[MLIR][FlatAffineConstraints][NFC] Move some static functions to be available to Presburger/

[Groverkss]

Summary

This patch moves some static functions from AffineStructures.cpp to
Presburger/Utils.cpp and some to be private members of FlatAffineConstraints
(which will later be moved to IntegerPolyhedron) to allow for a smoother
transition for moving FlatAffineConstraints math functionality to
Presburger/IntegerPolyhedron.

This patch is part of a series of patches for moving math functionality to
Presburger directory.

[arjunp]

Seems good to me. I would wait for a few days to land in case @bondhugula or others have any comments.

mlir/lib/Analysis/Presburger/Utils.cpp

2

I think the C++ is only used for headers, not for .cpp files which are anyway known to be C++.

[bondhugula]

mlir/include/mlir/Analysis/Presburger/Utils.h

24

I don't think a function with such a name should be exposed on mlir::. In fact, this can just be a method on FlatAffineCosntraints.

36

Likewise -- these shouldn't be exposed on mlir::.

[Groverkss]

Address Uday and Arjun's comments

[bondhugula]

mlir/include/mlir/Analysis/Presburger/Utils.h

17

I don't think you need this header. Please use forward declarations instead -- unnecessary header includes impact build time negatively.

20

namespace uses snake case I think -- can you check other examples in the code base?

mlir/lib/Analysis/Presburger/Utils.cpp

116

foundRepr should have been an ArrayRef. I understand you are just moving but this should be fixed.

[Groverkss]

• Address Uday's comments

[Groverkss]

Reverted because this change broke the mlir-nvidia build bot (but surprisingly passed the builds) https://lab.llvm.org/buildbot/#/builders/61/builds/19184

[Groverkss]

• Remove dependency of Presburger/Utils on FlatAffineConstraints

[Groverkss]

Removed dependency on FlatAffineConstraints for Presburger/Utils. This required moving a function from FlatAffineConstraints to IntegerPolyhedron. This will not affect any design since the next patch I plan to send would have moved it anyway.

[mehdi_amini]

In D115869#3209402, @Groverkss wrote:

Reverted because this change broke the mlir-nvidia build bot (but surprisingly passed the builds) https://lab.llvm.org/buildbot/#/builders/61/builds/19184

Thanks for the quick revert! Please think about mentioning the revert reason in the revert commit message itself.

[Groverkss]

In D115869#3209523, @mehdi_amini wrote:

In D115869#3209402, @Groverkss wrote:

Reverted because this change broke the mlir-nvidia build bot (but surprisingly passed the builds) https://lab.llvm.org/buildbot/#/builders/61/builds/19184

Thanks for the quick revert! Please think about mentioning the revert reason in the revert commit message itself.

Sorry. Will keep that in mind next time :)

[arjunp]

LGTM.

mlir/include/mlir/Analysis/AffineStructures.h

protected:
   /// 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();
 
+  /// Searches for a constraint with a non-zero coefficient at `colIdx` in
+  /// equality (isEq=true) or inequality (isEq=false) constraints.
+  /// Returns true and sets row found in search in `rowIdx`, false otherwise.
+  bool findConstraintWithNonZeroAt(unsigned colIdx, bool isEq,
+                                   unsigned *rowIdx) const;
+
+  /// Returns true if the pos^th column is all zero for both inequalities and
+  /// equalities.
+  bool isColZero(unsigned pos) const;
+
   /// 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

mlir/include/mlir/Analysis/Presburger/Utils.h

+//===- Utils.h - General utilities for Presburger library ------*- 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
+//
+//===----------------------------------------------------------------------===//
+//
+// Utility functions required by the Presburger Library.
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef MLIR_ANALYSIS_PRESBURGER_UTILS_H
+#define MLIR_ANALYSIS_PRESBURGER_UTILS_H
+
+#include "mlir/Support/LLVM.h"
+
+namespace mlir {
+
+class FlatAffineConstraints;
+
+namespace presburger_utils {
+
+/// Check if the pos^th identifier can be expressed as a floordiv of an affine
+/// function of other identifiers (where the divisor is a positive constant).
+/// `foundRepr` contains a boolean for each identifier indicating if the
+/// explicit representation for that identifier has already been computed.
+/// Returns the upper and lower bound inequalities using which the floordiv
+/// can be computed. If the representation could be computed, `dividend` and
+/// `denominator` are set. If the representation could not be computed,
+/// `llvm::None` is returned.
+Optional<std::pair<unsigned, unsigned>>
+computeSingleVarRepr(const FlatAffineConstraints &cst, ArrayRef<bool> foundRepr,
+                     unsigned pos, SmallVector<int64_t, 8> &dividend,
+                     unsigned &divisor);
+
+} // namespace presburger_utils
+} // namespace mlir
+
+#endif // MLIR_ANALYSIS_PRESBURGER_UTILS_H

mlir/lib/Analysis/AffineStructures.cpp

 //===- AffineStructures.cpp - MLIR Affine Structures Class-----------------===//
 //
 // 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 affine dialect ops.
 //
 //===----------------------------------------------------------------------===//
 
 #include "mlir/Analysis/AffineStructures.h"
 #include "mlir/Analysis/LinearTransform.h"
 #include "mlir/Analysis/Presburger/Simplex.h"
+#include "mlir/Analysis/Presburger/Utils.h"
 #include "mlir/Dialect/Affine/IR/AffineOps.h"
 #include "mlir/Dialect/Affine/IR/AffineValueMap.h"
 #include "mlir/Dialect/Arithmetic/IR/Arithmetic.h"
 #include "mlir/Dialect/StandardOps/IR/Ops.h"
 #include "mlir/IR/AffineExprVisitor.h"
 #include "mlir/IR/IntegerSet.h"
 #include "mlir/Support/LLVM.h"
 #include "mlir/Support/MathExtras.h"
void FlatAffineValueConstraints::addAffineIfOpDomain(AffineIfOp ifOp) {
   // from the ifOp into the current one.
   mergeAndAlignIdsWithOther(0, &cst);
   append(cst);
 }
 
 // Searches for a constraint with a non-zero coefficient at `colIdx` in
 // equality (isEq=true) or inequality (isEq=false) constraints.
 // Returns true and sets row found in search in `rowIdx`, false otherwise.
-static bool findConstraintWithNonZeroAt(const FlatAffineConstraints &cst,
-                                        unsigned colIdx, bool isEq,
-                                        unsigned *rowIdx) {
-  assert(colIdx < cst.getNumCols() && "position out of bounds");
+bool FlatAffineConstraints::findConstraintWithNonZeroAt(
+    unsigned colIdx, bool isEq, unsigned *rowIdx) const {
+  assert(colIdx < getNumCols() && "position out of bounds");
   auto at = [&](unsigned rowIdx) -> int64_t {
-    return isEq ? cst.atEq(rowIdx, colIdx) : cst.atIneq(rowIdx, colIdx);
+    return isEq ? atEq(rowIdx, colIdx) : atIneq(rowIdx, colIdx);
   };
-  unsigned e = isEq ? cst.getNumEqualities() : cst.getNumInequalities();
+  unsigned e = isEq ? getNumEqualities() : getNumInequalities();
   for (*rowIdx = 0; *rowIdx < e; ++(*rowIdx)) {
     if (at(*rowIdx) != 0) {
       return true;
     }
   }
   return false;
 }
 
bool FlatAffineConstraints::containsPoint(ArrayRef<int64_t> point) const {
   }
   for (unsigned i = 0, e = getNumInequalities(); i < e; ++i) {
     if (valueAt(getInequality(i), point) < 0)
       return false;
   }
   return true;
 }
 
-/// Check if the pos^th identifier can be represented as a division using upper
-/// bound inequality at position `ubIneq` and lower bound inequality at position
-/// `lbIneq`.
-///
-/// Let `id` be the pos^th identifier, then `id` is equivalent to
-/// `expr floordiv divisor` if there are constraints of the form:
-///      0 <= expr - divisor * id <= divisor - 1
-/// Rearranging, we have:
-///       divisor * id - expr + (divisor - 1) >= 0  <-- Lower bound for 'id'
-///      -divisor * id + expr                 >= 0  <-- Upper bound for 'id'
-///
-/// For example:
-///     32*k >= 16*i + j - 31                 <-- Lower bound for 'k'
-///     32*k  <= 16*i + j                     <-- Upper bound for 'k'
-///     expr = 16*i + j, divisor = 32
-///     k = ( 16*i + j ) floordiv 32
-///
-///     4q >= i + j - 2                       <-- Lower bound for 'q'
-///     4q <= i + j + 1                       <-- Upper bound for 'q'
-///     expr = i + j + 1, divisor = 4
-///     q = (i + j + 1) floordiv 4
-//
-/// This function also supports detecting divisions from bounds that are
-/// strictly tighter than the division bounds described above, since tighter
-/// bounds imply the division bounds. For example:
-///     4q - i - j + 2 >= 0                       <-- Lower bound for 'q'
-///    -4q + i + j     >= 0                       <-- Tight upper bound for 'q'
-///
-/// To extract floor divisions with tighter bounds, we assume that that the
-/// constraints are of the form:
-///     c <= expr - divisior * id <= divisor - 1, where 0 <= c <= divisor - 1
-/// Rearranging, we have:
-///     divisor * id - expr + (divisor - 1) >= 0  <-- Lower bound for 'id'
-///    -divisor * id + expr - c             >= 0  <-- Upper bound for 'id'
-///
-/// If successful, `expr` is set to dividend of the division and `divisor` is
-/// set to the denominator of the division.
-static LogicalResult getDivRepr(const FlatAffineConstraints &cst, unsigned pos,
-                                unsigned ubIneq, unsigned lbIneq,
-                                SmallVector<int64_t, 8> &expr,
-                                unsigned &divisor) {
-
-  assert(pos <= cst.getNumIds() && "Invalid identifier position");
-  assert(ubIneq <= cst.getNumInequalities() &&
-         "Invalid upper bound inequality position");
-  assert(lbIneq <= cst.getNumInequalities() &&
-         "Invalid upper bound inequality position");
-
-  // Extract divisor from the lower bound.
-  divisor = cst.atIneq(lbIneq, pos);
-
-  // First, check if the constraints are opposite of each other except the
-  // constant term.
-  unsigned i = 0, e = 0;
-  for (i = 0, e = cst.getNumIds(); i < e; ++i)
-    if (cst.atIneq(ubIneq, i) != -cst.atIneq(lbIneq, i))
-      break;
-
-  if (i < e)
-    return failure();
-
-  // Then, check if the constant term is of the proper form.
-  // Due to the form of the upper/lower bound inequalities, the sum of their
-  // constants is `divisor - 1 - c`. From this, we can extract c:
-  int64_t constantSum = cst.atIneq(lbIneq, cst.getNumCols() - 1) +
-                        cst.atIneq(ubIneq, cst.getNumCols() - 1);
-  int64_t c = divisor - 1 - constantSum;
-
-  // Check if `c` satisfies the condition `0 <= c <= divisor - 1`. This also
-  // implictly checks that `divisor` is positive.
-  if (!(c >= 0 && c <= divisor - 1))
-    return failure();
-
-  // The inequality pair can be used to extract the division.
-  // Set `expr` to the dividend of the division except the constant term, which
-  // is set below.
-  expr.resize(cst.getNumCols(), 0);
-  for (i = 0, e = cst.getNumIds(); i < e; ++i)
-    if (i != pos)
-      expr[i] = cst.atIneq(ubIneq, i);
-
-  // From the upper bound inequality's form, its constant term is equal to the
-  // constant term of `expr`, minus `c`. From this,
-  // constant term of `expr` = constant term of upper bound + `c`.
-  expr.back() = cst.atIneq(ubIneq, cst.getNumCols() - 1) + c;
-
-  return success();
-}
-
-/// Check if the pos^th identifier can be expressed as a floordiv of an affine
-/// function of other identifiers (where the divisor is a positive constant).
-/// `foundRepr` contains a boolean for each identifier indicating if the
-/// explicit representation for that identifier has already been computed.
-/// Returns the upper and lower bound inequalities using which the floordiv can
-/// be computed. If the representation could be computed, `dividend` and
-/// `denominator` are set. If the representation could not be computed,
-/// `llvm::None` is returned.
-static Optional<std::pair<unsigned, unsigned>>
-computeSingleVarRepr(const FlatAffineConstraints &cst,
-                     const SmallVector<bool, 8> &foundRepr, unsigned pos,
-                     SmallVector<int64_t, 8> &dividend, unsigned &divisor) {
-  assert(pos < cst.getNumIds() && "invalid position");
-  assert(foundRepr.size() == cst.getNumIds() &&
-         "Size of foundRepr does not match total number of variables");
-
-  SmallVector<unsigned, 4> lbIndices, ubIndices;
-  cst.getLowerAndUpperBoundIndices(pos, &lbIndices, &ubIndices);
-
-  for (unsigned ubPos : ubIndices) {
-    for (unsigned lbPos : lbIndices) {
-      // Attempt to get divison representation from ubPos, lbPos.
-      if (failed(getDivRepr(cst, pos, ubPos, lbPos, dividend, divisor)))
-        continue;
-
-      // Check if the inequalities depend on a variable for which
-      // an explicit representation has not been found yet.
-      // Exit to avoid circular dependencies between divisions.
-      unsigned c, f;
-      for (c = 0, f = cst.getNumIds(); c < f; ++c) {
-        if (c == pos)
-          continue;
-        if (!foundRepr[c] && dividend[c] != 0)
-          break;
-      }
-
-      // Expression can't be constructed as it depends on a yet unknown
-      // identifier.
-      // TODO: Visit/compute the identifiers in an order so that this doesn't
-      // happen. More complex but much more efficient.
-      if (c < f)
-        continue;
-
-      return std::make_pair(ubPos, lbPos);
-    }
-  }
-
-  return llvm::None;
-}
-
 void FlatAffineConstraints::getLocalReprs(
     std::vector<llvm::Optional<std::pair<unsigned, unsigned>>> &repr) const {
   std::vector<SmallVector<int64_t, 8>> dividends(getNumLocalIds());
   SmallVector<unsigned, 4> denominators(getNumLocalIds());
   getLocalReprs(dividends, denominators, repr);
 }
 
 void FlatAffineConstraints::getLocalReprs(
void FlatAffineConstraints::getLocalReprs(
   unsigned divOffset = getNumDimAndSymbolIds();
   bool changed;
   do {
     // Each time changed is true, at end of this iteration, one or more local
     // vars have been detected as floor divs.
     changed = false;
     for (unsigned i = 0, e = getNumLocalIds(); i < e; ++i) {
       if (!foundRepr[i + divOffset]) {
-        if (auto res = computeSingleVarRepr(*this, foundRepr, divOffset + i,
-                                            dividends[i], denominators[i])) {
+        if (auto res = presburger_utils::computeSingleVarRepr(
+                *this, foundRepr, divOffset + i, dividends[i],
+                denominators[i])) {
           foundRepr[i + divOffset] = true;
           repr[i] = res;
           changed = true;
         }
       }
     }
   } while (changed);
 
if (posStart >= posLimit)
     return 0;
 
   gcdTightenInequalities();
 
   unsigned pivotCol = 0;
   for (pivotCol = posStart; pivotCol < posLimit; ++pivotCol) {
     // Find a row which has a non-zero coefficient in column 'j'.
     unsigned pivotRow;
-    if (!findConstraintWithNonZeroAt(*this, pivotCol, /*isEq=*/true,
-                                     &pivotRow)) {
+    if (!findConstraintWithNonZeroAt(pivotCol, /*isEq=*/true, &pivotRow)) {
       // No pivot row in equalities with non-zero at 'pivotCol'.
-      if (!findConstraintWithNonZeroAt(*this, pivotCol, /*isEq=*/false,
-                                       &pivotRow)) {
+      if (!findConstraintWithNonZeroAt(pivotCol, /*isEq=*/false, &pivotRow)) {
         // If inequalities are also non-zero in 'pivotCol', it can be
         // eliminated.
         continue;
       }
       break;
     }
 
     // Eliminate identifier at 'pivotCol' from each equality row.
static bool detectAsFloorDiv(const FlatAffineConstraints &cst, unsigned pos,
   // Get upper-lower bound pair for this variable.
   SmallVector<bool, 8> foundRepr(cst.getNumIds(), false);
   for (unsigned i = 0, e = cst.getNumIds(); i < e; ++i)
     if (exprs[i])
       foundRepr[i] = true;
 
   SmallVector<int64_t, 8> dividend;
   unsigned divisor;
-  auto ulPair = computeSingleVarRepr(cst, foundRepr, pos, dividend, divisor);
+  auto ulPair = presburger_utils::computeSingleVarRepr(cst, foundRepr, pos,
+                                                       dividend, divisor);
 
   // No upper-lower bound pair found for this var.
   if (!ulPair)
     return false;
 
   // Construct the dividend expression.
   auto dividendExpr = getAffineConstantExpr(dividend.back(), context);
   for (unsigned c = 0, f = cst.getNumIds(); c < f; c++)
for (unsigned pos = 0; pos < getNumIds(); pos++) {
       // identifiers (divisor is a positive constant).
       if (detectAsFloorDiv(*this, pos, context, memo)) {
         changed = true;
         continue;
       }
 
       // Detect an identifier as an expression of other identifiers.
       unsigned idx;
-      if (!findConstraintWithNonZeroAt(*this, pos, /*isEq=*/true, &idx)) {
+      if (!findConstraintWithNonZeroAt(pos, /*isEq=*/true, &idx)) {
         continue;
       }
 
       // Build AffineExpr solving for identifier 'pos' in terms of all others.
       auto expr = getAffineConstantExpr(0, context);
       unsigned j, e;
       for (j = 0, e = getNumIds(); j < e; ++j) {
         if (j == pos)
void FlatAffineValueConstraints::getIneqAsAffineValueMap(
   SmallVector<Value, 4> operands;
   getValues(0, pos, &operands);
   SmallVector<Value, 4> trailingOperands;
   getValues(pos + 1, getNumDimAndSymbolIds(), &trailingOperands);
   operands.append(trailingOperands.begin(), trailingOperands.end());
   vmap.reset(AffineMap::get(numDims - 1, numSyms, boundExpr), operands);
 }
 
-/// Returns true if the pos^th column is all zero for both inequalities and
-/// equalities..
-static bool isColZero(const FlatAffineConstraints &cst, unsigned pos) {
+bool FlatAffineConstraints::isColZero(unsigned pos) const {
   unsigned rowPos;
-  return !findConstraintWithNonZeroAt(cst, pos, /*isEq=*/false, &rowPos) &&
-         !findConstraintWithNonZeroAt(cst, pos, /*isEq=*/true, &rowPos);
+  return !findConstraintWithNonZeroAt(pos, /*isEq=*/false, &rowPos) &&
+         !findConstraintWithNonZeroAt(pos, /*isEq=*/true, &rowPos);
 }
 
 IntegerSet FlatAffineConstraints::getAsIntegerSet(MLIRContext *context) const {
   if (getNumConstraints() == 0)
     // Return universal set (always true): 0 == 0.
     return IntegerSet::get(getNumDimIds(), getNumSymbolIds(),
                            getAffineConstantExpr(/*constant=*/0, context),
                            /*eqFlags=*/true);
 
   // Construct local references.
   SmallVector<AffineExpr, 8> memo(getNumIds(), AffineExpr());
 
   if (failed(computeLocalVars(*this, memo, context))) {
     // Check if the local variables without an explicit representation have
     // zero coefficients everywhere.
     SmallVector<unsigned> noLocalRepVars;
     unsigned numDimsSymbols = getNumDimAndSymbolIds();
     for (unsigned i = numDimsSymbols, e = getNumIds(); i < e; ++i) {
-      if (!memo[i] && !isColZero(*this, /*pos=*/i))
+      if (!memo[i] && !isColZero(/*pos=*/i))
         noLocalRepVars.push_back(i - numDimsSymbols);
     }
     if (!noLocalRepVars.empty()) {
       LLVM_DEBUG({
         llvm::dbgs() << "local variables at position(s) ";
         llvm::interleaveComma(noLocalRepVars, llvm::dbgs());
         llvm::dbgs() << " do not have an explicit representation in:\n";
         this->dump();

mlir/lib/Analysis/Presburger/CMakeLists.txt

 add_mlir_library(MLIRPresburger
   IntegerPolyhedron.cpp
   Matrix.cpp
   Simplex.cpp
+  Utils.cpp
 
   DEPENDS
   MLIRBuiltinLocationAttributesIncGen
 
   LINK_LIBS PUBLIC
   MLIRIR
   )

mlir/lib/Analysis/Presburger/Utils.cpp

+//===- Utils.cpp - General utilities for Presburger library ---------------===//
+//
+// 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
+//
+//===----------------------------------------------------------------------===//
+//
+// Utility functions required by the Presburger Library.
+//
+//===----------------------------------------------------------------------===//
+
+#include "mlir/Analysis/Presburger/Utils.h"
+#include "mlir/Analysis/AffineStructures.h"
+
+using namespace mlir;
+
+/// Check if the pos^th identifier can be represented as a division using upper
+/// bound inequality at position `ubIneq` and lower bound inequality at position
+/// `lbIneq`.
+///
+/// Let `id` be the pos^th identifier, then `id` is equivalent to
+/// `expr floordiv divisor` if there are constraints of the form:
+///      0 <= expr - divisor * id <= divisor - 1
+/// Rearranging, we have:
+///       divisor * id - expr + (divisor - 1) >= 0  <-- Lower bound for 'id'
+///      -divisor * id + expr                 >= 0  <-- Upper bound for 'id'
+///
+/// For example:
+///     32*k >= 16*i + j - 31                 <-- Lower bound for 'k'
+///     32*k  <= 16*i + j                     <-- Upper bound for 'k'
+///     expr = 16*i + j, divisor = 32
+///     k = ( 16*i + j ) floordiv 32
+///
+///     4q >= i + j - 2                       <-- Lower bound for 'q'
+///     4q <= i + j + 1                       <-- Upper bound for 'q'
+///     expr = i + j + 1, divisor = 4
+///     q = (i + j + 1) floordiv 4
+//
+/// This function also supports detecting divisions from bounds that are
+/// strictly tighter than the division bounds described above, since tighter
+/// bounds imply the division bounds. For example:
+///     4q - i - j + 2 >= 0                       <-- Lower bound for 'q'
+///    -4q + i + j     >= 0                       <-- Tight upper bound for 'q'
+///
+/// To extract floor divisions with tighter bounds, we assume that that the
+/// constraints are of the form:
+///     c <= expr - divisior * id <= divisor - 1, where 0 <= c <= divisor - 1
+/// Rearranging, we have:
+///     divisor * id - expr + (divisor - 1) >= 0  <-- Lower bound for 'id'
+///    -divisor * id + expr - c             >= 0  <-- Upper bound for 'id'
+///
+/// If successful, `expr` is set to dividend of the division and `divisor` is
+/// set to the denominator of the division.
+static LogicalResult getDivRepr(const FlatAffineConstraints &cst, unsigned pos,
+                                unsigned ubIneq, unsigned lbIneq,
+                                SmallVector<int64_t, 8> &expr,
+                                unsigned &divisor) {
+
+  assert(pos <= cst.getNumIds() && "Invalid identifier position");
+  assert(ubIneq <= cst.getNumInequalities() &&
+         "Invalid upper bound inequality position");
+  assert(lbIneq <= cst.getNumInequalities() &&
+         "Invalid upper bound inequality position");
+
+  // Extract divisor from the lower bound.
+  divisor = cst.atIneq(lbIneq, pos);
+
+  // First, check if the constraints are opposite of each other except the
+  // constant term.
+  unsigned i = 0, e = 0;
+  for (i = 0, e = cst.getNumIds(); i < e; ++i)
+    if (cst.atIneq(ubIneq, i) != -cst.atIneq(lbIneq, i))
+      break;
+
+  if (i < e)
+    return failure();
+
+  // Then, check if the constant term is of the proper form.
+  // Due to the form of the upper/lower bound inequalities, the sum of their
+  // constants is `divisor - 1 - c`. From this, we can extract c:
+  int64_t constantSum = cst.atIneq(lbIneq, cst.getNumCols() - 1) +
+                        cst.atIneq(ubIneq, cst.getNumCols() - 1);
+  int64_t c = divisor - 1 - constantSum;
+
+  // Check if `c` satisfies the condition `0 <= c <= divisor - 1`. This also
+  // implictly checks that `divisor` is positive.
+  if (!(c >= 0 && c <= divisor - 1))
+    return failure();
+
+  // The inequality pair can be used to extract the division.
+  // Set `expr` to the dividend of the division except the constant term, which
+  // is set below.
+  expr.resize(cst.getNumCols(), 0);
+  for (i = 0, e = cst.getNumIds(); i < e; ++i)
+    if (i != pos)
+      expr[i] = cst.atIneq(ubIneq, i);
+
+  // From the upper bound inequality's form, its constant term is equal to the
+  // constant term of `expr`, minus `c`. From this,
+  // constant term of `expr` = constant term of upper bound + `c`.
+  expr.back() = cst.atIneq(ubIneq, cst.getNumCols() - 1) + c;
+
+  return success();
+}
+
+/// Check if the pos^th identifier can be expressed as a floordiv of an affine
+/// function of other identifiers (where the divisor is a positive constant).
+/// `foundRepr` contains a boolean for each identifier indicating if the
+/// explicit representation for that identifier has already been computed.
+/// Returns the upper and lower bound inequalities using which the floordiv can
+/// be computed. If the representation could be computed, `dividend` and
+/// `denominator` are set. If the representation could not be computed,
+/// `llvm::None` is returned.
+Optional<std::pair<unsigned, unsigned>> presburger_utils::computeSingleVarRepr(
+    const FlatAffineConstraints &cst, ArrayRef<bool> foundRepr, unsigned pos,
+    SmallVector<int64_t, 8> &dividend, unsigned &divisor) {
+  assert(pos < cst.getNumIds() && "invalid position");
+  assert(foundRepr.size() == cst.getNumIds() &&
+         "Size of foundRepr does not match total number of variables");
+
+  SmallVector<unsigned, 4> lbIndices, ubIndices;
+  cst.getLowerAndUpperBoundIndices(pos, &lbIndices, &ubIndices);
+
+  for (unsigned ubPos : ubIndices) {
+    for (unsigned lbPos : lbIndices) {
+      // Attempt to get divison representation from ubPos, lbPos.
+      if (failed(getDivRepr(cst, pos, ubPos, lbPos, dividend, divisor)))
+        continue;
+
+      // Check if the inequalities depend on a variable for which
+      // an explicit representation has not been found yet.
+      // Exit to avoid circular dependencies between divisions.
+      unsigned c, f;
+      for (c = 0, f = cst.getNumIds(); c < f; ++c) {
+        if (c == pos)
+          continue;
+        if (!foundRepr[c] && dividend[c] != 0)
+          break;
+      }
+
+      // Expression can't be constructed as it depends on a yet unknown
+      // identifier.
+      // TODO: Visit/compute the identifiers in an order so that this doesn't
+      // happen. More complex but much more efficient.
+      if (c < f)
+        continue;
+
+      return std::make_pair(ubPos, lbPos);
+    }
+  }
+
+  return llvm::None;
+}