mlir/Analysis/AffineStructures.h

//===- AffineStructures.h - MLIR Affine Structures Class --------*- 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
//
//===----------------------------------------------------------------------===//
//
// Structures for affine/polyhedral analysis of ML functions.
//
//===----------------------------------------------------------------------===//

#ifndef MLIR_ANALYSIS_AFFINESTRUCTURES_H
#define MLIR_ANALYSIS_AFFINESTRUCTURES_H

#include "mlir/Analysis/Presburger/Matrix.h"
#include "mlir/IR/AffineExpr.h"
#include "mlir/IR/OpDefinition.h"
#include "mlir/Support/LogicalResult.h"

namespace mlir {

class AffineCondition;
class AffineForOp;
class AffineIfOp;
class AffineMap;
class AffineValueMap;
class IntegerSet;
class MLIRContext;
class Value;
class MemRefType;
struct MutableAffineMap;

/// A flat list of affine equalities and inequalities in the form.
/// Inequality: c_0*x_0 + c_1*x_1 + .... + c_{n-1}*x_{n-1} >= 0
/// Equality: c_0*x_0 + c_1*x_1 + .... + c_{n-1}*x_{n-1} == 0
///
/// FlatAffineConstraints stores coefficients in a contiguous buffer (one buffer
/// for equalities and one for inequalities). The size of each buffer is
/// numReservedCols * number of inequalities (or equalities). The reserved size
/// is numReservedCols * numReservedInequalities (or numReservedEqualities). A
/// coefficient (r, c) lives at the location numReservedCols * r + c in the
/// buffer. The extra space between getNumCols() and numReservedCols exists to
/// prevent frequent movement of data when adding columns, especially at the
/// end.
///
/// The identifiers x_0, x_1, ... appear in the order: dimensional identifiers,
/// symbolic identifiers, and local identifiers.  The local identifiers
/// correspond to local/internal variables created when converting from
/// AffineExpr's containing mod's and div's; they are thus needed to increase
/// representational power. Each local identifier is always (by construction) a
/// floordiv of a pure add/mul affine function of dimensional, symbolic, and
/// other local identifiers, in a non-mutually recursive way. Hence, every local
/// identifier can ultimately always be recovered as an affine function of
/// dimensional and symbolic identifiers (involving floordiv's); note however
/// that some floordiv combinations are converted to mod's by AffineExpr
/// construction.
///
class FlatAffineConstraints {
public:
  enum IdKind { Dimension, Symbol, Local };

  /// Constructs a constraint system reserving memory for the specified number
  /// of constraints and identifiers..
  FlatAffineConstraints(unsigned numReservedInequalities,
                        unsigned numReservedEqualities,
                        unsigned numReservedCols, unsigned numDims = 0,
                        unsigned numSymbols = 0, unsigned numLocals = 0,
                        ArrayRef<Optional<Value>> idArgs = {})
      : numIds(numDims + numSymbols + numLocals), numDims(numDims),
        numSymbols(numSymbols),
        equalities(0, numIds + 1, numReservedEqualities, numReservedCols),
        inequalities(0, numIds + 1, numReservedInequalities, numReservedCols) {
    assert(numReservedCols >= numIds + 1);
    assert(idArgs.empty() || idArgs.size() == numIds);
    ids.reserve(numReservedCols);
    if (idArgs.empty())
      ids.resize(numIds, None);
    else
      ids.append(idArgs.begin(), idArgs.end());
  }

  /// Constructs a constraint system with the specified number of
  /// dimensions and symbols.
  FlatAffineConstraints(unsigned numDims = 0, unsigned numSymbols = 0,
                        unsigned numLocals = 0,
                        ArrayRef<Optional<Value>> idArgs = {})
      : FlatAffineConstraints(/*numReservedInequalities=*/0,
                              /*numReservedEqualities=*/0,
                              /*numReservedCols=*/numDims + numSymbols +
                                  numLocals + 1,
                              numDims, numSymbols, numLocals, idArgs) {}

  /// Return a system with no constraints, i.e., one which is satisfied by all
  /// points.
  static FlatAffineConstraints getUniverse(unsigned numDims = 0,
                                           unsigned numSymbols = 0) {
    return FlatAffineConstraints(numDims, numSymbols);
  }

  /// Create a flat affine constraint system from an AffineValueMap or a list of
  /// these. The constructed system will only include equalities.
  explicit FlatAffineConstraints(const AffineValueMap &avm);
  explicit FlatAffineConstraints(ArrayRef<const AffineValueMap *> avmRef);

  /// Creates an affine constraint system from an IntegerSet.
  explicit FlatAffineConstraints(IntegerSet set);

  FlatAffineConstraints(ArrayRef<const AffineValueMap *> avmRef,
                        IntegerSet set);

  FlatAffineConstraints(const MutableAffineMap &map);

  ~FlatAffineConstraints() {}

  // Clears any existing data and reserves memory for the specified constraints.
  void reset(unsigned numReservedInequalities, unsigned numReservedEqualities,
             unsigned numReservedCols, unsigned numDims, unsigned numSymbols,
             unsigned numLocals = 0, ArrayRef<Value> idArgs = {});

  void reset(unsigned numDims = 0, unsigned numSymbols = 0,
             unsigned numLocals = 0, ArrayRef<Value> idArgs = {});

  /// Appends constraints from 'other' into this. This is equivalent to an
  /// intersection with no simplification of any sort attempted.
  void append(const FlatAffineConstraints &other);

  /// Checks for emptiness by performing variable elimination on all
  /// identifiers, running the GCD test on each equality constraint, and
  /// checking for invalid constraints. Returns true if the GCD test fails for
  /// any equality, or if any invalid constraints are discovered on any row.
  /// Returns false otherwise.
  bool isEmpty() const;

  /// Runs the GCD test on all equality constraints. Returns 'true' if this test
  /// fails on any equality. Returns 'false' otherwise.
  /// This test can be used to disprove the existence of a solution. If it
  /// returns true, no integer solution to the equality constraints can exist.
  bool isEmptyByGCDTest() const;

  /// Returns true if the set of constraints is found to have no solution,
  /// false if a solution exists. Uses the same algorithm as findIntegerSample.
  bool isIntegerEmpty() const;

  // Returns a matrix where each row is a vector along which the polytope is
  // bounded. The span of the returned vectors is guaranteed to contain all
  // such vectors. The returned vectors are NOT guaranteed to be linearly
  // independent. This function should not be called on empty sets.
  Matrix getBoundedDirections() const;

  /// Find an integer sample point satisfying the constraints using a
  /// branch and bound algorithm with generalized basis reduction, with some
  /// additional processing using Simplex for unbounded sets.
  ///
  /// Returns an integer sample point if one exists, or an empty Optional
  /// otherwise.
  Optional<SmallVector<int64_t, 8>> findIntegerSample() const;

  /// Returns true if the given point satisfies the constraints, or false
  /// otherwise.
  bool containsPoint(ArrayRef<int64_t> point) const;

  // Clones this object.
  std::unique_ptr<FlatAffineConstraints> clone() const;

  /// Returns the value at the specified equality row and column.
  inline int64_t atEq(unsigned i, unsigned j) const { return equalities(i, j); }
  inline int64_t &atEq(unsigned i, unsigned j) { return equalities(i, j); }

  inline int64_t atIneq(unsigned i, unsigned j) const {
    return inequalities(i, j);
  }

  inline int64_t &atIneq(unsigned i, unsigned j) { return inequalities(i, j); }

  /// Returns the number of columns in the constraint system.
  inline unsigned getNumCols() const { return numIds + 1; }

  inline unsigned getNumEqualities() const { return equalities.getNumRows(); }

  inline unsigned getNumInequalities() const {
    return inequalities.getNumRows();
  }

  inline unsigned getNumReservedEqualities() const {
    return equalities.getNumReservedRows();
  }

  inline unsigned getNumReservedInequalities() const {
    return inequalities.getNumReservedRows();
  }

  inline ArrayRef<int64_t> getEquality(unsigned idx) const {
    return equalities.getRow(idx);
  }

  inline ArrayRef<int64_t> getInequality(unsigned idx) const {
    return inequalities.getRow(idx);
  }

  /// Adds constraints (lower and upper bounds) for the specified 'affine.for'
  /// operation's Value using IR information stored in its bound maps. The
  /// right identifier is first looked up using forOp's Value. Asserts if the
  /// Value corresponding to the 'affine.for' operation isn't found in the
  /// constraint system. Returns failure for the yet unimplemented/unsupported
  /// cases.  Any new identifiers that are found in the bound operands of the
  /// 'affine.for' operation are added as trailing identifiers (either
  /// dimensional or symbolic depending on whether the operand is a valid
  /// symbol).
  //  TODO: add support for non-unit strides.
  LogicalResult addAffineForOpDomain(AffineForOp forOp);

  /// Adds constraints (lower and upper bounds) for each loop in the loop nest
  /// described by the bound maps 'lbMaps' and 'ubMaps' of a computation slice.
  /// Every pair ('lbMaps[i]', 'ubMaps[i]') describes the bounds of a loop in
  /// the nest, sorted outer-to-inner. 'operands' contains the bound operands
  /// for a single bound map. All the bound maps will use the same bound
  /// operands. Note that some loops described by a computation slice might not
  /// exist yet in the IR so the Value attached to those dimension identifiers
  /// might be empty. For that reason, this method doesn't perform Value
  /// look-ups to retrieve the dimension identifier positions. Instead, it
  /// assumes the position of the dim identifiers in the constraint system is
  /// the same as the position of the loop in the loop nest.
  LogicalResult addDomainFromSliceMaps(ArrayRef<AffineMap> lbMaps,
                                       ArrayRef<AffineMap> ubMaps,
                                       ArrayRef<Value> operands);

  /// Adds constraints imposed by the `affine.if` operation. These constraints
  /// are collected from the IntegerSet attached to the given `affine.if`
  /// instance argument (`ifOp`). It is asserted that:
  /// 1) The IntegerSet of the given `affine.if` instance should not contain
  /// semi-affine expressions,
  /// 2) The columns of the constraint system created from `ifOp` should match
  /// the columns in the current one regarding numbers and values.
  void addAffineIfOpDomain(AffineIfOp ifOp);

  /// Adds a lower or an upper bound for the identifier at the specified
  /// position with constraints being drawn from the specified bound map and
  /// operands. If `eq` is true, add a single equality equal to the bound map's
  /// first result expr.
  LogicalResult addLowerOrUpperBound(unsigned pos, AffineMap boundMap,
                                     ValueRange operands, bool eq,
                                     bool lower = true);

  /// Returns the bound for the identifier at `pos` from the inequality at
  /// `ineqPos` as a 1-d affine value map (affine map + operands). The returned
  /// affine value map can either be a lower bound or an upper bound depending
  /// on the sign of atIneq(ineqPos, pos). Asserts if the row at `ineqPos` does
  /// not involve the `pos`th identifier.
  void getIneqAsAffineValueMap(unsigned pos, unsigned ineqPos,
                               AffineValueMap &vmap,
                               MLIRContext *context) const;

  /// Returns the constraint system as an integer set. Returns a null integer
  /// set if the system has no constraints, or if an integer set couldn't be
  /// constructed as a result of a local variable's explicit representation not
  /// being known and such a local variable appearing in any of the constraints.
  IntegerSet getAsIntegerSet(MLIRContext *context) const;

  /// Computes the lower and upper bounds of the first 'num' dimensional
  /// identifiers (starting at 'offset') as an affine map of the remaining
  /// identifiers (dimensional and symbolic). This method is able to detect
  /// identifiers as floordiv's and mod's of affine expressions of other
  /// identifiers with respect to (positive) constants. Sets bound map to a
  /// null AffineMap if such a bound can't be found (or yet unimplemented).
  void getSliceBounds(unsigned offset, unsigned num, MLIRContext *context,
                      SmallVectorImpl<AffineMap> *lbMaps,
                      SmallVectorImpl<AffineMap> *ubMaps);

  /// Adds slice lower bounds represented by lower bounds in 'lbMaps' and upper
  /// bounds in 'ubMaps' to each identifier in the constraint system which has
  /// a value in 'values'. Note that both lower/upper bounds share the same
  /// operand list 'operands'.
  /// This function assumes 'values.size' == 'lbMaps.size' == 'ubMaps.size'.
  /// Note that both lower/upper bounds use operands from 'operands'.
  LogicalResult addSliceBounds(ArrayRef<Value> values,
                               ArrayRef<AffineMap> lbMaps,
                               ArrayRef<AffineMap> ubMaps,
                               ArrayRef<Value> operands);

  // Adds an inequality (>= 0) from the coefficients specified in inEq.
  void addInequality(ArrayRef<int64_t> inEq);
  // Adds an equality from the coefficients specified in eq.
  void addEquality(ArrayRef<int64_t> eq);

  /// Adds a constant lower bound constraint for the specified identifier.
  void addConstantLowerBound(unsigned pos, int64_t lb);
  /// Adds a constant upper bound constraint for the specified identifier.
  void addConstantUpperBound(unsigned pos, int64_t ub);

  /// Adds a new local identifier as the floordiv of an affine function of other
  /// identifiers, the coefficients of which are provided in 'dividend' and with
  /// respect to a positive constant 'divisor'. Two constraints are added to the
  /// system to capture equivalence with the floordiv:
  /// q = dividend floordiv c    <=>   c*q <= dividend <= c*q + c - 1.
  void addLocalFloorDiv(ArrayRef<int64_t> dividend, int64_t divisor);

  /// Adds a constant lower bound constraint for the specified expression.
  void addConstantLowerBound(ArrayRef<int64_t> expr, int64_t lb);
  /// Adds a constant upper bound constraint for the specified expression.
  void addConstantUpperBound(ArrayRef<int64_t> expr, int64_t ub);

  /// Sets the identifier at the specified position to a constant.
  void setIdToConstant(unsigned pos, int64_t val);

  /// Sets the identifier corresponding to the specified Value id to a
  /// constant. Asserts if the 'id' is not found.
  void setIdToConstant(Value id, int64_t val);

  /// Looks up the position of the identifier with the specified Value. Returns
  /// true if found (false otherwise). `pos' is set to the (column) position of
  /// the identifier.
  bool findId(Value id, unsigned *pos) const;

  /// Returns true if an identifier with the specified Value exists, false
  /// otherwise.
  bool containsId(Value id) const;

  /// Swap the posA^th identifier with the posB^th identifier.
  void swapId(unsigned posA, unsigned posB);

  // Add identifiers of the specified kind - specified positions are relative to
  // the kind of identifier. The coefficient column corresponding to the added
  // identifier is initialized to zero. 'id' is the Value corresponding to the
  // identifier that can optionally be provided.
  void addDimId(unsigned pos, Value id = nullptr);
  void addSymbolId(unsigned pos, Value id = nullptr);
  void addLocalId(unsigned pos);
  void addId(IdKind kind, unsigned pos, Value id = nullptr);

  /// Add the specified values as a dim or symbol id depending on its nature, if
  /// it already doesn't exist in the system. `id' has to be either a terminal
  /// symbol or a loop IV, i.e., it cannot be the result affine.apply of any
  /// symbols or loop IVs. The identifier is added to the end of the existing
  /// dims or symbols. Additional information on the identifier is extracted
  /// from the IR and added to the constraint system.
  void addInductionVarOrTerminalSymbol(Value id);

  /// Composes the affine value map with this FlatAffineConstrains, adding the
  /// results of the map as dimensions at the front [0, vMap->getNumResults())
  /// and with the dimensions set to the equalities specified by the value map.
  /// Returns failure if the composition fails (when vMap is a semi-affine map).
  /// The vMap's operand Value's are used to look up the right positions in
  /// the FlatAffineConstraints with which to associate. The dimensional and
  /// symbolic operands of vMap should match 1:1 (in the same order) with those
  /// of this constraint system, but the latter could have additional trailing
  /// operands.
  LogicalResult composeMap(const AffineValueMap *vMap);

  /// Composes an affine map whose dimensions match one to one to the
  /// dimensions of this FlatAffineConstraints. The results of the map 'other'
  /// are added as the leading dimensions of this constraint system. Returns
  /// failure if 'other' is a semi-affine map.
  LogicalResult composeMatchingMap(AffineMap other);

  /// Projects out (aka eliminates) 'num' identifiers starting at position
  /// 'pos'. The resulting constraint system is the shadow along the dimensions
  /// that still exist. This method may not always be integer exact.
  // TODO: deal with integer exactness when necessary - can return a value to
  // mark exactness for example.
  void projectOut(unsigned pos, unsigned num);
  inline void projectOut(unsigned pos) { return projectOut(pos, 1); }

  /// Projects out the identifier that is associate with Value .
  void projectOut(Value id);

  /// Removes the specified identifier from the system.
  void removeId(unsigned pos);

  void removeEquality(unsigned pos);
  void removeInequality(unsigned pos);

  /// Changes the partition between dimensions and symbols. Depending on the new
  /// symbol count, either a chunk of trailing dimensional identifiers becomes
  /// symbols, or some of the leading symbols become dimensions.
  void setDimSymbolSeparation(unsigned newSymbolCount);

  /// Changes all symbol identifiers which are loop IVs to dim identifiers.
  void convertLoopIVSymbolsToDims();

  /// Sets the values.size() identifiers starting at pos to the specified values
  /// and removes them.
  void setAndEliminate(unsigned pos, ArrayRef<int64_t> values);

  /// Tries to fold the specified identifier to a constant using a trivial
  /// equality detection; if successful, the constant is substituted for the
  /// identifier everywhere in the constraint system and then removed from the
  /// system.
  LogicalResult constantFoldId(unsigned pos);

  /// This method calls constantFoldId for the specified range of identifiers,
  /// 'num' identifiers starting at position 'pos'.
  void constantFoldIdRange(unsigned pos, unsigned num);

  /// Updates the constraints to be the smallest bounding (enclosing) box that
  /// contains the points of 'this' set and that of 'other', with the symbols
  /// being treated specially. For each of the dimensions, the min of the lower
  /// bounds (symbolic) and the max of the upper bounds (symbolic) is computed
  /// to determine such a bounding box. `other' is expected to have the same
  /// dimensional identifiers as this constraint system (in the same order).
  ///
  /// Eg: if 'this' is {0 <= d0 <= 127}, 'other' is {16 <= d0 <= 192}, the
  ///      output is {0 <= d0 <= 192}.
  /// 2) 'this' = {s0 + 5 <= d0 <= s0 + 20}, 'other' is {s0 + 1 <= d0 <= s0 +
  ///     9}, output = {s0 + 1 <= d0 <= s0 + 20}.
  /// 3) 'this' = {0 <= d0 <= 5, 1 <= d1 <= 9}, 'other' = {2 <= d0 <= 6, 5 <= d1
  ///     <= 15}, output = {0 <= d0 <= 6, 1 <= d1 <= 15}.
  LogicalResult unionBoundingBox(const FlatAffineConstraints &other);

  /// Returns 'true' if this constraint system and 'other' are in the same
  /// space, i.e., if they are associated with the same set of identifiers,
  /// appearing in the same order. Returns 'false' otherwise.
  bool areIdsAlignedWithOther(const FlatAffineConstraints &other);

  /// Merge and align the identifiers of 'this' and 'other' starting at
  /// 'offset', so that both constraint systems get the union of the contained
  /// identifiers that is dimension-wise and symbol-wise unique; both
  /// constraint systems are updated so that they have the union of all
  /// identifiers, with this's original identifiers appearing first followed by
  /// any of other's identifiers that didn't appear in 'this'. Local
  /// identifiers of each system are by design separate/local and are placed
  /// one after other (this's followed by other's).
  //  Eg: Input: 'this'  has ((%i %j) [%M %N])
  //             'other' has (%k, %j) [%P, %N, %M])
  //      Output: both 'this', 'other' have (%i, %j, %k) [%M, %N, %P]
  //
  void mergeAndAlignIdsWithOther(unsigned offset, FlatAffineConstraints *other);

  unsigned getNumConstraints() const {
    return getNumInequalities() + getNumEqualities();
  }
  inline unsigned getNumIds() const { return numIds; }
  inline unsigned getNumDimIds() const { return numDims; }
  inline unsigned getNumSymbolIds() const { return numSymbols; }
  inline unsigned getNumDimAndSymbolIds() const { return numDims + numSymbols; }
  inline unsigned getNumLocalIds() const {
    return numIds - numDims - numSymbols;
  }

  inline ArrayRef<Optional<Value>> getIds() const {
    return {ids.data(), ids.size()};
  }
  inline MutableArrayRef<Optional<Value>> getIds() {
    return {ids.data(), ids.size()};
  }

  /// Returns the optional Value corresponding to the pos^th identifier.
  inline Optional<Value> getId(unsigned pos) const { return ids[pos]; }
  inline Optional<Value> &getId(unsigned pos) { return ids[pos]; }

  /// Returns the Value associated with the pos^th identifier. Asserts if
  /// no Value identifier was associated.
  inline Value getIdValue(unsigned pos) const {
    assert(ids[pos].hasValue() && "identifier's Value not set");
    return ids[pos].getValue();
  }

  /// Returns the Values associated with identifiers in range [start, end).
  /// Asserts if no Value was associated with one of these identifiers.
  void getIdValues(unsigned start, unsigned end,
                   SmallVectorImpl<Value> *values) const {
    assert((start < numIds || start == end) && "invalid start position");
    assert(end <= numIds && "invalid end position");
    values->clear();
    values->reserve(end - start);
    for (unsigned i = start; i < end; i++) {
      values->push_back(getIdValue(i));
    }
  }
  inline void getAllIdValues(SmallVectorImpl<Value> *values) const {
    getIdValues(0, numIds, values);
  }

  /// Sets Value associated with the pos^th identifier.
  inline void setIdValue(unsigned pos, Value val) {
    assert(pos < numIds && "invalid id position");
    ids[pos] = val;
  }
  /// Sets Values associated with identifiers in the range [start, end).
  void setIdValues(unsigned start, unsigned end, ArrayRef<Value> values) {
    assert((start < numIds || end == start) && "invalid start position");
    assert(end <= numIds && "invalid end position");
    assert(values.size() == end - start);
    for (unsigned i = start; i < end; ++i)
      ids[i] = values[i - start];
  }

  /// Clears this list of constraints and copies other into it.
  void clearAndCopyFrom(const FlatAffineConstraints &other);

  /// Returns the smallest known constant bound for the extent of the specified
  /// identifier (pos^th), i.e., the smallest known constant that is greater
  /// than or equal to 'exclusive upper bound' - 'lower bound' of the
  /// identifier. This constant bound is guaranteed to be non-negative. Returns
  /// None if it's not a constant. This method employs trivial (low complexity /
  /// cost) checks and detection. Symbolic identifiers are treated specially,
  /// i.e., it looks for constant differences between affine expressions
  /// involving only the symbolic identifiers. `lb` and `ub` (along with the
  /// `boundFloorDivisor`) are set to represent the lower and upper bound
  /// associated with the constant difference: `lb`, `ub` have the coefficients,
  /// and boundFloorDivisor, their divisor. `minLbPos` and `minUbPos` if
  /// non-null are set to the position of the constant lower bound and upper
  /// bound respectively (to the same if they are from an equality). Ex: if the
  /// lower bound is [(s0 + s2 - 1) floordiv 32] for a system with three
  /// symbolic identifiers, *lb = [1, 0, 1], lbDivisor = 32. See comments at
  /// function definition for examples.
  Optional<int64_t> getConstantBoundOnDimSize(
      unsigned pos, SmallVectorImpl<int64_t> *lb = nullptr,
      int64_t *boundFloorDivisor = nullptr,
      SmallVectorImpl<int64_t> *ub = nullptr, unsigned *minLbPos = nullptr,
      unsigned *minUbPos = nullptr) const;

  /// Returns the constant lower bound for the pos^th identifier if there is
  /// one; None otherwise.
  Optional<int64_t> getConstantLowerBound(unsigned pos) const;

  /// Returns the constant upper bound for the pos^th identifier if there is
  /// one; None otherwise.
  Optional<int64_t> getConstantUpperBound(unsigned pos) const;

  /// Gets the lower and upper bound of the `offset` + `pos`th identifier
  /// treating [0, offset) U [offset + num, symStartPos) as dimensions and
  /// [symStartPos, getNumDimAndSymbolIds) as symbols, and `pos` lies in
  /// [0, num). The multi-dimensional maps in the returned pair represent the
  /// max and min of potentially multiple affine expressions. The upper bound is
  /// exclusive. `localExprs` holds pre-computed AffineExpr's for all local
  /// identifiers in the system.
  std::pair<AffineMap, AffineMap>
  getLowerAndUpperBound(unsigned pos, unsigned offset, unsigned num,
                        unsigned symStartPos, ArrayRef<AffineExpr> localExprs,
                        MLIRContext *context) const;

  /// Gather positions of all lower and upper bounds of the identifier at `pos`,
  /// and optionally any equalities on it. In addition, the bounds are to be
  /// independent of identifiers in position range [`offset`, `offset` + `num`).
  void
  getLowerAndUpperBoundIndices(unsigned pos,
                               SmallVectorImpl<unsigned> *lbIndices,
                               SmallVectorImpl<unsigned> *ubIndices,
                               SmallVectorImpl<unsigned> *eqIndices = nullptr,
                               unsigned offset = 0, unsigned num = 0) const;

  /// Removes constraints that are independent of (i.e., do not have a
  /// coefficient for) for identifiers in the range [pos, pos + num).
  void removeIndependentConstraints(unsigned pos, unsigned num);

  /// Returns true if the set can be trivially detected as being
  /// hyper-rectangular on the specified contiguous set of identifiers.
  bool isHyperRectangular(unsigned pos, unsigned num) const;

  /// Removes duplicate constraints, trivially true constraints, and constraints
  /// that can be detected as redundant as a result of differing only in their
  /// constant term part. A constraint of the form <non-negative constant> >= 0
  /// is considered trivially true. This method is a linear time method on the
  /// constraints, does a single scan, and updates in place. It also normalizes
  /// constraints by their GCD and performs GCD tightening on inequalities.
  void removeTrivialRedundancy();

  /// A more expensive check to detect redundant inequalities thatn
  /// removeTrivialRedundancy.
  void removeRedundantInequalities();

  /// Removes redundant constraints using Simplex. Although the algorithm can
  /// theoretically take exponential time in the worst case (rare), it is known
  /// to perform much better in the average case. If V is the number of vertices
  /// in the polytope and C is the number of constraints, the algorithm takes
  /// O(VC) time.
  void removeRedundantConstraints();

  // Removes all equalities and inequalities.
  void clearConstraints();

  void print(raw_ostream &os) const;
  void dump() const;

private:
  /// Returns false if the fields corresponding to various identifier counts, or
  /// equality/inequality buffer sizes aren't consistent; true otherwise. This
  /// is meant to be used within an assert internally.
  bool hasConsistentState() const;

  /// Checks all rows of equality/inequality constraints for trivial
  /// contradictions (for example: 1 == 0, 0 >= 1), which may have surfaced
  /// after elimination. Returns 'true' if an invalid constraint is found;
  /// 'false'otherwise.
  bool hasInvalidConstraint() const;

  /// Returns the constant lower bound bound if isLower is true, and the upper
  /// bound if isLower is false.
  template <bool isLower>
  Optional<int64_t> computeConstantLowerOrUpperBound(unsigned pos);

  // Eliminates a single identifier at 'position' from equality and inequality
  // constraints. Returns 'success' if the identifier was eliminated, and
  // 'failure' otherwise.
  inline LogicalResult gaussianEliminateId(unsigned position) {
    return success(gaussianEliminateIds(position, position + 1) == 1);
  }

  // Eliminates identifiers from equality and inequality constraints
  // in column range [posStart, posLimit).
  // Returns the number of variables eliminated.
  unsigned gaussianEliminateIds(unsigned posStart, unsigned posLimit);

  /// Eliminates identifier at the specified position using Fourier-Motzkin
  /// variable elimination, but uses Gaussian elimination if there is an
  /// equality involving that identifier. If the result of the elimination is
  /// integer exact, *isResultIntegerExact is set to true. If 'darkShadow' is
  /// set to true, a potential under approximation (subset) of the rational
  /// shadow / exact integer shadow is computed.
  // See implementation comments for more details.
  void fourierMotzkinEliminate(unsigned pos, bool darkShadow = false,
                               bool *isResultIntegerExact = nullptr);

  /// Tightens inequalities given that we are dealing with integer spaces. This
  /// is similar to the GCD test but applied to inequalities. The constant term
  /// can be reduced to the preceding multiple of the GCD of the coefficients,
  /// i.e.,
  ///  64*i - 100 >= 0  =>  64*i - 128 >= 0 (since 'i' is an integer). This is a
  /// fast method (linear in the number of coefficients).
  void gcdTightenInequalities();

  /// Normalized each constraints by the GCD of its coefficients.
  void normalizeConstraintsByGCD();

  /// Removes identifiers in the column range [idStart, idLimit), and copies any
  /// remaining valid data into place, updates member variables, and resizes
  /// arrays as needed.
  void removeIdRange(unsigned idStart, unsigned idLimit);

  /// Total number of identifiers.
  unsigned numIds;

  /// Number of identifiers corresponding to real dimensions.
  unsigned numDims;

  /// Number of identifiers corresponding to symbols (unknown but constant for
  /// analysis).
  unsigned numSymbols;

  /// Coefficients of affine equalities (in == 0 form).
  Matrix equalities;

  /// Coefficients of affine inequalities (in >= 0 form).
  Matrix inequalities;

  /// Values corresponding to the (column) identifiers of this constraint
  /// system appearing in the order the identifiers correspond to columns.
  /// Temporary ones or those that aren't associated to any Value are set to
  /// None.
  SmallVector<Optional<Value>, 8> ids;

  /// A parameter that controls detection of an unrealistic number of
  /// constraints. If the number of constraints is this many times the number of
  /// variables, we consider such a system out of line with the intended use
  /// case of FlatAffineConstraints.
  // The rationale for 32 is that in the typical simplest of cases, an
  // identifier is expected to have one lower bound and one upper bound
  // constraint. With a level of tiling or a connection to another identifier
  // through a div or mod, an extra pair of bounds gets added. As a limit, we
  // don't expect an identifier to have more than 32 lower/upper/equality
  // constraints. This is conservatively set low and can be raised if needed.
  constexpr static unsigned kExplosionFactor = 32;
};

/// Flattens 'expr' into 'flattenedExpr', which contains the coefficients of the
/// dimensions, symbols, and additional variables that represent floor divisions
/// of dimensions, symbols, and in turn other floor divisions.  Returns failure
/// if 'expr' could not be flattened (i.e., semi-affine is not yet handled).
/// 'cst' contains constraints that connect newly introduced local identifiers
/// to existing dimensional and symbolic identifiers. See documentation for
/// AffineExprFlattener on how mod's and div's are flattened.
LogicalResult getFlattenedAffineExpr(AffineExpr expr, unsigned numDims,
                                     unsigned numSymbols,
                                     SmallVectorImpl<int64_t> *flattenedExpr,
                                     FlatAffineConstraints *cst = nullptr);

/// Flattens the result expressions of the map to their corresponding flattened
/// forms and set in 'flattenedExprs'. Returns failure if any expression in the
/// map could not be flattened (i.e., semi-affine is not yet handled). 'cst'
/// contains constraints that connect newly introduced local identifiers to
/// existing dimensional and / symbolic identifiers. See documentation for
/// AffineExprFlattener on how mod's and div's are flattened. For all affine
/// expressions that share the same operands (like those of an affine map), this
/// method should be used instead of repeatedly calling getFlattenedAffineExpr
/// since local variables added to deal with div's and mod's will be reused
/// across expressions.
LogicalResult
getFlattenedAffineExprs(AffineMap map,
                        std::vector<SmallVector<int64_t, 8>> *flattenedExprs,
                        FlatAffineConstraints *cst = nullptr);
LogicalResult
getFlattenedAffineExprs(IntegerSet set,
                        std::vector<SmallVector<int64_t, 8>> *flattenedExprs,
                        FlatAffineConstraints *cst = nullptr);

} // end namespace mlir.

#endif // MLIR_ANALYSIS_AFFINESTRUCTURES_H