wasm-type.cpp

/*
 * Copyright 2017 WebAssembly Community Group participants
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *     http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

#include <algorithm>
#include <array>
#include <cassert>
#include <map>
#include <shared_mutex>
#include <sstream>
#include <unordered_map>
#include <unordered_set>
#include <variant>

#include "compiler-support.h"
#include "support/hash.h"
#include "support/insert_ordered.h"
#include "wasm-features.h"
#include "wasm-type-printing.h"
#include "wasm-type.h"

#define TRACE_CANONICALIZATION 0

#if TRACE_CANONICALIZATION
#include <iostream>
#endif

namespace wasm {

namespace {

using RecGroupInfo = std::vector<HeapType>;

struct HeapTypeInfo {
  using type_t = HeapType;
  // Used in assertions to ensure that temporary types don't leak into the
  // global store.
  bool isTemp = false;
  bool isOpen = false;
  Shareability share = Unshared;
  // The supertype of this HeapType, if it exists.
  HeapTypeInfo* supertype = nullptr;
  // The descriptor of this HeapType, if it exists.
  HeapTypeInfo* descriptor = nullptr;
  // The HeapType described by this one, if it exists.
  HeapTypeInfo* described = nullptr;
  // The recursion group of this type or null if the recursion group is trivial
  // (i.e. contains only this type).
  RecGroupInfo* recGroup = nullptr;
  size_t recGroupIndex = 0;
  HeapTypeKind kind;
  union {
    Signature signature;
    Continuation continuation;
    Struct struct_;
    Array array;
  };

  HeapTypeInfo(Signature sig) : kind(HeapTypeKind::Func), signature(sig) {}
  HeapTypeInfo(Continuation continuation)
    : kind(HeapTypeKind::Cont), continuation(continuation) {}
  HeapTypeInfo(const Struct& struct_)
    : kind(HeapTypeKind::Struct), struct_(struct_) {}
  HeapTypeInfo(Struct&& struct_)
    : kind(HeapTypeKind::Struct), struct_(std::move(struct_)) {}
  HeapTypeInfo(Array array) : kind(HeapTypeKind::Array), array(array) {}
  ~HeapTypeInfo();

  constexpr bool isSignature() const { return kind == HeapTypeKind::Func; }
  constexpr bool isContinuation() const { return kind == HeapTypeKind::Cont; }
  constexpr bool isStruct() const { return kind == HeapTypeKind::Struct; }
  constexpr bool isArray() const { return kind == HeapTypeKind::Array; }
  constexpr bool isData() const { return isStruct() || isArray(); }
};

// Helper for coinductively checking whether a pair of Types or HeapTypes are in
// a subtype relation.
struct SubTyper {
  bool isSubType(Type a, Type b);
  bool isSubType(HeapType a, HeapType b);
  bool isSubType(const Tuple& a, const Tuple& b);
  bool isSubType(const Field& a, const Field& b);
  bool isSubType(const Signature& a, const Signature& b);
  bool isSubType(const Continuation& a, const Continuation& b);
  bool isSubType(const Struct& a, const Struct& b);
  bool isSubType(const Array& a, const Array& b);
};

// Helper for finding the equirecursive least upper bound of two types.
// Helper for printing types.
struct TypePrinter {
  // The stream we are printing to.
  std::ostream& os;

  // The default generator state if no other generator is provided.
  std::optional<DefaultTypeNameGenerator> defaultGenerator;

  // The function we call to get HeapType names.
  HeapTypeNameGenerator generator;

  TypePrinter(std::ostream& os, HeapTypeNameGenerator generator)
    : os(os), defaultGenerator(), generator(generator) {}
  TypePrinter(std::ostream& os)
    : TypePrinter(
        os, [&](HeapType type) { return defaultGenerator->getNames(type); }) {
    defaultGenerator = DefaultTypeNameGenerator{};
  }

  void printHeapTypeName(HeapType type);

  std::ostream& print(Type type);
  std::ostream& print(HeapType type);
  std::ostream& print(const Tuple& tuple);
  std::ostream& print(const Field& field);
  std::ostream& print(const Signature& sig);
  std::ostream& print(const Continuation& cont);
  std::ostream& print(const Struct& struct_,
                      const std::unordered_map<Index, Name>& fieldNames);
  std::ostream& print(const Array& array);
};

struct RecGroupHasher {
  // `group` may or may not be canonical, but any other recursion group it
  // reaches must be canonical.
  RecGroup group;

  RecGroupHasher(RecGroup group) : group(group) {}

  // Perform the hash.
  size_t operator()() const;

  // `topLevelHash` is applied to the top-level group members and observes their
  // structure, while `hash(HeapType)` is applied to the children of group
  // members and does not observe their structure.
  size_t topLevelHash(HeapType type) const;
  size_t hash(Type type) const;
  size_t hash(HeapType type) const;
  size_t hash(const HeapTypeInfo& info) const;
  size_t hash(const Tuple& tuple) const;
  size_t hash(const Field& field) const;
  size_t hash(const Signature& sig) const;
  size_t hash(const Continuation& sig) const;
  size_t hash(const Struct& struct_) const;
  size_t hash(const Array& array) const;
};

struct RecGroupEquator {
  // `newGroup` may or may not be canonical, but `otherGroup` and any other
  // recursion group reachable by either of them must be canonical.
  RecGroup newGroup, otherGroup;

  RecGroupEquator(RecGroup newGroup, RecGroup otherGroup)
    : newGroup(newGroup), otherGroup(otherGroup) {}

  // Perform the comparison.
  bool operator()() const;

  // `topLevelEq` is applied to the top-level group members and observes their
  // structure, while `eq(HeapType)` is applied to the children of group members
  // and does not observe their structure.
  bool topLevelEq(HeapType a, HeapType b) const;
  bool eq(Type a, Type b) const;
  bool eq(HeapType a, HeapType b) const;
  bool eq(const HeapTypeInfo& a, const HeapTypeInfo& b) const;
  bool eq(const Tuple& a, const Tuple& b) const;
  bool eq(const Field& a, const Field& b) const;
  bool eq(const Signature& a, const Signature& b) const;
  bool eq(const Continuation& a, const Continuation& b) const;
  bool eq(const Struct& a, const Struct& b) const;
  bool eq(const Array& a, const Array& b) const;
};

// A wrapper around a RecGroup that provides equality and hashing based on the
// structure of the group such that isorecursively equivalent recursion groups
// will compare equal and will have the same hash. Assumes that all recursion
// groups reachable from this one have been canonicalized, except for the
// wrapped group itself.
struct RecGroupStructure {
  RecGroup group;
  bool operator==(const RecGroupStructure& other) const {
    return RecGroupEquator{group, other.group}();
  }
};

} // anonymous namespace
} // namespace wasm

namespace std {

template<> class hash<wasm::RecGroupStructure> {
public:
  size_t operator()(const wasm::RecGroupStructure& structure) const {
    return wasm::RecGroupHasher{structure.group}();
  }
};

template<typename T> class hash<reference_wrapper<const T>> {
public:
  size_t operator()(const reference_wrapper<const T>& ref) const {
    return hash<T>{}(ref.get());
  }
};

template<typename T> class equal_to<reference_wrapper<const T>> {
public:
  bool operator()(const reference_wrapper<const T>& a,
                  const reference_wrapper<const T>& b) const {
    return equal_to<T>{}(a.get(), b.get());
  }
};

} // namespace std

namespace wasm {
namespace {

HeapTypeInfo* getHeapTypeInfo(HeapType ht) {
  assert(!ht.isBasic());
  return (HeapTypeInfo*)(ht.getRawID());
}

HeapType asHeapType(std::unique_ptr<HeapTypeInfo>& info) {
  return HeapType(uintptr_t(info.get()));
}

bool isTemp(HeapType type) {
  return !type.isBasic() && getHeapTypeInfo(type)->isTemp;
}

// Generic utility for traversing type graphs. The inserted roots must live as
// long as the Walker because they are referenced by address. This base class
// only has logic for traversing type graphs; figuring out when to stop
// traversing the graph and doing useful work during the traversal is left to
// subclasses, which should override `scanType` and/or `scanHeapType`. Edges
// from reference types to the referenced heap types are not walked, so
// subclasses should handle referenced heap types when their reference types are
// visited.
template<typename Self> struct TypeGraphWalkerBase {
  void walkRoot(Type* type) {
    assert(taskList.empty());
    taskList.push_back(Task::scan(type));
    doWalk();
  }

  void walkRoot(HeapType* ht) {
    assert(taskList.empty());
    taskList.push_back(Task::scan(ht));
    doWalk();
  }

protected:
  Self& self() { return *static_cast<Self*>(this); }

  void scanType(Type* type) {
    if (type->isTuple()) {
      auto& types = const_cast<Tuple&>(type->getTuple());
      for (auto it = types.rbegin(); it != types.rend(); ++it) {
        taskList.push_back(Task::scan(&*it));
      }
    }
  }

  void scanHeapType(HeapType* ht) {
    if (ht->isBasic()) {
      return;
    }
    auto* info = getHeapTypeInfo(*ht);
    switch (info->kind) {
      case HeapTypeKind::Func:
        taskList.push_back(Task::scan(&info->signature.results));
        taskList.push_back(Task::scan(&info->signature.params));
        break;
      case HeapTypeKind::Cont:
        taskList.push_back(Task::scan(&info->continuation.type));
        break;
      case HeapTypeKind::Struct: {
        auto& fields = info->struct_.fields;
        for (auto field = fields.rbegin(); field != fields.rend(); ++field) {
          taskList.push_back(Task::scan(&field->type));
        }
        break;
      }
      case HeapTypeKind::Array:
        taskList.push_back(Task::scan(&info->array.element.type));
        break;
      case HeapTypeKind::Basic:
        WASM_UNREACHABLE("unexpected kind");
    }
  }

private:
  struct Task {
    enum Kind {
      ScanType,
      ScanHeapType,
    } kind;
    union {
      Type* type;
      HeapType* heapType;
    };
    static Task scan(Type* type) { return Task(type, ScanType); }
    static Task scan(HeapType* ht) { return Task(ht, ScanHeapType); }

  private:
    Task(Type* type, Kind kind) : kind(kind), type(type) {}
    Task(HeapType* ht, Kind kind) : kind(kind), heapType(ht) {}
  };

  std::vector<Task> taskList;

  void doWalk() {
    while (!taskList.empty()) {
      auto curr = taskList.back();
      taskList.pop_back();
      switch (curr.kind) {
        case Task::ScanType:
          self().scanType(curr.type);
          break;
        case Task::ScanHeapType:
          self().scanHeapType(curr.heapType);
          break;
      }
    }
  }
};

// A type graph walker that scans each each direct HeapType child of the root.
template<typename Self> struct HeapTypeChildWalker : TypeGraphWalkerBase<Self> {
  void scanType(Type* type) {
    isTopLevel = false;
    if (type->isRef()) {
      this->self().noteChild(type->getHeapType());
    } else {
      TypeGraphWalkerBase<Self>::scanType(type);
    }
  }

  void scanHeapType(HeapType* type) {
    if (isTopLevel) {
      isTopLevel = false;
      TypeGraphWalkerBase<Self>::scanHeapType(type);
    } else {
      this->self().noteChild(*type);
    }
  }

private:
  bool isTopLevel = true;
};

struct HeapTypeChildCollector : HeapTypeChildWalker<HeapTypeChildCollector> {
  std::vector<HeapType> children;
  void noteChild(HeapType type) { children.push_back(type); }
};

HeapType::BasicHeapType getBasicHeapSupertype(HeapType type) {
  if (type.isBasic()) {
    return HeapType::BasicHeapType(type.getID());
  }
  auto* info = getHeapTypeInfo(type);
  switch (info->kind) {
    case HeapTypeKind::Func:
      return HeapTypes::func.getBasic(info->share);
    case HeapTypeKind::Cont:
      return HeapTypes::cont.getBasic(info->share);
    case HeapTypeKind::Struct:
      return HeapTypes::struct_.getBasic(info->share);
    case HeapTypeKind::Array:
      return HeapTypes::array.getBasic(info->share);
    case HeapTypeKind::Basic:
      break;
  }
  WASM_UNREACHABLE("unexpected kind");
};

std::optional<HeapType> getBasicHeapTypeLUB(HeapType::BasicHeapType a,
                                            HeapType::BasicHeapType b) {
  if (a == b) {
    return a;
  }
  if (HeapType(a).getTop() != HeapType(b).getTop()) {
    return {};
  }
  if (HeapType(a).isBottom()) {
    return b;
  }
  if (HeapType(b).isBottom()) {
    return a;
  }
  // Canonicalize to have `a` be the lesser type.
  if (unsigned(a) > unsigned(b)) {
    std::swap(a, b);
  }
  auto bUnshared = HeapType(b).getBasic(Unshared);
  HeapType lubUnshared;
  switch (HeapType(a).getBasic(Unshared)) {
    case HeapType::ext:
      if (bUnshared != HeapType::string) {
        return std::nullopt;
      }
      lubUnshared = HeapType::ext;
      break;
    case HeapType::func:
    case HeapType::cont:
    case HeapType::exn:
      return std::nullopt;
    case HeapType::any:
      lubUnshared = HeapType::any;
      break;
    case HeapType::eq:
      if (bUnshared == HeapType::i31 || bUnshared == HeapType::struct_ ||
          bUnshared == HeapType::array) {
        lubUnshared = HeapType::eq;
      } else {
        lubUnshared = HeapType::any;
      }
      break;
    case HeapType::i31:
      if (bUnshared == HeapType::struct_ || bUnshared == HeapType::array) {
        lubUnshared = HeapType::eq;
      } else {
        lubUnshared = HeapType::any;
      }
      break;
    case HeapType::struct_:
      if (bUnshared == HeapType::array) {
        lubUnshared = HeapType::eq;
      } else {
        lubUnshared = HeapType::any;
      }
      break;
    case HeapType::array:
      lubUnshared = HeapType::any;
      break;
    case HeapType::string:
      // String has already been handled: we sorted before in a way that ensures
      // the type the string is compared to is of a higher index, which means it
      // is a bottom type (string is the last type that is not a bottom type),
      // but we have handled the case of either a or b being a bottom type
      // earlier already.
    case HeapType::none:
    case HeapType::noext:
    case HeapType::nofunc:
    case HeapType::nocont:
    case HeapType::noexn:
      // Bottom types already handled.
      WASM_UNREACHABLE("unexpected basic type");
  }
  auto share = HeapType(a).getShared();
  return {lubUnshared.getBasic(share)};
}

} // anonymous namespace

HeapTypeInfo::~HeapTypeInfo() {
  switch (kind) {
    case HeapTypeKind::Func:
      signature.~Signature();
      return;
    case HeapTypeKind::Cont:
      continuation.~Continuation();
      return;
    case HeapTypeKind::Struct:
      struct_.~Struct();
      return;
    case HeapTypeKind::Array:
      array.~Array();
      return;
    case HeapTypeKind::Basic:
      break;
  }
  WASM_UNREACHABLE("unexpected kind");
}

namespace {

struct TupleStore {
  std::recursive_mutex mutex;

  // Track unique_ptrs for constructed tuples to avoid leaks.
  std::vector<std::unique_ptr<Tuple>> constructedTuples;

  // Maps from constructed tuples to their canonical Type IDs.
  std::unordered_map<std::reference_wrapper<const Tuple>, uintptr_t> typeIDs;

  Type insert(const Tuple& info) { return doInsert(info); }
  Type insert(std::unique_ptr<Tuple>&& info) { return doInsert(info); }
  bool hasCanonical(const Tuple& info, Tuple& canonical);

  void clear() {
    typeIDs.clear();
    constructedTuples.clear();
  }

private:
  template<typename Ref> Type doInsert(Ref& tupleRef) {
    const Tuple& tuple = [&]() {
      if constexpr (std::is_same_v<Ref, const Tuple>) {
        return tupleRef;
      } else if constexpr (std::is_same_v<Ref, std::unique_ptr<Tuple>>) {
        return *tupleRef;
      }
    }();

    auto getPtr = [&]() -> std::unique_ptr<Tuple> {
      if constexpr (std::is_same_v<Ref, const Tuple>) {
        return std::make_unique<Tuple>(tupleRef);
      } else if constexpr (std::is_same_v<Ref, std::unique_ptr<Tuple>>) {
        return std::move(tupleRef);
      }
    };

    auto insertNew = [&]() {
      auto ptr = getPtr();
      TypeID id = uintptr_t(ptr.get()) | 1;
      assert(id > Type::_last_basic_type);
      typeIDs.insert({*ptr, id});
      constructedTuples.emplace_back(std::move(ptr));
      return Type(id);
    };

    // Turn e.g. singleton tuple into non-tuple.
    if (tuple.size() == 0) {
      return Type::none;
    }
    if (tuple.size() == 1) {
      return tuple[0];
    }

    std::lock_guard<std::recursive_mutex> lock(mutex);
    // Check whether we already have a type for this tuple.
    auto indexIt = typeIDs.find(std::cref(tuple));
    if (indexIt != typeIDs.end()) {
      return Type(indexIt->second);
    }
    // We do not have a type for this tuple already. Create one.
    return insertNew();
  }
};

static TupleStore globalTupleStore;

static std::vector<std::unique_ptr<HeapTypeInfo>> globalHeapTypeStore;
static std::recursive_mutex globalHeapTypeStoreMutex;

// Keep track of the constructed recursion groups.
struct RecGroupStore {
  std::mutex mutex;
  // Store the structures of all rec groups created so far so we can avoid
  // creating duplicates.
  std::unordered_set<RecGroupStructure> canonicalGroups;
  // Keep the `RecGroupInfos` for the nontrivial groups stored in
  // `canonicalGroups` alive.
  std::vector<std::unique_ptr<RecGroupInfo>> builtGroups;

  RecGroup insert(RecGroup group) {
    RecGroupStructure structure{group};
    auto [it, inserted] = canonicalGroups.insert(structure);
    if (inserted) {
      return group;
    } else {
      return it->group;
    }
  }

  RecGroup insert(std::unique_ptr<RecGroupInfo>&& info) {
    RecGroup group{uintptr_t(info.get())};
    auto canonical = insert(group);
    if (canonical == group) {
      builtGroups.emplace_back(std::move(info));
    }
    return canonical;
  }

  // Utility for canonicalizing HeapTypes with trivial recursion groups.
  HeapType insert(std::unique_ptr<HeapTypeInfo>&& info) {
    std::lock_guard<std::mutex> lock(mutex);
    assert(!info->recGroup && "Unexpected nontrivial rec group");
    auto group = asHeapType(info).getRecGroup();
    auto canonical = insert(group);
    if (group == canonical) {
      std::lock_guard<std::recursive_mutex> storeLock(globalHeapTypeStoreMutex);
      globalHeapTypeStore.emplace_back(std::move(info));
    }
    return canonical[0];
  }

  void clear() {
    canonicalGroups.clear();
    builtGroups.clear();
  }
};

static RecGroupStore globalRecGroupStore;

void validateTuple(const Tuple& tuple) {
#ifndef NDEBUG
  for (auto type : tuple) {
    assert(type.isSingle());
  }
#endif
}

} // anonymous namespace

void destroyAllTypesForTestingPurposesOnly() {
  globalTupleStore.clear();
  globalHeapTypeStore.clear();
  globalRecGroupStore.clear();
}

Type::Type(std::initializer_list<Type> types) : Type(Tuple(types)) {}

Type::Type(const Tuple& tuple) {
  validateTuple(tuple);
  new (this) Type(globalTupleStore.insert(tuple));
}

Type::Type(Tuple&& tuple) {
  new (this) Type(globalTupleStore.insert(std::move(tuple)));
}

bool Type::isDefaultable() const {
  // A variable can get a default value if its type is concrete (unreachable
  // and none have no values, hence no default), and if it's a reference, it
  // must be nullable.
  if (isTuple()) {
    for (auto t : *this) {
      if (!t.isDefaultable()) {
        return false;
      }
    }
    return true;
  }
  return isConcrete() && !isNonNullable();
}

unsigned Type::getByteSize() const {
  // TODO: alignment?
  auto getSingleByteSize = [](Type t) {
    switch (t.getBasic()) {
      case Type::i32:
      case Type::f32:
        return 4;
      case Type::i64:
      case Type::f64:
        return 8;
      case Type::v128:
        return 16;
      case Type::none:
      case Type::unreachable:
        break;
    }
    WASM_UNREACHABLE("invalid type");
  };

  if (isTuple()) {
    unsigned size = 0;
    for (const auto& t : *this) {
      size += getSingleByteSize(t);
    }
    return size;
  }
  return getSingleByteSize(*this);
}

unsigned Type::hasByteSize() const {
  auto hasSingleByteSize = [](Type t) { return t.isNumber(); };
  if (isTuple()) {
    for (const auto& t : *this) {
      if (!hasSingleByteSize(t)) {
        return false;
      }
    }
    return true;
  }
  return hasSingleByteSize(*this);
}

Type Type::reinterpret() const {
  assert(!isTuple() && "Unexpected tuple type");
  switch ((*begin()).getBasic()) {
    case Type::i32:
      return f32;
    case Type::i64:
      return f64;
    case Type::f32:
      return i32;
    case Type::f64:
      return i64;
    default:
      WASM_UNREACHABLE("invalid type");
  }
}

FeatureSet Type::getFeatures() const {
  auto getSingleFeatures = [](Type t) -> FeatureSet {
    if (t.isRef()) {
      return t.getHeapType().getFeatures();
    }

    switch (t.getBasic()) {
      case Type::v128:
        return FeatureSet::SIMD;
      default:
        return FeatureSet::MVP;
    }
  };

  if (isTuple()) {
    FeatureSet feats = FeatureSet::Multivalue;
    for (const auto& t : *this) {
      feats |= getSingleFeatures(t);
    }
    return feats;
  }
  return getSingleFeatures(*this);
}

Type Type::get(unsigned byteSize, bool float_) {
  if (byteSize < 4) {
    return Type::i32;
  }
  if (byteSize == 4) {
    return float_ ? Type::f32 : Type::i32;
  }
  if (byteSize == 8) {
    return float_ ? Type::f64 : Type::i64;
  }
  if (byteSize == 16) {
    return Type::v128;
  }
  WASM_UNREACHABLE("invalid size");
}

bool Type::isSubType(Type left, Type right) {
  // As an optimization, in the common case do not even construct a SubTyper.
  if (left == right) {
    return true;
  }
  return SubTyper().isSubType(left, right);
}

std::vector<HeapType> Type::getHeapTypeChildren() {
  HeapTypeChildCollector collector;
  collector.walkRoot(this);
  return collector.children;
}

bool Type::hasLeastUpperBound(Type a, Type b) {
  return getLeastUpperBound(a, b) != Type::none;
}

Type Type::getLeastUpperBound(Type a, Type b) {
  if (a == b) {
    return a;
  }
  if (a == Type::unreachable) {
    return b;
  }
  if (b == Type::unreachable) {
    return a;
  }
  if (a.isTuple() && b.isTuple()) {
    auto size = a.size();
    if (size != b.size()) {
      return Type::none;
    }
    std::vector<Type> elems;
    elems.reserve(size);
    for (size_t i = 0; i < size; ++i) {
      auto lub = Type::getLeastUpperBound(a[i], b[i]);
      if (lub == Type::none) {
        return Type::none;
      }
      elems.push_back(lub);
    }
    return Type(elems);
  }
  if (a.isRef() && b.isRef()) {
    if (auto heapType =
          HeapType::getLeastUpperBound(a.getHeapType(), b.getHeapType())) {
      auto nullability =
        (a.isNullable() || b.isNullable()) ? Nullable : NonNullable;
      return Type(*heapType, nullability);
    }
  }
  return Type::none;
}

Type Type::getGreatestLowerBound(Type a, Type b) {
  if (a == b) {
    return a;
  }
  if (a.isTuple() && b.isTuple() && a.size() == b.size()) {
    std::vector<Type> elems;
    size_t size = a.size();
    elems.reserve(size);
    for (size_t i = 0; i < size; ++i) {
      auto glb = Type::getGreatestLowerBound(a[i], b[i]);
      if (glb == Type::unreachable) {
        return Type::unreachable;
      }
      elems.push_back(glb);
    }
    return Tuple(elems);
  }
  if (!a.isRef() || !b.isRef()) {
    return Type::unreachable;
  }
  auto heapA = a.getHeapType();
  auto heapB = b.getHeapType();
  if (heapA.getBottom() != heapB.getBottom()) {
    return Type::unreachable;
  }
  auto nullability =
    (a.isNonNullable() || b.isNonNullable()) ? NonNullable : Nullable;
  HeapType heapType;
  if (HeapType::isSubType(heapA, heapB)) {
    heapType = heapA;
  } else if (HeapType::isSubType(heapB, heapA)) {
    heapType = heapB;
  } else {
    heapType = heapA.getBottom();
  }
  return Type(heapType, nullability);
}

const Type& Type::Iterator::operator*() const {
  if (parent->isTuple()) {
    return parent->getTuple()[index];
  } else {
    assert(index == 0 && *parent != Type::none && "Index out of bounds");
    return *parent;
  }
}

HeapType::HeapType(Signature sig) {
  new (this)
    HeapType(globalRecGroupStore.insert(std::make_unique<HeapTypeInfo>(sig)));
}

HeapType::HeapType(Continuation continuation) {
  new (this) HeapType(
    globalRecGroupStore.insert(std::make_unique<HeapTypeInfo>(continuation)));
}

HeapType::HeapType(const Struct& struct_) {
  new (this) HeapType(
    globalRecGroupStore.insert(std::make_unique<HeapTypeInfo>(struct_)));
}

HeapType::HeapType(Struct&& struct_) {
  new (this) HeapType(globalRecGroupStore.insert(
    std::make_unique<HeapTypeInfo>(std::move(struct_))));
}

HeapType::HeapType(Array array) {
  new (this)
    HeapType(globalRecGroupStore.insert(std::make_unique<HeapTypeInfo>(array)));
}

HeapTypeKind HeapType::getKind() const {
  if (isBasic()) {
    return HeapTypeKind::Basic;
  }
  return getHeapTypeInfo(*this)->kind;
}

bool HeapType::isOpen() const {
  if (isBasic()) {
    return false;
  } else {
    return getHeapTypeInfo(*this)->isOpen;
  }
}

Shareability HeapType::getShared() const {
  if (isBasic()) {
    return (id & SharedMask) != 0 ? Shared : Unshared;
  } else {
    return getHeapTypeInfo(*this)->share;
  }
}

Signature HeapType::getSignature() const {
  assert(isSignature());
  return getHeapTypeInfo(*this)->signature;
}

Continuation HeapType::getContinuation() const {
  assert(isContinuation());
  return getHeapTypeInfo(*this)->continuation;
}

const Struct& HeapType::getStruct() const {
  assert(isStruct());
  return getHeapTypeInfo(*this)->struct_;
}

Array HeapType::getArray() const {
  assert(isArray());
  return getHeapTypeInfo(*this)->array;
}

std::optional<HeapType> HeapType::getDeclaredSuperType() const {
  if (isBasic()) {
    return {};
  }
  HeapTypeInfo* super = getHeapTypeInfo(*this)->supertype;
  if (super != nullptr) {
    return HeapType(uintptr_t(super));
  }
  return {};
}

std::optional<HeapType> HeapType::getSuperType() const {
  auto ret = getDeclaredSuperType();
  if (ret) {
    return ret;
  }

  auto share = getShared();

  // There may be a basic supertype.
  if (isBasic()) {
    switch (getBasic(Unshared)) {
      case ext:
      case noext:
      case func:
      case nofunc:
      case cont:
      case nocont:
      case any:
      case none:
      case exn:
      case noexn:
        return {};
      case string:
        return HeapType(ext).getBasic(share);
      case eq:
        return HeapType(any).getBasic(share);
      case i31:
      case struct_:
      case array:
        return HeapType(eq).getBasic(share);
    }
  }

  auto* info = getHeapTypeInfo(*this);
  switch (info->kind) {
    case HeapTypeKind::Func:
      return HeapType(func).getBasic(share);
    case HeapTypeKind::Cont:
      return HeapType(cont).getBasic(share);
    case HeapTypeKind::Struct:
      return HeapType(struct_).getBasic(share);
    case HeapTypeKind::Array:
      return HeapType(array).getBasic(share);
    case HeapTypeKind::Basic:
      break;
  }
  WASM_UNREACHABLE("unexpected kind");
}

std::optional<HeapType> HeapType::getDescriptorType() const {
  if (isBasic()) {
    return std::nullopt;
  }
  if (auto* desc = getHeapTypeInfo(*this)->descriptor) {
    return HeapType(uintptr_t(desc));
  }
  return std::nullopt;
}

std::optional<HeapType> HeapType::getDescribedType() const {
  if (isBasic()) {
    return std::nullopt;
  }
  if (auto* desc = getHeapTypeInfo(*this)->described) {
    return HeapType(uintptr_t(desc));
  }
  return std::nullopt;
}

size_t HeapType::getDepth() const {
  size_t depth = 0;
  std::optional<HeapType> super;
  for (auto curr = *this; (super = curr.getDeclaredSuperType());
       curr = *super) {
    ++depth;
  }
  // In addition to the explicit supertypes we just traversed over, there is
  // implicit supertyping wrt basic types. A signature type always has one more
  // super, HeapType::func, etc.
  switch (getKind()) {
    case HeapTypeKind::Basic:
      // Some basic types have supers.
      switch (getBasic(Unshared)) {
        case HeapType::ext:
        case HeapType::func:
        case HeapType::cont:
        case HeapType::any:
        case HeapType::exn:
          break;
        case HeapType::eq:
        case HeapType::string:
          depth++;
          break;
        case HeapType::i31:
        case HeapType::struct_:
        case HeapType::array:
          depth += 2;
          break;
        case HeapType::none:
        case HeapType::nofunc:
        case HeapType::nocont:
        case HeapType::noext:
        case HeapType::noexn:
          // Bottom types are infinitely deep.
          depth = size_t(-1l);
      }
      break;
    case HeapTypeKind::Func:
    case HeapTypeKind::Cont:
      ++depth;
      break;
    case HeapTypeKind::Struct:
      // specific struct types <: struct <: eq <: any
      depth += 3;
      break;
    case HeapTypeKind::Array:
      // specific array types <: array <: eq <: any
      depth += 3;
      break;
  }
  return depth;
}

HeapType::BasicHeapType HeapType::getUnsharedBottom() const {
  if (isBasic()) {
    switch (getBasic(Unshared)) {
      case ext:
        return noext;
      case func:
        return nofunc;
      case cont:
        return nocont;
      case exn:
        return noexn;
      case any:
      case eq:
      case i31:
      case struct_:
      case array:
      case none:
        return none;
      case string:
      case noext:
        return noext;
      case nofunc:
        return nofunc;
      case nocont:
        return nocont;
      case noexn:
        return noexn;
    }
  }
  auto* info = getHeapTypeInfo(*this);
  switch (info->kind) {
    case HeapTypeKind::Func:
      return nofunc;
    case HeapTypeKind::Cont:
      return nocont;
    case HeapTypeKind::Struct:
    case HeapTypeKind::Array:
      return none;
    case HeapTypeKind::Basic:
      break;
  }
  WASM_UNREACHABLE("unexpected kind");
}

HeapType::BasicHeapType HeapType::getUnsharedTop() const {
  switch (getUnsharedBottom()) {
    case none:
      return any;
    case nofunc:
      return func;
    case nocont:
      return cont;
    case noext:
      return ext;
    case noexn:
      return exn;
    case ext:
    case func:
    case cont:
    case any:
    case eq:
    case i31:
    case struct_:
    case array:
    case exn:
    case string:
      break;
  }
  WASM_UNREACHABLE("unexpected type");
}

bool HeapType::isSubType(HeapType left, HeapType right) {
  // As an optimization, in the common case do not even construct a SubTyper.
  if (left == right) {
    return true;
  }
  return SubTyper().isSubType(left, right);
}

std::vector<Type> HeapType::getTypeChildren() const {
  switch (getKind()) {
    case HeapTypeKind::Basic:
      return {};
    case HeapTypeKind::Func: {
      std::vector<Type> children;
      auto sig = getSignature();
      for (auto tuple : {sig.params, sig.results}) {
        for (auto t : tuple) {
          children.push_back(t);
        }
      }
      return children;
    }
    case HeapTypeKind::Struct: {
      std::vector<Type> children;
      for (auto& field : getStruct().fields) {
        children.push_back(field.type);
      }
      return children;
    }
    case HeapTypeKind::Array:
      return {getArray().element.type};
    case HeapTypeKind::Cont:
      return {};
  }
  WASM_UNREACHABLE("unexpected kind");
}

std::vector<HeapType> HeapType::getHeapTypeChildren() const {
  HeapTypeChildCollector collector;
  collector.walkRoot(const_cast<HeapType*>(this));
  return collector.children;
}

std::vector<HeapType> HeapType::getReferencedHeapTypes() const {
  auto types = getHeapTypeChildren();
  if (auto super = getDeclaredSuperType()) {
    types.push_back(*super);
  }
  if (auto desc = getDescriptorType()) {
    types.push_back(*desc);
  }
  if (auto desc = getDescribedType()) {
    types.push_back(*desc);
  }
  return types;
}

std::optional<HeapType> HeapType::getLeastUpperBound(HeapType a, HeapType b) {
  if (a == b) {
    return a;
  }
  if (a.getBottom() != b.getBottom()) {
    return {};
  }
  if (a.isBottom()) {
    return b;
  }
  if (b.isBottom()) {
    return a;
  }
  if (a.isBasic() || b.isBasic()) {
    return getBasicHeapTypeLUB(getBasicHeapSupertype(a),
                               getBasicHeapSupertype(b));
  }
  if (a.with(Inexact) == b.with(Inexact)) {
    return a.with(Inexact);
  }

  auto* infoA = getHeapTypeInfo(a);
  auto* infoB = getHeapTypeInfo(b);

  if (infoA->kind != infoB->kind) {
    return getBasicHeapTypeLUB(getBasicHeapSupertype(a),
                               getBasicHeapSupertype(b));
  }

  // Walk up the subtype tree to find the LUB. Ascend the tree from both `a`
  // and `b` in lockstep. The first type we see for a second time must be the
  // LUB because there are no cycles and the only way to encounter a type
  // twice is for it to be on the path above both `a` and `b`.
  std::unordered_set<HeapTypeInfo*> seen;
  seen.insert(infoA);
  seen.insert(infoB);
  while (true) {
    auto* nextA = infoA->supertype;
    auto* nextB = infoB->supertype;
    if (nextA == nullptr && nextB == nullptr) {
      // Did not find a LUB in the subtype tree.
      return getBasicHeapTypeLUB(getBasicHeapSupertype(a),
                                 getBasicHeapSupertype(b));
    }
    if (nextA) {
      if (!seen.insert(nextA).second) {
        return HeapType(uintptr_t(nextA));
      }
      infoA = nextA;
    }
    if (nextB) {
      if (!seen.insert(nextB).second) {
        return HeapType(uintptr_t(nextB));
      }
      infoB = nextB;
    }
  }
}

// Recursion groups with single elements are encoded as that single element's
// type ID with the low bit set and other recursion groups are encoded with the
// address of the vector containing their members. These encodings are disjoint
// because the alignment of the vectors is greater than 1.
static_assert(alignof(std::vector<HeapType>) > 1);

RecGroup HeapType::getRecGroup() const {
  assert(!isBasic());
  if (auto* info = getHeapTypeInfo(*this)->recGroup) {
    return RecGroup(uintptr_t(info));
  } else {
    // Mark the low bit to signify that this is a trivial recursion group and
    // points to a heap type info rather than a vector of heap types.
    return RecGroup(getRawID() | 1);
  }
}

size_t HeapType::getRecGroupIndex() const {
  assert(!isBasic());
  return getHeapTypeInfo(*this)->recGroupIndex;
}

FeatureSet HeapType::getFeatures() const {
  // Collects features from a type + children.
  struct ReferenceFeatureCollector
    : HeapTypeChildWalker<ReferenceFeatureCollector> {
    FeatureSet feats = FeatureSet::None;

    void noteChild(HeapType heapType) {
      if (heapType.isShared()) {
        feats |= FeatureSet::SharedEverything;
      }

      if (heapType.isBasic()) {
        switch (heapType.getBasic(Unshared)) {
          case HeapType::ext:
          case HeapType::func:
            feats |= FeatureSet::ReferenceTypes;
            return;
          case HeapType::any:
          case HeapType::eq:
          case HeapType::i31:
          case HeapType::struct_:
          case HeapType::array:
          case HeapType::none:
            feats |= FeatureSet::ReferenceTypes | FeatureSet::GC;
            return;
          case HeapType::string:
            feats |= FeatureSet::ReferenceTypes | FeatureSet::Strings;
            return;
          case HeapType::noext:
          case HeapType::nofunc:
            // Technically introduced in GC, but used internally as part of
            // ref.null with just reference types.
            feats |= FeatureSet::ReferenceTypes;
            return;
          case HeapType::exn:
          case HeapType::noexn:
            feats |= FeatureSet::ExceptionHandling | FeatureSet::ReferenceTypes;
            return;
          case HeapType::cont:
          case HeapType::nocont:
            feats |= FeatureSet::StackSwitching;
            return;
        }
      }

      if (heapType.getRecGroup().size() > 1 ||
          heapType.getDeclaredSuperType() || heapType.isOpen()) {
        feats |= FeatureSet::ReferenceTypes | FeatureSet::GC;
      }

      if (heapType.getDescriptorType() || heapType.getDescribedType()) {
        feats |= FeatureSet::CustomDescriptors;
      }

      if (heapType.isStruct() || heapType.isArray()) {
        feats |= FeatureSet::ReferenceTypes | FeatureSet::GC;
      } else if (heapType.isSignature()) {
        // This is a function reference, which requires reference types and
        // possibly also multivalue (if it has multiple returns). Note that
        // technically typed function references also require GC, however,
        // we use these types internally regardless of the presence of GC
        // (in particular, since during load of the wasm we don't know the
        // features yet, so we apply the more refined types), so we don't
        // add that in any case here.
        feats |= FeatureSet::ReferenceTypes;
        auto sig = heapType.getSignature();
        if (sig.results.isTuple()) {
          feats |= FeatureSet::Multivalue;
        }
      } else if (heapType.isContinuation()) {
        feats |= FeatureSet::StackSwitching;
      }

      // In addition, scan their non-ref children, to add dependencies on
      // things like SIMD.
      for (auto child : heapType.getTypeChildren()) {
        if (!child.isRef()) {
          feats |= child.getFeatures();
        }
      }
    }
  };

  ReferenceFeatureCollector collector;
  // For internal reasons, the walkRoot/noteChild APIs all require non-const
  // pointers. We only use them to scan the type, so it is safe for us to
  // send |this| there from a |const| method.
  auto* unconst = const_cast<HeapType*>(this);
  collector.walkRoot(unconst);
  collector.noteChild(*unconst);
  return collector.feats;
}

HeapType RecGroup::Iterator::operator*() const {
  if (parent->id & 1) {
    // This is a trivial recursion group. Mask off the low bit to recover the
    // single HeapType.
    return {HeapType(parent->id & ~(uintptr_t)1)};
  } else {
    return (*(std::vector<HeapType>*)parent->id)[index];
  }
}

size_t RecGroup::size() const {
  if (id & 1) {
    return 1;
  } else {
    return ((std::vector<HeapType>*)id)->size();
  }
}

TypeNames DefaultTypeNameGenerator::getNames(HeapTypeDef type) {
  auto [it, inserted] = nameCache.insert({type, {}});
  if (inserted) {
    // Generate a new name for this type we have not previously seen.
    std::stringstream stream;
    switch (type.getKind()) {
      case HeapTypeKind::Func:
        stream << "func." << funcCount++;
        break;
      case HeapTypeKind::Struct:
        stream << "struct." << structCount++;
        break;
      case HeapTypeKind::Array:
        stream << "array." << arrayCount++;
        break;
      case HeapTypeKind::Cont:
        stream << "cont." << contCount++;
        break;
      case HeapTypeKind::Basic:
        WASM_UNREACHABLE("unexpected kind");
    }
    it->second = {stream.str(), {}};
  }
  return it->second;
}

template<typename T> static std::string genericToString(const T& t) {
  std::ostringstream ss;
  ss << t;
  return ss.str();
}
std::string Type::toString() const { return genericToString(*this); }
std::string HeapType::toString() const { return genericToString(*this); }
std::string Signature::toString() const { return genericToString(*this); }
std::string Continuation::toString() const { return genericToString(*this); }
std::string Struct::toString() const { return genericToString(*this); }
std::string Array::toString() const { return genericToString(*this); }

std::ostream& operator<<(std::ostream& os, Type type) {
  return TypePrinter(os).print(type);
}
std::ostream& operator<<(std::ostream& os, Type::Printed printed) {
  return TypePrinter(os, printed.generateName).print(Type(printed.typeID));
}
std::ostream& operator<<(std::ostream& os, HeapType type) {
  return TypePrinter(os).print(type);
}
std::ostream& operator<<(std::ostream& os, HeapType::Printed printed) {
  return TypePrinter(os, printed.generateName).print(HeapType(printed.typeID));
}
std::ostream& operator<<(std::ostream& os, Tuple tuple) {
  return TypePrinter(os).print(tuple);
}
std::ostream& operator<<(std::ostream& os, Signature sig) {
  return TypePrinter(os).print(sig);
}
std::ostream& operator<<(std::ostream& os, Continuation cont) {
  return TypePrinter(os).print(cont);
}
std::ostream& operator<<(std::ostream& os, Field field) {
  return TypePrinter(os).print(field);
}
std::ostream& operator<<(std::ostream& os, Struct struct_) {
  return TypePrinter(os).print(struct_);
}
std::ostream& operator<<(std::ostream& os, Array array) {
  return TypePrinter(os).print(array);
}
std::ostream& operator<<(std::ostream& os, TypeBuilder::ErrorReason reason) {
  switch (reason) {
    case TypeBuilder::ErrorReason::SelfSupertype:
      return os << "Heap type is a supertype of itself";
    case TypeBuilder::ErrorReason::InvalidSupertype:
      return os << "Heap type has an invalid supertype";
    case TypeBuilder::ErrorReason::ForwardSupertypeReference:
      return os << "Heap type has an undeclared supertype";
    case TypeBuilder::ErrorReason::ForwardChildReference:
      return os << "Heap type has an undeclared child";
    case TypeBuilder::ErrorReason::InvalidFuncType:
      return os << "Continuation has invalid function type";
    case TypeBuilder::ErrorReason::InvalidUnsharedField:
      return os << "Heap type has an invalid unshared field";
    case TypeBuilder::ErrorReason::NonStructDescribes:
      return os << "Describes clause on a non-struct type";
    case TypeBuilder::ErrorReason::ForwardDescribesReference:
      return os << "Describes clause is a forward reference";
    case TypeBuilder::ErrorReason::MismatchedDescribes:
      return os << "Described type is not a matching descriptor";
    case TypeBuilder::ErrorReason::NonStructDescriptor:
      return os << "Descriptor clause on a non-struct type";
    case TypeBuilder::ErrorReason::MismatchedDescriptor:
      return os << "Descriptor type does not describe heap type";
    case TypeBuilder::ErrorReason::InvalidUnsharedDescriptor:
      return os << "Heap type has an invalid unshared descriptor";
    case TypeBuilder::ErrorReason::InvalidUnsharedDescribes:
      return os << "Heap type describes an invalid unshared type";
  }
  WASM_UNREACHABLE("Unexpected error reason");
}

unsigned Field::getByteSize() const {
  if (type != Type::i32) {
    return type.getByteSize();
  }
  switch (packedType) {
    case Field::PackedType::i8:
      return 1;
    case Field::PackedType::i16:
      return 2;
    case Field::PackedType::not_packed:
      return 4;
  }
  WASM_UNREACHABLE("impossible packed type");
}

namespace {

bool SubTyper::isSubType(Type a, Type b) {
  if (a == b) {
    return true;
  }
  if (a == Type::unreachable) {
    return true;
  }
  if (a.isRef() && b.isRef()) {
    return (a.isNullable() == b.isNullable() || !a.isNullable()) &&
           isSubType(a.getHeapType(), b.getHeapType());
  }
  if (a.isTuple() && b.isTuple()) {
    return isSubType(a.getTuple(), b.getTuple());
  }
  return false;
}

bool SubTyper::isSubType(HeapType a, HeapType b) {
  // See:
  // https://github.com/WebAssembly/function-references/blob/master/proposals/function-references/Overview.md#subtyping
  // https://github.com/WebAssembly/gc/blob/master/proposals/gc/MVP.md#defined-types
  if (a == b || a.with(Inexact) == b) {
    return true;
  }
  if (a.isShared() != b.isShared()) {
    return false;
  }
  if (b.isBasic()) {
    auto aTop = a.getUnsharedTop();
    auto aUnshared = a.isBasic() ? a.getBasic(Unshared) : a;
    switch (b.getBasic(Unshared)) {
      case HeapType::ext:
        return aTop == HeapType::ext;
      case HeapType::func:
        return aTop == HeapType::func;
      case HeapType::cont:
        return aTop == HeapType::cont;
      case HeapType::exn:
        return aTop == HeapType::exn;
      case HeapType::any:
        return aTop == HeapType::any;
      case HeapType::eq:
        return aUnshared == HeapType::i31 || aUnshared == HeapType::none ||
               aUnshared == HeapType::struct_ || aUnshared == HeapType::array ||
               a.isStruct() || a.isArray();
      case HeapType::i31:
        return aUnshared == HeapType::none;
      case HeapType::string:
        return aUnshared == HeapType::noext;
      case HeapType::struct_:
        return aUnshared == HeapType::none || a.isStruct();
      case HeapType::array:
        return aUnshared == HeapType::none || a.isArray();
      case HeapType::none:
      case HeapType::noext:
      case HeapType::nofunc:
      case HeapType::nocont:
      case HeapType::noexn:
        return false;
    }
  }
  if (a.isBasic()) {
    // Basic HeapTypes are only subtypes of compound HeapTypes if they are
    // bottom types.
    return a == b.getBottom();
  }
  if (b.isExact()) {
    // The only subtypes of an exact type are itself and bottom, both of which
    // we have ruled out.
    return false;
  }
  // Subtyping must be declared rather than derived from structure, so we will
  // not recurse. TODO: optimize this search with some form of caching.
  HeapTypeInfo* curr = getHeapTypeInfo(a);
  while ((curr = curr->supertype)) {
    if (curr == getHeapTypeInfo(b)) {
      return true;
    }
  }
  return false;
}

bool SubTyper::isSubType(const Tuple& a, const Tuple& b) {
  if (a.size() != b.size()) {
    return false;
  }
  for (size_t i = 0; i < a.size(); ++i) {
    if (!isSubType(a[i], b[i])) {
      return false;
    }
  }
  return true;
}

bool SubTyper::isSubType(const Field& a, const Field& b) {
  if (a == b) {
    return true;
  }
  // Immutable fields can be subtypes.
  return a.mutable_ == Immutable && b.mutable_ == Immutable &&
         a.packedType == b.packedType && isSubType(a.type, b.type);
}

bool SubTyper::isSubType(const Signature& a, const Signature& b) {
  return isSubType(b.params, a.params) && isSubType(a.results, b.results);
}

bool SubTyper::isSubType(const Continuation& a, const Continuation& b) {
  return isSubType(a.type, b.type);
}

bool SubTyper::isSubType(const Struct& a, const Struct& b) {
  // There may be more fields on the left, but not fewer.
  if (a.fields.size() < b.fields.size()) {
    return false;
  }
  for (size_t i = 0; i < b.fields.size(); ++i) {
    if (!isSubType(a.fields[i], b.fields[i])) {
      return false;
    }
  }
  return true;
}

bool SubTyper::isSubType(const Array& a, const Array& b) {
  return isSubType(a.element, b.element);
}

void TypePrinter::printHeapTypeName(HeapType type) {
  if (type.isExact()) {
    os << "(exact ";
  }
  if (type.isBasic()) {
    print(type);
  } else {
    generator(type.with(Inexact)).name.print(os);
#if TRACE_CANONICALIZATION
    os << "(;" << ((type.with(Inexact).getID() >> 4) % 1000) << ";) ";
#endif
  }
  if (type.isExact()) {
    os << ')';
  }
}

std::ostream& TypePrinter::print(Type type) {
  if (type.isBasic()) {
    switch (type.getBasic()) {
      case Type::none:
        return os << "none";
      case Type::unreachable:
        return os << "unreachable";
      case Type::i32:
        return os << "i32";
      case Type::i64:
        return os << "i64";
      case Type::f32:
        return os << "f32";
      case Type::f64:
        return os << "f64";
      case Type::v128:
        return os << "v128";
    }
  }

#if TRACE_CANONICALIZATION
  os << "(;" << ((type.getID() >> 4) % 1000) << ";) ";
#endif
  if (type.isTuple()) {
    print(type.getTuple());
  } else if (type.isRef()) {
    auto heapType = type.getHeapType();
    if (type.isNullable() && heapType.isBasic() && !heapType.isShared()) {
      // Print shorthands for certain basic heap types.
      switch (heapType.getBasic(Unshared)) {
        case HeapType::ext:
          return os << "externref";
        case HeapType::func:
          return os << "funcref";
        case HeapType::cont:
          return os << "contref";
        case HeapType::any:
          return os << "anyref";
        case HeapType::eq:
          return os << "eqref";
        case HeapType::i31:
          return os << "i31ref";
        case HeapType::struct_:
          return os << "structref";
        case HeapType::array:
          return os << "arrayref";
        case HeapType::exn:
          return os << "exnref";
        case HeapType::string:
          return os << "stringref";
        case HeapType::none:
          return os << "nullref";
        case HeapType::noext:
          return os << "nullexternref";
        case HeapType::nofunc:
          return os << "nullfuncref";
        case HeapType::nocont:
          return os << "nullcontref";
        case HeapType::noexn:
          return os << "nullexnref";
      }
    }
    os << "(ref ";
    if (type.isNullable()) {
      os << "null ";
    }
    printHeapTypeName(heapType);
    os << ')';
  } else {
    WASM_UNREACHABLE("unexpected type");
  }
  return os;
}

std::ostream& TypePrinter::print(HeapType type) {
  if (type.isBasic()) {
    if (type.isShared()) {
      os << "(shared ";
    }
    switch (type.getBasic(Unshared)) {
      case HeapType::ext:
        os << "extern";
        break;
      case HeapType::func:
        os << "func";
        break;
      case HeapType::cont:
        os << "cont";
        break;
      case HeapType::any:
        os << "any";
        break;
      case HeapType::eq:
        os << "eq";
        break;
      case HeapType::i31:
        os << "i31";
        break;
      case HeapType::struct_:
        os << "struct";
        break;
      case HeapType::array:
        os << "array";
        break;
      case HeapType::exn:
        os << "exn";
        break;
      case HeapType::string:
        os << "string";
        break;
      case HeapType::none:
        os << "none";
        break;
      case HeapType::noext:
        os << "noextern";
        break;
      case HeapType::nofunc:
        os << "nofunc";
        break;
      case HeapType::nocont:
        os << "nocont";
        break;
      case HeapType::noexn:
        os << "noexn";
        break;
    }
    if (type.isShared()) {
      os << ')';
    }
    return os;
  }

  auto names = generator(type);

  os << "(type ";
  names.name.print(os) << ' ';

  if (isTemp(type)) {
    os << "(; temp ;) ";
  }

  bool useSub = false;
  auto super = type.getDeclaredSuperType();
  if (super || type.isOpen()) {
    useSub = true;
    os << "(sub ";
    if (!type.isOpen()) {
      os << "final ";
    }
    if (super) {
      printHeapTypeName(*super);
      os << ' ';
    }
  }
  if (type.isShared()) {
    os << "(shared ";
  }
  if (auto desc = type.getDescribedType()) {
    os << "(describes ";
    printHeapTypeName(*desc);
    os << ' ';
  }
  if (auto desc = type.getDescriptorType()) {
    os << "(descriptor ";
    printHeapTypeName(*desc);
    os << ' ';
  }
  switch (type.getKind()) {
    case HeapTypeKind::Func:
      print(type.getSignature());
      break;
    case HeapTypeKind::Struct:
      print(type.getStruct(), names.fieldNames);
      break;
    case HeapTypeKind::Array:
      print(type.getArray());
      break;
    case HeapTypeKind::Cont:
      print(type.getContinuation());
      break;
    case HeapTypeKind::Basic:
      WASM_UNREACHABLE("unexpected kind");
  }
  if (type.getDescriptorType()) {
    os << ')';
  }
  if (type.getDescribedType()) {
    os << ')';
  }
  if (type.isShared()) {
    os << ')';
  }
  if (useSub) {
    os << ')';
  }
  return os << ')';
}

std::ostream& TypePrinter::print(const Tuple& tuple) {
  os << "(tuple";
  for (Type type : tuple) {
    os << ' ';
    print(type);
  }
  return os << ')';
}

std::ostream& TypePrinter::print(const Field& field) {
  if (field.mutable_) {
    os << "(mut ";
  }
  if (field.isPacked()) {
    auto packedType = field.packedType;
    if (packedType == Field::PackedType::i8) {
      os << "i8";
    } else if (packedType == Field::PackedType::i16) {
      os << "i16";
    } else {
      WASM_UNREACHABLE("unexpected packed type");
    }
  } else {
    print(field.type);
  }
  if (field.mutable_) {
    os << ')';
  }
  return os;
}

std::ostream& TypePrinter::print(const Signature& sig) {
  auto printPrefixed = [&](const char* prefix, Type type) {
    os << '(' << prefix;
    for (Type t : type) {
      os << ' ';
      print(t);
    }
    os << ')';
  };

  os << "(func";
  if (sig.params.getID() != Type::none) {
    os << ' ';
    printPrefixed("param", sig.params);
  }
  if (sig.results.getID() != Type::none) {
    os << ' ';
    printPrefixed("result", sig.results);
  }
  return os << ')';
}

std::ostream& TypePrinter::print(const Continuation& continuation) {
  os << "(cont ";
  printHeapTypeName(continuation.type);
  return os << ')';
}

std::ostream&
TypePrinter::print(const Struct& struct_,
                   const std::unordered_map<Index, Name>& fieldNames) {
  os << "(struct";
  for (Index i = 0; i < struct_.fields.size(); ++i) {
    // TODO: move this to the function for printing fields.
    os << " (field ";
    if (auto it = fieldNames.find(i); it != fieldNames.end()) {
      it->second.print(os) << ' ';
    }
    print(struct_.fields[i]);
    os << ')';
  }
  return os << ")";
}

std::ostream& TypePrinter::print(const Array& array) {
  os << "(array ";
  print(array.element);
  return os << ')';
}

size_t RecGroupHasher::operator()() const {
  size_t digest = wasm::hash(group.size());
  for (auto type : group) {
    hash_combine(digest, topLevelHash(type));
  }
  return digest;
}

size_t RecGroupHasher::topLevelHash(HeapType type) const {
  size_t digest = wasm::hash(type.isBasic());
  if (type.isBasic()) {
    wasm::rehash(digest, type.getID());
  } else {
    hash_combine(digest, hash(*getHeapTypeInfo(type)));
  }
  return digest;
}

size_t RecGroupHasher::hash(Type type) const {
  size_t digest = wasm::hash(type.isBasic());
  if (type.isBasic()) {
    wasm::rehash(digest, type.getID());
    return digest;
  }
  wasm::rehash(digest, type.isTuple());
  if (type.isTuple()) {
    hash_combine(digest, hash(type.getTuple()));
    return digest;
  }
  assert(type.isRef());
  rehash(digest, type.getNullability());
  rehash(digest, hash(type.getHeapType()));
  return digest;
}

size_t RecGroupHasher::hash(HeapType type) const {
  // Do not recurse into the structure of this child type, but rather hash it as
  // an index into a rec group. Only take the rec group identity into account if
  // the child is not a member of the top-level group because in that case the
  // group may not be canonicalized yet.
  size_t digest = wasm::hash(type.isBasic());
  if (type.isBasic()) {
    wasm::rehash(digest, type.getID());
    return digest;
  }
  wasm::rehash(digest, type.isExact());
  wasm::rehash(digest, type.getRecGroupIndex());
  auto currGroup = type.getRecGroup();
  wasm::rehash(digest, currGroup != group);
  if (currGroup != group) {
    wasm::rehash(digest, currGroup.getID());
  }
  return digest;
}

size_t RecGroupHasher::hash(const HeapTypeInfo& info) const {
  size_t digest = wasm::hash(bool(info.supertype));
  wasm::rehash(digest, !!info.supertype);
  if (info.supertype) {
    hash_combine(digest, hash(HeapType(uintptr_t(info.supertype))));
  }
  wasm::rehash(digest, !!info.descriptor);
  if (info.descriptor) {
    hash_combine(digest, hash(HeapType(uintptr_t(info.descriptor))));
  }
  wasm::rehash(digest, !!info.described);
  if (info.described) {
    hash_combine(digest, hash(HeapType(uintptr_t(info.described))));
  }
  wasm::rehash(digest, info.isOpen);
  wasm::rehash(digest, info.share);
  wasm::rehash(digest, info.kind);
  switch (info.kind) {
    case HeapTypeKind::Func:
      hash_combine(digest, hash(info.signature));
      return digest;
    case HeapTypeKind::Cont:
      hash_combine(digest, hash(info.continuation));
      return digest;
    case HeapTypeKind::Struct:
      hash_combine(digest, hash(info.struct_));
      return digest;
    case HeapTypeKind::Array:
      hash_combine(digest, hash(info.array));
      return digest;
    case HeapTypeKind::Basic:
      break;
  }
  WASM_UNREACHABLE("unexpected kind");
}

size_t RecGroupHasher::hash(const Tuple& tuple) const {
  size_t digest = wasm::hash(tuple.size());
  for (auto type : tuple) {
    hash_combine(digest, hash(type));
  }
  return digest;
}

size_t RecGroupHasher::hash(const Field& field) const {
  size_t digest = wasm::hash(field.packedType);
  rehash(digest, field.mutable_);
  hash_combine(digest, hash(field.type));
  return digest;
}

size_t RecGroupHasher::hash(const Signature& sig) const {
  size_t digest = hash(sig.params);
  hash_combine(digest, hash(sig.results));
  return digest;
}

size_t RecGroupHasher::hash(const Continuation& continuation) const {
  // We throw in a magic constant to distinguish (cont $foo) from $foo
  size_t magic = 0xc0117;
  size_t digest = hash(continuation.type);
  rehash(digest, magic);
  return digest;
}

size_t RecGroupHasher::hash(const Struct& struct_) const {
  size_t digest = wasm::hash(struct_.fields.size());
  for (const auto& field : struct_.fields) {
    hash_combine(digest, hash(field));
  }
  return digest;
}

size_t RecGroupHasher::hash(const Array& array) const {
  return hash(array.element);
}

bool RecGroupEquator::operator()() const {
  if (newGroup == otherGroup) {
    return true;
  }
  // The rec groups are equivalent if they are piecewise equivalent.
  return std::equal(
    newGroup.begin(),
    newGroup.end(),
    otherGroup.begin(),
    otherGroup.end(),
    [&](const HeapType& a, const HeapType& b) { return topLevelEq(a, b); });
}

bool RecGroupEquator::topLevelEq(HeapType a, HeapType b) const {
  if (a == b) {
    return true;
  }
  if (a.isBasic() || b.isBasic()) {
    return false;
  }
  return eq(*getHeapTypeInfo(a), *getHeapTypeInfo(b));
}

bool RecGroupEquator::eq(Type a, Type b) const {
  if (a.isBasic() || b.isBasic()) {
    return a == b;
  }
  if (a.isTuple() && b.isTuple()) {
    return eq(a.getTuple(), b.getTuple());
  }
  if (a.isRef() && b.isRef()) {
    return a.getNullability() == b.getNullability() &&
           eq(a.getHeapType(), b.getHeapType());
  }
  return false;
}

bool RecGroupEquator::eq(HeapType a, HeapType b) const {
  // Do not recurse into the structure of children `a` and `b`, but check
  // whether their recursion groups and indices match. Since `newGroup` may not
  // be canonicalized, explicitly check whether `a` and `b` are in the
  // respective recursion groups of the respective top-level groups we are
  // comparing, in which case the structure is still equivalent.
  if (a.isBasic() || b.isBasic()) {
    return a == b;
  }
  if (a.getExactness() != b.getExactness()) {
    return false;
  }
  if (a.getRecGroupIndex() != b.getRecGroupIndex()) {
    return false;
  }
  auto groupA = a.getRecGroup();
  auto groupB = b.getRecGroup();
  bool selfRefA = groupA == newGroup;
  bool selfRefB = groupB == otherGroup;
  return (selfRefA && selfRefB) || (!selfRefA && !selfRefB && groupA == groupB);
}

bool RecGroupEquator::eq(const HeapTypeInfo& a, const HeapTypeInfo& b) const {
  if (!!a.supertype != !!b.supertype) {
    return false;
  }
  if (a.supertype) {
    HeapType superA(uintptr_t(a.supertype));
    HeapType superB(uintptr_t(b.supertype));
    if (!eq(superA, superB)) {
      return false;
    }
  }
  if (!!a.descriptor != !!b.descriptor) {
    return false;
  }
  if (a.descriptor) {
    HeapType descA(uintptr_t(a.descriptor));
    HeapType descB(uintptr_t(b.descriptor));
    if (!eq(descA, descB)) {
      return false;
    }
  }
  if (!!a.described != !!b.described) {
    return false;
  }
  if (a.described) {
    HeapType descA(uintptr_t(a.described));
    HeapType descB(uintptr_t(b.described));
    if (!eq(descA, descB)) {
      return false;
    }
  }
  if (a.isOpen != b.isOpen) {
    return false;
  }
  if (a.share != b.share) {
    return false;
  }
  if (a.kind != b.kind) {
    return false;
  }
  switch (a.kind) {
    case HeapTypeKind::Func:
      return eq(a.signature, b.signature);
    case HeapTypeKind::Cont:
      return eq(a.continuation, b.continuation);
    case HeapTypeKind::Struct:
      return eq(a.struct_, b.struct_);
    case HeapTypeKind::Array:
      return eq(a.array, b.array);
    case HeapTypeKind::Basic:
      break;
  }
  WASM_UNREACHABLE("unexpected kind");
}

bool RecGroupEquator::eq(const Tuple& a, const Tuple& b) const {
  return std::equal(
    a.begin(), a.end(), b.begin(), b.end(), [&](const Type& x, const Type& y) {
      return eq(x, y);
    });
}

bool RecGroupEquator::eq(const Field& a, const Field& b) const {
  return a.packedType == b.packedType && a.mutable_ == b.mutable_ &&
         eq(a.type, b.type);
}

bool RecGroupEquator::eq(const Signature& a, const Signature& b) const {
  return eq(a.params, b.params) && eq(a.results, b.results);
}

bool RecGroupEquator::eq(const Continuation& a, const Continuation& b) const {
  return eq(a.type, b.type);
}

bool RecGroupEquator::eq(const Struct& a, const Struct& b) const {
  return std::equal(a.fields.begin(),
                    a.fields.end(),
                    b.fields.begin(),
                    b.fields.end(),
                    [&](const Field& x, const Field& y) { return eq(x, y); });
}

bool RecGroupEquator::eq(const Array& a, const Array& b) const {
  return eq(a.element, b.element);
}

} // anonymous namespace

struct TypeBuilder::Impl {
  // Store of temporary tuples. Tuples that need to be canonicalized will be
  // copied into the global TupleStore.
  TupleStore tupleStore;

  // Store of temporary recursion groups, which will be moved to the global
  // collection of recursion groups as part of building.
  std::unordered_map<RecGroup, std::unique_ptr<RecGroupInfo>> recGroups;

  struct Entry {
    std::unique_ptr<HeapTypeInfo> info;
    bool initialized = false;
    Entry() {
      // We need to eagerly allocate the HeapTypeInfo so we have a TypeID to use
      // to refer to it before it is initialized. Arbitrarily choose a default
      // value.
      info = std::make_unique<HeapTypeInfo>(Signature());
      info->isTemp = true;
    }
    void set(HeapTypeInfo&& hti) {
      info->kind = hti.kind;
      switch (info->kind) {
        case HeapTypeKind::Func:
          info->signature = hti.signature;
          break;
        case HeapTypeKind::Cont:
          info->continuation = hti.continuation;
          break;
        case HeapTypeKind::Struct:
          info->struct_ = std::move(hti.struct_);
          break;
        case HeapTypeKind::Array:
          info->array = hti.array;
          break;
        case HeapTypeKind::Basic:
          WASM_UNREACHABLE("unexpected kind");
      }
      initialized = true;
    }
    HeapType get() { return HeapType(TypeID(info.get())); }
  };

  std::vector<Entry> entries;

  Impl(size_t n) : entries(n) {}
};

TypeBuilder::TypeBuilder(size_t n) {
  impl = std::make_unique<TypeBuilder::Impl>(n);
}

TypeBuilder::~TypeBuilder() = default;

TypeBuilder::TypeBuilder(TypeBuilder&& other) = default;
TypeBuilder& TypeBuilder::operator=(TypeBuilder&& other) = default;

void TypeBuilder::grow(size_t n) {
  assert(size() + n >= size());
  impl->entries.resize(size() + n);
}

size_t TypeBuilder::size() { return impl->entries.size(); }

void TypeBuilder::setHeapType(size_t i, Signature signature) {
  assert(i < size() && "index out of bounds");
  impl->entries[i].set(signature);
}

void TypeBuilder::setHeapType(size_t i, Continuation continuation) {
  assert(i < size() && "index out of bounds");
  impl->entries[i].set(continuation);
}

void TypeBuilder::setHeapType(size_t i, const Struct& struct_) {
  assert(i < size() && "index out of bounds");
  impl->entries[i].set(struct_);
}

void TypeBuilder::setHeapType(size_t i, Struct&& struct_) {
  assert(i < size() && "index out of bounds");
  impl->entries[i].set(std::move(struct_));
}

void TypeBuilder::setHeapType(size_t i, Array array) {
  assert(i < size() && "index out of bounds");
  impl->entries[i].set(array);
}

HeapTypeDef TypeBuilder::getTempHeapType(size_t i) {
  assert(i < size() && "index out of bounds");
  return impl->entries[i].get();
}

Type TypeBuilder::getTempTupleType(const Tuple& tuple) {
  return impl->tupleStore.insert(tuple);
}

Type TypeBuilder::getTempRefType(HeapType type, Nullability nullable) {
  return Type(type, nullable);
}

void TypeBuilder::setSubType(size_t i, std::optional<HeapType> super) {
  assert(i < size() && "index out of bounds");
  HeapTypeInfo* sub = impl->entries[i].info.get();
  sub->supertype = super ? getHeapTypeInfo(*super) : nullptr;
}
void TypeBuilder::setDescriptor(size_t i, std::optional<HeapType> desc) {
  assert(i < size() && "index out of bounds");
  HeapTypeInfo* info = impl->entries[i].info.get();
  info->descriptor = desc ? getHeapTypeInfo(*desc) : nullptr;
}
void TypeBuilder::setDescribed(size_t i, std::optional<HeapType> desc) {
  assert(i < size() && "index out of bounds");
  HeapTypeInfo* info = impl->entries[i].info.get();
  info->described = desc ? getHeapTypeInfo(*desc) : nullptr;
}

void TypeBuilder::createRecGroup(size_t index, size_t length) {
  assert(index <= size() && index + length <= size() && "group out of bounds");
  // Only materialize nontrivial recursion groups.
  if (length < 2) {
    return;
  }
  auto groupInfo = std::make_unique<RecGroupInfo>();
  groupInfo->reserve(length);
  for (size_t i = 0; i < length; ++i) {
    auto& info = impl->entries[index + i].info;
    assert(info->recGroup == nullptr && "group already assigned");
    groupInfo->push_back(asHeapType(info));
    info->recGroup = groupInfo.get();
    info->recGroupIndex = i;
  }
  impl->recGroups.insert(
    {RecGroup(uintptr_t(groupInfo.get())), std::move(groupInfo)});
}

void TypeBuilder::setOpen(size_t i, bool open) {
  assert(i < size() && "index out of bounds");
  impl->entries[i].info->isOpen = open;
}

void TypeBuilder::setShared(size_t i, Shareability share) {
  assert(i < size() && "index out of bounds");
  impl->entries[i].info->share = share;
}

namespace {

bool isValidSupertype(const HeapTypeInfo& sub, const HeapTypeInfo& super) {
  if (!super.isOpen) {
    return false;
  }
  if (sub.share != super.share) {
    return false;
  }
  if (sub.kind != super.kind) {
    return false;
  }
  if (sub.descriptor) {
    // A supertype of a type with a (descriptor $x) must either not have a
    // descriptor or have a (descriptor $y) where $y is the declared supertype
    // of $x.
    if (super.descriptor && sub.descriptor->supertype != super.descriptor) {
      return false;
    }
  } else {
    // A supertype of a type without a descriptor must also not have a
    // descriptor.
    if (super.descriptor) {
      return false;
    }
  }
  // A supertype of a type must have a describes clause iff the type has a
  // describes clause.
  if (bool(sub.described) != bool(super.described)) {
    return false;
  }
  SubTyper typer;
  switch (sub.kind) {
    case HeapTypeKind::Func:
      return typer.isSubType(sub.signature, super.signature);
    case HeapTypeKind::Cont:
      return typer.isSubType(sub.continuation, super.continuation);
    case HeapTypeKind::Struct:
      return typer.isSubType(sub.struct_, super.struct_);
    case HeapTypeKind::Array:
      return typer.isSubType(sub.array, super.array);
    case HeapTypeKind::Basic:
      break;
  }
  WASM_UNREACHABLE("unknown kind");
}

std::optional<TypeBuilder::ErrorReason>
validateType(HeapTypeInfo& info, std::unordered_set<HeapType>& seenTypes) {
  if (auto* super = info.supertype) {
    // The supertype must be canonical (i.e. defined in a previous rec group)
    // or have already been defined in this rec group.
    if (super->isTemp && !seenTypes.count(HeapType(uintptr_t(super)))) {
      return TypeBuilder::ErrorReason::ForwardSupertypeReference;
    }
    // The supertype must have a valid structure.
    if (!isValidSupertype(info, *super)) {
      return TypeBuilder::ErrorReason::InvalidSupertype;
    }
  }
  if (auto* desc = info.described) {
    if (info.kind != HeapTypeKind::Struct) {
      return TypeBuilder::ErrorReason::NonStructDescribes;
    }
    assert(desc->isTemp && "unexpected canonical described type");
    if (!seenTypes.count(HeapType(uintptr_t(desc)))) {
      return TypeBuilder::ErrorReason::ForwardDescribesReference;
    }
    if (desc->descriptor != &info) {
      return TypeBuilder::ErrorReason::MismatchedDescribes;
    }
  }
  if (auto* desc = info.descriptor) {
    if (info.kind != HeapTypeKind::Struct) {
      return TypeBuilder::ErrorReason::NonStructDescriptor;
    }
    if (desc->described != &info) {
      return TypeBuilder::ErrorReason::MismatchedDescriptor;
    }
  }
  if (info.isContinuation()) {
    if (!info.continuation.type.isSignature()) {
      return TypeBuilder::ErrorReason::InvalidFuncType;
    }
  }
  if (info.share == Shared) {
    if (info.described && info.described->share != Shared) {
      return TypeBuilder::ErrorReason::InvalidUnsharedDescribes;
    }
    if (info.descriptor && info.descriptor->share != Shared) {
      return TypeBuilder::ErrorReason::InvalidUnsharedDescriptor;
    }
    switch (info.kind) {
      case HeapTypeKind::Func:
        // TODO: Figure out and enforce shared function rules.
        break;
      case HeapTypeKind::Cont:
        if (!info.continuation.type.isShared()) {
          return TypeBuilder::ErrorReason::InvalidFuncType;
        }
        break;
      case HeapTypeKind::Struct:
        for (auto& field : info.struct_.fields) {
          if (field.type.isRef() && !field.type.getHeapType().isShared()) {
            return TypeBuilder::ErrorReason::InvalidUnsharedField;
          }
        }
        break;
      case HeapTypeKind::Array: {
        auto elem = info.array.element.type;
        if (elem.isRef() && !elem.getHeapType().isShared()) {
          return TypeBuilder::ErrorReason::InvalidUnsharedField;
        }
        break;
      }
      case HeapTypeKind::Basic:
        WASM_UNREACHABLE("unexpected kind");
    }
  }
  return std::nullopt;
}

void updateReferencedHeapTypes(
  std::unique_ptr<HeapTypeInfo>& info,
  const std::unordered_map<HeapType, HeapType>& canonicalized) {
  // Update the reference types that refer to canonicalized heap types to be
  // their canonical versions.
  struct ChildUpdater : TypeGraphWalkerBase<ChildUpdater> {
    const std::unordered_map<HeapType, HeapType>& canonicalized;
    bool isTopLevel = true;

    ChildUpdater(const std::unordered_map<HeapType, HeapType>& canonicalized)
      : canonicalized(canonicalized) {}

    void scanType(Type* type) {
      isTopLevel = false;
      if (type->isRef()) {
        auto ht = type->getHeapType();
        auto exact = ht.getExactness();
        ht = ht.with(Inexact);
        if (auto it = canonicalized.find(ht); it != canonicalized.end()) {
          *type = Type(it->second.with(exact), type->getNullability());
        }
      } else if (type->isTuple()) {
        TypeGraphWalkerBase<ChildUpdater>::scanType(type);
      }
    }

    void scanHeapType(HeapType* type) {
      assert(!type->isExact() && "unexpected exact type in definition");
      if (isTopLevel) {
        isTopLevel = false;
        TypeGraphWalkerBase<ChildUpdater>::scanHeapType(type);
      } else {
        if (auto it = canonicalized.find(*type); it != canonicalized.end()) {
          *type = it->second;
        }
      }
    }
  };

  // Update the children.
  ChildUpdater updater(canonicalized);
  auto root = asHeapType(info);
  updater.walkRoot(&root);

  // Update the supertype.
  if (info->supertype) {
    HeapType super(uintptr_t(info->supertype));
    if (auto it = canonicalized.find(super); it != canonicalized.end()) {
      info->supertype = getHeapTypeInfo(it->second);
    }
  }

  // Update the descriptor and described types.
  if (info->descriptor) {
    HeapType desc(uintptr_t(info->descriptor));
    if (auto it = canonicalized.find(desc); it != canonicalized.end()) {
      info->descriptor = getHeapTypeInfo(it->second);
    }
  }
  if (info->described) {
    HeapType desc(uintptr_t(info->described));
    if (auto it = canonicalized.find(desc); it != canonicalized.end()) {
      info->described = getHeapTypeInfo(it->second);
    }
  }
}

TypeBuilder::BuildResult
buildRecGroup(std::unique_ptr<RecGroupInfo>&& groupInfo,
              std::vector<std::unique_ptr<HeapTypeInfo>>&& typeInfos,
              std::unordered_map<HeapType, HeapType>& canonicalized) {
  // First, we need to replace any referenced temporary HeapTypes from
  // previously built groups with their canonicalized versions.
  for (auto& info : typeInfos) {
    updateReferencedHeapTypes(info, canonicalized);
  }

  // Collect the types and check validity.
  std::unordered_set<HeapType> seenTypes;
  for (size_t i = 0; i < typeInfos.size(); ++i) {
    auto& info = typeInfos[i];
    if (auto err = validateType(*info, seenTypes)) {
      return {TypeBuilder::Error{i, *err}};
    }
    seenTypes.insert(asHeapType(info));
  }

  // Check that children are either already canonical (i.e. defined in a
  // previous rec group) or are defined in this current rec group.
  for (size_t i = 0; i < typeInfos.size(); ++i) {
    auto type = asHeapType(typeInfos[i]);
    for (auto child : type.getHeapTypeChildren()) {
      HeapType rawChild(child.getRawID());
      if (isTemp(rawChild) && !seenTypes.count(rawChild)) {
        return {TypeBuilder::Error{
          i, TypeBuilder::ErrorReason::ForwardChildReference}};
      }
    }
  }

  // The rec group is valid, so we can try to move the group into the global rec
  // group store. If the returned canonical group is not the same as the input
  // group, then there is already a canonical version of this rec group. Lock
  // the global rec group store here to avoid leaking temporary types to other
  // threads that may be trying to build an identical group.
  auto group = asHeapType(typeInfos[0]).getRecGroup();
  std::lock_guard<std::mutex> lock(globalRecGroupStore.mutex);
  auto canonical = groupInfo ? globalRecGroupStore.insert(std::move(groupInfo))
                             : globalRecGroupStore.insert(group);
  if (group != canonical) {
    // Replace the non-canonical types with their canonical equivalents.
    assert(canonical.size() == group.size());
    for (size_t i = 0; i < group.size(); ++i) {
      canonicalized.insert({group[i], canonical[i]});
    }
    // Return the canonical types.
    return {std::vector<HeapTypeDef>(canonical.begin(), canonical.end())};
  }

  // The group was successfully moved to the global rec group store, so it is
  // now canonical. We need to move its heap types to the heap type store so
  // they become canonical as well.
  {
    std::lock_guard<std::recursive_mutex> lock(globalHeapTypeStoreMutex);
    for (auto& info : typeInfos) {
      info->isTemp = false;
      globalHeapTypeStore.emplace_back(std::move(info));
    }
  }

  std::vector<HeapTypeDef> results(group.begin(), group.end());

  // We need to make the tuples canonical as well, but right now there is no way
  // to move them to their global store, so we have to create new tuples and
  // replace the old ones.
  struct TupleUpdater : TypeGraphWalkerBase<TupleUpdater> {
    bool isTopLevel = true;
    void scanHeapType(HeapType* type) {
      // Only scan top-level heap types. Heap type children will either be
      // scanned separately or are already canonical.
      if (isTopLevel) {
        TypeGraphWalkerBase<TupleUpdater>::scanHeapType(type);
        isTopLevel = false;
      }
    }
    void scanType(Type* type) {
      if (type->isTuple()) {
        *type = globalTupleStore.insert(type->getTuple());
      }
    }
  };

  for (auto& type : results) {
    TupleUpdater().walkRoot(&type);
  }

  return {results};
}

} // anonymous namespace

TypeBuilder::BuildResult TypeBuilder::build() {
  size_t entryCount = impl->entries.size();

  std::vector<HeapTypeDef> results;
  results.reserve(entryCount);

  // Map temporary HeapTypes to their canonicalized versions so they can be
  // replaced in later rec groups.
  std::unordered_map<HeapType, HeapType> canonicalized;

  // Canonicalize each rec group, one at a time.
  size_t groupStart = 0;
  while (groupStart < entryCount) {
    auto group = asHeapType(impl->entries[groupStart].info).getRecGroup();

    std::unique_ptr<RecGroupInfo> groupInfo;
    if (auto it = impl->recGroups.find(group); it != impl->recGroups.end()) {
      groupInfo = std::move(it->second);
    }

    size_t groupSize = group.size();
    std::vector<std::unique_ptr<HeapTypeInfo>> typeInfos;
    typeInfos.reserve(groupSize);
    for (size_t i = 0; i < groupSize; ++i) {
      typeInfos.emplace_back(std::move(impl->entries[groupStart + i].info));
    }

    auto built =
      buildRecGroup(std::move(groupInfo), std::move(typeInfos), canonicalized);
    if (auto* error = built.getError()) {
      return {TypeBuilder::Error{groupStart + error->index, error->reason}};
    }

    assert(built->size() == groupSize);
    results.insert(results.end(), built->begin(), built->end());

    groupStart += groupSize;
  }

  return {results};
}

void TypeBuilder::dump() {
  std::vector<HeapType> types;
  for (size_t i = 0; i < size(); ++i) {
    types.push_back((*this)[i]);
  }
  IndexedTypeNameGenerator<DefaultTypeNameGenerator> print(types);

  std::optional<RecGroup> currGroup;
  for (auto type : types) {
    if (auto newGroup = type.getRecGroup(); newGroup != currGroup) {
      if (currGroup && currGroup->size() > 1) {
        std::cerr << ")\n";
      }
      if (newGroup.size() > 1) {
        std::cerr << "(rec\n";
      }
      currGroup = newGroup;
    }
    if (currGroup->size() > 1) {
      std::cerr << "  ";
    }
    std::cerr << print(type) << "\n";
  }
  if (currGroup && currGroup->size() > 1) {
    std::cerr << ")\n";
  }
}

std::unordered_set<HeapType> getIgnorablePublicTypes() {
  auto array8 = Array(Field(Field::i8, Mutable));
  auto array16 = Array(Field(Field::i16, Mutable));
  TypeBuilder builder(2);
  builder[0] = array8;
  builder[1] = array16;
  auto result = builder.build();
  assert(result);
  std::unordered_set<HeapType> ret;
  for (auto type : *result) {
    ret.insert(type);
  }
  return ret;
}

} // namespace wasm

namespace std {

template<> class hash<wasm::Tuple> {
public:
  size_t operator()(const wasm::TypeList& types) const {
    auto digest = wasm::hash(types.size());
    for (auto type : types) {
      wasm::rehash(digest, type);
    }
    return digest;
  }
};

template<> class hash<wasm::FieldList> {
public:
  size_t operator()(const wasm::FieldList& fields) const {
    auto digest = wasm::hash(fields.size());
    for (auto field : fields) {
      wasm::rehash(digest, field);
    }
    return digest;
  }
};

size_t hash<wasm::Type>::operator()(const wasm::Type& type) const {
  return wasm::hash(type.getID());
}

size_t hash<wasm::Signature>::operator()(const wasm::Signature& sig) const {
  auto digest = wasm::hash(sig.params);
  wasm::rehash(digest, sig.results);
  return digest;
}

size_t
hash<wasm::Continuation>::operator()(const wasm::Continuation& cont) const {
  // We throw in a magic constant to distinguish (cont $foo) from $foo
  auto magic = 0xc0117;
  auto digest = wasm::hash(cont.type);
  wasm::rehash(digest, magic);
  return digest;
}

size_t hash<wasm::Field>::operator()(const wasm::Field& field) const {
  auto digest = wasm::hash(field.type);
  wasm::rehash(digest, field.packedType);
  wasm::rehash(digest, field.mutable_);
  return digest;
}

size_t hash<wasm::Struct>::operator()(const wasm::Struct& struct_) const {
  return wasm::hash(struct_.fields);
}

size_t hash<wasm::Array>::operator()(const wasm::Array& array) const {
  return wasm::hash(array.element);
}

size_t hash<wasm::HeapType>::operator()(const wasm::HeapType& heapType) const {
  return wasm::hash(heapType.getID());
}

size_t hash<wasm::HeapTypeDef>::operator()(const wasm::HeapTypeDef& def) const {
  return wasm::hash(def.getID());
}

size_t hash<wasm::RecGroup>::operator()(const wasm::RecGroup& group) const {
  return wasm::hash(group.getID());
}

} // namespace std