PredicateSimplifier.cpp

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//===-- PredicateSimplifier.cpp - Path Sensitive Simplifier ---------------===//
//
//                     The LLVM Compiler Infrastructure
//
// This file is distributed under the University of Illinois Open Source
// License. See LICENSE.TXT for details.
//
//===----------------------------------------------------------------------===//
//
// Path-sensitive optimizer. In a branch where x == y, replace uses of
// x with y. Permits further optimization, such as the elimination of
// the unreachable call:
//
// void test(int *p, int *q)
// {
//   if (p != q)
//     return;
// 
//   if (*p != *q)
//     foo(); // unreachable
// }
//
//===----------------------------------------------------------------------===//
//
// The InequalityGraph focusses on four properties; equals, not equals,
// less-than and less-than-or-equals-to. The greater-than forms are also held
// just to allow walking from a lesser node to a greater one. These properties
// are stored in a lattice; LE can become LT or EQ, NE can become LT or GT.
//
// These relationships define a graph between values of the same type. Each
// Value is stored in a map table that retrieves the associated Node. This
// is how EQ relationships are stored; the map contains pointers from equal
// Value to the same node. The node contains a most canonical Value* form
// and the list of known relationships with other nodes.
//
// If two nodes are known to be inequal, then they will contain pointers to
// each other with an "NE" relationship. If node getNode(%x) is less than
// getNode(%y), then the %x node will contain <%y, GT> and %y will contain
// <%x, LT>. This allows us to tie nodes together into a graph like this:
//
//   %a < %b < %c < %d
//
// with four nodes representing the properties. The InequalityGraph provides
// querying with "isRelatedBy" and mutators "addEquality" and "addInequality".
// To find a relationship, we start with one of the nodes any binary search
// through its list to find where the relationships with the second node start.
// Then we iterate through those to find the first relationship that dominates
// our context node.
//
// To create these properties, we wait until a branch or switch instruction
// implies that a particular value is true (or false). The VRPSolver is
// responsible for analyzing the variable and seeing what new inferences
// can be made from each property. For example:
//
//   %P = icmp ne i32* %ptr, null
//   %a = and i1 %P, %Q
//   br i1 %a label %cond_true, label %cond_false
//
// For the true branch, the VRPSolver will start with %a EQ true and look at
// the definition of %a and find that it can infer that %P and %Q are both
// true. From %P being true, it can infer that %ptr NE null. For the false
// branch it can't infer anything from the "and" instruction.
//
// Besides branches, we can also infer properties from instruction that may
// have undefined behaviour in certain cases. For example, the dividend of
// a division may never be zero. After the division instruction, we may assume
// that the dividend is not equal to zero.
//
//===----------------------------------------------------------------------===//
//
// The ValueRanges class stores the known integer bounds of a Value. When we
// encounter i8 %a u< %b, the ValueRanges stores that %a = [1, 255] and
// %b = [0, 254].
//
// It never stores an empty range, because that means that the code is
// unreachable. It never stores a single-element range since that's an equality
// relationship and better stored in the InequalityGraph, nor an empty range
// since that is better stored in UnreachableBlocks.
//
//===----------------------------------------------------------------------===//

#define DEBUG_TYPE "predsimplify"
#include "llvm/Transforms/Scalar.h"
#include "llvm/Constants.h"
#include "llvm/DerivedTypes.h"
#include "llvm/Instructions.h"
#include "llvm/Pass.h"
#include "llvm/ADT/DepthFirstIterator.h"
#include "llvm/ADT/SetOperations.h"
#include "llvm/ADT/SetVector.h"
#include "llvm/ADT/Statistic.h"
#include "llvm/ADT/STLExtras.h"
#include "llvm/Analysis/Dominators.h"
#include "llvm/Assembly/Writer.h"
#include "llvm/Support/CFG.h"
#include "llvm/Support/Compiler.h"
#include "llvm/Support/ConstantRange.h"
#include "llvm/Support/Debug.h"
#include "llvm/Support/InstVisitor.h"
#include "llvm/Target/TargetData.h"
#include "llvm/Transforms/Utils/Local.h"
#include <algorithm>
#include <deque>
#include <stack>
using namespace llvm;

STATISTIC(NumVarsReplaced, "Number of argument substitutions");
STATISTIC(NumInstruction , "Number of instructions removed");
STATISTIC(NumSimple      , "Number of simple replacements");
STATISTIC(NumBlocks      , "Number of blocks marked unreachable");
STATISTIC(NumSnuggle     , "Number of comparisons snuggled");

namespace {
  class DomTreeDFS {
  public:
    class Node {
      friend class DomTreeDFS;
    public:
      typedef std::vector<Node *>::iterator       iterator;
      typedef std::vector<Node *>::const_iterator const_iterator;

      unsigned getDFSNumIn()  const { return DFSin;  }
      unsigned getDFSNumOut() const { return DFSout; }

      BasicBlock *getBlock() const { return BB; }

      iterator begin() { return Children.begin(); }
      iterator end()   { return Children.end();   }

      const_iterator begin() const { return Children.begin(); }
      const_iterator end()   const { return Children.end();   }

      bool dominates(const Node *N) const {
        return DFSin <= N->DFSin && DFSout >= N->DFSout;
      }

      bool DominatedBy(const Node *N) const {
        return N->dominates(this);
      }

      /// Sorts by the number of descendants. With this, you can iterate
      /// through a sorted list and the first matching entry is the most
      /// specific match for your basic block. The order provided is stable;
      /// DomTreeDFS::Nodes with the same number of descendants are sorted by
      /// DFS in number.
      bool operator<(const Node &N) const {
        unsigned   spread =   DFSout -   DFSin;
        unsigned N_spread = N.DFSout - N.DFSin;
        if (spread == N_spread) return DFSin < N.DFSin;
        return spread < N_spread;
      }
      bool operator>(const Node &N) const { return N < *this; }

    private:
      unsigned DFSin, DFSout;
      BasicBlock *BB;

      std::vector<Node *> Children;
    };

    // XXX: this may be slow. Instead of using "new" for each node, consider
    // putting them in a vector to keep them contiguous.
    explicit DomTreeDFS(DominatorTree *DT) {
      std::stack<std::pair<Node *, DomTreeNode *> > S;

      Entry = new Node;
      Entry->BB = DT->getRootNode()->getBlock();
      S.push(std::make_pair(Entry, DT->getRootNode()));

      NodeMap[Entry->BB] = Entry;

      while (!S.empty()) {
        std::pair<Node *, DomTreeNode *> &Pair = S.top();
        Node *N = Pair.first;
        DomTreeNode *DTNode = Pair.second;
        S.pop();

        for (DomTreeNode::iterator I = DTNode->begin(), E = DTNode->end();
             I != E; ++I) {
          Node *NewNode = new Node;
          NewNode->BB = (*I)->getBlock();
          N->Children.push_back(NewNode);
          S.push(std::make_pair(NewNode, *I));

          NodeMap[NewNode->BB] = NewNode;
        }
      }

      renumber();

#ifndef NDEBUG
      DEBUG(dump());
#endif
    }

#ifndef NDEBUG
    virtual
#endif
    ~DomTreeDFS() {
      std::stack<Node *> S;

      S.push(Entry);
      while (!S.empty()) {
        Node *N = S.top(); S.pop();

        for (Node::iterator I = N->begin(), E = N->end(); I != E; ++I)
          S.push(*I);

        delete N;
      }
    }

    /// getRootNode - This returns the entry node for the CFG of the function.
    Node *getRootNode() const { return Entry; }

    /// getNodeForBlock - return the node for the specified basic block.
    Node *getNodeForBlock(BasicBlock *BB) const {
      if (!NodeMap.count(BB)) return 0;
      return const_cast<DomTreeDFS*>(this)->NodeMap[BB];
    }

    /// dominates - returns true if the basic block for I1 dominates that of
    /// the basic block for I2. If the instructions belong to the same basic
    /// block, the instruction first instruction sequentially in the block is
    /// considered dominating.
    bool dominates(Instruction *I1, Instruction *I2) {
      BasicBlock *BB1 = I1->getParent(),
                 *BB2 = I2->getParent();
      if (BB1 == BB2) {
        if (isa<TerminatorInst>(I1)) return false;
        if (isa<TerminatorInst>(I2)) return true;
        if ( isa<PHINode>(I1) && !isa<PHINode>(I2)) return true;
        if (!isa<PHINode>(I1) &&  isa<PHINode>(I2)) return false;

        for (BasicBlock::const_iterator I = BB2->begin(), E = BB2->end();
             I != E; ++I) {
          if (&*I == I1) return true;
          else if (&*I == I2) return false;
        }
        assert(!"Instructions not found in parent BasicBlock?");
      } else {
        Node *Node1 = getNodeForBlock(BB1),
             *Node2 = getNodeForBlock(BB2);
        return Node1 && Node2 && Node1->dominates(Node2);
      }
      return false; // Not reached
    }

  private:
    /// renumber - calculates the depth first search numberings and applies
    /// them onto the nodes.
    void renumber() {
      std::stack<std::pair<Node *, Node::iterator> > S;
      unsigned n = 0;

      Entry->DFSin = ++n;
      S.push(std::make_pair(Entry, Entry->begin()));

      while (!S.empty()) {
        std::pair<Node *, Node::iterator> &Pair = S.top();
        Node *N = Pair.first;
        Node::iterator &I = Pair.second;

        if (I == N->end()) {
          N->DFSout = ++n;
          S.pop();
        } else {
          Node *Next = *I++;
          Next->DFSin = ++n;
          S.push(std::make_pair(Next, Next->begin()));
        }
      }
    }

#ifndef NDEBUG
    virtual void dump() const {
      dump(*cerr.stream());
    }

    void dump(std::ostream &os) const {
      os << "Predicate simplifier DomTreeDFS: \n";
      dump(Entry, 0, os);
      os << "\n\n";
    }

    void dump(Node *N, int depth, std::ostream &os) const {
      ++depth;
      for (int i = 0; i < depth; ++i) { os << " "; }
      os << "[" << depth << "] ";

      os << N->getBlock()->getName() << " (" << N->getDFSNumIn()
         << ", " << N->getDFSNumOut() << ")\n";

      for (Node::iterator I = N->begin(), E = N->end(); I != E; ++I)
        dump(*I, depth, os);
    }
#endif

    Node *Entry;
    std::map<BasicBlock *, Node *> NodeMap;
  };

  // SLT SGT ULT UGT EQ
  //   0   1   0   1  0 -- GT                  10
  //   0   1   0   1  1 -- GE                  11
  //   0   1   1   0  0 -- SGTULT              12
  //   0   1   1   0  1 -- SGEULE              13
  //   0   1   1   1  0 -- SGT                 14
  //   0   1   1   1  1 -- SGE                 15
  //   1   0   0   1  0 -- SLTUGT              18
  //   1   0   0   1  1 -- SLEUGE              19
  //   1   0   1   0  0 -- LT                  20
  //   1   0   1   0  1 -- LE                  21
  //   1   0   1   1  0 -- SLT                 22
  //   1   0   1   1  1 -- SLE                 23
  //   1   1   0   1  0 -- UGT                 26
  //   1   1   0   1  1 -- UGE                 27
  //   1   1   1   0  0 -- ULT                 28
  //   1   1   1   0  1 -- ULE                 29
  //   1   1   1   1  0 -- NE                  30
  enum LatticeBits {
    EQ_BIT = 1, UGT_BIT = 2, ULT_BIT = 4, SGT_BIT = 8, SLT_BIT = 16
  };
  enum LatticeVal {
    GT = SGT_BIT | UGT_BIT,
    GE = GT | EQ_BIT,
    LT = SLT_BIT | ULT_BIT,
    LE = LT | EQ_BIT,
    NE = SLT_BIT | SGT_BIT | ULT_BIT | UGT_BIT,
    SGTULT = SGT_BIT | ULT_BIT,
    SGEULE = SGTULT | EQ_BIT,
    SLTUGT = SLT_BIT | UGT_BIT,
    SLEUGE = SLTUGT | EQ_BIT,
    ULT = SLT_BIT | SGT_BIT | ULT_BIT,
    UGT = SLT_BIT | SGT_BIT | UGT_BIT,
    SLT = SLT_BIT | ULT_BIT | UGT_BIT,
    SGT = SGT_BIT | ULT_BIT | UGT_BIT,
    SLE = SLT | EQ_BIT,
    SGE = SGT | EQ_BIT,
    ULE = ULT | EQ_BIT,
    UGE = UGT | EQ_BIT
  };

#ifndef NDEBUG
  /// validPredicate - determines whether a given value is actually a lattice
  /// value. Only used in assertions or debugging.
  static bool validPredicate(LatticeVal LV) {
    switch (LV) {
      case GT: case GE: case LT: case LE: case NE:
      case SGTULT: case SGT: case SGEULE:
      case SLTUGT: case SLT: case SLEUGE:
      case ULT: case UGT:
      case SLE: case SGE: case ULE: case UGE:
        return true;
      default:
        return false;
    }
  }
#endif

  /// reversePredicate - reverse the direction of the inequality
  static LatticeVal reversePredicate(LatticeVal LV) {
    unsigned reverse = LV ^ (SLT_BIT|SGT_BIT|ULT_BIT|UGT_BIT); //preserve EQ_BIT

    if ((reverse & (SLT_BIT|SGT_BIT)) == 0)
      reverse |= (SLT_BIT|SGT_BIT);

    if ((reverse & (ULT_BIT|UGT_BIT)) == 0)
      reverse |= (ULT_BIT|UGT_BIT);

    LatticeVal Rev = static_cast<LatticeVal>(reverse);
    assert(validPredicate(Rev) && "Failed reversing predicate.");
    return Rev;
  }

  /// ValueNumbering stores the scope-specific value numbers for a given Value.
  class VISIBILITY_HIDDEN ValueNumbering {

    /// VNPair is a tuple of {Value, index number, DomTreeDFS::Node}. It
    /// includes the comparison operators necessary to allow you to store it
    /// in a sorted vector.
    class VISIBILITY_HIDDEN VNPair {
    public:
      Value *V;
      unsigned index;
      DomTreeDFS::Node *Subtree;

      VNPair(Value *V, unsigned index, DomTreeDFS::Node *Subtree)
        : V(V), index(index), Subtree(Subtree) {}

      bool operator==(const VNPair &RHS) const {
        return V == RHS.V && Subtree == RHS.Subtree;
      }

      bool operator<(const VNPair &RHS) const {
        if (V != RHS.V) return V < RHS.V;
        return *Subtree < *RHS.Subtree;
      }

      bool operator<(Value *RHS) const {
        return V < RHS;
      }

      bool operator>(Value *RHS) const {
        return V > RHS;
      }

      friend bool operator<(Value *RHS, const VNPair &pair) {
        return pair.operator>(RHS);
      }
    };

    typedef std::vector<VNPair> VNMapType;
    VNMapType VNMap;

    /// The canonical choice for value number at index.
    std::vector<Value *> Values;

    DomTreeDFS *DTDFS;

  public:
#ifndef NDEBUG
    virtual ~ValueNumbering() {}
    virtual void dump() {
      dump(*cerr.stream());
    }

    void dump(std::ostream &os) {
      for (unsigned i = 1; i <= Values.size(); ++i) {
        os << i << " = ";
        WriteAsOperand(os, Values[i-1]);
        os << " {";
        for (unsigned j = 0; j < VNMap.size(); ++j) {
          if (VNMap[j].index == i) {
            WriteAsOperand(os, VNMap[j].V);
            os << " (" << VNMap[j].Subtree->getDFSNumIn() << ")  ";
          }
        }
        os << "}\n";
      }
    }
#endif

    /// compare - returns true if V1 is a better canonical value than V2.
    bool compare(Value *V1, Value *V2) const {
      if (isa<Constant>(V1))
        return !isa<Constant>(V2);
      else if (isa<Constant>(V2))
        return false;
      else if (isa<Argument>(V1))
        return !isa<Argument>(V2);
      else if (isa<Argument>(V2))
        return false;

      Instruction *I1 = dyn_cast<Instruction>(V1);
      Instruction *I2 = dyn_cast<Instruction>(V2);

      if (!I1 || !I2)
        return V1->getNumUses() < V2->getNumUses();

      return DTDFS->dominates(I1, I2);
    }

    ValueNumbering(DomTreeDFS *DTDFS) : DTDFS(DTDFS) {}

    /// valueNumber - finds the value number for V under the Subtree. If
    /// there is no value number, returns zero.
    unsigned valueNumber(Value *V, DomTreeDFS::Node *Subtree) {
      if (!(isa<Constant>(V) || isa<Argument>(V) || isa<Instruction>(V))
          || V->getType() == Type::VoidTy) return 0;

      VNMapType::iterator E = VNMap.end();
      VNPair pair(V, 0, Subtree);
      VNMapType::iterator I = std::lower_bound(VNMap.begin(), E, pair);
      while (I != E && I->V == V) {
        if (I->Subtree->dominates(Subtree))
          return I->index;
        ++I;
      }
      return 0;
    }

    /// getOrInsertVN - always returns a value number, creating it if necessary.
    unsigned getOrInsertVN(Value *V, DomTreeDFS::Node *Subtree) {
      if (unsigned n = valueNumber(V, Subtree))
        return n;
      else
        return newVN(V);
    }

    /// newVN - creates a new value number. Value V must not already have a
    /// value number assigned.
    unsigned newVN(Value *V) {
      assert((isa<Constant>(V) || isa<Argument>(V) || isa<Instruction>(V)) &&
             "Bad Value for value numbering.");
      assert(V->getType() != Type::VoidTy && "Won't value number a void value");

      Values.push_back(V);

      VNPair pair = VNPair(V, Values.size(), DTDFS->getRootNode());
      VNMapType::iterator I = std::lower_bound(VNMap.begin(), VNMap.end(), pair);
      assert((I == VNMap.end() || value(I->index) != V) &&
             "Attempt to create a duplicate value number.");
      VNMap.insert(I, pair);

      return Values.size();
    }

    /// value - returns the Value associated with a value number.
    Value *value(unsigned index) const {
      assert(index != 0 && "Zero index is reserved for not found.");
      assert(index <= Values.size() && "Index out of range.");
      return Values[index-1];
    }

    /// canonicalize - return a Value that is equal to V under Subtree.
    Value *canonicalize(Value *V, DomTreeDFS::Node *Subtree) {
      if (isa<Constant>(V)) return V;

      if (unsigned n = valueNumber(V, Subtree))
        return value(n);
      else
        return V;
    }

    /// addEquality - adds that value V belongs to the set of equivalent
    /// values defined by value number n under Subtree.
    void addEquality(unsigned n, Value *V, DomTreeDFS::Node *Subtree) {
      assert(canonicalize(value(n), Subtree) == value(n) &&
             "Node's 'canonical' choice isn't best within this subtree.");

      // Suppose that we are given "%x -> node #1 (%y)". The problem is that
      // we may already have "%z -> node #2 (%x)" somewhere above us in the
      // graph. We need to find those edges and add "%z -> node #1 (%y)"
      // to keep the lookups canonical.

      std::vector<Value *> ToRepoint(1, V);

      if (unsigned Conflict = valueNumber(V, Subtree)) {
        for (VNMapType::iterator I = VNMap.begin(), E = VNMap.end();
             I != E; ++I) {
          if (I->index == Conflict && I->Subtree->dominates(Subtree))
            ToRepoint.push_back(I->V);
        }
      }

      for (std::vector<Value *>::iterator VI = ToRepoint.begin(),
           VE = ToRepoint.end(); VI != VE; ++VI) {
        Value *V = *VI;

        VNPair pair(V, n, Subtree);
        VNMapType::iterator B = VNMap.begin(), E = VNMap.end();
        VNMapType::iterator I = std::lower_bound(B, E, pair);
        if (I != E && I->V == V && I->Subtree == Subtree)
          I->index = n; // Update best choice
        else
          VNMap.insert(I, pair); // New Value

        // XXX: we currently don't have to worry about updating values with
        // more specific Subtrees, but we will need to for PHI node support.

#ifndef NDEBUG
        Value *V_n = value(n);
        if (isa<Constant>(V) && isa<Constant>(V_n)) {
          assert(V == V_n && "Constant equals different constant?");
        }
#endif
      }
    }

    /// remove - removes all references to value V.
    void remove(Value *V) {
      VNMapType::iterator B = VNMap.begin(), E = VNMap.end();
      VNPair pair(V, 0, DTDFS->getRootNode());
      VNMapType::iterator J = std::upper_bound(B, E, pair);
      VNMapType::iterator I = J;

      while (I != B && (I == E || I->V == V)) --I;

      VNMap.erase(I, J);
    }
  };

  /// The InequalityGraph stores the relationships between values.
  /// Each Value in the graph is assigned to a Node. Nodes are pointer
  /// comparable for equality. The caller is expected to maintain the logical
  /// consistency of the system.
  ///
  /// The InequalityGraph class may invalidate Node*s after any mutator call.
  /// @brief The InequalityGraph stores the relationships between values.
  class VISIBILITY_HIDDEN InequalityGraph {
    ValueNumbering &VN;
    DomTreeDFS::Node *TreeRoot;

    InequalityGraph();                  // DO NOT IMPLEMENT
    InequalityGraph(InequalityGraph &); // DO NOT IMPLEMENT
  public:
    InequalityGraph(ValueNumbering &VN, DomTreeDFS::Node *TreeRoot)
      : VN(VN), TreeRoot(TreeRoot) {}

    class Node;

    /// An Edge is contained inside a Node making one end of the edge implicit
    /// and contains a pointer to the other end. The edge contains a lattice
    /// value specifying the relationship and an DomTreeDFS::Node specifying
    /// the root in the dominator tree to which this edge applies.
    class VISIBILITY_HIDDEN Edge {
    public:
      Edge(unsigned T, LatticeVal V, DomTreeDFS::Node *ST)
        : To(T), LV(V), Subtree(ST) {}

      unsigned To;
      LatticeVal LV;
      DomTreeDFS::Node *Subtree;

      bool operator<(const Edge &edge) const {
        if (To != edge.To) return To < edge.To;
        return *Subtree < *edge.Subtree;
      }

      bool operator<(unsigned to) const {
        return To < to;
      }

      bool operator>(unsigned to) const {
        return To > to;
      }

      friend bool operator<(unsigned to, const Edge &edge) {
        return edge.operator>(to);
      }
    };

    /// A single node in the InequalityGraph. This stores the canonical Value
    /// for the node, as well as the relationships with the neighbours.
    ///
    /// @brief A single node in the InequalityGraph.
    class VISIBILITY_HIDDEN Node {
      friend class InequalityGraph;

      typedef SmallVector<Edge, 4> RelationsType;
      RelationsType Relations;

      // TODO: can this idea improve performance?
      //friend class std::vector<Node>;
      //Node(Node &N) { RelationsType.swap(N.RelationsType); }

    public:
      typedef RelationsType::iterator       iterator;
      typedef RelationsType::const_iterator const_iterator;

#ifndef NDEBUG
      virtual ~Node() {}
      virtual void dump() const {
        dump(*cerr.stream());
      }
    private:
      void dump(std::ostream &os) const {
        static const std::string names[32] =
          { "000000", "000001", "000002", "000003", "000004", "000005",
            "000006", "000007", "000008", "000009", "     >", "    >=",
            "  s>u<", "s>=u<=", "    s>", "   s>=", "000016", "000017",
            "  s<u>", "s<=u>=", "     <", "    <=", "    s<", "   s<=",
            "000024", "000025", "    u>", "   u>=", "    u<", "   u<=",
            "    !=", "000031" };
        for (Node::const_iterator NI = begin(), NE = end(); NI != NE; ++NI) {
          os << names[NI->LV] << " " << NI->To
             << " (" << NI->Subtree->getDFSNumIn() << "), ";
        }
      }
    public:
#endif

      iterator begin()             { return Relations.begin(); }
      iterator end()               { return Relations.end();   }
      const_iterator begin() const { return Relations.begin(); }
      const_iterator end()   const { return Relations.end();   }

      iterator find(unsigned n, DomTreeDFS::Node *Subtree) {
        iterator E = end();
        for (iterator I = std::lower_bound(begin(), E, n);
             I != E && I->To == n; ++I) {
          if (Subtree->DominatedBy(I->Subtree))
            return I;
        }
        return E;
      }

      const_iterator find(unsigned n, DomTreeDFS::Node *Subtree) const {
        const_iterator E = end();
        for (const_iterator I = std::lower_bound(begin(), E, n);
             I != E && I->To == n; ++I) {
          if (Subtree->DominatedBy(I->Subtree))
            return I;
        }
        return E;
      }

      /// update - updates the lattice value for a given node, creating a new
      /// entry if one doesn't exist. The new lattice value must not be
      /// inconsistent with any previously existing value.
      void update(unsigned n, LatticeVal R, DomTreeDFS::Node *Subtree) {
        assert(validPredicate(R) && "Invalid predicate.");

        Edge edge(n, R, Subtree);
        iterator B = begin(), E = end();
        iterator I = std::lower_bound(B, E, edge);

        iterator J = I;
        while (J != E && J->To == n) {
          if (Subtree->DominatedBy(J->Subtree))
            break;
          ++J;
        }

        if (J != E && J->To == n) {
          edge.LV = static_cast<LatticeVal>(J->LV & R);
          assert(validPredicate(edge.LV) && "Invalid union of lattice values.");

          if (edge.LV == J->LV)
            return; // This update adds nothing new.
        }

        if (I != B) {
          // We also have to tighten any edge beneath our update.
          for (iterator K = I - 1; K->To == n; --K) {
            if (K->Subtree->DominatedBy(Subtree)) {
              LatticeVal LV = static_cast<LatticeVal>(K->LV & edge.LV);
              assert(validPredicate(LV) && "Invalid union of lattice values");
              K->LV = LV;
            }
            if (K == B) break;
          }
        }

        // Insert new edge at Subtree if it isn't already there.
        if (I == E || I->To != n || Subtree != I->Subtree)
          Relations.insert(I, edge);
      }
    };

  private:

    std::vector<Node> Nodes;

  public:
    /// node - returns the node object at a given value number. The pointer
    /// returned may be invalidated on the next call to node().
    Node *node(unsigned index) {
      assert(VN.value(index)); // This triggers the necessary checks.
      if (Nodes.size() < index) Nodes.resize(index);
      return &Nodes[index-1];
    }

    /// isRelatedBy - true iff n1 op n2
    bool isRelatedBy(unsigned n1, unsigned n2, DomTreeDFS::Node *Subtree,
                     LatticeVal LV) {
      if (n1 == n2) return LV & EQ_BIT;

      Node *N1 = node(n1);
      Node::iterator I = N1->find(n2, Subtree), E = N1->end();
      if (I != E) return (I->LV & LV) == I->LV;

      return false;
    }

    // The add* methods assume that your input is logically valid and may 
    // assertion-fail or infinitely loop if you attempt a contradiction.

    /// addInequality - Sets n1 op n2.
    /// It is also an error to call this on an inequality that is already true.
    void addInequality(unsigned n1, unsigned n2, DomTreeDFS::Node *Subtree,
                       LatticeVal LV1) {
      assert(n1 != n2 && "A node can't be inequal to itself.");

      if (LV1 != NE)
        assert(!isRelatedBy(n1, n2, Subtree, reversePredicate(LV1)) &&
               "Contradictory inequality.");

      // Suppose we're adding %n1 < %n2. Find all the %a < %n1 and
      // add %a < %n2 too. This keeps the graph fully connected.
      if (LV1 != NE) {
        // Break up the relationship into signed and unsigned comparison parts.
        // If the signed parts of %a op1 %n1 match that of %n1 op2 %n2, and
        // op1 and op2 aren't NE, then add %a op3 %n2. The new relationship
        // should have the EQ_BIT iff it's set for both op1 and op2.

        unsigned LV1_s = LV1 & (SLT_BIT|SGT_BIT);
        unsigned LV1_u = LV1 & (ULT_BIT|UGT_BIT);

        for (Node::iterator I = node(n1)->begin(), E = node(n1)->end(); I != E; ++I) {
          if (I->LV != NE && I->To != n2) {

            DomTreeDFS::Node *Local_Subtree = NULL;
            if (Subtree->DominatedBy(I->Subtree))
              Local_Subtree = Subtree;
            else if (I->Subtree->DominatedBy(Subtree))
              Local_Subtree = I->Subtree;

            if (Local_Subtree) {
              unsigned new_relationship = 0;
              LatticeVal ILV = reversePredicate(I->LV);
              unsigned ILV_s = ILV & (SLT_BIT|SGT_BIT);
              unsigned ILV_u = ILV & (ULT_BIT|UGT_BIT);

              if (LV1_s != (SLT_BIT|SGT_BIT) && ILV_s == LV1_s)
                new_relationship |= ILV_s;
              if (LV1_u != (ULT_BIT|UGT_BIT) && ILV_u == LV1_u)
                new_relationship |= ILV_u;

              if (new_relationship) {
                if ((new_relationship & (SLT_BIT|SGT_BIT)) == 0)
                  new_relationship |= (SLT_BIT|SGT_BIT);
                if ((new_relationship & (ULT_BIT|UGT_BIT)) == 0)
                  new_relationship |= (ULT_BIT|UGT_BIT);
                if ((LV1 & EQ_BIT) && (ILV & EQ_BIT))
                  new_relationship |= EQ_BIT;

                LatticeVal NewLV = static_cast<LatticeVal>(new_relationship);

                node(I->To)->update(n2, NewLV, Local_Subtree);
                node(n2)->update(I->To, reversePredicate(NewLV), Local_Subtree);
              }
            }
          }
        }

        for (Node::iterator I = node(n2)->begin(), E = node(n2)->end(); I != E; ++I) {
          if (I->LV != NE && I->To != n1) {
            DomTreeDFS::Node *Local_Subtree = NULL;
            if (Subtree->DominatedBy(I->Subtree))
              Local_Subtree = Subtree;
            else if (I->Subtree->DominatedBy(Subtree))
              Local_Subtree = I->Subtree;

            if (Local_Subtree) {
              unsigned new_relationship = 0;
              unsigned ILV_s = I->LV & (SLT_BIT|SGT_BIT);
              unsigned ILV_u = I->LV & (ULT_BIT|UGT_BIT);

              if (LV1_s != (SLT_BIT|SGT_BIT) && ILV_s == LV1_s)
                new_relationship |= ILV_s;

              if (LV1_u != (ULT_BIT|UGT_BIT) && ILV_u == LV1_u)
                new_relationship |= ILV_u;

              if (new_relationship) {
                if ((new_relationship & (SLT_BIT|SGT_BIT)) == 0)
                  new_relationship |= (SLT_BIT|SGT_BIT);
                if ((new_relationship & (ULT_BIT|UGT_BIT)) == 0)
                  new_relationship |= (ULT_BIT|UGT_BIT);
                if ((LV1 & EQ_BIT) && (I->LV & EQ_BIT))
                  new_relationship |= EQ_BIT;

                LatticeVal NewLV = static_cast<LatticeVal>(new_relationship);

                node(n1)->update(I->To, NewLV, Local_Subtree);
                node(I->To)->update(n1, reversePredicate(NewLV), Local_Subtree);
              }
            }
          }
        }
      }

      node(n1)->update(n2, LV1, Subtree);
      node(n2)->update(n1, reversePredicate(LV1), Subtree);
    }

    /// remove - removes a node from the graph by removing all references to
    /// and from it.
    void remove(unsigned n) {
      Node *N = node(n);
      for (Node::iterator NI = N->begin(), NE = N->end(); NI != NE; ++NI) {
        Node::iterator Iter = node(NI->To)->find(n, TreeRoot);
        do {
          node(NI->To)->Relations.erase(Iter);
          Iter = node(NI->To)->find(n, TreeRoot);
        } while (Iter != node(NI->To)->end());
      }
      N->Relations.clear();
    }

#ifndef NDEBUG
    virtual ~InequalityGraph() {}
    virtual void