xref: /llvm-project/mlir/lib/Dialect/SparseTensor/IR/SparseTensorDialect.cpp (revision af6e1881e0791ac1ee611b62a3d12d9fb03ca142)

//===- SparseTensorDialect.cpp - Sparse tensor dialect implementation -----===//
//
// 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
//
//===----------------------------------------------------------------------===//

#include <utility>

#include "Detail/DimLvlMapParser.h"

#include "mlir/Dialect/SparseTensor/IR/Enums.h"
#include "mlir/Dialect/SparseTensor/IR/SparseTensor.h"
#include "mlir/Dialect/SparseTensor/IR/SparseTensorStorageLayout.h"
#include "mlir/Dialect/SparseTensor/IR/SparseTensorType.h"

#include "mlir/Dialect/Arith/IR/Arith.h"
#include "mlir/Dialect/Bufferization/IR/BufferizableOpInterface.h"
#include "mlir/Dialect/Complex/IR/Complex.h"
#include "mlir/Dialect/Utils/StaticValueUtils.h"
#include "mlir/IR/Builders.h"
#include "mlir/IR/DialectImplementation.h"
#include "mlir/IR/Matchers.h"
#include "mlir/IR/OpImplementation.h"
#include "mlir/IR/PatternMatch.h"
#include "llvm/ADT/Bitset.h"
#include "llvm/ADT/TypeSwitch.h"
#include "llvm/Support/FormatVariadic.h"

#define GET_ATTRDEF_CLASSES
#include "mlir/Dialect/SparseTensor/IR/SparseTensorAttrDefs.cpp.inc"
#include "mlir/Dialect/SparseTensor/IR/SparseTensorAttrEnums.cpp.inc"

// Forward declarations, following custom print/parsing methods are referenced
// by the generated code for SparseTensorTypes.td.
static mlir::ParseResult parseLevelRange(mlir::AsmParser &,
                                         mlir::sparse_tensor::Level &,
                                         mlir::sparse_tensor::Level &);
static void printLevelRange(mlir::AsmPrinter &, mlir::sparse_tensor::Level,
                            mlir::sparse_tensor::Level);

#define GET_TYPEDEF_CLASSES
#include "mlir/Dialect/SparseTensor/IR/SparseTensorTypes.cpp.inc"

using namespace mlir;
using namespace mlir::sparse_tensor;

// Support hashing LevelType such that SparseTensorEncodingAttr can be hashed as
// well.
namespace mlir::sparse_tensor {
llvm::hash_code hash_value(LevelType lt) {
  return llvm::hash_value(static_cast<uint64_t>(lt));
}
} // namespace mlir::sparse_tensor

//===----------------------------------------------------------------------===//
// Local Convenience Methods.
//===----------------------------------------------------------------------===//

static constexpr bool acceptBitWidth(unsigned bitWidth) {
  switch (bitWidth) {
  case 0:
  case 8:
  case 16:
  case 32:
  case 64:
    return true;
  default:
    return false;
  }
}

static SmallVector<Size>
getSparseFieldShape(const SparseTensorEncodingAttr enc,
                    std::optional<ArrayRef<int64_t>> dimShape) {
  assert(enc);
  // With only encoding, we can not determine the static shape for leading
  // batch levels, we therefore return a dynamic shape memref instead.
  SmallVector<int64_t> memrefShape(enc.getBatchLvlRank(), ShapedType::kDynamic);
  if (dimShape.has_value()) {
    // If the actual tensor shape is provided, we can then refine the leading
    // batch dimension.
    SmallVector<int64_t> lvlShape =
        enc.translateShape(*dimShape, CrdTransDirectionKind::dim2lvl);
    memrefShape.assign(lvlShape.begin(),
                       lvlShape.begin() + enc.getBatchLvlRank());
  }
  // Another dynamic dimension to store the sparse level.
  memrefShape.push_back(ShapedType::kDynamic);
  return memrefShape;
}

//===----------------------------------------------------------------------===//
// SparseTensorDialect StorageLayout.
//===----------------------------------------------------------------------===//

static constexpr Level kInvalidLevel = -1u;
static constexpr Level kInvalidFieldIndex = -1u;
static constexpr FieldIndex kDataFieldStartingIdx = 0;

void StorageLayout::foreachField(
    llvm::function_ref<bool(FieldIndex, SparseTensorFieldKind, Level,
                            LevelType)>
        callback) const {
  const auto lvlTypes = enc.getLvlTypes();
  const Level lvlRank = enc.getLvlRank();
  SmallVector<COOSegment> cooSegs = enc.getCOOSegments();
  FieldIndex fieldIdx = kDataFieldStartingIdx;

  ArrayRef cooSegsRef = cooSegs;
  // Per-level storage.
  for (Level l = 0; l < lvlRank; /*l += 1 or l += AoSCooLen*/) {
    const auto lt = lvlTypes[l];
    if (isWithPosLT(lt)) {
      if (!(callback(fieldIdx++, SparseTensorFieldKind::PosMemRef, l, lt)))
        return;
    }
    if (isWithCrdLT(lt)) {
      if (!(callback(fieldIdx++, SparseTensorFieldKind::CrdMemRef, l, lt)))
        return;
    }
    if (!cooSegsRef.empty() && cooSegsRef.front().isSegmentStart(l)) {
      if (!cooSegsRef.front().isSoA) {
        // AoS COO, all singletons are fused into one memrefs. Skips the entire
        // COO segement.
        l = cooSegsRef.front().lvlRange.second;
      } else {
        // SoA COO, each singleton level has one memref.
        l++;
      }
      // Expire handled COO segment.
      cooSegsRef = cooSegsRef.drop_front();
    } else {
      // Non COO levels.
      l++;
    }
  }
  // The values array.
  if (!(callback(fieldIdx++, SparseTensorFieldKind::ValMemRef, kInvalidLevel,
                 LevelFormat::Undef)))
    return;
  // Put metadata at the end.
  if (!(callback(fieldIdx++, SparseTensorFieldKind::StorageSpec, kInvalidLevel,
                 LevelFormat::Undef)))
    return;
}

void sparse_tensor::foreachFieldAndTypeInSparseTensor(
    SparseTensorType stt,
    llvm::function_ref<bool(Type, FieldIndex, SparseTensorFieldKind, Level,
                            LevelType)>
        callback) {
  assert(stt.hasEncoding());

  SmallVector<int64_t> memrefShape =
      getSparseFieldShape(stt.getEncoding(), stt.getDimShape());

  const Type specType = StorageSpecifierType::get(stt.getEncoding());
  // memref<[batch] x ? x pos>  positions
  const Type posMemType = MemRefType::get(memrefShape, stt.getPosType());
  // memref<[batch] x ? x crd>  coordinates
  const Type crdMemType = MemRefType::get(memrefShape, stt.getCrdType());
  // memref<[batch] x ? x eltType> values
  const Type valMemType = MemRefType::get(memrefShape, stt.getElementType());

  StorageLayout(stt).foreachField([specType, posMemType, crdMemType, valMemType,
                                   callback](FieldIndex fieldIdx,
                                             SparseTensorFieldKind fieldKind,
                                             Level lvl, LevelType lt) -> bool {
    switch (fieldKind) {
    case SparseTensorFieldKind::StorageSpec:
      return callback(specType, fieldIdx, fieldKind, lvl, lt);
    case SparseTensorFieldKind::PosMemRef:
      return callback(posMemType, fieldIdx, fieldKind, lvl, lt);
    case SparseTensorFieldKind::CrdMemRef:
      return callback(crdMemType, fieldIdx, fieldKind, lvl, lt);
    case SparseTensorFieldKind::ValMemRef:
      return callback(valMemType, fieldIdx, fieldKind, lvl, lt);
    };
    llvm_unreachable("unrecognized field kind");
  });
}

unsigned StorageLayout::getNumFields() const {
  unsigned numFields = 0;
  foreachField([&numFields](FieldIndex, SparseTensorFieldKind, Level,
                            LevelType) -> bool {
    numFields++;
    return true;
  });
  return numFields;
}

unsigned StorageLayout::getNumDataFields() const {
  unsigned numFields = 0; // one value memref
  foreachField([&numFields](FieldIndex fidx, SparseTensorFieldKind, Level,
                            LevelType) -> bool {
    if (fidx >= kDataFieldStartingIdx)
      numFields++;
    return true;
  });
  numFields -= 1; // the last field is StorageSpecifier
  assert(numFields == getNumFields() - kDataFieldStartingIdx - 1);
  return numFields;
}

std::pair<FieldIndex, unsigned>
StorageLayout::getFieldIndexAndStride(SparseTensorFieldKind kind,
                                      std::optional<Level> lvl) const {
  FieldIndex fieldIdx = kInvalidFieldIndex;
  unsigned stride = 1;
  if (kind == SparseTensorFieldKind::CrdMemRef) {
    assert(lvl.has_value());
    const Level cooStart = enc.getAoSCOOStart();
    const Level lvlRank = enc.getLvlRank();
    if (lvl.value() >= cooStart && lvl.value() < lvlRank) {
      lvl = cooStart;
      stride = lvlRank - cooStart;
    }
  }
  foreachField([lvl, kind, &fieldIdx](FieldIndex fIdx,
                                      SparseTensorFieldKind fKind, Level fLvl,
                                      LevelType lt) -> bool {
    if ((lvl && fLvl == lvl.value() && kind == fKind) ||
        (kind == fKind && fKind == SparseTensorFieldKind::ValMemRef)) {
      fieldIdx = fIdx;
      // Returns false to break the iteration.
      return false;
    }
    return true;
  });
  assert(fieldIdx != kInvalidFieldIndex);
  return std::pair<FieldIndex, unsigned>(fieldIdx, stride);
}

//===----------------------------------------------------------------------===//
// SparseTensorDialect Attribute Methods.
//===----------------------------------------------------------------------===//

std::optional<uint64_t> SparseTensorDimSliceAttr::getStatic(int64_t v) {
  return isDynamic(v) ? std::nullopt
                      : std::make_optional(static_cast<uint64_t>(v));
}

std::optional<uint64_t> SparseTensorDimSliceAttr::getStaticOffset() const {
  return getStatic(getOffset());
}

std::optional<uint64_t> SparseTensorDimSliceAttr::getStaticStride() const {
  return getStatic(getStride());
}

std::optional<uint64_t> SparseTensorDimSliceAttr::getStaticSize() const {
  return getStatic(getSize());
}

bool SparseTensorDimSliceAttr::isCompletelyDynamic() const {
  return isDynamic(getOffset()) && isDynamic(getStride()) &&
         isDynamic(getSize());
}

std::string SparseTensorDimSliceAttr::getStaticString(int64_t v) {
  return isDynamic(v) ? "?" : std::to_string(v);
}

void SparseTensorDimSliceAttr::print(llvm::raw_ostream &os) const {
  assert(getImpl() && "Uninitialized SparseTensorDimSliceAttr");
  os << '(';
  os << getStaticString(getOffset());
  os << ", ";
  os << getStaticString(getSize());
  os << ", ";
  os << getStaticString(getStride());
  os << ')';
}

void SparseTensorDimSliceAttr::print(AsmPrinter &printer) const {
  print(printer.getStream());
}

static ParseResult parseOptionalStaticSlice(int64_t &result,
                                            AsmParser &parser) {
  auto parseResult = parser.parseOptionalInteger(result);
  if (parseResult.has_value()) {
    if (parseResult.value().succeeded() && result < 0) {
      parser.emitError(
          parser.getCurrentLocation(),
          "expect positive value or ? for slice offset/size/stride");
      return failure();
    }
    return parseResult.value();
  }

  // Else, and '?' which represented dynamic slice
  result = SparseTensorDimSliceAttr::kDynamic;
  return parser.parseQuestion();
}

Attribute SparseTensorDimSliceAttr::parse(AsmParser &parser, Type type) {
  int64_t offset = kDynamic, size = kDynamic, stride = kDynamic;

  if (failed(parser.parseLParen()) ||
      failed(parseOptionalStaticSlice(offset, parser)) ||
      failed(parser.parseComma()) ||
      failed(parseOptionalStaticSlice(size, parser)) ||
      failed(parser.parseComma()) ||
      failed(parseOptionalStaticSlice(stride, parser)) ||
      failed(parser.parseRParen()))
    return {};

  return parser.getChecked<SparseTensorDimSliceAttr>(parser.getContext(),
                                                     offset, size, stride);
}

LogicalResult
SparseTensorDimSliceAttr::verify(function_ref<InFlightDiagnostic()> emitError,
                                 int64_t offset, int64_t size, int64_t stride) {
  if (!isDynamic(offset) && offset < 0)
    return emitError() << "expect non-negative value or ? for slice offset";
  if (!isDynamic(size) && size <= 0)
    return emitError() << "expect positive value or ? for slice size";
  if (!isDynamic(stride) && stride <= 0)
    return emitError() << "expect positive value or ? for slice stride";
  return success();
}

SparseTensorEncodingAttr
SparseTensorEncodingAttr::withDimToLvl(AffineMap dimToLvl) const {
  assert(getImpl() && "Uninitialized SparseTensorEncodingAttr");
  return SparseTensorEncodingAttr::get(
      getContext(), getLvlTypes(), dimToLvl, AffineMap(), getPosWidth(),
      getCrdWidth(), getExplicitVal(), getImplicitVal());
}

SparseTensorEncodingAttr
SparseTensorEncodingAttr::withDimToLvl(SparseTensorEncodingAttr enc) const {
  return withDimToLvl(enc ? enc.getDimToLvl() : AffineMap());
}

SparseTensorEncodingAttr SparseTensorEncodingAttr::withoutDimToLvl() const {
  return withDimToLvl(AffineMap());
}

SparseTensorEncodingAttr
SparseTensorEncodingAttr::withBitWidths(unsigned posWidth,
                                        unsigned crdWidth) const {
  assert(getImpl() && "Uninitialized SparseTensorEncodingAttr");
  return SparseTensorEncodingAttr::get(
      getContext(), getLvlTypes(), getDimToLvl(), getLvlToDim(), posWidth,
      crdWidth, getExplicitVal(), getImplicitVal());
}

SparseTensorEncodingAttr SparseTensorEncodingAttr::withoutBitWidths() const {
  return withBitWidths(0, 0);
}

SparseTensorEncodingAttr
SparseTensorEncodingAttr::withExplicitVal(Attribute explicitVal) const {
  assert(getImpl() && "Uninitialized SparseTensorEncodingAttr");
  return SparseTensorEncodingAttr::get(
      getContext(), getLvlTypes(), getDimToLvl(), getLvlToDim(), getPosWidth(),
      getCrdWidth(), explicitVal, getImplicitVal());
}

SparseTensorEncodingAttr SparseTensorEncodingAttr::withoutExplicitVal() const {
  return withExplicitVal(Attribute());
}

SparseTensorEncodingAttr
SparseTensorEncodingAttr::withImplicitVal(Attribute implicitVal) const {
  assert(getImpl() && "Uninitialized SparseTensorEncodingAttr");
  return SparseTensorEncodingAttr::get(
      getContext(), getLvlTypes(), getDimToLvl(), getLvlToDim(), getPosWidth(),
      getCrdWidth(), getExplicitVal(), implicitVal);
}

SparseTensorEncodingAttr SparseTensorEncodingAttr::withoutImplicitVal() const {
  return withImplicitVal(Attribute());
}

SparseTensorEncodingAttr SparseTensorEncodingAttr::withDimSlices(
    ArrayRef<SparseTensorDimSliceAttr> dimSlices) const {
  return SparseTensorEncodingAttr::get(
      getContext(), getLvlTypes(), getDimToLvl(), getLvlToDim(), getPosWidth(),
      getCrdWidth(), getExplicitVal(), getImplicitVal(), dimSlices);
}

SparseTensorEncodingAttr SparseTensorEncodingAttr::withoutDimSlices() const {
  return withDimSlices(ArrayRef<SparseTensorDimSliceAttr>{});
}

uint64_t SparseTensorEncodingAttr::getBatchLvlRank() const {
  ArrayRef<LevelType> lvlTypes = getLvlTypes();
  auto lastBatch = std::find_if(lvlTypes.rbegin(), lvlTypes.rend(), isBatchLT);
  return std::distance(lastBatch, lvlTypes.rend());
}

bool SparseTensorEncodingAttr::isAllDense() const {
  return !getImpl() || llvm::all_of(getLvlTypes(), isDenseLT);
}

bool SparseTensorEncodingAttr::isAllOrdered() const {
  return !getImpl() || llvm::all_of(getLvlTypes(), isOrderedLT);
}

Type SparseTensorEncodingAttr::getCrdElemType() const {
  if (!getImpl())
    return nullptr;
  if (getCrdWidth())
    return IntegerType::get(getContext(), getCrdWidth());
  return IndexType::get(getContext());
}

Type SparseTensorEncodingAttr::getPosElemType() const {
  if (!getImpl())
    return nullptr;
  if (getPosWidth())
    return IntegerType::get(getContext(), getPosWidth());
  return IndexType::get(getContext());
}

MemRefType SparseTensorEncodingAttr::getCrdMemRefType(
    std::optional<ArrayRef<int64_t>> dimShape) const {
  SmallVector<Size> shape = getSparseFieldShape(*this, dimShape);
  return MemRefType::get(shape, getCrdElemType());
}

MemRefType SparseTensorEncodingAttr::getPosMemRefType(
    std::optional<ArrayRef<int64_t>> dimShape) const {
  SmallVector<Size> shape = getSparseFieldShape(*this, dimShape);
  return MemRefType::get(shape, getPosElemType());
}

bool SparseTensorEncodingAttr::isIdentity() const {
  return !getImpl() || !getDimToLvl() || getDimToLvl().isIdentity();
}

bool SparseTensorEncodingAttr::isPermutation() const {
  return !getImpl() || !getDimToLvl() || getDimToLvl().isPermutation();
}

Dimension SparseTensorEncodingAttr::getDimRank() const {
  assert(getImpl() && "Uninitialized SparseTensorEncodingAttr");
  const auto dimToLvl = getDimToLvl();
  return dimToLvl ? dimToLvl.getNumDims() : getLvlRank();
}

Level SparseTensorEncodingAttr::getLvlRank() const {
  assert(getImpl() && "Uninitialized SparseTensorEncodingAttr");
  return getLvlTypes().size();
}

LevelType SparseTensorEncodingAttr::getLvlType(Level l) const {
  if (!getImpl())
    return LevelFormat::Batch;
  assert(l < getLvlRank() && "Level is out of bounds");
  return getLvlTypes()[l];
}

bool SparseTensorEncodingAttr::isSlice() const {
  assert(getImpl() && "Uninitialized SparseTensorEncodingAttr");
  return !getDimSlices().empty();
}

SparseTensorDimSliceAttr
SparseTensorEncodingAttr::getDimSlice(Dimension dim) const {
  assert(isSlice() && "Is not a slice");
  const auto dimSlices = getDimSlices();
  assert(dim < dimSlices.size() && "Dimension is out of bounds");
  return dimSlices[dim];
}

std::optional<uint64_t>
SparseTensorEncodingAttr::getStaticDimSliceOffset(Dimension dim) const {
  return getDimSlice(dim).getStaticOffset();
}

std::optional<uint64_t>
SparseTensorEncodingAttr::getStaticDimSliceStride(Dimension dim) const {
  return getDimSlice(dim).getStaticStride();
}

std::optional<uint64_t>
SparseTensorEncodingAttr::getStaticLvlSliceOffset(Level lvl) const {
  return getStaticDimSliceOffset(toDim(*this, lvl));
}

std::optional<uint64_t>
SparseTensorEncodingAttr::getStaticLvlSliceStride(Level lvl) const {
  return getStaticDimSliceStride(toDim(*this, lvl));
}

SmallVector<int64_t>
SparseTensorEncodingAttr::translateShape(ArrayRef<int64_t> srcShape,
                                         CrdTransDirectionKind dir) const {
  if (isIdentity())
    return SmallVector<int64_t>(srcShape);

  SmallVector<int64_t> ret;
  unsigned rank =
      dir == CrdTransDirectionKind::dim2lvl ? getLvlRank() : getDimRank();
  ret.reserve(rank);

  if (isPermutation()) {
    for (unsigned r = 0; r < rank; r++) {
      unsigned trans = dir == CrdTransDirectionKind::dim2lvl ? toDim(*this, r)
                                                             : toLvl(*this, r);
      ret.push_back(srcShape[trans]);
    }
    return ret;
  }

  // Handle non-permutation maps.
  AffineMap transMap =
      dir == CrdTransDirectionKind::dim2lvl ? getDimToLvl() : getLvlToDim();

  SmallVector<AffineExpr> dimRep;
  dimRep.reserve(srcShape.size());
  for (int64_t sz : srcShape) {
    if (!ShapedType::isDynamic(sz)) {
      // Push back the max coordinate for the given dimension/level size.
      dimRep.push_back(getAffineConstantExpr(sz - 1, getContext()));
    } else {
      // A dynamic size, use a AffineDimExpr to symbolize the value.
      dimRep.push_back(getAffineDimExpr(dimRep.size(), getContext()));
    }
  };

  for (AffineExpr exp : transMap.getResults()) {
    // Do constant propagation on the affine map.
    AffineExpr evalExp =
        simplifyAffineExpr(exp.replaceDims(dimRep), srcShape.size(), 0);
    // use llvm namespace here to avoid ambiguity
    if (auto c = llvm::dyn_cast<AffineConstantExpr>(evalExp)) {
      ret.push_back(c.getValue() + 1);
    } else {
      if (auto mod = llvm::dyn_cast<AffineBinaryOpExpr>(evalExp);
          mod && mod.getKind() == AffineExprKind::Mod) {
        // We can still infer a static bound for expressions in form
        // "d % constant" since d % constant \in [0, constant).
        if (auto bound = llvm::dyn_cast<AffineConstantExpr>(mod.getRHS())) {
          ret.push_back(bound.getValue());
          continue;
        }
      }
      ret.push_back(ShapedType::kDynamic);
    }
  }
  assert(ret.size() == rank);
  return ret;
}

ValueRange
SparseTensorEncodingAttr::translateCrds(OpBuilder &builder, Location loc,
                                        ValueRange crds,
                                        CrdTransDirectionKind dir) const {
  if (!getImpl())
    return crds;

  SmallVector<Type> retType(
      dir == CrdTransDirectionKind::lvl2dim ? getDimRank() : getLvlRank(),
      builder.getIndexType());
  auto transOp = builder.create<CrdTranslateOp>(loc, retType, crds, dir, *this);
  return transOp.getOutCrds();
}

Attribute SparseTensorEncodingAttr::parse(AsmParser &parser, Type type) {
  // Open "<{" part.
  if (failed(parser.parseLess()))
    return {};
  if (failed(parser.parseLBrace()))
    return {};

  // Process the data from the parsed dictionary value into struct-like data.
  SmallVector<LevelType> lvlTypes;
  SmallVector<SparseTensorDimSliceAttr> dimSlices;
  AffineMap dimToLvl = {};
  AffineMap lvlToDim = {};
  unsigned posWidth = 0;
  unsigned crdWidth = 0;
  Attribute explicitVal;
  Attribute implicitVal;
  StringRef attrName;
  SmallVector<StringRef, 5> keys = {"map", "posWidth", "crdWidth",
                                    "explicitVal", "implicitVal"};
  while (succeeded(parser.parseOptionalKeyword(&attrName))) {
    // Detect admissible keyword.
    auto *it = find(keys, attrName);
    if (it == keys.end()) {
      parser.emitError(parser.getNameLoc(), "unexpected key: ") << attrName;
      return {};
    }
    unsigned keyWordIndex = it - keys.begin();
    // Consume the `=` after keys
    if (failed(parser.parseEqual()))
      return {};
    // Dispatch on keyword.
    switch (keyWordIndex) {
    case 0: { // map
      ir_detail::DimLvlMapParser cParser(parser);
      auto res = cParser.parseDimLvlMap();
      if (failed(res))
        return {};
      const auto &dlm = *res;

      const Level lvlRank = dlm.getLvlRank();
      for (Level lvl = 0; lvl < lvlRank; lvl++)
        lvlTypes.push_back(dlm.getLvlType(lvl));

      const Dimension dimRank = dlm.getDimRank();
      for (Dimension dim = 0; dim < dimRank; dim++)
        dimSlices.push_back(dlm.getDimSlice(dim));
      // NOTE: the old syntax requires an all-or-nothing approach to
      // `dimSlices`; therefore, if any slice actually exists then we need
      // to convert null-DSA into default/nop DSA.
      const auto isDefined = [](SparseTensorDimSliceAttr slice) {
        return static_cast<bool>(slice.getImpl());
      };
      if (llvm::any_of(dimSlices, isDefined)) {
        const auto defaultSlice =
            SparseTensorDimSliceAttr::get(parser.getContext());
        for (Dimension dim = 0; dim < dimRank; dim++)
          if (!isDefined(dimSlices[dim]))
            dimSlices[dim] = defaultSlice;
      } else {
        dimSlices.clear();
      }

      dimToLvl = dlm.getDimToLvlMap(parser.getContext());
      lvlToDim = dlm.getLvlToDimMap(parser.getContext());
      break;
    }
    case 1: { // posWidth
      Attribute attr;
      if (failed(parser.parseAttribute(attr)))
        return {};
      auto intAttr = llvm::dyn_cast<IntegerAttr>(attr);
      if (!intAttr) {
        parser.emitError(parser.getNameLoc(),
                         "expected an integral position bitwidth");
        return {};
      }
      posWidth = intAttr.getInt();
      break;
    }
    case 2: { // crdWidth
      Attribute attr;
      if (failed(parser.parseAttribute(attr)))
        return {};
      auto intAttr = llvm::dyn_cast<IntegerAttr>(attr);
      if (!intAttr) {
        parser.emitError(parser.getNameLoc(),
                         "expected an integral index bitwidth");
        return {};
      }
      crdWidth = intAttr.getInt();
      break;
    }
    case 3: { // explicitVal
      Attribute attr;
      if (failed(parser.parseAttribute(attr)))
        return {};
      if (auto result = llvm::dyn_cast<FloatAttr>(attr)) {
        explicitVal = result;
      } else if (auto result = llvm::dyn_cast<IntegerAttr>(attr)) {
        explicitVal = result;
      } else if (auto result = llvm::dyn_cast<complex::NumberAttr>(attr)) {
        explicitVal = result;
      } else {
        parser.emitError(parser.getNameLoc(),
                         "expected a numeric value for explicitVal");
        return {};
      }
      break;
    }
    case 4: { // implicitVal
      Attribute attr;
      if (failed(parser.parseAttribute(attr)))
        return {};
      if (auto result = llvm::dyn_cast<FloatAttr>(attr)) {
        implicitVal = result;
      } else if (auto result = llvm::dyn_cast<IntegerAttr>(attr)) {
        implicitVal = result;
      } else if (auto result = llvm::dyn_cast<complex::NumberAttr>(attr)) {
        implicitVal = result;
      } else {
        parser.emitError(parser.getNameLoc(),
                         "expected a numeric value for implicitVal");
        return {};
      }
      break;
    }
    } // switch
    // Only last item can omit the comma.
    if (parser.parseOptionalComma().failed())
      break;
  }

  // Close "}>" part.
  if (failed(parser.parseRBrace()))
    return {};
  if (failed(parser.parseGreater()))
    return {};

  // Construct struct-like storage for attribute.
  if (!lvlToDim || lvlToDim.isEmpty()) {
    lvlToDim = inferLvlToDim(dimToLvl, parser.getContext());
  }
  return parser.getChecked<SparseTensorEncodingAttr>(
      parser.getContext(), lvlTypes, dimToLvl, lvlToDim, posWidth, crdWidth,
      explicitVal, implicitVal, dimSlices);
}

void SparseTensorEncodingAttr::print(AsmPrinter &printer) const {
  auto map = static_cast<AffineMap>(getDimToLvl());
  // Empty affine map indicates identity map
  if (!map)
    map = AffineMap::getMultiDimIdentityMap(getLvlTypes().size(), getContext());
  printer << "<{ map = ";
  printSymbols(map, printer);
  printer << '(';
  printDimensions(map, printer, getDimSlices());
  printer << ") -> (";
  printLevels(map, printer, getLvlTypes());
  printer << ')';
  // Print remaining members only for non-default values.
  if (getPosWidth())
    printer << ", posWidth = " << getPosWidth();
  if (getCrdWidth())
    printer << ", crdWidth = " << getCrdWidth();
  if (getExplicitVal()) {
    printer << ", explicitVal = " << getExplicitVal();
  }
  if (getImplicitVal())
    printer << ", implicitVal = " << getImplicitVal();
  printer << " }>";
}

void SparseTensorEncodingAttr::printSymbols(AffineMap &map,
                                            AsmPrinter &printer) const {
  if (map.getNumSymbols() == 0)
    return;
  printer << '[';
  for (unsigned i = 0, n = map.getNumSymbols() - 1; i < n; i++)
    printer << 's' << i << ", ";
  if (map.getNumSymbols() >= 1)
    printer << 's' << map.getNumSymbols() - 1;
  printer << ']';
}

void SparseTensorEncodingAttr::printDimensions(
    AffineMap &map, AsmPrinter &printer,
    ArrayRef<SparseTensorDimSliceAttr> dimSlices) const {
  if (!dimSlices.empty()) {
    for (unsigned i = 0, n = map.getNumDims() - 1; i < n; i++)
      printer << 'd' << i << " : " << dimSlices[i] << ", ";
    if (map.getNumDims() >= 1) {
      printer << 'd' << map.getNumDims() - 1 << " : "
              << dimSlices[map.getNumDims() - 1];
    }
  } else {
    for (unsigned i = 0, n = map.getNumDims() - 1; i < n; i++)
      printer << 'd' << i << ", ";
    if (map.getNumDims() >= 1)
      printer << 'd' << map.getNumDims() - 1;
  }
}

void SparseTensorEncodingAttr::printLevels(AffineMap &map, AsmPrinter &printer,
                                           ArrayRef<LevelType> lvlTypes) const {
  for (unsigned i = 0, n = map.getNumResults() - 1; i < n; i++) {
    map.getResult(i).print(printer.getStream());
    printer << " : " << toMLIRString(lvlTypes[i]) << ", ";
  }
  if (map.getNumResults() >= 1) {
    auto lastIndex = map.getNumResults() - 1;
    map.getResult(lastIndex).print(printer.getStream());
    printer << " : " << toMLIRString(lvlTypes[lastIndex]);
  }
}

LogicalResult SparseTensorEncodingAttr::verify(
    function_ref<InFlightDiagnostic()> emitError, ArrayRef<LevelType> lvlTypes,
    AffineMap dimToLvl, AffineMap lvlToDim, unsigned posWidth,
    unsigned crdWidth, Attribute explicitVal, Attribute implicitVal,
    ArrayRef<SparseTensorDimSliceAttr> dimSlices) {
  if (!acceptBitWidth(posWidth))
    return emitError() << "unexpected position bitwidth: " << posWidth;
  if (!acceptBitWidth(crdWidth))
    return emitError() << "unexpected coordinate bitwidth: " << crdWidth;

  // Verify every COO segment.
  auto *it = std::find_if(lvlTypes.begin(), lvlTypes.end(), isSingletonLT);
  while (it != lvlTypes.end()) {
    if (it == lvlTypes.begin() ||
        !(it - 1)->isa<LevelFormat::Compressed, LevelFormat::LooseCompressed>())
      return emitError() << "expected compressed or loose_compressed level "
                            "before singleton level";

    auto *curCOOEnd = std::find_if_not(it, lvlTypes.end(), isSingletonLT);
    if (!std::all_of(it, curCOOEnd,
                     [](LevelType i) { return isSingletonLT(i); }))
      return emitError() << "expected all singleton lvlTypes "
                            "following a singleton level";
    // We can potentially support mixed SoA/AoS singleton levels.
    if (!std::all_of(it, curCOOEnd, [it](LevelType i) {
          return it->isa<LevelPropNonDefault::SoA>() ==
                 i.isa<LevelPropNonDefault::SoA>();
        })) {
      return emitError() << "expected all singleton lvlTypes stored in the "
                            "same memory layout (SoA vs AoS).";
    }
    it = std::find_if(curCOOEnd, lvlTypes.end(), isSingletonLT);
  }

  auto lastBatch = std::find_if(lvlTypes.rbegin(), lvlTypes.rend(), isBatchLT);
  if (!std::all_of(lastBatch, lvlTypes.rend(), isBatchLT))
    return emitError() << "Batch lvlType can only be leading levels.";

  // SoA property can only be applied on singleton level.
  auto soaLvls = llvm::make_filter_range(lvlTypes, [](LevelType lt) {
    return lt.isa<LevelPropNonDefault::SoA>();
  });
  if (llvm::any_of(soaLvls, [](LevelType lt) {
        return !lt.isa<LevelFormat::Singleton>();
      })) {
    return emitError() << "SoA is only applicable to singleton lvlTypes.";
  }

  // TODO: audit formats that actually are supported by backend.
  if (auto it = std::find_if(lvlTypes.begin(), lvlTypes.end(), isNOutOfMLT);
      it != std::end(lvlTypes)) {
    if (it != lvlTypes.end() - 1)
      return emitError() << "expected n_out_of_m to be the last level type";
    if (!std::all_of(lvlTypes.begin(), it,
                     [](LevelType i) { return isDenseLT(i); }))
      return emitError() << "expected all dense lvlTypes "
                            "before a n_out_of_m level";
    if (dimToLvl && (dimToLvl.getNumDims() != dimToLvl.getNumResults())) {
      if (!isBlockSparsity(dimToLvl)) {
        return emitError()
               << "expected 1xm block structure for n_out_of_m level";
      }
      auto sizes = getBlockSize(dimToLvl);
      unsigned coefficient = 0;
      for (const auto &elem : sizes) {
        if (elem != 0) {
          if (elem != coefficient && coefficient != 0) {
            return emitError() << "expected only one blocked level "
                                  "with the same coefficients";
          }
          coefficient = elem;
        }
      }
      if (coefficient != getM(*it)) {
        return emitError() << "expected coeffiencts of Affine expressions "
                              "to be equal to m of n_out_of_m level";
      }
    }
  }
  // Before we can check that the level-rank is consistent/coherent
  // across all fields, we need to define it.  The source-of-truth for
  // the `getLvlRank` method is the length of the level-types array,
  // since it must always be provided and have full rank; therefore we
  // use that same source-of-truth here.
  const Level lvlRank = lvlTypes.size();
  if (lvlRank == 0)
    return emitError() << "expected a non-empty array for lvlTypes";
  // We save `dimRank` here because we'll also need it to verify `dimSlices`.
  const Dimension dimRank = dimToLvl ? dimToLvl.getNumDims() : lvlRank;
  if (dimToLvl) {
    if (dimToLvl.getNumResults() != lvlRank)
      return emitError()
             << "level-rank mismatch between dimToLvl and lvlTypes: "
             << dimToLvl.getNumResults() << " != " << lvlRank;
    auto inferRes = inferLvlToDim(dimToLvl, dimToLvl.getContext());
    // Symbols can't be inferred but are acceptable.
    if (!inferRes && dimToLvl.getNumSymbols() == 0)
      return emitError() << "failed to infer lvlToDim from dimToLvl";
    if (lvlToDim && (inferRes != lvlToDim))
      return emitError() << "expected lvlToDim to be an inverse of dimToLvl";
    if (dimRank > lvlRank)
      return emitError() << "unexpected dimToLvl mapping from " << dimRank
                         << " to " << lvlRank;
  }
  if (!dimSlices.empty()) {
    if (dimSlices.size() != dimRank)
      return emitError()
             << "dimension-rank mismatch between dimSlices and dimToLvl: "
             << dimSlices.size() << " != " << dimRank;
    // Compiler support for `dimSlices` currently requires that the two
    // ranks agree.  (However, it does allow `dimToLvl` to be a permutation.)
    if (dimRank != lvlRank)
      return emitError()
             << "dimSlices expected dimension-rank to match level-rank: "
             << dimRank << " != " << lvlRank;
  }
  return success();
}

LogicalResult SparseTensorEncodingAttr::verifyEncoding(
    ArrayRef<Size> dimShape, Type elementType,
    function_ref<InFlightDiagnostic()> emitError) const {
  // Check structural integrity.  In particular, this ensures that the
  // level-rank is coherent across all the fields.
  if (failed(verify(emitError, getLvlTypes(), getDimToLvl(), getLvlToDim(),
                    getPosWidth(), getCrdWidth(), getExplicitVal(),
                    getImplicitVal(), getDimSlices())))
    return failure();
  // Check integrity with tensor type specifics.  In particular, we
  // need only check that the dimension-rank of the tensor agrees with
  // the dimension-rank of the encoding.
  const Dimension dimRank = dimShape.size();
  if (dimRank == 0)
    return emitError() << "expected non-scalar sparse tensor";
  if (getDimRank() != dimRank)
    return emitError()
           << "dimension-rank mismatch between encoding and tensor shape: "
           << getDimRank() << " != " << dimRank;
  if (auto expVal = getExplicitVal()) {
    Type attrType = llvm::dyn_cast<TypedAttr>(expVal).getType();
    if (attrType != elementType) {
      return emitError() << "explicit value type mismatch between encoding and "
                         << "tensor element type: " << attrType
                         << " != " << elementType;
    }
  }
  if (auto impVal = getImplicitVal()) {
    Type attrType = llvm::dyn_cast<TypedAttr>(impVal).getType();
    if (attrType != elementType) {
      return emitError() << "implicit value type mismatch between encoding and "
                         << "tensor element type: " << attrType
                         << " != " << elementType;
    }
    // Currently, we only support zero as the implicit value.
    auto impFVal = llvm::dyn_cast<FloatAttr>(impVal);
    auto impIntVal = llvm::dyn_cast<IntegerAttr>(impVal);
    auto impComplexVal = llvm::dyn_cast<complex::NumberAttr>(impVal);
    if ((impFVal && impFVal.getValue().isNonZero()) ||
        (impIntVal && !impIntVal.getValue().isZero()) ||
        (impComplexVal && (impComplexVal.getImag().isNonZero() ||
                           impComplexVal.getReal().isNonZero()))) {
      return emitError() << "implicit value must be zero";
    }
  }
  return success();
}

Level mlir::sparse_tensor::SparseTensorEncodingAttr::getAoSCOOStart() const {
  SmallVector<COOSegment> coo = getCOOSegments();
  assert(coo.size() == 1 || coo.empty());
  if (!coo.empty() && coo.front().isAoS()) {
    return coo.front().lvlRange.first;
  }
  return getLvlRank();
}

SmallVector<COOSegment>
mlir::sparse_tensor::SparseTensorEncodingAttr::getCOOSegments() const {
  SmallVector<COOSegment> ret;
  if (getLvlRank() <= 1)
    return ret;

  ArrayRef<LevelType> lts = getLvlTypes();
  Level l = 0;
  while (l < getLvlRank()) {
    auto lt = lts[l];
    if (lt.isa<LevelFormat::Compressed, LevelFormat::LooseCompressed>()) {
      auto cur = lts.begin() + l;
      auto end = std::find_if(cur + 1, lts.end(), [](LevelType lt) {
        return !lt.isa<LevelFormat::Singleton>();
      });
      unsigned cooLen = std::distance(cur, end);
      if (cooLen > 1) {
        // To support mixed SoA/AoS COO, we should break the segment when the
        // storage scheme changes, for now we faithfully assume that all
        // consecutive singleton levels have the same storage format as verified
        // STEA.
        ret.push_back(COOSegment{std::make_pair(l, l + cooLen),
                                 lts[l + 1].isa<LevelPropNonDefault::SoA>()});
      }
      l += cooLen;
    } else {
      l++;
    }
  }
  return ret;
}

//===----------------------------------------------------------------------===//
// SparseTensorType Methods.
//===----------------------------------------------------------------------===//

bool mlir::sparse_tensor::SparseTensorType::isCOOType(Level startLvl,
                                                      bool isUnique) const {
  if (!hasEncoding())
    return false;
  if (!isCompressedLvl(startLvl) && !isLooseCompressedLvl(startLvl))
    return false;
  for (Level l = startLvl + 1; l < lvlRank; ++l)
    if (!isSingletonLvl(l))
      return false;
  // If isUnique is true, then make sure that the last level is unique,
  // that is, when lvlRank == 1, the only compressed level is unique,
  // and when lvlRank > 1, the last singleton is unique.
  return !isUnique || isUniqueLvl(lvlRank - 1);
}

RankedTensorType
mlir::sparse_tensor::SparseTensorType::getCOOType(bool ordered) const {
  SmallVector<LevelType> lvlTypes;
  lvlTypes.reserve(lvlRank);
  // A non-unique compressed level at beginning (unless this is
  // also the last level, then it is unique).
  lvlTypes.push_back(
      *buildLevelType(LevelFormat::Compressed, ordered, lvlRank == 1));
  if (lvlRank > 1) {
    // Followed by n-2 non-unique singleton levels.
    std::fill_n(std::back_inserter(lvlTypes), lvlRank - 2,
                *buildLevelType(LevelFormat::Singleton, ordered, false));
    // Ends by a unique singleton level.
    lvlTypes.push_back(*buildLevelType(LevelFormat::Singleton, ordered, true));
  }
  auto enc = SparseTensorEncodingAttr::get(
      getContext(), lvlTypes, getDimToLvl(), getLvlToDim(), getPosWidth(),
      getCrdWidth(), getExplicitVal(), getImplicitVal());
  return RankedTensorType::get(getDimShape(), getElementType(), enc);
}

//===----------------------------------------------------------------------===//
// Convenience Methods.
//===----------------------------------------------------------------------===//

SparseTensorEncodingAttr
mlir::sparse_tensor::getSparseTensorEncoding(Type type) {
  if (auto ttp = llvm::dyn_cast<RankedTensorType>(type))
    return llvm::dyn_cast_or_null<SparseTensorEncodingAttr>(ttp.getEncoding());
  if (auto mdtp = llvm::dyn_cast<StorageSpecifierType>(type))
    return mdtp.getEncoding();
  return nullptr;
}

AffineMap mlir::sparse_tensor::inferLvlToDim(AffineMap dimToLvl,
                                             MLIRContext *context) {
  auto map = static_cast<AffineMap>(dimToLvl);
  AffineMap lvlToDim;
  // Return an empty lvlToDim when inference is not successful.
  if (!map || map.getNumSymbols() != 0) {
    lvlToDim = AffineMap();
  } else if (map.isPermutation()) {
    lvlToDim = inversePermutation(map);
  } else if (isBlockSparsity(map)) {
    lvlToDim = inverseBlockSparsity(map, context);
  }
  return lvlToDim;
}

AffineMap mlir::sparse_tensor::inverseBlockSparsity(AffineMap dimToLvl,
                                                    MLIRContext *context) {
  SmallVector<AffineExpr> lvlExprs;
  auto numLvls = dimToLvl.getNumResults();
  lvlExprs.reserve(numLvls);
  // lvlExprComponents stores information of the floordiv and mod operations
  // applied to the same dimension, so as to build the lvlToDim map.
  std::map<unsigned, SmallVector<AffineExpr, 3>> lvlExprComponents;
  for (unsigned i = 0, n = numLvls; i < n; i++) {
    auto result = dimToLvl.getResult(i);
    if (auto binOp = dyn_cast<AffineBinaryOpExpr>(result)) {
      if (result.getKind() == AffineExprKind::FloorDiv) {
        // Position of the dimension in dimToLvl.
        auto pos = dyn_cast<AffineDimExpr>(binOp.getLHS()).getPosition();
        assert(lvlExprComponents.find(pos) == lvlExprComponents.end() &&
               "expected only one floordiv for each dimension");
        SmallVector<AffineExpr, 3> components;
        // Level variable for floordiv.
        components.push_back(getAffineDimExpr(i, context));
        // Multiplier.
        components.push_back(binOp.getRHS());
        // Map key is the position of the dimension.
        lvlExprComponents[pos] = components;
      } else if (result.getKind() == AffineExprKind::Mod) {
        auto pos = dyn_cast<AffineDimExpr>(binOp.getLHS()).getPosition();
        assert(lvlExprComponents.find(pos) != lvlExprComponents.end() &&
               "expected floordiv before mod");
        // Add level variable for mod to the same vector
        // of the corresponding floordiv.
        lvlExprComponents[pos].push_back(getAffineDimExpr(i, context));
      } else {
        assert(false && "expected floordiv or mod");
      }
    } else {
      lvlExprs.push_back(getAffineDimExpr(i, context));
    }
  }
  // Build lvlExprs from lvlExprComponents.
  // For example, for il = i floordiv 2 and ii = i mod 2, the components
  // would be [il, 2, ii]. It could be used to build the AffineExpr
  // i = il * 2 + ii in lvlToDim.
  for (auto &components : lvlExprComponents) {
    assert(components.second.size() == 3 &&
           "expected 3 components to build lvlExprs");
    auto mulOp = getAffineBinaryOpExpr(
        AffineExprKind::Mul, components.second[0], components.second[1]);
    auto addOp =
        getAffineBinaryOpExpr(AffineExprKind::Add, mulOp, components.second[2]);
    lvlExprs.push_back(addOp);
  }
  return dimToLvl.get(dimToLvl.getNumResults(), 0, lvlExprs, context);
}

SmallVector<unsigned> mlir::sparse_tensor::getBlockSize(AffineMap dimToLvl) {
  assert(isBlockSparsity(dimToLvl) &&
         "expected dimToLvl to be block sparsity for calling getBlockSize");
  SmallVector<unsigned> blockSize;
  for (auto result : dimToLvl.getResults()) {
    if (auto binOp = dyn_cast<AffineBinaryOpExpr>(result)) {
      if (result.getKind() == AffineExprKind::Mod) {
        blockSize.push_back(
            dyn_cast<AffineConstantExpr>(binOp.getRHS()).getValue());
      }
    } else {
      blockSize.push_back(0);
    }
  }
  return blockSize;
}

bool mlir::sparse_tensor::isBlockSparsity(AffineMap dimToLvl) {
  if (!dimToLvl)
    return false;
  std::map<unsigned, int64_t> coeffientMap;
  bool hasBlock = false;
  for (auto result : dimToLvl.getResults()) {
    if (auto binOp = dyn_cast<AffineBinaryOpExpr>(result)) {
      // Check for "dim op const".
      auto dimOp = dyn_cast<AffineDimExpr>(binOp.getLHS());
      auto conOp = dyn_cast<AffineConstantExpr>(binOp.getRHS());
      if (!dimOp || !conOp || conOp.getValue() <= 0)
        return false;
      // Inspect "dim / const" or "dim % const".
      auto pos = dimOp.getPosition();
      if (binOp.getKind() == AffineExprKind::FloorDiv) {
        // Expect only one floordiv for each dimension.
        auto [it, inserted] = coeffientMap.try_emplace(pos);
        if (!inserted)
          return false;
        // Record coefficient of the floordiv.
        it->second = conOp.getValue();
      } else if (binOp.getKind() == AffineExprKind::Mod) {
        // Expect floordiv before mod.
        auto it = coeffientMap.find(pos);
        if (it == coeffientMap.end())
          return false;
        // Expect mod to have the same coefficient as floordiv.
        if (conOp.getValue() != it->second)
          return false;
        hasBlock = true;
      } else {
        return false;
      }
    } else if (auto dimOp = dyn_cast<AffineDimExpr>(result)) {
      auto pos = dimOp.getPosition();
      // Expect dim to be unset.
      if (!coeffientMap.try_emplace(pos, 0).second)
        return false;
    } else {
      return false;
    }
  }
  return hasBlock;
}

bool mlir::sparse_tensor::hasAnyNonIdentityOperandsOrResults(Operation *op) {
  auto hasNonIdentityMap = [](Value v) {
    auto stt = tryGetSparseTensorType(v);
    return stt && !stt->isIdentity();
  };

  return llvm::any_of(op->getOperands(), hasNonIdentityMap) ||
         llvm::any_of(op->getResults(), hasNonIdentityMap);
}

Dimension mlir::sparse_tensor::toDim(SparseTensorEncodingAttr enc, Level l) {
  if (enc) {
    assert(enc.isPermutation() && "Non permutation map not supported");
    if (const auto dimToLvl = enc.getDimToLvl())
      return dimToLvl.getDimPosition(l);
  }
  return l;
}

Level mlir::sparse_tensor::toLvl(SparseTensorEncodingAttr enc, Dimension d) {
  if (enc) {
    assert(enc.isPermutation() && "Non permutation map not supported");
    if (const auto lvlToDim = enc.getLvlToDim())
      return lvlToDim.getDimPosition(d);
  }
  return d;
}

/// We normalized sparse tensor encoding attribute by always using
/// ordered/unique LT such that "compressed_nu_no" and "compressed_nu" (as well
/// as other variants) lead to the same storage specifier type, and stripping
/// irrelevant fields that do not alter the sparse tensor memory layout.
static SparseTensorEncodingAttr
getNormalizedEncodingForSpecifier(SparseTensorEncodingAttr enc) {
  SmallVector<LevelType> lts;
  for (auto lt : enc.getLvlTypes())
    lts.push_back(lt.stripStorageIrrelevantProperties());

  return SparseTensorEncodingAttr::get(
      enc.getContext(), lts,
      AffineMap(), // dimToLvl (irrelevant to storage specifier)
      AffineMap(), // lvlToDim (irrelevant to storage specifier)
      // Always use `index` for memSize and lvlSize instead of reusing
      // `getPosWidth` and `getCrdWidth`. It allows us to reuse the same SSA
      // value for different bitwidth, it also avoids casting between index and
      // integer (returned by DimOp)
      0, 0,
      Attribute(), // explicitVal (irrelevant to storage specifier)
      Attribute(), // implicitVal (irrelevant to storage specifier)
      enc.getDimSlices());
}

StorageSpecifierType
StorageSpecifierType::get(MLIRContext *ctx, SparseTensorEncodingAttr encoding) {
  return Base::get(ctx, getNormalizedEncodingForSpecifier(encoding));
}

StorageSpecifierType
StorageSpecifierType::getChecked(function_ref<InFlightDiagnostic()> emitError,
                                 MLIRContext *ctx,
                                 SparseTensorEncodingAttr encoding) {
  return Base::getChecked(emitError, ctx,
                          getNormalizedEncodingForSpecifier(encoding));
}

//===----------------------------------------------------------------------===//
// SparseTensorDialect Operations.
//===----------------------------------------------------------------------===//

static LogicalResult lvlIsInBounds(Level lvl, Value tensor) {
  return success(lvl < getSparseTensorType(tensor).getLvlRank());
}

static LogicalResult isMatchingWidth(Value mem, unsigned width) {
  const Type etp = getMemRefType(mem).getElementType();
  return success(width == 0 ? etp.isIndex() : etp.isInteger(width));
}

static LogicalResult verifySparsifierGetterSetter(
    StorageSpecifierKind mdKind, std::optional<Level> lvl,
    TypedValue<StorageSpecifierType> md, Operation *op) {
  if (mdKind == StorageSpecifierKind::ValMemSize && lvl) {
    return op->emitError(
        "redundant level argument for querying value memory size");
  }

  const auto enc = md.getType().getEncoding();
  const Level lvlRank = enc.getLvlRank();

  if (mdKind == StorageSpecifierKind::DimOffset ||
      mdKind == StorageSpecifierKind::DimStride)
    if (!enc.isSlice())
      return op->emitError("requested slice data on non-slice tensor");

  if (mdKind != StorageSpecifierKind::ValMemSize) {
    if (!lvl)
      return op->emitError("missing level argument");

    const Level l = lvl.value();
    if (l >= lvlRank)
      return op->emitError("requested level is out of bounds");

    if (mdKind == StorageSpecifierKind::PosMemSize && enc.isSingletonLvl(l))
      return op->emitError(
          "requested position memory size on a singleton level");
  }
  return success();
}

static Type getFieldElemType(SparseTensorType stt, SparseTensorFieldKind kind) {
  switch (kind) {
  case SparseTensorFieldKind::CrdMemRef:
    return stt.getCrdType();
  case SparseTensorFieldKind::PosMemRef:
    return stt.getPosType();
  case SparseTensorFieldKind::ValMemRef:
    return stt.getElementType();
  case SparseTensorFieldKind::StorageSpec:
    return nullptr;
  }
  llvm_unreachable("Unrecognizable FieldKind");
}

static LogicalResult verifyPackUnPack(Operation *op, bool requiresStaticShape,
                                      SparseTensorType stt,
                                      RankedTensorType valTp,
                                      TypeRange lvlTps) {
  if (requiresStaticShape && !stt.hasStaticDimShape())
    return op->emitError("the sparse-tensor must have static shape");
  if (!stt.hasEncoding())
    return op->emitError("the sparse-tensor must have an encoding attribute");

  // Verifies the trailing COO.
  Level cooStartLvl = stt.getAoSCOOStart();
  if (cooStartLvl < stt.getLvlRank()) {
    // We only supports trailing COO for now, must be the last input.
    auto cooTp = llvm::cast<ShapedType>(lvlTps.back());
    // The coordinates should be in shape of <? x rank>
    unsigned expCOORank = stt.getLvlRank() - cooStartLvl;
    if (cooTp.getRank() != 2 || expCOORank != cooTp.getShape().back()) {
      return op->emitError("input/output trailing COO level-ranks don't match");
    }
  }

  // Verifies that all types match.
  StorageLayout layout(stt.getEncoding());
  if (layout.getNumDataFields() != lvlTps.size() + 1) // plus one value memref
    return op->emitError("inconsistent number of fields between input/output");

  unsigned idx = 0;
  bool misMatch = false;
  layout.foreachField([&idx, &misMatch, stt, valTp,
                       lvlTps](FieldIndex fid, SparseTensorFieldKind fKind,
                               Level lvl, LevelType lt) -> bool {
    if (fKind == SparseTensorFieldKind::StorageSpec)
      return true;

    Type inputTp = nullptr;
    if (fKind == SparseTensorFieldKind::ValMemRef) {
      inputTp = valTp;
    } else {
      assert(fid == idx && stt.getLvlType(lvl) == lt);
      inputTp = lvlTps[idx++];
    }
    // The input element type and expected element type should match.
    Type inpElemTp = llvm::cast<TensorType>(inputTp).getElementType();
    Type expElemTp = getFieldElemType(stt, fKind);
    if (inpElemTp != expElemTp) {
      misMatch = true;
      return false; // to terminate the iteration
    }
    return true;
  });

  if (misMatch)
    return op->emitError("input/output element-types don't match");
  return success();
}

LogicalResult AssembleOp::verify() {
  RankedTensorType valuesTp = getValues().getType();
  const auto lvlsTp = getLevels().getTypes();
  const auto resTp = getSparseTensorType(getResult());
  return verifyPackUnPack(*this, true, resTp, valuesTp, lvlsTp);
}

LogicalResult DisassembleOp::verify() {
  if (getOutValues().getType() != getRetValues().getType())
    return emitError("output values and return value type mismatch");

  for (auto [ot, rt] : llvm::zip_equal(getOutLevels(), getRetLevels()))
    if (ot.getType() != rt.getType())
      return emitError("output levels and return levels type mismatch");

  RankedTensorType valuesTp = getRetValues().getType();
  const auto lvlsTp = getRetLevels().getTypes();
  const auto srcTp = getSparseTensorType(getTensor());
  return verifyPackUnPack(*this, false, srcTp, valuesTp, lvlsTp);
}

LogicalResult ConvertOp::verify() {
  RankedTensorType tp1 = getSource().getType();
  RankedTensorType tp2 = getDest().getType();
  if (tp1.getRank() != tp2.getRank())
    return emitError("unexpected conversion mismatch in rank");
  auto dstEnc =
      llvm::dyn_cast_or_null<SparseTensorEncodingAttr>(tp2.getEncoding());
  if (dstEnc && dstEnc.isSlice())
    return emitError("cannot convert to a sparse tensor slice");

  auto shape1 = tp1.getShape();
  auto shape2 = tp2.getShape();
  // Accept size matches between the source and the destination type
  // (e.g. 10 vs. 10, 10 vs. ?, or ? vs. ?), but reject direct mismatches or
  // matches that would need a runtime assert (e.g. 10 vs. 20 or ? vs. 10).
  for (Dimension d = 0, dimRank = tp1.getRank(); d < dimRank; d++)
    if (shape1[d] != shape2[d] && shape2[d] != ShapedType::kDynamic)
      return emitError("unexpected conversion mismatch in dimension ") << d;
  return success();
}

OpFoldResult ConvertOp::fold(FoldAdaptor adaptor) {
  if (getType() == getSource().getType())
    return getSource();
  return {};
}

bool ConvertOp::needsExtraSort() {
  SparseTensorType srcStt = getSparseTensorType(getSource());
  SparseTensorType dstStt = getSparseTensorType(getDest());

  // We do not need an extra sort when returning unordered sparse tensors or
  // dense tensor since dense tensor support random access.
  if (dstStt.isAllDense() || !dstStt.isAllOrdered())
    return false;

  if (srcStt.isAllOrdered() && dstStt.isAllOrdered() &&
      srcStt.hasSameDimToLvl(dstStt)) {
    return false;
  }

  // Source and dest tensors are ordered in different ways. We only do direct
  // dense to sparse conversion when the dense input is defined by a sparse
  // constant. Note that we can theoretically always directly convert from dense
  // inputs by rotating dense loops but it leads to bad cache locality and hurt
  // performance.
  if (auto constOp = getSource().getDefiningOp<arith::ConstantOp>())
    if (isa<SparseElementsAttr>(constOp.getValue()))
      return false;

  return true;
}

LogicalResult CrdTranslateOp::verify() {
  uint64_t inRank = getEncoder().getLvlRank();
  uint64_t outRank = getEncoder().getDimRank();

  if (getDirection() == CrdTransDirectionKind::dim2lvl)
    std::swap(inRank, outRank);

  if (inRank != getInCrds().size() || outRank != getOutCrds().size())
    return emitError("Coordinate rank mismatch with encoding");

  return success();
}

LogicalResult CrdTranslateOp::fold(FoldAdaptor adaptor,
                                   SmallVectorImpl<OpFoldResult> &results) {
  if (getEncoder().isIdentity()) {
    results.assign(getInCrds().begin(), getInCrds().end());
    return success();
  }
  if (getEncoder().isPermutation()) {
    AffineMap perm = getDirection() == CrdTransDirectionKind::dim2lvl
                         ? getEncoder().getDimToLvl()
                         : getEncoder().getLvlToDim();
    for (AffineExpr exp : perm.getResults())
      results.push_back(getInCrds()[cast<AffineDimExpr>(exp).getPosition()]);
    return success();
  }

  // Fuse dim2lvl/lvl2dim pairs.
  auto def = getInCrds()[0].getDefiningOp<CrdTranslateOp>();
  bool sameDef = def && llvm::all_of(getInCrds(), [def](Value v) {
                   return v.getDefiningOp() == def;
                 });
  if (!sameDef)
    return failure();

  bool oppositeDir = def.getDirection() != getDirection();
  bool sameOracle =
      def.getEncoder().getDimToLvl() == getEncoder().getDimToLvl();
  bool sameCount = def.getNumResults() == getInCrds().size();
  if (!oppositeDir || !sameOracle || !sameCount)
    return failure();

  // The definition produces the coordinates in the same order as the input
  // coordinates.
  bool sameOrder = llvm::all_of(llvm::zip_equal(def.getOutCrds(), getInCrds()),
                                [](auto valuePair) {
                                  auto [lhs, rhs] = valuePair;
                                  return lhs == rhs;
                                });

  if (!sameOrder)
    return failure();
  // l1 = dim2lvl (lvl2dim l0)
  // ==> l0
  results.append(def.getInCrds().begin(), def.getInCrds().end());
  return success();
}

void LvlOp::build(OpBuilder &builder, OperationState &state, Value source,
                  int64_t index) {
  Value val = builder.create<arith::ConstantIndexOp>(state.location, index);
  return build(builder, state, source, val);
}

LogicalResult LvlOp::verify() {
  if (std::optional<uint64_t> lvl = getConstantLvlIndex()) {
    auto stt = getSparseTensorType(getSource());
    if (static_cast<uint64_t>(lvl.value()) >= stt.getLvlRank())
      return emitError(
          "Level index exceeds the rank of the input sparse tensor");
  }
  return success();
}

std::optional<uint64_t> LvlOp::getConstantLvlIndex() {
  return getConstantIntValue(getIndex());
}

Speculation::Speculatability LvlOp::getSpeculatability() {
  auto constantIndex = getConstantLvlIndex();
  if (!constantIndex)
    return Speculation::NotSpeculatable;

  assert(constantIndex <
         cast<RankedTensorType>(getSource().getType()).getRank());
  return Speculation::Speculatable;
}

OpFoldResult LvlOp::fold(FoldAdaptor adaptor) {
  auto lvlIndex = llvm::dyn_cast_if_present<IntegerAttr>(adaptor.getIndex());
  if (!lvlIndex)
    return {};

  Level lvl = lvlIndex.getAPSInt().getZExtValue();
  auto stt = getSparseTensorType(getSource());
  if (lvl >= stt.getLvlRank()) {
    // Follows the same convention used by tensor.dim operation. Out of bound
    // indices produce undefined behavior but are still valid IR. Don't choke on
    // them.
    return {};
  }

  // Helper lambda to build an IndexAttr.
  auto getIndexAttr = [this](int64_t lvlSz) {
    return IntegerAttr::get(IndexType::get(getContext()), APInt(64, lvlSz));
  };

  SmallVector<Size> lvlShape = stt.getLvlShape();
  if (!ShapedType::isDynamic(lvlShape[lvl]))
    return getIndexAttr(lvlShape[lvl]);

  return {};
}

void ReinterpretMapOp::build(OpBuilder &odsBuilder, OperationState &odsState,
                             SparseTensorEncodingAttr dstEnc, Value source) {
  auto srcStt = getSparseTensorType(source);
  SmallVector<int64_t> srcLvlShape = srcStt.getLvlShape();
  SmallVector<int64_t> dstDimShape =
      dstEnc.translateShape(srcLvlShape, CrdTransDirectionKind::lvl2dim);
  auto dstTp =
      RankedTensorType::get(dstDimShape, srcStt.getElementType(), dstEnc);
  return build(odsBuilder, odsState, dstTp, source);
}

LogicalResult ReinterpretMapOp::verify() {
  auto srcStt = getSparseTensorType(getSource());
  auto dstStt = getSparseTensorType(getDest());
  ArrayRef<LevelType> srcLvlTps = srcStt.getLvlTypes();
  ArrayRef<LevelType> dstLvlTps = dstStt.getLvlTypes();

  if (srcLvlTps.size() != dstLvlTps.size())
    return emitError("Level rank mismatch between source/dest tensors");

  for (auto [srcLvlTp, dstLvlTp] : llvm::zip(srcLvlTps, dstLvlTps))
    if (srcLvlTp != dstLvlTp)
      return emitError("Level type mismatch between source/dest tensors");

  if (srcStt.getPosWidth() != dstStt.getPosWidth() ||
      srcStt.getCrdWidth() != dstStt.getCrdWidth()) {
    return emitError("Crd/Pos width mismatch between source/dest tensors");
  }

  if (srcStt.getElementType() != dstStt.getElementType())
    return emitError("Element type mismatch between source/dest tensors");

  SmallVector<Size> srcLvlShape = srcStt.getLvlShape();
  SmallVector<Size> dstLvlShape = dstStt.getLvlShape();
  for (auto [srcLvlSz, dstLvlSz] : llvm::zip(srcLvlShape, dstLvlShape)) {
    if (srcLvlSz != dstLvlSz) {
      // Should we allow one side to be dynamic size, e.g., <?x?> should be
      // compatible to <3x4>? For now, we require all the level sizes to be
      // *exactly* matched for simplicity.
      return emitError("Level size mismatch between source/dest tensors");
    }
  }

  return success();
}

OpFoldResult ReinterpretMapOp::fold(FoldAdaptor adaptor) {
  if (getSource().getType() == getDest().getType())
    return getSource();

  if (auto def = getSource().getDefiningOp<ReinterpretMapOp>()) {
    // A -> B, B -> A ==> A
    if (def.getSource().getType() == getDest().getType())
      return def.getSource();
  }
  return {};
}

template <typename ToBufferOp>
static LogicalResult inferSparseBufferType(ValueRange ops, DictionaryAttr attr,
                                           OpaqueProperties prop,
                                           RegionRange region,
                                           SmallVectorImpl<mlir::Type> &ret) {
  typename ToBufferOp::Adaptor adaptor(ops, attr, prop, region);
  SparseTensorType stt = getSparseTensorType(adaptor.getTensor());
  Type elemTp = nullptr;
  bool withStride = false;
  if constexpr (std::is_same_v<ToBufferOp, ToPositionsOp>) {
    elemTp = stt.getPosType();
  } else if constexpr (std::is_same_v<ToBufferOp, ToCoordinatesOp> ||
                       std::is_same_v<ToBufferOp, ToCoordinatesBufferOp>) {
    elemTp = stt.getCrdType();
    if constexpr (std::is_same_v<ToBufferOp, ToCoordinatesOp>)
      withStride = stt.getAoSCOOStart() <= adaptor.getLevel();
  } else if constexpr (std::is_same_v<ToBufferOp, ToValuesOp>) {
    elemTp = stt.getElementType();
  }

  assert(elemTp && "unhandled operation.");
  SmallVector<int64_t> bufShape = stt.getBatchLvlShape();
  bufShape.push_back(ShapedType::kDynamic);

  auto layout = withStride ? StridedLayoutAttr::StridedLayoutAttr::get(
                                 stt.getContext(), ShapedType::kDynamic,
                                 {ShapedType::kDynamic})
                           : StridedLayoutAttr();
  ret.emplace_back(MemRefType::get(bufShape, elemTp, layout));
  return success();
}

LogicalResult ToPositionsOp::verify() {
  auto stt = getSparseTensorType(getTensor());
  if (failed(lvlIsInBounds(getLevel(), getTensor())))
    return emitError("requested level is out of bounds");
  if (failed(isMatchingWidth(getResult(), stt.getPosWidth())))
    return emitError("unexpected type for positions");
  return success();
}

LogicalResult
ToPositionsOp::inferReturnTypes(MLIRContext *ctx, std::optional<Location> loc,
                                ValueRange ops, DictionaryAttr attr,
                                OpaqueProperties prop, RegionRange region,
                                SmallVectorImpl<mlir::Type> &ret) {
  return inferSparseBufferType<ToPositionsOp>(ops, attr, prop, region, ret);
}

LogicalResult ToCoordinatesOp::verify() {
  auto stt = getSparseTensorType(getTensor());
  if (failed(lvlIsInBounds(getLevel(), getTensor())))
    return emitError("requested level is out of bounds");
  if (failed(isMatchingWidth(getResult(), stt.getCrdWidth())))
    return emitError("unexpected type for coordinates");
  return success();
}

LogicalResult
ToCoordinatesOp::inferReturnTypes(MLIRContext *ctx, std::optional<Location> loc,
                                  ValueRange ops, DictionaryAttr attr,
                                  OpaqueProperties prop, RegionRange region,
                                  SmallVectorImpl<mlir::Type> &ret) {
  return inferSparseBufferType<ToCoordinatesOp>(ops, attr, prop, region, ret);
}

LogicalResult ToCoordinatesBufferOp::verify() {
  auto stt = getSparseTensorType(getTensor());
  if (stt.getAoSCOOStart() >= stt.getLvlRank())
    return emitError("expected sparse tensor with a COO region");
  return success();
}

LogicalResult ToCoordinatesBufferOp::inferReturnTypes(
    MLIRContext *ctx, std::optional<Location> loc, ValueRange ops,
    DictionaryAttr attr, OpaqueProperties prop, RegionRange region,
    SmallVectorImpl<mlir::Type> &ret) {
  return inferSparseBufferType<ToCoordinatesBufferOp>(ops, attr, prop, region,
                                                      ret);
}

LogicalResult ToValuesOp::verify() {
  auto stt = getSparseTensorType(getTensor());
  auto mtp = getMemRefType(getResult());
  if (stt.getElementType() != mtp.getElementType())
    return emitError("unexpected mismatch in element types");
  return success();
}

LogicalResult ToValuesOp::inferReturnTypes(MLIRContext *ctx,
                                           std::optional<Location> loc,
                                           ValueRange ops, DictionaryAttr attr,
                                           OpaqueProperties prop,
                                           RegionRange region,
                                           SmallVectorImpl<mlir::Type> &ret) {
  return inferSparseBufferType<ToValuesOp>(ops, attr, prop, region, ret);
}

LogicalResult ToSliceOffsetOp::verify() {
  auto rank = getSlice().getType().getRank();
  if (rank <= getDim().getSExtValue() || getDim().getSExtValue() < 0)
    return emitError("requested dimension out of bound");
  return success();
}

LogicalResult ToSliceStrideOp::verify() {
  auto rank = getSlice().getType().getRank();
  if (rank <= getDim().getSExtValue() || getDim().getSExtValue() < 0)
    return emitError("requested dimension out of bound");
  return success();
}

LogicalResult GetStorageSpecifierOp::verify() {
  return verifySparsifierGetterSetter(getSpecifierKind(), getLevel(),
                                      getSpecifier(), getOperation());
}

template <typename SpecifierOp>
static SetStorageSpecifierOp getSpecifierSetDef(SpecifierOp op) {
  return op.getSpecifier().template getDefiningOp<SetStorageSpecifierOp>();
}

OpFoldResult GetStorageSpecifierOp::fold(FoldAdaptor adaptor) {
  const StorageSpecifierKind kind = getSpecifierKind();
  const auto lvl = getLevel();
  for (auto op = getSpecifierSetDef(*this); op; op = getSpecifierSetDef(op))
    if (kind == op.getSpecifierKind() && lvl == op.getLevel())
      return op.getValue();
  return {};
}

LogicalResult SetStorageSpecifierOp::verify() {
  return verifySparsifierGetterSetter(getSpecifierKind(), getLevel(),
                                      getSpecifier(), getOperation());
}

template <class T>
static LogicalResult verifyNumBlockArgs(T *op, Region &region,
                                        const char *regionName,
                                        TypeRange inputTypes, Type outputType) {
  unsigned numArgs = region.getNumArguments();
  unsigned expectedNum = inputTypes.size();
  if (numArgs != expectedNum)
    return op->emitError() << regionName << " region must have exactly "
                           << expectedNum << " arguments";

  for (unsigned i = 0; i < numArgs; i++) {
    Type typ = region.getArgument(i).getType();
    if (typ != inputTypes[i])
      return op->emitError() << regionName << " region argument " << (i + 1)
                             << " type mismatch";
  }
  Operation *term = region.front().getTerminator();
  YieldOp yield = dyn_cast<YieldOp>(term);
  if (!yield)
    return op->emitError() << regionName
                           << " region must end with sparse_tensor.yield";
  if (!yield.hasSingleResult() ||
      yield.getSingleResult().getType() != outputType)
    return op->emitError() << regionName << " region yield type mismatch";

  return success();
}

LogicalResult BinaryOp::verify() {
  NamedAttrList attrs = (*this)->getAttrs();
  Type leftType = getX().getType();
  Type rightType = getY().getType();
  Type outputType = getOutput().getType();
  Region &overlap = getOverlapRegion();
  Region &left = getLeftRegion();
  Region &right = getRightRegion();

  // Check correct number of block arguments and return type for each
  // non-empty region.
  if (!overlap.empty()) {
    if (failed(verifyNumBlockArgs(this, overlap, "overlap",
                                  TypeRange{leftType, rightType}, outputType)))
      return failure();
  }
  if (!left.empty()) {
    if (failed(verifyNumBlockArgs(this, left, "left", TypeRange{leftType},
                                  outputType)))
      return failure();
  } else if (getLeftIdentity()) {
    if (leftType != outputType)
      return emitError("left=identity requires first argument to have the same "
                       "type as the output");
  }
  if (!right.empty()) {
    if (failed(verifyNumBlockArgs(this, right, "right", TypeRange{rightType},
                                  outputType)))
      return failure();
  } else if (getRightIdentity()) {
    if (rightType != outputType)
      return emitError("right=identity requires second argument to have the "
                       "same type as the output");
  }
  return success();
}

LogicalResult UnaryOp::verify() {
  Type inputType = getX().getType();
  Type outputType = getOutput().getType();

  // Check correct number of block arguments and return type for each
  // non-empty region.
  Region &present = getPresentRegion();
  if (!present.empty()) {
    if (failed(verifyNumBlockArgs(this, present, "present",
                                  TypeRange{inputType}, outputType)))
      return failure();
  }
  Region &absent = getAbsentRegion();
  if (!absent.empty()) {
    if (failed(verifyNumBlockArgs(this, absent, "absent", TypeRange{},
                                  outputType)))
      return failure();
    // Absent branch can only yield invariant values.
    Block *absentBlock = &absent.front();
    Block *parent = getOperation()->getBlock();
    Value absentVal =
        cast<YieldOp>(absentBlock->getTerminator()).getSingleResult();
    if (auto arg = dyn_cast<BlockArgument>(absentVal)) {
      if (arg.getOwner() == parent)
        return emitError("absent region cannot yield linalg argument");
    } else if (Operation *def = absentVal.getDefiningOp()) {
      if (!isa<arith::ConstantOp>(def) &&
          (def->getBlock() == absentBlock || def->getBlock() == parent))
        return emitError("absent region cannot yield locally computed value");
    }
  }
  return success();
}

bool ConcatenateOp::needsExtraSort() {
  SparseTensorType dstStt = getSparseTensorType(*this);
  if (dstStt.isAllDense() || !dstStt.isAllOrdered())
    return false;

  bool allSameOrdered = llvm::all_of(getInputs(), [dstStt](Value op) {
    return getSparseTensorType(op).hasSameDimToLvl(dstStt);
  });
  // TODO: When conDim != 0, as long as conDim corresponding to the first level
  // in all input/output buffers, and all input/output buffers have the same
  // dimToLvl, the tmp COO buffer is still unnecessary (e.g, concatenate
  // CSC matrices along column).
  bool directLowerable =
      allSameOrdered && getDimension() == 0 && dstStt.isIdentity();
  return !directLowerable;
}

LogicalResult ConcatenateOp::verify() {
  const auto dstTp = getSparseTensorType(*this);
  const Dimension concatDim = getDimension();
  const Dimension dimRank = dstTp.getDimRank();

  if (getInputs().size() <= 1)
    return emitError("Need at least two tensors to concatenate.");

  if (concatDim >= dimRank)
    return emitError(llvm::formatv(
        "Concat-dimension is out of bounds for dimension-rank ({0} >= {1})",
        concatDim, dimRank));

  for (const auto &it : llvm::enumerate(getInputs())) {
    const auto i = it.index();
    const auto srcTp = getSparseTensorType(it.value());
    if (srcTp.hasDynamicDimShape())
      return emitError(llvm::formatv("Input tensor ${0} has dynamic shape", i));
    const Dimension srcDimRank = srcTp.getDimRank();
    if (srcDimRank != dimRank)
      return emitError(
          llvm::formatv("Input tensor ${0} has a different rank (rank={1}) "
                        "from the output tensor (rank={2}).",
                        i, srcDimRank, dimRank));
  }

  for (Dimension d = 0; d < dimRank; d++) {
    const Size dstSh = dstTp.getDimShape()[d];
    if (d == concatDim) {
      if (!ShapedType::isDynamic(dstSh)) {
        // If we reach here, then all inputs have static shapes.  So we
        // can use `getDimShape()[d]` instead of `*getDynamicDimSize(d)`
        // to avoid redundant assertions in the loop.
        Size sumSz = 0;
        for (const auto src : getInputs())
          sumSz += getSparseTensorType(src).getDimShape()[d];
        // If all dimension are statically known, the sum of all the input
        // dimensions should be equal to the output dimension.
        if (sumSz != dstSh)
          return emitError(
              "The concatenation dimension of the output tensor should be the "
              "sum of all the concatenation dimensions of the input tensors.");
      }
    } else {
      Size prev = dstSh;
      for (const auto src : getInputs()) {
        const auto sh = getSparseTensorType(src).getDimShape()[d];
        if (!ShapedType::isDynamic(prev) && sh != prev)
          return emitError("All dimensions (expect for the concatenating one) "
                           "should be equal.");
        prev = sh;
      }
    }
  }

  return success();
}

void PushBackOp::build(OpBuilder &builder, OperationState &result,
                       Value curSize, Value inBuffer, Value value) {
  build(builder, result, curSize, inBuffer, value, Value());
}

LogicalResult PushBackOp::verify() {
  if (Value n = getN()) {
    std::optional<int64_t> nValue = getConstantIntValue(n);
    if (nValue && nValue.value() < 1)
      return emitOpError("n must be not less than 1");
  }
  return success();
}

LogicalResult CompressOp::verify() {
  const auto stt = getSparseTensorType(getTensor());
  if (stt.getLvlRank() != 1 + static_cast<Level>(getLvlCoords().size()))
    return emitOpError("incorrect number of coordinates");
  return success();
}

void ForeachOp::build(
    OpBuilder &builder, OperationState &result, Value tensor,
    ValueRange initArgs, AffineMapAttr order,
    function_ref<void(OpBuilder &, Location, ValueRange, Value, ValueRange)>
        bodyBuilder) {
  build(builder, result, initArgs.getTypes(), tensor, initArgs, order);
  // Builds foreach body.
  if (!bodyBuilder)
    return;
  const auto stt = getSparseTensorType(tensor);
  const Dimension dimRank = stt.getDimRank();

  // Starts with `dimRank`-many coordinates.
  SmallVector<Type> blockArgTypes(dimRank, builder.getIndexType());
  // Followed by one value.
  blockArgTypes.push_back(stt.getElementType());
  // Followed by the reduction variables.
  blockArgTypes.append(initArgs.getTypes().begin(), initArgs.getTypes().end());

  SmallVector<Location> blockArgLocs(blockArgTypes.size(), tensor.getLoc());

  OpBuilder::InsertionGuard guard(builder);
  auto &region = *result.regions.front();
  Block *bodyBlock =
      builder.createBlock(&region, region.end(), blockArgTypes, blockArgLocs);
  bodyBuilder(builder, result.location,
              bodyBlock->getArguments().slice(0, dimRank),
              bodyBlock->getArguments()[dimRank],
              bodyBlock->getArguments().drop_front(dimRank + 1));
}

LogicalResult ForeachOp::verify() {
  const auto t = getSparseTensorType(getTensor());
  const Dimension dimRank = t.getDimRank();
  const auto args = getBody()->getArguments();

  if (getOrder().has_value() && getOrder()->getNumDims() != t.getLvlRank())
    return emitError("Level traverse order does not match tensor's level rank");

  if (dimRank + 1 + getInitArgs().size() != args.size())
    return emitError("Unmatched number of arguments in the block");

  if (getNumResults() != getInitArgs().size())
    return emitError("Mismatch in number of init arguments and results");

  if (getResultTypes() != getInitArgs().getTypes())
    return emitError("Mismatch in types of init arguments and results");

  // Cannot mark this const, because the getters aren't.
  auto yield = cast<YieldOp>(getBody()->getTerminator());
  if (yield.getNumOperands() != getNumResults() ||
      yield.getOperands().getTypes() != getResultTypes())
    return emitError("Mismatch in types of yield values and results");

  const auto iTp = IndexType::get(getContext());
  for (Dimension d = 0; d < dimRank; d++)
    if (args[d].getType() != iTp)
      return emitError(
          llvm::formatv("Expecting Index type for argument at index {0}", d));

  const auto elemTp = t.getElementType();
  const auto valueTp = args[dimRank].getType();
  if (elemTp != valueTp)
    return emitError(
        llvm::formatv("Unmatched element type between input tensor and "
                      "block argument, expected:{0}, got: {1}",
                      elemTp, valueTp));
  return success();
}

OpFoldResult ReorderCOOOp::fold(FoldAdaptor adaptor) {
  if (getSparseTensorEncoding(getInputCoo().getType()) ==
      getSparseTensorEncoding(getResultCoo().getType()))
    return getInputCoo();

  return {};
}

LogicalResult ReorderCOOOp::verify() {
  SparseTensorType srcStt = getSparseTensorType(getInputCoo());
  SparseTensorType dstStt = getSparseTensorType(getResultCoo());

  if (!srcStt.isCOOType() || !dstStt.isCOOType())
    return emitError("Expected COO sparse tensors only");

  if (!srcStt.hasSameDimToLvl(dstStt))
    return emitError("Unmatched dim2lvl map between input and result COO");

  if (srcStt.getPosType() != dstStt.getPosType() ||
      srcStt.getCrdType() != dstStt.getCrdType() ||
      srcStt.getElementType() != dstStt.getElementType())
    return emitError("Unmatched storage format between input and result COO");

  return success();
}

LogicalResult ReduceOp::verify() {
  Type inputType = getX().getType();
  Region &formula = getRegion();
  return verifyNumBlockArgs(this, formula, "reduce",
                            TypeRange{inputType, inputType}, inputType);
}

LogicalResult SelectOp::verify() {
  Builder b(getContext());
  Type inputType = getX().getType();
  Type boolType = b.getI1Type();
  Region &formula = getRegion();
  return verifyNumBlockArgs(this, formula, "select", TypeRange{inputType},
                            boolType);
}

LogicalResult SortOp::verify() {
  AffineMap xPerm = getPermMap();
  uint64_t nx = xPerm.getNumDims();
  if (nx < 1)
    return emitError(llvm::formatv("Expected rank(perm_map) > 1, got {0}", nx));

  if (!xPerm.isPermutation())
    return emitError(
        llvm::formatv("Expected a permutation map, got {0}", xPerm));

  // We can't check the size of the buffers when n or buffer dimensions aren't
  // compile-time constants.
  std::optional<int64_t> cn = getConstantIntValue(getN());
  if (!cn)
    return success();

  // Verify dimensions.
  const auto checkDim = [&](Value v, Size minSize,
                            const char *message) -> LogicalResult {
    const Size sh = getMemRefType(v).getShape()[0];
    if (!ShapedType::isDynamic(sh) && sh < minSize)
      return emitError(
          llvm::formatv("{0} got {1} < {2}", message, sh, minSize));
    return success();
  };
  uint64_t n = cn.value();
  uint64_t ny = 0;
  if (auto nyAttr = getNyAttr())
    ny = nyAttr.getInt();
  if (failed(checkDim(getXy(), n * (nx + ny),
                      "Expected dimension(xy) >= n * (rank(perm_map) + ny)")))
    return failure();
  for (Value opnd : getYs())
    if (failed(checkDim(opnd, n, "Expected dimension(y) >= n")))
      return failure();

  return success();
}

//===----------------------------------------------------------------------===//
// Sparse Tensor Iteration Operations.
//===----------------------------------------------------------------------===//

IterSpaceType IteratorType::getIterSpaceType() const {
  return IterSpaceType::get(getContext(), getEncoding(), getLoLvl(),
                            getHiLvl());
}

IteratorType IterSpaceType::getIteratorType() const {
  return IteratorType::get(getContext(), getEncoding(), getLoLvl(), getHiLvl());
}

/// Parses a level range in the form "$lo `to` $hi"
/// or simply "$lo" if $hi - $lo = 1
static ParseResult parseLevelRange(AsmParser &parser, Level &lvlLo,
                                   Level &lvlHi) {
  if (parser.parseInteger(lvlLo))
    return failure();

  if (succeeded(parser.parseOptionalKeyword("to"))) {
    if (parser.parseInteger(lvlHi))
      return failure();
  } else {
    lvlHi = lvlLo + 1;
  }

  if (lvlHi <= lvlLo)
    return parser.emitError(parser.getNameLoc(),
                            "expect larger level upper bound than lower bound");

  return success();
}

/// Parses a level range in the form "$lo `to` $hi"
/// or simply "$lo" if $hi - $lo = 1
static ParseResult parseLevelRange(OpAsmParser &parser, IntegerAttr &lvlLoAttr,
                                   IntegerAttr &lvlHiAttr) {
  Level lvlLo, lvlHi;
  if (parseLevelRange(parser, lvlLo, lvlHi))
    return failure();

  lvlLoAttr = IntegerAttr::get(parser.getBuilder().getIndexType(), lvlLo);
  lvlHiAttr = IntegerAttr::get(parser.getBuilder().getIndexType(), lvlHi);
  return success();
}

/// Prints a level range in the form "$lo `to` $hi"
/// or simply "$lo" if $hi - $lo = 1
static void printLevelRange(AsmPrinter &p, Level lo, Level hi) {

  if (lo + 1 == hi)
    p << lo;
  else
    p << lo << " to " << hi;
}

/// Prints a level range in the form "$lo `to` $hi"
/// or simply "$lo" if $hi - $lo = 1
static void printLevelRange(OpAsmPrinter &p, Operation *, IntegerAttr lvlLo,
                            IntegerAttr lvlHi) {
  unsigned lo = lvlLo.getValue().getZExtValue();
  unsigned hi = lvlHi.getValue().getZExtValue();
  printLevelRange(p, lo, hi);
}

/// Parses a list of `optional` defined list in the form of
/// "(%val0, _, %val1, ...)", where `_` is used to annotate that the
/// corresponding value is not defined (e.g., to represent an undefined
/// coordinate in the sparse iteration space).
static ParseResult parseOptionalDefinedList(
    OpAsmParser &parser, OperationState &state, I64BitSet &definedSet,
    SmallVectorImpl<OpAsmParser::Argument> &definedArgs,
    unsigned maxCnt = std::numeric_limits<unsigned>::max(),
    OpAsmParser::Delimiter delimiter = OpAsmParser::Delimiter::Paren) {
  unsigned cnt = 0;
  ParseResult crdList =
      parser.parseCommaSeparatedList(delimiter, [&]() -> ParseResult {
        if (parser.parseOptionalKeyword("_")) {
          if (parser.parseArgument(definedArgs.emplace_back()))
            return failure();
          definedSet.set(cnt);
        }
        cnt += 1;
        return success();
      });

  if (cnt > maxCnt)
    return parser.emitError(parser.getNameLoc(),
                            "parsed more value than expected.");

  if (failed(crdList)) {
    return parser.emitError(
        parser.getNameLoc(),
        "expecting SSA value or \"_\" for level coordinates");
  }
  assert(definedArgs.size() == definedSet.count());
  return success();
}

static void printOptionalDefinedList(OpAsmPrinter &p, unsigned size,
                                     Block::BlockArgListType blocksArgs,
                                     I64BitSet definedSet) {
  if (definedSet.empty())
    return;

  for (unsigned i = 0; i < size; i++) {
    if (definedSet[i]) {
      p << blocksArgs.front();
      blocksArgs = blocksArgs.drop_front();
    } else {
      p << "_";
    }
    if (i != size - 1)
      p << ", ";
  }
  assert(blocksArgs.empty());
}

static ParseResult
parseUsedCoordList(OpAsmParser &parser, OperationState &state,
                   SmallVectorImpl<OpAsmParser::Argument> &coords) {
  // Parse "at(%crd0, _, ...)"
  I64BitSet crdUsedLvlSet;
  if (succeeded(parser.parseOptionalKeyword("at")) &&
      failed(parseOptionalDefinedList(parser, state, crdUsedLvlSet, coords)))
    return failure();

  // Always use IndexType for the coordinate.
  for (auto &coord : coords)
    coord.type = parser.getBuilder().getIndexType();

  // Set the CrdUsedLvl bitset.
  state.addAttribute("crdUsedLvls",
                     parser.getBuilder().getI64IntegerAttr(crdUsedLvlSet));
  return success();
}

static ParseResult
parseSparseIterateLoop(OpAsmParser &parser, OperationState &state,
                       SmallVectorImpl<OpAsmParser::Argument> &iterators,
                       SmallVectorImpl<OpAsmParser::Argument> &blockArgs) {
  SmallVector<OpAsmParser::UnresolvedOperand> spaces;
  SmallVector<OpAsmParser::UnresolvedOperand> initArgs;

  // Parse "%iters, ... in %spaces, ..."
  if (parser.parseArgumentList(iterators) || parser.parseKeyword("in") ||
      parser.parseOperandList(spaces))
    return failure();

  if (iterators.size() != spaces.size())
    return parser.emitError(
        parser.getNameLoc(),
        "mismatch in number of sparse iterators and sparse spaces");

  SmallVector<OpAsmParser::Argument> coords;
  if (failed(parseUsedCoordList(parser, state, coords)))
    return failure();
  size_t numCrds = coords.size();

  // Parse "iter_args(%arg = %init, ...)"
  bool hasIterArgs = succeeded(parser.parseOptionalKeyword("iter_args"));
  if (hasIterArgs)
    if (parser.parseAssignmentList(blockArgs, initArgs))
      return failure();

  blockArgs.append(coords);

  SmallVector<Type> iterSpaceTps;
  // parse ": sparse_tensor.iter_space -> ret"
  if (parser.parseColon() || parser.parseTypeList(iterSpaceTps))
    return failure();
  if (iterSpaceTps.size() != spaces.size())
    return parser.emitError(parser.getNameLoc(),
                            "mismatch in number of iteration space operands "
                            "and iteration space types");

  for (auto [it, tp] : llvm::zip_equal(iterators, iterSpaceTps)) {
    IterSpaceType spaceTp = llvm::dyn_cast<IterSpaceType>(tp);
    if (!spaceTp)
      return parser.emitError(parser.getNameLoc(),
                              "expected sparse_tensor.iter_space type for "
                              "iteration space operands");
    it.type = spaceTp.getIteratorType();
  }

  if (hasIterArgs)
    if (parser.parseArrowTypeList(state.types))
      return failure();

  // Resolves input operands.
  if (parser.resolveOperands(spaces, iterSpaceTps, parser.getNameLoc(),
                             state.operands))
    return failure();

  if (hasIterArgs) {
    // Strip off leading args that used for coordinates.
    MutableArrayRef args = MutableArrayRef(blockArgs).drop_back(numCrds);
    if (args.size() != initArgs.size() || args.size() != state.types.size()) {
      return parser.emitError(
          parser.getNameLoc(),
          "mismatch in number of iteration arguments and return values");
    }

    for (auto [it, init, tp] : llvm::zip_equal(args, initArgs, state.types)) {
      it.type = tp;
      if (parser.resolveOperand(init, tp, state.operands))
        return failure();
    }
  }
  return success();
}

static ParseResult
parseSparseCoIterateLoop(OpAsmParser &parser, OperationState &state,
                         SmallVectorImpl<Value> &spacesVals,
                         SmallVectorImpl<OpAsmParser::Argument> &blockArgs) {

  // Parse "(%spaces, ...)"
  SmallVector<OpAsmParser::UnresolvedOperand> spaces;
  if (parser.parseOperandList(spaces, OpAsmParser::Delimiter::Paren))
    return failure();

  SmallVector<OpAsmParser::Argument> coords;
  if (failed(parseUsedCoordList(parser, state, coords)))
    return failure();
  size_t numCrds = coords.size();

  // Parse "iter_args(%arg = %init, ...)"
  SmallVector<OpAsmParser::UnresolvedOperand> initArgs;
  bool hasIterArgs = succeeded(parser.parseOptionalKeyword("iter_args"));
  if (hasIterArgs)
    if (parser.parseAssignmentList(blockArgs, initArgs))
      return failure();
  blockArgs.append(coords);

  SmallVector<Type> iterSpaceTps;
  // parse ": (sparse_tensor.iter_space, ...) -> ret"
  if (parser.parseColon() || parser.parseLParen() ||
      parser.parseTypeList(iterSpaceTps) || parser.parseRParen())
    return failure();

  if (iterSpaceTps.size() != spaces.size())
    return parser.emitError(parser.getNameLoc(),
                            "mismatch in number of iteration space operands "
                            "and iteration space types");

  if (hasIterArgs)
    if (parser.parseArrowTypeList(state.types))
      return failure();

  // Resolves input sparse iteration spaces.
  if (parser.resolveOperands(spaces, iterSpaceTps, parser.getNameLoc(),
                             spacesVals))
    return failure();
  state.operands.append(spacesVals);

  if (hasIterArgs) {
    // Strip off trailing args that used for coordinates.
    MutableArrayRef args = MutableArrayRef(blockArgs).drop_back(numCrds);
    if (args.size() != initArgs.size() || args.size() != state.types.size()) {
      return parser.emitError(
          parser.getNameLoc(),
          "mismatch in number of iteration arguments and return values");
    }

    for (auto [it, init, tp] : llvm::zip_equal(args, initArgs, state.types)) {
      it.type = tp;
      if (parser.resolveOperand(init, tp, state.operands))
        return failure();
    }
  }
  return success();
}

LogicalResult ExtractIterSpaceOp::inferReturnTypes(
    MLIRContext *ctx, std::optional<Location> loc, ValueRange ops,
    DictionaryAttr attr, OpaqueProperties prop, RegionRange region,
    SmallVectorImpl<mlir::Type> &ret) {

  ExtractIterSpaceOp::Adaptor adaptor(ops, attr, prop, region);
  SparseTensorType stt = getSparseTensorType(adaptor.getTensor());
  ret.push_back(IterSpaceType::get(ctx, stt.getEncoding(), adaptor.getLoLvl(),
                                   adaptor.getHiLvl()));
  return success();
}

LogicalResult ExtractIterSpaceOp::verify() {
  if (getLoLvl() >= getHiLvl())
    return emitOpError("expected smaller level low than level high");

  TypedValue<IteratorType> pIter = getParentIter();
  if ((pIter && getLoLvl() == 0) || (!pIter && getLoLvl() != 0)) {
    return emitOpError(
        "parent iterator should be specified iff level lower bound equals 0");
  }

  if (pIter) {
    IterSpaceType spaceTp = getExtractedSpace().getType();
    if (pIter.getType().getEncoding() != spaceTp.getEncoding())
      return emitOpError(
          "mismatch in parent iterator encoding and iteration space encoding.");

    if (spaceTp.getLoLvl() != pIter.getType().getHiLvl())
      return emitOpError("parent iterator should be used to extract an "
                         "iteration space from a consecutive level.");
  }

  return success();
}

LogicalResult ExtractValOp::verify() {
  auto stt = getSparseTensorType(getTensor());
  auto itTp = getIterator().getType();

  if (stt.getEncoding() != itTp.getEncoding())
    return emitOpError("mismatch in tensor encoding and iterator encoding.");

  if (stt.getLvlRank() != itTp.getHiLvl())
    return emitOpError("must use last-level iterator to extract values. ");

  return success();
}

struct RemoveUnusedLvlCrds : public OpRewritePattern<IterateOp> {
  using OpRewritePattern::OpRewritePattern;

  LogicalResult matchAndRewrite(IterateOp iterateOp,
                                PatternRewriter &rewriter) const override {
    I64BitSet newUsedLvls(0);
    llvm::BitVector toRemove(iterateOp.getBody()->getNumArguments());
    for (unsigned i = 0, e = iterateOp.getSpaceDim(); i < e; i++) {
      if (auto crd = iterateOp.getLvlCrd(i)) {
        if (crd->getUsers().empty())
          toRemove.set(crd->getArgNumber());
        else
          newUsedLvls.set(i);
      }
    }

    // All coordinates are used.
    if (toRemove.none())
      return failure();

    rewriter.startOpModification(iterateOp);
    iterateOp.setCrdUsedLvls(newUsedLvls);
    iterateOp.getBody()->eraseArguments(toRemove);
    rewriter.finalizeOpModification(iterateOp);
    return success();
  }
};

void IterateOp::getCanonicalizationPatterns(mlir::RewritePatternSet &results,
                                            mlir::MLIRContext *context) {
  results.add<RemoveUnusedLvlCrds>(context);
}

void IterateOp::build(OpBuilder &builder, OperationState &odsState,
                      Value iterSpace, ValueRange initArgs) {
  unsigned rank = llvm::cast<IterSpaceType>(iterSpace.getType()).getSpaceDim();
  // All ones.
  I64BitSet set((1 << rank) - 1);
  return build(builder, odsState, iterSpace, initArgs, set);
}

void IterateOp::build(OpBuilder &builder, OperationState &odsState,
                      Value iterSpace, ValueRange initArgs,
                      I64BitSet crdUsedLvls) {
  OpBuilder::InsertionGuard guard(builder);

  odsState.addOperands(iterSpace);
  odsState.addOperands(initArgs);
  odsState.getOrAddProperties<Properties>().crdUsedLvls =
      builder.getIntegerAttr(builder.getIntegerType(64), crdUsedLvls);
  Region *bodyRegion = odsState.addRegion();
  odsState.addTypes(initArgs.getTypes());
  Block *bodyBlock = builder.createBlock(bodyRegion);

  // Starts with a list of user-provided loop arguments.
  for (Value v : initArgs)
    bodyBlock->addArgument(v.getType(), v.getLoc());

  // Follows by a list of used coordinates.
  for (unsigned i = 0, e = crdUsedLvls.count(); i < e; i++)
    bodyBlock->addArgument(builder.getIndexType(), odsState.location);

  // Ends with sparse iterator
  bodyBlock->addArgument(
      llvm::cast<IterSpaceType>(iterSpace.getType()).getIteratorType(),
      odsState.location);
}

ParseResult IterateOp::parse(OpAsmParser &parser, OperationState &result) {
  OpAsmParser::Argument iterator;
  OpAsmParser::UnresolvedOperand iterSpace;

  SmallVector<OpAsmParser::Argument> iters, iterArgs;
  if (parseSparseIterateLoop(parser, result, iters, iterArgs))
    return failure();
  if (iters.size() != 1)
    return parser.emitError(parser.getNameLoc(),
                            "expected only one iterator/iteration space");

  iterArgs.append(iters);
  Region *body = result.addRegion();
  if (parser.parseRegion(*body, iterArgs))
    return failure();

  IterateOp::ensureTerminator(*body, parser.getBuilder(), result.location);

  // Parse the optional attribute list.
  if (parser.parseOptionalAttrDict(result.attributes))
    return failure();

  return success();
}

/// Prints the initialization list in the form of
///   <prefix>(%inner = %outer, %inner2 = %outer2, <...>)
/// where 'inner' values are assumed to be region arguments and 'outer' values
/// are regular SSA values.
static void printInitializationList(OpAsmPrinter &p,
                                    Block::BlockArgListType blocksArgs,
                                    ValueRange initializers,
                                    StringRef prefix = "") {
  assert(blocksArgs.size() == initializers.size() &&
         "expected same length of arguments and initializers");
  if (initializers.empty())
    return;

  p << prefix << '(';
  llvm::interleaveComma(llvm::zip(blocksArgs, initializers), p, [&](auto it) {
    p << std::get<0>(it) << " = " << std::get<1>(it);
  });
  p << ")";
}

template <typename SparseLoopOp>
static LogicalResult verifySparseLoopOp(SparseLoopOp op) {
  if (op.getInitArgs().size() != op.getNumResults()) {
    return op.emitOpError(
        "mismatch in number of loop-carried values and defined values");
  }
  if (op.getCrdUsedLvls().max() > op.getSpaceDim())
    return op.emitOpError("required out-of-bound coordinates");

  return success();
}

LogicalResult IterateOp::verify() { return verifySparseLoopOp(*this); }
LogicalResult CoIterateOp::verify() { return verifySparseLoopOp(*this); }

void IterateOp::print(OpAsmPrinter &p) {
  p << " " << getIterator() << " in " << getIterSpace();
  if (!getCrdUsedLvls().empty()) {
    p << " at(";
    printOptionalDefinedList(p, getSpaceDim(), getCrds(), getCrdUsedLvls());
    p << ")";
  }
  printInitializationList(p, getRegionIterArgs(), getInitArgs(), " iter_args");

  p << " : " << getIterSpace().getType() << " ";
  if (!getInitArgs().empty())
    p.printArrowTypeList(getInitArgs().getTypes());

  p << " ";
  p.printRegion(getRegion(), /*printEntryBlockArgs=*/false,
                /*printBlockTerminators=*/!getInitArgs().empty());
}

LogicalResult IterateOp::verifyRegions() {
  if (getIterator().getType() != getIterSpace().getType().getIteratorType())
    return emitOpError("mismatch in iterator and iteration space type");
  if (getNumRegionIterArgs() != getNumResults())
    return emitOpError(
        "mismatch in number of basic block args and defined values");

  auto initArgs = getInitArgs();
  auto iterArgs = getRegionIterArgs();
  auto yieldVals = getYieldedValues();
  auto opResults = getResults();
  if (!llvm::all_equal({initArgs.size(), iterArgs.size(), yieldVals.size(),
                        opResults.size()})) {
    return emitOpError() << "number mismatch between iter args and results.";
  }

  for (auto [i, init, iter, yield, ret] :
       llvm::enumerate(initArgs, iterArgs, yieldVals, opResults)) {
    if (init.getType() != ret.getType())
      return emitOpError() << "types mismatch between " << i
                           << "th iter operand and defined value";
    if (iter.getType() != ret.getType())
      return emitOpError() << "types mismatch between " << i
                           << "th iter region arg and defined value";
    if (yield.getType() != ret.getType())
      return emitOpError() << "types mismatch between " << i
                           << "th yield value and defined value";
  }

  return success();
}

/// OpInterfaces' methods implemented by IterateOp.
SmallVector<Region *> IterateOp::getLoopRegions() { return {&getRegion()}; }

MutableArrayRef<OpOperand> IterateOp::getInitsMutable() {
  return getInitArgsMutable();
}

Block::BlockArgListType IterateOp::getRegionIterArgs() {
  return getRegion().getArguments().take_front(getNumRegionIterArgs());
}

std::optional<MutableArrayRef<OpOperand>> IterateOp::getYieldedValuesMutable() {
  return cast<sparse_tensor::YieldOp>(
             getRegion().getBlocks().front().getTerminator())
      .getResultsMutable();
}

std::optional<ResultRange> IterateOp::getLoopResults() { return getResults(); }

OperandRange IterateOp::getEntrySuccessorOperands(RegionBranchPoint point) {
  return getInitArgs();
}

void IterateOp::getSuccessorRegions(RegionBranchPoint point,
                                    SmallVectorImpl<RegionSuccessor> &regions) {
  // Both the operation itself and the region may be branching into the body
  // or back into the operation itself.
  regions.push_back(RegionSuccessor(&getRegion(), getRegionIterArgs()));
  // It is possible for loop not to enter the body.
  regions.push_back(RegionSuccessor(getResults()));
}

void CoIterateOp::build(OpBuilder &builder, OperationState &odsState,
                        ValueRange iterSpaces, ValueRange initArgs,
                        unsigned numCases) {
  unsigned rank =
      cast<IterSpaceType>(iterSpaces.front().getType()).getSpaceDim();
  // All ones.
  I64BitSet set((1 << rank) - 1);
  // Generates all-zero case bits (they only serve as placeholders), which are
  // supposed to be overriden later. We need to preallocate all the regions as
  // mlir::Region cannot be dynamically added later after the operation is
  // created.
  SmallVector<int64_t> caseBits(numCases, 0);
  ArrayAttr cases = builder.getI64ArrayAttr(caseBits);
  return CoIterateOp::build(builder, odsState, initArgs.getTypes(), iterSpaces,
                            initArgs, set, cases,
                            /*caseRegionsCount=*/numCases);
}

ParseResult CoIterateOp::parse(OpAsmParser &parser, OperationState &result) {

  SmallVector<Value> spaces;
  // The block argument list of each regions, it is arranged in the order of
  // ([used coordinate list], [loop iterations args], [sparse iterator list]).
  SmallVector<OpAsmParser::Argument> blockArgs;
  if (parseSparseCoIterateLoop(parser, result, spaces, blockArgs))
    return failure();

  result.addAttribute("operandSegmentSizes",
                      parser.getBuilder().getDenseI32ArrayAttr(
                          {static_cast<int32_t>(spaces.size()),
                           static_cast<int32_t>(result.types.size())}));

  SmallVector<Attribute> cases;
  while (succeeded(parser.parseOptionalKeyword("case"))) {
    // Parse one region per case.
    I64BitSet definedItSet;
    SmallVector<OpAsmParser::Argument> definedIts;
    if (parseOptionalDefinedList(parser, result, definedItSet, definedIts,
                                 spaces.size(), OpAsmParser::Delimiter::None))
      return failure();

    cases.push_back(parser.getBuilder().getI64IntegerAttr(definedItSet));

    for (auto [i, definedIdx] : llvm::enumerate(definedItSet.bits())) {
      // Resolve the iterator type based on the iteration space type.
      auto spaceTp = llvm::cast<IterSpaceType>(spaces[definedIdx].getType());
      definedIts[i].type = spaceTp.getIteratorType();
    }
    definedIts.insert(definedIts.begin(), blockArgs.begin(), blockArgs.end());
    Region *body = result.addRegion();
    if (parser.parseRegion(*body, definedIts))
      return failure();

    CoIterateOp::ensureTerminator(*body, parser.getBuilder(), result.location);
  }

  result.addAttribute("cases", ArrayAttr::get(parser.getContext(), cases));

  // Parse the optional attribute list.
  if (parser.parseOptionalAttrDict(result.attributes))
    return failure();

  return success();
}

void CoIterateOp::print(OpAsmPrinter &p) {
  p << " (";
  llvm::interleaveComma(getIterSpaces(), p, [&](auto s) { p << s; });
  p << ")";

  if (!getCrdUsedLvls().empty()) {
    p << " at(";
    printOptionalDefinedList(p, getSpaceDim(), getCrds(0), getCrdUsedLvls());
    p << ")";
  }

  printInitializationList(p, getRegionIterArgs(0), getInitArgs(), " iter_args");

  p << " : (" << getIterSpaces().getTypes() << ")";
  if (!getInitArgs().empty())
    p.printArrowTypeList(getInitArgs().getTypes());

  for (unsigned idx = 0, e = getRegions().size(); idx < e; idx++) {
    p.printNewline();
    p << "case ";
    printOptionalDefinedList(p, getIterSpaces().size(), getRegionIterators(idx),
                             getRegionDefinedSpace(idx));
    p << " ";
    p.printRegion(getRegion(idx), /*printEntryBlockArgs=*/false,
                  /*printBlockTerminators=*/!getInitArgs().empty());
  }
}

ValueRange CoIterateOp::getYieldedValues(unsigned regionIdx) {
  return cast<sparse_tensor::YieldOp>(
             getRegion(regionIdx).getBlocks().front().getTerminator())
      .getResults();
}

LogicalResult CoIterateOp::verifyRegions() {
  for (unsigned r = 0, e = getNumRegions(); r < e; r++) {
    if (getNumRegionIterArgs() != getNumResults())
      return emitOpError(
          "mismatch in number of basic block args and defined values");

    auto initArgs = getInitArgs();
    auto iterArgs = getRegionIterArgs(r);
    auto yieldVals = getYieldedValues(r);
    auto opResults = getResults();
    if (!llvm::all_equal({initArgs.size(), iterArgs.size(), yieldVals.size(),
                          opResults.size()})) {
      return emitOpError()
             << "number mismatch between iter args and results on " << r
             << "th region";
    }

    for (auto [i, init, iter, yield, ret] :
         llvm::enumerate(initArgs, iterArgs, yieldVals, opResults)) {
      if (init.getType() != ret.getType())
        return emitOpError()
               << "types mismatch between " << i
               << "th iter operand and defined value on " << r << "th region";
      if (iter.getType() != ret.getType())
        return emitOpError() << "types mismatch between " << i
                             << "th iter region arg and defined value on " << r
                             << "th region";
      if (yield.getType() != ret.getType())
        return emitOpError()
               << "types mismatch between " << i
               << "th yield value and defined value on " << r << "th region";
    }
  }

  auto cases = getRegionDefinedSpaces();
  llvm::SmallSetVector<uint64_t, 8> set(cases.begin(), cases.end());
  if (set.size() != getNumRegions())
    return emitOpError("contains duplicated cases.");

  return success();
}

SmallVector<Region *> CoIterateOp::getSubCasesOf(unsigned regionIdx) {
  SmallVector<Region *> ret;
  I64BitSet caseBit = getRegionDefinedSpace(regionIdx);
  for (Region &r : getCaseRegions())
    if (getRegionDefinedSpace(r.getRegionNumber()).isSubSetOf(caseBit))
      ret.push_back(&r);

  return ret;
}

//===----------------------------------------------------------------------===//
// Sparse Tensor Dialect Setups.
//===----------------------------------------------------------------------===//

/// Materialize a single constant operation from a given attribute value with
/// the desired resultant type.
Operation *SparseTensorDialect::materializeConstant(OpBuilder &builder,
                                                    Attribute value, Type type,
                                                    Location loc) {
  if (auto op = arith::ConstantOp::materialize(builder, value, type, loc))
    return op;
  return nullptr;
}

namespace {
struct SparseTensorAsmDialectInterface : public OpAsmDialectInterface {
  using OpAsmDialectInterface::OpAsmDialectInterface;

  AliasResult getAlias(Attribute attr, raw_ostream &os) const override {
    if (isa<SparseTensorEncodingAttr>(attr)) {
      os << "sparse";
      return AliasResult::OverridableAlias;
    }
    return AliasResult::NoAlias;
  }
};
} // namespace

void SparseTensorDialect::initialize() {
  addInterface<SparseTensorAsmDialectInterface>();
  addAttributes<
#define GET_ATTRDEF_LIST
#include "mlir/Dialect/SparseTensor/IR/SparseTensorAttrDefs.cpp.inc"
      >();
  addTypes<
#define GET_TYPEDEF_LIST
#include "mlir/Dialect/SparseTensor/IR/SparseTensorTypes.cpp.inc"
      >();
  addOperations<
#define GET_OP_LIST
#include "mlir/Dialect/SparseTensor/IR/SparseTensorOps.cpp.inc"
      >();
  declarePromisedInterfaces<
      bufferization::BufferizableOpInterface, ConcatenateOp, ConvertOp, LoadOp,
      NewOp, NumberOfEntriesOp, AssembleOp, DisassembleOp,
      ToCoordinatesBufferOp, ToCoordinatesOp, ToPositionsOp, ToValuesOp>();
}

#define GET_OP_CLASSES
#include "mlir/Dialect/SparseTensor/IR/SparseTensorOps.cpp.inc"

#include "mlir/Dialect/SparseTensor/IR/SparseTensorOpsDialect.cpp.inc"