LoopUtils.cpp

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//===- LoopUtils.cpp ---- Misc utilities for loop transformation ----------===//
//
// 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
//
//===----------------------------------------------------------------------===//
//
// This file implements miscellaneous loop transformation routines.
//
//===----------------------------------------------------------------------===//
 
#include "mlir/Dialect/Affine/LoopUtils.h"
#include "mlir/Analysis/SliceAnalysis.h"
#include "mlir/Dialect/Affine/Analysis/LoopAnalysis.h"
#include "mlir/Dialect/Affine/Analysis/Utils.h"
#include "mlir/Dialect/Affine/IR/AffineValueMap.h"
#include "mlir/Dialect/Affine/Utils.h"
#include "mlir/Dialect/Func/IR/FuncOps.h"
#include "mlir/Dialect/MemRef/IR/MemRef.h"
#include "mlir/Dialect/SCF/IR/SCF.h"
#include "mlir/IR/IRMapping.h"
#include "mlir/IR/IntegerSet.h"
#include "mlir/Transforms/GreedyPatternRewriteDriver.h"
#include "llvm/ADT/MapVector.h"
#include "llvm/ADT/SmallPtrSet.h"
#include "llvm/Support/Debug.h"
#include "llvm/Support/raw_ostream.h"
#include <optional>
 
#define DEBUG_TYPE "loop-utils"
 
using namespace mlir;
using namespace affine;
using namespace presburger;
using llvm::SmallMapVector;
 
/// Computes the cleanup loop lower bound of the loop being unrolled with
/// the specified unroll factor; this bound will also be upper bound of the main
/// part of the unrolled loop. Computes the bound as an AffineMap with its
/// operands or a null map when the trip count can't be expressed as an affine
/// expression.
static void
getCleanupLoopLowerBound(AffineForOp forOp, unsigned unrollFactor,
                         AffineMap &cleanupLbMap,
                         SmallVectorImpl<Value> &cleanupLbOperands) {
  AffineMap tripCountMap;
  SmallVector<Value, 4> tripCountOperands;
  getTripCountMapAndOperands(forOp, &tripCountMap, &tripCountOperands);
  // Trip count can't be computed.
  if (!tripCountMap) {
    cleanupLbMap = AffineMap();
    return;
  }
 
  OpBuilder b(forOp);
  auto lbMap = forOp.getLowerBoundMap();
  auto lb = b.create<AffineApplyOp>(forOp.getLoc(), lbMap,
                                    forOp.getLowerBoundOperands());
 
  // For each upper bound expr, get the range.
  // Eg: affine.for %i = lb to min (ub1, ub2),
  // where tripCountExprs yield (tr1, tr2), we create affine.apply's:
  // lb + tr1 - tr1 % ufactor, lb + tr2 - tr2 % ufactor; the results of all
  // these affine.apply's make up the cleanup loop lower bound.
  SmallVector<AffineExpr, 4> bumpExprs(tripCountMap.getNumResults());
  SmallVector<Value, 4> bumpValues(tripCountMap.getNumResults());
  int64_t step = forOp.getStepAsInt();
  for (unsigned i = 0, e = tripCountMap.getNumResults(); i < e; i++) {
    auto tripCountExpr = tripCountMap.getResult(i);
    bumpExprs[i] = (tripCountExpr - tripCountExpr % unrollFactor) * step;
    auto bumpMap = AffineMap::get(tripCountMap.getNumDims(),
                                  tripCountMap.getNumSymbols(), bumpExprs[i]);
    bumpValues[i] =
        b.create<AffineApplyOp>(forOp.getLoc(), bumpMap, tripCountOperands);
  }
 
  SmallVector<AffineExpr, 4> newUbExprs(tripCountMap.getNumResults());
  for (unsigned i = 0, e = bumpExprs.size(); i < e; i++)
    newUbExprs[i] = b.getAffineDimExpr(0) + b.getAffineDimExpr(i + 1);
 
  cleanupLbOperands.clear();
  cleanupLbOperands.push_back(lb);
  cleanupLbOperands.append(bumpValues.begin(), bumpValues.end());
  cleanupLbMap = AffineMap::get(1 + tripCountMap.getNumResults(), 0, newUbExprs,
                                b.getContext());
  // Simplify the cleanupLbMap + cleanupLbOperands.
  fullyComposeAffineMapAndOperands(&cleanupLbMap, &cleanupLbOperands);
  cleanupLbMap = simplifyAffineMap(cleanupLbMap);
  canonicalizeMapAndOperands(&cleanupLbMap, &cleanupLbOperands);
  // Remove any affine.apply's that became dead from the simplification above.
  for (auto v : bumpValues)
    if (v.use_empty())
      v.getDefiningOp()->erase();
 
  if (lb.use_empty())
    lb.erase();
}
 
/// Helper to replace uses of loop carried values (iter_args) and loop
/// yield values while promoting single iteration affine.for ops.
static void replaceIterArgsAndYieldResults(AffineForOp forOp) {
  // Replace uses of iter arguments with iter operands (initial values).
  auto iterOperands = forOp.getInits();
  auto iterArgs = forOp.getRegionIterArgs();
  for (auto e : llvm::zip(iterOperands, iterArgs))
    std::get<1>(e).replaceAllUsesWith(std::get<0>(e));
 
  // Replace uses of loop results with the values yielded by the loop.
  auto outerResults = forOp.getResults();
  auto innerResults = forOp.getBody()->getTerminator()->getOperands();
  for (auto e : llvm::zip(outerResults, innerResults))
    std::get<0>(e).replaceAllUsesWith(std::get<1>(e));
}
 
/// Promotes the loop body of a forOp to its containing block if the forOp
/// was known to have a single iteration.
LogicalResult mlir::affine::promoteIfSingleIteration(AffineForOp forOp) {
  std::optional<uint64_t> tripCount = getConstantTripCount(forOp);
  if (!tripCount || *tripCount != 1)
    return failure();
 
  // TODO: extend this for arbitrary affine bounds.
  if (forOp.getLowerBoundMap().getNumResults() != 1)
    return failure();
 
  // Replaces all IV uses to its single iteration value.
  auto iv = forOp.getInductionVar();
  auto *parentBlock = forOp->getBlock();
  if (!iv.use_empty()) {
    if (forOp.hasConstantLowerBound()) {
      OpBuilder topBuilder(forOp->getParentOfType<func::FuncOp>().getBody());
      auto constOp = topBuilder.create<arith::ConstantIndexOp>(
          forOp.getLoc(), forOp.getConstantLowerBound());
      iv.replaceAllUsesWith(constOp);
    } else {
      auto lbOperands = forOp.getLowerBoundOperands();
      auto lbMap = forOp.getLowerBoundMap();
      OpBuilder builder(forOp);
      if (lbMap == builder.getDimIdentityMap()) {
        // No need of generating an affine.apply.
        iv.replaceAllUsesWith(lbOperands[0]);
      } else {
        auto affineApplyOp =
            builder.create<AffineApplyOp>(forOp.getLoc(), lbMap, lbOperands);
        iv.replaceAllUsesWith(affineApplyOp);
      }
    }
  }
 
  replaceIterArgsAndYieldResults(forOp);
 
  // Move the loop body operations, except for its terminator, to the loop's
  // containing block.
  forOp.getBody()->back().erase();
  parentBlock->getOperations().splice(Block::iterator(forOp),
                                      forOp.getBody()->getOperations());
  forOp.erase();
  return success();
}
 
/// Generates an affine.for op with the specified lower and upper bounds
/// while generating the right IV remappings to realize shifts for operations in
/// its body. The operations that go into the loop body are specified in
/// opGroupQueue starting from the specified offset, and in that order. The
/// first element of the pair specifies the shift applied to that group of
/// operations; the shift is multiplied by the loop step before being applied.
/// Returns nullptr if the generated loop simplifies to a single iteration one.
static AffineForOp generateShiftedLoop(
    AffineMap lbMap, AffineMap ubMap,
    const std::vector<std::pair<uint64_t, ArrayRef<Operation *>>> &opGroupQueue,
    unsigned offset, AffineForOp srcForOp, OpBuilder b) {
  auto lbOperands = srcForOp.getLowerBoundOperands();
  auto ubOperands = srcForOp.getUpperBoundOperands();
 
  assert(lbMap.getNumInputs() == lbOperands.size());
  assert(ubMap.getNumInputs() == ubOperands.size());
 
  auto loopChunk =
      b.create<AffineForOp>(srcForOp.getLoc(), lbOperands, lbMap, ubOperands,
                            ubMap, srcForOp.getStepAsInt());
  auto loopChunkIV = loopChunk.getInductionVar();
  auto srcIV = srcForOp.getInductionVar();
 
  IRMapping operandMap;
 
  auto bodyBuilder = OpBuilder::atBlockTerminator(loopChunk.getBody());
  for (const auto &it : llvm::drop_begin(opGroupQueue, offset)) {
    uint64_t shift = it.first;
    auto ops = it.second;
    // All 'same shift' operations get added with their operands being
    // remapped to results of cloned operations, and their IV used remapped.
    // Generate the remapping if the shift is not zero: remappedIV = newIV -
    // shift.
    if (!srcIV.use_empty() && shift != 0) {
      auto ivRemap = bodyBuilder.create<AffineApplyOp>(
          srcForOp.getLoc(),
          bodyBuilder.getSingleDimShiftAffineMap(
              -static_cast<int64_t>(srcForOp.getStepAsInt() * shift)),
          loopChunkIV);
      operandMap.map(srcIV, ivRemap);
    } else {
      operandMap.map(srcIV, loopChunkIV);
    }
    for (auto *op : ops)
      bodyBuilder.clone(*op, operandMap);
  };
  if (succeeded(promoteIfSingleIteration(loopChunk)))
    return AffineForOp();
  return loopChunk;
}
 
// The skewing of operations with respect to one another can be used for
// example to allow overlap of asynchronous operations (such as DMA
// communication) with computation, or just relative shifting of operations
// for better register reuse, locality or parallelism. As such, the shifts are
// typically expected to be at most of the order of the number of operations.
// This method should not be used as a substitute for loop distribution/fission.
// This method uses an algorithm// in time linear in the number of operations
// in the body of the for loop - (using the 'sweep line' paradigm). This method
// asserts preservation of SSA dominance. A check for that as well as that for
// memory-based dependence preservation check rests with the users of this
// method.
LogicalResult mlir::affine::affineForOpBodySkew(AffineForOp forOp,
                                                ArrayRef<uint64_t> shifts,
                                                bool unrollPrologueEpilogue) {
  assert(forOp.getBody()->getOperations().size() == shifts.size() &&
         "too few/many shifts");
  if (forOp.getBody()->begin() == std::prev(forOp.getBody()->end()))
    return success();
 
  // If the trip counts aren't constant, we would need versioning and
  // conditional guards (or context information to prevent such versioning). The
  // better way to pipeline for such loops is to first tile them and extract
  // constant trip count "full tiles" before applying this.
  auto mayBeConstTripCount = getConstantTripCount(forOp);
  if (!mayBeConstTripCount) {
    LLVM_DEBUG(forOp.emitRemark("non-constant trip count loop not handled"));
    return success();
  }
  uint64_t tripCount = *mayBeConstTripCount;
 
  assert(isOpwiseShiftValid(forOp, shifts) &&
         "shifts will lead to an invalid transformation\n");
 
  int64_t step = forOp.getStepAsInt();
 
  unsigned numChildOps = shifts.size();
 
  // Do a linear time (counting) sort for the shifts.
  uint64_t maxShift = *llvm::max_element(shifts);
  if (maxShift >= numChildOps) {
    // Large shifts are not the typical use case.
    forOp.emitWarning("not shifting because shifts are unrealistically large");
    return success();
  }
 
  // An array of operation groups sorted by shift amount; each group has all
  // operations with the same shift in the order in which they appear in the
  // body of the 'affine.for' op.
  std::vector<std::vector<Operation *>> sortedOpGroups(maxShift + 1);
  unsigned pos = 0;
  for (auto &op : forOp.getBody()->without_terminator()) {
    auto shift = shifts[pos++];
    sortedOpGroups[shift].push_back(&op);
  }
 
  // Unless the shifts have a specific pattern (which actually would be the
  // common use case), prologue and epilogue are not meaningfully defined.
  // Nevertheless, if 'unrollPrologueEpilogue' is set, we will treat the first
  // loop generated as the prologue and the last as epilogue and unroll these
  // fully.
  AffineForOp prologue, epilogue;
 
  // Do a sweep over the sorted shifts while storing open groups in a
  // vector, and generating loop portions as necessary during the sweep. A block
  // of operations is paired with its shift.
  std::vector<std::pair<uint64_t, ArrayRef<Operation *>>> opGroupQueue;
 
  auto origLbMap = forOp.getLowerBoundMap();
  uint64_t lbShift = 0;
  OpBuilder b(forOp);
  for (uint64_t d = 0, e = sortedOpGroups.size(); d < e; ++d) {
    // If nothing is shifted by d, continue.
    if (sortedOpGroups[d].empty())
      continue;
    if (!opGroupQueue.empty()) {
      assert(d > 0 &&
             "Queue expected to be empty when the first block is found");
      // The interval for which the loop needs to be generated here is:
      // [lbShift, min(lbShift + tripCount, d)) and the body of the
      // loop needs to have all operations in opQueue in that order.
      AffineForOp res;
      if (lbShift + tripCount * step < d * step) {
        res = generateShiftedLoop(
            b.getShiftedAffineMap(origLbMap, lbShift),
            b.getShiftedAffineMap(origLbMap, lbShift + tripCount * step),
            opGroupQueue, /*offset=*/0, forOp, b);
        // Entire loop for the queued op groups generated, empty it.
        opGroupQueue.clear();
        lbShift += tripCount * step;
      } else {
        res = generateShiftedLoop(b.getShiftedAffineMap(origLbMap, lbShift),
                                  b.getShiftedAffineMap(origLbMap, d),
                                  opGroupQueue, /*offset=*/0, forOp, b);
        lbShift = d * step;
      }
 
      if (res) {
        // Simplify/canonicalize the affine.for.
        RewritePatternSet patterns(res.getContext());
        AffineForOp::getCanonicalizationPatterns(patterns, res.getContext());
        GreedyRewriteConfig config;
        config.strictMode = GreedyRewriteStrictness::ExistingOps;
        bool erased;
        (void)applyOpPatternsAndFold(res.getOperation(), std::move(patterns),
                                     config, /*changed=*/nullptr, &erased);
        if (!erased && !prologue)
          prologue = res;
        if (!erased)
          epilogue = res;
      }
    } else {
      // Start of first interval.
      lbShift = d * step;
    }
    // Augment the list of operations that get into the current open interval.
    opGroupQueue.emplace_back(d, sortedOpGroups[d]);
  }
 
  // Those operations groups left in the queue now need to be processed (FIFO)
  // and their loops completed.
  for (unsigned i = 0, e = opGroupQueue.size(); i < e; ++i) {
    uint64_t ubShift = (opGroupQueue[i].first + tripCount) * step;
    epilogue = generateShiftedLoop(b.getShiftedAffineMap(origLbMap, lbShift),
                                   b.getShiftedAffineMap(origLbMap, ubShift),
                                   opGroupQueue, /*offset=*/i, forOp, b);
    lbShift = ubShift;
    if (!prologue)
      prologue = epilogue;
  }
 
  // Erase the original for op.
  forOp.erase();
 
  if (unrollPrologueEpilogue && prologue)
    (void)loopUnrollFull(prologue);
  if (unrollPrologueEpilogue && !epilogue && epilogue != prologue)
    (void)loopUnrollFull(epilogue);
 
  return success();
}
 
/// Checks whether a loop nest is hyper-rectangular or not.
static LogicalResult
checkIfHyperRectangular(MutableArrayRef<AffineForOp> input) {
  FlatAffineValueConstraints cst;
  SmallVector<Operation *, 8> ops(input.begin(), input.end());
  // 0-d or 1-d is trivially hyper-rectangular.
  if (input.size() <= 1)
    return success();
  if (failed(getIndexSet(ops, &cst))) {
    LLVM_DEBUG(llvm::dbgs() << "Index set computation failed!\n");
    return failure();
  }
  if (!cst.isHyperRectangular(0, input.size())) {
    LLVM_DEBUG(llvm::dbgs()
               << "Non-hyperrectangular nests not supported for tiling!\n");
    return failure();
  }
  return success();
}
 
/// Check if the input nest is supported for tiling and whether tiling would be
/// legal or not.
template <typename t>
static LogicalResult performPreTilingChecks(MutableArrayRef<AffineForOp> input,
                                            ArrayRef<t> tileSizes) {
  assert(input.size() == tileSizes.size() && "Too few/many tile sizes");
 
  if (llvm::any_of(input,
                   [](AffineForOp op) { return op.getNumResults() > 0; })) {
    LLVM_DEBUG(llvm::dbgs()
               << "Cannot tile nest where a loop has yield values\n");
    return failure();
  }
 
  // Check if the supplied `for` ops are all successively nested.
  if (!isPerfectlyNested(input)) {
    LLVM_DEBUG(llvm::dbgs() << "input loops not perfectly nested");
    return failure();
  }
 
  //  TODO: handle non hyper-rectangular spaces.
  if (failed(checkIfHyperRectangular(input)))
    return failure();
 
  return success();
}
 
/// Move the loop body of AffineForOp 'src' from 'src' into the specified
/// location in destination's body, ignoring the terminator.
static void moveLoopBodyImpl(AffineForOp src, AffineForOp dest,
                             Block::iterator loc) {
  auto &ops = src.getBody()->getOperations();
  dest.getBody()->getOperations().splice(loc, ops, ops.begin(),
                                         std::prev(ops.end()));
}
 
/// Move the loop body of AffineForOp 'src' from 'src' to the start of dest
/// body.
static void moveLoopBody(AffineForOp src, AffineForOp dest) {
  moveLoopBodyImpl(src, dest, dest.getBody()->begin());
}
 
/// Constructs tiled loop nest, without setting the loop bounds and move the
/// body of the original loop nest to the tiled loop nest.
static void constructTiledLoopNest(MutableArrayRef<AffineForOp> origLoops,
                                   AffineForOp rootAffineForOp, unsigned width,
                                   MutableArrayRef<AffineForOp> tiledLoops) {
  Location loc = rootAffineForOp.getLoc();
 
  // The outermost among the loops as we add more..
  Operation *topLoop = rootAffineForOp.getOperation();
  AffineForOp innermostPointLoop;
 
  // Add intra-tile (or point) loops.
  for (unsigned i = 0; i < width; i++) {
    OpBuilder b(topLoop);
    // Loop bounds will be set later.
    AffineForOp pointLoop = b.create<AffineForOp>(loc, 0, 0);
    pointLoop.getBody()->getOperations().splice(
        pointLoop.getBody()->begin(), topLoop->getBlock()->getOperations(),
        topLoop);
    tiledLoops[2 * width - 1 - i] = pointLoop;
    topLoop = pointLoop.getOperation();
    if (i == 0)
      innermostPointLoop = pointLoop;
  }
 
  // Add tile space loops;
  for (unsigned i = width; i < 2 * width; i++) {
    OpBuilder b(topLoop);
    // Loop bounds will be set later.
    AffineForOp tileSpaceLoop = b.create<AffineForOp>(loc, 0, 0);
    tileSpaceLoop.getBody()->getOperations().splice(
        tileSpaceLoop.getBody()->begin(), topLoop->getBlock()->getOperations(),
        topLoop);
    tiledLoops[2 * width - i - 1] = tileSpaceLoop;
    topLoop = tileSpaceLoop.getOperation();
  }
 
  // Move the loop body of the original nest to the new one.
  moveLoopBody(origLoops.back(), innermostPointLoop);
}
 
/// Set lower and upper bounds of intra-tile loops for parametric tiling.
//  TODO: Handle non-constant lower bounds.
static void setIntraTileBoundsParametric(OpBuilder &b, AffineForOp origLoop,
                                         AffineForOp newInterTileLoop,
                                         AffineForOp newIntraTileLoop,
                                         Value tileSize) {
  // The lower bound for the intra-tile loop is represented by an affine map
  // as (%i, %t0)->((%i - %origlb) * %t0 + %origlb). Similarly, the upper bound
  // for the intra-tile loop is represented by an affine map as (%i, %t0)->((%i
  // - %origlb) * %t0) + (%t0 * %origLoopStep) + %origlb), where %i is loop IV
  // of the corresponding inter-tile loop, %t0 is the corresponding tiling
  // parameter, %origlb is lower bound and %origLoopStep is the loop step of the
  // corresponding inter-tile loop.
 
  assert(origLoop.hasConstantLowerBound() &&
         "expected input loops to have constant lower bound.");
 
  // Get lower bound of original loop as an affine expression.
  AffineExpr origLowerBoundExpr;
  origLowerBoundExpr =
      b.getAffineConstantExpr(origLoop.getConstantLowerBound());
 
  // Add dim operands from original lower/upper bound.
  SmallVector<Value, 4> lbOperands, ubOperands;
  AffineBound lb = origLoop.getLowerBound();
  AffineBound ub = origLoop.getUpperBound();
  lbOperands.reserve(lb.getNumOperands() + 2);
  ubOperands.reserve(ub.getNumOperands() + 2);
  AffineMap origLbMap = lb.getMap();
  AffineMap origUbMap = ub.getMap();
  for (unsigned j = 0, e = origLbMap.getNumDims(); j < e; ++j)
    lbOperands.push_back(lb.getOperand(j));
  for (unsigned j = 0, e = origUbMap.getNumDims(); j < e; ++j)
    ubOperands.push_back(ub.getOperand(j));
 
  // Add a new dim operand in lb/ubOperands corresponding to the origLoop
  // IV.
  lbOperands.push_back(newInterTileLoop.getInductionVar());
  ubOperands.push_back(newInterTileLoop.getInductionVar());
 
  // Get loop IV as an affine expression for lower/upper bound. Size of
  // lb/ubOperands is guaranteed to be atleast one.
  AffineExpr lbLoopIvExpr = b.getAffineDimExpr(lbOperands.size() - 1);
  AffineExpr ubLoopIvExpr = b.getAffineDimExpr(ubOperands.size() - 1);
 
  // Add symbol operands from original lower/upper bound.
  for (unsigned j = 0, e = origLbMap.getNumSymbols(); j < e; ++j)
    lbOperands.push_back(lb.getOperand(origLbMap.getNumDims() + j));
  for (unsigned j = 0, e = origUbMap.getNumSymbols(); j < e; ++j)
    ubOperands.push_back(ub.getOperand(origUbMap.getNumDims() + j));
 
  // Add a new symbol operand which is the tile size for this loop.
  lbOperands.push_back(tileSize);
  ubOperands.push_back(tileSize);
 
  SmallVector<AffineExpr, 4> lbBoundExprs;
  SmallVector<AffineExpr, 4> ubBoundExprs;
  lbBoundExprs.reserve(origLbMap.getNumResults());
  ubBoundExprs.reserve(origUbMap.getNumResults());
 
  // Get tiling parameter as an affine expression for lb/ub.
  AffineExpr lbTileParameter = b.getAffineSymbolExpr(origLbMap.getNumSymbols());
  AffineExpr ubTileParameter = b.getAffineSymbolExpr(origUbMap.getNumSymbols());
 
  // Insert lb as inter-tile ((loop IV - origlb) * tilingParameter) + origlb.
  lbBoundExprs.push_back(
      ((lbLoopIvExpr - origLowerBoundExpr) * lbTileParameter) +
      origLowerBoundExpr);
 
  // Get the origLoopStep as an affine expression.
  AffineExpr origLoopStep = b.getAffineConstantExpr(origLoop.getStepAsInt());
 
  // Insert ub as inter-tile ((loop IV - origlb) * tilingParameter) +
  // (tilingParameter * origLoopStep) + origlb.
  ubBoundExprs.push_back(
      ((ubLoopIvExpr - origLowerBoundExpr) * ubTileParameter) +
      (ubTileParameter * origLoopStep) + origLowerBoundExpr);
 
  ubBoundExprs.append(origUbMap.getResults().begin(),
                      origUbMap.getResults().end());
 
  AffineMap lbMap =
      AffineMap::get(origLbMap.getNumDims() + 1, origLbMap.getNumSymbols() + 1,
                     lbBoundExprs, b.getContext());
  newIntraTileLoop.setLowerBound(lbOperands, lbMap);
 
  AffineMap ubMap =
      AffineMap::get(origUbMap.getNumDims() + 1, origUbMap.getNumSymbols() + 1,
                     ubBoundExprs, b.getContext());
  newIntraTileLoop.setUpperBound(ubOperands, ubMap);
 
  // Original loop step must be preserved.
  newIntraTileLoop.setStep(origLoop.getStepAsInt());
}
 
/// Set lower and upper bounds of inter-tile loops for parametric tiling.
//  TODO: Handle non-constant lower bounds.
static void setInterTileBoundsParametric(OpBuilder &b, AffineForOp origLoop,
                                         AffineForOp newLoop, Value tileSize) {
  OperandRange newLbOperands = origLoop.getLowerBoundOperands();
 
  // The lower bounds for inter-tile loops are same as the corresponding lower
  // bounds of original loops.
  newLoop.setLowerBound(newLbOperands, origLoop.getLowerBoundMap());
 
  // The new upper bound map for inter-tile loops, assuming constant lower
  // bounds, are now originalLowerBound + ceildiv((originalUpperBound -
  // originalLowerBound), tiling parameter); where tiling parameter is the
  // respective tile size for that loop. For e.g. if the original ubmap was
  // ()->(1024), the new map will be
  // ()[s0]->(ceildiv((1024 -lb) % s0)), where s0 is the tiling parameter.
  // Therefore a new symbol operand is inserted in the map and the result
  // expression is overwritten.
 
  assert(origLoop.hasConstantLowerBound() &&
         "expected input loops to have constant lower bound.");
 
  // Get lower bound of original loop as an affine expression.
  AffineExpr origLowerBoundExpr;
  origLowerBoundExpr =
      b.getAffineConstantExpr(origLoop.getConstantLowerBound());
 
  // Add dim operands from original upper bound.
  SmallVector<Value, 4> ubOperands;
  AffineBound ub = origLoop.getUpperBound();
  ubOperands.reserve(ub.getNumOperands() + 1);
  AffineMap origUbMap = ub.getMap();
  for (unsigned j = 0, e = origUbMap.getNumDims(); j < e; ++j)
    ubOperands.push_back(ub.getOperand(j));
 
  // Add symbol operands from original upper bound.
  for (unsigned j = 0, e = origUbMap.getNumSymbols(); j < e; ++j)
    ubOperands.push_back(ub.getOperand(origUbMap.getNumDims() + j));
 
  // Add a new symbol operand which is the tile size for this loop.
  ubOperands.push_back(tileSize);
 
  // Get tiling parameter as an affine expression.
  AffineExpr tileParameter = b.getAffineSymbolExpr(origUbMap.getNumSymbols());
 
  SmallVector<AffineExpr, 4> boundExprs;
  boundExprs.reserve(origUbMap.getNumResults());
  int64_t origUpperBound;
  AffineExpr origUpperBoundExpr;
 
  // If upper bound for the original loop is constant, then the constant can
  // be obtained as an affine expression straight away.
  if (origLoop.hasConstantUpperBound()) {
    origUpperBound = origLoop.getConstantUpperBound();
 
    // Get original constant upper bound as an affine expression.
    origUpperBoundExpr = b.getAffineConstantExpr(origUpperBound);
 
    // Insert the bound as originalLowerBoundceildiv((originalUpperBound -
    // originalLowerBound), tilingParameter).
    boundExprs.push_back(
        origLowerBoundExpr +
        (origUpperBoundExpr - origLowerBoundExpr).ceilDiv(tileParameter));
  } else {
    // If upper bound for the original loop is not constant then two cases
    // are possible, although there handeling is the same, 1.) The result of
    // ubmap has only one result expression. For e.g.
    //    affine.for %i = 5 to %ub
    //
    // A symbol operand is added which represents the tiling parameter. The
    // new loop bounds here will be like ()[s0, s1] -> ((s0 - 5) ceildiv s1 + 5)
    // where 's0' is the original upper bound and 's1' is the tiling
    // parameter. 2.) When ubMap has more than one result expression. For e.g.
    //    #map0 = affine_map<()[s0, s1] -> (s0, s1)
    //    affine.for %i = 5 to min #map0()[%s0, %s1]
    //
    // A symbol operand is added which represents the tiling parameter. The
    // new loop bounds will be like ()[s0, s1, s2] -> ((s0 - 5) ceildiv s2 + 5,
    // (s1 -5) ceildiv s2 + 5), where s2 is the tiling parameter.
 
    // Insert the bounds as originalLowerBound + ceildiv((originalUpperBound -
    // originalLowerBound), tilingParameter).
    for (AffineExpr origUpperBoundExpr : origUbMap.getResults())
      boundExprs.push_back(
          origLowerBoundExpr +
          (origUpperBoundExpr - origLowerBoundExpr).ceilDiv(tileParameter));
  }
 
  AffineMap ubMap =
      AffineMap::get(origUbMap.getNumDims(), origUbMap.getNumSymbols() + 1,
                     boundExprs, b.getContext());
  newLoop.setUpperBound(ubOperands, ubMap);
 
  // Original loop step must be preserved.
  newLoop.setStep(origLoop.getStepAsInt());
}
 
/// Constructs and sets new loop bounds after tiling for the case of
/// hyper-rectangular index sets, where the bounds of one dimension do not
/// depend on other dimensions and tiling parameters are captured from SSA
/// values. Bounds of each dimension can thus be treated independently,
/// and deriving the new bounds is much simpler and faster than for the case of
/// tiling arbitrary polyhedral shapes.
static void constructParametricallyTiledIndexSetHyperRect(
    MutableArrayRef<AffineForOp> origLoops,
    MutableArrayRef<AffineForOp> newLoops, ArrayRef<Value> tileSizes) {
  assert(!origLoops.empty() && "expected atleast one loop in band");
  assert(origLoops.size() == tileSizes.size() &&
         "expected tiling parameter for each loop in band.");
 
  OpBuilder b(origLoops[0].getOperation());
  unsigned width = origLoops.size();
 
  // Set bounds for tile space loops.
  for (unsigned i = 0; i < width; ++i) {
    setInterTileBoundsParametric(b, origLoops[i], newLoops[i], tileSizes[i]);
  }
 
  // Set bounds for intra-tile loops.
  for (unsigned i = 0; i < width; ++i) {
    setIntraTileBoundsParametric(b, origLoops[i], newLoops[i],
                                 newLoops[i + width], tileSizes[i]);
  }
}
 
/// Constructs and sets new loop bounds after tiling for the case of
/// hyper-rectangular index sets, where the bounds of one dimension do not
/// depend on other dimensions. Bounds of each dimension can thus be treated
/// independently, and deriving the new bounds is much simpler and faster
/// than for the case of tiling arbitrary polyhedral shapes.
static void
constructTiledIndexSetHyperRect(MutableArrayRef<AffineForOp> origLoops,
                                MutableArrayRef<AffineForOp> newLoops,
                                ArrayRef<unsigned> tileSizes) {
  assert(!origLoops.empty());
  assert(origLoops.size() == tileSizes.size());
 
  OpBuilder b(origLoops[0].getOperation());
  unsigned width = origLoops.size();
 
  // Bounds for tile space loops.
  for (unsigned i = 0; i < width; i++) {
    OperandRange newLbOperands = origLoops[i].getLowerBoundOperands();
    OperandRange newUbOperands = origLoops[i].getUpperBoundOperands();
    newLoops[i].setLowerBound(newLbOperands, origLoops[i].getLowerBoundMap());
    newLoops[i].setUpperBound(newUbOperands, origLoops[i].getUpperBoundMap());
    // If the step size of original loop is x and tileSize is y then after
    // tiling the tile space loops' step size becomes x*y.
    newLoops[i].setStep(tileSizes[i] * origLoops[i].getStepAsInt());
  }
  // Bounds for intra-tile loops.
  for (unsigned i = 0; i < width; i++) {
    int64_t largestDiv = getLargestDivisorOfTripCount(origLoops[i]);
    std::optional<uint64_t> mayBeConstantCount =
        getConstantTripCount(origLoops[i]);
    // The lower bound is just the tile-space loop.
    AffineMap lbMap = b.getDimIdentityMap();
    newLoops[width + i].setLowerBound(
        /*operands=*/newLoops[i].getInductionVar(), lbMap);
    // The step sizes of intra-tile loops is just the original loops' step size.
    newLoops[width + i].setStep(origLoops[i].getStepAsInt());
 
    // Set the upper bound.
    if (mayBeConstantCount && *mayBeConstantCount < tileSizes[i]) {
      // Trip count is less than the tile size: upper bound is lower bound +
      // trip count * stepSize.
      AffineMap ubMap = b.getSingleDimShiftAffineMap(
          *mayBeConstantCount * origLoops[i].getStepAsInt());
      newLoops[width + i].setUpperBound(
          /*operands=*/newLoops[i].getInductionVar(), ubMap);
    } else if (largestDiv % tileSizes[i] != 0) {
      // Intra-tile loop ii goes from i to min(i + tileSize * stepSize, ub_i).
      // Construct the upper bound map; the operands are the original operands
      // with 'i' (tile-space loop) appended to it. The new upper bound map is
      // the original one with an additional expression i + tileSize * stepSize
      // appended.
 
      // Add dim operands from original upper bound.
      SmallVector<Value, 4> ubOperands;
      AffineBound ub = origLoops[i].getUpperBound();
      ubOperands.reserve(ub.getNumOperands() + 1);
      AffineMap origUbMap = ub.getMap();
      for (unsigned j = 0, e = origUbMap.getNumDims(); j < e; ++j)
        ubOperands.push_back(ub.getOperand(j));
 
      // Add dim operand for new loop upper bound.
      ubOperands.push_back(newLoops[i].getInductionVar());
 
      // Add symbol operands from original upper bound.
      for (unsigned j = 0, e = origUbMap.getNumSymbols(); j < e; ++j)
        ubOperands.push_back(ub.getOperand(origUbMap.getNumDims() + j));
 
      SmallVector<AffineExpr, 4> boundExprs;
      boundExprs.reserve(1 + origUbMap.getNumResults());
      AffineExpr dim = b.getAffineDimExpr(origUbMap.getNumDims());
      // The new upper bound map is the original one with an additional
      // expression i + tileSize * stepSize (of original loop) appended.
      boundExprs.push_back(dim + tileSizes[i] * origLoops[i].getStepAsInt());
      boundExprs.append(origUbMap.getResults().begin(),
                        origUbMap.getResults().end());
      AffineMap ubMap =
          AffineMap::get(origUbMap.getNumDims() + 1, origUbMap.getNumSymbols(),
                         boundExprs, b.getContext());
      newLoops[width + i].setUpperBound(/*operands=*/ubOperands, ubMap);
    } else {
      // No need of the min expression.
      AffineExpr dim = b.getAffineDimExpr(0);
      AffineMap ubMap = AffineMap::get(
          1, 0, dim + tileSizes[i] * origLoops[i].getStepAsInt());
      newLoops[width + i].setUpperBound(newLoops[i].getInductionVar(), ubMap);
    }
  }
}
 
LogicalResult
mlir::affine::tilePerfectlyNested(MutableArrayRef<AffineForOp> input,
                                  ArrayRef<unsigned> tileSizes,
                                  SmallVectorImpl<AffineForOp> *tiledNest) {
  if (input.empty())
    return success();
 
  if (failed(performPreTilingChecks(input, tileSizes)))
    return failure();
 
  MutableArrayRef<AffineForOp> origLoops = input;
  AffineForOp rootAffineForOp = origLoops[0];
 
  // Note that width is at least one since the band isn't empty.
  unsigned width = input.size();
  SmallVector<AffineForOp, 6> tiledLoops(2 * width);
 
  // Construct a tiled loop nest without setting their bounds. Bounds are
  // set later.
  constructTiledLoopNest(origLoops, rootAffineForOp, width, tiledLoops);
 
  SmallVector<Value, 8> origLoopIVs;
  extractForInductionVars(input, &origLoopIVs);
 
  // Set loop bounds for the tiled loop nest.
  constructTiledIndexSetHyperRect(origLoops, tiledLoops, tileSizes);
 
  // Replace original IVs with intra-tile loop IVs.
  for (unsigned i = 0; i < width; i++)
    origLoopIVs[i].replaceAllUsesWith(tiledLoops[i + width].getInductionVar());
 
  // Erase the old loop nest.
  rootAffineForOp.erase();
 
  if (tiledNest)
    *tiledNest = std::move(tiledLoops);
 
  return success();
}
 
/// Tiles the specified band of perfectly nested loops creating tile-space
/// loops and intra-tile loops, using SSA values as tiling parameters. A band
/// is a contiguous set of loops.
LogicalResult mlir::affine::tilePerfectlyNestedParametric(
    MutableArrayRef<AffineForOp> input, ArrayRef<Value> tileSizes,
    SmallVectorImpl<AffineForOp> *tiledNest) {
  if (input.empty())
    return success();
 
  if (failed(performPreTilingChecks(input, tileSizes)))
    return failure();
 
  MutableArrayRef<AffineForOp> origLoops = input;
  AffineForOp rootAffineForOp = origLoops[0];
  unsigned width = input.size();
  SmallVector<AffineForOp, 6> tiledLoops(2 * width);
 
  // Construct a tiled loop nest without setting their bounds. Bounds are
  // set later.
  constructTiledLoopNest(origLoops, rootAffineForOp, width, tiledLoops);
 
  SmallVector<Value, 8> origLoopIVs;
  extractForInductionVars(input, &origLoopIVs);
 
  // Set loop bounds for the tiled loop nest.
  constructParametricallyTiledIndexSetHyperRect(origLoops, tiledLoops,
                                                tileSizes);
 
  // Replace original IVs with intra-tile loop IVs.
  for (unsigned i = 0; i < width; i++)
    origLoopIVs[i].replaceAllUsesWith(tiledLoops[i + width].getInductionVar());
 
  // Erase the old loop nest.
  rootAffineForOp.erase();
 
  if (tiledNest)
    *tiledNest = std::move(tiledLoops);
 
  return success();
}
 
/// Get perfectly nested sequence of loops starting at root of loop nest
/// (the first op being another AffineFor, and the second op - a terminator).
/// A loop is perfectly nested iff: the first op in the loop's body is another
/// AffineForOp, and the second op is a terminator).
void mlir::affine::getPerfectlyNestedLoops(
    SmallVectorImpl<AffineForOp> &nestedLoops, AffineForOp root) {
  for (unsigned i = 0; i < std::numeric_limits<unsigned>::max(); ++i) {
    nestedLoops.push_back(root);
    Block &body = root.getRegion().front();
    if (body.begin() != std::prev(body.end(), 2))
      return;
 
    root = dyn_cast<AffineForOp>(&body.front());
    if (!root)
      return;
  }
}
 
/// Identify valid and profitable bands of loops to tile. This is currently just
/// a temporary placeholder to test the mechanics of tiled code generation.
/// Returns all maximal outermost perfect loop nests to tile.
void mlir::affine::getTileableBands(
    func::FuncOp f, std::vector<SmallVector<AffineForOp, 6>> *bands) {
  // Get maximal perfect nest of 'affine.for' insts starting from root
  // (inclusive).
  for (AffineForOp forOp : f.getOps<AffineForOp>()) {
    SmallVector<AffineForOp, 6> band;
    getPerfectlyNestedLoops(band, forOp);
    bands->push_back(band);
  }
}
 
/// Unrolls this loop completely.
LogicalResult mlir::affine::loopUnrollFull(AffineForOp forOp) {
  std::optional<uint64_t> mayBeConstantTripCount = getConstantTripCount(forOp);
  if (mayBeConstantTripCount.has_value()) {
    uint64_t tripCount = *mayBeConstantTripCount;
    if (tripCount == 0)
      return success();
    if (tripCount == 1)
      return promoteIfSingleIteration(forOp);
    return loopUnrollByFactor(forOp, tripCount);
  }
  return failure();
}
 
/// Unrolls this loop by the specified factor or by the trip count (if constant)
/// whichever is lower.
LogicalResult mlir::affine::loopUnrollUpToFactor(AffineForOp forOp,
                                                 uint64_t unrollFactor) {
  std::optional<uint64_t> mayBeConstantTripCount = getConstantTripCount(forOp);
  if (mayBeConstantTripCount.has_value() &&
      *mayBeConstantTripCount < unrollFactor)
    return loopUnrollByFactor(forOp, *mayBeConstantTripCount);
  return loopUnrollByFactor(forOp, unrollFactor);
}
 
/// Generates unrolled copies of AffineForOp 'loopBodyBlock', with associated
/// 'forOpIV' by 'unrollFactor', calling 'ivRemapFn' to remap 'forOpIV' for each
/// unrolled body. If specified, annotates the Ops in each unrolled iteration
/// using annotateFn.
static void generateUnrolledLoop(
    Block *loopBodyBlock, Value forOpIV, uint64_t unrollFactor,
    function_ref<Value(unsigned, Value, OpBuilder)> ivRemapFn,
    function_ref<void(unsigned, Operation *, OpBuilder)> annotateFn,
    ValueRange iterArgs, ValueRange yieldedValues) {
  // Builder to insert unrolled bodies just before the terminator of the body of
  // 'forOp'.
  auto builder = OpBuilder::atBlockTerminator(loopBodyBlock);
 
  if (!annotateFn)
    annotateFn = [](unsigned, Operation *, OpBuilder) {};
 
  // Keep a pointer to the last non-terminator operation in the original block
  // so that we know what to clone (since we are doing this in-place).
  Block::iterator srcBlockEnd = std::prev(loopBodyBlock->end(), 2);
 
  // Unroll the contents of 'forOp' (append unrollFactor - 1 additional copies).
  SmallVector<Value, 4> lastYielded(yieldedValues);
 
  for (unsigned i = 1; i < unrollFactor; i++) {
    IRMapping operandMap;
 
    // Prepare operand map.
    operandMap.map(iterArgs, lastYielded);
 
    // If the induction variable is used, create a remapping to the value for
    // this unrolled instance.
    if (!forOpIV.use_empty()) {
      Value ivUnroll = ivRemapFn(i, forOpIV, builder);
      operandMap.map(forOpIV, ivUnroll);
    }
 
    // Clone the original body of 'forOp'.
    for (auto it = loopBodyBlock->begin(); it != std::next(srcBlockEnd); it++) {
      Operation *clonedOp = builder.clone(*it, operandMap);
      annotateFn(i, clonedOp, builder);
    }
 
    // Update yielded values. If the yielded value is defined outside the
    // `loopBodyBlock` or if it is a BlockArgument then it won't be cloned, thus
    // the `lastYielded` value remains unchanged. Else, update the `lastYielded`
    // value with the clone corresponding to the yielded value.
    for (unsigned i = 0, e = lastYielded.size(); i < e; i++) {
      Operation *defOp = yieldedValues[i].getDefiningOp();
      if (defOp && defOp->getBlock() == loopBodyBlock)
        lastYielded[i] = operandMap.lookup(yieldedValues[i]);
    }
  }
 
  // Make sure we annotate the Ops in the original body. We do this last so that
  // any annotations are not copied into the cloned Ops above.
  for (auto it = loopBodyBlock->begin(); it != std::next(srcBlockEnd); it++)
    annotateFn(0, &*it, builder);
 
  // Update operands of the yield statement.
  loopBodyBlock->getTerminator()->setOperands(lastYielded);
}
 
/// Helper to generate cleanup loop for unroll or unroll-and-jam when the trip
/// count is not a multiple of `unrollFactor`.
static LogicalResult generateCleanupLoopForUnroll(AffineForOp forOp,
                                                  uint64_t unrollFactor) {
  // Insert the cleanup loop right after 'forOp'.
  OpBuilder builder(forOp->getBlock(), std::next(Block::iterator(forOp)));
  auto cleanupForOp = cast<AffineForOp>(builder.clone(*forOp));
 
  // Update uses of `forOp` results. `cleanupForOp` should use `forOp` result
  // and produce results for the original users of `forOp` results.
  auto results = forOp.getResults();
  auto cleanupResults = cleanupForOp.getResults();
  auto cleanupIterOperands = cleanupForOp.getInits();
 
  for (auto e : llvm::zip(results, cleanupResults, cleanupIterOperands)) {
    std::get<0>(e).replaceAllUsesWith(std::get<1>(e));
    cleanupForOp->replaceUsesOfWith(std::get<2>(e), std::get<0>(e));
  }
 
  AffineMap cleanupMap;
  SmallVector<Value, 4> cleanupOperands;
  getCleanupLoopLowerBound(forOp, unrollFactor, cleanupMap, cleanupOperands);
  if (!cleanupMap)
    return failure();
 
  cleanupForOp.setLowerBound(cleanupOperands, cleanupMap);
  // Promote the loop body up if this has turned into a single iteration loop.
  (void)promoteIfSingleIteration(cleanupForOp);
 
  // Adjust upper bound of the original loop; this is the same as the lower
  // bound of the cleanup loop.
  forOp.setUpperBound(cleanupOperands, cleanupMap);
  return success();
}
 
/// Unrolls this loop by the specified factor. Returns success if the loop
/// is successfully unrolled.
LogicalResult mlir::affine::loopUnrollByFactor(
    AffineForOp forOp, uint64_t unrollFactor,
    function_ref<void(unsigned, Operation *, OpBuilder)> annotateFn,
    bool cleanUpUnroll) {
  assert(unrollFactor > 0 && "unroll factor should be positive");
 
  std::optional<uint64_t> mayBeConstantTripCount = getConstantTripCount(forOp);
  if (unrollFactor == 1) {
    if (mayBeConstantTripCount && *mayBeConstantTripCount == 1 &&
        failed(promoteIfSingleIteration(forOp)))
      return failure();
    return success();
  }
 
  // Nothing in the loop body other than the terminator.
  if (llvm::hasSingleElement(forOp.getBody()->getOperations()))
    return success();
 
  // If the trip count is lower than the unroll factor, no unrolled body.
  if (mayBeConstantTripCount && *mayBeConstantTripCount < unrollFactor) {
    if (cleanUpUnroll) {
      // Unroll the cleanup loop if cleanUpUnroll is specified.
      return loopUnrollFull(forOp);
    }
 
    return failure();
  }
 
  // Generate the cleanup loop if trip count isn't a multiple of unrollFactor.
  if (getLargestDivisorOfTripCount(forOp) % unrollFactor != 0) {
    // Loops where the lower bound is a max expression or the upper bound is
    // a min expression and the trip count doesn't divide the unroll factor
    // can't be unrolled since the lower bound of the cleanup loop in such cases
    // cannot be expressed as an affine function or a max over affine functions.
    if (forOp.getLowerBoundMap().getNumResults() != 1 ||
        forOp.getUpperBoundMap().getNumResults() != 1)
      return failure();
    if (cleanUpUnroll)
      // Force unroll including cleanup loop
      return loopUnrollFull(forOp);
    if (failed(generateCleanupLoopForUnroll(forOp, unrollFactor)))
      assert(false && "cleanup loop lower bound map for single result lower "
                      "and upper bound maps can always be determined");
  }
 
  ValueRange iterArgs(forOp.getRegionIterArgs());
  auto yieldedValues = forOp.getBody()->getTerminator()->getOperands();
 
  // Scale the step of loop being unrolled by unroll factor.
  int64_t step = forOp.getStepAsInt();
  forOp.setStep(step * unrollFactor);
  generateUnrolledLoop(
      forOp.getBody(), forOp.getInductionVar(), unrollFactor,
      [&](unsigned i, Value iv, OpBuilder b) {
        // iv' = iv + i * step
        auto d0 = b.getAffineDimExpr(0);
        auto bumpMap = AffineMap::get(1, 0, d0 + i * step);
        return b.create<AffineApplyOp>(forOp.getLoc(), bumpMap, iv);
      },
      /*annotateFn=*/annotateFn,
      /*iterArgs=*/iterArgs, /*yieldedValues=*/yieldedValues);
 
  // Promote the loop body up if this has turned into a single iteration loop.
  (void)promoteIfSingleIteration(forOp);
  return success();
}
 
LogicalResult mlir::affine::loopUnrollJamUpToFactor(AffineForOp forOp,
                                                    uint64_t unrollJamFactor) {
  std::optional<uint64_t> mayBeConstantTripCount = getConstantTripCount(forOp);
  if (mayBeConstantTripCount.has_value() &&
      *mayBeConstantTripCount < unrollJamFactor)
    return loopUnrollJamByFactor(forOp, *mayBeConstantTripCount);
  return loopUnrollJamByFactor(forOp, unrollJamFactor);
}
 
/// Check if all control operands of all loops are defined outside of `forOp`
/// and return false if not.
static bool areInnerBoundsInvariant(AffineForOp forOp) {
  auto walkResult = forOp.walk([&](AffineForOp aForOp) {
    for (auto controlOperand : aForOp.getControlOperands()) {
      if (!forOp.isDefinedOutsideOfLoop(controlOperand))
        return WalkResult::interrupt();
    }
    return WalkResult::advance();
  });
  return !walkResult.wasInterrupted();
}
 
/// Unrolls and jams this loop by the specified factor.
LogicalResult mlir::affine::loopUnrollJamByFactor(AffineForOp forOp,
                                                  uint64_t unrollJamFactor) {
  assert(unrollJamFactor > 0 && "unroll jam factor should be positive");
 
  std::optional<uint64_t> mayBeConstantTripCount = getConstantTripCount(forOp);
  if (unrollJamFactor == 1) {
    if (mayBeConstantTripCount && *mayBeConstantTripCount == 1 &&
        failed(promoteIfSingleIteration(forOp)))
      return failure();
    return success();
  }
 
  // Nothing in the loop body other than the terminator.
  if (llvm::hasSingleElement(forOp.getBody()->getOperations()))
    return success();
 
  // If the trip count is lower than the unroll jam factor, no unroll jam.
  if (mayBeConstantTripCount && *mayBeConstantTripCount < unrollJamFactor) {
    LLVM_DEBUG(llvm::dbgs() << "[failed] trip count < unroll-jam factor\n");
    return failure();
  }
 
  // If any control operand of any inner loop of `forOp` is defined within
  // `forOp`, no unroll jam.
  if (!areInnerBoundsInvariant(forOp))
    return failure();
 
  // Gather all sub-blocks to jam upon the loop being unrolled.
  JamBlockGatherer<AffineForOp> jbg;
  jbg.walk(forOp);
  auto &subBlocks = jbg.subBlocks;
 
  // Collect loops with iter_args.
  SmallVector<AffineForOp, 4> loopsWithIterArgs;
  forOp.walk([&](AffineForOp aForOp) {
    if (aForOp.getNumIterOperands() > 0)
      loopsWithIterArgs.push_back(aForOp);
  });
 
  // Get supported reductions to be used for creating reduction ops at the end.
  SmallVector<LoopReduction> reductions;
  if (forOp.getNumIterOperands() > 0)
    getSupportedReductions(forOp, reductions);
 
  // Generate the cleanup loop if trip count isn't a multiple of
  // unrollJamFactor.
  if (getLargestDivisorOfTripCount(forOp) % unrollJamFactor != 0) {
    // Loops where the lower bound is a max expression or the upper bound is
    // a min expression and the trip count doesn't divide the unroll factor
    // can't be unrolled since the lower bound of the cleanup loop in such cases
    // cannot be expressed as an affine function or a max over affine functions.
    if (forOp.getLowerBoundMap().getNumResults() != 1 ||
        forOp.getUpperBoundMap().getNumResults() != 1)
      return failure();
    if (failed(generateCleanupLoopForUnroll(forOp, unrollJamFactor)))
      assert(false && "cleanup loop lower bound map for single result lower "
                      "and upper bound maps can always be determined");
  }
 
  // `operandMaps[i - 1]` carries old->new operand mapping for the ith unrolled
  // iteration. There are (`unrollJamFactor` - 1) iterations.
  SmallVector<IRMapping, 4> operandMaps(unrollJamFactor - 1);
 
  // For any loop with iter_args, replace it with a new loop that has
  // `unrollJamFactor` copies of its iterOperands, iter_args and yield
  // operands.
  SmallVector<AffineForOp, 4> newLoopsWithIterArgs;
  IRRewriter rewriter(forOp.getContext());
  for (AffineForOp oldForOp : loopsWithIterArgs) {
    SmallVector<Value> dupIterOperands, dupYieldOperands;
    ValueRange oldIterOperands = oldForOp.getInits();
    ValueRange oldIterArgs = oldForOp.getRegionIterArgs();
    ValueRange oldYieldOperands =
        cast<AffineYieldOp>(oldForOp.getBody()->getTerminator()).getOperands();
    // Get additional iterOperands, iterArgs, and yield operands. We will
    // fix iterOperands and yield operands after cloning of sub-blocks.
    for (unsigned i = unrollJamFactor - 1; i >= 1; --i) {
      dupIterOperands.append(oldIterOperands.begin(), oldIterOperands.end());
      dupYieldOperands.append(oldYieldOperands.begin(), oldYieldOperands.end());
    }
    // Create a new loop with additional iterOperands, iter_args and yield
    // operands. This new loop will take the loop body of the original loop.
    bool forOpReplaced = oldForOp == forOp;
    AffineForOp newForOp =
        cast<AffineForOp>(*oldForOp.replaceWithAdditionalYields(
            rewriter, dupIterOperands, /*replaceInitOperandUsesInLoop=*/false,
            [&](OpBuilder &b, Location loc, ArrayRef<BlockArgument> newBbArgs) {
              return dupYieldOperands;
            }));
    newLoopsWithIterArgs.push_back(newForOp);
    // `forOp` has been replaced with a new loop.
    if (forOpReplaced)
      forOp = newForOp;
    // Update `operandMaps` for `newForOp` iterArgs and results.
    ValueRange newIterArgs = newForOp.getRegionIterArgs();
    unsigned oldNumIterArgs = oldIterArgs.size();
    ValueRange newResults = newForOp.getResults();
    unsigned oldNumResults = newResults.size() / unrollJamFactor;
    assert(oldNumIterArgs == oldNumResults &&
           "oldNumIterArgs must be the same as oldNumResults");
    for (unsigned i = unrollJamFactor - 1; i >= 1; --i) {
      for (unsigned j = 0; j < oldNumIterArgs; ++j) {
        // `newForOp` has `unrollJamFactor` - 1 new sets of iterArgs and
        // results. Update `operandMaps[i - 1]` to map old iterArgs and results
        // to those in the `i`th new set.
        operandMaps[i - 1].map(newIterArgs[j],
                               newIterArgs[i * oldNumIterArgs + j]);
        operandMaps[i - 1].map(newResults[j],
                               newResults[i * oldNumResults + j]);
      }
    }
  }
 
  // Scale the step of loop being unroll-jammed by the unroll-jam factor.
  int64_t step = forOp.getStepAsInt();
  forOp.setStep(step * unrollJamFactor);
 
  auto forOpIV = forOp.getInductionVar();
  // Unroll and jam (appends unrollJamFactor - 1 additional copies).
  for (unsigned i = unrollJamFactor - 1; i >= 1; --i) {
    for (auto &subBlock : subBlocks) {
      // Builder to insert unroll-jammed bodies. Insert right at the end of
      // sub-block.
      OpBuilder builder(subBlock.first->getBlock(), std::next(subBlock.second));
 
      // If the induction variable is used, create a remapping to the value for
      // this unrolled instance.
      if (!forOpIV.use_empty()) {
        // iv' = iv + i * step, i = 1 to unrollJamFactor-1.
        auto d0 = builder.getAffineDimExpr(0);
        auto bumpMap = AffineMap::get(1, 0, d0 + i * step);
        auto ivUnroll =
            builder.create<AffineApplyOp>(forOp.getLoc(), bumpMap, forOpIV);
        operandMaps[i - 1].map(forOpIV, ivUnroll);
      }
      // Clone the sub-block being unroll-jammed.
      for (auto it = subBlock.first; it != std::next(subBlock.second); ++it)
        builder.clone(*it, operandMaps[i - 1]);
    }
    // Fix iterOperands and yield op operands of newly created loops.
    for (auto newForOp : newLoopsWithIterArgs) {
      unsigned oldNumIterOperands =
          newForOp.getNumIterOperands() / unrollJamFactor;
      unsigned numControlOperands = newForOp.getNumControlOperands();
      auto yieldOp = cast<AffineYieldOp>(newForOp.getBody()->getTerminator());
      unsigned oldNumYieldOperands = yieldOp.getNumOperands() / unrollJamFactor;
      assert(oldNumIterOperands == oldNumYieldOperands &&
             "oldNumIterOperands must be the same as oldNumYieldOperands");
      for (unsigned j = 0; j < oldNumIterOperands; ++j) {
        // The `i`th duplication of an old iterOperand or yield op operand
        // needs to be replaced with a mapped value from `operandMaps[i - 1]`
        // if such mapped value exists.
        newForOp.setOperand(numControlOperands + i * oldNumIterOperands + j,
                            operandMaps[i - 1].lookupOrDefault(
                                newForOp.getOperand(numControlOperands + j)));
        yieldOp.setOperand(
            i * oldNumYieldOperands + j,
            operandMaps[i - 1].lookupOrDefault(yieldOp.getOperand(j)));
      }
    }
  }
  if (forOp.getNumResults() > 0) {
    // Create reduction ops to combine every `unrollJamFactor` related results
    // into one value. For example, for %0:2 = affine.for ... and addf, we add
    // %1 = arith.addf %0#0, %0#1, and replace the following uses of %0#0 with
    // %1.
    rewriter.setInsertionPointAfter(forOp);
    auto loc = forOp.getLoc();
    unsigned oldNumResults = forOp.getNumResults() / unrollJamFactor;
    for (LoopReduction &reduction : reductions) {
      unsigned pos = reduction.iterArgPosition;
      Value lhs = forOp.getResult(pos);
      Value rhs;
      SmallPtrSet<Operation *, 4> newOps;
      for (unsigned i = unrollJamFactor - 1; i >= 1; --i) {
        rhs = forOp.getResult(i * oldNumResults + pos);
        // Create ops based on reduction type.
        lhs = arith::getReductionOp(reduction.kind, rewriter, loc, lhs, rhs);
        if (!lhs)
          return failure();
        Operation *op = lhs.getDefiningOp();
        assert(op && "Reduction op should have been created");
        newOps.insert(op);
      }
      // Replace all uses except those in newly created reduction ops.
      forOp.getResult(pos).replaceAllUsesExcept(lhs, newOps);
    }
  }
 
  // Promote the loop body up if this has turned into a single iteration loop.
  (void)promoteIfSingleIteration(forOp);
  return success();
}
 
/// Performs loop interchange on 'forOpA' and 'forOpB', where 'forOpB' is
/// nested within 'forOpA' as the only non-terminator operation in its block.
void mlir::affine::interchangeLoops(AffineForOp forOpA, AffineForOp forOpB) {
  assert(&*forOpA.getBody()->begin() == forOpB.getOperation());
  auto &forOpABody = forOpA.getBody()->getOperations();
  auto &forOpBBody = forOpB.getBody()->getOperations();
 
  // 1) Splice forOpA's non-terminator operations (which is just forOpB) just
  // before forOpA (in ForOpA's parent's block) this should leave 'forOpA's
  // body containing only the terminator.
  forOpA->getBlock()->getOperations().splice(Block::iterator(forOpA),
                                             forOpABody, forOpABody.begin(),
                                             std::prev(forOpABody.end()));
  // 2) Splice forOpB's non-terminator operations into the beginning of forOpA's
  // body (this leaves forOpB's body containing only the terminator).
  forOpABody.splice(forOpABody.begin(), forOpBBody, forOpBBody.begin(),
                    std::prev(forOpBBody.end()));
  // 3) Splice forOpA into the beginning of forOpB's body.
  forOpBBody.splice(forOpBBody.begin(), forOpA->getBlock()->getOperations(),
                    Block::iterator(forOpA));
}
 
// Checks each dependence component against the permutation to see if the
// desired loop interchange would violate dependences by making the
// dependence component lexicographically negative.
static bool checkLoopInterchangeDependences(
    const std::vector<SmallVector<DependenceComponent, 2>> &depCompsVec,
    ArrayRef<AffineForOp> loops, ArrayRef<unsigned> loopPermMap) {
  // Invert permutation map.
  unsigned maxLoopDepth = loops.size();
  SmallVector<unsigned, 4> loopPermMapInv;
  loopPermMapInv.resize(maxLoopDepth);
  for (unsigned i = 0; i < maxLoopDepth; ++i)
    loopPermMapInv[loopPermMap[i]] = i;
 
  // Check each dependence component against the permutation to see if the
  // desired loop interchange permutation would make the dependence vectors
  // lexicographically negative.
  // Example 1: [-1, 1][0, 0]
  // Example 2: [0, 0][-1, 1]
  for (const auto &depComps : depCompsVec) {
    assert(depComps.size() >= maxLoopDepth);
    // Check if the first non-zero dependence component is positive.
    // This iterates through loops in the desired order.
    for (unsigned j = 0; j < maxLoopDepth; ++j) {
      unsigned permIndex = loopPermMapInv[j];
      assert(depComps[permIndex].lb);
      int64_t depCompLb = *depComps[permIndex].lb;
      if (depCompLb > 0)
        break;
      if (depCompLb < 0)
        return false;
    }
  }
  return true;
}
 
/// Checks if the loop interchange permutation 'loopPermMap' of the perfectly
/// nested sequence of loops in 'loops' would violate dependences.
bool mlir::affine::isValidLoopInterchangePermutation(
    ArrayRef<AffineForOp> loops, ArrayRef<unsigned> loopPermMap) {
  // Gather dependence components for dependences between all ops in loop nest
  // rooted at 'loops[0]', at loop depths in range [1, maxLoopDepth].
  assert(loopPermMap.size() == loops.size());
  unsigned maxLoopDepth = loops.size();
  std::vector<SmallVector<DependenceComponent, 2>> depCompsVec;
  getDependenceComponents(loops[0], maxLoopDepth, &depCompsVec);
  return checkLoopInterchangeDependences(depCompsVec, loops, loopPermMap);
}
 
/// Returns true if `loops` is a perfectly nested loop nest, where loops appear
/// in it from outermost to innermost.
bool LLVM_ATTRIBUTE_UNUSED
mlir::affine::isPerfectlyNested(ArrayRef<AffineForOp> loops) {
  assert(!loops.empty() && "no loops provided");
 
  // We already know that the block can't be empty.
  auto hasTwoElements = [](Block *block) {
    auto secondOpIt = std::next(block->begin());
    return secondOpIt != block->end() && &*secondOpIt == &block->back();
  };
 
  auto enclosingLoop = loops.front();
  for (auto loop : loops.drop_front()) {
    auto parentForOp = dyn_cast<AffineForOp>(loop->getParentOp());
    // parentForOp's body should be just this loop and the terminator.
    if (parentForOp != enclosingLoop || !hasTwoElements(parentForOp.getBody()))
      return false;
    enclosingLoop = loop;
  }
  return true;
}
 
// input[i] should move from position i -> permMap[i]. Returns the position in
// `input` that becomes the new outermost loop.
unsigned mlir::affine::permuteLoops(ArrayRef<AffineForOp> input,
                                    ArrayRef<unsigned> permMap) {
  assert(input.size() == permMap.size() && "invalid permutation map size");
  // Check whether the permutation spec is valid. This is a small vector - we'll
  // just sort and check if it's iota.
  SmallVector<unsigned, 4> checkPermMap(permMap);
  llvm::sort(checkPermMap);
  if (llvm::any_of(llvm::enumerate(checkPermMap),
                   [](const auto &en) { return en.value() != en.index(); }))
    assert(false && "invalid permutation map");
 
  // Nothing to do.
  if (input.size() < 2)
    return 0;
 
  assert(isPerfectlyNested(input) && "input not perfectly nested");
 
  // Compute the inverse mapping, invPermMap: since input[i] goes to position
  // permMap[i], position i of the permuted nest is at input[invPermMap[i]].
  SmallVector<std::pair<unsigned, unsigned>, 4> invPermMap;
  for (unsigned i = 0, e = input.size(); i < e; ++i)
    invPermMap.push_back({permMap[i], i});
  llvm::sort(invPermMap);
 
  // Move the innermost loop body to the loop that would be the innermost in the
  // permuted nest (only if the innermost loop is going to change).
  if (permMap.back() != input.size() - 1) {
    Block *destBody = ((AffineForOp)input[invPermMap.back().second]).getBody();
    Block *srcBody = ((AffineForOp)input.back()).getBody();
    destBody->getOperations().splice(destBody->begin(),
                                     srcBody->getOperations(), srcBody->begin(),
                                     std::prev(srcBody->end()));
  }
 
  // We'll move each loop in `input` in the reverse order so that its body is
  // empty when we are moving it; this incurs zero copies and no erasing.
  for (int i = input.size() - 1; i >= 0; --i) {
    // If this has to become the outermost loop after permutation, add it to the
    // parent block of the original root.
    if (permMap[i] == 0) {
      // If the root remains the same, nothing to do.
      if (i == 0)
        continue;
      // Make input[i] the new outermost loop moving it into parentBlock.
      auto *parentBlock = input[0]->getBlock();
      parentBlock->getOperations().splice(Block::iterator(input[0]),
                                          input[i]->getBlock()->getOperations(),
                                          Block::iterator(input[i]));
      continue;
    }
 
    // If the parent in the permuted order is the same as in the original,
    // nothing to do.
    unsigned parentPosInInput = invPermMap[permMap[i] - 1].second;
    if (i > 0 && static_cast<unsigned>(i - 1) == parentPosInInput)
      continue;
 
    // Move input[i] to its surrounding loop in the transformed nest.
    auto *destBody = ((AffineForOp)input[parentPosInInput]).getBody();
    destBody->getOperations().splice(destBody->begin(),
                                     input[i]->getBlock()->getOperations(),
                                     Block::iterator(input[i]));
  }
 
  return invPermMap[0].second;
}
 
// Sinks all sequential loops to the innermost levels (while preserving
// relative order among them) and moves all parallel loops to the
// outermost (while again preserving relative order among them).
AffineForOp mlir::affine::sinkSequentialLoops(AffineForOp forOp) {
  SmallVector<AffineForOp, 4> loops;
  getPerfectlyNestedLoops(loops, forOp);
  if (loops.size() < 2)
    return forOp;
 
  // Gather dependence components for dependences between all ops in loop nest
  // rooted at 'loops[0]', at loop depths in range [1, maxLoopDepth].
  unsigned maxLoopDepth = loops.size();
  std::vector<SmallVector<DependenceComponent, 2>> depCompsVec;
  getDependenceComponents(loops[0], maxLoopDepth, &depCompsVec);
 
  // Mark loops as either parallel or sequential.
  SmallVector<bool, 8> isParallelLoop(maxLoopDepth, true);
  for (auto &depComps : depCompsVec) {
    assert(depComps.size() >= maxLoopDepth);
    for (unsigned j = 0; j < maxLoopDepth; ++j) {
      DependenceComponent &depComp = depComps[j];
      assert(depComp.lb.has_value() && depComp.ub.has_value());
      if (*depComp.lb != 0 || *depComp.ub != 0)
        isParallelLoop[j] = false;
    }
  }
 
  unsigned numParallelLoops = llvm::count(isParallelLoop, true);
 
  // Compute permutation of loops that sinks sequential loops (and thus raises
  // parallel loops) while preserving relative order.
  SmallVector<unsigned, 4> loopPermMap(maxLoopDepth);
  unsigned nextSequentialLoop = numParallelLoops;
  unsigned nextParallelLoop = 0;
  for (unsigned i = 0; i < maxLoopDepth; ++i) {
    if (isParallelLoop[i]) {
      loopPermMap[i] = nextParallelLoop++;
    } else {
      loopPermMap[i] = nextSequentialLoop++;
    }
  }
 
  // Check if permutation 'loopPermMap' would violate dependences.
  if (!checkLoopInterchangeDependences(depCompsVec, loops, loopPermMap))
    return forOp;
  // Perform loop interchange according to permutation 'loopPermMap'.
  unsigned loopNestRootIndex = permuteLoops(loops, loopPermMap);
  return loops[loopNestRootIndex];
}
 
// Factors out common behavior to add a new `iv` (resp. `iv` + `offset`) to the
// lower (resp. upper) loop bound. When called for both the lower and upper
// bounds, the resulting IR resembles:
//
// ```mlir
//    affine.for %i = max (`iv, ...) to min (`iv` + `offset`) {
//      ...
//    }
// ```
static void augmentMapAndBounds(OpBuilder &b, Value iv, AffineMap *map,
                                SmallVector<Value, 4> *operands,
                                int64_t offset = 0) {
  auto bounds = llvm::to_vector<4>(map->getResults());
  bounds.push_back(b.getAffineDimExpr(map->getNumDims()) + offset);
  operands->insert(operands->begin() + map->getNumDims(), iv);
  *map = AffineMap::get(map->getNumDims() + 1, map->getNumSymbols(), bounds,
                        b.getContext());
  canonicalizeMapAndOperands(map, operands);
}
 
// Stripmines `forOp` by `factor` and sinks it under each of the `targets`.
// Stripmine-sink is a primitive building block for generalized tiling of
// imperfectly nested loops.
// This transformation is purely mechanical and does not check legality,
// profitability or even structural correctness. It is the user's
// responsibility to specify `targets` that are dominated by `forOp`.
// Returns the new AffineForOps, one per `targets`, nested immediately under
// each of the `targets`.
static SmallVector<AffineForOp, 8>
stripmineSink(AffineForOp forOp, uint64_t factor,
              ArrayRef<AffineForOp> targets) {
  auto originalStep = forOp.getStepAsInt();
  auto scaledStep = originalStep * factor;
  forOp.setStep(scaledStep);
 
  OpBuilder b(forOp->getBlock(), std::next(Block::iterator(forOp)));
 
  // Lower-bound map creation.
  auto lbMap = forOp.getLowerBoundMap();
  SmallVector<Value, 4> lbOperands(forOp.getLowerBoundOperands());
  augmentMapAndBounds(b, forOp.getInductionVar(), &lbMap, &lbOperands);
 
  // Upper-bound map creation.
  auto ubMap = forOp.getUpperBoundMap();
  SmallVector<Value, 4> ubOperands(forOp.getUpperBoundOperands());
  augmentMapAndBounds(b, forOp.getInductionVar(), &ubMap, &ubOperands,
                      /*offset=*/scaledStep);
 
  auto iv = forOp.getInductionVar();
  SmallVector<AffineForOp, 8> innerLoops;
  for (auto t : targets) {
    // Insert newForOp before the terminator of `t`.
    auto b = OpBuilder::atBlockTerminator(t.getBody());
    auto newForOp = b.create<AffineForOp>(t.getLoc(), lbOperands, lbMap,
                                          ubOperands, ubMap, originalStep);
    auto begin = t.getBody()->begin();
    // Skip terminator and `newForOp` which is just before the terminator.
    auto nOps = t.getBody()->getOperations().size() - 2;
    newForOp.getBody()->getOperations().splice(
        newForOp.getBody()->getOperations().begin(),
        t.getBody()->getOperations(), begin, std::next(begin, nOps));
    replaceAllUsesInRegionWith(iv, newForOp.getInductionVar(),
                               newForOp.getRegion());
    innerLoops.push_back(newForOp);
  }
 
  return innerLoops;
}
 
// Stripmines a `forOp` by `factor` and sinks it under a single `target`.
// Returns the new AffineForOps, nested immediately under `target`.
template <typename SizeType>
static AffineForOp stripmineSink(AffineForOp forOp, SizeType factor,
                                 AffineForOp target) {
  // TODO: Use cheap structural assertions that targets are nested under
  // forOp and that targets are not nested under each other when DominanceInfo
  // exposes the capability. It seems overkill to construct a whole function
  // dominance tree at this point.
  auto res = stripmineSink(forOp, factor, ArrayRef<AffineForOp>(target));
  assert(res.size() == 1 && "Expected 1 inner forOp");
  return res[0];
}
 
SmallVector<SmallVector<AffineForOp, 8>, 8>
mlir::affine::tile(ArrayRef<AffineForOp> forOps, ArrayRef<uint64_t> sizes,
                   ArrayRef<AffineForOp> targets) {
  SmallVector<SmallVector<AffineForOp, 8>, 8> res;
  SmallVector<AffineForOp, 8> currentTargets(targets);
  for (auto it : llvm::zip(forOps, sizes)) {
    auto step = stripmineSink(std::get<0>(it), std::get<1>(it), currentTargets);
    res.push_back(step);
    currentTargets = step;
  }
  return res;
}
 
SmallVector<AffineForOp, 8> mlir::affine::tile(ArrayRef<AffineForOp> forOps,
                                               ArrayRef<uint64_t> sizes,
                                               AffineForOp target) {
  SmallVector<AffineForOp, 8> res;
  for (auto loops : tile(forOps, sizes, ArrayRef<AffineForOp>(target))) {
    assert(loops.size() == 1);
    res.push_back(loops[0]);
  }
  return res;
}
 
LogicalResult mlir::affine::coalesceLoops(MutableArrayRef<AffineForOp> loops) {
  if (loops.size() < 2)
    return success();
 
  AffineForOp innermost = loops.back();
  AffineForOp outermost = loops.front();
  AffineBound ub = outermost.getUpperBound();
  AffineMap origUbMap = ub.getMap();
  Location loc = outermost.getLoc();
  OpBuilder builder(outermost);
  for (AffineForOp loop : loops) {
    // We only work on normalized loops.
    if (loop.getStepAsInt() != 1 || !loop.hasConstantLowerBound() ||
        loop.getConstantLowerBound() != 0)
      return failure();
  }
  SmallVector<Value, 4> upperBoundSymbols;
  SmallVector<Value, 4> ubOperands(ub.getOperands().begin(),
                                   ub.getOperands().end());
 
  // 1. Store the upper bound of the outermost loop in a variable.
  Value prev;
  if (!llvm::hasSingleElement(origUbMap.getResults()))
    prev = builder.create<AffineMinOp>(loc, origUbMap, ubOperands);
  else
    prev = builder.create<AffineApplyOp>(loc, origUbMap, ubOperands);
  upperBoundSymbols.push_back(prev);
 
  // 2. Emit code computing the upper bound of the coalesced loop as product of
  // the number of iterations of all loops.
  for (AffineForOp loop : loops.drop_front()) {
    ub = loop.getUpperBound();
    origUbMap = ub.getMap();
    ubOperands = ub.getOperands();
    Value upperBound;
    // If upper bound map has more than one result, take their minimum.
    if (!llvm::hasSingleElement(origUbMap.getResults()))
      upperBound = builder.create<AffineMinOp>(loc, origUbMap, ubOperands);
    else
      upperBound = builder.create<AffineApplyOp>(loc, origUbMap, ubOperands);
    upperBoundSymbols.push_back(upperBound);
    SmallVector<Value, 4> operands;
    operands.push_back(prev);
    operands.push_back(upperBound);
    // Maintain running product of loop upper bounds.
    prev = builder.create<AffineApplyOp>(
        loc,
        AffineMap::get(/*dimCount=*/1,
                       /*symbolCount=*/1,
                       builder.getAffineDimExpr(0) *
                           builder.getAffineSymbolExpr(0)),
        operands);
  }
  // Set upper bound of the coalesced loop.
  AffineMap newUbMap = AffineMap::get(
      /*dimCount=*/0,
      /*symbolCount=*/1, builder.getAffineSymbolExpr(0), builder.getContext());
  outermost.setUpperBound(prev, newUbMap);
 
  builder.setInsertionPointToStart(outermost.getBody());
 
  // 3. Remap induction variables. For each original loop, the value of the
  // induction variable can be obtained by dividing the induction variable of
  // the linearized loop by the total number of iterations of the loops nested
  // in it modulo the number of iterations in this loop (remove the values
  // related to the outer loops):
  //   iv_i = floordiv(iv_linear, product-of-loop-ranges-until-i) mod range_i.
  // Compute these iteratively from the innermost loop by creating a "running
  // quotient" of division by the range.
  Value previous = outermost.getInductionVar();
  for (unsigned idx = loops.size(); idx > 0; --idx) {
    if (idx != loops.size()) {
      SmallVector<Value, 4> operands;
      operands.push_back(previous);
      operands.push_back(upperBoundSymbols[idx]);
      previous = builder.create<AffineApplyOp>(
          loc,
          AffineMap::get(
              /*dimCount=*/1, /*symbolCount=*/1,
              builder.getAffineDimExpr(0).floorDiv(
                  builder.getAffineSymbolExpr(0))),
          operands);
    }
    // Modified value of the induction variables of the nested loops after
    // coalescing.
    Value inductionVariable;
    if (idx == 1) {
      inductionVariable = previous;
    } else {
      SmallVector<Value, 4> applyOperands;
      applyOperands.push_back(previous);
      applyOperands.push_back(upperBoundSymbols[idx - 1]);
      inductionVariable = builder.create<AffineApplyOp>(
          loc,
          AffineMap::get(
              /*dimCount=*/1, /*symbolCount=*/1,
              builder.getAffineDimExpr(0) % builder.getAffineSymbolExpr(0)),
          applyOperands);
    }
    replaceAllUsesInRegionWith(loops[idx - 1].getInductionVar(),
                               inductionVariable, loops.back().getRegion());
  }
 
  // 4. Move the operations from the innermost just above the second-outermost
  // loop, delete the extra terminator and the second-outermost loop.
  AffineForOp secondOutermostLoop = loops[1];
  innermost.getBody()->back().erase();
  outermost.getBody()->getOperations().splice(
      Block::iterator(secondOutermostLoop.getOperation()),
      innermost.getBody()->getOperations());
  secondOutermostLoop.erase();
  return success();
}
 
void mlir::affine::mapLoopToProcessorIds(scf::ForOp forOp,
                                         ArrayRef<Value> processorId,
                                         ArrayRef<Value> numProcessors) {
  assert(processorId.size() == numProcessors.size());
  if (processorId.empty())
    return;
 
  OpBuilder b(forOp);
  Location loc(forOp.getLoc());
  AffineExpr lhs, rhs;
  bindSymbols(forOp.getContext(), lhs, rhs);
  auto mulMap = AffineMap::get(0, 2, lhs * rhs);
  auto addMap = AffineMap::get(0, 2, lhs + rhs);
 
  Value linearIndex = processorId.front();
  for (unsigned i = 1, e = processorId.size(); i < e; ++i) {
    auto mulApplyOp = b.create<AffineApplyOp>(
        loc, mulMap, ValueRange{linearIndex, numProcessors[i]});
    linearIndex = b.create<AffineApplyOp>(
        loc, addMap, ValueRange{mulApplyOp, processorId[i]});
  }
 
  auto mulApplyOp = b.create<AffineApplyOp>(
      loc, mulMap, ValueRange{linearIndex, forOp.getStep()});
  Value lb = b.create<AffineApplyOp>(
      loc, addMap, ValueRange{mulApplyOp, forOp.getLowerBound()});
  forOp.setLowerBound(lb);
 
  Value step = forOp.getStep();
  for (auto numProcs : numProcessors)
    step = b.create<AffineApplyOp>(loc, mulMap, ValueRange{numProcs, step});
  forOp.setStep(step);
}
 
/// Given a memref region, determine the lowest depth at which transfers can be
/// placed for it, and return the corresponding block, start and end positions
/// in the block for placing incoming (read) and outgoing (write) copies
/// respectively. The lowest depth depends on whether the region being accessed
/// is hoistable with respect to one or more immediately surrounding loops.
static void
findHighestBlockForPlacement(const MemRefRegion &region, Block &block,
                             Block::iterator &begin, Block::iterator &end,
                             Block **copyPlacementBlock,
                             Block::iterator *copyInPlacementStart,
                             Block::iterator *copyOutPlacementStart) {
  const auto *cst = region.getConstraints();
  SmallVector<Value, 4> symbols;
  cst->getValues(cst->getNumDimVars(), cst->getNumDimAndSymbolVars(), &symbols);
 
  SmallVector<AffineForOp, 4> enclosingFors;
  getAffineForIVs(*block.begin(), &enclosingFors);
  // Walk up loop parents till we find an IV on which this region is
  // symbolic/variant.
  auto it = enclosingFors.rbegin();
  for (auto e = enclosingFors.rend(); it != e; ++it) {
    // TODO: also need to be checking this for regions symbols that
    // aren't loop IVs, whether we are within their resp. defs' dominance scope.
    if (llvm::is_contained(symbols, it->getInductionVar()))
      break;
  }
 
  if (it != enclosingFors.rbegin()) {
    auto lastInvariantIV = *std::prev(it);
    *copyInPlacementStart = Block::iterator(lastInvariantIV.getOperation());
    *copyOutPlacementStart = std::next(*copyInPlacementStart);
    *copyPlacementBlock = lastInvariantIV->getBlock();
  } else {
    *copyInPlacementStart = begin;
    *copyOutPlacementStart = end;
    *copyPlacementBlock = &block;
  }
}
 
// Info comprising stride and number of elements transferred every stride.
struct StrideInfo {
  int64_t stride;
  int64_t numEltPerStride;
};
 
/// Returns striding information for a copy/transfer of this region with
/// potentially multiple striding levels from outermost to innermost. For an
/// n-dimensional region, there can be at most n-1 levels of striding
/// successively nested.
//  TODO: make this work with non-identity layout maps.
static void getMultiLevelStrides(const MemRefRegion &region,
                                 ArrayRef<int64_t> bufferShape,
                                 SmallVectorImpl<StrideInfo> *strideInfos) {
  if (bufferShape.size() <= 1)
    return;
 
  int64_t numEltPerStride = 1;
  int64_t stride = 1;
  for (int d = bufferShape.size() - 1; d >= 1; d--) {
    int64_t dimSize = cast<MemRefType>(region.memref.getType()).getDimSize(d);
    stride *= dimSize;
    numEltPerStride *= bufferShape[d];
    // A stride is needed only if the region has a shorter extent than the
    // memref along the dimension *and* has an extent greater than one along the
    // next major dimension.
    if (bufferShape[d] < dimSize && bufferShape[d - 1] > 1) {
      strideInfos->push_back({stride, numEltPerStride});
    }
  }
}
 
/// Generates a point-wise copy from/to `memref' to/from `fastMemRef' and
/// returns the outermost AffineForOp of the copy loop nest. `lbMaps` and
/// `ubMaps` along with `lbOperands` and `ubOperands` hold the lower and upper
/// bound information for the copy loop nest. `fastBufOffsets` contain the
/// expressions to be subtracted out from the respective copy loop iterators in
/// order to index the fast buffer. If `copyOut' is true, generates a copy-out;
/// otherwise a copy-in. Builder `b` should be set to the point the copy nest is
/// inserted.
//
/// The copy-in nest is generated as follows as an example for a 2-d region:
/// for x = ...
///   for y = ...
///     fast_buf[x - offset_x][y - offset_y] = memref[x][y]
///
static AffineForOp
generatePointWiseCopy(Location loc, Value memref, Value fastMemRef,
                      ArrayRef<AffineMap> lbMaps, ArrayRef<Value> lbOperands,
                      ArrayRef<AffineMap> ubMaps, ArrayRef<Value> ubOperands,
                      ArrayRef<AffineExpr> fastBufOffsets, bool isCopyOut,
                      OpBuilder b) {
  assert(llvm::all_of(lbMaps, [&](AffineMap lbMap) {
    return lbMap.getNumInputs() == lbOperands.size();
  }));
  assert(llvm::all_of(ubMaps, [&](AffineMap ubMap) {
    return ubMap.getNumInputs() == ubOperands.size();
  }));
 
  unsigned rank = cast<MemRefType>(memref.getType()).getRank();
  assert(lbMaps.size() == rank && "wrong number of lb maps");
  assert(ubMaps.size() == rank && "wrong number of ub maps");
 
  SmallVector<Value, 4> memIndices;
  SmallVector<AffineExpr, 4> fastBufExprs;
  SmallVector<Value, 4> fastBufMapOperands;
  AffineForOp copyNestRoot;
  SmallVector<AffineApplyOp, 4> mayBeDeadApplys;
  for (unsigned d = 0; d < rank; ++d) {
    auto forOp = createCanonicalizedAffineForOp(b, loc, lbOperands, lbMaps[d],
                                                ubOperands, ubMaps[d]);
    if (d == 0)
      copyNestRoot = forOp;
 
    b = OpBuilder::atBlockTerminator(forOp.getBody());
 
    auto fastBufOffsetMap =
        AffineMap::get(lbOperands.size(), 0, fastBufOffsets[d]);
    auto offset = b.create<AffineApplyOp>(loc, fastBufOffsetMap, lbOperands);
 
    // Construct the subscript for the fast memref being copied into/from:
    // x - offset_x.
    fastBufExprs.push_back(b.getAffineDimExpr(2 * d + 1) -
                           b.getAffineDimExpr(2 * d));
    fastBufMapOperands.push_back(offset);
    fastBufMapOperands.push_back(forOp.getInductionVar());
    mayBeDeadApplys.push_back(offset);
 
    // Subscript for the slow memref being copied.
    memIndices.push_back(forOp.getInductionVar());
  }
 
  auto fastBufMap =
      AffineMap::get(2 * rank, /*symbolCount=*/0, fastBufExprs, b.getContext());
  fullyComposeAffineMapAndOperands(&fastBufMap, &fastBufMapOperands);
  fastBufMap = simplifyAffineMap(fastBufMap);
  canonicalizeMapAndOperands(&fastBufMap, &fastBufMapOperands);
 
  // Drop any dead affine.applys.
  for (auto applyOp : mayBeDeadApplys)
    if (applyOp.use_empty())
      applyOp.erase();
 
  if (!isCopyOut) {
    // Copy in.
    auto load = b.create<AffineLoadOp>(loc, memref, memIndices);
    b.create<AffineStoreOp>(loc, load, fastMemRef, fastBufMap,
                            fastBufMapOperands);
    return copyNestRoot;
  }
 
  // Copy out.
  auto load =
      b.create<AffineLoadOp>(loc, fastMemRef, fastBufMap, fastBufMapOperands);
  b.create<AffineStoreOp>(loc, load, memref, memIndices);
  return copyNestRoot;
}
 
static InFlightDiagnostic LLVM_ATTRIBUTE_UNUSED
emitRemarkForBlock(Block &block) {
  return block.getParentOp()->emitRemark();
}
 
/// Creates a buffer in the faster memory space for the specified memref region;
/// generates a copy from the lower memory space to this one, and replaces all
/// loads/stores in the block range [`begin', `end') of `block' to load/store
/// from that buffer. Returns failure if copies could not be generated due to
/// yet unimplemented cases. `copyInPlacementStart` and `copyOutPlacementStart`
/// in copyPlacementBlock specify the insertion points where the incoming copies
/// and outgoing copies, respectively, should be inserted (the insertion happens
/// right before the insertion point). Since `begin` can itself be invalidated
/// due to the memref rewriting done from this method, the output argument
/// `nBegin` is set to its replacement (set to `begin` if no invalidation
/// happens). Since outgoing copies could have  been inserted at `end`, the
/// output argument `nEnd` is set to the new end. `sizeInBytes` is set to the
/// size of the fast buffer allocated.
static LogicalResult generateCopy(
    const MemRefRegion &region, Block *block, Block::iterator begin,
    Block::iterator end, Block *copyPlacementBlock,
    Block::iterator copyInPlacementStart, Block::iterator copyOutPlacementStart,
    const AffineCopyOptions &copyOptions, DenseMap<Value, Value> &fastBufferMap,
    DenseSet<Operation *> &copyNests, uint64_t *sizeInBytes,
    Block::iterator *nBegin, Block::iterator *nEnd) {
  *nBegin = begin;
  *nEnd = end;
 
  func::FuncOp f = begin->getParentOfType<func::FuncOp>();
  OpBuilder topBuilder(f.getBody());
  Value zeroIndex = topBuilder.create<arith::ConstantIndexOp>(f.getLoc(), 0);
 
  *sizeInBytes = 0;
 
  if (begin == end)
    return success();
 
  // Is the copy out point at the end of the block where we are doing
  // explicit copying.
  bool isCopyOutAtEndOfBlock = (end == copyOutPlacementStart);
 
  // Copies for read regions are going to be inserted at 'begin'.
  OpBuilder prologue(copyPlacementBlock, copyInPlacementStart);
  // Copies for write regions are going to be inserted at 'end'.
  OpBuilder epilogue(copyPlacementBlock, copyOutPlacementStart);
  OpBuilder &b = region.isWrite() ? epilogue : prologue;
 
  // Builder to create constants at the top level.
  auto func = copyPlacementBlock->getParent()->getParentOfType<func::FuncOp>();
  OpBuilder top(func.getBody());
 
  auto loc = region.loc;
  auto memref = region.memref;
  auto memRefType = cast<MemRefType>(memref.getType());
 
  if (!memRefType.getLayout().isIdentity()) {
    LLVM_DEBUG(llvm::dbgs() << "Non-identity layout map not yet supported\n");
    return failure();
  }
 
  // Indices to use for the copying.
  // Indices for the original memref being copied from/to.
  SmallVector<Value, 4> memIndices;
  // Indices for the faster buffer being copied into/from.
  SmallVector<Value, 4> bufIndices;
 
  unsigned rank = memRefType.getRank();
  SmallVector<int64_t, 4> fastBufferShape;
 
  // Compute the extents of the buffer.
  std::vector<SmallVector<int64_t, 4>> lbs;
  SmallVector<int64_t, 8> lbDivisors;
  lbs.reserve(rank);
  std::optional<int64_t> numElements = region.getConstantBoundingSizeAndShape(
      &fastBufferShape, &lbs, &lbDivisors);
  if (!numElements) {
    LLVM_DEBUG(llvm::dbgs() << "Non-constant region size not supported\n");
    return failure();
  }
 
  if (*numElements == 0) {
    LLVM_DEBUG(llvm::dbgs() << "Nothing to copy\n");
    return success();
  }
 
  SmallVector<AffineMap, 4> lbMaps(rank), ubMaps(rank);
  for (unsigned i = 0; i < rank; ++i)
    region.getLowerAndUpperBound(i, lbMaps[i], ubMaps[i]);
 
  const FlatAffineValueConstraints *cst = region.getConstraints();
  // 'regionSymbols' hold values that this memory region is symbolic/parametric
  // on; these typically include loop IVs surrounding the level at which the
  // copy generation is being done or other valid symbols in MLIR.
  SmallVector<Value, 8> regionSymbols;
  cst->getValues(rank, cst->getNumVars(), &regionSymbols);
 
  // Construct the index expressions for the fast memory buffer. The index
  // expression for a particular dimension of the fast buffer is obtained by
  // subtracting out the lower bound on the original memref's data region
  // along the corresponding dimension.
 
  // Index start offsets for faster memory buffer relative to the original.
  SmallVector<AffineExpr, 4> fastBufOffsets;
  fastBufOffsets.reserve(rank);
  for (unsigned d = 0; d < rank; d++) {
    assert(lbs[d].size() == cst->getNumCols() - rank && "incorrect bound size");
 
    AffineExpr offset = top.getAffineConstantExpr(0);
    for (unsigned j = 0, e = cst->getNumCols() - rank - 1; j < e; j++)
      offset = offset + lbs[d][j] * top.getAffineDimExpr(j);
    assert(lbDivisors[d] > 0);
    offset =
        (offset + lbs[d][cst->getNumCols() - 1 - rank]).floorDiv(lbDivisors[d]);
 
    // Set copy start location for this dimension in the lower memory space
    // memref.
    if (auto caf = dyn_cast<AffineConstantExpr>(offset)) {
      auto indexVal = caf.getValue();
      if (indexVal == 0) {
        memIndices.push_back(zeroIndex);
      } else {
        memIndices.push_back(
            top.create<arith::ConstantIndexOp>(loc, indexVal).getResult());
      }
    } else {
      // The coordinate for the start location is just the lower bound along the
      // corresponding dimension on the memory region (stored in 'offset').
      auto map = AffineMap::get(
          cst->getNumDimVars() + cst->getNumSymbolVars() - rank, 0, offset);
      memIndices.push_back(b.create<AffineApplyOp>(loc, map, regionSymbols));
    }
    // The fast buffer is copied into at location zero; addressing is relative.
    bufIndices.push_back(zeroIndex);
 
    // Record the offsets since they are needed to remap the memory accesses of
    // the original memref further below.
    fastBufOffsets.push_back(offset);
  }
 
  // The faster memory space buffer.
  Value fastMemRef;
 
  // Check if a buffer was already created.
  bool existingBuf = fastBufferMap.count(memref) > 0;
  if (!existingBuf) {
    AffineMap fastBufferLayout = b.getMultiDimIdentityMap(rank);
    auto fastMemRefType =
        MemRefType::get(fastBufferShape, memRefType.getElementType(),
                        fastBufferLayout, copyOptions.fastMemorySpace);
 
    // Create the fast memory space buffer just before the 'affine.for'
    // operation.
    fastMemRef =
        prologue.create<memref::AllocOp>(loc, fastMemRefType).getResult();
    // Record it.
    fastBufferMap[memref] = fastMemRef;
    // fastMemRefType is a constant shaped memref.
    auto maySizeInBytes = getIntOrFloatMemRefSizeInBytes(fastMemRefType);
    // We don't account for things of unknown size.
    *sizeInBytes = maySizeInBytes.value_or(0);
 
    LLVM_DEBUG(emitRemarkForBlock(*block)
               << "Creating fast buffer of type " << fastMemRefType
               << " and size " << llvm::divideCeil(*sizeInBytes, 1024)
               << " KiB\n");
  } else {
    // Reuse the one already created.
    fastMemRef = fastBufferMap[memref];
  }
 
  auto numElementsSSA = top.create<arith::ConstantIndexOp>(loc, *numElements);
 
  Value dmaStride;
  Value numEltPerDmaStride;
  if (copyOptions.generateDma) {
    SmallVector<StrideInfo, 4> dmaStrideInfos;
    getMultiLevelStrides(region, fastBufferShape, &dmaStrideInfos);
 
    // TODO: use all stride levels once DmaStartOp is extended for
    // multi-level strides.
    if (dmaStrideInfos.size() > 1) {
      LLVM_DEBUG(llvm::dbgs() << "Only up to one level of stride supported\n");
      return failure();
    }
 
    if (!dmaStrideInfos.empty()) {
      dmaStride =
          top.create<arith::ConstantIndexOp>(loc, dmaStrideInfos[0].stride);
      numEltPerDmaStride = top.create<arith::ConstantIndexOp>(
          loc, dmaStrideInfos[0].numEltPerStride);
    }
  }
 
  // Record the last operation where we want the memref replacement to end. We
  // later do the memref replacement only in [begin, postDomFilter] so
  // that the original memref's used in the data movement code themselves don't
  // get replaced.
  auto postDomFilter = std::prev(end);
 
  // Create fully composed affine maps for each memref.
  auto memAffineMap = b.getMultiDimIdentityMap(memIndices.size());
  fullyComposeAffineMapAndOperands(&memAffineMap, &memIndices);
  auto bufAffineMap = b.getMultiDimIdentityMap(bufIndices.size());
  fullyComposeAffineMapAndOperands(&bufAffineMap, &bufIndices);
 
  if (!copyOptions.generateDma) {
    // Point-wise copy generation.
    auto copyNest =
        generatePointWiseCopy(loc, memref, fastMemRef, lbMaps,
                              /*lbOperands=*/regionSymbols, ubMaps,
                              /*ubOperands=*/regionSymbols, fastBufOffsets,
                              /*isCopyOut=*/region.isWrite(), b);
 
    // Record this so that we can skip it from yet another copy.
    copyNests.insert(copyNest);
 
    // Since new ops are being appended (for copy out's), adjust the end to
    // mark end of block range being processed if necessary.
    if (region.isWrite() && isCopyOutAtEndOfBlock)
      *nEnd = Block::iterator(copyNest.getOperation());
  } else {
    // DMA generation.
    // Create a tag (single element 1-d memref) for the DMA.
    auto tagMemRefType = MemRefType::get({1}, top.getIntegerType(32), {},
                                         copyOptions.tagMemorySpace);
    auto tagMemRef = prologue.create<memref::AllocOp>(loc, tagMemRefType);
 
    SmallVector<Value, 4> tagIndices({zeroIndex});
    auto tagAffineMap = b.getMultiDimIdentityMap(tagIndices.size());
    fullyComposeAffineMapAndOperands(&tagAffineMap, &tagIndices);
    if (!region.isWrite()) {
      // DMA non-blocking read from original buffer to fast buffer.
      b.create<AffineDmaStartOp>(loc, memref, memAffineMap, memIndices,
                                 fastMemRef, bufAffineMap, bufIndices,
                                 tagMemRef, tagAffineMap, tagIndices,
                                 numElementsSSA, dmaStride, numEltPerDmaStride);
    } else {
      // DMA non-blocking write from fast buffer to the original memref.
      auto op = b.create<AffineDmaStartOp>(
          loc, fastMemRef, bufAffineMap, bufIndices, memref, memAffineMap,
          memIndices, tagMemRef, tagAffineMap, tagIndices, numElementsSSA,
          dmaStride, numEltPerDmaStride);
      // Since new ops may be appended at 'end' (for outgoing DMAs), adjust the
      // end to mark end of block range being processed.
      if (isCopyOutAtEndOfBlock)
        *nEnd = Block::iterator(op.getOperation());
    }
 
    // Matching DMA wait to block on completion; tag always has a 0 index.
    b.create<AffineDmaWaitOp>(loc, tagMemRef, tagAffineMap, zeroIndex,
                              numElementsSSA);
 
    // Generate dealloc for the tag.
    auto tagDeallocOp = epilogue.create<memref::DeallocOp>(loc, tagMemRef);
    if (*nEnd == end && isCopyOutAtEndOfBlock)
      // Since new ops are being appended (for outgoing DMAs), adjust the end to
      // mark end of range of the original.
      *nEnd = Block::iterator(tagDeallocOp.getOperation());
  }
 
  // Generate dealloc for the buffer.
  if (!existingBuf) {
    auto bufDeallocOp = epilogue.create<memref::DeallocOp>(loc, fastMemRef);
    // When generating pointwise copies, `nEnd' has to be set to deallocOp on
    // the fast buffer (since it marks the new end insertion point).
    if (!copyOptions.generateDma && *nEnd == end && isCopyOutAtEndOfBlock)
      *nEnd = Block::iterator(bufDeallocOp.getOperation());
  }
 
  // Replace all uses of the old memref with the faster one while remapping
  // access indices (subtracting out lower bound offsets for each dimension).
  // Ex: to replace load %A[%i, %j] with load %Abuf[%i - %iT, %j - %jT],
  // index remap will be (%i, %j) -> (%i - %iT, %j - %jT),
  // i.e., affine.apply (d0, d1, d2, d3) -> (d2-d0, d3-d1) (%iT, %jT, %i, %j),
  // and (%iT, %jT) will be the 'extraOperands' for 'rep all memref uses with'.
  // d2, d3 correspond to the original indices (%i, %j).
  SmallVector<AffineExpr, 4> remapExprs;
  remapExprs.reserve(rank);
  for (unsigned i = 0; i < rank; i++) {
    // The starting operands of indexRemap will be regionSymbols (the symbols on
    // which the memref region is parametric); then those corresponding to
    // the memref's original indices follow.
    auto dimExpr = b.getAffineDimExpr(regionSymbols.size() + i);
    remapExprs.push_back(dimExpr - fastBufOffsets[i]);
  }
  auto indexRemap = AffineMap::get(regionSymbols.size() + rank, 0, remapExprs,
                                   b.getContext());
 
  // Record the begin since it may be invalidated by memref replacement.
  Block::iterator prevOfBegin;
  bool isBeginAtStartOfBlock = (begin == block->begin());
  if (!isBeginAtStartOfBlock)
    prevOfBegin = std::prev(begin);
 
  // *Only* those uses within the range [begin, end) of 'block' are replaced.
  (void)replaceAllMemRefUsesWith(memref, fastMemRef,
                                 /*extraIndices=*/{}, indexRemap,
                                 /*extraOperands=*/regionSymbols,
                                 /*symbolOperands=*/{},
                                 /*domOpFilter=*/&*begin,
                                 /*postDomOpFilter=*/&*postDomFilter);
 
  *nBegin = isBeginAtStartOfBlock ? block->begin() : std::next(prevOfBegin);
 
  return success();
}
 
/// Construct the memref region to just include the entire memref. Returns false
/// dynamic shaped memref's for now. `numParamLoopIVs` is the number of
/// enclosing loop IVs of `op` (starting from the outermost) that the region
/// is parametric on.
static bool getFullMemRefAsRegion(Operation *op, unsigned numParamLoopIVs,
                                  MemRefRegion *region) {
  unsigned rank;
  if (auto loadOp = dyn_cast<AffineLoadOp>(op)) {
    rank = loadOp.getMemRefType().getRank();
    region->memref = loadOp.getMemRef();
    region->setWrite(false);
  } else if (auto storeOp = dyn_cast<AffineStoreOp>(op)) {
    rank = storeOp.getMemRefType().getRank();
    region->memref = storeOp.getMemRef();
    region->setWrite(true);
  } else {
    assert(false && "expected load or store op");
    return false;
  }
  auto memRefType = cast<MemRefType>(region->memref.getType());
  if (!memRefType.hasStaticShape())
    return false;
 
  auto *regionCst = region->getConstraints();
 
  // Just get the first numSymbols IVs, which the memref region is parametric
  // on.
  SmallVector<AffineForOp, 4> ivs;
  getAffineForIVs(*op, &ivs);
  ivs.resize(numParamLoopIVs);
  SmallVector<Value, 4> symbols;
  extractForInductionVars(ivs, &symbols);
  *regionCst = FlatAffineValueConstraints(rank, numParamLoopIVs, 0);
  regionCst->setValues(rank, rank + numParamLoopIVs, symbols);
 
  // Memref dim sizes provide the bounds.
  for (unsigned d = 0; d < rank; d++) {
    auto dimSize = memRefType.getDimSize(d);
    assert(dimSize > 0 && "filtered dynamic shapes above");
    regionCst->addBound(BoundType::LB, d, 0);
    regionCst->addBound(BoundType::UB, d, dimSize - 1);
  }
  return true;
}
 
LogicalResult
mlir::affine::affineDataCopyGenerate(Block::iterator begin, Block::iterator end,
                                     const AffineCopyOptions &copyOptions,
                                     std::optional<Value> filterMemRef,
                                     DenseSet<Operation *> &copyNests) {
  if (begin == end)
    return success();
 
  assert(begin->getBlock() == std::prev(end)->getBlock() &&
         "Inconsistent block begin/end args");
  assert(end != end->getBlock()->end() && "end can't be the block terminator");
 
  Block *block = begin->getBlock();
 
  // Copies will be generated for this depth, i.e., symbolic in all loops
  // surrounding the this block range.
  unsigned copyDepth = getNestingDepth(&*begin);
 
  LLVM_DEBUG(llvm::dbgs() << "Generating copies at depth " << copyDepth
                          << "\n");
  LLVM_DEBUG(llvm::dbgs() << "from begin: " << *begin << "\n");
  LLVM_DEBUG(llvm::dbgs() << "to inclusive end: " << *std::prev(end) << "\n");
 
  // List of memory regions to copy for. We need a map vector to have a
  // guaranteed iteration order to write test cases. CHECK-DAG doesn't help here
  // since the alloc's for example are identical except for the SSA id.
  SmallMapVector<Value, std::unique_ptr<MemRefRegion>, 4> readRegions;
  SmallMapVector<Value, std::unique_ptr<MemRefRegion>, 4> writeRegions;
 
  // Map from original memref's to the fast buffers that their accesses are
  // replaced with.
  DenseMap<Value, Value> fastBufferMap;
 
  // To check for errors when walking the block.
  bool error = false;
 
  // Walk this range of operations  to gather all memory regions.
  block->walk(begin, end, [&](Operation *opInst) {
    // Gather regions to allocate to buffers in faster memory space.
    if (auto loadOp = dyn_cast<AffineLoadOp>(opInst)) {
      if ((filterMemRef.has_value() && filterMemRef != loadOp.getMemRef()) ||
          (loadOp.getMemRefType().getMemorySpaceAsInt() !=
           copyOptions.slowMemorySpace))
        return;
    } else if (auto storeOp = dyn_cast<AffineStoreOp>(opInst)) {
      if ((filterMemRef.has_value() && filterMemRef != storeOp.getMemRef()) ||
          storeOp.getMemRefType().getMemorySpaceAsInt() !=
              copyOptions.slowMemorySpace)
        return;
    } else {
      // Neither load nor a store op.
      return;
    }
 
    // Compute the MemRefRegion accessed.
    auto region = std::make_unique<MemRefRegion>(opInst->getLoc());
    if (failed(region->compute(opInst, copyDepth, /*sliceState=*/nullptr,
                               /*addMemRefDimBounds=*/false))) {
      LLVM_DEBUG(llvm::dbgs()
                 << "Error obtaining memory region: semi-affine maps?\n");
      LLVM_DEBUG(llvm::dbgs() << "over-approximating to the entire memref\n");
      if (!getFullMemRefAsRegion(opInst, copyDepth, region.get())) {
        LLVM_DEBUG(
            opInst->emitError("non-constant memref sizes not yet supported"));
        error = true;
        return;
      }
    }
 
    // Each memref has a single buffer associated with it irrespective of how
    // many load's and store's happen on it.
    // TODO: in the future, when regions don't intersect and satisfy
    // other properties (based on load/store regions), we could consider
    // multiple buffers per memref.
 
    // Add to the appropriate region if it's not already in it, or take a
    // bounding box union with the existing one if it's already in there.
    // Note that a memref may have both read and write regions - so update the
    // region in the other list if one exists (write in case of read and vice
    // versa) since there is a single bounding box for a memref across all reads
    // and writes that happen on it.
 
    // Attempts to update; returns true if 'region' exists in targetRegions.
    auto updateRegion =
        [&](const SmallMapVector<Value, std::unique_ptr<MemRefRegion>, 4>
                &targetRegions) {
          const auto *const it = targetRegions.find(region->memref);
          if (it == targetRegions.end())
            return false;
 
          // Perform a union with the existing region.
          if (failed(it->second->unionBoundingBox(*region))) {
            LLVM_DEBUG(llvm::dbgs()
                       << "Memory region bounding box failed; "
                          "over-approximating to the entire memref\n");
            // If the union fails, we will overapproximate.
            if (!getFullMemRefAsRegion(opInst, copyDepth, region.get())) {
              LLVM_DEBUG(opInst->emitError(
                  "non-constant memref sizes not yet supported"));
              error = true;
              return true;
            }
            it->second->getConstraints()->clearAndCopyFrom(
                *region->getConstraints());
          } else {
            // Union was computed and stored in 'it->second': copy to 'region'.
            region->getConstraints()->clearAndCopyFrom(
                *it->second->getConstraints());
          }
          return true;
        };
 
    bool existsInRead = updateRegion(readRegions);
    if (error)
      return;
    bool existsInWrite = updateRegion(writeRegions);
    if (error)
      return;
 
    // Finally add it to the region list.
    if (region->isWrite() && !existsInWrite) {
      writeRegions[region->memref] = std::move(region);
    } else if (!region->isWrite() && !existsInRead) {
      readRegions[region->memref] = std::move(region);
    }
  });
 
  if (error) {
    LLVM_DEBUG(begin->emitError(
        "copy generation failed for one or more memref's in this block\n"));
    return failure();
  }
 
  uint64_t totalCopyBuffersSizeInBytes = 0;
  bool ret = true;
  auto processRegions =
      [&](const SmallMapVector<Value, std::unique_ptr<MemRefRegion>, 4>
              &regions) {
        for (const auto &regionEntry : regions) {
          // For each region, hoist copy in/out past all hoistable
          // 'affine.for's.
          Block::iterator copyInPlacementStart, copyOutPlacementStart;
          Block *copyPlacementBlock;
          findHighestBlockForPlacement(
              *regionEntry.second, *block, begin, end, &copyPlacementBlock,
              &copyInPlacementStart, &copyOutPlacementStart);
 
          uint64_t sizeInBytes;
          Block::iterator nBegin, nEnd;
          LogicalResult iRet = generateCopy(
              *regionEntry.second, block, begin, end, copyPlacementBlock,
              copyInPlacementStart, copyOutPlacementStart, copyOptions,
              fastBufferMap, copyNests, &sizeInBytes, &nBegin, &nEnd);
          if (succeeded(iRet)) {
            // begin/end could have been invalidated, and need update.
            begin = nBegin;
            end = nEnd;
            totalCopyBuffersSizeInBytes += sizeInBytes;
          }
          ret = ret & succeeded(iRet);
        }
      };
  processRegions(readRegions);
  processRegions(writeRegions);
 
  if (!ret) {
    LLVM_DEBUG(begin->emitError(
        "copy generation failed for one or more memref's in this block\n"));
    return failure();
  }
 
  // For a range of operations, a note will be emitted at the caller.
  AffineForOp forOp;
  if (llvm::DebugFlag && (forOp = dyn_cast<AffineForOp>(&*begin))) {
    LLVM_DEBUG(forOp.emitRemark()
               << llvm::divideCeil(totalCopyBuffersSizeInBytes, 1024)
               << " KiB of copy buffers in fast memory space for this block");
  }
 
  if (totalCopyBuffersSizeInBytes > copyOptions.fastMemCapacityBytes) {
    block->getParentOp()->emitWarning(
        "total size of all copy buffers' for this block exceeds fast memory "
        "capacity");
  }
 
  return success();
}
 
// A convenience version of affineDataCopyGenerate for all ops in the body of
// an AffineForOp.
LogicalResult mlir::affine::affineDataCopyGenerate(
    AffineForOp forOp, const AffineCopyOptions &copyOptions,
    std::optional<Value> filterMemRef, DenseSet<Operation *> &copyNests) {
  return affineDataCopyGenerate(forOp.getBody()->begin(),
                                std::prev(forOp.getBody()->end()), copyOptions,
                                filterMemRef, copyNests);
}
 
LogicalResult mlir::affine::generateCopyForMemRegion(
    const MemRefRegion &memrefRegion, Operation *analyzedOp,
    const AffineCopyOptions &copyOptions, CopyGenerateResult &result) {
  Block *block = analyzedOp->getBlock();
  auto begin = analyzedOp->getIterator();
  auto end = std::next(begin);
  DenseMap<Value, Value> fastBufferMap;
  DenseSet<Operation *> copyNests;
 
  auto err = generateCopy(memrefRegion, block, begin, end, block, begin, end,
                          copyOptions, fastBufferMap, copyNests,
                          &result.sizeInBytes, &begin, &end);
  if (failed(err))
    return err;
 
  const auto &en = fastBufferMap.find(memrefRegion.memref);
  // In some cases (empty loops), no copy generation would have happened.
  if (en == fastBufferMap.end())
    return failure();
  result.alloc = en->second.getDefiningOp();
  assert(result.alloc && "fast buffer expected to be locally allocated");
  assert(copyNests.size() <= 1 && "At most one copy nest is expected.");
  result.copyNest = copyNests.empty() ? nullptr : *copyNests.begin();
  return success();
}
 
/// Gathers all AffineForOps in 'block' at 'currLoopDepth' in 'depthToLoops'.
static void
gatherLoopsInBlock(Block *block, unsigned currLoopDepth,
                   std::vector<SmallVector<AffineForOp, 2>> &depthToLoops) {
  // Add a new empty level to output if it doesn't exist level already.
  assert(currLoopDepth <= depthToLoops.size() && "Unexpected currLoopDepth");
  if (currLoopDepth == depthToLoops.size())
    depthToLoops.emplace_back();
 
  for (auto &op : *block) {
    if (auto forOp = dyn_cast<AffineForOp>(op)) {
      depthToLoops[currLoopDepth].push_back(forOp);
      gatherLoopsInBlock(forOp.getBody(), currLoopDepth + 1, depthToLoops);
    }
  }
}
 
/// Gathers all AffineForOps in 'func.func' grouped by loop depth.
void mlir::affine::gatherLoops(
    func::FuncOp func, std::vector<SmallVector<AffineForOp, 2>> &depthToLoops) {
  for (auto &block : func)
    gatherLoopsInBlock(&block, /*currLoopDepth=*/0, depthToLoops);
 
  // Remove last loop level from output since it's empty.
  if (!depthToLoops.empty()) {
    assert(depthToLoops.back().empty() && "Last loop level is not empty?");
    depthToLoops.pop_back();
  }
}
 
AffineForOp mlir::affine::createCanonicalizedAffineForOp(
    OpBuilder b, Location loc, ValueRange lbOperands, AffineMap lbMap,
    ValueRange ubOperands, AffineMap ubMap, int64_t step) {
  SmallVector<Value, 4> lowerOperands(lbOperands);
  SmallVector<Value, 4> upperOperands(ubOperands);
 
  fullyComposeAffineMapAndOperands(&lbMap, &lowerOperands);
  canonicalizeMapAndOperands(&lbMap, &lowerOperands);
  lbMap = removeDuplicateExprs(lbMap);
  fullyComposeAffineMapAndOperands(&ubMap, &upperOperands);
  canonicalizeMapAndOperands(&ubMap, &upperOperands);
  ubMap = removeDuplicateExprs(ubMap);
 
  return b.create<AffineForOp>(loc, lowerOperands, lbMap, upperOperands, ubMap,
                               step);
}
 
/// Creates an AffineIfOp that encodes the conditional to choose between
/// the constant trip count version and an unknown trip count version of this
/// nest of loops. This is used to separate partial and full tiles if `loops`
/// has the intra-tile loops. The affine.if op is inserted at the builder
/// insertion point of `b`.
static AffineIfOp createSeparationCondition(MutableArrayRef<AffineForOp> loops,
                                            OpBuilder b) {
  if (loops.empty())
    return nullptr;
 
  auto *context = loops[0].getContext();
 
  FlatAffineValueConstraints cst;
  SmallVector<Operation *, 8> ops;
  llvm::append_range(ops, loops);
  (void)getIndexSet(ops, &cst);
 
  // Remove constraints that are independent of these loop IVs.
  cst.removeIndependentConstraints(/*pos=*/0, /*num=*/loops.size());
 
  // Construct the constraint set representing the guard for full tiles. The
  // lower bound (and upper bound) corresponding to the full tile should be
  // larger (and resp. smaller) than any other lower (or upper bound).
  SmallVector<int64_t, 8> fullTileLb, fullTileUb;
  for (auto loop : loops) {
    (void)loop;
    // TODO: Non-unit stride is not an issue to generalize to.
    assert(loop.getStepAsInt() == 1 && "point loop step expected to be one");
    // Mark everything symbols for the purpose of finding a constant diff pair.
    cst.setDimSymbolSeparation(/*newSymbolCount=*/cst.getNumDimAndSymbolVars() -
                               1);
    unsigned fullTileLbPos, fullTileUbPos;
    if (!cst.getConstantBoundOnDimSize(0, /*lb=*/nullptr,
                                       /*boundFloorDivisor=*/nullptr,
                                       /*ub=*/nullptr, &fullTileLbPos,
                                       &fullTileUbPos)) {
      LLVM_DEBUG(llvm::dbgs() << "Can't get constant diff pair for a loop\n");
      return nullptr;
    }
 
    SmallVector<unsigned, 4> lbIndices, ubIndices;
    cst.getLowerAndUpperBoundIndices(/*pos=*/0, &lbIndices, &ubIndices);
 
    auto fLb = cst.getInequality(fullTileLbPos);
    auto fUb = cst.getInequality(fullTileUbPos);
    fullTileLb.assign(fLb.begin(), fLb.end());
    fullTileUb.assign(fUb.begin(), fUb.end());
 
    // Full tile lower bound should be >= than any other lower bound.
    for (auto lbIndex : lbIndices)
      for (unsigned i = 0, e = cst.getNumCols(); i < e; ++i)
        cst.atIneq(lbIndex, i) = fullTileLb[i] - cst.atIneq(lbIndex, i);
 
    // Full tile upper bound should be <= any other upper bound.
    for (auto ubIndex : ubIndices)
      for (unsigned i = 0, e = cst.getNumCols(); i < e; ++i)
        cst.atIneq(ubIndex, i) -= fullTileUb[i];
 
    cst.removeVar(0);
  }
 
  // The previous step leads to all zeros for the full tile lb and ub position
  // itself; remove those and any other duplicates / trivial redundancies.
  cst.removeTrivialRedundancy();
 
  // Turn everything into dims conservatively since we earlier turned all
  // trailing ids past point loop IV into symbols. Some of these could be outer
  // loop IVs; we'll canonicalize anyway.
  cst.setDimSymbolSeparation(0);
 
  IntegerSet ifCondSet = cst.getAsIntegerSet(context);
  // ifCondSet can be null if cst was empty -- this can happen if all loops
  // in the nest have constant trip counts.
  if (!ifCondSet)
    return nullptr;
 
  SmallVector<Value, 4> setOperands;
  cst.getValues(0, cst.getNumDimAndSymbolVars(), &setOperands);
  canonicalizeSetAndOperands(&ifCondSet, &setOperands);
  return b.create<AffineIfOp>(loops[0].getLoc(), ifCondSet, setOperands,
                              /*withElseRegion=*/true);
}
 
/// Create the full tile loop nest (along with its body).
static LogicalResult
createFullTiles(MutableArrayRef<AffineForOp> inputNest,
                SmallVectorImpl<AffineForOp> &fullTileLoops, OpBuilder b) {
  fullTileLoops.reserve(inputNest.size());
 
  // For each loop in the original nest identify a lower/upper bound pair such
  // that their difference is a constant.
  FlatAffineValueConstraints cst;
  for (auto loop : inputNest) {
    // TODO: straightforward to generalize to a non-unit stride.
    if (loop.getStepAsInt() != 1) {
      LLVM_DEBUG(llvm::dbgs()
                 << "[tile separation] non-unit stride not implemented\n");
      return failure();
    }
    SmallVector<Operation *, 1> loopOp{loop.getOperation()};
    (void)getIndexSet(loopOp, &cst);
    // We will mark everything other than this loop IV as symbol for getting a
    // pair of <lb, ub> with a constant difference.
    cst.setDimSymbolSeparation(cst.getNumDimAndSymbolVars() - 1);
    unsigned lbPos, ubPos;
    if (!cst.getConstantBoundOnDimSize(/*pos=*/0, /*lb=*/nullptr,
                                       /*boundFloorDivisor=*/nullptr,
                                       /*ub=*/nullptr, &lbPos, &ubPos) ||
        lbPos == ubPos) {
      LLVM_DEBUG(llvm::dbgs() << "[tile separation] Can't get constant diff / "
                                 "equalities not yet handled\n");
      return failure();
    }
 
    // Set all variables as dimensions uniformly since some of those marked as
    // symbols above could be outer loop IVs (corresponding tile space IVs).
    cst.setDimSymbolSeparation(/*newSymbolCount=*/0);
 
    AffineValueMap lbVmap, ubVmap;
    cst.getIneqAsAffineValueMap(/*pos=*/0, lbPos, lbVmap, b.getContext());
    cst.getIneqAsAffineValueMap(/*pos=*/0, ubPos, ubVmap, b.getContext());
    AffineForOp fullTileLoop = createCanonicalizedAffineForOp(
        b, loop.getLoc(), lbVmap.getOperands(), lbVmap.getAffineMap(),
        ubVmap.getOperands(), ubVmap.getAffineMap());
    b = OpBuilder::atBlockTerminator(fullTileLoop.getBody());
    fullTileLoops.push_back(fullTileLoop);
  }
 
  // Add the body for the full tile loop nest.
  IRMapping operandMap;
  for (const auto &loopEn : llvm::enumerate(inputNest))
    operandMap.map(loopEn.value().getInductionVar(),
                   fullTileLoops[loopEn.index()].getInductionVar());
  b = OpBuilder::atBlockTerminator(fullTileLoops.back().getBody());
  for (auto &op : inputNest.back().getBody()->without_terminator())
    b.clone(op, operandMap);
  return success();
}
 
LogicalResult
mlir::affine::separateFullTiles(MutableArrayRef<AffineForOp> inputNest,
                                SmallVectorImpl<AffineForOp> *fullTileNest) {
  if (inputNest.empty())
    return success();
 
  auto firstLoop = inputNest[0];
 
  // Each successive for op has to be nested in the other.
  auto prevLoop = firstLoop;
  for (auto loop : inputNest.drop_front(1)) {
    assert(loop->getParentOp() == prevLoop && "input not contiguously nested");
    prevLoop = loop;
  }
 
  // Create the full tile loop nest.
  SmallVector<AffineForOp, 4> fullTileLoops;
  OpBuilder b(firstLoop);
  if (failed(createFullTiles(inputNest, fullTileLoops, b))) {
    if (!fullTileLoops.empty())
      fullTileLoops.front().erase();
    return failure();
  }
 
  // Create and insert the version select right before the root of the nest.
  b = OpBuilder(firstLoop);
  AffineIfOp ifOp = createSeparationCondition(inputNest, b);
  if (!ifOp) {
    fullTileLoops.front().erase();
    LLVM_DEBUG(llvm::dbgs() << "All tiles are full tiles, or failure creating "
                               "separation condition\n");
    return failure();
  }
 
  // Move the full tile into the then block.
  Block *thenBlock = ifOp.getThenBlock();
  AffineForOp outermostFullTileLoop = fullTileLoops[0];
  thenBlock->getOperations().splice(
      std::prev(thenBlock->end()),
      outermostFullTileLoop->getBlock()->getOperations(),
      Block::iterator(outermostFullTileLoop));
 
  // Move the partial tile into the else block. The partial tile is the same as
  // the original loop nest.
  Block *elseBlock = ifOp.getElseBlock();
  elseBlock->getOperations().splice(std::prev(elseBlock->end()),
                                    firstLoop->getBlock()->getOperations(),
                                    Block::iterator(firstLoop));
 
  if (fullTileNest)
    *fullTileNest = std::move(fullTileLoops);
 
  return success();
}
 
LogicalResult affine::coalescePerfectlyNestedAffineLoops(AffineForOp op) {
  LogicalResult result(failure());
  SmallVector<AffineForOp> loops;
  getPerfectlyNestedLoops(loops, op);
  if (loops.size() <= 1)
    return success();
 
  // Look for a band of loops that can be coalesced, i.e. perfectly nested
  // loops with bounds defined above some loop.
  // 1. For each loop, find above which parent loop its operands are
  // defined.
  SmallVector<unsigned> operandsDefinedAbove(loops.size());
  for (unsigned i = 0, e = loops.size(); i < e; ++i) {
    operandsDefinedAbove[i] = i;
    for (unsigned j = 0; j < i; ++j) {
      if (areValuesDefinedAbove(loops[i].getOperands(), loops[j].getRegion())) {
        operandsDefinedAbove[i] = j;
        break;
      }
    }
  }
 
  // 2. Identify bands of loops such that the operands of all of them are
  // defined above the first loop in the band.  Traverse the nest bottom-up
  // so that modifications don't invalidate the inner loops.
  for (unsigned end = loops.size(); end > 0; --end) {
    unsigned start = 0;
    for (; start < end - 1; ++start) {
      auto maxPos =
          *std::max_element(std::next(operandsDefinedAbove.begin(), start),
                            std::next(operandsDefinedAbove.begin(), end));
      if (maxPos > start)
        continue;
      assert(maxPos == start &&
             "expected loop bounds to be known at the start of the band");
      auto band = llvm::MutableArrayRef(loops.data() + start, end - start);
      if (succeeded(coalesceLoops(band)))
        result = success();
      break;
    }
    // If a band was found and transformed, keep looking at the loops above
    // the outermost transformed loop.
    if (start != end - 1)
      end = start + 1;
  }
  return result;
}