}

  @Override
  public Set<E> outEdges(N node) {
    return nodeInvalidatableSet(checkedConnections(node).outEdges(), node);
  }

  @Override
  public Set<N> predecessors(N node) {
    return nodeInvalidatableSet(checkedConnections(node).predecessors(), node);
  }

  @Override
  public Set<N> successors(N node) {
    return nodeInvalidatableSet(checkedConnections(node).successors(), node);
  }

  @Override
  ElementOrder<N> nodeOrder();

  /**
   * Returns an {@link ElementOrder} that specifies the order of iteration for the elements of
   * {@link #edges()}, {@link #adjacentNodes(Object)}, {@link #predecessors(Object)}, {@link
   * #successors(Object)} and {@link #incidentEdges(Object)}.
   *
   * @since 29.0
   */
  @Override
  ElementOrder<N> incidentEdgeOrder();

  //
  // Element-level accessors

  }

  private static final int ENDPOINT = -2;

  // TODO(user): predecessors and successors should be collocated (reducing cache misses).
  // Might also explore collocating all of [hash, next, predecessor, successor] fields of an
  // entry in a *single* long[], though that reduces the maximum size of the set by a factor of 2

  /**
   * Pointer to the predecessor of an entry in insertion order. ENDPOINT indicates a node is the

  }

  private static final int ENDPOINT = -2;

  // TODO(user): predecessors and successors should be collocated (reducing cache misses).
  // Might also explore collocating all of [hash, next, predecessor, successor] fields of an
  // entry in a *single* long[], though that reduces the maximum size of the set by a factor of 2

  /**
   * Pointer to the predecessor of an entry in insertion order. ENDPOINT indicates a node is the

  @Override
  ElementOrder<N> nodeOrder();

  /**
   * Returns an {@link ElementOrder} that specifies the order of iteration for the elements of
   * {@link #edges()}, {@link #adjacentNodes(Object)}, {@link #predecessors(Object)}, {@link
   * #successors(Object)} and {@link #incidentEdges(Object)}.
   *
   * @since 29.0
   */
  @Override
  ElementOrder<N> incidentEdgeOrder();

  //
  // Element-level accessors

  @Override
  ElementOrder<N> nodeOrder();

  /**
   * Returns an {@link ElementOrder} that specifies the order of iteration for the elements of
   * {@link #edges()}, {@link #adjacentNodes(Object)}, {@link #predecessors(Object)}, {@link
   * #successors(Object)} and {@link #incidentEdges(Object)}.
   *
   * @since 29.0
   */
  @Override
  ElementOrder<N> incidentEdgeOrder();

  //
  // Element-level accessors

      return llvm::join(id_vec, ",");
    };
    func.walk([&](Operation* op) {
      if (!analysis.DirectControlPredecessors(op).empty()) {
        op->emitRemark("Predecessors: ")
            << "{" << join_ids(analysis.DirectControlPredecessors(op)) << "}";
      }
      if (!analysis.DirectControlSuccessors(op).empty()) {
        op->emitRemark("Successors: ")

    for (Operation* op : ops_to_process) {
      if (predecessors) {
        for (Value operand : op->getOperands()) {
          // Stop at the block boundary.
          if (mlir::isa<BlockArgument>(operand)) continue;

          Operation* predecessor = operand.getDefiningOp();
          if (!operations->contains(predecessor) &&
              !ops_to_avoid.contains(predecessor)) {
            new_ops.insert(operand.getDefiningOp());

    auto predecessors = analysis.DirectControlPredecessors(&cluster_op);
    if (!predecessors.empty() && !IsEmbeddingOp(&cluster_op)) {
      bool skip = false;
      for (Operation* predecessor : llvm::reverse(predecessors)) {
        if (IsEmbeddingOp(predecessor)) continue;
        skip = !head_outside_compiled_ops.contains(predecessor);
        break;
      }
      if (skip) continue;

//
// The implementation here borrows the framework of the prove pass.
//
//	1, Traverse all blocks of function f to find If blocks.
//	2,   For any If block b, traverse all its predecessors to find If blocks.
//	3,     For any If block predecessor p, update relationship p->b.
//	4,     Traverse all successors of b.
//	5,       For any successor s of b, try to update relationship b->s, if a