--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/polly/lib/Support/ISLTools.cpp	Thu Feb 13 15:10:13 2020 +0900
@@ -0,0 +1,870 @@
+//===------ ISLTools.cpp ----------------------------------------*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+//
+// Tools, utilities, helpers and extensions useful in conjunction with the
+// Integer Set Library (isl).
+//
+//===----------------------------------------------------------------------===//
+
+#include "polly/Support/ISLTools.h"
+#include "llvm/Support/raw_ostream.h"
+#include <cassert>
+#include <vector>
+
+using namespace polly;
+
+namespace {
+/// Create a map that shifts one dimension by an offset.
+///
+/// Example:
+/// makeShiftDimAff({ [i0, i1] -> [o0, o1] }, 1, -2)
+///   = { [i0, i1] -> [i0, i1 - 1] }
+///
+/// @param Space  The map space of the result. Must have equal number of in- and
+///               out-dimensions.
+/// @param Pos    Position to shift.
+/// @param Amount Value added to the shifted dimension.
+///
+/// @return An isl_multi_aff for the map with this shifted dimension.
+isl::multi_aff makeShiftDimAff(isl::space Space, int Pos, int Amount) {
+  auto Identity = isl::multi_aff::identity(Space);
+  if (Amount == 0)
+    return Identity;
+  auto ShiftAff = Identity.get_aff(Pos);
+  ShiftAff = ShiftAff.set_constant_si(Amount);
+  return Identity.set_aff(Pos, ShiftAff);
+}
+
+/// Construct a map that swaps two nested tuples.
+///
+/// @param FromSpace1 { Space1[] }
+/// @param FromSpace2 { Space2[] }
+///
+/// @return { [Space1[] -> Space2[]] -> [Space2[] -> Space1[]] }
+isl::basic_map makeTupleSwapBasicMap(isl::space FromSpace1,
+                                     isl::space FromSpace2) {
+  // Fast-path on out-of-quota.
+  if (!FromSpace1 || !FromSpace2)
+    return {};
+
+  assert(FromSpace1.is_set());
+  assert(FromSpace2.is_set());
+
+  unsigned Dims1 = FromSpace1.dim(isl::dim::set);
+  unsigned Dims2 = FromSpace2.dim(isl::dim::set);
+
+  isl::space FromSpace =
+      FromSpace1.map_from_domain_and_range(FromSpace2).wrap();
+  isl::space ToSpace = FromSpace2.map_from_domain_and_range(FromSpace1).wrap();
+  isl::space MapSpace = FromSpace.map_from_domain_and_range(ToSpace);
+
+  isl::basic_map Result = isl::basic_map::universe(MapSpace);
+  for (unsigned i = 0u; i < Dims1; i += 1)
+    Result = Result.equate(isl::dim::in, i, isl::dim::out, Dims2 + i);
+  for (unsigned i = 0u; i < Dims2; i += 1) {
+    Result = Result.equate(isl::dim::in, Dims1 + i, isl::dim::out, i);
+  }
+
+  return Result;
+}
+
+/// Like makeTupleSwapBasicMap(isl::space,isl::space), but returns
+/// an isl_map.
+isl::map makeTupleSwapMap(isl::space FromSpace1, isl::space FromSpace2) {
+  isl::basic_map BMapResult = makeTupleSwapBasicMap(FromSpace1, FromSpace2);
+  return isl::map(BMapResult);
+}
+} // anonymous namespace
+
+isl::map polly::beforeScatter(isl::map Map, bool Strict) {
+  isl::space RangeSpace = Map.get_space().range();
+  isl::map ScatterRel =
+      Strict ? isl::map::lex_gt(RangeSpace) : isl::map::lex_ge(RangeSpace);
+  return Map.apply_range(ScatterRel);
+}
+
+isl::union_map polly::beforeScatter(isl::union_map UMap, bool Strict) {
+  isl::union_map Result = isl::union_map::empty(UMap.get_space());
+
+  for (isl::map Map : UMap.get_map_list()) {
+    isl::map After = beforeScatter(Map, Strict);
+    Result = Result.add_map(After);
+  }
+
+  return Result;
+}
+
+isl::map polly::afterScatter(isl::map Map, bool Strict) {
+  isl::space RangeSpace = Map.get_space().range();
+  isl::map ScatterRel =
+      Strict ? isl::map::lex_lt(RangeSpace) : isl::map::lex_le(RangeSpace);
+  return Map.apply_range(ScatterRel);
+}
+
+isl::union_map polly::afterScatter(const isl::union_map &UMap, bool Strict) {
+  isl::union_map Result = isl::union_map::empty(UMap.get_space());
+  for (isl::map Map : UMap.get_map_list()) {
+    isl::map After = afterScatter(Map, Strict);
+    Result = Result.add_map(After);
+  }
+  return Result;
+}
+
+isl::map polly::betweenScatter(isl::map From, isl::map To, bool InclFrom,
+                               bool InclTo) {
+  isl::map AfterFrom = afterScatter(From, !InclFrom);
+  isl::map BeforeTo = beforeScatter(To, !InclTo);
+
+  return AfterFrom.intersect(BeforeTo);
+}
+
+isl::union_map polly::betweenScatter(isl::union_map From, isl::union_map To,
+                                     bool InclFrom, bool InclTo) {
+  isl::union_map AfterFrom = afterScatter(From, !InclFrom);
+  isl::union_map BeforeTo = beforeScatter(To, !InclTo);
+
+  return AfterFrom.intersect(BeforeTo);
+}
+
+isl::map polly::singleton(isl::union_map UMap, isl::space ExpectedSpace) {
+  if (!UMap)
+    return nullptr;
+
+  if (isl_union_map_n_map(UMap.get()) == 0)
+    return isl::map::empty(ExpectedSpace);
+
+  isl::map Result = isl::map::from_union_map(UMap);
+  assert(!Result || Result.get_space().has_equal_tuples(ExpectedSpace));
+
+  return Result;
+}
+
+isl::set polly::singleton(isl::union_set USet, isl::space ExpectedSpace) {
+  if (!USet)
+    return nullptr;
+
+  if (isl_union_set_n_set(USet.get()) == 0)
+    return isl::set::empty(ExpectedSpace);
+
+  isl::set Result(USet);
+  assert(!Result || Result.get_space().has_equal_tuples(ExpectedSpace));
+
+  return Result;
+}
+
+unsigned polly::getNumScatterDims(const isl::union_map &Schedule) {
+  unsigned Dims = 0;
+  for (isl::map Map : Schedule.get_map_list())
+    Dims = std::max(Dims, Map.dim(isl::dim::out));
+  return Dims;
+}
+
+isl::space polly::getScatterSpace(const isl::union_map &Schedule) {
+  if (!Schedule)
+    return nullptr;
+  unsigned Dims = getNumScatterDims(Schedule);
+  isl::space ScatterSpace = Schedule.get_space().set_from_params();
+  return ScatterSpace.add_dims(isl::dim::set, Dims);
+}
+
+isl::union_map polly::makeIdentityMap(const isl::union_set &USet,
+                                      bool RestrictDomain) {
+  isl::union_map Result = isl::union_map::empty(USet.get_space());
+  for (isl::set Set : USet.get_set_list()) {
+    isl::map IdentityMap = isl::map::identity(Set.get_space().map_from_set());
+    if (RestrictDomain)
+      IdentityMap = IdentityMap.intersect_domain(Set);
+    Result = Result.add_map(IdentityMap);
+  }
+  return Result;
+}
+
+isl::map polly::reverseDomain(isl::map Map) {
+  isl::space DomSpace = Map.get_space().domain().unwrap();
+  isl::space Space1 = DomSpace.domain();
+  isl::space Space2 = DomSpace.range();
+  isl::map Swap = makeTupleSwapMap(Space1, Space2);
+  return Map.apply_domain(Swap);
+}
+
+isl::union_map polly::reverseDomain(const isl::union_map &UMap) {
+  isl::union_map Result = isl::union_map::empty(UMap.get_space());
+  for (isl::map Map : UMap.get_map_list()) {
+    auto Reversed = reverseDomain(std::move(Map));
+    Result = Result.add_map(Reversed);
+  }
+  return Result;
+}
+
+isl::set polly::shiftDim(isl::set Set, int Pos, int Amount) {
+  int NumDims = Set.dim(isl::dim::set);
+  if (Pos < 0)
+    Pos = NumDims + Pos;
+  assert(Pos < NumDims && "Dimension index must be in range");
+  isl::space Space = Set.get_space();
+  Space = Space.map_from_domain_and_range(Space);
+  isl::multi_aff Translator = makeShiftDimAff(Space, Pos, Amount);
+  isl::map TranslatorMap = isl::map::from_multi_aff(Translator);
+  return Set.apply(TranslatorMap);
+}
+
+isl::union_set polly::shiftDim(isl::union_set USet, int Pos, int Amount) {
+  isl::union_set Result = isl::union_set::empty(USet.get_space());
+  for (isl::set Set : USet.get_set_list()) {
+    isl::set Shifted = shiftDim(Set, Pos, Amount);
+    Result = Result.add_set(Shifted);
+  }
+  return Result;
+}
+
+isl::map polly::shiftDim(isl::map Map, isl::dim Dim, int Pos, int Amount) {
+  int NumDims = Map.dim(Dim);
+  if (Pos < 0)
+    Pos = NumDims + Pos;
+  assert(Pos < NumDims && "Dimension index must be in range");
+  isl::space Space = Map.get_space();
+  switch (Dim) {
+  case isl::dim::in:
+    Space = Space.domain();
+    break;
+  case isl::dim::out:
+    Space = Space.range();
+    break;
+  default:
+    llvm_unreachable("Unsupported value for 'dim'");
+  }
+  Space = Space.map_from_domain_and_range(Space);
+  isl::multi_aff Translator = makeShiftDimAff(Space, Pos, Amount);
+  isl::map TranslatorMap = isl::map::from_multi_aff(Translator);
+  switch (Dim) {
+  case isl::dim::in:
+    return Map.apply_domain(TranslatorMap);
+  case isl::dim::out:
+    return Map.apply_range(TranslatorMap);
+  default:
+    llvm_unreachable("Unsupported value for 'dim'");
+  }
+}
+
+isl::union_map polly::shiftDim(isl::union_map UMap, isl::dim Dim, int Pos,
+                               int Amount) {
+  isl::union_map Result = isl::union_map::empty(UMap.get_space());
+
+  for (isl::map Map : UMap.get_map_list()) {
+    isl::map Shifted = shiftDim(Map, Dim, Pos, Amount);
+    Result = Result.add_map(Shifted);
+  }
+  return Result;
+}
+
+void polly::simplify(isl::set &Set) {
+  Set = isl::manage(isl_set_compute_divs(Set.copy()));
+  Set = Set.detect_equalities();
+  Set = Set.coalesce();
+}
+
+void polly::simplify(isl::union_set &USet) {
+  USet = isl::manage(isl_union_set_compute_divs(USet.copy()));
+  USet = USet.detect_equalities();
+  USet = USet.coalesce();
+}
+
+void polly::simplify(isl::map &Map) {
+  Map = isl::manage(isl_map_compute_divs(Map.copy()));
+  Map = Map.detect_equalities();
+  Map = Map.coalesce();
+}
+
+void polly::simplify(isl::union_map &UMap) {
+  UMap = isl::manage(isl_union_map_compute_divs(UMap.copy()));
+  UMap = UMap.detect_equalities();
+  UMap = UMap.coalesce();
+}
+
+isl::union_map polly::computeReachingWrite(isl::union_map Schedule,
+                                           isl::union_map Writes, bool Reverse,
+                                           bool InclPrevDef, bool InclNextDef) {
+
+  // { Scatter[] }
+  isl::space ScatterSpace = getScatterSpace(Schedule);
+
+  // { ScatterRead[] -> ScatterWrite[] }
+  isl::map Relation;
+  if (Reverse)
+    Relation = InclPrevDef ? isl::map::lex_lt(ScatterSpace)
+                           : isl::map::lex_le(ScatterSpace);
+  else
+    Relation = InclNextDef ? isl::map::lex_gt(ScatterSpace)
+                           : isl::map::lex_ge(ScatterSpace);
+
+  // { ScatterWrite[] -> [ScatterRead[] -> ScatterWrite[]] }
+  isl::map RelationMap = Relation.range_map().reverse();
+
+  // { Element[] -> ScatterWrite[] }
+  isl::union_map WriteAction = Schedule.apply_domain(Writes);
+
+  // { ScatterWrite[] -> Element[] }
+  isl::union_map WriteActionRev = WriteAction.reverse();
+
+  // { Element[] -> [ScatterUse[] -> ScatterWrite[]] }
+  isl::union_map DefSchedRelation =
+      isl::union_map(RelationMap).apply_domain(WriteActionRev);
+
+  // For each element, at every point in time, map to the times of previous
+  // definitions. { [Element[] -> ScatterRead[]] -> ScatterWrite[] }
+  isl::union_map ReachableWrites = DefSchedRelation.uncurry();
+  if (Reverse)
+    ReachableWrites = ReachableWrites.lexmin();
+  else
+    ReachableWrites = ReachableWrites.lexmax();
+
+  // { [Element[] -> ScatterWrite[]] -> ScatterWrite[] }
+  isl::union_map SelfUse = WriteAction.range_map();
+
+  if (InclPrevDef && InclNextDef) {
+    // Add the Def itself to the solution.
+    ReachableWrites = ReachableWrites.unite(SelfUse).coalesce();
+  } else if (!InclPrevDef && !InclNextDef) {
+    // Remove Def itself from the solution.
+    ReachableWrites = ReachableWrites.subtract(SelfUse);
+  }
+
+  // { [Element[] -> ScatterRead[]] -> Domain[] }
+  return ReachableWrites.apply_range(Schedule.reverse());
+}
+
+isl::union_map
+polly::computeArrayUnused(isl::union_map Schedule, isl::union_map Writes,
+                          isl::union_map Reads, bool ReadEltInSameInst,
+                          bool IncludeLastRead, bool IncludeWrite) {
+  // { Element[] -> Scatter[] }
+  isl::union_map ReadActions = Schedule.apply_domain(Reads);
+  isl::union_map WriteActions = Schedule.apply_domain(Writes);
+
+  // { [Element[] -> DomainWrite[]] -> Scatter[] }
+  isl::union_map EltDomWrites =
+      Writes.reverse().range_map().apply_range(Schedule);
+
+  // { [Element[] -> Scatter[]] -> DomainWrite[] }
+  isl::union_map ReachingOverwrite = computeReachingWrite(
+      Schedule, Writes, true, ReadEltInSameInst, !ReadEltInSameInst);
+
+  // { [Element[] -> Scatter[]] -> DomainWrite[] }
+  isl::union_map ReadsOverwritten =
+      ReachingOverwrite.intersect_domain(ReadActions.wrap());
+
+  // { [Element[] -> DomainWrite[]] -> Scatter[] }
+  isl::union_map ReadsOverwrittenRotated =
+      reverseDomain(ReadsOverwritten).curry().reverse();
+  isl::union_map LastOverwrittenRead = ReadsOverwrittenRotated.lexmax();
+
+  // { [Element[] -> DomainWrite[]] -> Scatter[] }
+  isl::union_map BetweenLastReadOverwrite = betweenScatter(
+      LastOverwrittenRead, EltDomWrites, IncludeLastRead, IncludeWrite);
+
+  // { [Element[] -> Scatter[]] -> DomainWrite[] }
+  isl::union_map ReachingOverwriteZone = computeReachingWrite(
+      Schedule, Writes, true, IncludeLastRead, IncludeWrite);
+
+  // { [Element[] -> DomainWrite[]] -> Scatter[] }
+  isl::union_map ReachingOverwriteRotated =
+      reverseDomain(ReachingOverwriteZone).curry().reverse();
+
+  // { [Element[] -> DomainWrite[]] -> Scatter[] }
+  isl::union_map WritesWithoutReads = ReachingOverwriteRotated.subtract_domain(
+      ReadsOverwrittenRotated.domain());
+
+  return BetweenLastReadOverwrite.unite(WritesWithoutReads)
+      .domain_factor_domain();
+}
+
+isl::union_set polly::convertZoneToTimepoints(isl::union_set Zone,
+                                              bool InclStart, bool InclEnd) {
+  if (!InclStart && InclEnd)
+    return Zone;
+
+  auto ShiftedZone = shiftDim(Zone, -1, -1);
+  if (InclStart && !InclEnd)
+    return ShiftedZone;
+  else if (!InclStart && !InclEnd)
+    return Zone.intersect(ShiftedZone);
+
+  assert(InclStart && InclEnd);
+  return Zone.unite(ShiftedZone);
+}
+
+isl::union_map polly::convertZoneToTimepoints(isl::union_map Zone, isl::dim Dim,
+                                              bool InclStart, bool InclEnd) {
+  if (!InclStart && InclEnd)
+    return Zone;
+
+  auto ShiftedZone = shiftDim(Zone, Dim, -1, -1);
+  if (InclStart && !InclEnd)
+    return ShiftedZone;
+  else if (!InclStart && !InclEnd)
+    return Zone.intersect(ShiftedZone);
+
+  assert(InclStart && InclEnd);
+  return Zone.unite(ShiftedZone);
+}
+
+isl::map polly::convertZoneToTimepoints(isl::map Zone, isl::dim Dim,
+                                        bool InclStart, bool InclEnd) {
+  if (!InclStart && InclEnd)
+    return Zone;
+
+  auto ShiftedZone = shiftDim(Zone, Dim, -1, -1);
+  if (InclStart && !InclEnd)
+    return ShiftedZone;
+  else if (!InclStart && !InclEnd)
+    return Zone.intersect(ShiftedZone);
+
+  assert(InclStart && InclEnd);
+  return Zone.unite(ShiftedZone);
+}
+
+isl::map polly::distributeDomain(isl::map Map) {
+  // Note that we cannot take Map apart into { Domain[] -> Range1[] } and {
+  // Domain[] -> Range2[] } and combine again. We would loose any relation
+  // between Range1[] and Range2[] that is not also a constraint to Domain[].
+
+  isl::space Space = Map.get_space();
+  isl::space DomainSpace = Space.domain();
+  unsigned DomainDims = DomainSpace.dim(isl::dim::set);
+  isl::space RangeSpace = Space.range().unwrap();
+  isl::space Range1Space = RangeSpace.domain();
+  unsigned Range1Dims = Range1Space.dim(isl::dim::set);
+  isl::space Range2Space = RangeSpace.range();
+  unsigned Range2Dims = Range2Space.dim(isl::dim::set);
+
+  isl::space OutputSpace =
+      DomainSpace.map_from_domain_and_range(Range1Space)
+          .wrap()
+          .map_from_domain_and_range(
+              DomainSpace.map_from_domain_and_range(Range2Space).wrap());
+
+  isl::basic_map Translator = isl::basic_map::universe(
+      Space.wrap().map_from_domain_and_range(OutputSpace.wrap()));
+
+  for (unsigned i = 0; i < DomainDims; i += 1) {
+    Translator = Translator.equate(isl::dim::in, i, isl::dim::out, i);
+    Translator = Translator.equate(isl::dim::in, i, isl::dim::out,
+                                   DomainDims + Range1Dims + i);
+  }
+  for (unsigned i = 0; i < Range1Dims; i += 1)
+    Translator = Translator.equate(isl::dim::in, DomainDims + i, isl::dim::out,
+                                   DomainDims + i);
+  for (unsigned i = 0; i < Range2Dims; i += 1)
+    Translator = Translator.equate(isl::dim::in, DomainDims + Range1Dims + i,
+                                   isl::dim::out,
+                                   DomainDims + Range1Dims + DomainDims + i);
+
+  return Map.wrap().apply(Translator).unwrap();
+}
+
+isl::union_map polly::distributeDomain(isl::union_map UMap) {
+  isl::union_map Result = isl::union_map::empty(UMap.get_space());
+  for (isl::map Map : UMap.get_map_list()) {
+    auto Distributed = distributeDomain(Map);
+    Result = Result.add_map(Distributed);
+  }
+  return Result;
+}
+
+isl::union_map polly::liftDomains(isl::union_map UMap, isl::union_set Factor) {
+
+  // { Factor[] -> Factor[] }
+  isl::union_map Factors = makeIdentityMap(Factor, true);
+
+  return Factors.product(UMap);
+}
+
+isl::union_map polly::applyDomainRange(isl::union_map UMap,
+                                       isl::union_map Func) {
+  // This implementation creates unnecessary cross products of the
+  // DomainDomain[] and Func. An alternative implementation could reverse
+  // domain+uncurry,apply Func to what now is the domain, then undo the
+  // preparing transformation. Another alternative implementation could create a
+  // translator map for each piece.
+
+  // { DomainDomain[] }
+  isl::union_set DomainDomain = UMap.domain().unwrap().domain();
+
+  // { [DomainDomain[] -> DomainRange[]] -> [DomainDomain[] -> NewDomainRange[]]
+  // }
+  isl::union_map LifetedFunc = liftDomains(std::move(Func), DomainDomain);
+
+  return UMap.apply_domain(LifetedFunc);
+}
+
+isl::map polly::intersectRange(isl::map Map, isl::union_set Range) {
+  isl::set RangeSet = Range.extract_set(Map.get_space().range());
+  return Map.intersect_range(RangeSet);
+}
+
+isl::map polly::subtractParams(isl::map Map, isl::set Params) {
+  auto MapSpace = Map.get_space();
+  auto ParamsMap = isl::map::universe(MapSpace).intersect_params(Params);
+  return Map.subtract(ParamsMap);
+}
+
+isl::val polly::getConstant(isl::pw_aff PwAff, bool Max, bool Min) {
+  assert(!Max || !Min); // Cannot return min and max at the same time.
+  isl::val Result;
+  isl::stat Stat = PwAff.foreach_piece(
+      [=, &Result](isl::set Set, isl::aff Aff) -> isl::stat {
+        if (Result && Result.is_nan())
+          return isl::stat::ok();
+
+        // TODO: If Min/Max, we can also determine a minimum/maximum value if
+        // Set is constant-bounded.
+        if (!Aff.is_cst()) {
+          Result = isl::val::nan(Aff.get_ctx());
+          return isl::stat::error();
+        }
+
+        isl::val ThisVal = Aff.get_constant_val();
+        if (!Result) {
+          Result = ThisVal;
+          return isl::stat::ok();
+        }
+
+        if (Result.eq(ThisVal))
+          return isl::stat::ok();
+
+        if (Max && ThisVal.gt(Result)) {
+          Result = ThisVal;
+          return isl::stat::ok();
+        }
+
+        if (Min && ThisVal.lt(Result)) {
+          Result = ThisVal;
+          return isl::stat::ok();
+        }
+
+        // Not compatible
+        Result = isl::val::nan(Aff.get_ctx());
+        return isl::stat::error();
+      });
+
+  if (Stat.is_error())
+    return {};
+
+  return Result;
+}
+
+#if !defined(NDEBUG) || defined(LLVM_ENABLE_DUMP)
+static void foreachPoint(const isl::set &Set,
+                         const std::function<void(isl::point P)> &F) {
+  Set.foreach_point([&](isl::point P) -> isl::stat {
+    F(P);
+    return isl::stat::ok();
+  });
+}
+
+static void foreachPoint(isl::basic_set BSet,
+                         const std::function<void(isl::point P)> &F) {
+  foreachPoint(isl::set(BSet), F);
+}
+
+/// Determine the sorting order of the sets @p A and @p B without considering
+/// the space structure.
+///
+/// Ordering is based on the lower bounds of the set's dimensions. First
+/// dimensions are considered first.
+static int flatCompare(const isl::basic_set &A, const isl::basic_set &B) {
+  unsigned ALen = A.dim(isl::dim::set);
+  unsigned BLen = B.dim(isl::dim::set);
+  unsigned Len = std::min(ALen, BLen);
+
+  for (unsigned i = 0; i < Len; i += 1) {
+    isl::basic_set ADim =
+        A.project_out(isl::dim::param, 0, A.dim(isl::dim::param))
+            .project_out(isl::dim::set, i + 1, ALen - i - 1)
+            .project_out(isl::dim::set, 0, i);
+    isl::basic_set BDim =
+        B.project_out(isl::dim::param, 0, B.dim(isl::dim::param))
+            .project_out(isl::dim::set, i + 1, BLen - i - 1)
+            .project_out(isl::dim::set, 0, i);
+
+    isl::basic_set AHull = isl::set(ADim).convex_hull();
+    isl::basic_set BHull = isl::set(BDim).convex_hull();
+
+    bool ALowerBounded =
+        bool(isl::set(AHull).dim_has_any_lower_bound(isl::dim::set, 0));
+    bool BLowerBounded =
+        bool(isl::set(BHull).dim_has_any_lower_bound(isl::dim::set, 0));
+
+    int BoundedCompare = BLowerBounded - ALowerBounded;
+    if (BoundedCompare != 0)
+      return BoundedCompare;
+
+    if (!ALowerBounded || !BLowerBounded)
+      continue;
+
+    isl::pw_aff AMin = isl::set(ADim).dim_min(0);
+    isl::pw_aff BMin = isl::set(BDim).dim_min(0);
+
+    isl::val AMinVal = polly::getConstant(AMin, false, true);
+    isl::val BMinVal = polly::getConstant(BMin, false, true);
+
+    int MinCompare = AMinVal.sub(BMinVal).sgn();
+    if (MinCompare != 0)
+      return MinCompare;
+  }
+
+  // If all the dimensions' lower bounds are equal or incomparable, sort based
+  // on the number of dimensions.
+  return ALen - BLen;
+}
+
+/// Compare the sets @p A and @p B according to their nested space structure.
+/// Returns 0 if the structure is considered equal.
+/// If @p ConsiderTupleLen is false, the number of dimensions in a tuple are
+/// ignored, i.e. a tuple with the same name but different number of dimensions
+/// are considered equal.
+static int structureCompare(const isl::space &ASpace, const isl::space &BSpace,
+                            bool ConsiderTupleLen) {
+  int WrappingCompare = bool(ASpace.is_wrapping()) - bool(BSpace.is_wrapping());
+  if (WrappingCompare != 0)
+    return WrappingCompare;
+
+  if (ASpace.is_wrapping() && BSpace.is_wrapping()) {
+    isl::space AMap = ASpace.unwrap();
+    isl::space BMap = BSpace.unwrap();
+
+    int FirstResult =
+        structureCompare(AMap.domain(), BMap.domain(), ConsiderTupleLen);
+    if (FirstResult != 0)
+      return FirstResult;
+
+    return structureCompare(AMap.range(), BMap.range(), ConsiderTupleLen);
+  }
+
+  std::string AName;
+  if (ASpace.has_tuple_name(isl::dim::set))
+    AName = ASpace.get_tuple_name(isl::dim::set);
+
+  std::string BName;
+  if (BSpace.has_tuple_name(isl::dim::set))
+    BName = BSpace.get_tuple_name(isl::dim::set);
+
+  int NameCompare = AName.compare(BName);
+  if (NameCompare != 0)
+    return NameCompare;
+
+  if (ConsiderTupleLen) {
+    int LenCompare = BSpace.dim(isl::dim::set) - ASpace.dim(isl::dim::set);
+    if (LenCompare != 0)
+      return LenCompare;
+  }
+
+  return 0;
+}
+
+/// Compare the sets @p A and @p B according to their nested space structure. If
+/// the structure is the same, sort using the dimension lower bounds.
+/// Returns an std::sort compatible bool.
+static bool orderComparer(const isl::basic_set &A, const isl::basic_set &B) {
+  isl::space ASpace = A.get_space();
+  isl::space BSpace = B.get_space();
+
+  // Ignoring number of dimensions first ensures that structures with same tuple
+  // names, but different number of dimensions are still sorted close together.
+  int TupleNestingCompare = structureCompare(ASpace, BSpace, false);
+  if (TupleNestingCompare != 0)
+    return TupleNestingCompare < 0;
+
+  int TupleCompare = structureCompare(ASpace, BSpace, true);
+  if (TupleCompare != 0)
+    return TupleCompare < 0;
+
+  return flatCompare(A, B) < 0;
+}
+
+/// Print a string representation of @p USet to @p OS.
+///
+/// The pieces of @p USet are printed in a sorted order. Spaces with equal or
+/// similar nesting structure are printed together. Compared to isl's own
+/// printing function the uses the structure itself as base of the sorting, not
+/// a hash of it. It ensures that e.g. maps spaces with same domain structure
+/// are printed together. Set pieces with same structure are printed in order of
+/// their lower bounds.
+///
+/// @param USet     Polyhedra to print.
+/// @param OS       Target stream.
+/// @param Simplify Whether to simplify the polyhedron before printing.
+/// @param IsMap    Whether @p USet is a wrapped map. If true, sets are
+///                 unwrapped before printing to again appear as a map.
+static void printSortedPolyhedra(isl::union_set USet, llvm::raw_ostream &OS,
+                                 bool Simplify, bool IsMap) {
+  if (!USet) {
+    OS << "<null>\n";
+    return;
+  }
+
+  if (Simplify)
+    simplify(USet);
+
+  // Get all the polyhedra.
+  std::vector<isl::basic_set> BSets;
+
+  for (isl::set Set : USet.get_set_list()) {
+    for (isl::basic_set BSet : Set.get_basic_set_list()) {
+      BSets.push_back(BSet);
+    }
+  }
+
+  if (BSets.empty()) {
+    OS << "{\n}\n";
+    return;
+  }
+
+  // Sort the polyhedra.
+  llvm::sort(BSets, orderComparer);
+
+  // Print the polyhedra.
+  bool First = true;
+  for (const isl::basic_set &BSet : BSets) {
+    std::string Str;
+    if (IsMap)
+      Str = isl::map(BSet.unwrap()).to_str();
+    else
+      Str = isl::set(BSet).to_str();
+    size_t OpenPos = Str.find_first_of('{');
+    assert(OpenPos != std::string::npos);
+    size_t ClosePos = Str.find_last_of('}');
+    assert(ClosePos != std::string::npos);
+
+    if (First)
+      OS << llvm::StringRef(Str).substr(0, OpenPos + 1) << "\n ";
+    else
+      OS << ";\n ";
+
+    OS << llvm::StringRef(Str).substr(OpenPos + 1, ClosePos - OpenPos - 2);
+    First = false;
+  }
+  assert(!First);
+  OS << "\n}\n";
+}
+
+static void recursiveExpand(isl::basic_set BSet, int Dim, isl::set &Expanded) {
+  int Dims = BSet.dim(isl::dim::set);
+  if (Dim >= Dims) {
+    Expanded = Expanded.unite(BSet);
+    return;
+  }
+
+  isl::basic_set DimOnly =
+      BSet.project_out(isl::dim::param, 0, BSet.dim(isl::dim::param))
+          .project_out(isl::dim::set, Dim + 1, Dims - Dim - 1)
+          .project_out(isl::dim::set, 0, Dim);
+  if (!DimOnly.is_bounded()) {
+    recursiveExpand(BSet, Dim + 1, Expanded);
+    return;
+  }
+
+  foreachPoint(DimOnly, [&, Dim](isl::point P) {
+    isl::val Val = P.get_coordinate_val(isl::dim::set, 0);
+    isl::basic_set FixBSet = BSet.fix_val(isl::dim::set, Dim, Val);
+    recursiveExpand(FixBSet, Dim + 1, Expanded);
+  });
+}
+
+/// Make each point of a set explicit.
+///
+/// "Expanding" makes each point a set contains explicit. That is, the result is
+/// a set of singleton polyhedra. Unbounded dimensions are not expanded.
+///
+/// Example:
+///   { [i] : 0 <= i < 2 }
+/// is expanded to:
+///   { [0]; [1] }
+static isl::set expand(const isl::set &Set) {
+  isl::set Expanded = isl::set::empty(Set.get_space());
+  for (isl::basic_set BSet : Set.get_basic_set_list())
+    recursiveExpand(BSet, 0, Expanded);
+  return Expanded;
+}
+
+/// Expand all points of a union set explicit.
+///
+/// @see expand(const isl::set)
+static isl::union_set expand(const isl::union_set &USet) {
+  isl::union_set Expanded = isl::union_set::empty(USet.get_space());
+  for (isl::set Set : USet.get_set_list()) {
+    isl::set SetExpanded = expand(Set);
+    Expanded = Expanded.add_set(SetExpanded);
+  }
+  return Expanded;
+}
+
+LLVM_DUMP_METHOD void polly::dumpPw(const isl::set &Set) {
+  printSortedPolyhedra(Set, llvm::errs(), true, false);
+}
+
+LLVM_DUMP_METHOD void polly::dumpPw(const isl::map &Map) {
+  printSortedPolyhedra(Map.wrap(), llvm::errs(), true, true);
+}
+
+LLVM_DUMP_METHOD void polly::dumpPw(const isl::union_set &USet) {
+  printSortedPolyhedra(USet, llvm::errs(), true, false);
+}
+
+LLVM_DUMP_METHOD void polly::dumpPw(const isl::union_map &UMap) {
+  printSortedPolyhedra(UMap.wrap(), llvm::errs(), true, true);
+}
+
+LLVM_DUMP_METHOD void polly::dumpPw(__isl_keep isl_set *Set) {
+  dumpPw(isl::manage_copy(Set));
+}
+
+LLVM_DUMP_METHOD void polly::dumpPw(__isl_keep isl_map *Map) {
+  dumpPw(isl::manage_copy(Map));
+}
+
+LLVM_DUMP_METHOD void polly::dumpPw(__isl_keep isl_union_set *USet) {
+  dumpPw(isl::manage_copy(USet));
+}
+
+LLVM_DUMP_METHOD void polly::dumpPw(__isl_keep isl_union_map *UMap) {
+  dumpPw(isl::manage_copy(UMap));
+}
+
+LLVM_DUMP_METHOD void polly::dumpExpanded(const isl::set &Set) {
+  printSortedPolyhedra(expand(Set), llvm::errs(), false, false);
+}
+
+LLVM_DUMP_METHOD void polly::dumpExpanded(const isl::map &Map) {
+  printSortedPolyhedra(expand(Map.wrap()), llvm::errs(), false, true);
+}
+
+LLVM_DUMP_METHOD void polly::dumpExpanded(const isl::union_set &USet) {
+  printSortedPolyhedra(expand(USet), llvm::errs(), false, false);
+}
+
+LLVM_DUMP_METHOD void polly::dumpExpanded(const isl::union_map &UMap) {
+  printSortedPolyhedra(expand(UMap.wrap()), llvm::errs(), false, true);
+}
+
+LLVM_DUMP_METHOD void polly::dumpExpanded(__isl_keep isl_set *Set) {
+  dumpExpanded(isl::manage_copy(Set));
+}
+
+LLVM_DUMP_METHOD void polly::dumpExpanded(__isl_keep isl_map *Map) {
+  dumpExpanded(isl::manage_copy(Map));
+}
+
+LLVM_DUMP_METHOD void polly::dumpExpanded(__isl_keep isl_union_set *USet) {
+  dumpExpanded(isl::manage_copy(USet));
+}
+
+LLVM_DUMP_METHOD void polly::dumpExpanded(__isl_keep isl_union_map *UMap) {
+  dumpExpanded(isl::manage_copy(UMap));
+}
+#endif