> { S_1[i, j, k] -> [t0, t1, k, i, j] :
+	exists (e0 = [(-t0 + i)/16], e1 = [(-t1 + j)/16]:
+	16e0 = -t0 + i and 16e1 = -t1 + j and k >= 0 and j >= 0 and
+	j <= -1 + nj and i >= 0 and i <= -1 + ni and k <= -1 + nk and
+	ni >= 1 and nj >= 1 and nk >= 1 and j <= 63 and t1 >= 0 and
+	i <= 63 and k <= 15 and t0 >= 0 and t1 <= 15 and t0 <= 15) }
+[t0, t1] -> { : 0 <= t0, t1 <= 15 }
+{ [t0, t1, i5, i6, i7] -> unroll[x] : x >= 3}
diff --git a/polly/lib/External/isl/test_inputs/codegen/hoist2.c b/polly/lib/External/isl/test_inputs/codegen/hoist2.c
new file mode 100644
index 00000000000..24cfc639166
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/hoist2.c
@@ -0,0 +1,3 @@
+for (int c0 = 1; c0 <= 5; c0 += 1)
+  for (int c1 = t1 - 64 * b + 64; c1 <= min(70, -((c0 - 1) % 2) - c0 + 73); c1 += 64)
+    A(c0, 64 * b + c1 - 8);
diff --git a/polly/lib/External/isl/test_inputs/codegen/hoist2.in b/polly/lib/External/isl/test_inputs/codegen/hoist2.in
new file mode 100644
index 00000000000..6a29a026f20
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/hoist2.in
@@ -0,0 +1,5 @@
+# Check that the constraints hoisted from the inner loop
+# do not end up involving the inner loop iterator.
+[t1, b] -> { A[i1, i2] -> [i1, 8 - 64b + i2] : exists (e0, e1 = [(-8 + t1 - i2)/64]: 64e1 = -8 + t1 - i2 and i2 >= 1 and i2 <= 127 and 2e0 >= -3 + i1 and 2e0 >= -1 - i1 and 2e0 <= 8 - i1 and 2e0 <= 6 + i1 and 2e0 >= -65 - 64b + i2 and 2e0 >= -1 + 64b - i2 and e0 <= 1 and e0 >= 0 and 2e0 <= 62 + 64b - i2 and b <= 1 and b >= 0 and i1 >= 1 and i1 <= 2046 and t1 >= 5 and t1 <= 8) }
+[t1, b] -> { : b >= 0 and b <= 1 and t1 >= 5 and t1 <= 8 }
+[t1] -> { [i0, i1, i5, a] -> atomic[x]}
diff --git a/polly/lib/External/isl/test_inputs/codegen/lu.c b/polly/lib/External/isl/test_inputs/codegen/lu.c
new file mode 100644
index 00000000000..ecbb895d3ac
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/lu.c
@@ -0,0 +1,17 @@
+for (int c0 = 0; c0 < n - 1; c0 += 32)
+  for (int c1 = c0; c1 < n; c1 += 32)
+    for (int c2 = c0; c2 < n; c2 += 32) {
+      if (c1 >= c0 + 32) {
+        for (int c3 = c0; c3 <= min(c0 + 31, c2 + 30); c3 += 1)
+          for (int c4 = c1; c4 <= min(n - 1, c1 + 31); c4 += 1)
+            for (int c5 = max(c2, c3 + 1); c5 <= min(n - 1, c2 + 31); c5 += 1)
+              S_6(c3, c4, c5);
+      } else
+        for (int c3 = c0; c3 <= min(min(n - 2, c0 + 31), c2 + 30); c3 += 1) {
+          for (int c5 = max(c2, c3 + 1); c5 <= min(n - 1, c2 + 31); c5 += 1)
+            S_2(c3, c5);
+          for (int c4 = c3 + 1; c4 <= min(n - 1, c0 + 31); c4 += 1)
+            for (int c5 = max(c2, c3 + 1); c5 <= min(n - 1, c2 + 31); c5 += 1)
+              S_6(c3, c4, c5);
+        }
+    }
diff --git a/polly/lib/External/isl/test_inputs/codegen/lu.in b/polly/lib/External/isl/test_inputs/codegen/lu.in
new file mode 100644
index 00000000000..ea884f59d60
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/lu.in
@@ -0,0 +1,4 @@
+# Check that the stride of the second loop is properly detected
+[n] -> { S_2[k, j] -> [o0, o0, o2, k, k, j, 1] : exists (e0 = floor((o2)/32), e1 = floor((o0)/32): 32e0 = o2 and 32e1 = o0 and o0 <= k and o0 >= -31 + k and k >= 0 and j <= -1 + n and o2 <= j and o2 >= -31 + j and j >= 1 + k); S_6[k, i, j] -> [o0, o1, o2, k, i, j, 0] : exists (e0 = floor((o0)/32), e1 = floor((o1)/32), e2 = floor((o2)/32): 32e0 = o0 and 32e1 = o1 and 32e2 = o2 and o0 <= k and o0 >= -31 + k and o1 <= i and o1 >= -31 + i and o2 <= j and o2 >= -31 + j and k >= 0 and i >= 1 + k and j <= -1 + n and j >= 1 + k and i <= -1 + n) }
+{  :  }
+{ [a,b,c,d,e,f,g] -> atomic[x] : x < 3; [a,b,c,d,e,f,g] -> separate[x] : x >= 3  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/mod.c b/polly/lib/External/isl/test_inputs/codegen/mod.c
new file mode 100644
index 00000000000..dcd8319f4ca
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/mod.c
@@ -0,0 +1,2 @@
+if (2 * (n % 100) == 3 * (m % 200))
+  A();
diff --git a/polly/lib/External/isl/test_inputs/codegen/mod.in b/polly/lib/External/isl/test_inputs/codegen/mod.in
new file mode 100644
index 00000000000..7a04c5a4158
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/mod.in
@@ -0,0 +1,4 @@
+# check that modulo constraint is generated correctly
+[n, m] -> { A[] -> [] : 2 * (n % 100) = 3 * (m % 200) }
+[n, m] -> { : m, n >= 0 }
+{}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/README b/polly/lib/External/isl/test_inputs/codegen/omega/README
new file mode 100644
index 00000000000..80e27fd462d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/README
@@ -0,0 +1,5 @@
+The tests in this directory have been adapted from the corresponding omega+
+test cases.
+The options have been derived semi-automatically and may not always
+correspond to the intended meaning of the specified "effort" in the omega+
+test cases.
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/basics-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/basics-0.c
new file mode 100644
index 00000000000..53995e9e89e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/basics-0.c
@@ -0,0 +1,8 @@
+{
+  for (int c0 = 5; c0 <= 8; c0 += 1)
+    s0(c0);
+  for (int c0 = 10; c0 <= 16; c0 += 2)
+    s0(c0);
+  for (int c0 = 20; c0 <= 25; c0 += 1)
+    s0(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/basics-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/basics-0.in
new file mode 100644
index 00000000000..025da3d30d6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/basics-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1] -> [In_1] : (In_1 >= 5 and In_1 <= 8) or (exists (e0 = [(In_1)/2]: 2e0 = In_1 and In_1 >= 10 and In_1 <= 16)) or (In_1 >= 20 and In_1 <= 25) }
+{  :  }
+{ [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/basics-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/basics-1.c
new file mode 100644
index 00000000000..d3d69704165
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/basics-1.c
@@ -0,0 +1,3 @@
+for (int c0 = -9; c0 <= 9; c0 += 1)
+  for (int c1 = max(1, -c0 + 1); c1 <= min(10, -c0 + 10); c1 += 1)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/basics-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/basics-1.in
new file mode 100644
index 00000000000..82e109311bc
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/basics-1.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2] -> [In_1, In_2] : In_2 >= 1 - In_1 and In_2 >= 1 and In_2 <= 10 - In_1 and In_2 <= 10 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/chosol-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/chosol-0.c
new file mode 100644
index 00000000000..464ee493499
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/chosol-0.c
@@ -0,0 +1,9 @@
+{
+  for (int c1 = 2; c1 <= n; c1 += 1)
+    s0(c1);
+  for (int c1 = 1; c1 < n; c1 += 1) {
+    for (int c3 = c1 + 1; c3 <= n; c3 += 1)
+      s1(c3, c1);
+    s2(c1 + 1);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/chosol-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/chosol-0.in
new file mode 100644
index 00000000000..81fde503f74
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/chosol-0.in
@@ -0,0 +1,3 @@
+[n] -> { s0[i] -> [0, i, 0, 0] : i >= 2 and i <= n; s1[i, j] -> [1, j, 0, i] : j >= 1 and j <= -1 + i and i <= n; s2[i] -> [1, -1 + i, 1, 0] : i >= 2 and i <= n }
+{  :  }
+[n] -> { [i0, i1, i2, i3] -> atomic[o0] : o0 <= 2; [i0, i1, i2, i3] -> separate[o0] : o0 >= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/chosol-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/chosol-1.c
new file mode 100644
index 00000000000..464ee493499
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/chosol-1.c
@@ -0,0 +1,9 @@
+{
+  for (int c1 = 2; c1 <= n; c1 += 1)
+    s0(c1);
+  for (int c1 = 1; c1 < n; c1 += 1) {
+    for (int c3 = c1 + 1; c3 <= n; c3 += 1)
+      s1(c3, c1);
+    s2(c1 + 1);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/chosol-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/chosol-1.in
new file mode 100644
index 00000000000..b5abf2917e8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/chosol-1.in
@@ -0,0 +1,3 @@
+[n] -> { s0[i] -> [0, i, 0, 0] : i >= 2 and i <= n; s1[i, j] -> [1, j, 0, i] : j >= 1 and j <= -1 + i and i <= n; s2[i] -> [1, -1 + i, 1, 0] : i >= 2 and i <= n }
+{  :  }
+[n] -> { [i0, i1, i2, i3] -> atomic[o0] : o0 <= 1; [i0, i1, i2, i3] -> separate[o0] : o0 >= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-0.c
new file mode 100644
index 00000000000..6340134fcdb
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-0.c
@@ -0,0 +1,7 @@
+for (int c0 = 1; c0 <= 8; c0 += 1)
+  for (int c1 = 0; c1 <= 7; c1 += 1) {
+    if (c0 >= 2 && c0 <= 6 && c1 <= 4)
+      s1(c0, c1);
+    if (c1 + 1 >= c0)
+      s0(c0, c1);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-0.in
new file mode 100644
index 00000000000..a3fa3b1652d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-0.in
@@ -0,0 +1,3 @@
+{ s1[In_1, In_2] -> [In_1, In_2] : In_1 >= 2 and In_1 <= 6 and In_2 >= 0 and In_2 <= 4; s0[In_1, In_2] -> [In_1, In_2] : In_1 >= 1 and In_2 >= -1 + In_1 and In_2 <= 7 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 1; [i0, i1] -> separate[o0] : o0 >= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-1.c
new file mode 100644
index 00000000000..f4ec0e9c066
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-1.c
@@ -0,0 +1,15 @@
+for (int c0 = 1; c0 <= 8; c0 += 1) {
+  if (c0 >= 2) {
+    if (c0 <= 6)
+      for (int c1 = 0; c1 < c0 - 1; c1 += 1)
+        s1(c0, c1);
+    for (int c1 = c0 - 1; c1 <= 4; c1 += 1) {
+      s1(c0, c1);
+      s0(c0, c1);
+    }
+    for (int c1 = max(5, c0 - 1); c1 <= 7; c1 += 1)
+      s0(c0, c1);
+  } else
+    for (int c1 = 0; c1 <= 7; c1 += 1)
+      s0(1, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-1.in
new file mode 100644
index 00000000000..c6f7fa54a91
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-1.in
@@ -0,0 +1,3 @@
+{ s1[In_1, In_2] -> [In_1, In_2] : In_1 >= 2 and In_1 <= 6 and In_2 >= 0 and In_2 <= 4; s0[In_1, In_2] -> [In_1, In_2] : In_1 >= 1 and In_2 >= -1 + In_1 and In_2 <= 7 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-2.c b/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-2.c
new file mode 100644
index 00000000000..f92e58368a5
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-2.c
@@ -0,0 +1,17 @@
+{
+  for (int c1 = 0; c1 <= 7; c1 += 1)
+    s0(1, c1);
+  for (int c0 = 2; c0 <= 6; c0 += 1) {
+    for (int c1 = 0; c1 < c0 - 1; c1 += 1)
+      s1(c0, c1);
+    for (int c1 = c0 - 1; c1 <= 4; c1 += 1) {
+      s1(c0, c1);
+      s0(c0, c1);
+    }
+    for (int c1 = 5; c1 <= 7; c1 += 1)
+      s0(c0, c1);
+  }
+  for (int c0 = 7; c0 <= 8; c0 += 1)
+    for (int c1 = c0 - 1; c1 <= 7; c1 += 1)
+      s0(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-2.in b/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-2.in
new file mode 100644
index 00000000000..03a7b5df3a3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/code_gen-2.in
@@ -0,0 +1,3 @@
+{ s1[In_1, In_2] -> [In_1, In_2] : In_1 >= 2 and In_1 <= 6 and In_2 >= 0 and In_2 <= 4; s0[In_1, In_2] -> [In_1, In_2] : In_1 >= 1 and In_2 >= -1 + In_1 and In_2 <= 7 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= -1; [i0, i1] -> separate[o0] : o0 >= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/collard-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/collard-0.c
new file mode 100644
index 00000000000..413fbda482f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/collard-0.c
@@ -0,0 +1,16 @@
+{
+  for (int c4 = 1; c4 <= n; c4 += 1)
+    s2(c4);
+  for (int c1 = 1; c1 < n; c1 += 1) {
+    for (int c4 = 0; c4 < n - c1; c4 += 1)
+      s0(c1, n - c4);
+    for (int c3 = 0; c3 < n - c1; c3 += 1)
+      for (int c4 = c1 + 1; c4 <= n; c4 += 1)
+        s1(c1, n - c3, c4);
+  }
+  for (int c1 = 1; c1 <= n; c1 += 1) {
+    s4(c1);
+    for (int c3 = c1 + 1; c3 <= n; c3 += 1)
+      s3(c3, c1);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/collard-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/collard-0.in
new file mode 100644
index 00000000000..d803da7574c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/collard-0.in
@@ -0,0 +1,3 @@
+[n] -> { s1[i, j, k] -> [1, i, 1, n - j, k] : j >= 1 + i and k >= 1 + i and i >= 1 and j <= n and k <= n; s2[i] -> [0, 0, 0, 0, i] : i >= 1 and i <= n; s4[i] -> [2, i, 0, 0, 0] : i >= 1 and i <= n; s0[i, j] -> [1, i, 0, n - i, n - j] : i >= 1 and j >= 1 + i and j <= n; s3[i, j] -> [2, j, 1, i, j] : j >= 1 and j <= -1 + i and i <= n }
+{  :  }
+[n] -> { [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 4; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/dagstuhl1-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/dagstuhl1-0.c
new file mode 100644
index 00000000000..3e403ddcd33
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/dagstuhl1-0.c
@@ -0,0 +1,2 @@
+for (int c0 = 0; c0 <= 99; c0 += 1)
+  s0(c0 % 10, c0 / 10);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/dagstuhl1-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/dagstuhl1-0.in
new file mode 100644
index 00000000000..0c2b107517a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/dagstuhl1-0.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [i + 10j] : i >= 0 and i <= 9 and j >= 0 and j <= 9 }
+{  :  }
+{ [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/dagstuhl1-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/dagstuhl1-1.c
new file mode 100644
index 00000000000..a3e4458f146
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/dagstuhl1-1.c
@@ -0,0 +1,2 @@
+for (int c0 = 0; c0 <= 99; c0 += 1)
+  s0(c0, c0 % 10, (c0 + 10) / 10 - 1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/dagstuhl1-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/dagstuhl1-1.in
new file mode 100644
index 00000000000..e0c72739736
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/dagstuhl1-1.in
@@ -0,0 +1,3 @@
+{s0[p,i,j] -> [p,i,j] : 0 <= i,j <= 9 && p = i+10j}
+{  :  }
+{ [p,i,j] -> separate[o0] : o0 >= 2; [p,i,j] -> atomic[o0] : o0 <= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/fc1-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/fc1-0.c
new file mode 100644
index 00000000000..fbec13a719b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/fc1-0.c
@@ -0,0 +1,7 @@
+for (int c0 = 0; c0 < n - 1; c0 += 1) {
+  for (int c3 = 0; c3 < n - c0 - 1; c3 += 1)
+    s0(c0 + 1, n - c3);
+  for (int c3 = 0; c3 < n - c0 - 1; c3 += 1)
+    for (int c6 = c0 + 2; c6 <= n; c6 += 1)
+      s1(c0 + 1, n - c3, c6);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/fc1-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/fc1-0.in
new file mode 100644
index 00000000000..688f9580d9b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/fc1-0.in
@@ -0,0 +1,3 @@
+[n] -> { s0[i, j] -> [-1 + i, 0, n - i, n - j] : i >= 1 and j >= 1 + i and j <= n; s1[i, j, k] -> [-1 + i, 1, n - i, n - j] : j >= 1 + i and k >= 1 + i and i >= 1 and j <= n and k <= n }
+{  :  }
+[n] -> { [i0, i1, i2, i3] -> atomic[o0] : o0 <= 2; [i0, i1, i2, i3] -> separate[o0] : o0 >= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/fc1-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/fc1-1.c
new file mode 100644
index 00000000000..768a04fbc4d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/fc1-1.c
@@ -0,0 +1,17 @@
+{
+  for (int c3 = 1; c3 <= n; c3 += 1)
+    s2(c3);
+  for (int c0 = 0; c0 < n - 1; c0 += 1) {
+    for (int c3 = 0; c3 < n - c0 - 1; c3 += 1)
+      s0(c0 + 1, n - c3);
+    for (int c3 = 0; c3 < n - c0 - 1; c3 += 1)
+      for (int c6 = c0 + 2; c6 <= n; c6 += 1)
+        s1(c0 + 1, n - c3, c6);
+  }
+  for (int c0 = n - 1; c0 < 2 * n - 1; c0 += 1) {
+    if (c0 >= n)
+      for (int c2 = -n + c0 + 2; c2 <= n; c2 += 1)
+        s3(c2, -n + c0 + 1);
+    s4(-n + c0 + 2);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/fc1-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/fc1-1.in
new file mode 100644
index 00000000000..c9135fff76c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/fc1-1.in
@@ -0,0 +1,3 @@
+[n] -> { s1[i, j, k] -> [-1 + i, 1, n - i, n - j] : j >= 1 + i and k >= 1 + i and i >= 1 and j <= n and k <= n; s3[i, j] -> [-1 + n + j, 0, i, j] : j >= 1 and j <= -1 + i and i <= n; s4[i] -> [-2 + n + i, 1, 0, 0] : i >= 1 and i <= n; s0[i, j] -> [-1 + i, 0, n - i, n - j] : i >= 1 and j >= 1 + i and j <= n; s2[i] -> [0, 0, 0, i] : i >= 1 and i <= n }
+{  :  }
+[n] -> { [i0, i1, i2, i3] -> atomic[o0] : o0 <= 2; [i0, i1, i2, i3] -> separate[o0] : o0 >= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/fc1-2.c b/polly/lib/External/isl/test_inputs/codegen/omega/fc1-2.c
new file mode 100644
index 00000000000..768a04fbc4d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/fc1-2.c
@@ -0,0 +1,17 @@
+{
+  for (int c3 = 1; c3 <= n; c3 += 1)
+    s2(c3);
+  for (int c0 = 0; c0 < n - 1; c0 += 1) {
+    for (int c3 = 0; c3 < n - c0 - 1; c3 += 1)
+      s0(c0 + 1, n - c3);
+    for (int c3 = 0; c3 < n - c0 - 1; c3 += 1)
+      for (int c6 = c0 + 2; c6 <= n; c6 += 1)
+        s1(c0 + 1, n - c3, c6);
+  }
+  for (int c0 = n - 1; c0 < 2 * n - 1; c0 += 1) {
+    if (c0 >= n)
+      for (int c2 = -n + c0 + 2; c2 <= n; c2 += 1)
+        s3(c2, -n + c0 + 1);
+    s4(-n + c0 + 2);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/fc1-2.in b/polly/lib/External/isl/test_inputs/codegen/omega/fc1-2.in
new file mode 100644
index 00000000000..6d9a1626a5b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/fc1-2.in
@@ -0,0 +1,3 @@
+[n] -> { s1[i, j, k] -> [-1 + i, 1, n - i, n - j] : j >= 1 + i and k >= 1 + i and i >= 1 and j <= n and k <= n; s3[i, j] -> [-1 + n + j, 0, i, j] : j >= 1 and j <= -1 + i and i <= n; s4[i] -> [-2 + n + i, 1, 0, 0] : i >= 1 and i <= n; s0[i, j] -> [-1 + i, 0, n - i, n - j] : i >= 1 and j >= 1 + i and j <= n; s2[i] -> [0, 0, 0, i] : i >= 1 and i <= n }
+{  :  }
+[n] -> { [i0, i1, i2, i3] -> atomic[o0] : o0 <= 1; [i0, i1, i2, i3] -> separate[o0] : o0 >= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/fc2-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/fc2-0.c
new file mode 100644
index 00000000000..fbec13a719b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/fc2-0.c
@@ -0,0 +1,7 @@
+for (int c0 = 0; c0 < n - 1; c0 += 1) {
+  for (int c3 = 0; c3 < n - c0 - 1; c3 += 1)
+    s0(c0 + 1, n - c3);
+  for (int c3 = 0; c3 < n - c0 - 1; c3 += 1)
+    for (int c6 = c0 + 2; c6 <= n; c6 += 1)
+      s1(c0 + 1, n - c3, c6);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/fc2-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/fc2-0.in
new file mode 100644
index 00000000000..688f9580d9b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/fc2-0.in
@@ -0,0 +1,3 @@
+[n] -> { s0[i, j] -> [-1 + i, 0, n - i, n - j] : i >= 1 and j >= 1 + i and j <= n; s1[i, j, k] -> [-1 + i, 1, n - i, n - j] : j >= 1 + i and k >= 1 + i and i >= 1 and j <= n and k <= n }
+{  :  }
+[n] -> { [i0, i1, i2, i3] -> atomic[o0] : o0 <= 2; [i0, i1, i2, i3] -> separate[o0] : o0 >= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/fc2-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/fc2-1.c
new file mode 100644
index 00000000000..768a04fbc4d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/fc2-1.c
@@ -0,0 +1,17 @@
+{
+  for (int c3 = 1; c3 <= n; c3 += 1)
+    s2(c3);
+  for (int c0 = 0; c0 < n - 1; c0 += 1) {
+    for (int c3 = 0; c3 < n - c0 - 1; c3 += 1)
+      s0(c0 + 1, n - c3);
+    for (int c3 = 0; c3 < n - c0 - 1; c3 += 1)
+      for (int c6 = c0 + 2; c6 <= n; c6 += 1)
+        s1(c0 + 1, n - c3, c6);
+  }
+  for (int c0 = n - 1; c0 < 2 * n - 1; c0 += 1) {
+    if (c0 >= n)
+      for (int c2 = -n + c0 + 2; c2 <= n; c2 += 1)
+        s3(c2, -n + c0 + 1);
+    s4(-n + c0 + 2);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/fc2-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/fc2-1.in
new file mode 100644
index 00000000000..c9135fff76c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/fc2-1.in
@@ -0,0 +1,3 @@
+[n] -> { s1[i, j, k] -> [-1 + i, 1, n - i, n - j] : j >= 1 + i and k >= 1 + i and i >= 1 and j <= n and k <= n; s3[i, j] -> [-1 + n + j, 0, i, j] : j >= 1 and j <= -1 + i and i <= n; s4[i] -> [-2 + n + i, 1, 0, 0] : i >= 1 and i <= n; s0[i, j] -> [-1 + i, 0, n - i, n - j] : i >= 1 and j >= 1 + i and j <= n; s2[i] -> [0, 0, 0, i] : i >= 1 and i <= n }
+{  :  }
+[n] -> { [i0, i1, i2, i3] -> atomic[o0] : o0 <= 2; [i0, i1, i2, i3] -> separate[o0] : o0 >= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-0.c
new file mode 100644
index 00000000000..db7f43707c3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-0.c
@@ -0,0 +1,2 @@
+for (int c0 = 4 * floord(m - 1, 12) + 4; c0 <= floord(n, 3); c0 += 4)
+  s0(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-0.in
new file mode 100644
index 00000000000..45faa57c37d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-0.in
@@ -0,0 +1,3 @@
+[m, n] -> { s0[In_1] -> [In_1] : exists (e0 = [(In_1)/4]: 4e0 = In_1 and 3In_1 >= m and 3In_1 <= n) }
+{  :  }
+[m, n] -> { [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-1.c
new file mode 100644
index 00000000000..bc879829073
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-1.c
@@ -0,0 +1,2 @@
+for (int c0 = floord(m, 4); c0 <= n; c0 += 1)
+  s0(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-1.in
new file mode 100644
index 00000000000..7ae72abb93e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-1.in
@@ -0,0 +1,3 @@
+[m, n] -> { s0[In_1] -> [In_1] : 4In_1 >= -3 + m and In_1 <= n }
+{  :  }
+[m, n] -> { [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-2.c b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-2.c
new file mode 100644
index 00000000000..ff5f3c33745
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-2.c
@@ -0,0 +1,2 @@
+for (int c0 = 4 * floord(m, 4); c0 <= n; c0 += 1)
+  s0(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-2.in b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-2.in
new file mode 100644
index 00000000000..73e1b8eb554
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-2.in
@@ -0,0 +1,3 @@
+[m, n] -> { s0[In_1] -> [In_1] : exists (e0 = [(m)/4]: 4e0 <= m and 4e0 >= -3 + m and 4e0 <= In_1 and In_1 <= n) }
+{  :  }
+[m, n] -> { [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-3.c b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-3.c
new file mode 100644
index 00000000000..e3c9728de91
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-3.c
@@ -0,0 +1,2 @@
+for (int c0 = 3 * floord(m, 3) + 4 * floord(m, 4); c0 <= n; c0 += 1)
+  s0(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-3.in b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-3.in
new file mode 100644
index 00000000000..1ff7e2c3861
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-3.in
@@ -0,0 +1,3 @@
+[m, n] -> { s0[In_1] -> [In_1] : exists (e0 = [(m)/3], e1 = [(m)/4]: 4e1 <= m and 3e0 <= m and 4e1 >= -3 + m and 3e0 >= -2 + m and In_1 <= n and 4e1 <= In_1 - 3e0) }
+{  :  }
+[m, n] -> { [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-4.c b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-4.c
new file mode 100644
index 00000000000..dc6ba0ae941
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-4.c
@@ -0,0 +1,3 @@
+if (n >= 3 * floord(n + 1, 3))
+  for (int c0 = m; c0 <= 5 * floord(n + 1, 3); c0 += 1)
+    s0(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-4.in b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-4.in
new file mode 100644
index 00000000000..7a726c39379
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-4.in
@@ -0,0 +1,3 @@
+[n, m] -> { s0[In_1] -> [In_1] : exists (e0 = [(1 + n)/3]: In_1 >= m and 5e0 >= In_1 and 3e0 <= n and 3e0 >= -1 + n) }
+{  :  }
+[n, m] -> { [i0] -> atomic[o0] : o0 <= -1; [i0] -> separate[o0] : o0 >= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-5.c b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-5.c
new file mode 100644
index 00000000000..3e9f47d4ccc
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-5.c
@@ -0,0 +1,2 @@
+for (int c0 = 4 * floord(m, 32); c0 <= n; c0 += 1)
+  s0(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-5.in b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-5.in
new file mode 100644
index 00000000000..cc24250ddaa
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-5.in
@@ -0,0 +1,3 @@
+[m, n] -> { s0[In_1] -> [In_1] : exists (e0 = [(m)/32]: 32e0 <= m and 32e0 >= -31 + m and 4e0 <= In_1 and In_1 <= n) }
+{  :  }
+[m, n] -> { [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-6.c b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-6.c
new file mode 100644
index 00000000000..743b0c938c5
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-6.c
@@ -0,0 +1,3 @@
+if (m >= 8 * floord(m + 1, 8))
+  for (int c0 = 4 * floord(m + 1, 32); c0 <= n; c0 += 1)
+    s0(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-6.in b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-6.in
new file mode 100644
index 00000000000..78d2be97e2a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/floor_bound-6.in
@@ -0,0 +1,3 @@
+[m, n] -> { s0[In_1] -> [In_1] : exists (e0 = [(1 + m)/8], e1 = [(e0)/4]: 8e0 <= m and 8e0 >= -6 + m and 4e1 <= In_1 and In_1 <= n and 32e1 <= 1 + m and 32e1 >= -30 + m) }
+{  :  }
+[m, n] -> { [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/gc-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/gc-0.c
new file mode 100644
index 00000000000..d8234a378ce
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/gc-0.c
@@ -0,0 +1,2 @@
+for (int c0 = 2; c0 <= 8; c0 += 2)
+  s0(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/gc-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/gc-0.in
new file mode 100644
index 00000000000..063e35361f1
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/gc-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1] -> [In_1] : exists (e0 = [(In_1)/2]: 2e0 = In_1 and In_1 >= 2 and In_1 <= 8) }
+{  :  }
+{ [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/ge-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/ge-0.c
new file mode 100644
index 00000000000..1c22a5d9b0a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/ge-0.c
@@ -0,0 +1,7 @@
+for (int c0 = 2; c0 <= n; c0 += 1)
+  for (int c1 = 1; c1 <= n; c1 += 1) {
+    for (int c3 = 1; c3 < min(c0, c1); c3 += 1)
+      s1(c3, c0, c1);
+    if (c0 >= c1 + 1)
+      s0(c1, c0);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/ge-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/ge-0.in
new file mode 100644
index 00000000000..edd7c36ee61
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/ge-0.in
@@ -0,0 +1,3 @@
+[n] -> { s0[k, i] -> [i, k, 1, 0] : k >= 1 and i >= 1 + k and i <= n; s1[k, i, j] -> [i, j, 0, k] : i >= 1 + k and j >= 1 + k and k >= 1 and i <= n and j <= n }
+{  :  }
+[n] -> { [i0, i1, i2, i3] -> atomic[o0] : o0 <= 2; [i0, i1, i2, i3] -> separate[o0] : o0 >= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/ge-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/ge-1.c
new file mode 100644
index 00000000000..1c22a5d9b0a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/ge-1.c
@@ -0,0 +1,7 @@
+for (int c0 = 2; c0 <= n; c0 += 1)
+  for (int c1 = 1; c1 <= n; c1 += 1) {
+    for (int c3 = 1; c3 < min(c0, c1); c3 += 1)
+      s1(c3, c0, c1);
+    if (c0 >= c1 + 1)
+      s0(c1, c0);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/ge-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/ge-1.in
new file mode 100644
index 00000000000..4a08142d506
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/ge-1.in
@@ -0,0 +1,3 @@
+[n] -> { s0[k, i] -> [i, k, 1, 0] : k >= 1 and i >= 1 + k and i <= n; s1[k, i, j] -> [i, j, 0, k] : i >= 1 + k and j >= 1 + k and k >= 1 and i <= n and j <= n }
+{  :  }
+[n] -> { [i0, i1, i2, i3] -> atomic[o0] : o0 <= 1; [i0, i1, i2, i3] -> separate[o0] : o0 >= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/gist-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/gist-0.c
new file mode 100644
index 00000000000..4608d61d8db
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/gist-0.c
@@ -0,0 +1,3 @@
+for (int c0 = 1; c0 <= n; c0 += 4)
+  for (int c1 = c0; c1 <= n; c1 += 3)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/gist-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/gist-0.in
new file mode 100644
index 00000000000..b2f3366f8d3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/gist-0.in
@@ -0,0 +1,3 @@
+[n] -> { s0[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(-1 + In_1)/4], e1 = [(-3 - In_1 + 4In_2)/12]: 4e0 = -1 + In_1 and 12e1 = -3 - In_1 + 4In_2 and In_1 >= 1 and In_2 >= In_1 and In_2 <= n) }
+{  :  }
+[n] -> { [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/gist-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/gist-1.c
new file mode 100644
index 00000000000..8f1c692c8ea
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/gist-1.c
@@ -0,0 +1,3 @@
+for (int c0 = 1; c0 <= n; c0 += 4)
+  for (int c1 = c0; c1 <= n; c1 += 8)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/gist-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/gist-1.in
new file mode 100644
index 00000000000..ecbf8e7b06e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/gist-1.in
@@ -0,0 +1,3 @@
+[n] -> { s0[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(-1 + In_1)/4], e1 = [(-In_1 + In_2)/8]: 4e0 = -1 + In_1 and 8e1 = -In_1 + In_2 and In_1 >= 1 and In_2 >= In_1 and In_2 <= n) }
+{  :  }
+[n] -> { [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/gist-2.c b/polly/lib/External/isl/test_inputs/codegen/omega/gist-2.c
new file mode 100644
index 00000000000..317498782d4
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/gist-2.c
@@ -0,0 +1,3 @@
+for (int c0 = 1; c0 <= n; c0 += 256)
+  for (int c1 = c0; c1 <= n; c1 += 8)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/gist-2.in b/polly/lib/External/isl/test_inputs/codegen/omega/gist-2.in
new file mode 100644
index 00000000000..63edb52b5f7
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/gist-2.in
@@ -0,0 +1,3 @@
+[n] -> { s0[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(-1 + In_1)/256], e1 = [(-1 + In_2)/8]: 256e0 = -1 + In_1 and 8e1 = -1 + In_2 and In_1 >= 1 and In_2 >= In_1 and In_2 <= n) }
+{  :  }
+[n] -> { [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/gist-3.c b/polly/lib/External/isl/test_inputs/codegen/omega/gist-3.c
new file mode 100644
index 00000000000..513f385c3c5
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/gist-3.c
@@ -0,0 +1,3 @@
+for (int c0 = 1; c0 < n; c0 += 4)
+  for (int c1 = c0 + 1; c1 <= n; c1 += 6)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/gist-3.in b/polly/lib/External/isl/test_inputs/codegen/omega/gist-3.in
new file mode 100644
index 00000000000..43cac1243b8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/gist-3.in
@@ -0,0 +1,3 @@
+[n] -> { s0[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(-1 + In_1)/4], e1 = [(-1 - In_1 + In_2)/6]: 4e0 = -1 + In_1 and 6e1 = -1 - In_1 + In_2 and In_1 >= 1 and In_2 >= 1 + In_1 and In_2 <= n) }
+{  :  }
+[n] -> { [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/gist-4.c b/polly/lib/External/isl/test_inputs/codegen/omega/gist-4.c
new file mode 100644
index 00000000000..a6f6fd001b0
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/gist-4.c
@@ -0,0 +1,3 @@
+for (int c0 = 1; c0 <= n; c0 += 6)
+  for (int c1 = c0; c1 <= n; c1 += 4)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/gist-4.in b/polly/lib/External/isl/test_inputs/codegen/omega/gist-4.in
new file mode 100644
index 00000000000..f67f837d598
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/gist-4.in
@@ -0,0 +1,3 @@
+[n] -> { s0[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(-1 + In_1)/6], e1 = [(-2 - In_1 + 3In_2)/12]: 6e0 = -1 + In_1 and 12e1 = -2 - In_1 + 3In_2 and In_1 >= 1 and In_2 >= In_1 and In_2 <= n) }
+{  :  }
+[n] -> { [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/gist-5.c b/polly/lib/External/isl/test_inputs/codegen/omega/gist-5.c
new file mode 100644
index 00000000000..c2df61fb4e1
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/gist-5.c
@@ -0,0 +1,3 @@
+for (int c0 = 1; c0 <= n; c0 += 12)
+  for (int c1 = c0; c1 <= n; c1 += 8)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/gist-5.in b/polly/lib/External/isl/test_inputs/codegen/omega/gist-5.in
new file mode 100644
index 00000000000..8b192c132fd
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/gist-5.in
@@ -0,0 +1,3 @@
+[n] -> { s0[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(-1 + In_1)/12], e1 = [(-2 - In_1 + 3In_2)/24]: 12e0 = -1 + In_1 and 24e1 = -2 - In_1 + 3In_2 and In_1 >= 1 and In_2 >= In_1 and In_2 <= n) }
+{  :  }
+[n] -> { [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/guard1-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/guard1-0.c
new file mode 100644
index 00000000000..555d644f561
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/guard1-0.c
@@ -0,0 +1,2 @@
+if ((n - m + 2) % 3 == 0)
+  s0(n, m);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/guard1-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/guard1-0.in
new file mode 100644
index 00000000000..52964e26489
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/guard1-0.in
@@ -0,0 +1,3 @@
+[n, m] -> { s0[n, m] -> [n, m] : exists (e0 = [(-2 - n + m)/3]: 3e0 = -2 - n + m) }
+{  :  }
+[n, m] -> { [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/guard1-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/guard1-1.c
new file mode 100644
index 00000000000..335c1b838fe
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/guard1-1.c
@@ -0,0 +1,2 @@
+if ((n - m + 1) % 2 == 0)
+  s0(n, m);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/guard1-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/guard1-1.in
new file mode 100644
index 00000000000..141f585d358
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/guard1-1.in
@@ -0,0 +1,3 @@
+[n, m] -> { s0[n, m] -> [n, m] : exists (e0 = [(-1 - n + m)/2]: 2e0 = -1 - n + m) }
+{  :  }
+[n, m] -> { [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/hpf-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/hpf-0.c
new file mode 100644
index 00000000000..c676558941f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/hpf-0.c
@@ -0,0 +1,4 @@
+if (P2 >= 0 && P2 <= 3 && P1 == P2)
+  for (int c0 = 0; c0 <= min(2, -P2 + 4); c0 += 1)
+    for (int c2 = (-P2 - c0 + 6) % 3; c2 <= 3; c2 += 3)
+      s0(c0, c0, c2, c2);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/hpf-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/hpf-0.in
new file mode 100644
index 00000000000..e697792ea66
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/hpf-0.in
@@ -0,0 +1,3 @@
+[P1, P2] -> { s0[In_1, In_1, In_3, In_3] -> [In_1, In_1, In_3, In_3] : exists (e0 = [(-2P2 - 2In_1 + In_3)/3]: P1 = P2 and 3e0 = -2P2 - 2In_1 + In_3 and P2 >= 0 and P2 <= 3 and In_1 <= 4 - P2 and In_1 >= 0 and In_1 <= 2 and In_3 >= 0 and In_3 <= 3) }
+{  :  }
+[P2, P1] -> { [i0, i1, i2, i3] -> separate[o0] : o0 >= 3; [i0, i1, i2, i3] -> atomic[o0] : o0 <= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/if_then-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-0.c
new file mode 100644
index 00000000000..bc8d3370a6d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-0.c
@@ -0,0 +1,12 @@
+if (m <= 1) {
+  for (int c0 = 1; c0 <= n; c0 += 1)
+    for (int c1 = 1; c1 <= n; c1 += 1)
+      s2(c0, c1);
+} else if (n >= m + 1) {
+  for (int c0 = 1; c0 <= n; c0 += 1)
+    for (int c1 = 1; c1 <= n; c1 += 1)
+      s0(c0, c1);
+} else
+  for (int c0 = 1; c0 <= n; c0 += 1)
+    for (int c1 = 1; c1 <= n; c1 += 1)
+      s1(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/if_then-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-0.in
new file mode 100644
index 00000000000..93f3f7c2bc8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-0.in
@@ -0,0 +1,3 @@
+[n, m] -> { s2[In_1, In_2] -> [In_1, In_2] : In_1 >= 1 and In_1 <= n and In_2 >= 1 and In_2 <= n and m <= 1; s0[In_1, In_2] -> [In_1, In_2] : m >= 2 and m <= -1 + n and In_1 >= 1 and In_1 <= n and In_2 >= 1 and In_2 <= n; s1[In_1, In_2] -> [In_1, In_2] : In_1 <= n and In_2 <= n and m >= n and In_1 >= 1 and In_2 >= 1 and m >= 2 }
+{  :  }
+[n, m] -> { [i0, i1] -> atomic[o0] : o0 <= 1; [i0, i1] -> separate[o0] : o0 >= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/if_then-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-1.c
new file mode 100644
index 00000000000..18ca3702186
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-1.c
@@ -0,0 +1,9 @@
+for (int c0 = 1; c0 <= 100; c0 += 1) {
+  if (n >= 2)
+    s0(c0);
+  for (int c1 = 1; c1 <= 100; c1 += 1) {
+    if (n >= 2)
+      s1(c0, c1);
+    s2(c0, c1);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/if_then-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-1.in
new file mode 100644
index 00000000000..0258439fff7
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-1.in
@@ -0,0 +1,3 @@
+[n] -> { s0[In_1] -> [In_1,0] : In_1 >= 1 and In_1 <= 100 and n >= 2; s1[In_1, In_2] -> [In_1, In_2] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 and n >= 2; s2[In_1, In_2] -> [In_1, In_2] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 }
+{  :  }
+[n] -> { [i0,i1] -> separate[o0] : o0 >= 2; [i0,i1] -> atomic[o0] : o0 <= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/if_then-2.c b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-2.c
new file mode 100644
index 00000000000..d1ce579aa97
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-2.c
@@ -0,0 +1,11 @@
+for (int c0 = 1; c0 <= 100; c0 += 1) {
+  if (n >= 2) {
+    s0(c0);
+    for (int c1 = 1; c1 <= 100; c1 += 1) {
+      s1(c0, c1);
+      s2(c0, c1);
+    }
+  } else
+    for (int c1 = 1; c1 <= 100; c1 += 1)
+      s2(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/if_then-2.in b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-2.in
new file mode 100644
index 00000000000..0a220ee3a54
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-2.in
@@ -0,0 +1,3 @@
+[n] -> { s0[In_1] -> [In_1,0] : In_1 >= 1 and In_1 <= 100 and n >= 2; s1[In_1, In_2] -> [In_1, In_2] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 and n >= 2; s2[In_1, In_2] -> [In_1, In_2] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 }
+{  :  }
+[n] -> { [i0,i1] -> separate[o0] : o0 >= 1; [i0,i1] -> atomic[o0] : o0 <= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/if_then-3.c b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-3.c
new file mode 100644
index 00000000000..570cbe5556f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-3.c
@@ -0,0 +1,12 @@
+if (n >= 2) {
+  for (int c0 = 1; c0 <= 100; c0 += 1) {
+    s0(c0);
+    for (int c1 = 1; c1 <= 100; c1 += 1) {
+      s1(c0, c1);
+      s2(c0, c1);
+    }
+  }
+} else
+  for (int c0 = 1; c0 <= 100; c0 += 1)
+    for (int c1 = 1; c1 <= 100; c1 += 1)
+      s2(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/if_then-3.in b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-3.in
new file mode 100644
index 00000000000..a0315f1a7b1
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-3.in
@@ -0,0 +1,3 @@
+[n] -> { s0[In_1] -> [In_1,0] : In_1 >= 1 and In_1 <= 100 and n >= 2; s1[In_1, In_2] -> [In_1, In_2] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 and n >= 2; s2[In_1, In_2] -> [In_1, In_2] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 }
+{  :  }
+[n] -> { [i0,i1] -> separate[o0] : o0 >= 0; [i0,i1] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/if_then-4.c b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-4.c
new file mode 100644
index 00000000000..fa81628ade8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-4.c
@@ -0,0 +1,7 @@
+for (int c0 = 4; c0 <= 100; c0 += 4) {
+  for (int c1 = 1; c1 <= 100; c1 += 1)
+    s0(c0, c1);
+  if (c0 >= 8 && c0 <= 96)
+    for (int c1 = 10; c1 <= 100; c1 += 1)
+      s1(c0 + 2, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/if_then-4.in b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-4.in
new file mode 100644
index 00000000000..30dc8dc24d2
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-4.in
@@ -0,0 +1,3 @@
+{ s1[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(-2 + In_1)/4]: 4e0 = -2 + In_1 and In_1 >= 10 and In_1 <= 98 and In_2 >= 10 and In_2 <= 100); s0[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(In_1)/4]: 4e0 = In_1 and In_1 >= 4 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100) }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 1; [i0, i1] -> separate[o0] : o0 >= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/if_then-5.c b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-5.c
new file mode 100644
index 00000000000..fa81628ade8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-5.c
@@ -0,0 +1,7 @@
+for (int c0 = 4; c0 <= 100; c0 += 4) {
+  for (int c1 = 1; c1 <= 100; c1 += 1)
+    s0(c0, c1);
+  if (c0 >= 8 && c0 <= 96)
+    for (int c1 = 10; c1 <= 100; c1 += 1)
+      s1(c0 + 2, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/if_then-5.in b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-5.in
new file mode 100644
index 00000000000..87a40f4733a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/if_then-5.in
@@ -0,0 +1,3 @@
+{ s1[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(-2 + In_1)/4]: 4e0 = -2 + In_1 and In_1 >= 10 and In_1 <= 98 and In_2 >= 10 and In_2 <= 100); s0[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(In_1)/4]: 4e0 = In_1 and In_1 >= 4 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100) }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter1-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/iter1-0.c
new file mode 100644
index 00000000000..c15c86b2f6b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter1-0.c
@@ -0,0 +1,2 @@
+for (int c0 = 2; c0 <= 9; c0 += 1)
+  s0(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter1-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/iter1-0.in
new file mode 100644
index 00000000000..8b876786027
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter1-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1] -> [In_1] : In_1 >= 2 and In_1 <= 9 }
+{  :  }
+{ [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter2-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/iter2-0.c
new file mode 100644
index 00000000000..5506929a106
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter2-0.c
@@ -0,0 +1,3 @@
+for (int c0 = 1; c0 <= 10; c0 += 1)
+  for (int c1 = 10; c1 <= 100; c1 += 1)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter2-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/iter2-0.in
new file mode 100644
index 00000000000..16ea0df1805
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter2-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2] -> [In_1, In_2] : In_1 >= 1 and In_1 <= 10 and In_2 >= 10 and In_2 <= 100 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter3-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/iter3-0.c
new file mode 100644
index 00000000000..43b6e3aa0f6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter3-0.c
@@ -0,0 +1,3 @@
+for (int c0 = 2; c0 <= 8; c0 += 1)
+  for (int c1 = c0 + 1; c1 <= 9; c1 += 1)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter3-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/iter3-0.in
new file mode 100644
index 00000000000..fc26493e28e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter3-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2] -> [In_1, In_2] : In_1 >= 2 and In_2 >= 1 + In_1 and In_2 <= 9 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter4-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/iter4-0.c
new file mode 100644
index 00000000000..65e37023ff3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter4-0.c
@@ -0,0 +1,3 @@
+for (int c0 = 2; c0 <= 9; c0 += 1)
+  for (int c1 = c0 + 1; c1 <= 2 * c0; c1 += 1)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter4-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/iter4-0.in
new file mode 100644
index 00000000000..94001d5251d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter4-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2] -> [In_1, In_2] : In_1 >= 2 and In_2 >= 1 + In_1 and In_2 <= 2In_1 and In_1 <= 9 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter5-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/iter5-0.c
new file mode 100644
index 00000000000..66c7c0971f8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter5-0.c
@@ -0,0 +1,3 @@
+for (int c0 = 2; c0 <= 9; c0 += 1)
+  for (int c1 = c0 + 1; c1 <= min(16, 2 * c0); c1 += 1)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter5-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/iter5-0.in
new file mode 100644
index 00000000000..1e4f739447e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter5-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2] -> [In_1, In_2] : In_1 >= 2 and In_2 >= 1 + In_1 and In_2 <= 2In_1 and In_2 <= 16 and In_1 <= 9 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter6-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/iter6-0.c
new file mode 100644
index 00000000000..a0346b69cdd
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter6-0.c
@@ -0,0 +1,3 @@
+for (int c0 = 1; c0 <= 5; c0 += 1)
+  for (int c1 = 12; c1 <= 17; c1 += 1)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter6-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/iter6-0.in
new file mode 100644
index 00000000000..1b8ed2b3c04
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter6-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2] -> [In_1, In_2] : In_1 >= 1 and In_1 <= 5 and In_2 >= 12 and In_2 <= 17 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter6-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/iter6-1.c
new file mode 100644
index 00000000000..daccd39be7a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter6-1.c
@@ -0,0 +1,2 @@
+for (int c0 = 46; c0 <= 70; c0 += 12)
+  s0(c0, (17 * c0 - 170) / 12);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter6-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/iter6-1.in
new file mode 100644
index 00000000000..4ee2ad6608d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter6-1.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2] -> [In_1, o1] : 12In_2 = -170 + 17In_1 and 12o1 = -170 + 17In_1 and In_1 >= 46 and In_1 <= 70 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter7-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/iter7-0.c
new file mode 100644
index 00000000000..2606890886e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter7-0.c
@@ -0,0 +1,2 @@
+for (int c0 = 1; c0 <= 3; c0 += 2)
+  s0(c0, (-3 * c0 + 15) / 2);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter7-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/iter7-0.in
new file mode 100644
index 00000000000..90d3efb7de0
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter7-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2] -> [In_1, o1] : 2In_2 = 15 - 3In_1 and 2o1 = 15 - 3In_1 and In_1 <= 3 and In_1 >= 1 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter8-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/iter8-0.c
new file mode 100644
index 00000000000..a33fc49b76a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter8-0.c
@@ -0,0 +1,2 @@
+for (int c0 = max(exprVar2 + 1, exprVar2 + 8 * floord(-exprVar2 + exprVar1 - 1, 8) + 9); c0 <= 16; c0 += 8)
+  s0(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter8-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/iter8-0.in
new file mode 100644
index 00000000000..682928048df
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter8-0.in
@@ -0,0 +1,3 @@
+[exprVar2, exprVar3, exprVar1] -> { s0[In_1] -> [In_1] : exists (e0 = [(-1 - exprVar2 + In_1)/8]: exprVar3 = 0 and 8e0 = -1 - exprVar2 + In_1 and exprVar1 >= 1 and In_1 >= 1 + exprVar1 and In_1 <= 16 and In_1 >= 1 + exprVar2) }
+[exprVar3, exprVar2, exprVar1] -> {  : exists (e0: exprVar3 = 0 and 8e0 >= -15 + exprVar2 and exprVar2 <= 15 and exprVar1 >= 1 and 8e0 <= exprVar2 - exprVar1) }
+[exprVar2, exprVar3, exprVar1] -> { [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter9-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/iter9-0.c
new file mode 100644
index 00000000000..9b8522ba7d8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter9-0.c
@@ -0,0 +1,11 @@
+for (int c0 = 1; c0 <= 15; c0 += 1) {
+  if (((-exprVar1 + 15) % 8) + c0 <= 15) {
+    s4(c0);
+    s0(c0);
+    s3(c0);
+    s2(c0);
+    s1(c0);
+  }
+  if (((-exprVar1 + 15) % 8) + c0 <= 15 || (c0 >= exprVar1 + 1 && (exprVar1 - c0 + 1) % 8 == 0))
+    s5(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/iter9-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/iter9-0.in
new file mode 100644
index 00000000000..9596d73727c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/iter9-0.in
@@ -0,0 +1,3 @@
+[exprVar2, exprVar1] -> { s3[In_1] -> [In_1] : exists (e0: exprVar2 = 0 and 8e0 >= -15 + exprVar1 and exprVar1 <= 15 and In_1 >= 1 and 8e0 <= exprVar1 - In_1); s4[In_1] -> [In_1] : exists (e0: exprVar2 = 0 and 8e0 >= -15 + exprVar1 and exprVar1 <= 15 and In_1 >= 1 and 8e0 <= exprVar1 - In_1); s1[In_1] -> [In_1] : exists (e0: exprVar2 = 0 and 8e0 >= -15 + exprVar1 and exprVar1 <= 15 and In_1 >= 1 and 8e0 <= exprVar1 - In_1); s5[In_1] -> [In_1] : (exists (e0: exprVar2 = 0 and 8e0 >= -15 + exprVar1 and exprVar1 <= 15 and In_1 >= 1 and 8e0 <= exprVar1 - In_1)) or (exists (e0 = [(-1 - exprVar1 + In_1)/8]: exprVar2 = 0 and 8e0 = -1 - exprVar1 + In_1 and In_1 >= 1 + exprVar1 and In_1 >= 1 and In_1 <= 15)); s0[In_1] -> [In_1] : exists (e0: exprVar2 = 0 and 8e0 >= -15 + exprVar1 and exprVar1 <= 15 and In_1 >= 1 and 8e0 <= exprVar1 - In_1); s2[In_1] -> [In_1] : exists (e0: exprVar2 = 0 and 8e0 >= -15 + exprVar1 and exprVar1 <= 15 and In_1 >= 1 and 8e0 <= exprVar1 - In_1) }
+[exprVar2, exprVar1] -> {  : exprVar2 = 0 and exprVar1 <= 15 }
+[exprVar2, exprVar1] -> { [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lefur00-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/lefur00-0.c
new file mode 100644
index 00000000000..aab26208811
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lefur00-0.c
@@ -0,0 +1,5 @@
+for (int c0 = 0; c0 <= 15; c0 += 1)
+  for (int c1 = max(2 * c0 - 15, c0 / 2); c1 <= min(15, c0 + 1); c1 += 1)
+    for (int c2 = max(max(max(1, 67 * c0 - (c0 + 1) / 3), 67 * c1 - (c1 + 2) / 3), 133 * c0 - 67 * c1 + (c0 + c1 + 1) / 3 - 66); c2 <= min(min(1000, 100 * c0 + 99), 133 * c0 - 67 * c1 + floord(c0 + c1 - 1, 3) + 133); c2 += 1)
+      for (int c3 = max(max(c2, 200 * c0 - c2), 100 * c1 + (c2 + 1) / 2); c3 <= min(min(2 * c2 + 1, 200 * c0 - c2 + 199), 100 * c1 + (c2 + 1) / 2 + 99); c3 += 1)
+        s0(c0, c1, c2, c3);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lefur00-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/lefur00-0.in
new file mode 100644
index 00000000000..cf99c811b7e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lefur00-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4] -> [In_1, In_2, In_3, In_4] : In_3 >= 1 and In_4 >= In_3 and In_4 <= 1 + 2In_3 and In_3 <= 1000 and In_4 >= 200In_1 - In_3 and In_4 <= 199 + 200In_1 - In_3 and 2In_4 >= 200In_2 + In_3 and 2In_4 <= 199 + 200In_2 + In_3 }
+{  :  }
+{ [i0, i1, i2, i3] -> atomic[o0] : o0 <= 2; [i0, i1, i2, i3] -> separate[o0] : o0 >= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lefur01-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/lefur01-0.c
new file mode 100644
index 00000000000..aab26208811
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lefur01-0.c
@@ -0,0 +1,5 @@
+for (int c0 = 0; c0 <= 15; c0 += 1)
+  for (int c1 = max(2 * c0 - 15, c0 / 2); c1 <= min(15, c0 + 1); c1 += 1)
+    for (int c2 = max(max(max(1, 67 * c0 - (c0 + 1) / 3), 67 * c1 - (c1 + 2) / 3), 133 * c0 - 67 * c1 + (c0 + c1 + 1) / 3 - 66); c2 <= min(min(1000, 100 * c0 + 99), 133 * c0 - 67 * c1 + floord(c0 + c1 - 1, 3) + 133); c2 += 1)
+      for (int c3 = max(max(c2, 200 * c0 - c2), 100 * c1 + (c2 + 1) / 2); c3 <= min(min(2 * c2 + 1, 200 * c0 - c2 + 199), 100 * c1 + (c2 + 1) / 2 + 99); c3 += 1)
+        s0(c0, c1, c2, c3);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lefur01-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/lefur01-0.in
new file mode 100644
index 00000000000..cf99c811b7e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lefur01-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4] -> [In_1, In_2, In_3, In_4] : In_3 >= 1 and In_4 >= In_3 and In_4 <= 1 + 2In_3 and In_3 <= 1000 and In_4 >= 200In_1 - In_3 and In_4 <= 199 + 200In_1 - In_3 and 2In_4 >= 200In_2 + In_3 and 2In_4 <= 199 + 200In_2 + In_3 }
+{  :  }
+{ [i0, i1, i2, i3] -> atomic[o0] : o0 <= 2; [i0, i1, i2, i3] -> separate[o0] : o0 >= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lefur01-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/lefur01-1.c
new file mode 100644
index 00000000000..aab26208811
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lefur01-1.c
@@ -0,0 +1,5 @@
+for (int c0 = 0; c0 <= 15; c0 += 1)
+  for (int c1 = max(2 * c0 - 15, c0 / 2); c1 <= min(15, c0 + 1); c1 += 1)
+    for (int c2 = max(max(max(1, 67 * c0 - (c0 + 1) / 3), 67 * c1 - (c1 + 2) / 3), 133 * c0 - 67 * c1 + (c0 + c1 + 1) / 3 - 66); c2 <= min(min(1000, 100 * c0 + 99), 133 * c0 - 67 * c1 + floord(c0 + c1 - 1, 3) + 133); c2 += 1)
+      for (int c3 = max(max(c2, 200 * c0 - c2), 100 * c1 + (c2 + 1) / 2); c3 <= min(min(2 * c2 + 1, 200 * c0 - c2 + 199), 100 * c1 + (c2 + 1) / 2 + 99); c3 += 1)
+        s0(c0, c1, c2, c3);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lefur01-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/lefur01-1.in
new file mode 100644
index 00000000000..bc9611db469
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lefur01-1.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4] -> [In_1, In_2, In_3, In_4] : In_3 >= 1 and In_4 >= In_3 and In_4 <= 1 + 2In_3 and In_3 <= 1000 and In_4 >= 200In_1 - In_3 and In_4 <= 199 + 200In_1 - In_3 and 2In_4 >= 200In_2 + In_3 and 2In_4 <= 199 + 200In_2 + In_3 }
+{  :  }
+{ [i0, i1, i2, i3] -> atomic[o0] : o0 <= 1; [i0, i1, i2, i3] -> separate[o0] : o0 >= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lefur03-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/lefur03-0.c
new file mode 100644
index 00000000000..05ac73f3a44
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lefur03-0.c
@@ -0,0 +1,7 @@
+for (int c0 = 0; c0 <= 3; c0 += 1)
+  for (int c1 = max(2 * c0 - 3, c0 / 2); c1 <= min(3, c0 + 1); c1 += 1)
+    for (int c2 = c0; c2 <= min(min(3, 2 * c0 - c1 + 1), 3 * c1 + 2); c2 += 1)
+      for (int c3 = max(max(max(0, c1 - (-c1 + 3) / 3), c0 - (-c2 + 3) / 3), c2 + floord(3 * c1 - c2 - 1, 6)); c3 <= min(3, c0 + 1); c3 += 1)
+        for (int c4 = max(max(max(max(-200 * c1 + 400 * c3 - 199, 250 * c3 + 1), 667 * c0 - 333 * c1 - (c0 + c1 + 3) / 3 - 332), 333 * c1 + c1 / 3), 333 * c2 + (c2 + 1) / 3); c4 <= min(min(min(min(1000, 500 * c0 + 499), -200 * c1 + 400 * c3 + 400), 333 * c2 - (-c2 + 3) / 3 + 333), 333 * c3 - (-c3 + 3) / 3 + 334); c4 += 1)
+          for (int c5 = max(max(max(c4, 1000 * c0 - c4), 1000 * c3 - 2 * c4 + 2), 500 * c1 + (c4 + 1) / 2); c5 <= min(min(min(2 * c4 + 1, 1000 * c0 - c4 + 999), 1000 * c3 - 2 * c4 + 1001), 500 * c1 + (c4 + 1) / 2 + 499); c5 += 1)
+            s0(c0, c1, c2, c3, c4, c5);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lefur03-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/lefur03-0.in
new file mode 100644
index 00000000000..890f413e743
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lefur03-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4, In_5, In_6] -> [In_1, In_2, In_3, In_4, In_5, In_6] : In_6 >= In_5 and In_6 <= 1 + 2In_5 and In_5 <= 1000 and In_6 >= 1000In_1 - In_5 and In_6 <= 999 + 1000In_1 - In_5 and In_6 >= 2 + 1000In_4 - 2In_5 and In_6 <= 1001 + 1000In_4 - 2In_5 and In_4 >= 0 and 2In_6 >= 1000In_2 + In_5 and 2In_6 <= 999 + 1000In_2 + In_5 and 3In_5 >= -1 + 1000In_3 and 3In_5 <= 998 + 1000In_3 }
+{  :  }
+{ [i0, i1, i2, i3, i4, i5] -> separate[o0] : o0 >= 5; [i0, i1, i2, i3, i4, i5] -> atomic[o0] : o0 <= 4 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lefur04-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/lefur04-0.c
new file mode 100644
index 00000000000..124d0f15cb5
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lefur04-0.c
@@ -0,0 +1,8 @@
+for (int c0 = 0; c0 <= 3; c0 += 1)
+  for (int c1 = max(2 * c0 - 3, c0 / 2); c1 <= min(3, c0 + 1); c1 += 1)
+    for (int c2 = c0; c2 <= min(min(3, 2 * c0 - c1 + 1), 3 * c1 + 2); c2 += 1)
+      for (int c3 = max(max(max(c1 - (-c1 + 3) / 3, c0 - (-c2 + 3) / 3), c2 - (c2 + 2) / 3), c2 + floord(3 * c1 - c2 - 1, 6)); c3 <= min(3, c0 + c2 / 3 + 1); c3 += 1)
+        for (int c5 = max(max(max(max(0, 2 * c3 - 4), c1 - (-c1 + 3) / 3), c2 - (c2 + 3) / 3), c3 - (c3 + 3) / 3); c5 <= min(min(c1 + 1, c3), -c2 + 2 * c3 - (c2 + 3) / 3 + 2); c5 += 1)
+          for (int c6 = max(max(max(max(max(-200 * c1 + 400 * c3 - 199, 250 * c3 + 1), 1000 * c0 - 500 * c5 - 501), 667 * c0 - 333 * c1 - (c0 + c1 + 3) / 3 - 332), 333 * c1 + c1 / 3), 333 * c2 + (c2 + 1) / 3); c6 <= min(min(min(min(min(min(1000, 500 * c0 + 499), -200 * c1 + 400 * c3 + 400), 500 * c5 + 501), 1000 * c0 - 500 * c5 + 997), 333 * c2 - (-c2 + 3) / 3 + 333), 333 * c3 - (-c3 + 3) / 3 + 334); c6 += 1)
+            for (int c7 = max(max(max(max(500 * c5 + 2, c6), 1000 * c0 - c6), 1000 * c3 - 2 * c6 + 2), 500 * c1 + (c6 + 1) / 2); c7 <= min(min(min(min(500 * c5 + 501, 2 * c6 + 1), 1000 * c0 - c6 + 999), 1000 * c3 - 2 * c6 + 1001), 500 * c1 + (c6 + 1) / 2 + 499); c7 += 1)
+              s0(c0, c1, c2, c3, c2 / 3, c5, c6, c7);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lefur04-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/lefur04-0.in
new file mode 100644
index 00000000000..c50a0da4c51
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lefur04-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4, In_5, In_6, In_7, In_8] -> [In_1, In_2, In_3, In_4, In_5, In_6, In_7, In_8] : In_7 >= 1000In_5 and In_8 >= In_7 and In_8 <= 501 + 500In_6 and In_8 <= 1 + 2In_7 and In_7 <= 999 + 1000In_5 and In_7 <= 1000 and In_8 >= 1000In_1 - In_7 and In_8 <= 999 + 1000In_1 - In_7 and 2In_8 >= 1000In_2 + In_7 and 2In_8 <= 999 + 1000In_2 + In_7 and 3In_7 >= -1 + 1000In_3 and 3In_7 <= 998 + 1000In_3 and In_8 >= 2 + 500In_6 and In_6 >= 0 and In_8 >= 2 + 1000In_4 - 2In_7 and In_8 <= 1001 + 1000In_4 - 2In_7 }
+{  :  }
+{ [i0, i1, i2, i3, i4, i5, i6, i7] -> atomic[o0] : o0 <= 6; [i0, i1, i2, i3, i4, i5, i6, i7] -> separate[o0] : o0 >= 7 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift1-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-0.c
new file mode 100644
index 00000000000..f7b1ae9ae7b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-0.c
@@ -0,0 +1,9 @@
+for (int c0 = 1; c0 <= 100; c0 += 1)
+  for (int c1 = 1; c1 <= 100; c1 += 1)
+    for (int c2 = 1; c2 <= 100; c2 += 1)
+      for (int c3 = 1; c3 <= 100; c3 += 1)
+        for (int c4 = 1; c4 <= 100; c4 += 1) {
+          s1(c0, c1, c2, c3, c4);
+          if (c0 <= 60)
+            s0(c0, c1, c2, c3, c4);
+        }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift1-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-0.in
new file mode 100644
index 00000000000..31ae11ad533
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 60 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100; s1[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100 }
+{  :  }
+{ [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 5; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= 4 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift1-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-1.c
new file mode 100644
index 00000000000..3910e4f264d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-1.c
@@ -0,0 +1,13 @@
+for (int c0 = 1; c0 <= 100; c0 += 1)
+  for (int c1 = 1; c1 <= 100; c1 += 1)
+    for (int c2 = 1; c2 <= 100; c2 += 1)
+      for (int c3 = 1; c3 <= 100; c3 += 1) {
+        if (c0 >= 61) {
+          for (int c4 = 1; c4 <= 100; c4 += 1)
+            s1(c0, c1, c2, c3, c4);
+        } else
+          for (int c4 = 1; c4 <= 100; c4 += 1) {
+            s1(c0, c1, c2, c3, c4);
+            s0(c0, c1, c2, c3, c4);
+          }
+      }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift1-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-1.in
new file mode 100644
index 00000000000..44de4acbb94
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-1.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 60 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100; s1[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100 }
+{  :  }
+{ [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 4; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift1-2.c b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-2.c
new file mode 100644
index 00000000000..8737f3c50bf
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-2.c
@@ -0,0 +1,14 @@
+for (int c0 = 1; c0 <= 100; c0 += 1)
+  for (int c1 = 1; c1 <= 100; c1 += 1)
+    for (int c2 = 1; c2 <= 100; c2 += 1) {
+      if (c0 >= 61) {
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1)
+            s1(c0, c1, c2, c3, c4);
+      } else
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1) {
+            s1(c0, c1, c2, c3, c4);
+            s0(c0, c1, c2, c3, c4);
+          }
+    }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift1-2.in b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-2.in
new file mode 100644
index 00000000000..241fbb96293
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-2.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 60 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100; s1[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100 }
+{  :  }
+{ [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 3; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift1-3.c b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-3.c
new file mode 100644
index 00000000000..4d73f89d63d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-3.c
@@ -0,0 +1,15 @@
+for (int c0 = 1; c0 <= 100; c0 += 1)
+  for (int c1 = 1; c1 <= 100; c1 += 1) {
+    if (c0 >= 61) {
+      for (int c2 = 1; c2 <= 100; c2 += 1)
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1)
+            s1(c0, c1, c2, c3, c4);
+    } else
+      for (int c2 = 1; c2 <= 100; c2 += 1)
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1) {
+            s1(c0, c1, c2, c3, c4);
+            s0(c0, c1, c2, c3, c4);
+          }
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift1-3.in b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-3.in
new file mode 100644
index 00000000000..a50a9ece72c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-3.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 60 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100; s1[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100 }
+{  :  }
+{ [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 2; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift1-4.c b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-4.c
new file mode 100644
index 00000000000..7d2de29261c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-4.c
@@ -0,0 +1,16 @@
+for (int c0 = 1; c0 <= 100; c0 += 1) {
+  if (c0 >= 61) {
+    for (int c1 = 1; c1 <= 100; c1 += 1)
+      for (int c2 = 1; c2 <= 100; c2 += 1)
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1)
+            s1(c0, c1, c2, c3, c4);
+  } else
+    for (int c1 = 1; c1 <= 100; c1 += 1)
+      for (int c2 = 1; c2 <= 100; c2 += 1)
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1) {
+            s1(c0, c1, c2, c3, c4);
+            s0(c0, c1, c2, c3, c4);
+          }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift1-4.in b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-4.in
new file mode 100644
index 00000000000..db7d85c8eed
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-4.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 60 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100; s1[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100 }
+{  :  }
+{ [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 1; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift1-5.c b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-5.c
new file mode 100644
index 00000000000..2a24f683e66
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-5.c
@@ -0,0 +1,16 @@
+{
+  for (int c0 = 1; c0 <= 60; c0 += 1)
+    for (int c1 = 1; c1 <= 100; c1 += 1)
+      for (int c2 = 1; c2 <= 100; c2 += 1)
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1) {
+            s1(c0, c1, c2, c3, c4);
+            s0(c0, c1, c2, c3, c4);
+          }
+  for (int c0 = 61; c0 <= 100; c0 += 1)
+    for (int c1 = 1; c1 <= 100; c1 += 1)
+      for (int c2 = 1; c2 <= 100; c2 += 1)
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1)
+            s1(c0, c1, c2, c3, c4);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift1-5.in b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-5.in
new file mode 100644
index 00000000000..2f9d5786569
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift1-5.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 60 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100; s1[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100 }
+{  :  }
+{ [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 0; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift2-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-0.c
new file mode 100644
index 00000000000..181d6e58fc3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-0.c
@@ -0,0 +1,9 @@
+for (int c0 = 1; c0 <= 100; c0 += 1)
+  for (int c1 = 1; c1 <= 100; c1 += 1)
+    for (int c2 = 1; c2 <= 100; c2 += 1)
+      for (int c3 = 1; c3 <= 100; c3 += 1)
+        for (int c4 = 1; c4 <= 100; c4 += 1) {
+          s1(c0, c1, c2, c3, c4);
+          if (c0 >= 5 && c0 <= 60)
+            s0(c0, c1, c2, c3, c4);
+        }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift2-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-0.in
new file mode 100644
index 00000000000..5df0444130e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 5 and In_1 <= 60 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100; s1[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100 }
+{  :  }
+{ [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 5; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= 4 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift2-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-1.c
new file mode 100644
index 00000000000..f914743ecc2
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-1.c
@@ -0,0 +1,16 @@
+for (int c0 = 1; c0 <= 100; c0 += 1)
+  for (int c1 = 1; c1 <= 100; c1 += 1)
+    for (int c2 = 1; c2 <= 100; c2 += 1)
+      for (int c3 = 1; c3 <= 100; c3 += 1) {
+        if (c0 >= 61) {
+          for (int c4 = 1; c4 <= 100; c4 += 1)
+            s1(c0, c1, c2, c3, c4);
+        } else if (c0 <= 4) {
+          for (int c4 = 1; c4 <= 100; c4 += 1)
+            s1(c0, c1, c2, c3, c4);
+        } else
+          for (int c4 = 1; c4 <= 100; c4 += 1) {
+            s1(c0, c1, c2, c3, c4);
+            s0(c0, c1, c2, c3, c4);
+          }
+      }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift2-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-1.in
new file mode 100644
index 00000000000..7ef2a8d9a9e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-1.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 5 and In_1 <= 60 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100; s1[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100 }
+{  :  }
+{ [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 4; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift2-2.c b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-2.c
new file mode 100644
index 00000000000..8b6dbafc507
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-2.c
@@ -0,0 +1,18 @@
+for (int c0 = 1; c0 <= 100; c0 += 1)
+  for (int c1 = 1; c1 <= 100; c1 += 1)
+    for (int c2 = 1; c2 <= 100; c2 += 1) {
+      if (c0 >= 61) {
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1)
+            s1(c0, c1, c2, c3, c4);
+      } else if (c0 <= 4) {
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1)
+            s1(c0, c1, c2, c3, c4);
+      } else
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1) {
+            s1(c0, c1, c2, c3, c4);
+            s0(c0, c1, c2, c3, c4);
+          }
+    }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift2-2.in b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-2.in
new file mode 100644
index 00000000000..3e9a9782460
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-2.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 5 and In_1 <= 60 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100; s1[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100 }
+{  :  }
+{ [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 3; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift2-3.c b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-3.c
new file mode 100644
index 00000000000..1a5954a8081
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-3.c
@@ -0,0 +1,20 @@
+for (int c0 = 1; c0 <= 100; c0 += 1)
+  for (int c1 = 1; c1 <= 100; c1 += 1) {
+    if (c0 >= 61) {
+      for (int c2 = 1; c2 <= 100; c2 += 1)
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1)
+            s1(c0, c1, c2, c3, c4);
+    } else if (c0 <= 4) {
+      for (int c2 = 1; c2 <= 100; c2 += 1)
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1)
+            s1(c0, c1, c2, c3, c4);
+    } else
+      for (int c2 = 1; c2 <= 100; c2 += 1)
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1) {
+            s1(c0, c1, c2, c3, c4);
+            s0(c0, c1, c2, c3, c4);
+          }
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift2-3.in b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-3.in
new file mode 100644
index 00000000000..0580f079dbc
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-3.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 5 and In_1 <= 60 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100; s1[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100 }
+{  :  }
+{ [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 2; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift2-4.c b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-4.c
new file mode 100644
index 00000000000..d1d04d6507d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-4.c
@@ -0,0 +1,22 @@
+for (int c0 = 1; c0 <= 100; c0 += 1) {
+  if (c0 >= 61) {
+    for (int c1 = 1; c1 <= 100; c1 += 1)
+      for (int c2 = 1; c2 <= 100; c2 += 1)
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1)
+            s1(c0, c1, c2, c3, c4);
+  } else if (c0 <= 4) {
+    for (int c1 = 1; c1 <= 100; c1 += 1)
+      for (int c2 = 1; c2 <= 100; c2 += 1)
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1)
+            s1(c0, c1, c2, c3, c4);
+  } else
+    for (int c1 = 1; c1 <= 100; c1 += 1)
+      for (int c2 = 1; c2 <= 100; c2 += 1)
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1) {
+            s1(c0, c1, c2, c3, c4);
+            s0(c0, c1, c2, c3, c4);
+          }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift2-4.in b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-4.in
new file mode 100644
index 00000000000..d64462ae11b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-4.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 5 and In_1 <= 60 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100; s1[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100 }
+{  :  }
+{ [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 1; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift2-5.c b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-5.c
new file mode 100644
index 00000000000..16bd6dbafd7
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-5.c
@@ -0,0 +1,22 @@
+{
+  for (int c0 = 1; c0 <= 4; c0 += 1)
+    for (int c1 = 1; c1 <= 100; c1 += 1)
+      for (int c2 = 1; c2 <= 100; c2 += 1)
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1)
+            s1(c0, c1, c2, c3, c4);
+  for (int c0 = 5; c0 <= 60; c0 += 1)
+    for (int c1 = 1; c1 <= 100; c1 += 1)
+      for (int c2 = 1; c2 <= 100; c2 += 1)
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1) {
+            s1(c0, c1, c2, c3, c4);
+            s0(c0, c1, c2, c3, c4);
+          }
+  for (int c0 = 61; c0 <= 100; c0 += 1)
+    for (int c1 = 1; c1 <= 100; c1 += 1)
+      for (int c2 = 1; c2 <= 100; c2 += 1)
+        for (int c3 = 1; c3 <= 100; c3 += 1)
+          for (int c4 = 1; c4 <= 100; c4 += 1)
+            s1(c0, c1, c2, c3, c4);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lift2-5.in b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-5.in
new file mode 100644
index 00000000000..37a57befecd
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lift2-5.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 5 and In_1 <= 60 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100; s1[In_1, In_2, In_3, In_4, In_5] -> [In_1, In_2, In_3, In_4, In_5] : In_1 >= 1 and In_1 <= 100 and In_2 >= 1 and In_2 <= 100 and In_3 >= 1 and In_3 <= 100 and In_4 >= 1 and In_4 <= 100 and In_5 >= 1 and In_5 <= 100 }
+{  :  }
+{ [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 0; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/lu-0.c
new file mode 100644
index 00000000000..e71f47bbd58
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu-0.c
@@ -0,0 +1,10 @@
+for (int c0 = 1; c0 < n; c0 += 64)
+  for (int c1 = c0 - 1; c1 <= n; c1 += 64)
+    for (int c2 = c0; c2 <= n; c2 += 1) {
+      for (int c3 = c0; c3 <= min(min(c0 + 63, c1 + 62), c2 - 1); c3 += 1)
+        for (int c4 = max(c1, c3 + 1); c4 <= min(n, c1 + 63); c4 += 1)
+          s1(c3, c4, c2);
+      if (c0 + 63 >= c2)
+        for (int c4 = max(c1, c2 + 1); c4 <= min(n, c1 + 63); c4 += 1)
+          s0(c2, c4);
+    }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/lu-0.in
new file mode 100644
index 00000000000..510d677c350
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu-0.in
@@ -0,0 +1,3 @@
+[n] -> { s1[k, i, j] -> [t1, t2, j, k, i] : exists (e0 = [(-1 + t1)/64], e1 = [(t2)/64]: 64e0 = -1 + t1 and 64e1 = t2 and t1 >= -63 + k and t1 <= k and t2 >= -63 + i and t2 <= i and i >= 1 + k and j >= 1 + k and k >= 1 and i <= n and j <= n); s0[k, i] -> [t1, t2, k, k, i] : exists (e0 = [(-1 + t1)/64], e1 = [(t2)/64]: 64e0 = -1 + t1 and 64e1 = t2 and t1 >= -63 + k and t1 <= k and t2 >= -63 + i and t2 <= i and k >= 1 and i >= 1 + k and i <= n) }
+{  :  }
+[n] -> { [t1, t2, i2, i3, i4] -> separate[o0] : o0 >= 5; [t1, t2, i2, i3, i4] -> atomic[o0] : o0 <= 4 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/lu-1.c
new file mode 100644
index 00000000000..e71f47bbd58
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu-1.c
@@ -0,0 +1,10 @@
+for (int c0 = 1; c0 < n; c0 += 64)
+  for (int c1 = c0 - 1; c1 <= n; c1 += 64)
+    for (int c2 = c0; c2 <= n; c2 += 1) {
+      for (int c3 = c0; c3 <= min(min(c0 + 63, c1 + 62), c2 - 1); c3 += 1)
+        for (int c4 = max(c1, c3 + 1); c4 <= min(n, c1 + 63); c4 += 1)
+          s1(c3, c4, c2);
+      if (c0 + 63 >= c2)
+        for (int c4 = max(c1, c2 + 1); c4 <= min(n, c1 + 63); c4 += 1)
+          s0(c2, c4);
+    }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/lu-1.in
new file mode 100644
index 00000000000..2e8f7efa535
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu-1.in
@@ -0,0 +1,3 @@
+[n] -> { s1[k, i, j] -> [t1, t2, j, k, i] : exists (e0 = [(-1 + t1)/64], e1 = [(t2)/64]: 64e0 = -1 + t1 and 64e1 = t2 and t1 >= -63 + k and t1 <= k and t2 >= -63 + i and t2 <= i and i >= 1 + k and j >= 1 + k and k >= 1 and i <= n and j <= n); s0[k, i] -> [t1, t2, k, k, i] : exists (e0 = [(-1 + t1)/64], e1 = [(t2)/64]: 64e0 = -1 + t1 and 64e1 = t2 and t1 >= -63 + k and t1 <= k and t2 >= -63 + i and t2 <= i and k >= 1 and i >= 1 + k and i <= n) }
+{  :  }
+[n] -> { [t1, t2, i2, i3, i4] -> separate[o0] : o0 >= 4; [t1, t2, i2, i3, i4] -> atomic[o0] : o0 <= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu-2.c b/polly/lib/External/isl/test_inputs/codegen/omega/lu-2.c
new file mode 100644
index 00000000000..e71f47bbd58
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu-2.c
@@ -0,0 +1,10 @@
+for (int c0 = 1; c0 < n; c0 += 64)
+  for (int c1 = c0 - 1; c1 <= n; c1 += 64)
+    for (int c2 = c0; c2 <= n; c2 += 1) {
+      for (int c3 = c0; c3 <= min(min(c0 + 63, c1 + 62), c2 - 1); c3 += 1)
+        for (int c4 = max(c1, c3 + 1); c4 <= min(n, c1 + 63); c4 += 1)
+          s1(c3, c4, c2);
+      if (c0 + 63 >= c2)
+        for (int c4 = max(c1, c2 + 1); c4 <= min(n, c1 + 63); c4 += 1)
+          s0(c2, c4);
+    }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu-2.in b/polly/lib/External/isl/test_inputs/codegen/omega/lu-2.in
new file mode 100644
index 00000000000..ab2ec412b06
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu-2.in
@@ -0,0 +1,3 @@
+[n] -> { s1[k, i, j] -> [t1, t2, j, k, i] : exists (e0 = [(-1 + t1)/64], e1 = [(t2)/64]: 64e0 = -1 + t1 and 64e1 = t2 and t1 >= -63 + k and t1 <= k and t2 >= -63 + i and t2 <= i and i >= 1 + k and j >= 1 + k and k >= 1 and i <= n and j <= n); s0[k, i] -> [t1, t2, k, k, i] : exists (e0 = [(-1 + t1)/64], e1 = [(t2)/64]: 64e0 = -1 + t1 and 64e1 = t2 and t1 >= -63 + k and t1 <= k and t2 >= -63 + i and t2 <= i and k >= 1 and i >= 1 + k and i <= n) }
+{  :  }
+[n] -> { [t1, t2, i2, i3, i4] -> separate[o0] : o0 >= 3; [t1, t2, i2, i3, i4] -> atomic[o0] : o0 <= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu-3.c b/polly/lib/External/isl/test_inputs/codegen/omega/lu-3.c
new file mode 100644
index 00000000000..3fd0d4eb543
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu-3.c
@@ -0,0 +1,14 @@
+for (int c0 = 1; c0 < n; c0 += 64)
+  for (int c1 = c0 - 1; c1 <= n; c1 += 64) {
+    for (int c2 = c0; c2 <= min(n, c0 + 63); c2 += 1) {
+      for (int c3 = c0; c3 <= min(c1 + 62, c2 - 1); c3 += 1)
+        for (int c4 = max(c1, c3 + 1); c4 <= min(n, c1 + 63); c4 += 1)
+          s1(c3, c4, c2);
+      for (int c4 = max(c1, c2 + 1); c4 <= min(n, c1 + 63); c4 += 1)
+        s0(c2, c4);
+    }
+    for (int c2 = c0 + 64; c2 <= n; c2 += 1)
+      for (int c3 = c0; c3 <= min(c0 + 63, c1 + 62); c3 += 1)
+        for (int c4 = max(c1, c3 + 1); c4 <= min(n, c1 + 63); c4 += 1)
+          s1(c3, c4, c2);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu-3.in b/polly/lib/External/isl/test_inputs/codegen/omega/lu-3.in
new file mode 100644
index 00000000000..817db46120a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu-3.in
@@ -0,0 +1,3 @@
+[n] -> { s1[k, i, j] -> [t1, t2, j, k, i] : exists (e0 = [(-1 + t1)/64], e1 = [(t2)/64]: 64e0 = -1 + t1 and 64e1 = t2 and t1 >= -63 + k and t1 <= k and t2 >= -63 + i and t2 <= i and i >= 1 + k and j >= 1 + k and k >= 1 and i <= n and j <= n); s0[k, i] -> [t1, t2, k, k, i] : exists (e0 = [(-1 + t1)/64], e1 = [(t2)/64]: 64e0 = -1 + t1 and 64e1 = t2 and t1 >= -63 + k and t1 <= k and t2 >= -63 + i and t2 <= i and k >= 1 and i >= 1 + k and i <= n) }
+{  :  }
+[n] -> { [t1, t2, i2, i3, i4] -> separate[o0] : o0 >= 2; [t1, t2, i2, i3, i4] -> atomic[o0] : o0 <= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-0.c
new file mode 100644
index 00000000000..da2571ac530
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-0.c
@@ -0,0 +1,7 @@
+for (int c0 = 1; c0 <= n; c0 += 1)
+  for (int c1 = 2; c1 <= n; c1 += 1) {
+    for (int c3 = 1; c3 < min(c0, c1); c3 += 1)
+      s1(c3, c1, c0);
+    if (c1 >= c0 + 1)
+      s0(c0, c1);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-0.in
new file mode 100644
index 00000000000..5aae58cfb57
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-0.in
@@ -0,0 +1,3 @@
+[n] -> { s0[k, j] -> [k, j, 1, 0] : k >= 1 and j >= 1 + k and j <= n; s1[k, j, i] -> [i, j, 0, k] : j >= 1 + k and i >= 1 + k and k >= 1 and j <= n and i <= n }
+{  :  }
+[n] -> { [i0, i1, i2, i3] -> atomic[o0] : o0 <= 2; [i0, i1, i2, i3] -> separate[o0] : o0 >= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-1.c
new file mode 100644
index 00000000000..da2571ac530
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-1.c
@@ -0,0 +1,7 @@
+for (int c0 = 1; c0 <= n; c0 += 1)
+  for (int c1 = 2; c1 <= n; c1 += 1) {
+    for (int c3 = 1; c3 < min(c0, c1); c3 += 1)
+      s1(c3, c1, c0);
+    if (c1 >= c0 + 1)
+      s0(c0, c1);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-1.in
new file mode 100644
index 00000000000..06a860c43fe
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-1.in
@@ -0,0 +1,3 @@
+[n] -> { s0[k, j] -> [k, j, 1, 0] : k >= 1 and j >= 1 + k and j <= n; s1[k, j, i] -> [i, j, 0, k] : j >= 1 + k and i >= 1 + k and k >= 1 and j <= n and i <= n }
+{  :  }
+[n] -> { [i0, i1, i2, i3] -> atomic[o0] : o0 <= 1; [i0, i1, i2, i3] -> separate[o0] : o0 >= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-2.c b/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-2.c
new file mode 100644
index 00000000000..561ae1104b5
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-2.c
@@ -0,0 +1,11 @@
+if (n >= 2)
+  for (int c0 = 1; c0 <= n; c0 += 1) {
+    for (int c1 = 2; c1 <= c0; c1 += 1)
+      for (int c3 = 1; c3 < c1; c3 += 1)
+        s1(c3, c1, c0);
+    for (int c1 = c0 + 1; c1 <= n; c1 += 1) {
+      for (int c3 = 1; c3 < c0; c3 += 1)
+        s1(c3, c1, c0);
+      s0(c0, c1);
+    }
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-2.in b/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-2.in
new file mode 100644
index 00000000000..6ad8a4497e4
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu_ijk-2.in
@@ -0,0 +1,3 @@
+[n] -> { s0[k, j] -> [k, j, 1, 0] : k >= 1 and j >= 1 + k and j <= n; s1[k, j, i] -> [i, j, 0, k] : j >= 1 + k and i >= 1 + k and k >= 1 and j <= n and i <= n }
+{  :  }
+[n] -> { [i0, i1, i2, i3] -> atomic[o0] : o0 <= 0; [i0, i1, i2, i3] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu_spmd-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/lu_spmd-0.c
new file mode 100644
index 00000000000..4c8c72ad594
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu_spmd-0.c
@@ -0,0 +1,12 @@
+if (ub >= lb)
+  for (int c0 = 1; c0 <= ub; c0 += 1)
+    for (int c1 = c0; c1 <= n; c1 += 1) {
+      if (c0 >= lb && c1 >= c0 + 1) {
+        s0(c0, c1);
+        if (n >= ub + 1)
+          s2(c0, c1);
+      } else if (lb >= c0 + 1)
+        s3(c0, c1, lb, c0, c1);
+      for (int c3 = max(lb, c0); c3 <= ub; c3 += 1)
+        s1(c0, c1, c3);
+    }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu_spmd-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/lu_spmd-0.in
new file mode 100644
index 00000000000..0dd09538192
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu_spmd-0.in
@@ -0,0 +1,3 @@
+[n, lb, ub] -> { s1[k, i, j] -> [k, i, 1, j, 0, 0, 0, 0] : k >= 1 and j >= k and j <= n and j <= ub and i >= k and i <= n and j >= lb; s3[k, i, lb, k, i] -> [k, i, 1, lb, -1, k, i, 0] : k >= 1 and k <= -1 + lb and lb <= n and ub >= lb and i >= k and i <= n; s0[k, i] -> [k, i, 0, 0, 0, 0, 0, 0] : k >= 1 and k >= lb and i >= 1 + k and i <= n and k <= ub; s2[k, i] -> [k, i, 0, 0, 1, 0, 0, 0] : k >= 1 and k >= lb and k <= ub and ub <= -1 + n and i >= 1 + k and i <= n }
+[lb, n, ub] -> {  : ub <= n and lb >= 1 }
+[n, lb, ub] -> { [i0, i1, i2, i3, i4, i5, i6, i7] -> atomic[o0] : o0 <= 7; [i0, i1, i2, i3, i4, i5, i6, i7] -> separate[o0] : o0 >= 8 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu_spmd-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/lu_spmd-1.c
new file mode 100644
index 00000000000..4c8c72ad594
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu_spmd-1.c
@@ -0,0 +1,12 @@
+if (ub >= lb)
+  for (int c0 = 1; c0 <= ub; c0 += 1)
+    for (int c1 = c0; c1 <= n; c1 += 1) {
+      if (c0 >= lb && c1 >= c0 + 1) {
+        s0(c0, c1);
+        if (n >= ub + 1)
+          s2(c0, c1);
+      } else if (lb >= c0 + 1)
+        s3(c0, c1, lb, c0, c1);
+      for (int c3 = max(lb, c0); c3 <= ub; c3 += 1)
+        s1(c0, c1, c3);
+    }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/lu_spmd-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/lu_spmd-1.in
new file mode 100644
index 00000000000..5d628befe3e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/lu_spmd-1.in
@@ -0,0 +1,3 @@
+[n, lb, ub] -> { s1[k, i, j] -> [k, i, 1, j, 0, 0, 0, 0] : k >= 1 and j >= k and j <= n and j <= ub and i >= k and i <= n and j >= lb; s3[k, i, lb, k, i] -> [k, i, 1, lb, -1, k, i, 0] : k >= 1 and k <= -1 + lb and lb <= n and ub >= lb and i >= k and i <= n; s0[k, i] -> [k, i, 0, 0, 0, 0, 0, 0] : k >= 1 and k >= lb and i >= 1 + k and i <= n and k <= ub; s2[k, i] -> [k, i, 0, 0, 1, 0, 0, 0] : k >= 1 and k >= lb and k <= ub and ub <= -1 + n and i >= 1 + k and i <= n }
+[lb, n, ub] -> {  : ub <= n and lb >= 1 }
+[n, lb, ub] -> { [i0, i1, i2, i3, i4, i5, i6, i7] -> atomic[o0] : o0 <= 6; [i0, i1, i2, i3, i4, i5, i6, i7] -> separate[o0] : o0 >= 7 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m1-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/m1-0.c
new file mode 100644
index 00000000000..bb149359f93
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m1-0.c
@@ -0,0 +1,6 @@
+for (int c0 = 1; c0 <= 9; c0 += 1)
+  for (int c1 = 1; c1 <= 9; c1 += 1) {
+    s0(c0, c1);
+    if (c0 == 5)
+      s1(5, c1);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m1-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/m1-0.in
new file mode 100644
index 00000000000..ca2b8a9b2c0
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m1-0.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [i, j, 0] : i >= 1 and i <= 9 and j >= 1 and j <= 9; s1[5, j] -> [5, j, 1] : j >= 1 and j <= 9 }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 2; [i0, i1, i2] -> atomic[o0] : o0 <= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m1-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/m1-1.c
new file mode 100644
index 00000000000..07b91f794e0
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m1-1.c
@@ -0,0 +1,13 @@
+for (int c0 = 1; c0 <= 9; c0 += 1) {
+  if (c0 >= 6) {
+    for (int c1 = 1; c1 <= 9; c1 += 1)
+      s0(c0, c1);
+  } else if (c0 <= 4) {
+    for (int c1 = 1; c1 <= 9; c1 += 1)
+      s0(c0, c1);
+  } else
+    for (int c1 = 1; c1 <= 9; c1 += 1) {
+      s0(5, c1);
+      s1(5, c1);
+    }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m1-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/m1-1.in
new file mode 100644
index 00000000000..08dbf45ed10
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m1-1.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [i, j, 0] : i >= 1 and i <= 9 and j >= 1 and j <= 9; s1[5, j] -> [5, j, 1] : j >= 1 and j <= 9 }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 1; [i0, i1, i2] -> atomic[o0] : o0 <= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m10-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/m10-0.c
new file mode 100644
index 00000000000..2b51c7b9106
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m10-0.c
@@ -0,0 +1,7 @@
+for (int c0 = 1; c0 <= 18; c0 += 1)
+  for (int c1 = 1; c1 <= 9; c1 += 1) {
+    if (c0 % 2 == 0)
+      s0(c1, c0 / 2);
+    if (c0 <= 9)
+      s1(c1, c0);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m10-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/m10-0.in
new file mode 100644
index 00000000000..3046811e743
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m10-0.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [4j, i, 0] : i >= 1 and i <= 9 and j >= 1 and j <= 9; s1[i, j] -> [2j, i, 1] : i >= 1 and i <= 9 and j >= 1 and j <= 9 }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 2; [i0, i1, i2] -> atomic[o0] : o0 <= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m10-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/m10-1.c
new file mode 100644
index 00000000000..b3f34f6ec93
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m10-1.c
@@ -0,0 +1,14 @@
+for (int c0 = 1; c0 <= 18; c0 += 1) {
+  if (c0 >= 2 && c0 <= 9) {
+    for (int c1 = 1; c1 <= 9; c1 += 1) {
+      if (c0 % 2 == 0)
+        s0(c1, c0 / 2);
+      s1(c1, c0);
+    }
+  } else if (c0 == 1) {
+    for (int c1 = 1; c1 <= 9; c1 += 1)
+      s1(c1, 1);
+  } else if (c0 % 2 == 0)
+    for (int c1 = 1; c1 <= 9; c1 += 1)
+      s0(c1, c0 / 2);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m10-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/m10-1.in
new file mode 100644
index 00000000000..81dfc3bbdea
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m10-1.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [4j, i, 0] : i >= 1 and i <= 9 and j >= 1 and j <= 9; s1[i, j] -> [2j, i, 1] : i >= 1 and i <= 9 and j >= 1 and j <= 9 }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 1; [i0, i1, i2] -> atomic[o0] : o0 <= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m11-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/m11-0.c
new file mode 100644
index 00000000000..1ecb4a53289
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m11-0.c
@@ -0,0 +1,6 @@
+for (int c0 = 1; c0 <= min(4, floord(2 * m - 1, 17) + 1); c0 += 1)
+  for (int c1 = 1; c1 <= 2; c1 += 1)
+    for (int c2 = 0; c2 <= min(2, -c0 - c1 + (2 * m + 3 * c0 + 10 * c1 + 6) / 20 + 1); c2 += 1)
+      for (int c3 = 8 * c0 + (c0 + 1) / 2 - 8; c3 <= min(min(30, m - 5 * c1 - 10 * c2 + 5), 8 * c0 + c0 / 2); c3 += 1)
+        for (int c4 = 5 * c1 + 10 * c2 - 4; c4 <= min(5 * c1 + 10 * c2, m - c3 + 1); c4 += 1)
+          s0(c0, c1, c2, c3, c4, -9 * c0 + c3 + c0 / 2 + 9, -5 * c1 - 5 * c2 + c4 + 5);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m11-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/m11-0.in
new file mode 100644
index 00000000000..d85fcf2734b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m11-0.in
@@ -0,0 +1,3 @@
+[m] -> { s0[In_1, In_2, In_3, In_4, In_5, In_6, 5 - 5In_2 - 5In_3 + In_5] -> [In_1, In_2, In_3, In_4, In_5, In_6, 5 - 5In_2 - 5In_3 + In_5] : In_2 >= 1 and 2In_3 >= 1 - In_2 and In_2 <= 2 and 2In_3 <= 6 - In_2 and In_4 <= 30 and In_1 >= 1 and 2In_6 <= 18 - 17In_1 + 2In_4 and 2In_6 >= 17 - 17In_1 + 2In_4 and In_5 <= 5In_2 + 10In_3 and In_5 >= -4 + 5In_2 + 10In_3 and 2In_4 <= 17In_1 and 2In_4 >= -16 + 17In_1 and In_5 <= 1 + m - In_4 }
+{  :  }
+[m] -> { [i0, i1, i2, i3, i4, i5, i6] -> atomic[o0] : o0 <= 5; [i0, i1, i2, i3, i4, i5, i6] -> separate[o0] : o0 >= 6 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m12-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/m12-0.c
new file mode 100644
index 00000000000..2dc6ab1fea6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m12-0.c
@@ -0,0 +1,3 @@
+for (int c1 = 1; c1 <= n; c1 += 1)
+  for (int c2 = 1; c2 <= m; c2 += 1)
+    s0(1, c1, c2, 0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m12-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/m12-0.in
new file mode 100644
index 00000000000..0c988ab110c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m12-0.in
@@ -0,0 +1,3 @@
+[m, n] -> { s0[1, In_2, In_3, 0] -> [1, In_2, In_3, 0] : In_3 >= 1 and In_3 <= m and In_2 >= 1 and In_2 <= n }
+{  :  }
+[m, n] -> { [i0, i1, i2, i3] -> separate[o0] : o0 >= 3; [i0, i1, i2, i3] -> atomic[o0] : o0 <= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m12-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/m12-1.c
new file mode 100644
index 00000000000..eba7c8a6170
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m12-1.c
@@ -0,0 +1,25 @@
+{
+  for (int c1 = 1; c1 <= n; c1 += 1)
+    for (int c2 = 1; c2 <= m; c2 += 1) {
+      s0(1, c1, c2, 0);
+      s1(1, c1, c2, 0);
+    }
+  for (int c1 = 1; c1 <= n; c1 += 1) {
+    s3(2, c1, 0, 0);
+    s2(2, c1, 0, 0);
+  }
+  for (int c1 = 1; c1 <= m; c1 += 1) {
+    for (int c3 = 1; c3 <= n; c3 += 1) {
+      s5(3, c1, 1, c3);
+      s4(3, c1, 1, c3);
+    }
+    for (int c3 = 1; c3 <= n; c3 += 1) {
+      s7(3, c1, 2, c3);
+      s6(3, c1, 2, c3);
+    }
+  }
+  for (int c1 = 1; c1 <= m; c1 += 1) {
+    s8(4, c1, 0, 0);
+    s9(4, c1, 0, 0);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m12-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/m12-1.in
new file mode 100644
index 00000000000..355075faf68
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m12-1.in
@@ -0,0 +1,3 @@
+[m, n] -> { s1[1, In_2, In_3, 0] -> [1, In_2, In_3, 0] : In_3 >= 1 and In_3 <= m and In_2 >= 1 and In_2 <= n; s2[2, In_2, 0, 0] -> [2, In_2, 0, 0] : In_2 >= 1 and In_2 <= n; s3[2, In_2, 0, 0] -> [2, In_2, 0, 0] : In_2 >= 1 and In_2 <= n; s8[4, In_2, 0, 0] -> [4, In_2, 0, 0] : In_2 >= 1 and In_2 <= m; s0[1, In_2, In_3, 0] -> [1, In_2, In_3, 0] : In_3 >= 1 and In_3 <= m and In_2 >= 1 and In_2 <= n; s7[3, In_2, 2, In_4] -> [3, In_2, 2, In_4] : In_4 >= 1 and In_4 <= n and In_2 >= 1 and In_2 <= m; s4[3, In_2, 1, In_4] -> [3, In_2, 1, In_4] : In_4 >= 1 and In_4 <= n and In_2 >= 1 and In_2 <= m; s6[3, In_2, 2, In_4] -> [3, In_2, 2, In_4] : In_4 >= 1 and In_4 <= n and In_2 >= 1 and In_2 <= m; s9[4, In_2, 0, 0] -> [4, In_2, 0, 0] : In_2 >= 1 and In_2 <= m; s5[3, In_2, 1, In_4] -> [3, In_2, 1, In_4] : In_4 >= 1 and In_4 <= n and In_2 >= 1 and In_2 <= m }
+{  :  }
+[m, n] -> { [i0, i1, i2, i3] -> separate[o0] : o0 >= 3; [i0, i1, i2, i3] -> atomic[o0] : o0 <= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m2-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/m2-0.c
new file mode 100644
index 00000000000..669f51a7d14
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m2-0.c
@@ -0,0 +1,11 @@
+for (int c0 = 2; c0 <= 9; c0 += 1) {
+  if (c0 >= 5) {
+    s1(c0, 1);
+    for (int c1 = 2; c1 <= 9; c1 += 1) {
+      s1(c0, c1);
+      s0(c0, c1);
+    }
+  } else
+    for (int c1 = 2; c1 <= 9; c1 += 1)
+      s0(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m2-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/m2-0.in
new file mode 100644
index 00000000000..ae7780e0403
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m2-0.in
@@ -0,0 +1,3 @@
+{ s1[In_1, In_2] -> [In_1, In_2] : In_1 >= 5 and In_1 <= 9 and In_2 >= 1 and In_2 <= 9; s0[In_1, In_2] -> [In_1, In_2] : In_1 >= 2 and In_1 <= 9 and In_2 >= 2 and In_2 <= 9 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m2-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/m2-1.c
new file mode 100644
index 00000000000..09255e0c510
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m2-1.c
@@ -0,0 +1,12 @@
+{
+  for (int c0 = 2; c0 <= 4; c0 += 1)
+    for (int c1 = 2; c1 <= 9; c1 += 1)
+      s0(c0, c1);
+  for (int c0 = 5; c0 <= 9; c0 += 1) {
+    s1(c0, 1);
+    for (int c1 = 2; c1 <= 9; c1 += 1) {
+      s1(c0, c1);
+      s0(c0, c1);
+    }
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m2-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/m2-1.in
new file mode 100644
index 00000000000..32302bb11e6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m2-1.in
@@ -0,0 +1,3 @@
+{ s1[In_1, In_2] -> [In_1, In_2] : In_1 >= 5 and In_1 <= 9 and In_2 >= 1 and In_2 <= 9; s0[In_1, In_2] -> [In_1, In_2] : In_1 >= 2 and In_1 <= 9 and In_2 >= 2 and In_2 <= 9 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= -1; [i0, i1] -> separate[o0] : o0 >= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m3-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/m3-0.c
new file mode 100644
index 00000000000..d3d69704165
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m3-0.c
@@ -0,0 +1,3 @@
+for (int c0 = -9; c0 <= 9; c0 += 1)
+  for (int c1 = max(1, -c0 + 1); c1 <= min(10, -c0 + 10); c1 += 1)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m3-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/m3-0.in
new file mode 100644
index 00000000000..82e109311bc
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m3-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2] -> [In_1, In_2] : In_2 >= 1 - In_1 and In_2 >= 1 and In_2 <= 10 - In_1 and In_2 <= 10 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m4-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/m4-0.c
new file mode 100644
index 00000000000..a5280732a46
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m4-0.c
@@ -0,0 +1,5 @@
+for (int c0 = 1; c0 <= 9; c0 += 1)
+  for (int c1 = 1; c1 <= 9; c1 += 1) {
+    s0(c1, c0);
+    s1(c1, c0);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m4-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/m4-0.in
new file mode 100644
index 00000000000..b7912a306b6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m4-0.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [j, i, 0] : i >= 1 and i <= 9 and j >= 1 and j <= 9; s1[i, j] -> [j, i, 1] : i >= 1 and i <= 9 and j >= 1 and j <= 9 }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 2; [i0, i1, i2] -> atomic[o0] : o0 <= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m4-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/m4-1.c
new file mode 100644
index 00000000000..a5280732a46
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m4-1.c
@@ -0,0 +1,5 @@
+for (int c0 = 1; c0 <= 9; c0 += 1)
+  for (int c1 = 1; c1 <= 9; c1 += 1) {
+    s0(c1, c0);
+    s1(c1, c0);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m4-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/m4-1.in
new file mode 100644
index 00000000000..c9e17d93466
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m4-1.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [j, i, 0] : i >= 1 and i <= 9 and j >= 1 and j <= 9; s1[i, j] -> [j, i, 1] : i >= 1 and i <= 9 and j >= 1 and j <= 9 }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 1; [i0, i1, i2] -> atomic[o0] : o0 <= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m7-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/m7-0.c
new file mode 100644
index 00000000000..d9f93d96fcd
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m7-0.c
@@ -0,0 +1,6 @@
+for (int c0 = 1; c0 <= 9; c0 += 1)
+  for (int c1 = 1; c1 <= 9; c1 += 1) {
+    s0(c1, c0);
+    if (c0 % 2 == 0)
+      s1(c1, c0);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m7-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/m7-0.in
new file mode 100644
index 00000000000..45a185043bc
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m7-0.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [j, i, 0] : i >= 1 and i <= 9 and j >= 1 and j <= 9; s1[i, j] -> [j, i, 1] : exists (e0 = [(j)/2]: 2e0 = j and i >= 1 and i <= 9 and j >= 2 and j <= 8) }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 2; [i0, i1, i2] -> atomic[o0] : o0 <= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m7-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/m7-1.c
new file mode 100644
index 00000000000..a01856e4b4d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m7-1.c
@@ -0,0 +1,10 @@
+for (int c0 = 1; c0 <= 9; c0 += 1) {
+  if (c0 % 2 == 0) {
+    for (int c1 = 1; c1 <= 9; c1 += 1) {
+      s0(c1, c0);
+      s1(c1, c0);
+    }
+  } else
+    for (int c1 = 1; c1 <= 9; c1 += 1)
+      s0(c1, c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m7-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/m7-1.in
new file mode 100644
index 00000000000..f4b6f6ffce8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m7-1.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [j, i, 0] : i >= 1 and i <= 9 and j >= 1 and j <= 9; s1[i, j] -> [j, i, 1] : exists (e0 = [(j)/2]: 2e0 = j and i >= 1 and i <= 9 and j >= 2 and j <= 8) }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 1; [i0, i1, i2] -> atomic[o0] : o0 <= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m8-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/m8-0.c
new file mode 100644
index 00000000000..9aaed9e9636
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m8-0.c
@@ -0,0 +1,6 @@
+for (int c0 = 2; c0 <= 8; c0 += 2)
+  for (int c1 = 1; c1 <= 9; c1 += 1) {
+    if (c0 % 4 == 0)
+      s0(c1, c0);
+    s1(c1, c0);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m8-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/m8-0.in
new file mode 100644
index 00000000000..e9a0e39646b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m8-0.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [j, i, 0] : exists (e0 = [(j)/4]: 4e0 = j and i >= 1 and i <= 9 and j >= 4 and j <= 8); s1[i, j] -> [j, i, 1] : exists (e0 = [(j)/2]: 2e0 = j and i >= 1 and i <= 9 and j >= 2 and j <= 8) }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 2; [i0, i1, i2] -> atomic[o0] : o0 <= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m8-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/m8-1.c
new file mode 100644
index 00000000000..51c77daeca4
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m8-1.c
@@ -0,0 +1,10 @@
+for (int c0 = 2; c0 <= 8; c0 += 2) {
+  if (c0 % 4 == 0) {
+    for (int c1 = 1; c1 <= 9; c1 += 1) {
+      s0(c1, c0);
+      s1(c1, c0);
+    }
+  } else
+    for (int c1 = 1; c1 <= 9; c1 += 1)
+      s1(c1, c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m8-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/m8-1.in
new file mode 100644
index 00000000000..80e57e499db
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m8-1.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [j, i, 0] : exists (e0 = [(j)/4]: 4e0 = j and i >= 1 and i <= 9 and j >= 4 and j <= 8); s1[i, j] -> [j, i, 1] : exists (e0 = [(j)/2]: 2e0 = j and i >= 1 and i <= 9 and j >= 2 and j <= 8) }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 1; [i0, i1, i2] -> atomic[o0] : o0 <= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m9-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/m9-0.c
new file mode 100644
index 00000000000..a5280732a46
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m9-0.c
@@ -0,0 +1,5 @@
+for (int c0 = 1; c0 <= 9; c0 += 1)
+  for (int c1 = 1; c1 <= 9; c1 += 1) {
+    s0(c1, c0);
+    s1(c1, c0);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m9-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/m9-0.in
new file mode 100644
index 00000000000..fc38bf2e486
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m9-0.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [2j, i, 0] : i >= 1 and i <= 9 and j >= 1 and j <= 9; s1[i, j] -> [2j, i, 1] : i >= 1 and i <= 9 and j >= 1 and j <= 9 }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 2; [i0, i1, i2] -> atomic[o0] : o0 <= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m9-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/m9-1.c
new file mode 100644
index 00000000000..a5280732a46
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m9-1.c
@@ -0,0 +1,5 @@
+for (int c0 = 1; c0 <= 9; c0 += 1)
+  for (int c1 = 1; c1 <= 9; c1 += 1) {
+    s0(c1, c0);
+    s1(c1, c0);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/m9-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/m9-1.in
new file mode 100644
index 00000000000..398e3260abf
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/m9-1.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [2j, i, 0] : i >= 1 and i <= 9 and j >= 1 and j <= 9; s1[i, j] -> [2j, i, 1] : i >= 1 and i <= 9 and j >= 1 and j <= 9 }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 1; [i0, i1, i2] -> atomic[o0] : o0 <= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/olda-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/olda-0.c
new file mode 100644
index 00000000000..b05f991ad1f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/olda-0.c
@@ -0,0 +1,8 @@
+for (int c0 = 1; c0 <= morb; c0 += 1)
+  for (int c1 = 1; c1 <= np; c1 += 1)
+    for (int c2 = 1; c2 <= np; c2 += 1) {
+      if (c2 >= c1)
+        s0(c2, c1, c0);
+      if (c1 >= c2)
+        s1(c1, c2, c0);
+    }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/olda-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/olda-0.in
new file mode 100644
index 00000000000..88f86f558c5
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/olda-0.in
@@ -0,0 +1,3 @@
+[np, morb] -> { s0[mp, mq, mi] -> [mi, mq, mp, 0] : mq >= 1 and mq <= mp and mp <= np and mi >= 1 and mi <= morb; s1[mp, mq, mi] -> [mi, mp, mq, 1] : mq >= 1 and mq <= mp and mp <= np and mi >= 1 and mi <= morb }
+{  :  }
+[np, morb] -> { [i0, i1, i2, i3] -> atomic[o0] : o0 <= 2; [i0, i1, i2, i3] -> separate[o0] : o0 >= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/olda-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/olda-1.c
new file mode 100644
index 00000000000..4b3098068e4
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/olda-1.c
@@ -0,0 +1,9 @@
+for (int c0 = 1; c0 <= morb; c0 += 1)
+  for (int c1 = 1; c1 <= np; c1 += 1) {
+    for (int c2 = 1; c2 < c1; c2 += 1)
+      s1(c1, c2, c0);
+    s0(c1, c1, c0);
+    s1(c1, c1, c0);
+    for (int c2 = c1 + 1; c2 <= np; c2 += 1)
+      s0(c2, c1, c0);
+  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/olda-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/olda-1.in
new file mode 100644
index 00000000000..a3a387fbac4
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/olda-1.in
@@ -0,0 +1,3 @@
+[np, morb] -> { s0[mp, mq, mi] -> [mi, mq, mp, 0] : mq >= 1 and mq <= mp and mp <= np and mi >= 1 and mi <= morb; s1[mp, mq, mi] -> [mi, mp, mq, 1] : mq >= 1 and mq <= mp and mp <= np and mi >= 1 and mi <= morb }
+{  :  }
+[np, morb] -> { [i0, i1, i2, i3] -> atomic[o0] : o0 <= 1; [i0, i1, i2, i3] -> separate[o0] : o0 >= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/p.delft-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/p.delft-0.c
new file mode 100644
index 00000000000..c676558941f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/p.delft-0.c
@@ -0,0 +1,4 @@
+if (P2 >= 0 && P2 <= 3 && P1 == P2)
+  for (int c0 = 0; c0 <= min(2, -P2 + 4); c0 += 1)
+    for (int c2 = (-P2 - c0 + 6) % 3; c2 <= 3; c2 += 3)
+      s0(c0, c0, c2, c2);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/p.delft-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/p.delft-0.in
new file mode 100644
index 00000000000..5b277a12d78
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/p.delft-0.in
@@ -0,0 +1,3 @@
+[P2, P1] -> { s0[In_1, In_1, In_3, In_3] -> [In_1, In_1, In_3, In_3] : exists (e0 = [(-2P2 - 2In_1 + In_3)/3]: P1 = P2 and 3e0 = -2P2 - 2In_1 + In_3 and P2 >= 0 and P2 <= 3 and In_1 <= 4 - P2 and In_1 >= 0 and In_1 <= 2 and In_3 >= 0 and In_3 <= 3) }
+{  :  }
+[P2, P1] -> { [i0, i1, i2, i3] -> separate[o0] : o0 >= 3; [i0, i1, i2, i3] -> atomic[o0] : o0 <= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/p.delft2-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/p.delft2-0.c
new file mode 100644
index 00000000000..ad5a4cab0d6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/p.delft2-0.c
@@ -0,0 +1,10 @@
+if (P1 >= 0 && P1 <= 3 && P2 >= 0 && P2 <= 3)
+  for (int c0 = P1 - 1; c0 <= 3; c0 += 1)
+    for (int c2 = 0; c2 <= 7; c2 += 1)
+      for (int c3 = 0; c3 <= 7; c3 += 1)
+        if ((5 * P2 + 2 * c3) % 9 <= 3) {
+          if (P1 >= 1 && c0 + 1 == P1 && (5 * P1 + 2 * c2) % 9 <= 2) {
+            s0(P1 - 1, P2, c2, c3, ((5 * P1 + 2 * c2) % 9) + 1, (-4 * P2 + 2 * c3 + 9) % 9);
+          } else if (P1 == 0 && c0 == 3 && c2 % 4 == 0)
+            s0(3, P2, c2, c3, (-c2 / 4) + 3, (-4 * P2 + 2 * c3 + 9) % 9);
+        }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/p.delft2-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/p.delft2-0.in
new file mode 100644
index 00000000000..f44b9bfdaa9
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/p.delft2-0.in
@@ -0,0 +1,3 @@
+[P1, P2] -> { s0[In_1, P2, In_3, In_4, In_5, In_6] -> [In_1, P2, In_3, In_4, In_5, In_6] : (exists (e0 = [(8 + 4In_1 + 16In_3 + In_5)/9], e1 = [(12 - 4P1 + 9e0)/16], e2 = [(-2In_1 - 2In_3 + In_5)/3], e3 = [(-5P2 - 2In_4 + In_6)/9]: 3e2 = -2In_1 - 2In_3 + In_5 and 9e3 = -5P2 - 2In_4 + In_6 and P1 >= 0 and In_1 >= 1 + P1 and In_1 <= 3 and P2 >= 0 and P2 <= 3 and In_6 >= 0 and In_6 <= 3 and In_5 >= 0 and In_5 <= 3 and In_5 >= 1 - 4In_1 - 16In_3 and In_5 <= 126 - 4In_1 - 16In_3 and In_6 <= 126 - 4P2 - 16In_4 and 16e1 <= -4P1 + 9e0 and 2In_6 <= P2 + 4In_4 and 9e0 <= 3 + 4In_1 + 16In_3 + In_5 and 9e0 >= 4In_1 + 16In_3 + In_5 and 16e1 >= -3 - 4P1 + 9e0)) or (exists (e0 = [(8 + 4In_1 + 16In_3 + In_5)/9], e1 = [(12 - 4P1 + 9e0)/16], e2 = [(-2In_1 - 2In_3 + In_5)/3], e3 = [(-5P2 - 2In_4 + In_6)/9]: 3e2 = -2In_1 - 2In_3 + In_5 and 9e3 = -5P2 - 2In_4 + In_6 and In_1 >= 0 and In_1 <= -1 + P1 and P1 <= 3 and In_6 >= 0 and In_6 <= 3 and In_6 <= 1 + 2In_4 and P2 >= 0 and P2 <= 3 and In_5 >= 0 and In_5 <= 3 and In_5 >= 1 - 4In_1 - 16In_3 and In_5 <= 126 - 4In_1 - 16In_3 and In_6 <= 126 - 4P2 - 16In_4 and 16e1 <= -4P1 + 9e0 and 9e0 <= 3 + 4In_1 + 16In_3 + In_5 and 9e0 >= 4In_1 + 16In_3 + In_5 and 16e1 >= -3 - 4P1 + 9e0)) }
+{  :  }
+[P1, P2] -> { [i0, i1, i2, i3, i4, i5] -> atomic[o0] : o0 <= 4; [i0, i1, i2, i3, i4, i5] -> separate[o0] : o0 >= 5 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/p6-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/p6-0.c
new file mode 100644
index 00000000000..53995e9e89e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/p6-0.c
@@ -0,0 +1,8 @@
+{
+  for (int c0 = 5; c0 <= 8; c0 += 1)
+    s0(c0);
+  for (int c0 = 10; c0 <= 16; c0 += 2)
+    s0(c0);
+  for (int c0 = 20; c0 <= 25; c0 += 1)
+    s0(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/p6-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/p6-0.in
new file mode 100644
index 00000000000..025da3d30d6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/p6-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1] -> [In_1] : (In_1 >= 5 and In_1 <= 8) or (exists (e0 = [(In_1)/2]: 2e0 = In_1 and In_1 >= 10 and In_1 <= 16)) or (In_1 >= 20 and In_1 <= 25) }
+{  :  }
+{ [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/p6-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/p6-1.c
new file mode 100644
index 00000000000..d3d69704165
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/p6-1.c
@@ -0,0 +1,3 @@
+for (int c0 = -9; c0 <= 9; c0 += 1)
+  for (int c1 = max(1, -c0 + 1); c1 <= min(10, -c0 + 10); c1 += 1)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/p6-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/p6-1.in
new file mode 100644
index 00000000000..82e109311bc
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/p6-1.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2] -> [In_1, In_2] : In_2 >= 1 - In_1 and In_2 >= 1 and In_2 <= 10 - In_1 and In_2 <= 10 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride1-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/stride1-0.c
new file mode 100644
index 00000000000..c65c2e39fbc
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride1-0.c
@@ -0,0 +1,2 @@
+for (int c0 = 3; c0 <= 9; c0 += 3)
+  s0(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride1-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/stride1-0.in
new file mode 100644
index 00000000000..e09908713f1
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride1-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1] -> [In_1] : exists (e0 = [(In_1)/3]: 3e0 = In_1 and In_1 >= 3 and In_1 <= 9) }
+{  :  }
+{ [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride2-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/stride2-0.c
new file mode 100644
index 00000000000..0a7e8e7464d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride2-0.c
@@ -0,0 +1,3 @@
+for (int c0 = 0; c0 <= n; c0 += 32)
+  for (int c1 = c0; c1 <= min(n, c0 + 31); c1 += 1)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride2-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/stride2-0.in
new file mode 100644
index 00000000000..73dc13d60e4
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride2-0.in
@@ -0,0 +1,3 @@
+[n] -> { s0[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(In_1)/32]: 32e0 = In_1 and In_2 <= 31 + In_1 and In_1 >= 0 and In_2 >= In_1 and In_2 <= n) }
+{  :  }
+[n] -> { [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride3-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/stride3-0.c
new file mode 100644
index 00000000000..8913c802cc0
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride3-0.c
@@ -0,0 +1,3 @@
+for (int c0 = 3; c0 <= n; c0 += 32)
+  for (int c1 = c0; c1 <= min(n, c0 + 31); c1 += 1)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride3-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/stride3-0.in
new file mode 100644
index 00000000000..360d7d01de1
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride3-0.in
@@ -0,0 +1,3 @@
+[n] -> { s0[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(-3 + In_1)/32]: 32e0 = -3 + In_1 and In_2 <= 31 + In_1 and In_1 >= 3 and In_2 >= In_1 and In_2 <= n) }
+{  :  }
+[n] -> { [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride4-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/stride4-0.c
new file mode 100644
index 00000000000..ed62b592ee2
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride4-0.c
@@ -0,0 +1,2 @@
+for (int c0 = 18; c0 <= 98; c0 += 5)
+  s0(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride4-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/stride4-0.in
new file mode 100644
index 00000000000..8779c3ceffa
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride4-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1] -> [In_1] : exists (e0 = [(-3 + In_1)/5]: 5e0 = -3 + In_1 and In_1 >= 18 and In_1 <= 98) }
+{  :  }
+{ [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride5-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/stride5-0.c
new file mode 100644
index 00000000000..2eeeb5a7b54
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride5-0.c
@@ -0,0 +1,3 @@
+for (int c0 = 2; c0 <= min(100, -2 * n + 400); c0 += 2)
+  for (int c1 = 2 * n + c0; c1 <= 400; c1 += 2)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride5-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/stride5-0.in
new file mode 100644
index 00000000000..c26c5476b3c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride5-0.in
@@ -0,0 +1,3 @@
+[n] -> { s0[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(In_1)/2], e1 = [(In_2)/2]: 2e0 = In_1 and 2e1 = In_2 and In_1 >= 2 and In_1 <= 100 and In_2 <= 400 and In_2 >= 2n + In_1) }
+{  :  }
+[n] -> { [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride6-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/stride6-0.c
new file mode 100644
index 00000000000..f4df469cf62
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride6-0.c
@@ -0,0 +1,3 @@
+for (int c0 = 1; c0 <= 101; c0 += 1)
+  for (int c1 = -((c0 - 1) % 2) + c0 + 1; c1 <= 400; c1 += 2)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride6-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/stride6-0.in
new file mode 100644
index 00000000000..16fda744249
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride6-0.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(In_2)/2]: 2e0 = In_2 and In_1 >= 1 and In_2 >= In_1 and In_2 <= 400 and In_1 <= 101) }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride6-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/stride6-1.c
new file mode 100644
index 00000000000..070028f0504
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride6-1.c
@@ -0,0 +1,3 @@
+for (int c0 = 2; c0 <= 100; c0 += 2)
+  for (int c1 = c0; c1 <= 400; c1 += 2)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride6-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/stride6-1.in
new file mode 100644
index 00000000000..91eaab26a32
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride6-1.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(In_1)/2], e1 = [(In_2)/2]: 2e0 = In_1 and 2e1 = In_2 and In_1 >= 2 and In_2 >= In_1 and In_2 <= 400 and In_1 <= 100) }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride6-2.c b/polly/lib/External/isl/test_inputs/codegen/omega/stride6-2.c
new file mode 100644
index 00000000000..070028f0504
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride6-2.c
@@ -0,0 +1,3 @@
+for (int c0 = 2; c0 <= 100; c0 += 2)
+  for (int c1 = c0; c1 <= 400; c1 += 2)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride6-2.in b/polly/lib/External/isl/test_inputs/codegen/omega/stride6-2.in
new file mode 100644
index 00000000000..91eaab26a32
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride6-2.in
@@ -0,0 +1,3 @@
+{ s0[In_1, In_2] -> [In_1, In_2] : exists (e0 = [(In_1)/2], e1 = [(In_2)/2]: 2e0 = In_1 and 2e1 = In_2 and In_1 >= 2 and In_2 >= In_1 and In_2 <= 400 and In_1 <= 100) }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride7-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/stride7-0.c
new file mode 100644
index 00000000000..7fe17588f3d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride7-0.c
@@ -0,0 +1,14 @@
+for (int c0 = 1; c0 <= 36; c0 += 1) {
+  if (c0 <= 3) {
+    for (int c1 = 1; c1 <= 9; c1 += 1)
+      s1(c1, c0);
+  } else if (c0 <= 9) {
+    for (int c1 = 1; c1 <= 9; c1 += 1) {
+      if (c0 % 4 == 0)
+        s0(c1, c0 / 4);
+      s1(c1, c0);
+    }
+  } else if (c0 % 4 == 0)
+    for (int c1 = 1; c1 <= 9; c1 += 1)
+      s0(c1, c0 / 4);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride7-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/stride7-0.in
new file mode 100644
index 00000000000..b84f2045fde
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride7-0.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [4j, i, 0] : i >= 1 and i <= 9 and j >= 1 and j <= 9; s1[i, j] -> [j, i, 1] : i >= 1 and i <= 9 and j >= 1 and j <= 9 }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 1; [i0, i1, i2] -> atomic[o0] : o0 <= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride7-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/stride7-1.c
new file mode 100644
index 00000000000..e79d58deb10
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride7-1.c
@@ -0,0 +1,14 @@
+{
+  for (int c0 = 1; c0 <= 3; c0 += 1)
+    for (int c1 = 1; c1 <= 9; c1 += 1)
+      s1(c1, c0);
+  for (int c0 = 4; c0 <= 9; c0 += 1)
+    for (int c1 = 1; c1 <= 9; c1 += 1) {
+      if (c0 % 4 == 0)
+        s0(c1, c0 / 4);
+      s1(c1, c0);
+    }
+  for (int c0 = 3; c0 <= 9; c0 += 1)
+    for (int c1 = 1; c1 <= 9; c1 += 1)
+      s0(c1, c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/stride7-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/stride7-1.in
new file mode 100644
index 00000000000..3ec82b9c39a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/stride7-1.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [4j, i, 0] : i >= 1 and i <= 9 and j >= 1 and j <= 9; s1[i, j] -> [j, i, 1] : i >= 1 and i <= 9 and j >= 1 and j <= 9 }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 0; [i0, i1, i2] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/substitution-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-0.c
new file mode 100644
index 00000000000..b12c07d3751
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-0.c
@@ -0,0 +1,3 @@
+for (int c0 = 0; c0 <= 10; c0 += 1)
+  for (int c1 = max(2 * c0 - 4, c0); c1 <= min(2 * c0, c0 + 6); c1 += 1)
+    s0(2 * c0 - c1, -c0 + c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/substitution-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-0.in
new file mode 100644
index 00000000000..f4af71649b6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-0.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [i + j, i + 2j] : i >= 0 and i <= 4 and j >= 0 and j <= 6 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/substitution-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-1.c
new file mode 100644
index 00000000000..486514c0433
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-1.c
@@ -0,0 +1,3 @@
+for (int c0 = 0; c0 <= 14; c0 += 1)
+  for (int c1 = max(2 * c0 - 12, -c0 + 3 * floord(c0 - 1, 2) + 3); c1 <= min(2 * c0, c0 / 2 + 9); c1 += 3)
+    s0((2 * c0 - c1) / 3, (-c0 + 2 * c1) / 3);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/substitution-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-1.in
new file mode 100644
index 00000000000..504b220fbef
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-1.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [2i + j, i + 2j] : i >= 0 and i <= 4 and j >= 0 and j <= 6 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/substitution-2.c b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-2.c
new file mode 100644
index 00000000000..9748950ae52
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-2.c
@@ -0,0 +1,2 @@
+for (int c0 = -3; c0 <= 96; c0 += 1)
+  s0(c0, c0 + 4);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/substitution-2.in b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-2.in
new file mode 100644
index 00000000000..bc9af0c0063
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-2.in
@@ -0,0 +1,3 @@
+{ s0[i, 4 + i] -> [i, 4 + i] : i >= -3 and i <= 96 }
+{  :  }
+{ [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/substitution-3.c b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-3.c
new file mode 100644
index 00000000000..86c197f7a95
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-3.c
@@ -0,0 +1 @@
+s0(n + 19);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/substitution-3.in b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-3.in
new file mode 100644
index 00000000000..ea1ac1ab914
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-3.in
@@ -0,0 +1,3 @@
+[n] -> { s0[19 + n] -> [19 + n] }
+{  :  }
+[n] -> { [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/substitution-4.c b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-4.c
new file mode 100644
index 00000000000..27b3199c0c3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-4.c
@@ -0,0 +1 @@
+s0(n + 1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/substitution-4.in b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-4.in
new file mode 100644
index 00000000000..ecb07cd85bd
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/substitution-4.in
@@ -0,0 +1,3 @@
+[n] -> { s0[i] -> [i] : exists (e0 = [(-1 - n + i)/18]: 18e0 = -1 - n + i and i <= 16 + n and i >= 1 + n) }
+{  :  }
+[n] -> { [i0] -> separate[o0] : o0 >= 0; [i0] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-0.c
new file mode 100644
index 00000000000..bb3ef7fa252
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-0.c
@@ -0,0 +1,4 @@
+for (int c0 = 1; c0 <= min(n, 2 * b - 1); c0 += 1)
+  for (int c1 = max(-n + 1, -b + 1); c1 <= min(b - c0, n - c0); c1 += 1)
+    for (int c2 = max(1, c0 + c1); c2 <= min(n, n + c1); c2 += 1)
+      s0(-c0 - c1 + c2 + 1, -c1 + c2, c2);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-0.in
new file mode 100644
index 00000000000..28a7deb1f05
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-0.in
@@ -0,0 +1,3 @@
+[n, b] -> { s0[i, j, k] -> [1 - i + j, -j + k, k] : i >= 1 and j >= i and j <= n and k >= 1 and k <= n and k <= -1 + b + i and k >= 1 - b + j }
+{  :  }
+[n, b] -> { [i0, i1, i2] -> separate[o0] : o0 >= 2; [i0, i1, i2] -> atomic[o0] : o0 <= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-1.c
new file mode 100644
index 00000000000..22ec83ec7ee
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-1.c
@@ -0,0 +1,4 @@
+for (int c0 = 1; c0 <= min(n, 2 * b - 1); c0 += 1)
+  for (int c1 = -b + 1; c1 <= b - c0; c1 += 1)
+    for (int c2 = max(1, c0 + c1); c2 <= min(n, n + c1); c2 += 1)
+      s0(-c0 - c1 + c2 + 1, -c1 + c2, c2);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-1.in
new file mode 100644
index 00000000000..496aebe2e8f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-1.in
@@ -0,0 +1,3 @@
+[n, b] -> { s0[i, j, k] -> [1 - i + j, -j + k, k] : i >= 1 and j >= i and j <= n and k >= 1 and k <= n and k <= -1 + b + i and k >= 1 - b + j }
+[b, n] -> {  : b >= 1 and n >= b }
+[n, b] -> { [i0, i1, i2] -> separate[o0] : o0 >= 2; [i0, i1, i2] -> atomic[o0] : o0 <= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-2.c b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-2.c
new file mode 100644
index 00000000000..bb3ef7fa252
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-2.c
@@ -0,0 +1,4 @@
+for (int c0 = 1; c0 <= min(n, 2 * b - 1); c0 += 1)
+  for (int c1 = max(-n + 1, -b + 1); c1 <= min(b - c0, n - c0); c1 += 1)
+    for (int c2 = max(1, c0 + c1); c2 <= min(n, n + c1); c2 += 1)
+      s0(-c0 - c1 + c2 + 1, -c1 + c2, c2);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-2.in b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-2.in
new file mode 100644
index 00000000000..28a7deb1f05
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-2.in
@@ -0,0 +1,3 @@
+[n, b] -> { s0[i, j, k] -> [1 - i + j, -j + k, k] : i >= 1 and j >= i and j <= n and k >= 1 and k <= n and k <= -1 + b + i and k >= 1 - b + j }
+{  :  }
+[n, b] -> { [i0, i1, i2] -> separate[o0] : o0 >= 2; [i0, i1, i2] -> atomic[o0] : o0 <= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-3.c b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-3.c
new file mode 100644
index 00000000000..22ec83ec7ee
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-3.c
@@ -0,0 +1,4 @@
+for (int c0 = 1; c0 <= min(n, 2 * b - 1); c0 += 1)
+  for (int c1 = -b + 1; c1 <= b - c0; c1 += 1)
+    for (int c2 = max(1, c0 + c1); c2 <= min(n, n + c1); c2 += 1)
+      s0(-c0 - c1 + c2 + 1, -c1 + c2, c2);
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-3.in b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-3.in
new file mode 100644
index 00000000000..496aebe2e8f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/syr2k-3.in
@@ -0,0 +1,3 @@
+[n, b] -> { s0[i, j, k] -> [1 - i + j, -j + k, k] : i >= 1 and j >= i and j <= n and k >= 1 and k <= n and k <= -1 + b + i and k >= 1 - b + j }
+[b, n] -> {  : b >= 1 and n >= b }
+[n, b] -> { [i0, i1, i2] -> separate[o0] : o0 >= 2; [i0, i1, i2] -> atomic[o0] : o0 <= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-check-sblock-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-check-sblock-0.c
new file mode 100644
index 00000000000..c70e95bcb41
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-check-sblock-0.c
@@ -0,0 +1,14 @@
+{
+  for (int c1 = 0; c1 <= 1; c1 += 1) {
+    if (c1 == 1) {
+      s0(1, 1, 1, 0, 0);
+      s0(1, 1, 1, N - 1, 0);
+    } else
+      for (int c3 = 0; c3 < N; c3 += 1)
+        s0(1, 0, 1, c3, 0);
+  }
+  for (int c1 = 0; c1 <= floord(T - 1, 1000); c1 += 1)
+    for (int c2 = 1000 * c1 + 1; c2 <= min(N + T - 3, N + 1000 * c1 + 997); c2 += 1)
+      for (int c3 = max(0, -N - 1000 * c1 + c2 + 2); c3 <= min(min(999, T - 1000 * c1 - 1), -1000 * c1 + c2 - 1); c3 += 1)
+        s1(2, 1000 * c1 + c3, 1, -1000 * c1 + c2 - c3, 1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-check-sblock-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-check-sblock-0.in
new file mode 100644
index 00000000000..b1f92ea71a7
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-check-sblock-0.in
@@ -0,0 +1,3 @@
+[T, N] -> { s1[2, t, 1, i, 1] -> [2, tb, t + i, t - 1000tb, 0] : 1000tb <= t and 1000tb >= -999 + t and i >= 1 and i <= -2 + N and t >= 0 and t <= -1 + T; s0[1, 0, 1, In_4, 0] -> [1, 0, 1, In_4, 0] : In_4 >= 0 and In_4 <= -1 + N; s0[1, 1, 1, 0, 0] -> [1, 1, 1, 0, 0]; s0[1, 1, 1, -1 + N, 0] -> [1, 1, 1, -1 + N, 0] }
+[T, N] -> {  : T >= 0 and N >= 4 }
+[N] -> { [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 3; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-check0-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-check0-0.c
new file mode 100644
index 00000000000..5cf4bc9f5c8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-check0-0.c
@@ -0,0 +1,13 @@
+{
+  for (int c1 = 0; c1 < N; c1 += 1)
+    s0(1, c1, 1, 0, 0);
+  for (int c1 = 0; c1 <= floord(T - 1, 500); c1 += 1)
+    for (int c2 = 1000 * c1; c2 <= min(N + 2 * T - 3, N + 1000 * c1 + 997); c2 += 1) {
+      for (int c3 = max(0, -((N + c2) % 2) - N - 1000 * c1 + c2 + 2); c3 <= min(min(998, 2 * T - 1000 * c1 - 2), -1000 * c1 + c2 - 2); c3 += 2) {
+        s1(2, 1000 * c1 + c3, 0, -1000 * c1 + c2 - c3, 1);
+        s2(2, 1000 * c1 + c3 + 1, 0, -1000 * c1 + c2 - c3 - 1, 1);
+      }
+      if (2 * T >= c2 + 1 && 1000 * c1 + 999 >= c2)
+        s1(2, -(c2 % 2) + c2, 0, c2 % 2, 1);
+    }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-check0-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-check0-0.in
new file mode 100644
index 00000000000..ce585b0836c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-check0-0.in
@@ -0,0 +1,3 @@
+[T, N] -> { s1[2, t, 0, i, 1] -> [2, tb, t + i, t - 1000tb, 1] : exists (e0 = [(t - 1000tb)/2]: 2e0 = t - 1000tb and 1000tb <= t and 1000tb >= -999 + t and i >= 0 and i <= -1 + N and t >= 0 and t <= -2 + 2T); s0[1, In_2, 1, 0, 0] -> [1, In_2, 1, 0, 0] : In_2 >= 0 and In_2 <= -1 + N; s2[2, t, 0, i, 1] -> [2, tb, t + i, t - 1000tb, 1] : exists (e0 = [(-1 + t - 1000tb)/2]: 2e0 = -1 + t - 1000tb and 1000tb <= t and 1000tb >= -999 + t and i >= 1 and i <= -2 + N and t >= 1 and t <= -1 + 2T) }
+[T, N] -> {  : T >= 0 and N >= 4 }
+[N] -> { [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 3; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-mp-i_ts-m_b-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-mp-i_ts-m_b-0.c
new file mode 100644
index 00000000000..41e3ae6094e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-mp-i_ts-m_b-0.c
@@ -0,0 +1,35 @@
+{
+  for (int c1 = -1; c1 < T; c1 += 1)
+    for (int c2 = 0; c2 < N; c2 += 1) {
+      if (c1 == -1) {
+        s0(1, -1, c2, 0, 0);
+      } else if (c2 == 0) {
+        s0(1, c1, 0, 0, 0);
+      } else if (c2 + 1 == N)
+        s0(1, c1, N - 1, 0, 0);
+    }
+  for (int c1 = 0; c1 <= floord(T - 1, 500); c1 += 1) {
+    for (int c3 = -((c1 + 9) / 8) + 2; c3 <= floord(N - 500 * c1 - 3, 4000) + 1; c3 += 1)
+      for (int c4 = max(500 * c1 + 1, 1000 * c1 + 4000 * c3 - 3999); c4 <= min(min(N + T - 3, 1000 * c1 + 4000 * c3 - 3000), 2 * N - 4000 * c3 + 3995); c4 += 1)
+        for (int c5 = max(0, -N - 500 * c1 + c4 + 2); c5 <= min(min(T - 500 * c1 - 1, -500 * c1 + c4 - 1), -500 * c1 - 2000 * c3 + (c4 + 1) / 2 + 1999); c5 += 1)
+          s1(2, 500 * c1 + c5, 1, -500 * c1 + c4 - c5, 1);
+    for (int c3 = max(-((T + 4000) / 4000) + 2, -((c1 + 9) / 8) + 2); c3 <= floord(N - 500 * c1 - 3, 4000) + 1; c3 += 1)
+      for (int c4 = max(1000 * c1 + 4000 * c3 - 3999, -4000 * c3 + 4000); c4 <= min(min(2 * T + 4000 * c3 - 4000, 1000 * c1 + 4000 * c3 - 3000), 2 * N - 4000 * c3 + 3995); c4 += 1)
+        s2(2, -2000 * c3 + (c4 + 1) / 2 + 1999, 1, 2000 * c3 + c4 - (c4 + 1) / 2 - 1999, 1);
+    for (int c3 = -((c1 + 7) / 8) + 1; c3 <= min(floord(N + T - 1000 * c1 - 1004, 4000) + 1, floord(N - 500 * c1 - 504, 4000) + 1); c3 += 1)
+      for (int c4 = max(500 * c1 + 1, 1000 * c1 + 4000 * c3 - 2999); c4 <= min(min(N + T - 3, N + 500 * c1 + 497), 1000 * c1 + 4000 * c3); c4 += 1)
+        for (int c5 = max(0, -N - 500 * c1 + c4 + 2); c5 <= min(min(499, T - 500 * c1 - 1), -500 * c1 + c4 - 1); c5 += 1)
+          s3(2, 500 * c1 + c5, 1, -500 * c1 + c4 - c5, 1);
+    for (int c3 = max(-((T + 4000) / 4000) + 1, -((c1 + 9) / 8) + 1); c3 <= floord(N - 500 * c1 - 3, 4000); c3 += 1)
+      for (int c4 = max(-4000 * c3, 1000 * c1 + 4000 * c3 + 1); c4 <= min(min(2 * N - 4000 * c3 - 5, 2 * T + 4000 * c3), 1000 * c1 + 4000 * c3 + 1000); c4 += 1)
+        s4(2, -2000 * c3 + (c4 + 1) / 2 - 1, 1, 2000 * c3 + c4 - (c4 + 1) / 2 + 1, 1);
+    for (int c3 = -((c1 + 8) / 8) + 1; c3 <= min(floord(N + T - 1000 * c1 - 4, 4000), floord(N - 500 * c1 + 496, 4000)); c3 += 1)
+      for (int c4 = max(1000 * c1 + 4000 * c3 + 1, -4000 * c3 + 2); c4 <= min(min(min(N + T - 3, N + 500 * c1 + 497), 2 * T + 4000 * c3 - 2), 1000 * c1 + 4000 * c3 + 998); c4 += 1)
+        for (int c5 = max(-N - 500 * c1 + c4 + 2, -500 * c1 - 2000 * c3 + (c4 + 1) / 2); c5 <= min(min(499, T - 500 * c1 - 1), -500 * c1 + c4 - 1); c5 += 1)
+          s5(2, 500 * c1 + c5, 1, -500 * c1 + c4 - c5, 1);
+  }
+  if (T >= 1)
+    for (int c3 = -((T + 3998) / 4000) + 1; c3 <= floord(N - T - 2, 4000) + 1; c3 += 1)
+      for (int c4 = max(T, 2 * T + 4000 * c3 - 4001); c4 < min(N + T - 2, 2 * T + 4000 * c3 - 1); c4 += 1)
+        s6(2, T - 1, 1, -T + c4 + 1, 1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-mp-i_ts-m_b-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-mp-i_ts-m_b-0.in
new file mode 100644
index 00000000000..0f86a1c38b9
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-mp-i_ts-m_b-0.in
@@ -0,0 +1,3 @@
+[T, N] -> { s1[2, t, 1, i, 1] -> [2, tb, 1, proc, t + i, t - 500tb, 0] : 4000proc >= 3000 + t + i - 1000tb and 500tb <= t and 4000proc <= 3999 - t + i and i >= 1 and i <= -2 + N and t >= 0 and t <= -1 + T; s0[1, -1, c, 0, 0] -> [1, -1, c, 0, 0, 0, 0] : c >= 0 and c <= -1 + N; s0[1, b, 0, 0, 0] -> [1, b, 0, 0, 0, 0, 0] : b >= 0 and b <= -1 + T; s0[1, b, -1 + N, 0, 0] -> [1, b, -1 + N, 0, 0, 0, 0] : b >= 0 and b <= -1 + T; s6[2, -1 + T, 1, i, 1] -> [3, tb, 7, proc, -1 + T + i, -1 + T - 500tb, 0] : 500tb <= -1 + T and 500tb >= -500 + T and 4000proc >= 1 - T + i and 4000proc <= 4000 - T + i and i >= 1 and i <= -2 + N and T >= 1; s3[2, t, 1, i, 1] -> [2, tb, 3, proc, t + i, t - 500tb, 0] : 500tb <= t and 500tb >= -499 + t and 4000proc <= 2999 + t + i - 1000tb and 4000proc >= t + i - 1000tb and i >= 1 and i <= -2 + N and t >= 0 and t <= -1 + T; s2[2, t, 1, i, 1] -> [2, tb, 2, proc, t + i, t - 500tb, 0] : 500tb <= t and 500tb >= -499 + t and 4000proc <= 3999 - t + i and 4000proc >= 3998 - t + i and i >= 1 and i <= -2 + N and t >= 0 and t <= -1 + T; s4[2, t, 1, i, 1] -> [2, tb, 4, Out_4, t + i, t - 500tb, 0] : 500tb <= t and 500tb >= -499 + t and 4000Out_4 <= -1 - t + i and 4000Out_4 >= -2 - t + i and i >= 1 and i <= -2 + N and t >= 0 and t <= -1 + T; s5[2, t, 1, i, 1] -> [2, tb, 5, proc, t + i, t - 500tb, 0] : 500tb >= -499 + t and 4000proc <= -1 + t + i - 1000tb and 4000proc >= -t + i and i >= 1 and i <= -2 + N and t >= 0 and t <= -1 + T }
+[T, N] -> {  : T >= 0 and N >= 4 }
+[N, T] -> { [i0, i1, i2, i3, i4, i5, i6] -> atomic[o0] : o0 <= 5; [i0, i1, i2, i3, i4, i5, i6] -> separate[o0] : o0 >= 6 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-orig0-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-orig0-0.c
new file mode 100644
index 00000000000..9eff756c5bd
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-orig0-0.c
@@ -0,0 +1,10 @@
+{
+  for (int c1 = 0; c1 < N; c1 += 1)
+    s0(1, c1, 1, 0, 0);
+  for (int c1 = 0; c1 < T; c1 += 1) {
+    for (int c3 = 0; c3 < N; c3 += 1)
+      s1(2, c1, 0, c3, 1);
+    for (int c3 = 1; c3 < N - 1; c3 += 1)
+      s2(2, c1, 1, c3, 1);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-orig0-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-orig0-0.in
new file mode 100644
index 00000000000..153ba7e9bfe
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/ts1d-orig0-0.in
@@ -0,0 +1,3 @@
+[T, N] -> { s1[2, In_2, 0, In_4, 1] -> [2, In_2, 0, In_4, 1] : In_4 >= 0 and In_4 <= -1 + N and In_2 >= 0 and In_2 <= -1 + T; s0[1, In_2, 1, 0, 0] -> [1, In_2, 1, 0, 0] : In_2 >= 0 and In_2 <= -1 + N; s2[2, In_2, 1, In_4, 1] -> [2, In_2, 1, In_4, 1] : In_4 >= 1 and In_4 <= -2 + N and In_2 >= 0 and In_2 <= -1 + T }
+[T, N] -> {  : T >= 0 and N >= 4 }
+[N] -> { [i0, i1, i2, i3, i4] -> separate[o0] : o0 >= 4; [i0, i1, i2, i3, i4] -> atomic[o0] : o0 <= 3 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak1-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/wak1-0.c
new file mode 100644
index 00000000000..2c310e8d2ee
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak1-0.c
@@ -0,0 +1,27 @@
+{
+  for (int c0 = a2; c0 <= min(min(a1 - 1, a3 - 1), b2); c0 += 1)
+    s1(c0);
+  for (int c0 = a1; c0 <= min(b1, a3 - 1); c0 += 1) {
+    s0(c0);
+    if (c0 >= a2 && b2 >= c0)
+      s1(c0);
+  }
+  for (int c0 = max(max(a1, b1 + 1), a2); c0 <= min(a3 - 1, b2); c0 += 1)
+    s1(c0);
+  for (int c0 = a3; c0 <= b3; c0 += 1) {
+    if (c0 >= a1 && b1 >= c0)
+      s0(c0);
+    if (c0 >= a2 && b2 >= c0)
+      s1(c0);
+    s2(c0);
+  }
+  for (int c0 = max(max(a3, b3 + 1), a2); c0 <= min(a1 - 1, b2); c0 += 1)
+    s1(c0);
+  for (int c0 = max(max(a1, a3), b3 + 1); c0 <= b1; c0 += 1) {
+    s0(c0);
+    if (c0 >= a2 && b2 >= c0)
+      s1(c0);
+  }
+  for (int c0 = max(max(max(max(a1, b1 + 1), a3), b3 + 1), a2); c0 <= b2; c0 += 1)
+    s1(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak1-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/wak1-0.in
new file mode 100644
index 00000000000..4cf37cb3192
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak1-0.in
@@ -0,0 +1,3 @@
+[a3, b3, a2, b2, a1, b1] -> { s2[i] -> [i, 2] : i >= a3 and i <= b3; s0[i] -> [i, 0] : i >= a1 and i <= b1; s1[i] -> [i, 1] : i >= a2 and i <= b2 }
+{  :  }
+[a1, b1] -> { [i0, i1] -> separate[o0] : o0 >= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak1-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/wak1-1.c
new file mode 100644
index 00000000000..7fdda1dd58c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak1-1.c
@@ -0,0 +1,55 @@
+{
+  for (int c0 = a2; c0 <= min(min(a1 - 1, a3 - 1), b2); c0 += 1)
+    s1(c0);
+  for (int c0 = a3; c0 <= min(min(a1 - 1, b3), a2 - 1); c0 += 1)
+    s2(c0);
+  for (int c0 = max(a3, a2); c0 <= min(min(a1 - 1, b3), b2); c0 += 1) {
+    s1(c0);
+    s2(c0);
+  }
+  for (int c0 = a1; c0 <= min(min(b1, a3 - 1), a2 - 1); c0 += 1)
+    s0(c0);
+  for (int c0 = max(a1, a2); c0 <= min(min(b1, a3 - 1), b2); c0 += 1) {
+    s0(c0);
+    s1(c0);
+  }
+  for (int c0 = max(max(a1, b1 + 1), a2); c0 <= min(a3 - 1, b2); c0 += 1)
+    s1(c0);
+  for (int c0 = max(a1, a3); c0 <= min(min(b1, b3), a2 - 1); c0 += 1) {
+    s0(c0);
+    s2(c0);
+  }
+  for (int c0 = max(max(a1, b1 + 1), a3); c0 <= min(b3, a2 - 1); c0 += 1)
+    s2(c0);
+  for (int c0 = max(max(a1, a3), a2); c0 <= min(min(b1, b3), b2); c0 += 1) {
+    s0(c0);
+    s1(c0);
+    s2(c0);
+  }
+  for (int c0 = max(max(max(a1, b1 + 1), a3), a2); c0 <= min(b3, b2); c0 += 1) {
+    s1(c0);
+    s2(c0);
+  }
+  for (int c0 = max(max(a3, a2), b2 + 1); c0 <= min(a1 - 1, b3); c0 += 1)
+    s2(c0);
+  for (int c0 = max(max(a1, a2), b2 + 1); c0 <= min(b1, a3 - 1); c0 += 1)
+    s0(c0);
+  for (int c0 = max(max(max(a1, a3), a2), b2 + 1); c0 <= min(b1, b3); c0 += 1) {
+    s0(c0);
+    s2(c0);
+  }
+  for (int c0 = max(max(max(max(a1, b1 + 1), a3), a2), b2 + 1); c0 <= b3; c0 += 1)
+    s2(c0);
+  for (int c0 = max(max(a3, b3 + 1), a2); c0 <= min(a1 - 1, b2); c0 += 1)
+    s1(c0);
+  for (int c0 = max(max(a1, a3), b3 + 1); c0 <= min(b1, a2 - 1); c0 += 1)
+    s0(c0);
+  for (int c0 = max(max(max(a1, a3), b3 + 1), a2); c0 <= min(b1, b2); c0 += 1) {
+    s0(c0);
+    s1(c0);
+  }
+  for (int c0 = max(max(max(max(a1, b1 + 1), a3), b3 + 1), a2); c0 <= b2; c0 += 1)
+    s1(c0);
+  for (int c0 = max(max(max(max(a1, a3), b3 + 1), a2), b2 + 1); c0 <= b1; c0 += 1)
+    s0(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak1-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/wak1-1.in
new file mode 100644
index 00000000000..75c96d49895
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak1-1.in
@@ -0,0 +1,3 @@
+[a3, b3, a2, b2, a1, b1] -> { s2[i] -> [i, 2] : i >= a3 and i <= b3; s0[i] -> [i, 0] : i >= a1 and i <= b1; s1[i] -> [i, 1] : i >= a2 and i <= b2 }
+{  :  }
+[a1, b1] -> { [i0, i1] -> separate[o0] : o0 >= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak2-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/wak2-0.c
new file mode 100644
index 00000000000..3f51dbca273
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak2-0.c
@@ -0,0 +1,26 @@
+{
+  for (int c0 = a1; c0 <= min(b1, a2 - 1); c0 += 1)
+    for (int c1_0 = c1; c1_0 <= d1; c1_0 += 1)
+      s0(c0, c1_0);
+  if (c2 >= d2 + 1) {
+    for (int c0 = max(a1, a2); c0 <= min(b1, b2); c0 += 1)
+      for (int c1_0 = c1; c1_0 <= d1; c1_0 += 1)
+        s0(c0, c1_0);
+  } else
+    for (int c0 = a2; c0 <= b2; c0 += 1) {
+      if (c0 >= a1 && b1 >= c0)
+        for (int c1_0 = c1; c1_0 <= min(d1, c2 - 1); c1_0 += 1)
+          s0(c0, c1_0);
+      for (int c1_0 = c2; c1_0 <= d2; c1_0 += 1) {
+        if (c0 >= a1 && b1 >= c0 && c1_0 >= c1 && d1 >= c1_0)
+          s0(c0, c1_0);
+        s1(c0, c1_0);
+      }
+      if (c0 >= a1 && b1 >= c0)
+        for (int c1_0 = max(c1, d2 + 1); c1_0 <= d1; c1_0 += 1)
+          s0(c0, c1_0);
+    }
+  for (int c0 = max(max(a1, a2), b2 + 1); c0 <= b1; c0 += 1)
+    for (int c1_0 = c1; c1_0 <= d1; c1_0 += 1)
+      s0(c0, c1_0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak2-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/wak2-0.in
new file mode 100644
index 00000000000..da7a6953bf3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak2-0.in
@@ -0,0 +1,3 @@
+[a2, b2, c2, d2, a1, b1, c1, d1] -> { s0[i, j] -> [i, j, 0] : i >= a1 and i <= b1 and j >= c1 and j <= d1; s1[i, j] -> [i, j, 1] : i >= a2 and i <= b2 and j >= c2 and j <= d2 }
+{  :  }
+[a1, b1, c1, d1] -> { [i0, i1, i2] -> separate[o0] : o0 >= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak2-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/wak2-1.c
new file mode 100644
index 00000000000..cc20357718a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak2-1.c
@@ -0,0 +1,35 @@
+{
+  for (int c0 = a1; c0 <= min(b1, a2 - 1); c0 += 1)
+    for (int c1_0 = c1; c1_0 <= d1; c1_0 += 1)
+      s0(c0, c1_0);
+  if (c2 >= d2 + 1) {
+    for (int c0 = max(a1, a2); c0 <= min(b1, b2); c0 += 1)
+      for (int c1_0 = c1; c1_0 <= d1; c1_0 += 1)
+        s0(c0, c1_0);
+  } else
+    for (int c0 = a2; c0 <= b2; c0 += 1) {
+      if (a1 >= c0 + 1) {
+        for (int c1_0 = c2; c1_0 <= d2; c1_0 += 1)
+          s1(c0, c1_0);
+      } else if (c0 >= b1 + 1) {
+        for (int c1_0 = c2; c1_0 <= d2; c1_0 += 1)
+          s1(c0, c1_0);
+      } else {
+        for (int c1_0 = c1; c1_0 <= min(d1, c2 - 1); c1_0 += 1)
+          s0(c0, c1_0);
+        for (int c1_0 = c2; c1_0 <= min(c1 - 1, d2); c1_0 += 1)
+          s1(c0, c1_0);
+        for (int c1_0 = max(c1, c2); c1_0 <= min(d1, d2); c1_0 += 1) {
+          s0(c0, c1_0);
+          s1(c0, c1_0);
+        }
+        for (int c1_0 = max(c1, d2 + 1); c1_0 <= d1; c1_0 += 1)
+          s0(c0, c1_0);
+        for (int c1_0 = max(max(c1, d1 + 1), c2); c1_0 <= d2; c1_0 += 1)
+          s1(c0, c1_0);
+      }
+    }
+  for (int c0 = max(max(a1, a2), b2 + 1); c0 <= b1; c0 += 1)
+    for (int c1_0 = c1; c1_0 <= d1; c1_0 += 1)
+      s0(c0, c1_0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak2-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/wak2-1.in
new file mode 100644
index 00000000000..c92ac3c82c8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak2-1.in
@@ -0,0 +1,3 @@
+[a2, b2, c2, d2, a1, b1, c1, d1] -> { s0[i, j] -> [i, j, 0] : i >= a1 and i <= b1 and j >= c1 and j <= d1; s1[i, j] -> [i, j, 1] : i >= a2 and i <= b2 and j >= c2 and j <= d2 }
+{  :  }
+[a1, b1, c1, d1] -> { [i0, i1, i2] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak3-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/wak3-0.c
new file mode 100644
index 00000000000..06be04aae78
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak3-0.c
@@ -0,0 +1,8 @@
+for (int c0 = a; c0 <= b + 20; c0 += 1) {
+  if (b >= c0)
+    s0(c0);
+  if (c0 >= a + 10 && b + 10 >= c0)
+    s1(c0);
+  if (c0 >= a + 20)
+    s2(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak3-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/wak3-0.in
new file mode 100644
index 00000000000..a4c4de9c88c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak3-0.in
@@ -0,0 +1,3 @@
+[a, b] -> { s2[i] -> [i, 2] : i >= 20 + a and i <= 20 + b; s0[i] -> [i, 0] : i >= a and i <= b; s1[i] -> [i, 1] : i >= 10 + a and i <= 10 + b }
+{  :  }
+[a, b] -> { [i0, i1] -> atomic[o0] : o0 <= 0; [i0, i1] -> separate[o0] : o0 >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak3-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/wak3-1.c
new file mode 100644
index 00000000000..3045b2195cb
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak3-1.c
@@ -0,0 +1,21 @@
+{
+  for (int c0 = a; c0 <= min(a + 9, b); c0 += 1)
+    s0(c0);
+  for (int c0 = a + 10; c0 <= min(a + 19, b); c0 += 1) {
+    s0(c0);
+    s1(c0);
+  }
+  for (int c0 = max(a + 10, b + 1); c0 <= min(a + 19, b + 10); c0 += 1)
+    s1(c0);
+  for (int c0 = a + 20; c0 <= b; c0 += 1) {
+    s0(c0);
+    s1(c0);
+    s2(c0);
+  }
+  for (int c0 = max(a + 20, b + 1); c0 <= b + 10; c0 += 1) {
+    s1(c0);
+    s2(c0);
+  }
+  for (int c0 = max(a + 20, b + 11); c0 <= b + 20; c0 += 1)
+    s2(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak3-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/wak3-1.in
new file mode 100644
index 00000000000..ae319e5ddeb
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak3-1.in
@@ -0,0 +1,3 @@
+[a, b] -> { s2[i] -> [i, 2] : i >= 20 + a and i <= 20 + b; s0[i] -> [i, 0] : i >= a and i <= b; s1[i] -> [i, 1] : i >= 10 + a and i <= 10 + b }
+{  :  }
+[a, b] -> { [i0, i1] -> atomic[o0] : o0 <= -1; [i0, i1] -> separate[o0] : o0 >= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak4-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/wak4-0.c
new file mode 100644
index 00000000000..5cefe4be4d3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak4-0.c
@@ -0,0 +1,4 @@
+for (int c0 = max(max(max(max(a1, a2), a3), a4), a5); c0 <= min(min(min(min(b1, b2), b3), b4), b5); c0 += 1) {
+  s0(c0);
+  s1(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak4-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/wak4-0.in
new file mode 100644
index 00000000000..9862df21458
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak4-0.in
@@ -0,0 +1,3 @@
+[a1, a2, a3, a4, a5, b1, b2, b3, b4, b5] -> { s0[i] -> [i, 0] : i >= a1 and i >= a2 and i >= a3 and i >= a4 and i >= a5 and i <= b1 and i <= b2 and i <= b3 and i <= b4 and i <= b5; s1[i] -> [i, 1] : i >= a1 and i >= a2 and i >= a3 and i >= a4 and i >= a5 and i <= b1 and i <= b2 and i <= b3 and i <= b4 and i <= b5 }
+{  :  }
+[a1, a2, a3, a4, a5, b1, b2, b3, b4, b5] -> { [i0, i1] -> separate[o0] : o0 >= 1; [i0, i1] -> atomic[o0] : o0 <= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak4-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/wak4-1.c
new file mode 100644
index 00000000000..5cefe4be4d3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak4-1.c
@@ -0,0 +1,4 @@
+for (int c0 = max(max(max(max(a1, a2), a3), a4), a5); c0 <= min(min(min(min(b1, b2), b3), b4), b5); c0 += 1) {
+  s0(c0);
+  s1(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/wak4-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/wak4-1.in
new file mode 100644
index 00000000000..076cdd1f48a
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/wak4-1.in
@@ -0,0 +1,3 @@
+[a1, a2, a3, a4, a5, b1, b2, b3, b4, b5] -> { s0[i] -> [i, 0] : i >= a1 and i >= a2 and i >= a3 and i >= a4 and i >= a5 and i <= b1 and i <= b2 and i <= b3 and i <= b4 and i <= b5; s1[i] -> [i, 1] : i >= a1 and i >= a2 and i >= a3 and i >= a4 and i >= a5 and i <= b1 and i <= b2 and i <= b3 and i <= b4 and i <= b5 }
+{  :  }
+[a1, a2, a3, a4, a5, b1, b2, b3, b4, b5] -> { [i0, i1] -> separate[o0] : o0 >= 0; [i0, i1] -> atomic[o0] : o0 <= -1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/x-0.c b/polly/lib/External/isl/test_inputs/codegen/omega/x-0.c
new file mode 100644
index 00000000000..f890920d709
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/x-0.c
@@ -0,0 +1,14 @@
+for (int c0 = 1; c0 <= 11; c0 += 1) {
+  for (int c1 = max(1, c0 - 3); c1 <= min(c0, -c0 + 8); c1 += 1)
+    s1(c1, c0 - c1 + 1);
+  for (int c1 = max(1, -c0 + 9); c1 <= min(c0 - 4, -c0 + 12); c1 += 1)
+    s0(c1, c0 + c1 - 8);
+  for (int c1 = max(c0 - 3, -c0 + 9); c1 <= min(c0, -c0 + 12); c1 += 1) {
+    s0(c1, c0 + c1 - 8);
+    s1(c1, c0 - c1 + 1);
+  }
+  for (int c1 = max(c0 - 3, -c0 + 13); c1 <= min(8, c0); c1 += 1)
+    s1(c1, c0 - c1 + 1);
+  for (int c1 = max(c0 + 1, -c0 + 9); c1 <= min(8, -c0 + 12); c1 += 1)
+    s0(c1, c0 + c1 - 8);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/x-0.in b/polly/lib/External/isl/test_inputs/codegen/omega/x-0.in
new file mode 100644
index 00000000000..24340efe4d8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/x-0.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [8 - i + j, i, 0] : i >= 1 and i <= 8 and j >= 1 and j <= 4; s1[i, j] -> [-1 + i + j, i, 1] : i >= 1 and i <= 8 and j >= 1 and j <= 4 }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 1; [i0, i1, i2] -> atomic[o0] : o0 <= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/x-1.c b/polly/lib/External/isl/test_inputs/codegen/omega/x-1.c
new file mode 100644
index 00000000000..f890920d709
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/x-1.c
@@ -0,0 +1,14 @@
+for (int c0 = 1; c0 <= 11; c0 += 1) {
+  for (int c1 = max(1, c0 - 3); c1 <= min(c0, -c0 + 8); c1 += 1)
+    s1(c1, c0 - c1 + 1);
+  for (int c1 = max(1, -c0 + 9); c1 <= min(c0 - 4, -c0 + 12); c1 += 1)
+    s0(c1, c0 + c1 - 8);
+  for (int c1 = max(c0 - 3, -c0 + 9); c1 <= min(c0, -c0 + 12); c1 += 1) {
+    s0(c1, c0 + c1 - 8);
+    s1(c1, c0 - c1 + 1);
+  }
+  for (int c1 = max(c0 - 3, -c0 + 13); c1 <= min(8, c0); c1 += 1)
+    s1(c1, c0 - c1 + 1);
+  for (int c1 = max(c0 + 1, -c0 + 9); c1 <= min(8, -c0 + 12); c1 += 1)
+    s0(c1, c0 + c1 - 8);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/omega/x-1.in b/polly/lib/External/isl/test_inputs/codegen/omega/x-1.in
new file mode 100644
index 00000000000..24340efe4d8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/omega/x-1.in
@@ -0,0 +1,3 @@
+{ s0[i, j] -> [8 - i + j, i, 0] : i >= 1 and i <= 8 and j >= 1 and j <= 4; s1[i, j] -> [-1 + i + j, i, 1] : i >= 1 and i <= 8 and j >= 1 and j <= 4 }
+{  :  }
+{ [i0, i1, i2] -> separate[o0] : o0 >= 1; [i0, i1, i2] -> atomic[o0] : o0 <= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/pldi2012/README b/polly/lib/External/isl/test_inputs/codegen/pldi2012/README
new file mode 100644
index 00000000000..316e8387782
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/pldi2012/README
@@ -0,0 +1,2 @@
+These examples are taken from the "Polyhedra Scanning Revisited" paper
+by Chun Chen.
diff --git a/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_b.c b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_b.c
new file mode 100644
index 00000000000..18ca3702186
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_b.c
@@ -0,0 +1,9 @@
+for (int c0 = 1; c0 <= 100; c0 += 1) {
+  if (n >= 2)
+    s0(c0);
+  for (int c1 = 1; c1 <= 100; c1 += 1) {
+    if (n >= 2)
+      s1(c0, c1);
+    s2(c0, c1);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_b.in b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_b.in
new file mode 100644
index 00000000000..93149176bc4
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_b.in
@@ -0,0 +1,5 @@
+[n] -> { s0[i] -> [i,0] : 1 <= i <= 100 and n > 1;
+	 s1[i,j] -> [i,j] : 1 <= i,j <= 100 and n > 1;
+	 s2[i,j] -> [i,j] : 1 <= i,j <= 100 }
+[n] -> {  :  }
+[n] -> { [i,j] -> separate[x] : x >= 2 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_c.c b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_c.c
new file mode 100644
index 00000000000..d1ce579aa97
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_c.c
@@ -0,0 +1,11 @@
+for (int c0 = 1; c0 <= 100; c0 += 1) {
+  if (n >= 2) {
+    s0(c0);
+    for (int c1 = 1; c1 <= 100; c1 += 1) {
+      s1(c0, c1);
+      s2(c0, c1);
+    }
+  } else
+    for (int c1 = 1; c1 <= 100; c1 += 1)
+      s2(c0, c1);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_c.in b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_c.in
new file mode 100644
index 00000000000..06a32663ec8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_c.in
@@ -0,0 +1,5 @@
+[n] -> { s0[i] -> [i,0] : 1 <= i <= 100 and n > 1;
+	 s1[i,j] -> [i,j] : 1 <= i,j <= 100 and n > 1;
+	 s2[i,j] -> [i,j] : 1 <= i,j <= 100 }
+[n] -> {  :  }
+[n] -> { [i,j] -> separate[x] : x >= 1 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_d.c b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_d.c
new file mode 100644
index 00000000000..570cbe5556f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_d.c
@@ -0,0 +1,12 @@
+if (n >= 2) {
+  for (int c0 = 1; c0 <= 100; c0 += 1) {
+    s0(c0);
+    for (int c1 = 1; c1 <= 100; c1 += 1) {
+      s1(c0, c1);
+      s2(c0, c1);
+    }
+  }
+} else
+  for (int c0 = 1; c0 <= 100; c0 += 1)
+    for (int c1 = 1; c1 <= 100; c1 += 1)
+      s2(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_d.in b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_d.in
new file mode 100644
index 00000000000..339e39d8cee
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure7_d.in
@@ -0,0 +1,5 @@
+[n] -> { s0[i] -> [i,0] : 1 <= i <= 100 and n > 1;
+	 s1[i,j] -> [i,j] : 1 <= i,j <= 100 and n > 1;
+	 s2[i,j] -> [i,j] : 1 <= i,j <= 100 }
+[n] -> {  :  }
+[n] -> { [i,j] -> separate[x] : x >= 0 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure8_a.c b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure8_a.c
new file mode 100644
index 00000000000..4608d61d8db
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure8_a.c
@@ -0,0 +1,3 @@
+for (int c0 = 1; c0 <= n; c0 += 4)
+  for (int c1 = c0; c1 <= n; c1 += 3)
+    s0(c0, c1);
diff --git a/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure8_a.in b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure8_a.in
new file mode 100644
index 00000000000..df6543cce22
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure8_a.in
@@ -0,0 +1,4 @@
+[n] -> { s0[i,j] -> [i,j] : exists alpha, beta: 1 <= i <= n and i <= j <= n and
+			    i = 1 + 4 alpha and j = i + 3 beta}
+[n] -> { : }
+[n] -> {}
diff --git a/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure8_b.c b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure8_b.c
new file mode 100644
index 00000000000..d7fa3cd8a37
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure8_b.c
@@ -0,0 +1,8 @@
+{
+  for (int c0 = 2; c0 < n - 1; c0 += 4) {
+    s1(c0);
+    s0(c0 + 2);
+  }
+  if (n >= 2 && n % 4 >= 2)
+    s1(-(n % 4) + n + 2);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure8_b.in b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure8_b.in
new file mode 100644
index 00000000000..2effb0a09cd
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/pldi2012/figure8_b.in
@@ -0,0 +1,4 @@
+[n] -> { s0[i] -> [i] : exists alpha: 1 <= i <= n and i = 4 alpha;
+	 s1[i] -> [i] : exists alpha: 1 <= i <= n and i = 4 alpha + 2 }
+[n] -> { : }
+[n] -> { }
diff --git a/polly/lib/External/isl/test_inputs/codegen/roman.c b/polly/lib/External/isl/test_inputs/codegen/roman.c
new file mode 100644
index 00000000000..1dd43c52012
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/roman.c
@@ -0,0 +1,32 @@
+{
+  for (int c1 = 0; c1 <= min(np1 - i, -i + 1); c1 += 1) {
+    S_9(c1);
+    S_12(c1);
+  }
+  for (int c1 = max(0, -i + 2); c1 <= -((-np1 + i + 4294967295) % 4294967296) + 4294967295; c1 += 1) {
+    S_9(c1);
+    S_10(c1);
+    for (int c3 = 0; c3 <= min(19, i + c1 - 3); c3 += 1) {
+      S_15(c1, c3);
+      for (int c5 = 0; c5 < c3; c5 += 1) {
+        S_16(c1, c3, c5);
+        S_17(c1, c3, c5);
+      }
+      S_16(c1, c3, c3);
+      S_18(c1, c3);
+      S_24(c1, c3);
+      S_19(c1, c3);
+    }
+    if (np1 >= i && i + c1 <= 21) {
+      S_15(c1, i + c1 - 2);
+      for (int c5 = 0; c5 < i + c1 - 2; c5 += 1) {
+        S_16(c1, i + c1 - 2, c5);
+        S_17(c1, i + c1 - 2, c5);
+      }
+      S_16(c1, i + c1 - 2, i + c1 - 2);
+      S_18(c1, i + c1 - 2);
+      S_24(c1, i + c1 - 2);
+    }
+    S_12(c1);
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/roman.in b/polly/lib/External/isl/test_inputs/codegen/roman.in
new file mode 100644
index 00000000000..55182a6efdb
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/roman.in
@@ -0,0 +1,5 @@
+# Older versions of isl would get confused on this input due to disappearing
+# div constraints.
+[np1, i] -> { S_17[i0, i1, i2] -> [0, i0, 2, i1, 1, i2, 1] : exists (e0 = [(np1 - i)/4294967296], e1 = [(-2 + i + i0)/4294967296], e2 = [(i1)/4294967296]: i0 >= 0 and 4294967296e0 <= np1 - i and 4294967296e0 >= -4294967295 + np1 - i and 4294967296e0 <= np1 - i - i0 and i0 <= 20 and i1 >= 0 and 4294967296e1 <= -2 + i + i0 and 4294967296e1 >= -4294967297 + i + i0 and 4294967296e1 <= -2 + i + i0 - i1 and i1 <= 19 and i2 >= 0 and 4294967296e2 <= i1 and 4294967296e2 >= -4294967295 + i1 and 4294967296e2 <= i1 - i2 and i2 <= 19 and i0 >= 2 - i and i2 <= -1 + i1); S_18[i0, i1] -> [0, i0, 2, i1, 2, 0, 0] : exists (e0 = [(np1 - i)/4294967296], e1 = [(-2 + i + i0)/4294967296]: i0 >= 0 and 4294967296e0 <= np1 - i and 4294967296e0 >= -4294967295 + np1 - i and 4294967296e0 <= np1 - i - i0 and i0 <= 20 and i1 >= 0 and 4294967296e1 <= -2 + i + i0 and 4294967296e1 >= -4294967297 + i + i0 and 4294967296e1 <= -2 + i + i0 - i1 and i1 <= 19 and i0 >= 2 - i); S_24[i0, i1] -> [0, i0, 2, i1, 3, 0, 0] : exists (e0 = [(np1 - i)/4294967296], e1 = [(-2 + i + i0)/4294967296]: i0 >= 0 and 4294967296e0 <= np1 - i and 4294967296e0 >= -4294967295 + np1 - i and 4294967296e0 <= np1 - i - i0 and i0 <= 20 and i1 >= 0 and 4294967296e1 <= -2 + i + i0 and 4294967296e1 >= -4294967297 + i + i0 and 4294967296e1 <= -2 + i + i0 - i1 and i1 <= 19 and i0 >= 2 - i); S_15[i0, i1] -> [0, i0, 2, i1, 0, 0, 0] : exists (e0 = [(np1 - i)/4294967296], e1 = [(-2 + i + i0)/4294967296]: i0 >= 0 and 4294967296e0 <= np1 - i and 4294967296e0 >= -4294967295 + np1 - i and 4294967296e0 <= np1 - i - i0 and i0 <= 20 and i1 >= 0 and 4294967296e1 <= -2 + i + i0 and 4294967296e1 >= -4294967297 + i + i0 and 4294967296e1 <= -2 + i + i0 - i1 and i1 <= 19 and i0 >= 2 - i); S_9[i0] -> [0, i0, 0, 0, 0, 0, 0] : exists (e0 = [(np1 - i)/4294967296]: i0 >= 0 and 4294967296e0 <= np1 - i and 4294967296e0 >= -4294967295 + np1 - i and 4294967296e0 <= np1 - i - i0 and i0 <= 20); S_10[i0] -> [0, i0, 1, 0, 0, 0, 0] : exists (e0 = [(np1 - i)/4294967296]: i0 >= 0 and 4294967296e0 <= np1 - i and 4294967296e0 >= -4294967295 + np1 - i and 4294967296e0 <= np1 - i - i0 and i0 <= 20 and i0 >= 2 - i); S_19[i0, i1] -> [0, i0, 2, i1, 4, 0, 0] : exists (e0 = [(np1 - i)/4294967296], e1 = [(-2 + i + i0)/4294967296]: i0 >= 0 and 4294967296e0 <= np1 - i and 4294967296e0 >= -4294967295 + np1 - i and 4294967296e0 <= np1 - i - i0 and i0 <= 20 and i1 >= 0 and 4294967296e1 <= -2 + i + i0 and 4294967296e1 >= -4294967297 + i + i0 and 4294967296e1 <= -2 + i + i0 - i1 and i1 <= 19 and i0 >= 2 - i and i1 <= -3 + i + i0); S_12[i0] -> [0, i0, 3, 0, 0, 0, 0] : exists (e0 = [(np1 - i)/4294967296]: i0 >= 0 and 4294967296e0 <= np1 - i and 4294967296e0 >= -4294967295 + np1 - i and 4294967296e0 <= np1 - i - i0 and i0 <= 20); S_16[i0, i1, i2] -> [0, i0, 2, i1, 1, i2, 0] : exists (e0 = [(np1 - i)/4294967296], e1 = [(-2 + i + i0)/4294967296], e2 = [(i1)/4294967296]: i0 >= 0 and 4294967296e0 <= np1 - i and 4294967296e0 >= -4294967295 + np1 - i and 4294967296e0 <= np1 - i - i0 and i0 <= 20 and i1 >= 0 and 4294967296e1 <= -2 + i + i0 and 4294967296e1 >= -4294967297 + i + i0 and 4294967296e1 <= -2 + i + i0 - i1 and i1 <= 19 and i2 >= 0 and 4294967296e2 <= i1 and 4294967296e2 >= -4294967295 + i1 and 4294967296e2 <= i1 - i2 and i2 <= 19 and i0 >= 2 - i) }
+[np1, i] -> {  : exists (e0 = [(np1 - i)/4294967296]: 4294967296e0 <= np1 - i and 4294967296e0 >= -20 + np1 - i and np1 >= -2147483648 and np1 <= 2147483647 and i >= -2147483648 and i <= 2147483647) }
+[np1, i] -> { [i0, i1, i2, i3, i4, i5, i6] -> separate[o0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/separate.c b/polly/lib/External/isl/test_inputs/codegen/separate.c
new file mode 100644
index 00000000000..042e3e77c6c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/separate.c
@@ -0,0 +1,8 @@
+{
+  a(0);
+  for (int c0 = 1; c0 <= 9; c0 += 1) {
+    b(c0 - 1);
+    a(c0);
+  }
+  b(9);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/separate.in b/polly/lib/External/isl/test_inputs/codegen/separate.in
new file mode 100644
index 00000000000..58486c3d356
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/separate.in
@@ -0,0 +1,3 @@
+{ a[i] -> [i] : 0 <= i < 10; b[i] -> [i+1] : 0 <= i < 10 }
+{ : }
+{ [i] -> separate[x] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/separate2.c b/polly/lib/External/isl/test_inputs/codegen/separate2.c
new file mode 100644
index 00000000000..81668215c9e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/separate2.c
@@ -0,0 +1,9 @@
+if ((length - 1) % 16 <= 14)
+  for (int c0 = 0; c0 <= 1; c0 += 1)
+    for (int c5 = 0; c5 <= 31; c5 += 1)
+      for (int c6 = max(0, 2 * ((length - 1) % 16) + 2 * c5 - 60); c6 <= 30; c6 += 1) {
+        if (length + c5 >= ((length - 1) % 32) + 2 && (length - 1) % 32 >= c5 && 2 * ((length - 1) % 32) + c6 >= 2 * c5 && 2 * c5 + 30 >= 2 * ((length - 1) % 32) + c6 && 2 * ((length - 1) % 32) + c6 == 2 * ((length - 1) % 16) + 2 * c5 && (2 * c5 - c6) % 32 == 0)
+          S_3(c0, 0, (c6 / 2) - ((length - 1) % 16) + length - 1);
+        if (length <= 16 && length >= c5 + 1 && c6 >= 1 && length >= c6)
+          S_0(c0, c5, c6 - 1);
+      }
diff --git a/polly/lib/External/isl/test_inputs/codegen/separate2.in b/polly/lib/External/isl/test_inputs/codegen/separate2.in
new file mode 100644
index 00000000000..8f4ebf1a5b9
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/separate2.in
@@ -0,0 +1,4 @@
+# Check that rational affine expressions are printer properly.
+[tsteps, length] -> { S_0[iter, i, j] -> [iter, 0, o2, o3, 0, o5, o6, 4] : exists (e0 = [(o2)/32], e1 = [(o3)/32], e2 = [(-i + o5)/32], e3 = [(-31 + j - o6)/32]: tsteps = 2 and 32e0 = o2 and 32e1 = o3 and 32e2 = -i + o5 and 32e3 = -31 + j - o6 and o2 <= i and o2 >= -31 + i and o3 <= 1 + j and o3 >= -30 + j and o5 >= 0 and o5 <= 31 and o6 >= 0 and o6 <= 31 and i <= -1 + length and i >= 0 and iter >= 0 and iter <= 1 and j <= -1 + length and j >= 0 and o2 >= -31 + length and o3 >= -30 + 2length); S_3[iter, 0, j] -> [iter, 0, o2, o3, o4, o5, o6, 2] : exists (e0 = [(o2)/32], e1 = [(o3)/32], e2 = [(o4)/32], e3 = [(-2o5 + o6)/32], e4 = [(j - o5)/32]: tsteps = 2 and 32e0 = o2 and 32e1 = o3 and 32e2 = o4 and 32e3 = -2o5 + o6 and 32e4 = j - o5 and iter <= 1 and j <= -1 + length and o2 <= j and o2 >= -31 + j and o3 <= 2j and o3 >= -30 + 2j and o4 >= 0 and o4 <= 31 and o5 >= 0 and o5 <= 31 and o6 >= 0 and o6 <= 30 and j >= 1 and iter >= 0 and o2 >= -31 + length and o3 >= -30 + 2length) }
+[tsteps, length] -> {  : length >= 1 and length <= 1024 and tsteps = 2 }
+{ [o0,o1,o2,o3,o4,o5,o6,o7] -> separate[x] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/separation_class.c b/polly/lib/External/isl/test_inputs/codegen/separation_class.c
new file mode 100644
index 00000000000..daac0d76a88
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/separation_class.c
@@ -0,0 +1,17 @@
+{
+  for (int c0 = 0; c0 <= 8; c0 += 1) {
+    for (int c1 = 0; c1 <= -c0 + 8; c1 += 1)
+      for (int c2 = 10 * c0; c2 <= 10 * c0 + 9; c2 += 1)
+        for (int c3 = 10 * c1; c3 <= 10 * c1 + 9; c3 += 1)
+          A(c2, c3);
+    for (int c1 = -c0 + 9; c1 <= -c0 + 10; c1 += 1)
+      for (int c2 = 10 * c0; c2 <= min(10 * c0 + 9, -10 * c1 + 100); c2 += 1)
+        for (int c3 = 10 * c1; c3 <= min(10 * c1 + 9, -c2 + 100); c3 += 1)
+          A(c2, c3);
+  }
+  for (int c0 = 9; c0 <= 10; c0 += 1)
+    for (int c1 = 0; c1 <= -c0 + 10; c1 += 1)
+      for (int c2 = 10 * c0; c2 <= min(10 * c0 + 9, -10 * c1 + 100); c2 += 1)
+        for (int c3 = 10 * c1; c3 <= min(10 * c1 + 9, -c2 + 100); c3 += 1)
+          A(c2, c3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/separation_class.in b/polly/lib/External/isl/test_inputs/codegen/separation_class.in
new file mode 100644
index 00000000000..f42ea27fa56
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/separation_class.in
@@ -0,0 +1,6 @@
+{ A[i,j] -> [([i/10]),[j/10],i,j] : 0 <= i,j and i + j <= 100 }
+{ : }
+{ [a,b,c,d] -> separation_class[[0]->[0]] :
+	exists b': 0 <= 10a,10b' and 10a+9+10b'+9 <= 100;
+  [a,b,c,d] -> separation_class[[1]->[0]] :
+	0 <= 10a,10b and 10a+9+10b+9 <= 100 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/separation_class2.c b/polly/lib/External/isl/test_inputs/codegen/separation_class2.c
new file mode 100644
index 00000000000..0fdbc3238e3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/separation_class2.c
@@ -0,0 +1,15 @@
+{
+  for (int c0 = 0; c0 < -(n % 8) + n; c0 += 8) {
+    for (int c1 = 0; c1 < -(n % 8) + n; c1 += 8)
+      for (int c2 = 0; c2 <= 7; c2 += 1)
+        for (int c3 = 0; c3 <= 7; c3 += 1)
+          A(c0 + c2, c1 + c3);
+    for (int c2 = 0; c2 <= 7; c2 += 1)
+      for (int c3 = 0; c3 < n % 8; c3 += 1)
+        A(c0 + c2, -((n - 1) % 8) + n + c3 - 1);
+  }
+  for (int c1 = 0; c1 < n; c1 += 8)
+    for (int c2 = 0; c2 < n % 8; c2 += 1)
+      for (int c3 = 0; c3 <= min(7, n - c1 - 1); c3 += 1)
+        A(-((n - 1) % 8) + n + c2 - 1, c1 + c3);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/separation_class2.in b/polly/lib/External/isl/test_inputs/codegen/separation_class2.in
new file mode 100644
index 00000000000..5469626ae7e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/separation_class2.in
@@ -0,0 +1,3 @@
+[n] -> { A[i,j] -> [it,jt, ip, jp] : 0 <= i,j < n and ip = i % 8 and it = i - ip and jp = j % 8 and jt = j - jp}
+[n] -> { : n >= 10}
+[n] -> { [it, jt, ip, jp] -> separation_class[[x]->[1]]: (exists id, jd: 0 <= x <= 3 and it < n - id and jt < n - jd and id = n %8 and jd = n %8)}
diff --git a/polly/lib/External/isl/test_inputs/codegen/separation_class3.c b/polly/lib/External/isl/test_inputs/codegen/separation_class3.c
new file mode 100644
index 00000000000..d21b19c29bc
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/separation_class3.c
@@ -0,0 +1,29 @@
+for (int c0 = 0; c0 <= 4; c0 += 1) {
+  if (c0 == 0) {
+    S_0(0, 4);
+  } else {
+    S_0(2 * c0 - 1, 1);
+    if (c0 == 4) {
+      for (int c6 = 3; c6 <= 5; c6 += 1)
+        S_0(7, c6);
+    } else
+      for (int c4 = 2 * c0 - 1; c4 <= 2 * c0; c4 += 1)
+        for (int c6 = -2 * c0 + c4 + 4; c6 <= 2 * c0 - c4 + 4; c6 += 1)
+          S_0(c4, c6);
+  }
+  for (int c4 = max(0, 2 * c0 - 1); c4 <= min(7, 2 * c0); c4 += 1)
+    for (int c6 = -2 * c0 + c4 + 8; c6 <= 8; c6 += 1)
+      S_0(c4, c6);
+  if (c0 >= 1 && c0 <= 3) {
+    for (int c2 = 0; c2 <= 1; c2 += 1)
+      for (int c4 = 2 * c0 - 1; c4 <= 2 * c0; c4 += 1)
+        for (int c6 = 2 * c0 + 4 * c2 - c4 + 1; c6 <= -2 * c0 + 4 * c2 + c4 + 3; c6 += 1)
+          S_0(c4, c6);
+  } else if (c0 == 4) {
+    for (int c2 = 0; c2 <= 1; c2 += 1)
+      S_0(7, 4 * c2 + 2);
+  } else
+    for (int c2 = 0; c2 <= 1; c2 += 1)
+      for (int c6 = 4 * c2 + 1; c6 <= 4 * c2 + 3; c6 += 1)
+        S_0(0, c6);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/separation_class3.in b/polly/lib/External/isl/test_inputs/codegen/separation_class3.in
new file mode 100644
index 00000000000..1aaf2517c18
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/separation_class3.in
@@ -0,0 +1,4 @@
+{ S_0[t, i] -> [o0, 1, o2, 0, t, 0, i] : 4o2 >= -4 + t + i - 2o0 and 4o2 >= -3 - t + i + 2o0 and 2o0 <= 1 + t and 2o0 >= t and 4o2 <= -1 + t + i - 2o0 and t >= 0 and t <= 7 and i >= 1 and i <= 8; S_0[t, i] -> [o0, 0, o2, 0, t, 0, i] : 4o2 >= t + i - 2o0 and 4o2 <= -t + i + 2o0 and 2o0 <= 1 + t and 2o0 >= t and t >= 0 and t <= 7 and i >= 1 and i <= 8 }
+{:}
+{ [i0, 1, i2, i3, i4, i5, i6] -> separation_class[[2] -> [0]] : i2 <= 1 and i2 >= 0 and i0 <= 3 and i0 >= 1; [i0, 0, 1, i3, i4, i5, i6] -> separation_class[[2] -> [0]] : i0 <= 3 and i0 >= 1; [i0, i1, i2, i3, i4, i5, i6] -> unroll[1] }
+
diff --git a/polly/lib/External/isl/test_inputs/codegen/separation_class4.c b/polly/lib/External/isl/test_inputs/codegen/separation_class4.c
new file mode 100644
index 00000000000..2ff23642e8d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/separation_class4.c
@@ -0,0 +1,20 @@
+for (int c0 = 0; c0 <= 128; c0 += 1) {
+  if (c0 <= 127) {
+    if (c0 == 0) {
+      for (int c3 = 0; c3 <= 1; c3 += 1)
+        for (int c5 = c3 + 58; c5 <= -c3 + 61; c5 += 1)
+          S_0(c3, c5);
+    } else
+      for (int c2 = 1; c2 <= 2; c2 += 1)
+        for (int c3 = max(4 * c0 - 2, 4 * c0 + 6 * c2 - 12); c3 <= min(4 * c0 + 1, 4 * c0 + 6 * c2 - 7); c3 += 1)
+          for (int c5 = max(4 * c0 - c3 + 57, -4 * c0 + c3 + 58); c5 <= min(4 * c0 - c3 + 61, -4 * c0 + c3 + 62); c5 += 1)
+            S_0(c3, c5);
+    for (int c2 = 1; c2 <= 2; c2 += 1)
+      for (int c3 = max(4 * c0, 4 * c0 + 6 * c2 - 10); c3 <= min(4 * c0 + 3, 4 * c0 + 6 * c2 - 5); c3 += 1)
+        for (int c5 = max(-4 * c0 + c3 + 59, 4 * c0 - c3 + 62); c5 <= min(-4 * c0 + c3 + 63, 4 * c0 - c3 + 66); c5 += 1)
+          S_0(c3, c5);
+  } else
+    for (int c3 = 510; c3 <= 511; c3 += 1)
+      for (int c5 = -c3 + 569; c5 < c3 - 449; c5 += 1)
+        S_0(c3, c5);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/separation_class4.in b/polly/lib/External/isl/test_inputs/codegen/separation_class4.in
new file mode 100644
index 00000000000..c04dd79bf23
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/separation_class4.in
@@ -0,0 +1,10 @@
+# Check that isl is not confused by the combination of separation classes
+# and unroll.
+{ S_0[t, i] -> [o0, 1, o9, t] : 4o0 >= -3 + t and 4o0 <= t and i >= 60 and i <= 65 and 6o9 >= 5 + t - 4o0 and 6o9 <= 10 + t - 4o0 and 4o0 <= -62 + t + i and 4o0 >= 59 + t - i and o0 >= 0 and o0 <= 127 and t <= 511 and t >= 0 and 4o0 >= -66 + t + i and 4o0 <= 63 + t - i;
+S_0[t, i] -> [o0, 0, o9, t] : 4o0 >= -1 + t and 4o0 <= 2 + t and i >= 57 and i <= 62 and 6o9 >= 7 + t - 4o0 and 6o9 <= 12 + t - 4o0 and t >= 0 and t <= 511 and 4o0 <= -57 + t + i and 4o0 >= 58 + t - i and o0 >= 0 and o0 <= 128 and 4o0 >= -61 + t + i and 4o0 <= 62 + t - i }
+{ : }
+{ [i0, i1, i2, t] -> unroll[1];
+[i0, 1, i2, t] -> separation_class[[1] -> [0]]
+	: 0 <= i0 <= 127;
+[i0, 0, i2, t] -> separation_class[[1] -> [0]]
+	: 1 <= i0 <= 127}
diff --git a/polly/lib/External/isl/test_inputs/codegen/shift.c b/polly/lib/External/isl/test_inputs/codegen/shift.c
new file mode 100644
index 00000000000..f4156bca99b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/shift.c
@@ -0,0 +1,4 @@
+for (int c0 = 0; c0 <= 9; c0 += 1) {
+  A(c0);
+  B(c0);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/shift.in b/polly/lib/External/isl/test_inputs/codegen/shift.in
new file mode 100644
index 00000000000..702c5275621
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/shift.in
@@ -0,0 +1,3 @@
+{ A[i] -> [2i]: 0 <= i < 10; B[i] -> [2i+1] : 0 <= i < 10 }
+{ : }
+{ }
diff --git a/polly/lib/External/isl/test_inputs/codegen/shift2.c b/polly/lib/External/isl/test_inputs/codegen/shift2.c
new file mode 100644
index 00000000000..30ea633b674
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/shift2.c
@@ -0,0 +1,43 @@
+for (int c0 = 0; c0 <= 1; c0 += 1) {
+  for (int c2 = 0; c2 <= length; c2 += 32) {
+    if (length >= c2 + 1) {
+      for (int c3 = 0; c3 <= length; c3 += 32) {
+        for (int c5 = 0; c5 <= min(31, length - c2 - 1); c5 += 1) {
+          for (int c6 = max(0, -c3 + 1); c6 <= min(min(31, length - c3), 2 * c2 - c3 + 2 * c5 - 1); c6 += 1)
+            S_0(c0, c2 + c5, c3 + c6 - 1);
+          if (c2 + c5 >= 1 && 2 * c2 + 2 * c5 >= c3 && c3 + 30 >= 2 * c2 + 2 * c5) {
+            S_3(c0, 0, c2 + c5);
+            if (length >= 2 * c2 + 2 * c5)
+              S_0(c0, c2 + c5, 2 * c2 + 2 * c5 - 1);
+          }
+          for (int c6 = max(0, 2 * c2 - c3 + 2 * c5 + 1); c6 <= min(31, length - c3); c6 += 1)
+            S_0(c0, c2 + c5, c3 + c6 - 1);
+        }
+        if (length <= 15 && c2 == 0 && c3 == 0)
+          S_4(c0);
+        if (c3 >= 2 * c2 && 2 * c2 + 32 >= c3)
+          for (int c4 = 1; c4 <= min(min(31, length - 2), (c3 / 2) + 14); c4 += 1)
+            for (int c5 = max((c3 / 2) - c2, -c2 + c4 + 1); c5 <= min(length - c2 - 1, (c3 / 2) - c2 + 15); c5 += 1)
+              S_3(c0, c4, c2 + c5);
+      }
+      for (int c3 = max(2 * c2, -(length % 32) + length + 32); c3 <= min(2 * length - 2, 2 * c2 + 62); c3 += 32)
+        for (int c4 = 0; c4 <= min(31, length - 2); c4 += 1) {
+          for (int c5 = max((c3 / 2) - c2, -c2 + c4 + 1); c5 <= min(length - c2 - 1, (c3 / 2) - c2 + 15); c5 += 1)
+            S_3(c0, c4, c2 + c5);
+          if (c3 + 30 >= 2 * length && c4 == 0)
+            S_4(c0);
+        }
+      if (c2 + 16 == length && (length - 16) % 32 == 0)
+        S_4(c0);
+    } else if (length == 0) {
+      S_4(c0);
+    } else
+      S_4(c0);
+  }
+  for (int c1 = 32; c1 < length - 1; c1 += 32)
+    for (int c2 = c1; c2 < length; c2 += 32)
+      for (int c3 = c2; c3 <= min(length - 1, c2 + 31); c3 += 16)
+        for (int c4 = 0; c4 <= min(min(31, length - c1 - 2), -c1 + c3 + 14); c4 += 1)
+          for (int c5 = max(-c2 + c3, c1 - c2 + c4 + 1); c5 <= min(length - c2 - 1, -c2 + c3 + 15); c5 += 1)
+            S_3(c0, c1 + c4, c2 + c5);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/shift2.in b/polly/lib/External/isl/test_inputs/codegen/shift2.in
new file mode 100644
index 00000000000..3c05fac3687
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/shift2.in
@@ -0,0 +1,5 @@
+# Check that the shifting code is not confused by domains that
+# have a non-obviously fixed value.
+[tsteps, length] -> { S_4[iter] -> [iter, 0, o2, o3, 0, o5, o6, 3] : exists (e0 = [(o2)/32], e1 = [(o3)/32], e2 = [(-length + o5)/32], e3 = [(-2length + o6)/32]: tsteps = 2 and 32e0 = o2 and 32e1 = o3 and 32e2 = -length + o5 and 32e3 = -2length + o6 and o2 <= length and o2 >= -31 + length and o3 <= 2length and o3 >= -30 + 2length and o5 >= 0 and o5 <= 31 and o6 >= 0 and o6 <= 30 and iter <= 1 and iter >= 0); S_3[iter, i, j] -> [iter, o1, o2, o3, o4, o5, o6, 2] : exists (e0 = [(o1)/32], e1 = [(o2)/32], e2 = [(o3)/32], e3 = [(-i + o4)/32], e4 = [(-j + o5)/32], e5 = [(-2j + o6)/32]: tsteps = 2 and 32e0 = o1 and 32e1 = o2 and 32e2 = o3 and 32e3 = -i + o4 and 32e4 = -j + o5 and 32e5 = -2j + o6 and o1 <= i and o1 >= -31 + i and o2 <= j and o2 >= -31 + j and o3 <= 2j and o3 >= -30 + 2j and o4 >= 0 and o4 <= 31 and o5 >= 0 and o5 <= 31 and o6 >= 0 and o6 <= 30 and j >= 1 + i and i >= 0 and iter <= 1 and iter >= 0 and j <= -1 + length); S_0[iter, i, j] -> [iter, 0, o2, o3, 0, o5, o6, 4] : exists (e0 = [(o2)/32], e1 = [(o3)/32], e2 = [(-i + o5)/32], e3 = [(-31 + j - o6)/32]: tsteps = 2 and 32e0 = o2 and 32e1 = o3 and 32e2 = -i + o5 and 32e3 = -31 + j - o6 and o2 <= i and o2 >= -31 + i and o3 <= 1 + j and o3 >= -30 + j and o5 >= 0 and o5 <= 31 and o6 >= 0 and o6 <= 31 and i <= -1 + length and i >= 0 and iter >= 0 and iter <= 1 and j <= -1 + length and j >= 0) }
+[tsteps, length] -> {  : length >= 0 and length <= 1024 and tsteps = 2 }
+{ }
diff --git a/polly/lib/External/isl/test_inputs/codegen/shift_unroll.c b/polly/lib/External/isl/test_inputs/codegen/shift_unroll.c
new file mode 100644
index 00000000000..2a38286272c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/shift_unroll.c
@@ -0,0 +1,14 @@
+for (int c0 = 0; c0 <= 9; c0 += 1) {
+  A(c0, 0);
+  A(c0, 1);
+  A(c0, 2);
+  A(c0, 3);
+  A(c0, 4);
+  A(c0, 5);
+  A(c0, 6);
+  A(c0, 7);
+  A(c0, 8);
+  A(c0, 9);
+  for (int c2 = 0; c2 <= 9; c2 += 1)
+    B(c0, c2);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/shift_unroll.in b/polly/lib/External/isl/test_inputs/codegen/shift_unroll.in
new file mode 100644
index 00000000000..4b773c27ac3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/shift_unroll.in
@@ -0,0 +1,3 @@
+{ A[i,j] -> [2i,0,j]: 0 <= i,j < 10; B[i,j] -> [2i+1,1,j] : 0 <= i,j < 10 }
+{ : }
+{ [i,0,j] -> unroll[2] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/single_valued.c b/polly/lib/External/isl/test_inputs/codegen/single_valued.c
new file mode 100644
index 00000000000..e071002b79d
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/single_valued.c
@@ -0,0 +1,2 @@
+if (2 * ((t1 - 1) % 64) + 8 >= t1)
+  S(-(2 * ((t1 - 1) % 64)) + t1 + 126);
diff --git a/polly/lib/External/isl/test_inputs/codegen/single_valued.in b/polly/lib/External/isl/test_inputs/codegen/single_valued.in
new file mode 100644
index 00000000000..d729942077e
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/single_valued.in
@@ -0,0 +1,5 @@
+# Check that isl recognizes that the inverse schedule is single-valued
+# and does not end up in an infinite recursion.
+[t1] -> {S[c2] -> [c2]: t1 <= c2 <= 134 and (c2+t1) % 128 = 0 and c2 > 0}
+[t1] -> {: t1 > 0}
+[t1] -> {}
diff --git a/polly/lib/External/isl/test_inputs/codegen/stride.c b/polly/lib/External/isl/test_inputs/codegen/stride.c
new file mode 100644
index 00000000000..5b688292cc5
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/stride.c
@@ -0,0 +1,6 @@
+for (int c0 = 0; c0 <= 100; c0 += 2) {
+  for (int c3 = 0; c3 <= 100; c3 += 1)
+    A(c0, c3);
+  for (int c2 = 0; c2 <= 100; c2 += 1)
+    B(c0, c2);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/stride.in b/polly/lib/External/isl/test_inputs/codegen/stride.in
new file mode 100644
index 00000000000..b01c56617d7
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/stride.in
@@ -0,0 +1,6 @@
+# Check that we find a common stride on the first dimension
+# even if it is imposed by different inner dimensions
+{ A[i,k] -> [i,0,j,k] : 0 <= i,k <= 100 and i = 2 j;
+  B[i,k] -> [i,1,k,j] : 0 <= i,k <= 100 and i = 2 j }
+{ : }
+{ }
diff --git a/polly/lib/External/isl/test_inputs/codegen/stride5.c b/polly/lib/External/isl/test_inputs/codegen/stride5.c
new file mode 100644
index 00000000000..ee89e367598
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/stride5.c
@@ -0,0 +1,3 @@
+if (n % 2 == 0)
+  for (int c0 = (n / 2) + 2 * floord(-n - 1, 4) + 2; c0 <= 100; c0 += 2)
+    S(c0);
diff --git a/polly/lib/External/isl/test_inputs/codegen/stride5.in b/polly/lib/External/isl/test_inputs/codegen/stride5.in
new file mode 100644
index 00000000000..a67a80276e4
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/stride5.in
@@ -0,0 +1,3 @@
+[n] -> { S[t] -> [t] : exists e : 2 t - n = 4e and 0 <= t <= 100 }
+[n] -> { : }
+{ }
diff --git a/polly/lib/External/isl/test_inputs/codegen/stride6.c b/polly/lib/External/isl/test_inputs/codegen/stride6.c
new file mode 100644
index 00000000000..c09e8b5441f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/stride6.c
@@ -0,0 +1,4 @@
+for (int c1 = -1024; c1 <= 0; c1 += 32)
+  for (int c2 = max(-((niter - 1) % 32) + niter - 1, -((niter - c1) % 32) + niter - c1 - 32); c2 <= min(niter + 1022, niter - c1 - 1); c2 += 32)
+    for (int c5 = max(max(0, -c1 - 1023), niter - c1 - c2 - 32); c5 <= min(min(31, -c1), niter - c1 - c2 - 1); c5 += 1)
+      S_4(niter - 1, -c1 - c5);
diff --git a/polly/lib/External/isl/test_inputs/codegen/stride6.in b/polly/lib/External/isl/test_inputs/codegen/stride6.in
new file mode 100644
index 00000000000..86ca54a92c6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/stride6.in
@@ -0,0 +1,3 @@
+[niter] -> { S_4[-1 + niter, i] -> [o0, o1, o2, o3, o4, o5, o6, o7, 4] : exists (e0 = [(o0)/32], e1 = [(o1)/32], e2 = [(o2)/32], e3 = [(o3)/32], e4 = [(-31i + o5)/32], e5 = [(-i - o4 + o6)/32], e6 = [(-o4 + o7)/32], e7 = [(-1 + niter - o4)/32]: 32e0 = o0 and 32e1 = o1 and 32e2 = o2 and 32e3 = o3 and 32e4 = -31i + o5 and 32e5 = -i - o4 + o6 and 32e6 = -o4 + o7 and 32e7 = -1 + niter - o4 and o0 <= -1 + niter and o0 >= -32 + niter and o1 <= -i and o1 >= -31 - i and o2 <= -1 + niter + i and o2 >= -32 + niter + i and o3 <= 1023 + niter and o3 >= 992 + niter and o4 >= 0 and o4 <= 31 and o5 >= 0 and o5 <= 31 and o6 >= 0 and o6 <= 31 and o7 >= 0 and o7 <= 31 and i <= 1023 and i >= 0 and niter >= 1) }
+[niter] -> {  : niter <= 8192 and niter >= 1 }
+[niter] -> {  }
diff --git a/polly/lib/External/isl/test_inputs/codegen/stride7.c b/polly/lib/External/isl/test_inputs/codegen/stride7.c
new file mode 100644
index 00000000000..b64c66d3810
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/stride7.c
@@ -0,0 +1,6 @@
+for (int c0 = 2; c0 <= 200; c0 += 64) {
+  for (int c2 = c0 - 1; c2 <= 120; c2 += 1)
+    s2(c0, c2);
+  for (int c2 = 122; c2 <= c0 + 62; c2 += 1)
+    s4(c0, c2);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/stride7.in b/polly/lib/External/isl/test_inputs/codegen/stride7.in
new file mode 100644
index 00000000000..cac9caca480
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/stride7.in
@@ -0,0 +1,7 @@
+# Check that no redundant guards are introduced
+{ s4[a, b] -> [a, 2, b] : exists (e0 = floor((-2 + a)/64):
+    64e0 = -2 + a and a <= 200 and b <= 62 + a and b >= 122);
+  s2[a, b] -> [a, 2, b] : exists (e0 = floor((-2 + a)/64):
+    64e0 = -2 + a and a >= 2 and b <= 120 and b >= -1 + a and a <= 100) }
+{ : }
+{ }
diff --git a/polly/lib/External/isl/test_inputs/codegen/unroll.c b/polly/lib/External/isl/test_inputs/codegen/unroll.c
new file mode 100644
index 00000000000..63594bb6bc7
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/unroll.c
@@ -0,0 +1,5 @@
+{
+  A(0);
+  A(100000000);
+  A(200000000);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/unroll.in b/polly/lib/External/isl/test_inputs/codegen/unroll.in
new file mode 100644
index 00000000000..65561e2b8de
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/unroll.in
@@ -0,0 +1,5 @@
+# Test that unrolling takes into account stride constraints.
+# If it didn't, it would run essentially forever on this example.
+[n] -> { A[i] -> [i] : exists a : i = 100000000 a and 0 <= a <= 2 }
+{:}
+{ [i] -> unroll[0] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/unroll10.c b/polly/lib/External/isl/test_inputs/codegen/unroll10.c
new file mode 100644
index 00000000000..061954ef628
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/unroll10.c
@@ -0,0 +1,29 @@
+if (m >= 1 && n >= 1) {
+  A(0);
+  if (m >= 2 && n >= 2) {
+    A(1);
+    if (m >= 3 && n >= 3) {
+      A(2);
+      if (m >= 4 && n >= 4) {
+        A(3);
+        if (m >= 5 && n >= 5) {
+          A(4);
+          if (m >= 6 && n >= 6) {
+            A(5);
+            if (m >= 7 && n >= 7) {
+              A(6);
+              if (m >= 8 && n >= 8) {
+                A(7);
+                if (m >= 9 && n >= 9) {
+                  A(8);
+                  if (m >= 10 && n >= 10)
+                    A(9);
+                }
+              }
+            }
+          }
+        }
+      }
+    }
+  }
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/unroll10.in b/polly/lib/External/isl/test_inputs/codegen/unroll10.in
new file mode 100644
index 00000000000..5890c4eaaa0
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/unroll10.in
@@ -0,0 +1,4 @@
+# Check that all information is taken into account while trying to unroll
+[m,n] -> { A[i] -> [i] : 0 <= i < n,m }
+[m,n] -> { : m <= 10 or n <= 10 }
+{ [i] -> unroll[x] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/unroll2.c b/polly/lib/External/isl/test_inputs/codegen/unroll2.c
new file mode 100644
index 00000000000..7ae38be1cb1
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/unroll2.c
@@ -0,0 +1,11 @@
+{
+  A(0);
+  A(1);
+  A(2);
+  A(3);
+  for (int c0 = 4; c0 <= 99996; c0 += 1)
+    A(c0);
+  A(99997);
+  A(99998);
+  A(99999);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/unroll2.in b/polly/lib/External/isl/test_inputs/codegen/unroll2.in
new file mode 100644
index 00000000000..0fb36ab06dc
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/unroll2.in
@@ -0,0 +1,5 @@
+# Check that the different disjuncts in the unroll option
+# are handled separately.
+{ A[i] -> [i] : 0 <= i < 100000 }
+{ : }
+{ [i] -> unroll[0] : i < 4 or i > 99996 }
diff --git a/polly/lib/External/isl/test_inputs/codegen/unroll3.c b/polly/lib/External/isl/test_inputs/codegen/unroll3.c
new file mode 100644
index 00000000000..95a30ba7ccc
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/unroll3.c
@@ -0,0 +1,2 @@
+if ((t1 + 121) % 128 <= 123)
+  write_shared_A(((t1 + 125) % 128) - 3);
diff --git a/polly/lib/External/isl/test_inputs/codegen/unroll3.in b/polly/lib/External/isl/test_inputs/codegen/unroll3.in
new file mode 100644
index 00000000000..098e5010577
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/unroll3.in
@@ -0,0 +1,6 @@
+# Check that the entire schedule is completely unrolled and
+# in particular that no spurious loop is introduced.
+[t1] -> { write_shared_A[i2] -> [1, 3, 6 + i2, 0, t1] : (exists (e0 = [(-6 + t1 - i2)/128]: 128e0 = -6 + t1 - i2 and i2 <= 122 and i2 >= 1 and t1 >= 0 and t1 <= 127)) or (exists (e0 = [(-6 + t1 - i2)/128]: 128e0 = -6 + t1 - i2 and i2 >= 123 and i2 <= 124 and t1 <= 127 and t1 >= 0 )) }
+[t1] -> { : t1 >= 0 and t1 <= 127 }
+[t1] -> { [i0, i1, i2, i3, i4] -> unroll[o0] }
+
diff --git a/polly/lib/External/isl/test_inputs/codegen/unroll4.c b/polly/lib/External/isl/test_inputs/codegen/unroll4.c
new file mode 100644
index 00000000000..8e2ac05ca97
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/unroll4.c
@@ -0,0 +1,22 @@
+{
+  write_shared_A(3, ((t1 + 3) % 4) + 1, ((t2 + 31) % 32) + 1);
+  if (t2 >= 1 && t2 <= 2 && t1 % 3 == 0)
+    write_shared_A(3, (-t1 / 3) + 4, t2 + 32);
+  {
+    int c3 = t2 >= 2 && ((t1 + 3) % 4) + 1 >= t2 ? t2 + 32 : ((t2 + 30) % 32) + 2;
+    if (c3 == t2 + 32 || (c3 == t2 && t2 >= ((t1 + 3) % 4) + ((t2 - 1) % 2) + 2))
+      write_shared_A(3, ((t1 + 3) % 4) + 5, c3);
+  }
+  if (t1 >= 1 && t2 >= t1 + 1 && t2 <= 4)
+    write_shared_A(3, t1 + 4, t2 + 32);
+  write_shared_A(4, ((t1 + 3) % 4) + 1, ((t2 + 31) % 32) + 1);
+  if (t2 >= 1 && t2 <= 2 && t1 % 3 == 0)
+    write_shared_A(4, (-t1 / 3) + 4, t2 + 32);
+  {
+    int c3 = t2 >= 2 && ((t1 + 3) % 4) + 1 >= t2 ? t2 + 32 : ((t2 + 30) % 32) + 2;
+    if (c3 == t2 + 32 || (c3 == t2 && t2 >= ((t1 + 3) % 4) + ((t2 - 1) % 2) + 2))
+      write_shared_A(4, ((t1 + 3) % 4) + 5, c3);
+  }
+  if (t1 >= 1 && t2 >= t1 + 1 && t2 <= 4)
+    write_shared_A(4, t1 + 4, t2 + 32);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/unroll4.in b/polly/lib/External/isl/test_inputs/codegen/unroll4.in
new file mode 100644
index 00000000000..d8be866f948
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/unroll4.in
@@ -0,0 +1,5 @@
+# Check that the generated code does not contain two declarations
+# of the same variable in the same scope.
+[t1, t2, g] -> { write_shared_A[a, b, c] -> [0, a, b, c] : exists (e0, e1 = [(-t1 + b)/4], e2 = [(-t2 + c)/32]: 4e1 = -t1 + b and 32e2 = -t2 + c and e0 <= 2 + 3g and e0 >= 3g and a <= 4 and a >= 3 and t2 >= 0 and t1 <= 3 and 2e0 >= 5 - c + 6g and 2e0 <= 36 - c + 6g and 2e0 >= 5 - b + 6g and 2e0 <= 8 - b + 6g and 2e0 <= 638 - c and 2e0 <= 638 - b and 2e0 >= 2 - a + 6g and 2e0 >= -8 + a + 6g and 2e0 <= 1 + a + 6g and 2e0 <= 11 - a + 6g and e0 >= 0 and e0 <= 254 and t1 >= 0 and t2 <= 31 and b >= 1 and b <= 126 and c >= 1 and c <= 126 and g <= 3 and g >= 0) }
+[t1, t2, g] -> { : g <= 3 and g >= 0 and t1 >= 0 and t1 <= 3 and t2 >= 0 and t2 <= 5 }
+[t1, t2] -> { [i0, i1, i2, i3] -> unroll[x] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/unroll6.c b/polly/lib/External/isl/test_inputs/codegen/unroll6.c
new file mode 100644
index 00000000000..1d8cc5eb771
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/unroll6.c
@@ -0,0 +1,8 @@
+{
+  if (g >= 0 && nn >= 128 * g + 6 && nn >= ((t1 + 127) % 128) + 128 * g + 3)
+    for (int c1 = 393214; c1 < nn - 1; c1 += 393216)
+      A(c1, ((t1 + 127) % 128) + 128 * g + 1, ((t1 + 127) % 128) + 1);
+  if (t1 >= 1 && t1 <= 2 && nn >= t1 + 128 * g + 130 && t1 + 128 * g >= -127)
+    for (int c1 = 393214; c1 < nn - 1; c1 += 393216)
+      A(c1, t1 + 128 * g + 128, t1 + 128);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/unroll6.in b/polly/lib/External/isl/test_inputs/codegen/unroll6.in
new file mode 100644
index 00000000000..b4338e331f9
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/unroll6.in
@@ -0,0 +1,7 @@
+# Check that the right lower bound is chosen for unrolling.
+# Older versions of isl would pick a lower bound that resulted
+# in a number of slices that exceeds the maximal value of an integer
+# and then only generated code for a truncated number (zero) of slices.
+[nn, t1, g] -> { A[a, b, c] -> [c] : exists (e0 = [(2 + a)/393216], e1 = [(t1 - c)/128]: 128g = b - c and 393216e0 = 2 + a and 128e1 = t1 - c and c <= 130 and c >= 6 - nn + b and c <= 128 + b and nn >= 137 and t1 >= 0 and c >= 1 and a <= -2 + nn and a >= 1 and nn <= 9223372036854775807 and b >= 1 and b <= -2 + nn and t1 <= 127) }
+[nn, t1, g] -> { : nn <= 9223372036854775807 and nn >= 137 and t1 >= 0 and t1 <= 127 }
+{ [c] -> unroll[x] }
diff --git a/polly/lib/External/isl/test_inputs/codegen/unroll7.c b/polly/lib/External/isl/test_inputs/codegen/unroll7.c
new file mode 100644
index 00000000000..8392185f71c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/unroll7.c
@@ -0,0 +1,10 @@
+{
+  S(0, 0);
+  S(0, 3);
+  S(0, 4);
+  S(1, 1);
+  S(1, 4);
+  S(2, 2);
+  S(3, 3);
+  S(4, 4);
+}
diff --git a/polly/lib/External/isl/test_inputs/codegen/unroll7.in b/polly/lib/External/isl/test_inputs/codegen/unroll7.in
new file mode 100644
index 00000000000..494e5975564
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/codegen/unroll7.in
@@ -0,0 +1,5 @@
+# Check that some code is generated.
+# Older versions of isl would abort on unknown divs.
+{ S[i,j] -> [i,j]: exists (alpha, beta: j=i+4alpha +3beta and 0 <= alpha < 24 and 0 <= beta and 0 <= i,j < 5) }
+{ : }
+{ [i,j] -> unroll[x] }
diff --git a/polly/lib/External/isl/test_inputs/convex0.polylib b/polly/lib/External/isl/test_inputs/convex0.polylib
new file mode 100644
index 00000000000..cbc4d3b6ceb
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex0.polylib
@@ -0,0 +1,11 @@
+2 3
+1 1 0
+1 -1 1
+
+2 3
+1 1 -1
+1 -1 2
+
+2 3
+1 1 0
+1 -1 2
diff --git a/polly/lib/External/isl/test_inputs/convex1.polylib b/polly/lib/External/isl/test_inputs/convex1.polylib
new file mode 100644
index 00000000000..b563d8de7ff
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex1.polylib
@@ -0,0 +1,17 @@
+# {j,N | 0<=j<=N-1; 2<=N}
+4 4
+1   1  0   0
+1  -1  1  -1
+1   0  1  -2
+1   0  0   1
+# {j, N | 1<=j<=N; 1<=N}
+4 4
+1  1  0  -1
+1 -1  1   0
+1  0  1  -1
+1  0  0   1
+# {j,N | 0<=j<=N; 2<=j+N}
+3 4
+   1    1    1   -2
+   1    1    0    0
+   1   -1    1    0
diff --git a/polly/lib/External/isl/test_inputs/convex10.polylib b/polly/lib/External/isl/test_inputs/convex10.polylib
new file mode 100644
index 00000000000..3d58cbf70e1
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex10.polylib
@@ -0,0 +1,17 @@
+3 4
+1 54 1 -4
+1 2 -1 58
+1 0 -1 6
+
+4 4
+1 54 1 -4
+1 2 -1 58
+1 0 1 -7
+1 -4 1 0
+
+4 4
+1  54    1   -4
+1   2   -1   58
+1   0   -1  116
+1   0    0    1
+
diff --git a/polly/lib/External/isl/test_inputs/convex11.polylib b/polly/lib/External/isl/test_inputs/convex11.polylib
new file mode 100644
index 00000000000..f664114d048
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex11.polylib
@@ -0,0 +1,14 @@
+3 4
+1 0 -1 6
+1 -1 1 1
+1 1 1 -10
+
+3 4
+1 1 0 -4
+1 -1 -1 8
+1 -1 1 1
+
+3 4
+1 0 -1 6
+1 1 0 -4
+1 -1 1 1
diff --git a/polly/lib/External/isl/test_inputs/convex12.polylib b/polly/lib/External/isl/test_inputs/convex12.polylib
new file mode 100644
index 00000000000..e4766307477
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex12.polylib
@@ -0,0 +1,12 @@
+3 5
+1   0    0    1    1
+1   0    1    0    1
+1  -1   -1    0   -2
+
+3 5
+1   0    0    1    2
+1   1   -1    0    0
+1   1    0    0   -1
+
+1 5
+1   0    0    1    2
diff --git a/polly/lib/External/isl/test_inputs/convex13.polylib b/polly/lib/External/isl/test_inputs/convex13.polylib
new file mode 100644
index 00000000000..1b3591200f8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex13.polylib
@@ -0,0 +1,17 @@
+3 5
+1   0    0   -1    3
+1   0   -1    0    2
+1   1    1    1   -4
+
+3 5
+1   0    0    1    0
+1   1    0    0   -1
+1   1    2    0    1
+
+6 5
+1   3    2    0   -1
+1   3    0    2   -3
+1   1    0    1   -1
+1   1    1    1    0
+1   1    1    0    0
+1   1    0    0    1
diff --git a/polly/lib/External/isl/test_inputs/convex14.polylib b/polly/lib/External/isl/test_inputs/convex14.polylib
new file mode 100644
index 00000000000..caaa8f5d316
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex14.polylib
@@ -0,0 +1,14 @@
+3 4
+0 1 0 2
+1 0 1 0
+1 0 -1 2
+
+3 4
+1 1 0 0
+1 0 1 0
+1 0 -1 2
+
+3 4
+1 1 0 2
+1 0 1 0
+1 0 -1 2
diff --git a/polly/lib/External/isl/test_inputs/convex15.polylib b/polly/lib/External/isl/test_inputs/convex15.polylib
new file mode 100644
index 00000000000..0118fa861db
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex15.polylib
@@ -0,0 +1,66 @@
+17 8
+1    -1    -8     0    16     0     0    37
+1     1     0   -48     0     2     0    -3
+1     0   -16   -32    16     1     0    14
+1    -1    24     0     0     1     0    18
+1    -1     8    16     0     0     1    21
+1     0     0   -16     0     1     1    -2
+1     1    32    16   -32     0     0    -1
+1    -1    16    16     0     0     0    28
+1     1    -8   -32     0     1     0    -1
+1     0     0     0     0     1     0    -1
+1     0    16    16   -16     0     1    -1
+1     1     8     0   -16     0     0     0
+1     0     3     2    -2     0     0     0
+1     0     1     2    -1     0     0     0
+1     0    -1    -1     1     0     0     0
+1    -1     8     0     0     1     2     4
+1    -1   -24   -32    32     1     0    36
+
+13 8
+1    -1     0     0     0     1     3    -4
+1     1     0   -48     0     2     0    -2
+1     0     0     0     0     1     0    -1
+1     0    -8     0     0     0     1    -1
+1     0     3     2    -2     0     0     0
+1     1   -16   -16     0     0     0     0
+1     1   -24     0     0     0     0     0
+1     0     1     0     0     0     0     0
+1     0    -3    -2     2     0     0     1
+1    -1     0    16     0     0     2    13
+1    -1    24     0     0     1     0    20
+1    -1    16    16     0     0     0    29
+1    -1     0    48     0     0     0    45
+
+31 8
+   1    0    1    0    0    0    0    0 
+   1    0    0  -16    0    1    1   -2 
+   1    0    0    0    0    1    0   -1 
+   1   -1    8    0    0    1    2    4 
+   1    0    3    2   -2    0    0    0 
+   1   -1   24    0    0    1    0   20 
+   1    1    0  -48    0    2    0   -2 
+   1   -1  -24  -32   32    1    0   36 
+   1    0    0    0    0    0    1   -1 
+   1   -1   24   64  -16    0    0   45 
+   1  -15  120  112    0   15   38   52 
+   1    1   24   32  -32    0    0    0 
+   1    0   -2   -2    2    0    0    1 
+   1   -1    8   16    0    0    1   21 
+   1  -15  120  352    0    0   23  307 
+   1    1   -8  -32    0    1    0   -1 
+   1    1   -8    0    0    0    0    0 
+   1    1   -8  -16    0    0    0    0 
+   1    0   16   16  -16    0    1   -1 
+   1   -1   16   16    0    0    0   29 
+   1   -1   -8    0   16    0    0   37 
+   1   -1    8   32    0    0    0   37 
+   1    1    8    0  -16    0    0    0 
+   1  -15  360  592 -240    0   23  307 
+   1   -1   -6    2   14    0    2   20 
+   1  -15  360  352 -240   15   38   52 
+   1   -1    8   48    0    0    0   45 
+   1    0  -16  -32   16    1    0   14 
+   1   -1   -6  -14   14    1    3    3 
+   1    1  -38  -78   30    2    0   13 
+   1    1   -3  -50    2    2    0   -1 
diff --git a/polly/lib/External/isl/test_inputs/convex2.polylib b/polly/lib/External/isl/test_inputs/convex2.polylib
new file mode 100644
index 00000000000..0bfd737291c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex2.polylib
@@ -0,0 +1,24 @@
+# {i,j,N | 1<=i<=N; 0<=j<=N-1; 2<=N}
+6 5
+1  1  0  0 -1
+1 -1  0  1  0
+1  0  1  0  0
+1  0 -1  1 -1 
+1  0  0  1 -2
+1  0  0  0  1
+# {i,j,N | 1<=i<=N; 1<=j<=N; 2<=N}
+6 5
+1  1  0  0 -1
+1 -1  0  1  0
+1  0  1  0 -1
+1  0 -1  1  0
+1  0  0  1 -2
+1  0  0  0  1
+# {i,j,N | 1<=i<=N; 0<=j<=N; 2<=N}
+6 5
+   1    0    0    1   -2
+   1   -1    0    1    0
+   1    0   -1    1    0
+   1    1    0    0   -1
+   1    0    1    0    0
+   1    0    0    0    1
diff --git a/polly/lib/External/isl/test_inputs/convex3.polylib b/polly/lib/External/isl/test_inputs/convex3.polylib
new file mode 100644
index 00000000000..ea612c6b81f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex3.polylib
@@ -0,0 +1,10 @@
+1 4
+1 1 1 -6
+
+3 4
+1 1 1 -3
+1 1 0 -5
+1 -1 0 10
+
+1 4
+1 1 1 -3
diff --git a/polly/lib/External/isl/test_inputs/convex4.polylib b/polly/lib/External/isl/test_inputs/convex4.polylib
new file mode 100644
index 00000000000..0c086538a66
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex4.polylib
@@ -0,0 +1,9 @@
+1 4
+1 1 1 -6
+
+2 4
+0 1 0 -1
+0 0 1 -4
+
+1 4
+1 1 1 -5
diff --git a/polly/lib/External/isl/test_inputs/convex5.polylib b/polly/lib/External/isl/test_inputs/convex5.polylib
new file mode 100644
index 00000000000..3aae7c26bcf
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex5.polylib
@@ -0,0 +1,12 @@
+2 4
+0 1 0 -2
+0 0 1 -6
+
+2 4
+0 1 0 -1
+0 0 1 -4
+
+3 4
+0 -2 1 -2
+1  1 0 -1
+1 -1 0  2
diff --git a/polly/lib/External/isl/test_inputs/convex6.polylib b/polly/lib/External/isl/test_inputs/convex6.polylib
new file mode 100644
index 00000000000..1bdb4e1c9dd
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex6.polylib
@@ -0,0 +1,17 @@
+3 4
+1   1    1   -2
+1  -1    1    2
+1   0   -1    2
+
+3 4
+1   0    1   -1
+1   1   -1    1
+1  -1   -1    5
+
+6 4
+1  -1    0    4
+1   1    0    0
+1   1    2   -2
+1  -1    2    2
+1   1   -2    4
+1  -1   -2    8
diff --git a/polly/lib/External/isl/test_inputs/convex7.polylib b/polly/lib/External/isl/test_inputs/convex7.polylib
new file mode 100644
index 00000000000..70eb483175b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex7.polylib
@@ -0,0 +1,9 @@
+1 4
+0 0 1 0
+
+2 4
+1 1 -1 1
+1 -1 -1 1
+
+1 4
+1 0 -1 1
diff --git a/polly/lib/External/isl/test_inputs/convex8.polylib b/polly/lib/External/isl/test_inputs/convex8.polylib
new file mode 100644
index 00000000000..ea1b757108c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex8.polylib
@@ -0,0 +1,24 @@
+4 5
+1   1    1    1    0
+1   0   -1    0    0
+1  -1    0    0    2
+1   1    1   -1    0
+
+4 5
+1  -1    1    0    2
+1   1   -2   -2   -1
+1  -1    0    2    3
+1   1    0    0   -1
+
+10 5
+1   1    0    1    0
+1   1    1    0    0
+1   0    1    1    2
+1  -3    1   -1    8
+1  -3    1    1    8
+1   0    1   -1    2
+1   1    0   -1    0
+1   1   -2   -1    0
+1  -1   -3    2    6
+1   1   -5   -2    2
+
diff --git a/polly/lib/External/isl/test_inputs/convex9.polylib b/polly/lib/External/isl/test_inputs/convex9.polylib
new file mode 100644
index 00000000000..f68fca097a9
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/convex9.polylib
@@ -0,0 +1,14 @@
+4 4
+1 1 0 0
+1 -1 0 1
+1 0 1 0
+1 0 -1 10
+
+2 4
+1 1 0 -10
+0 0 -1 5
+
+3 4
+1 1 0 0
+1 0 1 0
+1 0 -1 10
diff --git a/polly/lib/External/isl/test_inputs/devos.pwqp b/polly/lib/External/isl/test_inputs/devos.pwqp
new file mode 100644
index 00000000000..b4525446904
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/devos.pwqp
@@ -0,0 +1 @@
+[U] -> { [i0] -> ((1/3 * U + 2/3 * i0) - [(U + 2i0)/3]) : 2i0 >= -3 - U and 2i0 <= -U and U >= 0 and U <= 10 }
diff --git a/polly/lib/External/isl/test_inputs/equality1.pwqp b/polly/lib/External/isl/test_inputs/equality1.pwqp
new file mode 100644
index 00000000000..eb16a4ba7b8
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/equality1.pwqp
@@ -0,0 +1 @@
+[n] -> { [x] -> 1 + [(x+1)/3] : exists a : x = 3a +1 && 0 <= x <= n }
diff --git a/polly/lib/External/isl/test_inputs/equality2.pwqp b/polly/lib/External/isl/test_inputs/equality2.pwqp
new file mode 100644
index 00000000000..1629a6549ea
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/equality2.pwqp
@@ -0,0 +1 @@
+[n] -> { [x,y] -> x^2 * y : n = 2x + 4y and 0 <= x,y <= 10 }
diff --git a/polly/lib/External/isl/test_inputs/equality3.pwqp b/polly/lib/External/isl/test_inputs/equality3.pwqp
new file mode 100644
index 00000000000..c6f8c3acf19
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/equality3.pwqp
@@ -0,0 +1 @@
+[m,n] -> { [x,y] -> x^2 * y : n = 2x + 4y and 0 <= x,y <= 10 and 3 n = 5 m }
diff --git a/polly/lib/External/isl/test_inputs/equality4.pwqp b/polly/lib/External/isl/test_inputs/equality4.pwqp
new file mode 100644
index 00000000000..49da2e7eff6
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/equality4.pwqp
@@ -0,0 +1 @@
+[m,n] -> { [x,y] -> x^2 * y + m + 13 * n: n = 2x + 4y and 0 <= x,y <= 10 and 3 n = 5 m }
diff --git a/polly/lib/External/isl/test_inputs/equality5.pwqp b/polly/lib/External/isl/test_inputs/equality5.pwqp
new file mode 100644
index 00000000000..09cb7520113
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/equality5.pwqp
@@ -0,0 +1 @@
+[m,n] -> { [x,y,z] -> x^2 * y + z + m + 13 * n: n = 2x + 4y and 0 <= x,y <= 10 and 3 n = 5 m and z = x + y }
diff --git a/polly/lib/External/isl/test_inputs/esced.pip b/polly/lib/External/isl/test_inputs/esced.pip
new file mode 100644
index 00000000000..dbf56ffa0fd
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/esced.pip
@@ -0,0 +1,27 @@
+0 2
+
+-1
+
+16 18
+1 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0
+1 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 0
+1 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 0
+1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0
+1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
+1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
+
+
+0 0 0 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 0
+0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 0
+
+0 -1 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0
+
+0 0 0 0 0 0 -1 0 0 0 0 0 1 -1 0 -1 0 0
+0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0
+0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0
+
+0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0
+0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0
+
+1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1
+1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1
diff --git a/polly/lib/External/isl/test_inputs/ex.pip b/polly/lib/External/isl/test_inputs/ex.pip
new file mode 100644
index 00000000000..a405450b2ad
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/ex.pip
@@ -0,0 +1,9 @@
+1 5
+1 -1  1  1  0 
+
+-1
+
+3 7
+1  0 -1  0  1  0  0 
+1 -1  0  0  0  1  0 
+1  1  1 -1  0  0  0 
diff --git a/polly/lib/External/isl/test_inputs/ex2.pip b/polly/lib/External/isl/test_inputs/ex2.pip
new file mode 100644
index 00000000000..bb59848c8f2
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/ex2.pip
@@ -0,0 +1,9 @@
+1 5
+1 -1 1 1 0
+
+-1
+
+3 7
+1 0 -1 0 1 0 0
+1 -1 0 0 0 1 0
+1 1 1 -1 0 0 0
diff --git a/polly/lib/External/isl/test_inputs/faddeev.pwqp b/polly/lib/External/isl/test_inputs/faddeev.pwqp
new file mode 100644
index 00000000000..e7db61d1bc5
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/faddeev.pwqp
@@ -0,0 +1 @@
+[N] -> { [i, j, k] -> (((4 + 6 * N + 2 * N^2) + (-2 - 2 * N) * j) + ((-2 - N) + j) * k) : j = 1 + i and k = 1 + i and i >= 3 and N <= 100 and i <= N and N >= 10 }
diff --git a/polly/lib/External/isl/test_inputs/fimmel.pip b/polly/lib/External/isl/test_inputs/fimmel.pip
new file mode 100644
index 00000000000..a6dee417c02
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/fimmel.pip
@@ -0,0 +1,12 @@
+0 4
+
+-1
+
+7 6
+1 2  6 0 0 -9
+1 5 -3 0 0 0
+1 2 -10 0 0 15
+1 -2 6 0 0 -3
+1 -2 -6 0 0 17
+1 0 1 -1 0 0
+1 1 0 0 -1 0
diff --git a/polly/lib/External/isl/test_inputs/gist1.polylib b/polly/lib/External/isl/test_inputs/gist1.polylib
new file mode 100644
index 00000000000..802a4eb6ff2
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/gist1.polylib
@@ -0,0 +1,14 @@
+4 5
+0 1 0 0 -1
+0 0 1 0 1
+0 0 0 1 -3
+1 0 0 0 1
+
+4 5
+0 1 0 0 -1
+0 0 1 1 -2
+1 0 0 1 0
+1 0 0 -1 3
+
+1 5
+0 0 1 0 1
diff --git a/polly/lib/External/isl/test_inputs/linearExample.pwqp b/polly/lib/External/isl/test_inputs/linearExample.pwqp
new file mode 100644
index 00000000000..24c53946a0b
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/linearExample.pwqp
@@ -0,0 +1 @@
+[N, M, L] -> { [i, j, k] -> ((1/2 * i + 5 * j) + 1/7 * k) : i >= 0 and k >= -N + i and k >= -M - j and j <= L + i and L >= 0 and L >= -M }
diff --git a/polly/lib/External/isl/test_inputs/max.pip b/polly/lib/External/isl/test_inputs/max.pip
new file mode 100644
index 00000000000..e8af57b9f51
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/max.pip
@@ -0,0 +1,9 @@
+0 3
+
+-1
+
+4 5
+1 -1  0  1  0
+1  0 -1  1  0
+1 -1  3 -2 12
+1  2 -1 -1  3
diff --git a/polly/lib/External/isl/test_inputs/neg.pwqp b/polly/lib/External/isl/test_inputs/neg.pwqp
new file mode 100644
index 00000000000..596a7d77d09
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/neg.pwqp
@@ -0,0 +1 @@
+[n] -> { [i0] -> i0^2 : i0 >= -20 - n and i0 <= n and i0 <= -1 and n >= 0 }
diff --git a/polly/lib/External/isl/test_inputs/negative.pip b/polly/lib/External/isl/test_inputs/negative.pip
new file mode 100644
index 00000000000..45090a59230
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/negative.pip
@@ -0,0 +1,9 @@
+1 3
+#  n  1
+1  1 -1 # n >= 1
+-1
+
+2 4
+#  i  n  1
+1  1  0  1 # i >= -1
+1 -1  1  0 # i <= n
diff --git a/polly/lib/External/isl/test_inputs/philippe.pwqp b/polly/lib/External/isl/test_inputs/philippe.pwqp
new file mode 100644
index 00000000000..1c56e7af707
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/philippe.pwqp
@@ -0,0 +1 @@
+[N] -> { [i, j] -> ((1/2 * i + 1/2 * i^2) + j) : i <= N and j >= 0 and j <= i }
diff --git a/polly/lib/External/isl/test_inputs/philippe3vars.pwqp b/polly/lib/External/isl/test_inputs/philippe3vars.pwqp
new file mode 100644
index 00000000000..8d074962578
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/philippe3vars.pwqp
@@ -0,0 +1 @@
+[N] -> { [i, j, k] -> (((1/2 * i + 1/2 * i^2) + j) + k^3) : i >= 0 and k >= -N + i and k >= -j and j <= i }
diff --git a/polly/lib/External/isl/test_inputs/philippe3vars3pars.pwqp b/polly/lib/External/isl/test_inputs/philippe3vars3pars.pwqp
new file mode 100644
index 00000000000..f81b8cc5a68
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/philippe3vars3pars.pwqp
@@ -0,0 +1 @@
+[N, M, L] -> { [i, j, k] -> (((1/2 * i + 1/2 * i^2) + j) + k^3) : i >= 0 and k >= -N + i and k >= -M - j and j <= L + i and L >= 0 and L >= -M }
diff --git a/polly/lib/External/isl/test_inputs/philippeNeg.pwqp b/polly/lib/External/isl/test_inputs/philippeNeg.pwqp
new file mode 100644
index 00000000000..24dc805ed18
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/philippeNeg.pwqp
@@ -0,0 +1 @@
+[N] -> { [i, j] -> ((1/2 * i + 1/2 * i^2) + j) : i <= N and j >= -1 and j <= i }
diff --git a/polly/lib/External/isl/test_inputs/philippePolynomialCoeff.pwqp b/polly/lib/External/isl/test_inputs/philippePolynomialCoeff.pwqp
new file mode 100644
index 00000000000..e6327c78f35
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/philippePolynomialCoeff.pwqp
@@ -0,0 +1 @@
+[N, M] -> { [i, j] -> ((N * i + (1/5 * N + N^2) * i^2) + 5 * j) : i <= N and j >= 0 and j <= i and M >= 0 }
diff --git a/polly/lib/External/isl/test_inputs/philippePolynomialCoeff1P.pwqp b/polly/lib/External/isl/test_inputs/philippePolynomialCoeff1P.pwqp
new file mode 100644
index 00000000000..ae01d2fcb87
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/philippePolynomialCoeff1P.pwqp
@@ -0,0 +1 @@
+[N] -> { [i, j] -> ((N * i + (1/5 * N + N^2) * i^2) + 5 * j) : i <= N and j >= 0 and j <= i }
diff --git a/polly/lib/External/isl/test_inputs/product.pwqp b/polly/lib/External/isl/test_inputs/product.pwqp
new file mode 100644
index 00000000000..ee48b859ee3
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/product.pwqp
@@ -0,0 +1 @@
+[N] -> { [i0, i1, i2] -> (i0^3 * i1^2 + N * i1 * i2) : i0 >= 0 and i0 <= N and i1 >= 0 and i1 <= N and i2 >= 0 and i2 <= N }
diff --git a/polly/lib/External/isl/test_inputs/seghir-vd.pip b/polly/lib/External/isl/test_inputs/seghir-vd.pip
new file mode 100644
index 00000000000..b5395fbdb38
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/seghir-vd.pip
@@ -0,0 +1,17 @@
+0 6
+
+-1
+
+9 8
+ 0 0 0 1 1 0 0 2
+ 1 2 1 0 0 1 0 0
+ 1 0 1 0 -1 0 0 -1
+ 1 -2 -1 0 0 0 0 -1
+ 1 7 3 0 0 0 0 -1
+ 1 -6 -4 0 1 0 3 1
+ 1 -7 -3 0 0 1 6 4
+ 1 0 0 0 0 0 1 0
+ 1 0 0 0 0 0 0 1
+
+Urs_parms
+Urs_unknowns
diff --git a/polly/lib/External/isl/test_inputs/set.omega b/polly/lib/External/isl/test_inputs/set.omega
new file mode 100644
index 00000000000..ac8485fb9a4
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/set.omega
@@ -0,0 +1 @@
+{[y]: Exists ( alpha : 2alpha = y)}
diff --git a/polly/lib/External/isl/test_inputs/small.pip b/polly/lib/External/isl/test_inputs/small.pip
new file mode 100644
index 00000000000..59557d262bd
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/small.pip
@@ -0,0 +1,9 @@
+0 2
+
+-1
+
+4 4
+1  1  0  0
+1  0  1  0
+1  1 -3 12
+1 -2  1  3
diff --git a/polly/lib/External/isl/test_inputs/sor1d.pip b/polly/lib/External/isl/test_inputs/sor1d.pip
new file mode 100644
index 00000000000..1bef89e5bf1
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/sor1d.pip
@@ -0,0 +1,28 @@
+2 4
+ 1	1	0	0
+ 1	0	1	0
+
+-1
+
+20 8
+ 
+ 0	-1	0	0	0	0	0	2
+ 0	0	-1	0	0	0	0	1
+ 0	0	0	-1	0	0	0	2
+ 0	0	0	0	-1	0	0	4
+ 1	0	0	0	1	0	0	-2
+ 1	-2	0	2	1	0	0	-4
+ 1	0	0	0	-1	0	1	-1
+ 1	2	0	-2	-1	0	0	5
+ 1	0	0	1	0	0	0	-1
+ 1	0	-2	1	0	0	0	0
+ 1	-2	0	2	0	0	1	-5
+ 1	0	0	-1	0	1	0	0
+ 1	0	2	-1	0	0	0	1
+ 1	2	0	-2	0	0	0	3
+ 1	0	1	0	0	0	0	0
+ 1	-2	4	0	0	0	1	-3
+ 1	0	-2	0	0	1	0	0
+ 1	2	-4	0	0	0	0	3
+ 1	2	0	0	0	0	0	1
+ 1	-2	0	0	0	2	1	-5
diff --git a/polly/lib/External/isl/test_inputs/split.pwqp b/polly/lib/External/isl/test_inputs/split.pwqp
new file mode 100644
index 00000000000..1804563600c
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/split.pwqp
@@ -0,0 +1 @@
+[n] -> { [x] -> -1 + [(x+5)/7] : -n - 20 <= x <= n }
diff --git a/polly/lib/External/isl/test_inputs/square.pip b/polly/lib/External/isl/test_inputs/square.pip
new file mode 100644
index 00000000000..7bb3f0c90b5
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/square.pip
@@ -0,0 +1,9 @@
+0 3
+
+-1
+
+4 5
+1  1  0  0  0
+1 -1  0  1  0
+1  0  1  0  0
+1  0 -1  1  0
diff --git a/polly/lib/External/isl/test_inputs/sven.pip b/polly/lib/External/isl/test_inputs/sven.pip
new file mode 100644
index 00000000000..86027691036
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/sven.pip
@@ -0,0 +1,7 @@
+0 3
+
+-1
+
+2 3
+1  1 -4
+1 -1 10
diff --git a/polly/lib/External/isl/test_inputs/test3Deg3Var.pwqp b/polly/lib/External/isl/test_inputs/test3Deg3Var.pwqp
new file mode 100644
index 00000000000..d9a9ea9da01
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/test3Deg3Var.pwqp
@@ -0,0 +1 @@
+[p] -> { [n, m] -> (n + n^3) : n >= 1 and m >= n and m <= p }
diff --git a/polly/lib/External/isl/test_inputs/tobi.pip b/polly/lib/External/isl/test_inputs/tobi.pip
new file mode 100644
index 00000000000..c31beae1b0f
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/tobi.pip
@@ -0,0 +1,15 @@
+2 3
+1 1 -281
+1 -1 14000
+
+-1
+
+6 6
+0  -392     0     8    -1     0
+0   392     8     0     1     0
+1    -1     0     0     0     0
+1     1     0     0     0    35
+1   392     0     0     1     0
+1  -392     0     0    -1   280
+
+Urs_unknowns
diff --git a/polly/lib/External/isl/test_inputs/toplas.pwqp b/polly/lib/External/isl/test_inputs/toplas.pwqp
new file mode 100644
index 00000000000..9c0999540b2
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/toplas.pwqp
@@ -0,0 +1 @@
+[n] -> { [i0, i1] -> (((4 * n - n^2) + (-3/2 + 2 * n) * i0 - 1/2 * i0^2) - i1) : i1 >= -1 + 3n - i0 and i1 >= -1 + 2n - i0 and i0 >= 0 and i1 <= -2 + 4n - i0 and i0 <= -2 + 4n and i0 <= -1 + 3n and i1 >= 0 and i1 <= -1 + n }
diff --git a/polly/lib/External/isl/test_inputs/unexpanded.pwqp b/polly/lib/External/isl/test_inputs/unexpanded.pwqp
new file mode 100644
index 00000000000..5626d3baa41
--- /dev/null
+++ b/polly/lib/External/isl/test_inputs/unexpanded.pwqp
@@ -0,0 +1 @@
+{ [x, y] -> ((x - x^2) * y + (-x + x^2) * y^2) : x >= 0 and x <= 2 and y >= 0 and y <= 2 }
diff --git a/polly/lib/Makefile b/polly/lib/Makefile
index eb9ed8557ec..49e9333d9f0 100644
--- a/polly/lib/Makefile
+++ b/polly/lib/Makefile
@@ -11,6 +11,7 @@ LOADABLE_MODULE = 1
 # TODO: Export symbols for RTTI or EH?
 
 CPP.Flags += $(POLLY_INC)
+C.Flags += $(POLLY_CFLAGS)
 LD.Flags += $(POLLY_LD) $(POLLY_LIB)
 
 include $(LEVEL)/Makefile.config
@@ -32,6 +33,82 @@ POLLY_JSON_FILES= JSON/json_reader.cpp \
 		  JSON/json_value.cpp \
 		  JSON/json_writer.cpp
 
+ISL_FILES= External/isl/basis_reduction_tab.c \
+           External/isl/isl_aff.c \
+           External/isl/isl_affine_hull.c \
+           External/isl/isl_arg.c \
+           External/isl/isl_ast_build.c \
+           External/isl/isl_ast_build_expr.c \
+           External/isl/isl_ast.c \
+           External/isl/isl_ast_codegen.c \
+           External/isl/isl_ast_graft.c \
+           External/isl/isl_band.c \
+           External/isl/isl_bernstein.c \
+           External/isl/isl_blk.c \
+           External/isl/isl_bound.c \
+           External/isl/isl_coalesce.c \
+           External/isl/isl_constraint.c \
+           External/isl/isl_convex_hull.c \
+           External/isl/isl_ctx.c \
+           External/isl/isl_deprecated.c \
+           External/isl/isl_dim.c \
+           External/isl/isl_dim_map.c \
+           External/isl/isl_equalities.c \
+           External/isl/isl_factorization.c \
+           External/isl/isl_farkas.c \
+           External/isl/isl_flow.c \
+           External/isl/isl_fold.c \
+           External/isl/isl_hash.c \
+           External/isl/isl_id.c \
+           External/isl/isl_id_to_ast_expr.c \
+           External/isl/isl_id_to_pw_aff.c \
+           External/isl/isl_ilp.c \
+           External/isl/isl_imath.c \
+           External/isl/isl_input.c \
+           External/isl/isl_local_space.c \
+           External/isl/isl_lp.c \
+           External/isl/isl_map.c \
+           External/isl/isl_map_simplify.c \
+           External/isl/isl_map_subtract.c \
+           External/isl/isl_map_to_basic_set.c \
+           External/isl/isl_mat.c \
+           External/isl/isl_morph.c \
+           External/isl/isl_obj.c \
+           External/isl/isl_options.c \
+           External/isl/isl_output.c \
+           External/isl/isl_point.c \
+           External/isl/isl_polynomial.c \
+           External/isl/isl_printer.c \
+           External/isl/isl_range.c \
+           External/isl/isl_reordering.c \
+           External/isl/isl_sample.c \
+           External/isl/isl_scan.c \
+           External/isl/isl_schedule.c \
+           External/isl/isl_scheduler.c \
+           External/isl/isl_seq.c \
+           External/isl/isl_set_list.c \
+           External/isl/isl_sort.c \
+           External/isl/isl_space.c \
+           External/isl/isl_stream.c \
+           External/isl/isl_tab.c \
+           External/isl/isl_tab_pip.c \
+           External/isl/isl_tarjan.c \
+           External/isl/isl_transitive_closure.c \
+           External/isl/isl_union_map.c \
+           External/isl/isl_val.c \
+           External/isl/isl_val_imath.c \
+           External/isl/isl_vec.c \
+           External/isl/isl_version.c \
+           External/isl/isl_vertices.c \
+           External/isl/print.c \
+           External/isl/imath/gmp_compat.c \
+           External/isl/imath/imath.c \
+           External/isl/imath/imdrover.c \
+           External/isl/imath/imrat.c \
+           External/isl/imath/iprime.c \
+           External/isl/imath/pi.c \
+           External/isl/imath/rsamath.c
+
 SOURCES= Polly.cpp \
 	 Support/GICHelper.cpp \
 	 Support/SCEVValidator.cpp \
@@ -58,7 +135,8 @@ SOURCES= Polly.cpp \
 	 ${GPGPU_FILES} \
 	 ${ISL_CODEGEN_FILES} \
 	 ${POLLY_JSON_FILES} \
-	 ${POLLY_PLUTO_FILES}
+	 ${POLLY_PLUTO_FILES} \
+	 ${ISL_FILES}
 
 #
 # Include Makefile.common so we know what to do.